University of Economics, Prague Faculty of Finance and ... · University of Economics, Prague...

University of Economics, Prague

Faculty of Finance and Accounting

Department of Banking and InsuranceField of study: Financial Engineering

Interest Rate Modelling and Forecasting:Macro-Finance Approach

Author of the Master Thesis: Bc. Adam Kucera

Supervisor of the Master Thesis: doc. RNDr. Jirı Witzany, Ph.D.

Year of Defence: 2014

Declaration of Authorship

The author hereby declares that the thesis ”Interest Rate Modelling and Fore-

casting: Macro-Finance Approach” was compiled independently by him, using

only the resources and literature properly marked and included in the attached

list of references.

Prague, May 27, 2014

Signature

Acknowledgments

The author is grateful to doc. RNDr. Jirı Witzany, Ph.D. for his comments

and support during writing the thesis. Equal thanks are admitted to all others

inspiring the author during his studies.

Abstract

The thesis compares various approaches to the term structure of interest rates

modelling. Several models are built, following two general frameworks: a dy-

namic Nelson-Siegel approach and an affine class of models. Based on an eval-

uation of dynamic properties of the estimated models, particularly in terms of

impulse-responses and a forecasting performance, effects of an explicit inclusion

of macroeconomic variables into the models are tested. The thesis shows, that

the benefit of such macro-finance extension of the models is varying in time, and

also differs for both approaches. However, it is shown that the models can be

considered as complementary, as the particular approaches are differently use-

ful under various macroeconomic conditions and financial markets situations.

Moreover, unusually long maturities are included into the term structure of

interest rates, and some of the models are shown to be able to forecast these

maturities as well, particularly in certain periods of time.

JEL Classification C38, C51, C58, E43, E47

Keywords Interest Rate, Yield Curve, Macro-Finance

Model, Affine Model, Nelson-Siegel

Author’s e-mail [email protected]

Supervisor’s e-mail [email protected]

http://ideas.repec.org/j/C38.html



http://ideas.repec.org/j/E43.html


mailto:[email protected]


Abstrakt

Diplomova prace porovnava ruzne prıstupy k modelovanı casove struktury

urokovych mer. V praci je sledovano nekolik modelu, odvozenych ze dvou

obecnych skupin: dynamicke interpretace Nelson-Siegel parametrizace, a afinnı

trıdy modelu. Na zaklade zhodnocenı dynamickych vlastnostı odhadnutych

modelu, vychazejıcıch zejmena z porovnanı impulse-response funkcı a predpoved-

nıch schopnostı modelu, je nasledne testovan prınos prımeho zahrnutı makroeko-

nomickych promennych do modelu. Prace ukazuje, ze prınos takoveho makro-

financnıho rozsırenı modelu se menı v case, a lisı se take pro obe skupiny

modelu. Nicmene je ukazano, ze modely lze povazovat za vzajemne se doplnujıcı,

jelikoz jednotlive prıstupy jsou odlisne uzitecne v ruznych makroekonomickych

a financnıch podmınkach. Pomocı nekterych modelu lze, zejmena v jistych

casovych obdobıch, zaroven predpovıdat i urokove mıry pro velmi dlouhe splat-

nosti.

Klasifikace JEL C38, C51, C58, E43, E47

Klıcova slova urokova mıra, vynosova krivka, makro-

financnı model, afinnı model, Nelson-Siegel

E-mail autora [email protected]

E-mail vedoucıho prace [email protected]








Contents

List of Tables viii

List of Figures x

Acronyms xii

1 Introduction 1

2 Basic Definitions and Notations 3

2.1 Bond Market, Yield and Interest Rate . . . . . . . . . . . . . . 3

2.2 Term Structure of Interest Rates . . . . . . . . . . . . . . . . . 9

2.3 Term Structure Models . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Description of Models 20

3.1 Factors and Principal Component Analysis . . . . . . . . . . . . 20

3.2 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Dynamic Nelson-Siegel Approach . . . . . . . . . . . . . . . . . 23

3.4 Affine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Estimation 39

4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Dynamic Nelson-Siegel Approach . . . . . . . . . . . . . . . . . 50

4.4 Affine Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Performance Evaluation 68

5.1 In-Sample Characteristics . . . . . . . . . . . . . . . . . . . . . 68

5.2 Predictive Performance . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Comparison with Similar Studies . . . . . . . . . . . . . . . . . 79

Contents vii

6 Conclusion 84

Bibliography 89

A Estimated Parameters I

B Predictions XXIX

List of Tables

2.1 Latent-Factors-Only Models . . . . . . . . . . . . . . . . . . . . 17

2.2 Macro-Finance Models . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Dynamic Nelson-Siegel Models . . . . . . . . . . . . . . . . . . . 30

4.1 ADF Test Results - Yields . . . . . . . . . . . . . . . . . . . . . 44

4.2 Variances and Correlation Matrix (reduced) . . . . . . . . . . . 44

4.3 Variance Explained by Principal Components . . . . . . . . . . 45

4.4 Eigenvectors Related to Principal Components . . . . . . . . . . 46

4.5 ADF Tests Results - Principal Components . . . . . . . . . . . . 46

4.6 ADF Test Results - Macroeconomic Variables . . . . . . . . . . 49

4.7 Random Walk Estimation & Forecasts . . . . . . . . . . . . . . 51

4.8 Nelson-Siegel RSS for Various λ Values . . . . . . . . . . . . . . 52

4.9 ADF Test Results - Latent Factors (βs) . . . . . . . . . . . . . . 54

4.10 NS-L-A Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.11 NS-L-B Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.12 NS-M-A Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.13 NS-M-B Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.14 AF-L Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.15 AF-M Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1 In-Sample Fit Results . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Predictions - Total Square Error . . . . . . . . . . . . . . . . . . 73

5.3 Prediction Rankings - Random Walk . . . . . . . . . . . . . . . 75

5.4 Prediction Rankings - NS-L-A . . . . . . . . . . . . . . . . . . . 75

5.5 Prediction Rankings - NS-L-B . . . . . . . . . . . . . . . . . . . 76

5.6 Prediction Rankings - NS-M-A . . . . . . . . . . . . . . . . . . . 76

5.7 Prediction Rankings - NS-M-B . . . . . . . . . . . . . . . . . . . 77

5.8 Prediction Rankings - AF-L . . . . . . . . . . . . . . . . . . . . 77

List of Tables ix

5.9 Prediction Rankings - AF-M . . . . . . . . . . . . . . . . . . . . 78

5.10 Predictions - RMSE . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.1 Estimation results for equation beta1A: . . . . . . . . . . . . . . II

A.2 Estimation results for equation beta2A: . . . . . . . . . . . . . . II

A.3 Estimation results for equation beta3A: . . . . . . . . . . . . . . III

A.4 Estimation results for equation beta1B: . . . . . . . . . . . . . . V

A.5 Estimation results for equation beta2B: . . . . . . . . . . . . . . V

A.6 Estimation results for equation beta3B: . . . . . . . . . . . . . . VI

A.7 Estimation results for equation beta1A: . . . . . . . . . . . . . . IX

A.8 Estimation results for equation beta2A: . . . . . . . . . . . . . . X

A.9 Estimation results for equation beta3A: . . . . . . . . . . . . . . XI

A.10 Estimation results for equation IPI-A: . . . . . . . . . . . . . . . XII

A.11 Estimation results for equation CPI-A: . . . . . . . . . . . . . . XIII

A.12 Estimation results for equation M1-A: . . . . . . . . . . . . . . . XIV

A.13 Estimation results for equation USDI-A: . . . . . . . . . . . . . XV

A.14 Estimation results for equation beta1B: . . . . . . . . . . . . . . XVIII

A.15 Estimation results for equation beta2B: . . . . . . . . . . . . . . XIX

A.16 Estimation results for equation beta3B: . . . . . . . . . . . . . . XX

A.17 Estimation results for equation IPI-B: . . . . . . . . . . . . . . . XXI

A.18 Estimation results for equation CPI-B: . . . . . . . . . . . . . . XXII

A.19 Estimation results for equation M1-B: . . . . . . . . . . . . . . . XXIII

A.20 Estimation results for equation USDI-B: . . . . . . . . . . . . . XXIV

List of Figures

3.1 Relationship of the Factors . . . . . . . . . . . . . . . . . . . . . 21

3.2 Inclusion of the Factors in the Models . . . . . . . . . . . . . . . 22

3.3 Slope - Factor Loading for Various λ Values . . . . . . . . . . . 25

3.4 Curvature - Factor Loading for Various λ Values . . . . . . . . . 26

4.1 Term Structure: 1993-2002 . . . . . . . . . . . . . . . . . . . . . 41

4.2 Term Structure: 2003-2012 . . . . . . . . . . . . . . . . . . . . . 41

4.3 Yields Time Series . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 ACF and PACF of Yields . . . . . . . . . . . . . . . . . . . . . 43

4.5 Principal Components Time Series . . . . . . . . . . . . . . . . 45

4.6 ACF and PACF of Principal Components . . . . . . . . . . . . 46

4.7 Macro Variables Time Series . . . . . . . . . . . . . . . . . . . . 49

4.8 ACF and PACF of Macro-Variables . . . . . . . . . . . . . . . . 50

4.9 Fitted and Observed Values - Nelson-Siegel for λA . . . . . . . . 52

4.10 Development of βs - Nelson-Siegel for λA . . . . . . . . . . . . . 53

4.11 Fitted and Observed Values - Nelson-Siegel for λB . . . . . . . . 53

4.12 Development of βs - Nelson-Siegel for λB . . . . . . . . . . . . . 54

4.13 IRF of NS-L-A and NS-L-B . . . . . . . . . . . . . . . . . . . . 58

4.14 IRF of NS-M-A and NS-M-B: part 1 . . . . . . . . . . . . . . . 60

4.15 IRF of NS-M-A and NS-M-B: part 2 . . . . . . . . . . . . . . . 61

4.16 IRF of AF-L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.17 Fitted and Observed Values - AF-L . . . . . . . . . . . . . . . . 64

4.18 IRF of AF-M . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.19 Fitted and Observed Values - AF-M . . . . . . . . . . . . . . . . 67

5.1 3M Yields Forecasting: One Month Prediction Horizon . . . . . 79

5.2 3M Yields Forecasting: Six Months Prediction Horizon . . . . . 80

5.3 3M Yields Forecasting: One Year Prediction Horizon . . . . . . 80

5.4 3Y Yields Forecasting: One Month Prediction Horizon . . . . . 81

List of Figures xi

5.5 3Y Yields Forecasting: Six Months Prediction Horizon . . . . . 81

5.6 3Y Yields Forecasting: One Year Prediction Horizon . . . . . . 82

5.7 30Y Yields Forecasting: One Month Prediction Horizon . . . . . 82

5.8 30Y Yields Forecasting: Six Months Prediction Horizon . . . . . 83

5.9 30Y Yields Forecasting: One Year Prediction Horizon . . . . . . 83

A.1 NS-L-A estimation results . . . . . . . . . . . . . . . . . . . . . I

A.2 NS-L-B estimation results . . . . . . . . . . . . . . . . . . . . . IV

A.3 NS-M-A estimation results - latent variables . . . . . . . . . . . VII

A.4 NS-M-A estimation results - macroeconomic variables . . . . . . VIII

A.5 NS-M-B estimation results - latent variables . . . . . . . . . . . XVI

A.6 NS-M-B estimation results - macroeconomic variables . . . . . . XVII

A.7 AF-L estimation results . . . . . . . . . . . . . . . . . . . . . . XXV

A.8 AF-M estimation results . . . . . . . . . . . . . . . . . . . . . . XXVIII

B.1 Predictions for maturity 3M . . . . . . . . . . . . . . . . . . . . XXIX

B.2 Predictions for maturities 6M, 1Y and 2Y . . . . . . . . . . . . XXX

B.3 Predictions for maturities 3Y, 5Y and 7Y . . . . . . . . . . . . . XXXI

B.4 Predictions for maturities 10Y, 20Y and 30Y . . . . . . . . . . . XXXII

Acronyms

ACF Autocorrelation Function

ADF Augmented Dickey-Fuller

AIC Akaike Information Criterion

CPI Consumers Price Index

DSGE Dynamic Stochastic General Equilibrium

FED Federal Reserve Board of Governors

HJM Heath-Jarrow-Morton Model

HQIC Hannah-Quinn Intormation Criterion

IPI Industrial Production Index

IRF Impulse-Response Function

LMM LIBOR Market Model

M1 Monetary Aggregate M1

OLS Ordinary Least Squares

PACF Partial Autocorrelation Function

PCA Principal Component Analysis

RMSE Root Mean Square Error

RSS Residual Sum of Squares

SIC Schwarz Information Criterion

USDI U.S. Dollar Index

VAR Vector Autoregression

Chapter 1

Introduction

The global financial crisis, which completely broke out at the moment the large

investment bank Lehman Brothers declared bankruptcy in September 2008, has

fully shown a large weakness of both macroeconomic models used in the central

banks and the approach of the financial markets players to the asset valuation

and the risk management: in their models, the areas of macroeconomics and

finance were not explicitly considered as interconnected. The problem has

been known already before the crisis, but only the huge collapse attracted

the necessary attention to it. Consequently, so-called macro-finance models,

incorporating both financial principles and the macroeconomic dynamics, are

becoming increasingly popular. The expected benefit of such models can be

well illustrated in terms of the pre-crisis situation. An explicit inclusion of the

macroeconomic dynamics into the models underlying the investment and risk-

management decisions could maintain a reasonable level of the risk-aversion on

the global markets, as the banks and hedge funds would be possibly able to

identify the pre-crisis boom as related to the over-heated economy and hence

very unstable. On the other hand, enlarging the macroeconomic models used

in central banks by the financial assets valuation principles might help the

monetary authority to synchronize its policy steps with the financial markets

situation, and to react on the economic reality swiftly, not ex post.

During the last fifteen years, a large progress in the macro-finance modelling

has been done, with the most important milestones mentioned throughout the

further text. The thesis compares the most frequent approaches to the term

structure modelling, both with and without an explicit inclusion of the macroe-

conomic variables. Through the models specification, estimation, and results

evaluation, the work intends to contribute to the macro-finance field in a few

1. Introduction 2

respects, reacting on some weaknesses of preceding studies:

First, most of the models are estimated using a data sample starting far

before 1990, usually covering most of the second half of the 20th century. How-

ever, since the models include global macroeconomic relations resulting from

behaviour of the economic subjects (most importantly the central banks in this

case), it is sometimes argued that such a long horizon could provide models not

resisting the Lucas Critique. Consequently, the thesis examines, whether it is

possible to estimate the models using only on the recent data, for which the cen-

tral banks objective functions can be considered as stable under the paradigm

of the inflation targeting. Second, the models rarely include maturities longer

than ten years. The thesis consequently examines, whether the models are

able to fit and predict also the longest part of the yield curve. Third, it can be

expected that individual models produce different results, in terms of their per-

formance, under various macroeconomic and financial markets conditions. For

this reason, the performance of the models is analysed also dynamically over a

certain period of time, allowing to examine the strength of various models in

different parts of the business cycle.

When performing the analysis, the thesis is structured in the following way:

The second chapter provides necessary basis. It starts with a definition of

the core terms and the notation used thorough the work, simultaneously ex-

plaining both theory and mathematics of the basic fixed income instruments

— bonds. It also introduces motivations and approaches related to the in-

terest rates modelling. The theoretical part is finished with an outline of the

quantitative methods used in the thesis.

The third chapter starts with a discussion of the factors underlying the

interest rate dynamics. Then it describes in a detail two general frameworks of

the term structure models — a dynamic Nelson-Siegel approach and affine-class

models. The theoretical background behind each of the groups is explained,

and the models are then built in both latent-factors-only and macro-finance

forms. For each model, an approach used for its estimation is also outlined.

The fourth chapter covers description of the used data and the estimation

of the models. Figures and tables are extensively used in order to illustrate

results of the estimations. The fifth chapter describes the properties of the

estimated models, both in terms of impulse-response functions and a predictive

performance. Moreover, the development of the accuracy of models in time is

inspected. The chapter also includes a comparison of the results with similar

studies. Finally, the last chapter summarizes findings of the thesis.

Chapter 2

Basic Definitions and Notations

2.1 Bond Market, Yield and Interest Rate

The thesis deals with a dynamic behavior and relationship of both macroeco-

nomic and financial variables. A brief and concise description of the variables

is the necessary first step, together with the introduction of the notation used

throughout the work.

The financial instruments underlying the topic of the thesis are bonds. A

bond can be defined as ”any interest-bearing or discounted government or cor-

porate security that obligates the issuer to pay the bondholder a specified sum of

money, usually at specific intervals, and to repay the principal amount of the

loan at maturity.” (Downes & Goodman 1998, pg. 59). An important special

type of bonds is a zero-coupon bond, which pays no cash flow except for the

principal amount at the maturity of the bond. Zero-coupon bonds are almost

always traded with a discount, i.e. with their price smaller than the principal

amount, whereas coupon-bearing bonds can be traded for a price either below

or above the principal amount.

The bond market is a place where a bond demand and a bond supply

meet. The demand side is formed by investors motivated to obtain a return

from the amount they are offering to fund-seekers creating the bond supply.

Such definition is valid for the primary bond market, where the bonds are

underwritten. Nevertheless, a secondary market also exists, where the supply

of bonds is formed by the subjects that are re-selling the bonds once purchased,

i.e. investors exiting the bond market. From a simplified point of view, the

secondary bond supply may be considered as a negative bond demand, therefore

it is possible to describe both bond markets together.

2. Basic Definitions and Notations 4

Generally, a motivation of the investors is to realize a yield. They enter the

bond market only if the yield resulting from the bond purchase, after adjusting

for the risk related to the investment, is higher then a risk-adjusted yield of

different investment opportunities. Contrary, the fund seekers enter the bond

market in order to minimize their costs of funding: to get the lowest possible

interest rate. The terms yield and interest rate can not be considered as equiv-

alent terms — basically not because of the method of their calculation, but

because of the economic motivation related to them. Consequently, as notes

Choudhry (2011), the terms yield curve and term structure of interest rates, as

defined below, are in general not exactly the same and a precise description of

them is necessary for an exact specification of the subject of the thesis.

Yield

Yield (in general) represents a rate of an increase or a decrease of resources that

a subject (investor) holds for purposes other than consumption. It is usually

expressed as relative change during a period of time. Investors want to invest

their resources into such an asset, real or financial, that offers maximal yield

within given risk category — alternatively, the investor wants to maximize

the risk-adjusted yield. One of the possibilities, where to invest, is the bond

market. Investors enter the bond market having an intuition about the yield

they want to achieve (required yield), which is usually different from the actual

yield realized on the market — an investor confirms a trade only if his required

yield is lower or the same as the observed market yield. The required yield is

a function of many variables, with the most important being:

time factor ρ — the ”core” compensation required by the investors for post-

poning their consumption into the future

expected inflation E [π] — included as the investor is concerned with the real

yield regardless to the changes of the price level

risk premium ξ — required for the uncertainty related to the investment; it may

be further split into:

� market risk premium ξM — related to the fact, that market condi-

tions may change, which would have an impact on the price of the

bond


� liquidity risk premium ξL — related to the risk that investor will not

be able to sell the bond on the secondary market without any costs

(time, transaction costs)

� credit risk premium ξC — reflecting the fact there is non-zero prob-

ability the counter-party will not be able to meet its obligations

related to the bond

Individual required yield yreq is then a function of the variables mentioned

above:

yreq = f (ρ, ξM , ξL, ξC , E [π]) (2.1)

The function 2.1 is unique for each investor. Moreover, it differs among various

instruments, since these are related to different risks and time dimension. To

make the analysis simple and consistent, the study focuses on a single instru-

ment: government bonds1. However, there will be bonds of various maturities

used in the thesis — the variance in the required yield for bonds differing only

in their maturity is often called term premium. The individual demand for

bonds is then determined by the difference between the required yield and the

yield observed on the bond market, together with other factors: individual

wealth, income, taxation, institutional factors (law enforcement, political sta-

bility, etc.). Moreover, the demand for bonds is also influenced by the yields

on alternative markets – a stock market, a foreign currencies market, and a

real assets (commodities, real estate,...) market. The aggregate bond demand

is then the sum of all individual demands.

Let Pc,t(τ) be a price2 of a coupon-bearing bond at a time t, maturing at

time T = t+ τ , with n coupon payments with rates c1, c2, . . . cn paid at times

t1, t2, . . . tn. If the bond is held until its maturity, its yield (yytm) per one

period is a function of the variables (parameters):

yytm = f (Pc,t(τ), t, τ, ci, ti) for i=1...n (2.2)

1It can be assumed that results of the thesis might be generalized for the whole bondmarket (particularly also corporate or municipal bonds), in case the higher risk premium ishandled properly.

2In the study, price will be always expressed as a percentage of the nominal value – it isnot necessary to deal with either the nominal value of the bond or the absolute value of thecoupons.


and can be defined as a solution of the following equation.

Pc,t(τ) =n∑i=1

ci

(1 + yytm)ti−t+

1

(1 + yytm)τ(2.3)

Usually, a few simplifications may be used: all the coupons are assumed to

be identical c = c1 = c2 = ... = cn, paid in regular periods starting one period

after t with the last coupon paid with maturity. If the maturity is expressed

in years, the period between two coupon payments is equal to τ/n. The price

can be consequently expressed in a simpler way:

Pc,t(τ) =n∑i=1

c

(1 + yytm)iτn

+1

(1 + yytm)τ(2.4)

The bond price can be further split into a so-called clean price and the accrued

interest. However, since only the zero coupon bonds will be analysed in the

further work, it is not necessary to describe the general coupon bonds into more

detail. The Equation 2.4 implicitly assumes that the interest period is identical

with the coupon period. However, it can be also convenient to express the

Pc,t(τ) using an infinitely small interest period, i.e. continuous compounding:

Pc,t(τ) =n∑i=1

ce−yytmiτn + e−yytmτ (2.5)

In case the coupon rate c = 0, the bond is called zero-coupon bond, denoted

as Pt(τ)(omitting the c subscript), and the yield to maturity can be expressed

directly from either Equation 2.4:

yytm = Pt(τ)−1τ − 1 (2.6)

or Equation 2.5:

yytm = − lnPt(τ)

τ(2.7)

Generally, the bonds with various maturities, having all other parameters

identical, have to be considered as different bonds, since the inputs to the

Equation 2.1 are not the same - their yields differ in the term premium. The

function of yield to maturity with respect to the maturity is then called a yield

curve, which will be described in detail below.


Interest Rate

The fund seekers, which form the supply side of the bond market, are trying

to find the cheapest way how to finance their needs. The cost of borrowing

the funds (i.e. price of the free funds) is represented by the interest rate. The

bigger the interest rate is, the more costly it is for a borrower to get funds, and

the more profitable it is for a creditor to provide them.

In case of the coupon bond paying n coupons, it is useful to split the cash

flows and replicate the bond by a set of n zero-coupon bonds (the one with

the longest maturity including also the notional), which are sold to n investors.

Since each of the investor faces different maturity, they require different yields

- put differently, the fund seeker pays different interest rate for each single

cash-flow. The price of the bond, i.e. the amount the fund-seeker obtains

when issuing the bond, is correspondingly determined as the future cash flows

discounted by individual interest rates:

Pc,t(τ) =c[

1 + rt(1τn

)] 1τn

+ . . .+c[

1 + rt

((n−1)τn

)] (n−1)τn

+1 + c

[1 + rt (τ)]τ(2.8)

Pc,t(τ) =n∑i=1

c[1 + rt

(iτn

)] iτn

+1

(1 + rt(τ))τ(2.9)

where rt(iτn

)is the interest rate set in the time t for an amount repayable in

time t+ iτn

, i = 1 . . . n.

Obviously, the Equation 2.9 is very similar to the Equation 2.4. However,

the forces changing the bond supply are different. The borrowers will indi-

vidually calculate interest rate they are willing to pay, depending on costs of

different funding resources (bank loans, shares etc.), and their credit capacity

related to their real investments opportunities. The bond will be issued only if

its realization on the market is not more expensive than the acceptable interest

rate.

In case of a zero-coupon bond, the Equation 2.9 (plugging 0 for c) allows

an analytical expression of rt(τ), which is equal to the right-hand side of the

Equation 2.6:

rt(τ) = Pt(τ)−1τ − 1 (2.10)


The same holds for continuous compounding and Equation 2.7:

rt(τ) = − lnPt(τ)

τ(2.11)

For purpose of construction of various interest rate models, it is necessary

to present also the concept of an instantaneous spot interest rate rt. It can

be defined as an interest rate of a loan or a bond repayable after an infinitely

small period:

rt = limτ→0

rt(τ) (2.12)

Another common step is to define the forward rate ft(T1, T2), which can be

expressed as interest rate set in time t, for period beginning at time T1 = t+ τ1

and ending at T2 = t + τ2, i.e. τ2 > τ1. In order to ensure no arbitrage

possibilities on the market, following equation must be fulfilled:

(1 + rt(τ2))τ2 = (1 + rt(τ1))

τ1 (1 + ft(T1, T2))(τ2−τ1) (2.13)

ft(T1, T2) =

[(1 + rt(τ2))

τ2

(1 + rt(τ1))τ1

]τ2−τ1− 1 (2.14)

Similarly for continuous compounding:

eτ2rt(τ2) = eτ1rt(τ1)e(τ2−τ1)ft(T1,T2) (2.15)

ft(T1, T2) =τ2rt(τ2)− τ1rt(τ1)

τ2 − τ1(2.16)

Also for the forward rate, the instantaneous version can be defined as a rate

for an infinitely short period between T1 a T2:

ft(T1) = limT2→T1

ft(T1, T2) (2.17)

For T1 equal to t, the forward rate is identical to the spot rate. Since

both forward rates and bond prices (zero-coupon bonds with notional equal

to one currency unit) are given directly by the spot interest rates (and vice

versa), knowledge only one of these three sets of variables provides knowledge

of all of them. For this reason, focusing directly on the continuous compound-

ing (discrete-time compounding would provide equivalent form) and combining

equations 2.11 and 2.15, following relationships between forward rates and bond

prices can be expressed in terms of k forward rates (respectively k− 1 forward


rates and a spot rate):

Pt(τ) = e−[τ1ft(t,T1)+(τ2−τ1)ft(T1,T2)+...+(τ−τk−1)ft(Tk−1,Tk)] (2.18)

In case the k will grow infinitely for fixed τ , i.e. the length of the (Tj−1, Tj)

intervals will became infinitesimally small and all the forward rates in the Equa-

tion 2.18 will become instantaneous, the price of a bond can be expressed as:

Pt(τ) = e−∫ t+τt ft(s)ds (2.19)

2.2 Term Structure of Interest Rates

As has been described in the previous section, a price of a bond Pc,t(τ) is deter-

mined by the interaction of the bond demand, i.e. investors maximizing their

yield to maturity yytm, and the bond supply — fund seekers minimizing their

costs represented by a set of interest rates {rt(τi)}ni=1. However, an individual

subject does not influence the price significantly, as he or she can be assumed to

be a price-taker. From his point of view, the market prices are given, implying

both yields and interest rates of various maturities.

The term structure of interest rates is a function of rt(τ) with respect to

τ — the maturity. The interest rates of various maturities usually differ, and

the graphic representation of the term structure can consequently reach vari-

ous shapes - it can be upward-sloping, flat, downward-sloping, or even partly

upward- and partly downward-sloping. The theoretical background of various

shapes will be explained below.

Similarly to the term structure of interest rates, the yield curve can be

expressed as a function o yield (yytm) with respect to the maturity of the bond.

Generally, the yield curve and the term structure of interest rates are different:

a bond with a price Pc,t(τ) implies whole set {rt(τi)}ni=1 related to the cash flows

resulting from the bond, whereas the single yield for the investor is determined

directly. Consequently, the yield (and resulting yield curve) is also influenced

by the structure of the cash flows, and can be seen as a sort of average of the

situation over the whole term structure. For this reason, it is more suitable to

use the term structure of interest rates, since each interest rate is related to

single cash flow.

Both terms — the yield curve and the term structure of interest rates — are

similar only in two cases: when the interest rate is the same for all maturities


(both the term structure and the yield curve are absolutely flat), and when the

bonds used for its construction are zero-coupon bonds. The former condition is

purely theoretical, but the latter can be often met, since the zero-coupon bonds

are often used in this way. For this reason, the terms will be used equivalently

in further text, always assuming they refer to the term structure of interest

rates (or yields) constructed from the zero coupon bonds.

Yield Curve Theories

During the time, there were many attempts to offer a theoretical explanation for

the shapes of the term structure of interest rates. The most important theories,

examining the yield curve from different points of view, are the expectations

hypothesis, the liquidity preference hypothesis, the market segmentation hy-

pothesis and the preferred habitat hypothesis. Concise and brief description of

them can be found in Cox et al. (1985), which is followed also below:

The expectations hypothesis3 considers the current observed forward rates as

an unbiased estimator of the future spot rates. After plugging the Equation 2.19

into the Equation 2.11, it becomes obvious that the rt(τ) is an average of the

forward rates over the period until maturity τ . Consequently, the upward-

sloping term structure of interest rates is explained by the expected growth of

the (instantaneous) spot rates; similarly for the other possible shapes of the

yield curve. However, this explanation coincides with the empirical fact that

the yield curve is upward-sloping most of the time, which would provide sys-

tematic error to the expectations. Moreover, the expectation hypothesis does

not incorporate any difference in risk among various maturities (i.e. assumes

zero term premium). These facts imply that the forward rates cannot be an

unbiased estimator of the future spot rates under the real probability measure4,

and the pure expectations hypothesis is insufficient.

Basic principles of the liquidity preference hypothesis have been stated al-

ready in the work of Hicks (1946), who improved the expectations hypothesis

by the risk related to various maturities. Due to the risk aversion, the in-

vestors require a positive risk premium ξ as entering the Equation 2.1, which

makes the bond price relatively smaller. Consequently, as obvious from the

Equation 2.19, this is equivalent with the instantaneous forward rates being

3Various types of the expectation hypothesis are described for example in Malek (2005).4Nevertheless, the forward rates can be used as an estimate of the future spot rates

assuming we are in a risk-neutral world, e.g. the expected value is given under the risk-neutral measure.


relatively higher than the expected spot rates. Since the long-term bonds are

related to a relatively higher risk accepted by the investors, they demand a

higher risk premium. Because of this, the long-term yields are relatively higher

than the short-term, which sufficiently explains the prevailing upward slope of

the yield curve.

Different point of view offers the market segmentation hypothesis, mentioned

by Culberison (1957). It assumes that the investors on markets of bond of var-

ious maturities are strictly different, not moving between the markets. The

supply and demand for these bonds are therefore not related among the mar-

kets, and there is no reason for the yields to be equal. This hypothesis does

not directly explain, why the term structure of interest rates is usually upward-

sloping, however, it can offer an intuition why the interest rates for various

maturities can be distinctly different (which is useful especially at the times of

financial crises).

Finally, the preferred habitat hypothesis, introduced by Modigliani & Sutch

(1966), joins all the other theories together, using the positive aspect of each of

them. It is able to both explain the prevailing positive slope of the yield curve

and offer a reasoning for its different shapes. All the theories are, rather than

competing, together forming a complex view on the mechanisms underlying the

shape of the term structure of interest rates.

Term Structure Construction

An important issue related to the analysis of the term structure of interest

rates is its construction. Since the yield curve (as a continuous function of the

maturity) is not fully observed and there can be identified only a small number

(usually not more than 20) of points — yields of zero bonds of several maturities

— on the market, the problem has to be handled properly. It is necessary

to use mathematical and/or statistical techniques to fit these observed values

with a continuous function, preferably defined on (0,∞). The most common

approaches to the zero-bond yield curve construction, described into more detail

for example in Filipovic (2009), are:

Bootstrapping is based on the coupon bond prices observed on market, be-

ginning with a calculation of one-year interest rate rt(1) from one-year

(zero-coupon) bond. The rate is then inserted into the Equation 2.9 for

a two-year coupon bond, leaving the two-year interest rate rt(2) the only

unknown variable. The process is recursively repeated, obtaining interest


rates of higher maturities based on the knowledge of the rates of shorter

maturities. The approach is, however, producing an output which may

be biased, because the coupon-bond and zero-coupon-bond markets are

not perfect substitutes, particularly due to different cash-flows structure

resulting from the bonds. This restricts the bootstrapping to be used as

a theoretical tool rather than practical.

Interpolation can be considered as purely mathematical method. Using polyno-

mial interpolation, single polynomial function is used to fit all observed

maturities (poles), which may, however, offer unsatisfactory results par-

ticularly for both extremely short and long maturities. A different method

is the spline interpolation, which uses polynomials of lower orders on mul-

tiple sub-intervals. Most frequently used are either a cubic spline func-

tion, creating function continuous up to its second order, or a B-spline

interpolation, with splines constructed as a weighed sum of recursively

defined base-spline functions.

Nelson-Siegel approach, defined by Nelson & Siegel (1987) and sometimes used

in its extended version offered by Svensson (1994), is perhaps the most

frequently used method for the yield curve construction, mainly for its

relative simplicity. The approach is based on a specific functional form

including four parameters, which can be used for fitting the observed

yields. The method is described into detail in the Section 3.3, together

with the most common estimation technique.

Stochastic interest rates models represent distinctly different approach. Com-

pared to the static character of the previous methods, the interest rate

models try to capture dynamics if interest rates, typically focusing on the

short rate dynamics in continuous time. The model is first specified, and

a bond price is derived, defined as a function of the maturity and various

parameters (capturing dynamics of the latent factors, i.e. usually drift,

volatility and the mean-reversion level of the short rate). After the model

is calibrated on the observed bond prices, it can be used to price bonds

of any maturity, which is equivalent to the calculation of the entire term

structure (see the Equation 2.11). Stochastic models will be in the thesis

represented by an affine-class model, described in the Section 3.4


2.3 Term Structure Models

Purposes for Term Structure Modelling

To be able to describe and classify various models of the terms structure of

interest rates, it is first necessary to distinguish between two different general

motivations for the interest rates modelling, as each of them requires a different

approach:

1. Financial assets valuation. Time t value V (t) of any financial asset

(typically a financial derivative) can be determined as an expected value

of an uncertain future value V (T ) discounted to the present:

V (t) =EP [V (T )]

eτyreq(2.20)

where τ = T − t is the time to maturity of the asset and the P subscript

denotes the expectations under the real probability measure. It is difficult

to model yreq, since it is from its definition individual for each investor and

depends on particular instrument, maturity, time, market situation etc.

Moreover, the determination of the expected value EP [V (T )] is equally

demanding. For this reason, so-called risk-neutral valuation has been de-

veloped. This approach is built on an assumption, that the market is

arbitrage-free and complete, which ensures there exists a unique measure

(risk-neutral), equivalent to the original. Under this new measure, the

expected rate of return of assets is equal to the risk-free rate. Conse-

quently, instead of expressing the expected future value under the real

probability measure P and discounting by a risk-adjusted rate yreq, the

risk-neutral probability measure Q is used and the value of the instrument

is obtained as the expected future value under this risk-neutral measure

discounted simply by the risk-free rate, which is assumed to be equal to

the instantaneous interest rate rt.5

V (t) =EQ [V (T )]

eτrt(2.21)

Assuming the risk-free interest rate is not fixed, but floating over time,

5Relation of the original real-world and the equivalent risk-neutral measures is basedon Girsanov theorem and related use Radon-Nikodym derivative, which are described forexample in Musiela & Rutkowski (2005), and will be used also in the Section 3.4.


the Equation 2.21 can be generalized into the form:

V (t) = EQ

[V (T )

e∫ Tt rsds

](2.22)

More precisely, instead of using an abstract ”discount factor” approach,

it is better to talk about a so-called numeraire. Numeraire is an asset in

terms of which the values of other assets are computed. Typical example

of a numeraire is the money market account M(t), having at the time

t = 0 value M(0) = 1, which is accrued by the risk-free instantaneous

interest rate, using continuous compounding. The value of the money

market in time t > 0 is then:

M(t) = 1 ∗ e∫ t0 rsds (2.23)

Under the risk-neutral measure, the original asset V (t) expressed in terms

of the numeraire (i.e. discounted by the money market account, there-

fore called also as a discounted process) has to be a martingale. For this

reason, the term risk-neutral measure is sometimes replaced by a term

equivalent martingale measure. Utilizing properties of martingales, ex-

pected future value of the discounted process is equal to its present value

(still under the risk-neutral measure Q):

V (t)

e∫ t0 rsds

= EQ

[V (T )

e∫ T0 rsds

](2.24)

which is equivalent to the Equation 2.22.

Specifically for the bond, it is useful to utilize the fact that the value of

any bond at maturity is equal to its notional (i.e. V (T ) = P (T ) = 1 in

terms of relative price). In order to maintain consistency in the notation

throughout the thesis, the time t price of a zero-coupon bond maturing

at the time T = t+ τ , with τ standing for time to maturity of the bond,

will be henceforth noted Pt(τ). The Equation 2.22 then simplifies into

the form:

Pt(τ) = EQ

[e−

∫ t+τt rsds

](2.25)

Obviously, in case of bonds, the stochastic evolution of the instanta-

neous interest rate rt is the only variable to be modelled in order to


capture dynamics of bond prices. Through the relationship captured in

the Equation 2.11, the whole term structure can be simply derived from

the calculated bond prices of various maturities.

2. Portfolio management. The previous approach observes the actual sit-

uation on the market, according to it models the risk-neutral dynamics of

the interest rates, and determines the value of a derivative, a bond or the

term structure of interest rates. Contrary to it, modelling of interest rates

for portfolio management purposes is based on an analysis of the mar-

kets in the context of their historical development and, moreover, offers

predictions of the future development. Where the risk-neutral approach

considers the whole market as an arbitrage-free system, the portfolio ap-

proach focuses on individual financial assets and tries to determine, which

of them should be included into (or excluded from) the portfolio, in order

to maximize future value of the portfolio, or, respectively, to minimize loss

resulting from holding particular instruments. The portfolio management

requires an analysis of the real-world probability dynamics of the vari-

ables. Instead of calibrating the models on the actual data, an estimation

of the time series parameters, typically using autoreggresive models in-

cluding the Vector Autoregression (VAR), is a frequent approach to term

structure modelling under this motivation. Consequently, the continu-

ous dynamics is for the estimation purposes often replaced by a discrete

specification of the models.

Since the aim of the thesis is to model the real dynamics of the interest

rates, rather than to price financial assets, the latter motivation can be picked

as the crucial for the thesis.

Interest Rates Models Classification

In order to be able to evaluate the results of the performance comparison of the

models introduced in Chapter 3 correctly, it is necessary to classify the models

first. As points out Rudebusch (2008), the term structure of interest rates can

be generally modelled in three ways:

From the financial point of view, modelling the short rate as a function of latent

factors, and longer rates through an introduction of the risk premium.


Focusing on the macroeconomic relations only, which considers the short rate to

be a product of the macroeconomic dynamics, and the long rates to be

given by expectations of the short rates.6

Combination of both, which yields the macro-finance models.

Detailed classification of the models is outlined in the Table 2.1 and Ta-

ble 2.2, following De Pooter et al. (2007), Filipovic (2009), Witzany (2012),

Malek (2005) and Stork (2012). The classification already follows the fact that

the thesis is focused on capturing the real-world dynamics. The first family

of models, latent-factors-only models, is divided into two groups. Statistical

models7 capture the development of yields of various maturities without a di-

rect inclusion of relationships between various maturities, which is what the

no-arbitrage models focus on, employing the no-arbitrage condition, when ex-

pressing the longer rates in terms of the short rate.8

Short rate models, as a category within the no-arbitrage models, are try-

ing to capture the dynamics of the instantaneous interest rate, which is then

used to determine bond prices directly from the Equation 2.25. The short

rate dynamics is given specified as a function of several latent factors — state

variables. A frequent approach is to set the short rate itself as the only one

latent factor, whose dynamics is then given explicitly by a certain stochastic

differential equation. However, particularly for the purposes of the forecast-

ing, more then one factor approach is often used (usually two or three factors

are included). Moreover, for computational purposes, the functional form is

usually set as affine. An affine three-factor model is further described in the

Section 3.4. Second category within the no-arbitrage class are so-called term-

structure models, which focus on the forward rates, either instantaneous as uses

Heath-Jarrow-Morton Model (HJM) developed by Heath et al. (1992), or for a

given periodicity, as captures LIBOR Market Model (LMM) based on Brace

et al. (1997).

It is necessary to note, that the borders between the models classes are

not strict: Dynamic Nelson-Siegel approach has also its no-arbitrage version,

6Since these models are purely macroeconomic, not considering the interest rate as afinancial variable, they will not be further discussed.

7Represented by the dynamic Nelson-Siegel model introduced by Diebold & Li (2006) andfurther described in the Section 3.3.

8Sometimes, the no-arbitrage models are considered to be only a sub-group of the shortrate models, specific in their ability to fit the observed data perfectly, thanks to enough freeparameters. Such classification is meaningful for purposes of the risk-neutral calibration, butnot for capturing the real-world dynamics. For this reason, the no-arbitrage models will inthe thesis include all models utilizing the no-arbitrage conditions.


some of the models (for instance Ho & Lee 1986 or Hull & White 1990) can

be considered as part of either HJM framework or a as short rate model. How-

ever, the classification offered here still offers a view on the basic characteristic

properties of the models.

Table 2.1: Latent-Factors-Only Models

Group Category Type Example

Statistical Dynamic Nelson-Siegel

No-ArbitrageShort Rate

One-Factor Vasicek (1977)Multi-Factor Fong & Vasicek (1991)

Term StructureCont. Compound. HJM

Simple Compound. LMM

Source: author’s own

The second family of the models are the macro-finance models, which, in

contrast with the latent-factors-only models, explicitly include macroeconomic

variables when modelling the interest rates dynamics. Rudebusch (2008) rec-

ognizes three possible approaches to their construction:

1. First, following the work of Ang & Piazzesi (2003) as a cornerstone in this

field, an affine no-arbitrage latent-factors-only model in a discrete-time

specification is modified. This modification is made quite simply: the

vector containing the latent state variables, which is assumed to follow a

VAR process, is enriched by the observed macroeconomic variables.

2. Second, a Dynamic Stochastic General Equilibrium (DSGE) model serves

as basis. In this model, the stochastic discount factors is directly derived

from the households’ optimization problem. This group of models is

distinctly different from the others, since it uses purely macroeconomic

framework to derive values of financial variables, whereas in the all other

cases, the originally financial models are extended by the macroeconomic

variables.

3. Third, the dynamic Nelson-Siegel approach, as mentioned above, may be

utilized, as proposed by Diebold et al. (2006). The way of the macro-

variables inclusion is similar to the approach followed in the first point:

the vectors of macroeconomic and latent variables are merged to ensure

the joint dynamics within the VAR process.


Moreover, the macro-finance models differ also in the character of the macroe-

conomic variables included. Two general approaches are possible: either ex-

plicit macro-variables time-series enter the model, or indices of key macroeco-

nomic areas, built for example by using Principal Component Analysis (PCA),

represent the macro-dynamics instead. This issue is further described in the

Chapter 4, when describing the data used for the practical analysis.

Table 2.2: Macro-Finance Models

Approach Factors’ Dynamics Time-Series Character

Affine model extension Non-structural: VARExplicitIndices

DSGE framework Structural: optimization Explicit

Nelson-Siegel extension Non-structural: VARExplicitIndices


In further chapters, the thesis will focus on a comparison of the performance

of various models, especially in terms of the estimation properties and the

forecasting ability. Two latent-factors-only models will be used, including both

the no-arbitrage approach represented by a three-factor affine model, and the

statistical dynamic Nelson-Siegel framework. Similarly, two related macro-

finance models will be introduced and their ability to beat the predictive power

of the latent-factors-only models will be tested —the macro-finance models will

be built as extensions of the affine and Nelson-Siegel latent-factors-only models.

In these models, only the explicit inclusion of the macroeconomic variables will

be considered.

2.4 Methodology

The thesis will continue in the following way: First, in the Chapter 3, the men-

tioned models will be described into a detail. When dealing with the dynamic

Nelson-Siegel framework, particular focus will be set on the possibilities to ob-

tain the optimal values of the parameters. Contrary, for the affine models, the

most challenging task will be to derive properly the discrete-time specification

of the model, following relevant literature and utilizing the fundamentals of the

financial mathematics and the stochastic calculus.

Second, after describing the data used for the analysis and splitting them


into training and testing samples, PCA will be used to reduce the dimensionality

of the yields. Afterwards, the models themselves will be estimated, using several

techniques:

� time series analysis, utilizing the Box & Jenkins (1970) methodology,

particularly the reduced-form VAR;

� Ordinary Least Squares (OLS) method;

� and numerical iterative methods.

Third, the estimations will be evaluated. The features of the estimations

themselves will be described, using Residual Sum of Squares (RSS) to compare

the models in terms of their ability to fit the observed values, and Impulse-

Response Function (IRF) to depict the qualitative properties of the estimations.

The estimations will be also used to construct predictions, and the forecasting

performance will be compared by calculating the total squared predictive error.

Finally, all the models will be re-estimated for a shorter data samples, and

the time development of the forecasting ability will be inspected. In this case,

the Root Mean Square Error (RMSE) will be used as the quantitative measure.

At the end, the results will be compared with outcomes of similar studies.

In the thesis, MATLAB and R-Studio are used when estimating the models

and producing the forecasts, as well as for construction of various charts.

Chapter 3

Description of Models

3.1 Factors and Principal Component Analysis

One of the biggest difficulties related to the yield curve modelling and fore-

casting seems to be the fact, that it is necessary to capture the dynamics of

many maturities (usually 10-15). This might be, due to the resulting over-

parametrization of a model, quite a problematic task. The common approach

is to model the dynamics of several latent (i.e. unobservable) factors, and de-

rive relations of the yields to these factors. For purposes of the financial assets

valuation, the task can be well simplified by calibrating a one-factor model,

typically Vasicek (1977) or Hull & White (1990) model. Since the dynamics

of rates of longer maturities are given by the no-arbitrage condition (applied

under a risk-neutral probability measure) in these cases, the model is able to

capture the dynamics using only one source of uncertainty – the short rate.

However, for purposes of a dynamic real-world analysis (as an opposite to

the risk-neutral pricing) and particularly forecasting, this approach might be

considered as inefficientRather than fitting the observed market situation, the

model should in this case also incorporate the exact relationship between the

yield curve shifts and changes of the factors driving these movements — since

various parts of the term structure have a different sensitivity to changes of

various underlying factors, it is necessary to allow multiple sources of the risk

to enter the models. It has been shown by Litterman & Scheinkman (1991)

that three factors are perfectly able to explain the dynamics of the whole yield

curve. The nature of these factors can be obtained by the PCA of yields, which

allows an elegant reduction of the dimension1 while setting a useful basis for

1It is implied directly by the nature of the PCA, that there does not exits any linear

3. Description of Models 21

building macro-finance models. The logic can be illustrated by the Figure 3.1.

Figure 3.1: Relationship of the Factors

Source: author’s own.

Representation of the entire yield curve by the three factors — principal

components (usually called Level, Slope and Curvature of the yield curve) —

can be considered as distinctively accurate. Litterman & Scheinkman (1991)

have shown that the three factors are able to explain more than 98% of the

total variance.2 The use of level, slope and curvature as the factors underlying

the yield curve movements is therefore beneficial in two ways:

� Only three variables are modelled, which solves the over-parametrization

problem.

� The new variables are tightly related to real macroeconomic and financial

factors driving the dynamics of the term structure.

The latter has been illustrated by Diebold & Li (2006), who have shown that

the level (the first component) is reflecting long-term inflation expectations,

whereas the slope (the second component) is connected to the real activity

and short-term inflation and growth expectations. The third component is

sometimes believed (Kollar 2011) to be related to the expectations of economic

growth and inflation in the medium time horizon.

Assuming the market is efficient, as defined by Fama (1970), market prices

(i.e. also bond yields ) should always capture all relevant available informa-

tion, including development of the macroeconomic variables. Consequently, for

purposes of the financial assets pricing, capturing dynamics of the latent fac-

tors (level, slope and curvature) is perfectly sufficient. However, such approach

does not include the particular relations of the macroeconomic variables and

transformation of the original variables incorporating more information (i.e. variance ofthe original data) than the first principal component, and a linear transformation with thetransformation vector orthogonal to the vector of the first transformation, which would beable to include more information than the second principal component, etc.

2A similar analysis will be performed also in the practical part of the thesis.


the term structure of interest rates. This can be a shortcoming for economic

subjects — typically central banks or governments — assessing an impact of

particular monetary or fiscal policy steps on the interest rates of various ma-

turities. This is where the macro-finance models are particularly useful, when

adding the macroeconomic variables directly into the models - their explicit

inclusion allows a simple analysis of the impact of monetary (or fiscal) policy

steps on the interest rates and its further propagation into the whole economy.

The difference between the latent-factors-only and the macro-finance models is

depicted by the Figure 3.2.

Figure 3.2: Inclusion of the Factors in the Models


Four different models will be introduced in the following text, focusing on

their nature, derivation and an approach to their estimation and construction

of predictions.

Random walk will serve as a simple baseline, which the other models will be

assumed to outperform.

Dynamic Nelson-Siegel model, i.e. a model based on a specific utilization of the

Nelson-Siegel framework, will be estimated in two ways:

1. A simple dynamic version of the Nelson-Siegel representation of the

yield curve, including only the latent variables.

2. A macro-finance model explicitly including macro-variables into the

vector of factors driving the movements of the term structure of

interest rates.

Affine model, based on the no-arbitrage assumption, will be also built in two

forms, following the same logic:

1. An affine model including the three latent factors only.


2. And again its macro-finance extension with macro-variables included

in the vector of the factors — state variables.

3.2 Random Walk

To be able to assess the performance of both latent-factors-only and macro-

finance models, it is necessary to introduce a simple model serving as a baseline.

The yields of most maturities can be regarded as nonstationary, as will be shown

in the Section 4.1. For this reason, a random walk could be regarded as the most

simple baseline. Moreover, considering the random walk as a process the prices

are assumed to follow under the efficient market hypothesis (in compliance with

Fama 1965), testing the ability of other models to outperform the random walk

will implicitly create a naıve test of the market efficiency itself.3

The random walk (without a constant) can be written as:

rt (τ) = rt−1 (τ) + at,τ (3.1)

where at,τ represents a white noise process, i.e.

E [at,τ ] = 0

var [at,τ ] = σ2a,τ

cov [at,τ , as,τ ] = 0 for s 6= t

Predictions resulting from the random walk are very simple to obtain - they

are equal to the latest observation:

E [rt+1 (τ)] = E [rt (τ)] + E [at+1,τ ] = rt (τ) (3.2)

3.3 Dynamic Nelson-Siegel Approach

Basic Description

A pivotal work in this area is Diebold & Li (2006). Authors try to react on poor

results of no-arbitrage models in terms of predictive performance, assuming that

3It is, however, necessary to note, that the author is not aiming at proving or refusing thehypotheses of Eugene Fama, which are definitely going far behind the extent of the thesis.


leaving the no-arbitrage restrictions (used in the Section 3.4 when constructing

the affine model) may lead to more accurate forecasts.

The first step to take when building the model is to describe the basic

Nelson-Siegel framework, based on the work of Nelson & Siegel (1987). Authors

offered a statistical approach to the estimation of the term structure of interest

rates, which quickly became popular, mainly for its relative simplicity. The

main building block of the framework is a representation of the yield curve as

a function of the maturity in the following form:

r (τ) = β1 + β2

(1− e−λτ

λτ

)+ β3

(1− e−λτ

λτ− e−λτ

)(3.3)

where τ = T − t represents the time to maturity and β1, β2, β3 and λ are the

parameters to be estimated. Later, the Equation 3.3 was extended by Svensson

(1994), including an extra term to enhance the flexibility of the function when

fitting the term structure; however, the original form is often considered to be

flexible enough, which will be assumed also henceforth. The resulting estimated

term structure does not fit the observed values exactly — consequently the

bond prices implied by the Nelson-Siegel estimation may slightly deviate from

the observed ones. However, the specific exponential form of the function as

defined by the Equation 3.3 ensures substantial flexibility, which makes the

approach attractive.4

The three indexed beta parameters are of a special interest,5 since they can

be (and often are) considered as representatives of the main characteristics of

the term structure of interest rates — the latent factors:

� β1 is common for all maturities, so it represents the level of the term

structure.

� The expression directly following β2 (i.e. its factor loading) is decreasing

with growing maturity (approaching 1 with the maturity decreasing to

0, respectively going to 0 with the maturity approaching +∞), assuming

λ is positive. Positive β2 therefore implies short maturities being bigger

than the long ones and vice versa — and β2 can be therefore interpreted

as a negative slope of the term structure. The development of the β2

4Compared to the restricted ability of Vasicek (1977) or Cox et al. (1985) models to fitthe observed term structure of interest rates (Malek 2005).

5Whereas λ is set in order to ensure either an optimal shape of the yield curve or the bestfit of the original and model-implied yields, as will be described below.


factor loading, for maturities between 0 and 15 and various values of

lambdλ a, is outlined by the Figure 3.3

Figure 3.3: Slope - Factor Loading for Various λ Values

0 1 2 5 10 20 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

maturity

fact

or lo

adin

g

0.10.20.512510100


� The β3 factor loading is positive for positive λ and approaches zero for

maturities reaching either 0 or +∞. The maturity, for which the factor

loading is in its maximum, depends on the value of λ: with growing λ,

the maturity in which the maximum is obtained is decreasing. Conse-

quently, the third expression can be considered as a location of a ”hump”

in the term structure, whereas the β3 itself represents extent of the cur-

vatures. Setting the λ positive, yields of medium maturities will be rel-

atively higher than the short or long rates. Possible values of the factor

loading are illustrated by the Figure 3.4.

The interpretation of the β parameters is very close to the interpretation

of the first three principal components of the yield curve, as explained above.

Nelson-Siegel parametrization consequently offers an elegant and simple ap-

proach how to use the benefits of the Litterman & Scheinkman (1991) findings

in practice. The favourable properties of the Nelson-Siegel representation con-

firm also Diebold & Li (2006), when noting that this approach keeps the key

properties of the term structure, mostly in terms of implied forward rates and

the ability to fit well many possible shapes of the yield curve observed at the

market.

When fitting the yield curve on the observed data, one of the frequent

approaches is to estimate separately the β parameters, using OLS and assuming


Figure 3.4: Curvature - Factor Loading for Various λ Values

0 1 2 5 10 20 30−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

maturity

fact

or lo

adin

g

0.10.20.512510100


the λ parameter to be fixed, then evaluate the accuracy of the fit, and find such

a λ maximizing the accuracy (can be measured by R-square of the OLS or by

the information criterion of the model).

Dynamic Version and Macro-Finance Extension

Diebold & Li (2006) introduced a new view on the Nelson-Siegel function.

Considering the β parameters as time-varying latent factors, the dynamics of

the yield curve can be derived from the dynamics of these latent factors. The

model is described by two equations (Diebold et al. 2006):

rt (τ) = β1,t + β2,t

(1− e−λτ

λτ

)+ β3,t

(1− e−λτ

λτ− e−λτ

)+ εr,t (τ) (3.4)

βt = α+ Γβt−1 + εβ,t (3.5)

where βt is a 3× 1 vector consisting of βt,1, βt,2 and βt,3; the scalar error terms

of the yields εr,t (τ) are forming a m × 1 vector error term εr,t (where m is

number of maturities τ in the sample), which is believed to follow N (0,Σr)

distribution with Σr being a m ×m covariance matrix; εβ,t is a 3 × 1 vector

error term related to the stochastic process of βt, assumed to follow aN (0,Σβ)

distribution (with Σβ of a dimension 3 × 3), and α and Γ are a 3 × 1 vector,

respectively a 3× 3 matrix of parameters.

The Equation 3.5 is simply a representation of a three-dimensional VAR

process with one lag, which can be considered as the most simple way how to


capture the development of βt, and will be used also in the thesis. One benefit of

the VAR model used in the Equation 3.5 is that the construction of the resulting

macro-finance model is very simple. The 3 × 1 vector of latent variables βt

including the latent factors level, slope and curvature can be enriched by a k-

dimensional vector θt including k macroeconomic variables, to form a (3 + k)×1

vector (henceforth denoted ηt). The original three-dimensional VAR process

will be modified into a (3+k)-dimensional (following Diebold et al. 2006):

[βt

θt

]= µ+ Φ

[βt−1

θt−1

]+ εη,t (3.6)

ηt = µ+ Φηt−1 + εη,t (3.7)

where µ, Φ and εη,t are similar to α, Γ and εβ,t, differing only in their dimen-

sions.

The macro-variables included in the vector θ do not influence the yields

directly (i.e. they are not included in the Equation 3.4), but only through

the impact on the development of the latent β factors, which then govern the

yields themselves. The macroeconomic variables are considered as exogenous,

and the macro-finance relationship is therefore only one-sided6, lacking the

reverse causality in the direction from the interest rates to the macroeconomic

variables. This simplification can reduce the performance of the model, as

a consequence of an omission of some transition channels, nevertheless the

simplicity of resulting model can be beneficial in terms of a robustness and a

good estimation performance, especially when focusing on the predictive power.

Estimation and Forecasting

Diebold & Li (2006), or similarly Kollar (2011) for the Czech yield curve,

proceed in the following way: First, for each period, the Nelson-Siegel function

is fitted on the observed maturities — i.e. the parameters of the Equation 3.4

are estimated for each period separately; β coefficients are allowed to vary,

whereas λ is kept fixed. Afterwards, the estimated β parameters, representing

the latent factors — level, slope and curvature — enter the model capturing

their dynamics: a reduced-form VAR model defined in the Equation 3.5 is

estimated, using a least-squares method. After an evaluation of the error term

6Which will be followed also in all other models included in the thesis


of the model in terms of diagnostic tests, properties of the estimated model can

be analysed, typically using IRF, to comment on the dynamics implied by the

model.

Different approach present Diebold et al. (2006), using the very nature of

the described model — it is a typical representative of a State-Space model

in the most basic form. Observed values are determined by unobservable (la-

tent) factors, which are following VAR(1) process. Following the methodology

of these models (for example in Pichler 2007), the Equation 3.4 can be called a

measurement equation, since it represents the process how the latent factors are

related to the measurable variables (yields); consequently, εr,t is a measurement

error. The Equation 3.5 is a state (transition) equation, which describes be-

havior of the latent variables, and includes the state equation errors εβ,t. Since

the relationship is linear, the two steps can be jointly estimated by Kalman

filter (Kalman 1960), which is a special method for constructing and using the

likelihood function in the State-Space models framework, even when the least

squares methods are not applicable (Pichler 2007).

The macro-finance extension is then constructed rather easily. The β pa-

rameters enter the Equation 3.6 instead of the original one, regardless to which

of the two approaches to the estimation is used - no change of the estimation

methods is necessary. A shock into a macroeconomic variable (either external

or resulting from a policy-making process) is directly (with a one-period lag)

propagated into the other macroeconomic variables and simultaneously into

the latent factors, which can be well depicted by the IRF. The latent factors

then directly determine the new shape of the yield curve.

After the models are estimated, the forecasts can be constructed by step-by-

step iterating the β (or η in the macro-finance model) vector through the transi-

tion equation 3.5 (respectively 3.7), utilizing the fact that E [εβ,t] = E [εη,t] = 0.

The forecasts of the yields themselves then result from the measure Equa-

tion 3.4.

It is useful to describe, returning to the start of the estimation, how to

obtain the fixed λ parameter. Since the parameter determines the maturity in

which the curvature factor is maximal (see Figure 3.4), Diebold & Li (2006)

simply argue the optimal λ should ensure the largest curvature for maturities

2-3 years, which are, according to the authors, the most appropriate from the

empirical point of view. More specifically, the authors find such a λ maximizing

the curvatureβ3 factor loading for 30 months maturity.

However, because the data scope analysed in the thesis includes quite vari-


able shapes of the yield curve (particularly after the 2009 crisis), as well as

a different ”hump” location for different periods, the approach to set λ as

mentioned above could be considered as unreasonable. Instead, λ may be de-

termined so that it ensures an optimal fit of estimated and observed values.

The procedure can be described in the following way:

1. An arbitrary λ1 is chosen.

2. For each period t, the β vector is obtained by a simple OLS method, using

the maturities in the given period as the observations:

βt =(XTX

)−1XTrt (3.8)

where:

βt =

β1,t

β2,t

β3,t

,

rt =

rt (τ1)

rt (τ2)...

rt (τm)

,

X =

1 1−e−λ1τ1

λ1τ11−e−λ1τ1

λ1τ1− e−λ1τ1

1 1−e−λ1τ2

λ1τ21−e−λ1τ2

λ12− e−λ1τ2

......

...

1 1−e−λ1τm

λ1τm1−e−λ1τm

λ1τm− e−λ1τm

with m = number of maturities in the sample.

3. Then, the difference between the estimated and observed values (residuals

et) is calculated as:

et = rt − rt = rt −Xβt (3.9)

4. Finally, returning to λ, the numerical methods are used to find its optimal

value, i.e. minimizing the residual sum of squares (RSS) over all periods:

minλ

n∑t=1

etTet (3.10)

where n is the number of the periods included in the sample.


To conclude, there will be multiple models belonging to the dynamic Nelson-

Siegel family estimated in the Section 4.3, and their results will be compared.

The different approaches are captured by the Table 3.1.

Table 3.1: Dynamic Nelson-Siegel Models

model name type λ setting

NS-L-A latent-factors-only curvature locationNS-L-B latent-factors-only optimal fitNS-M-A macro-finance curvature locationNS-M-B macro-finance optimal fit


3.4 Affine Model

Basic Description

Apart from the dynamic Nelson-Siegel approach, model introduced in this sec-

tion is using no-arbitrage assumption when determining the bond prices and

the term structure of interest rates itself. As explained in the previous chapter,

under the risk-neutral measure, the value of a bond is given directly by the

expected development of the instantaneous (short) rate:

Pt(τ) = EQ

[e−

∫ t+τt rsds

](3.11)

where EQ denotes the expectations under the risk-neutral measure. To be

able to explain the dynamics of the bond prices (and hence the whole term

structure) in time, it is necessary (as notes Piazzesi 2009) to solve two related

issues:

� to capture the dynamics of the short rate, and

� to link risk-neutral and real probability measures.

To deal with both tasks, an affine model will be used, as one of the most

frequent approaches to the interest rate modelling. Following the work of Duffie

& Kan (1996), the term structure model is called affine if the dynamics of the

short rate rt is an affine function of the state variables:

rt = a0 + aT1Lt (3.12)


where Lt is a l-dimensional vector of the state variables (or latent factors), a0

is a scalar parameter and a1 is a l-dimensional vector of parameters. In case

l = 1, the a one-factor (usually short rate) model is obtained, otherwise it can

be called as a multi-factor model. Using a simple example of Vasicek (1977)

model, a0 = 0, l = 1, a1 = 1 and the only latent factor is consequently the short

rate itself, whose evolution is then given by a particular stochastic differential

equation.

More generally, for both one-factor and multi-factor models, the dynamics

of Lt is given by the following equation:

dLt = K1 (k2 −Lt) dt+ ΣLStdW t (3.13)

where K1 and ΣL are l× l matrices and k2 is a l× 1 vector of parameters, and

St is a diagonal l × l matrix with elements being a function of Lt. Moreover,

W t is a l-dimensional Brownian motion, assuming the model is described still

under the risk-neutral measure Q. The diagonal elements (noting Sii the i-th

diagonal element) of the St matrix can be expressed in the form:

Sii =√b1,i + bT2,iLt (3.14)

Where b1,i is a scalar parameter and b2,i is a l × 1 vector of parameters.

The affine models further assume there are certain restrictions on the par-

ticular form of the Equation 3.13 (which are outlined for example in Malek

2005 for a one-factor model). Duffie & Kan (1996) show that in that case,

price of a bond can be expressed as an exponentially affine function in terms

of the state factors:

Pt(τ) = eB1(τ)+BT2 (τ)Lt (3.15)

Where B1(τ) and B2(τ) are deterministic functions of a maturity, i.e. for a

given maturity τ , they represent coefficients (a scalar and a vector)7. Moreover,

the functions can be obtained directly by solving a set of ordinary differential

equations (for a one-factor model again in Malek 2005).

7Sometimes, there is used a negative sign before B2(τ) term in the Equation 3.15, butthe results must be necessarily equivalent regardless to the chosen form.


Utilizing the Equation 2.11, the interest rates (for the maturity τ > 0) can

be expressed, after plugging for Pt(τ) from the Equation 3.15, in the form:

rt(τ) = − lnPt(τ)

τ= −B1(τ)

τ− B

T2 (τ)Ltτ

= C1(τ) +CT2 (τ)Lt (3.16)

It is important to remind, that all the above-defined parameters are related

to the dynamics under the risk-neutral measure. Conversion of the risk-neutral

measure may be performed using the already mentioned Girsanov theorem,

which is defined for example in Shreve (2004). Assuming λ is a l-dimensional

vector of market prices of risk8 related to the l state variables, the Girsanov

theorem implies that

dW t = dW t + λtdt (3.17)

where W t is a Brownian motion under the real probability measure. Then, the

Equation 3.13 can be, after plugging in for expression dW t from Equation 3.17,

rewritten under the real probability measure as

dLt = K1 (k2 −Lt) dt+ ΣLStλtdt+ ΣLStdW t (3.18)

Duffie & Kan (1996) assume, that the market price of risk is a function of the

state variables volatility, specifically:

λt = Sth (3.19)

where h is a l × 1 vector of constant parameters. After plugging into the

Equation 3.18 and using the definition of the St matrix, the dynamics of the

state variables modifies in the following way:

dLt = K1 (k2 −Lt) dt+ ΣLS2thdt+ ΣLStdW t (3.20)

dLt = M 1 (m2 −Lt) dt+ ΣLStdW t (3.21)

where M 1 and m2 are of the same dimensions as K1 and k2, and are directly

given by K1, k2, ΣL, h and parameters b1,i and b2,i from the St matrix, as

shows for example Kladıvko (2011).

8λ symbol has been used in a different context in dynamic Nelson-Siegel models, the no-tation is though not adjusted in order to comply with the common notation in the literature.


Consequently, after fitting the model under the risk-neutral measure, the

real-world behavior of the latent factors can be, thanks to the mentioned specific

relation of the market prices of risk and latent factors, obtained quite easily.

However, for example Duffee (2002) considers this restriction as too strict, and

offers an affine model allowing more realistic dynamics of the market price of

risk.

Discrete-Time Specification

Similarly to other studies, it may be convenient to capture the whole dynamics

in a discrete time. Simultaneously, following Ang & Piazzesi (2003), the general

affine model as described above will be given a specific form, allowing to obtain

the parameters for both Equation 3.16 and dynamics of the latent factors. In

the model, three latent factors are used, which is perfectly enough (see the

discussion about Litterman & Scheinkman (1991) findings included above).

First, it is necessary to specify the dynamics of the underlying latent factors

Lt, i.e. a discrete-time analogy to the equation Equation 3.13. Being still

inspired by Ang & Piazzesi (2003), the VAR process is used in this respect.

Including only one lag into the VAR equation, the resulting model can be truly

seen as an analogy of the continuous-time process (admitting that the following

notation is not an exact discrete approximation, but rather an intuition)9:

∆Lt = K1 (k2 −Lt) ∆t+ ΣL∆W t

Lt+1 = Lt +K1 (k2 −Lt) (t+ 1− t) + ΣL

(W t+1 − W t

)Lt+1 = K1k2 + (1−K1)Lt + ΣLεt+1

Lt+1 = γ0 + Γ1Lt + ΣLεt+1 (3.22)

where γ0 is a l×1 vector and Γ1 represents a l×l matrix of parameters, whereas

εt+1 is a l-dimensional random term assumed to follow N(0, I). Moreover, ΣL

term is in this case a l×l matrix of coefficients. It is also necessary to determine

the specific dynamics of the market price of risk. The l-dimensional vector λt

9In the model, the specific form of the volatility as announced by Duffie & Kan (1996) anddescribed above is not utilized. For this reason, the volatility coefficients ΣLSt are replacedin the notation simply by ΣL in order to simplify the functional form, keeping in mind thatthe ΣL term is different in both cases.


is here assumed to be affine in the state variables:

λt = λ0 + λ1Lt (3.23)

with λ0 being a l-dimensional vector and λ1 being a l× l matrix of parameters.

The discrete-time specification influences also the way the bond price is

expressed in terms of expectations under the risk-neutral measure. In the

discrete time, the bond price may be defined, analogically to the Equation 3.11,

looking one period forward:

Pt(τ + 1) = EQ

[Pt+1(τ)

ert

](3.24)

Obviously, it is necessary to convert the risk-neutral measure into the real-

world. This can be done via the Radon-Nikodym derivative, which is, together

with the Girsanov theorem, one of the basic building blocks of the risk-neutral

dynamics. Following steps are based mostly on Ang & Piazzesi (2003), Gisiger

(2010), Haugh (2010), Shreve (2004) and Malek (2005), employing the basics

of the stochastic calculus and the risk-neutral probability approach.10

The Radon-Nikodym derivative Z of the risk-neutral measure Q with re-

spect to real-world measure P can be defined as:

Z =dQ

dP(3.25)

It is further possible to define a Radon-Nikodym process Zt, which is assumed

to be a martingale under the P -measure:

Zt = EP [Z|Ft] (3.26)

where Ft is a filtration.

Consequently, using conditional expectations and employing the definition

of Z, following holds for any Ft+1-measurable random variable Xt:

EQ [Xt+1|Ft] =EP[dQdPXt+1|Ft+1

]EP[dQdP|Ft] =

EP [Zt+1Xt+1|Ft+1]

Zt(3.27)

The discrete-time dynamics of the Radon-Nikodym proces Zt is specified as a

10Since a complex description of the stochastic calculus and the risk neutrality issues wouldrequire a far larger space than possible to use in the thesis, only the most relevant facts andissues will be mentioned, referring to the included literature in the other cases.


log-normal process (Ang & Piazzesi 2003)

Zt+1 = Zte− 1

2λTt λt−λTt εt+1 (3.28)

The last assumption is a definition of a nominal pricing kernel Mt+1:

Mt+1 = e−rtZt+1

Zt(3.29)

Mt+1 = e−rt−12λTt λt−λTt εt+1 (3.30)

The second equation has been obtained by plugging in for Zt+1 from the Equa-

tion 3.28. The pricing kernel (called also a stochastic discount factor) is a key

(random) variable used to price (discount) the nominal assets in economy - un-

der the real probability measure. Using a naıve intuition, the pricing kernel may

be seen as a representation of the denominator from the Equation 2.20. In other

words, returning to the very beginning of the thesis, the general thought of an

individual required yield yreq as introduced in Equation 2.1, which depends on

a risk-free time factor ρ, an expected inflation E [π] and a risk premium ξ, is

explicitly formulated by the Equation 3.30, including the risk-free rate rt and

the market price of risk λt as exact specifications of (ρ+ E [π]) and ξ.

The specification of the pricing kernel as defined above, as well as the men-

tioned intuition behind, is implying a recursive equation for bond prices:

Pt(τ + 1) = EP [Mt+1Pt+1(τ)|Ft] (3.31)

which can be proven directly by merging equations 3.24, 3.27 and 3.29, replacing

Xt and Xt+1 by Pt(τ + 1) and Pt+1(τ):

Pt(τ + 1) = EQ

[Pt+1(τ)

ert

∣∣∣∣Ft]= EP

[Zt+1Pt+1(τ)

ertZt

∣∣∣∣Ft]= EP [Mt+1Pt+1(τ)|Ft]

Obviously, the Equation 3.31 is the representation of the Equation 3.24 under

the real probability measure, which is exactly what was aimed to obtain.

The only variable determining (recursively) the bond price Pt(τ+1) is hence

the pricing kernel Mt+1, specified by the Equation 3.30. On the other hand, the

pricing kernel is, apart from the random term εt+1, driven by the development


of the short rate rt and the market price of risk λt. Since both these variables

are expressed as an affine function of the latent factors Lt (see equations 3.12

and 3.23), the bond price itself can be obtained as a function of the latent

factors only (apart from the maturity).

Moreover, Ang & Piazzesi (2003) show, that the functions Pt(τ + 1) =

fτ (Lt), which are recursively defined for bonds of growing maturities, are ex-

ponentially affine, which proves that the model belongs to the affine class of

models. The functions B1(τ) and B2(τ) from the Equation 3.15 themselves

can be then shown (see Appendix A. from Ang & Piazzesi (2003) for a proof)

to be in the following recursive form:

B1(1) = −a0B2(1) = −a1 (3.32)

and

B1(τ + 1) = B1(τ) +BT2 (τ) (γ0 −ΣLλ0) +

1

2BT

2 (τ)ΣLΣTLB2(τ)− a0

BT2 (τ + 1) = BT

2 (τ) (Γ1 −ΣLλ1)− aT1 (3.33)

where the coefficients are those used in the equations 3.12, 3.22 and 3.23.

The interest rates can be the simply expressed as affine functions of the

latent factors, with C1(τ) and C2(τ) obtained simply from B1(τ) and B2(τ),

as defined in the Equation 3.16. Consequently, the whole model can be written

as

rt(τ) = C1(τ) +CT2 (τ)Lt + εr,t(τ) (3.34)

Lt = γ0 + Γ1Lt−1 + ΣLεt (3.35)

where the inclusion of εr,t(τ), assumed to be N(0, σt(τ)) distributed, is moti-

vated by allowing a possibility that there may be a random disturbances in the

relation of the latent factors and the interest rates.

Macro-Finance Extension

As point out De Pooter et al. (2007), the equations 3.34 and 3.35 are very

similar to the equations 3.4 and 3.5. This allows to give them the same inter-

pretation: they can be considered as measure and transition equations forming


a simple state-space model. Moreover, the transformation of the model, in-

cluding macroeconomic variables, is based on the same idea: the l-dimensional

vector Lt in the Equation 3.35 is extended by k macroeconomic variables, form-

ing a (l + k)-dimensional vector V t. However, being inspired by the approach

used by Ang & Piazzesi (2003), there is one important difference compared

to the dynamic Nelson-Siegel approach: the macroeconomic variables influence

the dynamics of the yields directly. In other words, macro-factors enter also the

Equation 3.34, whereas in the case of the previous group of model, the Equa-

tion 3.4 remained unchanged, with the macro-variables influencing the yields

only indirectly through the joint dynamics with the latent factors, as capture

equations 3.6 or 3.7.

The whole affine macro-finance model can be written in the form:

rt(τ) = C1(τ) + CT2 (τ)V t + εr,t(τ) (3.36)

V t = γ0 + Γ1V t−1 + ΣMεt (3.37)

where

V t =

[Lt

θt

](3.38)

with θt being the k-dimensional vector of added macroeconomic variables, εt

being a (k + l)-dimensional random error assumed to be N(0, I) distributed,

and other vectors and matrices being the extended versions of similar terms

from the equations 3.34 and 3.35. When compared to the latent-factors-only

form, different will be not only the VAR parameters γ0, Γ1 and ΣM , but since

they enter the Equation 3.33, also C1(τ) and C2 will differ.

Estimation and Forecasting

For the estimation of the models, an approach following Ang et al. (2006) and

De Pooter et al. (2007) will be used. Focusing on the latent-factors-only model,

the estimation includes several steps:

� First, the latent factors will be assumed to be represented by the first

three principal components of the yields. Based on these variables, the

parameters of theEquation 3.35 will be estimated - γ0 and Γ1 will be


obtained by the least squares method, and ΣL will be calculated from

the covariance matrix of the residuals, using Choleski decomposition.

� Second, the parameters a0 and a1 from Equation 3.12 will be calculated

by OLS, using the shortest observed yield as a proxy for the short rate.

� Last step is to obtain values of λ0 and λ1 such that the resulting yields

from the Equation 3.34 are as close to the observed yields as possible.

That means that after iterative plugging in for λ0 and λ1 into the Equa-

tion 3.32, using values of other parameters obtained from the previous

steps, B1 and B2 are calculated recursively for each maturity, as well as

C1 and C2. The fitted yields are then obtained from:

rt(τ) = C1(τ) + CT

2 (τ)Lt (3.39)

and λ0 and λ1 will result from:

minλ0,λ1

n∑t=1

m∑i=1

[rt(τi)− rt(τi)]2 (3.40)

where m is a number of maturities and n a number of periods included

in the sample.

For the macro-finance version, the estimation process will be exactly iden-

tical, using the vector V t instead of Lt. Finally, forecasts of the yield curve

may be constructed, using the estimated values of parameters. First, by itera-

tive plugging in for Lt (or V t) into an estimation of the Equation 3.35 (3.37),

forecasts of Lt+1 (V t+1) will be obtained. The values of yields predicted by the

model then result directly from an estimation of the Equation 3.34 (3.36).

Chapter 4

Estimation

4.1 Data

Interest Rates Data

In this thesis, U.S. zero-coupon bond yields are used to form the term structure

data, similarly to other studies. The data include end-of-month yields for bonds

of 10 maturities1: American Government Bills of three and six months and one

year maturities, US Notes (2, 3, 5, 7 and 10 years) and, finally, 20- and 30-years

US Government Bonds. In the period between February, 2002 and February,

2006, there were no data released for the longest maturity bonds (30Y), but

the U.S. Treasury offers calculated adjustments allowing to obtain data for the

30-years US bonds by extrapolating the 20-years bonds yields (U.S. Treasury

2014), which is used also in the thesis.

The data are obtained from the FRED (2014) database. The monthly

frequency was chosen to comply with the frequency of the macroeconomic vari-

ables, for which the monthly periodicity is the most frequent possible, as will

be discussed below. The data are already obtained in terms of yields (per

annum). More specifically, the FRED (2014) database uses constant maturity

rates, which are calculated by U.S. Treasury by interpolating the yield curve for

the non-inflation-indexed Treasury securities2. Approaches to the yield curve

1Originally, 11 maturities were included into the analysis, adding also one-month U.S.Bills; however, restricted availability of the data for this maturity before year 2001 resultedin a weaker performance of the models, and moreover, originally analysed period became tooshort to allow the comparison of the performance of the models under different macroeco-nomic conditions.

2The issue of obtaining the constant maturity rates is described into the more detail inthe footnotes at the end of FED (2014).

4. Estimation 40

construction from the available data is theoretically described in the Chapter 2,

and since it is not in the centre of attention of the thesis, the interest rates data

are used passively, without further discussion or adjustments.

The data include period starting in October, 1993 and ending in February,

2014, which covers 245 observed end-of-month dates. This particular period

was chosen for a few reasons:

� The period is long enough to provide the data allowing to use the chosen

modelling techniques.

� Simultaneously, the period is short enough to ensure it is eligible to as-

sume there were no systematic changes in terms of a theoretical function

describing the optimal reaction of the economic subjects on the economic

conditions, which is necessary for the stability of the macroeconomic re-

lationships, as requires the famous Lucas (1976) Critique. This fact is

very important particularly for the behavior of the central bank, which

can be assumed to operate under the inflation targeting monetary policy

paradigm during the whole period.

� Moreover, the economic conditions themselves have been changing during

the period, including both periods of economic expansions and shrinkages.

This allows to compare the performance of the models in various parts of

the business cycle.

The range of data will be split into a training sample, containing the data

until the end of 2012 (231 periods), and a testing sample including remaining

14 months. In further data analysis, only the training sample will be used,

and based on it, the models will be estimated and evaluated in terms of the

estimation properties. The testing sample will be reintroduced in the Chap-

ter 5, when will be used for an evaluation of the forecasting performance of the

models.

Afterwards, the same data will be the used again, estimating the models

on a shorter sample and rolling the estimation-forecasting process through the

whole data range. This facilitates comparison of the forecasting ability of the

models in different periods, as it can be expected that the benefit of adding the

macro-variables will vary with changing economic situation.

It is useful to outline the development of the term structure of interest rate

during the in-sample period 1993-2012. Figures 4.1 and 4.2 illustrate the key

shapes of the yield curve of the U.S. Government Bond yields in this period.

4. Estimation 41

Constant maturity yields of the 10 maturities mentioned above are marked

as a points, and a simple linear interpolation has been used to connect the

yields between the points, and draw simple yield curves. To comment on

Figure 4.1: Term Structure: 1993-2002

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

maturity (years)

yiel

d (%

)

12−199312−199512−1998

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

maturity (years)

yiel

d (%

)

12−200012−200112−2002

Source: FRED (2014) database, author’s computations

Figure 4.2: Term Structure: 2003-2012

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

maturity (years)

yiel

d (%

)

12−200312−200512−2006

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

maturity (years)

yiel

d (%

)

12−200812−201012−2012

Source: FRED (2014) database, author’s computations

the development, the relation of the yield curve and the business cycle is quite

obvious. A typical upward-sloping term structure from the first half of the

first observed decade has gradually changed to a flat-shaped, which gradated

in 2000 as a rather unusual downward-sloping curve. Afterwards, a period of

a crisis (so-called ”dot.com” crisis) pushed the short end of the curve radically

down, which gave the term structure again the typical upward-sloping shape.

Similar scenario appeared in the second 10-year period captured by the Fig-

ure 4.2. After an economic expansion accompanied by the short rate remaining

at a low levels, the Federal Reserve Board of Governors (FED) has tightened

the monetary policy during years 2005-2007 as a reaction on the over-heated

economy. This resulted in a significant flattening of the yield curve, again

preceding an economic crisis, which came in 2008. Afterwards, the short rate

4. Estimation 42

dropped again, making the term structure largely upward sloping. The severity

of the crisis have caused the longest rates to decrease gradually, however the

upward-sloped shape has remained until the end of the observed (in-sample)

period.

The development of the term structure dynamics can be illustrated also by

the Figure 4.3 including time series of yields of maturity 3 months (a proxy for

the short rate), 3 years, 10 years and 30 years (long rate).

Figure 4.3: Yields Time Series

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−20120

1

2

3

4

5

6

7

8

time

yiel

d (%

)

3M 3Y10Y30Y

Source: FRED (2014) database

The graphic representation suggests for nonstacionarity of the yields, which

can be supported by the Autocorrelation Function (ACF) and the Partial Au-

tocorrelation Function (PACF) of the yields (the maturities 3M, 3Y, 10Y and

30Y again chosen as representatives), as shows the Figure 4.4. However, re-

turning to the Figure 4.3, it seems that for the longer maturities, a linear

deterministic trend can be identified. This is proven by using the Augmented

Dickey-Fuller (ADF) test, with a linear trend and a constant.The null hypoth-

esis is a non-stationarity in this case. Results are included in the Table 4.1:

whereas the non-stationarity could not be rejected for the short and medium

maturities, opposite holds for the longest maturity, which is, according to the

test, clearly stationary (assuming a presence of the linear trend).

The yields are also highly correlated, with the correlation gradually de-

creasing for opposite parts of the yield curve. A reduced correlation matrix,

4. Estimation 43

Figure 4.4: ACF and PACF of Yields

0.0 0.5 1.0 1.5

0.0

0.6

Lag

AC

F

ACF − 3M yields

0.5 1.0 1.5

−0.

20.

41.

0

Lag

Par

tial A

CF

PACF − 3M yields

0.0 0.5 1.0 1.5

0.0

0.6

Lag

AC

F

ACF − 3Y yields

0.5 1.0 1.50.

00.

6

Lag

Par

tial A

CF

PACF − 3Y yields

0.0 0.5 1.0 1.5

0.0

0.6

Lag

AC

F

ACF − 10Y yields

0.5 1.0 1.5

0.0

0.6

Lag

Par

tial A

CF

PACF − 10Y yields

0.0 0.5 1.0 1.5

0.0

0.6

Lag

AC

F

ACF − 30Y yields

0.5 1.0 1.5

0.0

0.6

Lag

Par

tial A

CF

PACF − 30Y yields

Source: author’s computations

including again the four representative maturities, is provided in the Table 4.2.

The correlation between the shortest (3M) and the longest (30Y) rate is the

smallest in the whole (unrestricted) correlation matrix, and still not negligible,

which proves the high interconnection among the rates of various maturities.

The variance of the yields is decreasing with a growing maturity.

4. Estimation 44

Table 4.1: ADF Test Results - Yields

maturity test statistic p-value H0 rejected (α = 5%)

3M -2.997 0.157 NO3Y -2.346 0.431 NO10Y -3.322 0.068 NO30Y -4.140 < 0.010 YES


Table 4.2: Variances and Correlation Matrix (reduced)

maturity variance st.dev. correlation matrix3M 3Y 10Y 30Y

3M 4.616 2.148 1.000 0.954 0.846 0.7453Y 4.187 2.046 0.954 1.000 0.957 0.88410Y 2.114 1.454 0.846 0.957 1.000 0.97730Y 1.397 1.182 0.745 0.884 0.977 1.000


Principal Components of Yields

It is necessary to verify, using the PCA, whether the yields analysed in the thesis

possess the ability to be explained by a few factors, as described above. Fol-

lowing the Section 3.1, the evaluation will be focused on the explanatory power

of the first three principal components. Time series of the first three principal

components are displayed in the Figure 4.5, showing their development in the

whole training sample.

The components are already de-centred. Further analysis shows, that the

first three components are able to explain over 99% of the total variance of

the original data, which is absolutely satisfying and allows to neglect the other

components (Table 4.3).

Another Table 4.4 displays the eigenvectors of the yields, which represent

the linear transformation coefficients used for the calculation of the principal

components from the original variables. The first eigenvector has all its ele-

ments approximately similar, which means that the first principal component

is based on a sort-of-average of the yields in each period. The second eigen-

vector has increasing elements, which makes the second principal component

small for high short rates and low long rates, and vice versa. Finally, the third

column of the Table 4.4 has positive elements for either long or short rates,

4. Estimation 45

Figure 4.5: Principal Components Time Series

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

−10

−8

−6

−4

−2

0

2

4

6

8

10

time

valu

es o

f PC

s

PC1 − LevelPC2 − SlopePC3 − Curvature


Table 4.3: Variance Explained by Principal Components

PC cum.var. (%)

1 94.86052 99.64493 99.89294 99.94915 99.9723...

...9 99.998610 100.0000


but negative for the middle rates. Consequently, this proves the interpretation

of the principal components mentioned above (except for the signs), the first

component representing a level, second one interpreted as a slope, and the third

one related to a negative curvature of the term structure.

Moreover, using again the basic time series diagnostics, captured in the

Figure 4.6 and the Table 4.5, the first two principal components can be regarded

as nonstationary, whereas for the third component, the nonstationarity can be

rejected on a 95% significance level.

4. Estimation 46

Table 4.4: Eigenvectors Related to Principal Components

EV1 EV2 EV3 EV4 EV5 EV6 EV7 EV8 EV9 EV10

0.37 -0.41 0.52 -0.41 -0.37 -0.11 -0.26 -0.22 0.02 -0.010.37 -0.38 0.20 0.06 0.39 0.05 0.52 0.44 0.24 0.020.38 -0.26 -0.08 0.46 0.36 0.08 -0.37 -0.20 -0.50 0.090.38 -0.08 -0.39 0.27 -0.33 0.13 0.19 -0.32 0.30 -0.520.36 0.05 -0.40 0.00 -0.38 0.02 -0.05 0.22 0.11 0.710.32 0.22 -0.25 -0.33 0.08 -0.21 -0.37 0.54 -0.13 -0.440.28 0.30 -0.09 -0.43 0.09 0.00 0.52 -0.32 -0.50 0.050.24 0.35 0.05 -0.03 0.44 -0.40 -0.17 -0.37 0.52 0.150.21 0.42 0.30 -0.03 0.05 0.80 -0.17 0.05 0.13 0.000.18 0.42 0.45 0.51 -0.33 -0.35 0.16 0.18 -0.18 -0.04


Figure 4.6: ACF and PACF of Principal Components

0 5 10 15 20

0.0

0.6

Lag

AC

F

ACF − PC1

5 10 15 20

−0.

20.

41.

0

Lag

Par

tial A

CF

PACF − PC1

0 5 10 15 20

0.0

0.6

Lag

AC

F

ACF − PC2

5 10 15 20

−0.

20.

4

Lag

Par

tial A

CF

PACF − PC2

0 5 10 15 20

−0.

20.

6

Lag

AC

F

ACF − PC3

5 10 15 20

−0.

20.

4

Lag

Par

tial A

CF

PACF − PC3


Table 4.5: ADF Tests Results - Principal Components

component test statistic p-value H0 rejected (α = 5%)

PC1 -2.3560 0.4263 NOPC2 -2.9699 0.1684 NOPC3 -3.8723 0.0162 YES


Macroeconomic Data

Important task of the macro-finance modelling is to decide, which macroeco-

nomic time series should be used to enter the models. When choosing the

4. Estimation 47

appropriate data, it is important to stick to the following points ensuring the

resulting models will be well-designed:

� It is necessary to synchronize the frequency of macroeconomic and finan-

cial data. For the thesis, it means to use the monthly periodicity, which

excludes the use of many series (particularly gross domestic product data,

which are released on a quarterly basis).

� Many authors (for example Choudhry 2011 or Diebold & Li 2006) men-

tion the relation of the yield curve behavior to the certain macroeco-

nomic dynamics. Authors either link the yields to several key variables

(Diebold & Li 2006, Kollar 2011), or to the factors extracted from a large

set of macroeconomic variables (commonly using the PCA) - for example

De Pooter et al. (2007), Monch (2006) or Ang & Piazzesi (2003). Regard-

less to the approach, the finally chosen variables often address following

aspects of the economy:

– business cycle position (production gap, level of unemployment etc.)

– domestic prices dynamics (expected and historical inflation in terms

of Consumers Price Index (CPI) or Producers Price Index, central

bank inflation target)

– monetary aggregates (money stock, reserves)

– foreign sector (exchange rates, foreign production, foreign interest

rates)

� Another caveat of including the macro variables is the fact that they

are usually released with a time delay, which has to be considered when

evaluating the predictive performance.

� As notes De Pooter et al. (2007), it is useful to work with the variables

in terms of their annual growth rates; the monthly growth rates tend to

be largely volatile, which might reduce their explanatory power.

Reflecting these facts, following variables will be used as representatives of

the key channels, through which the macro-economy impacts the term structure

of interest rates3:

3Inspired mostly by De Pooter et al. (2007) and Kollar (2011).

4. Estimation 48

Industrial Production Index (IPI) can be assumed to include the dynamics of the

business cycle situation of the economy. It moves in a similar way as the

growth of the gross domestic product, and, moreover, is calculated with

a monthly periodicity.

Consumer Price Index (CPI) is used as the most common measure of price changes.

Monetary Aggregate M1 (M1) can be believed to reflect both an intensity of the

monetary policy supportive/restrictive behavior and a response of the

financial institutions to it.4

U.S. Dollar Index (USDI), for the purposes of the thesis calculated as a trade-

weighted average of the exchange rates of the U.S.Dollar to the other

main global currencies, will be expected to represent the real effective

exchange rate of the currency.

In all cases, the annual growth rates are used. Sometimes, gap values (de-

viances from a long-term equilibria) are used in this causes, which is, however,

not suitable for purposes of the forecasting, due to a difficult determination of

the equilibrium value for the most recent variables - typically when the Hodrick-

Prescott filter is used to determine the equilibrium. Moreover, in case of M1,

the log-differences are used instead. Also, following the mentioned delay in the

data release, all macro-variables will be used in a one-period lag5. The time

series of the chosen variables captures theFigure 4.7.

When evaluating the time series properties, namely a stationarity of the

macro variables, the non-stationarity can be (on 95% significance level) re-

jected for all of them, using both the ACF functions for a graphical intuition

(Figure 4.8) and the ADF test for an exact evaluation (Table 4.6).

4A typical example of relevance of this variable is the situation in the Czech Republicat the end of the year 2013, when a side-product of the central bank interventions was aninflow of the additional money on the inter-bank market, which caused a downside shift ofthe Czech government bond yield curve.

5However, in the further text, in order to avoid a misleading indexing in the models, theone-period-lagged macroeconomic variables will be indexed as contemporary (i.e. Xt). Thiscan be supported by an argument, that instead of changing of the indexes, the interpretationis modified: variable Xt will be assumed to represent the value of the variable X released inthe time t.

4. Estimation 49

Figure 4.7: Macro Variables Time Series

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−20

−15

−10

−5

0

5

10

15

time

annu

al g

row

th r

ate

(%)

IPI (y−o−y change)

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−4

−2

0

2

4

6

8

time

annu

al g

row

th r

ate

(%)

CPI (y−o−y change)

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−10

−5

0

5

10

15

20

25

time

annu

al g

row

th r

ate

(%)

M1 (y−o−y log−difference)

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−20

−15

−10

−5

0

5

10

15

20

25

time

annu

al g

row

th r

ate

(%)

USD index (y−o−y change)


Table 4.6: ADF Test Results - Macroeconomic Variables

variable test statistic p-value H0 rejected (α = 5%)

IPI -4.8301 < 0.0100 YESCPI -4.3134 < 0.0100 YESM1 -4.1549 < 0.0100 YES

USDI -3.9866 0.0105 YES


4.2 Random Walk

The random walk serves as a baseline model assumed to be outperformed in

its forecasting ability by the more sophisticated models used in the thesis. For

all maturities, an estimate of the standard deviation of the random errors and

forecasts for 14 periods are calculated. The point forecasts are set simply as

the latest observed values; moreover, forecasting intervals are included as well,

calculated on the 95% significance level. Results are captured in the Table 4.7.

The confidence interval is quickly widening for growing forecast horizon,

which allows the future interest rates for most of the maturities to become

negative at the given significance level. This is an unpleasant result of such

4. Estimation 50

Figure 4.8: ACF and PACF of Macro-Variables

0.0 0.5 1.0 1.5

0.0

0.6

Lag

AC

F

ACF − IPI

0.5 1.0 1.5

−0.

20.

41.

0

Lag

Par

tial A

CF

PACF − IPI

0.0 0.5 1.0 1.5

−0.

20.

41.

0

Lag

AC

F

ACF − CPI

0.5 1.0 1.5

−0.

40.

4

LagP

artia

l AC

F

PACF − CPI

0.0 0.5 1.0 1.5

0.0

0.6

Lag

AC

F

ACF − M1

0.5 1.0 1.5

−0.

20.

41.

0

Lag

Par

tial A

CF

PACF − M1

0.0 0.5 1.0 1.5

0.0

0.6

Lag

AC

F

ACF − USD index

0.5 1.0 1.5

−0.

20.

4

Lag

Par

tial A

CF

PACF − USD index


simply constructed forecasts, and for the more sophisticated models estimated

in the thesis, it will be examined, whether they perform better in this respect.

4.3 Dynamic Nelson-Siegel Approach

In the Section 3.3, there have been mentioned two possible ways how to esti-

mate the dynamic Nelson-Siegel model. In the thesis, the two-step approach

will be used, considering the Equation 3.4 (measure equation) and the Equa-

tion 3.5 (transition equation) separately, which enables a detailed discussion

of the properties of the βt dynamics resulting from the first step. First, the

model including only the latent factors will be estimated. Afterwards, the es-

4. Estimation 51

Table 4.7: Random Walk Estimation & Forecasts

mat SD(a) PF1 LB1 UB1 PF6 LB6 UB6 PF14 LB14 UB14

3M 0.2303 0.05 -0.40 0.50 0.05 -1.06 1.16 0.05 -1.64 1.746M 0.2239 0.11 -0.33 0.55 0.11 -0.96 1.18 0.11 -1.53 1.751Y 0.2376 0.16 -0.31 0.63 0.16 -0.98 1.30 0.16 -1.58 1.902Y 0.2722 0.25 -0.28 0.78 0.25 -1.06 1.56 0.25 -1.75 2.253Y 0.2860 0.36 -0.20 0.92 0.36 -1.01 1.73 0.36 -1.74 2.465Y 0.2923 0.72 0.15 1.29 0.72 -0.68 2.12 0.72 -1.42 2.867Y 0.2869 1.18 0.62 1.74 1.18 -0.20 2.56 1.18 -0.92 3.28

10Y 0.2763 1.78 1.24 2.32 1.78 0.45 3.11 1.78 -0.25 3.8120Y 0.2511 2.54 2.05 3.03 2.54 1.33 3.75 2.54 0.70 4.3830Y 0.2415 2.95 2.47 3.42 2.95 1.79 4.11 2.95 1.18 4.72

mat = maturity; SD(a) = estimated standard deviation of the white noise;PF = point forecast; UB/LB = upper/lower bound on 95% confidence level;adjacent numbers represent prediction horizon in months


timation will be replicated, extending the vector of the state factors by the

macroeconomic variables.

However, the very first task is to find the value of λ parameter, which plays

an important role in the Equation 3.4. Using the approach of Diebold & Li

(2006), maximizing the curvature for the maturity 30 months, the λA will result

directly from:

∂(

1−e−λAτλAτ

− e−λAτ)

∂τ

∣∣∣∣∣∣τ=2.5

= 0

After differentiating and plugging in for τ = 2.5, the λA is obtained from:

− 1

2.52λA+

2.5λAe−2.5λA + e−2.5λA

2.52λA+ λAe

−2.5λA = 0

Solving the equation numerically, the final result is:

λA.= 0.7173

Diebold & Li (2006) are using the maturity expressed in months instead, which

results in a significantly lower λA; however, after the value of λA is divided by

12, the results are similar.

The alternative approach is to obtain the optimal λ parameter value from

solving the expressions 3.8, 3.9 and 3.10. Some outputs of the minimization

problem6 are captured in the Table 4.8, the optimal λB has been found at the

6The minimization problem resulted in a single minima on the interval (0,170). For bigger

4. Estimation 52

value 0.5264.

Table 4.8: Nelson-Siegel RSS for Various λ Values

λB λA

λ: 0.001 0.1 0.5 0.5264 0.6 0.7173 1 10 100

RSS: 88.65 41.86 17.34 17.21 18.24 23.71 50.07 642.4 1070.1


The preciseness of the fit is outlined by the Figure 4.9 for λA, respectively

by the Figure 4.11 for λB. Similarly, the development of the β parameters in

time is captured by the Figure 4.10 (and the Figure 4.12, respectively).

Figure 4.9: Fitted and Observed Values - Nelson-Siegel for λA

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)

3M fitted3M observed

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)

3Y fitted3Y observed

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)


12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)



As apparent from the figures, the level factor (β1) is almost equivalent in

both cases, whereas the other two factors slightly differ. Despite the fact they

have a similar interpretation, the βs resulting from the dynamic Nelson-Seigel

model differ from the principal components of the yields (Figure 4.5). The

values of λ, the matrix XTX is approaching singularity. It can be, however, intuitivelyassumed that the large values, related with an extremely fast exponential decay of the Nelson-Siegel function, cannot fit the values effectively, and the minimum can be hence consideredas the global minima for all positive λ. The optimal value has been obtained numerically inMATLAB.

4. Estimation 53

Figure 4.10: Development of βs - Nelson-Siegel for λA

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

−8

−6

−4

−2

0

2

4

6

8

time

beta

s

beta1beta2beta3


Figure 4.11: Fitted and Observed Values - Nelson-Siegel for λB

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)

3M fitted3M observed

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)


12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)


12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

0

1

2

3

4

5

6

7

8

time

yiel

d (%

)



level factor β1 is corresponding to an infinitely long maturity, and is decreasing

more slowly in time compared to the first principal component, calculated as

an almost-average of the rates. Consequently, the second and the third factors

play relatively more important role in the dynamic Nelson-Siegel model than

in case of the principal components.

Similarly to the preceding variables, also the βs resulting from this model

4. Estimation 54

Figure 4.12: Development of βs - Nelson-Siegel for λB

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012

−8

−6

−4

−2

0

2

4

6

8

time

beta

s

beta1beta2beta3


should be described in terms of the time series properties. The results of the

ADF tests are in the Table 4.9. For both λA and λB, β1 has been rejected

to be non-stationary, which was not possible for either β2 or β3 on the 95%

significance level.

Table 4.9: ADF Test Results - Latent Factors (βs)

variable test statistic p-value H0 rejected (α = 5%)

β1,A -3.7844 0.0206 YESβ2,A -2.8315 0.2266 NOβ3,A -2.6064 0.3212 NOβ1,B -3.6519 0.0292 YESβ2,B -2.7954 0.2417 NOβ3,B -3.1884 0.0907 NO


Latent-Factors-Only Models

After the βs are extracted, the estimation of the model itself can be done. Be-

fore it, it is important to note, that the VAR model is used despite the fact,

that some of the variables (namely β2 and β3) were not rejected to be non-

stationary. However, Sims et al. (1990) has shown, that the use of the VAR

model for the datasets containing both stationary and non-stationary variables

is still possible without a loss of the model consistency or performing a spu-

4. Estimation 55

rious regression.7 Moreover, returning to the PCA of the yields, the key first

principal component, explaining more than 94% of variability of the original

data, is related to the latent factor β1, which is stationary for both λs. Finally,

the two non-stationary β2 and β3 variables are still assumed to have a mean-

reverting character, since there is only a limited range the yield curve can move

to (not considering any extreme situation in the economy, e.g. hyperinflation).

Since there are only stationary variables in the set of macro-variables, this

argumentation holds for both latent-factors-only and macro-finance models.

The time-varying vectors βA and βB already estimated for the two different

values of λ can be directly used for an estimation of the Equation 3.5, i.e. αA

and ΓA, respectively αB and ΓB. Maximizing the information criteria up to

the lag 12, the Akaike Information Criterion (AIC) is maximized for lag 4,

whereas Schwarz Information Criterion (SIC) and Hannah-Quinn Intormation

Criterion (HQIC) selected the one period lag as the most appropriate for both

models. The latter will be used in order to minimize the number of parameters

of the models. Moreover, the VAR model is estimated in the basic set-up

including both a constant and a trend (as a consequence of the interest rates

gradually decreasing for the last decades), and a necessity of these elements

will be examined.

For a better orientation, the estimated model is rewritten again, with the

measure equation in the matrix form, and including also the trend term:

rt = Xβt (4.1)

βt = α+ Γβt−1 + δt (4.2)

Since the dataset contains 10 maturities for each period, the dimensions are

following: rt is a 10 × 1 vector; X is a 10 × 3 matrix defined directly by the

maturities and the chosen λ; βt is a 3×1 vector of latent variables8; α and δ are

3× 1 vectors, whereas Γ is a 3× 3 matrix of estimated parameters; t is a scalar

denoting time. Moreover, εr,t and εβ,t are resulting 10× 1 and 3× 1 vectors of

residuals with estimates of the covariance matrices Σr and Σβ, respectively.

Focusing on the model using λA, i.e. model NS-L-A (from classification in

Table 3.1), after estimating the VAR(1) process, the transition equation residual

diagnostics9 has shown a serious serial correlation of the error terms - using the

7This argument was adopted from Kollar (2011).8Since in the two-step estimation procedure the βt vector is estimated explicitly, it is

noted by the ”hat” symbol9The residual analysis is focusing only on the transition equation error terms diagnostics,

4. Estimation 56

Breusch-Godfrey LM-statistic, the null hypothesis of absence of the serial cor-

relation was rejected on a significance level higher than 99,9%. This indicates a

wrong specification of the model, which might be resolved by adding more lags

to the models. In this case, the four-lagged version VAR(4) (as offered by the

AIC) is sufficient to resolve the problem (unable to reject the null hypothesis of

absence of the serial correlation of the Breusch-Godfrey test even on the 90%

significance level), changing the βt process estimation into the form:

βt = α+ Γ1βt−1 + Γ2βt−2 + Γ3βt−3 + Γ4βt−4 + δt (4.3)

The estimated model still indicates problems with a normality (which may be

considered as typical for the financial data models), and a related rejection of

the null hypothesis of homoscedasticity (using Engle’s ARCH test). On the

other hand, the VAR model can be considered as stable, with no roots of the

characteristic polynomial lying outside the unit circle. Since the mentioned

problems can be expected to be present for all estimated models, the model

will be, for purposes of a relative comparison of the predictive performance,

used in this form.

The estimated parameters (i.e. vectors and matrices of parameters) are

included in the Appendix A, as well as the Figure A.1 capturing the actual and

fitted values resulting from the VAR model, a plot of residuals and their ACF

and PACF.

The second model NS-L-B, equivalent to the previous except for the λB used

in this case, results in very similar estimation properties as the first model. The

VAR(4) model is again used due to the serial autocorrelation in the error term of

the VAR(1) model; the non-normality and a slight heretoscedasticity are present

as well, but the model is still stable in terms of the unit roots. The estimated

parameters are in the Appendix A, with the actual and fitted values, residuals

dynamics and their ACF and PACF captured in the Figure A.2.

Using the estimated models, values for future 14 periods may be calculated.

This forecasting will proceed analogically to the two steps of estimations (but

in the opposite direction):

� First, the βt+h,A, for h = {1, 2...14} will be calculated by iterative plug-

since the measure equation error term has in the case of the two-step estimation rather atechnical character, and was sufficiently described in the section dealing with the estimationof λ parameter.

4. Estimation 57

ging into the Equation 4.3 with the parameters estimated for the NS-L-A

model.

� Then, pre-multiplying the βt+h,A vector by the XA matrix, the forecasts

of rt+h are obtained.

� The process is replicated also using the estimation of the transition equa-

tion NS-L-B, βt+h,B and XB.

Some of the estimations are displayed in the tables 4.10 and 4.11.

Table 4.10: NS-L-A Forecasts

PF1 LB1 UB1 PF6 LB6 UB6 PF14 LB14 UB14

β1,A 3.19 2.71 3.66 3.44 2.51 4.37 3.39 2.42 4.37β2,A -2.74 -3.4 -2.09 -2.99 -4.6 -1.39 -3.06 -5.48 -0.64β3,A -5.47 -6.91 -4.04 -5.1 -8.06 -2.14 -5.64 -8.98 -2.3

3M 0.24 -0.95 1.43 0.3 -2.34 2.93 0.15 -3.31 3.66M 0.11 -1.12 1.33 0.21 -2.49 2.9 0.03 -3.45 3.511Y -0.01 -1.27 1.26 0.16 -2.59 2.9 -0.06 -3.51 3.392Y 0.13 -1.11 1.37 0.36 -2.29 3.01 0.12 -3.12 3.363Y 0.45 -0.72 1.61 0.71 -1.75 3.17 0.48 -2.47 3.435Y 1.11 0.11 2.11 1.39 -0.69 3.47 1.19 -1.25 3.647Y 1.6 0.72 2.47 1.88 0.06 3.69 1.71 -0.38 3.8

10Y 2.05 1.28 2.81 2.32 0.76 3.88 2.19 0.41 3.9620Y 2.61 1.99 3.23 2.88 1.63 4.13 2.79 1.41 4.1730Y 2.8 2.23 3.37 3.07 1.93 4.21 2.99 1.75 4.23

notes: see Table 4.7


Since the model is largely over-parametrized, a discussion of the values of

elements of the α and δ vectors and Γ matrices is not very useful. However,

it may be interesting to examine the impulse-response functions resulting from

the models — Figure 4.13. The implications resulting from the estimations and

the IRF, as well as the accuracy of the forecasts, will be into a detail described

in the Chapter 5 in the context of other models’ estimation and performance.

Macro-Finance Models

The macro-finance models, based on the dynamic Nelson-Siegel approach, will

utilize the already estimated λA and λB parameters, as well as βt,A, respectively

βt,B time-varying vectors. Different will be the transition equation, described

by the Equation 3.7, with the ηt including seven elements - three β latent

4. Estimation 58

Table 4.11: NS-L-B Forecasts


β1,B 3.44 2.98 3.9 3.66 2.77 4.55 3.63 2.68 4.58β2,B -3.18 -3.82 -2.55 -3.43 -5.1 -1.76 -3.56 -6.09 -1.04β3,B -4.71 -6.15 -3.27 -4.19 -6.96 -1.42 -4.64 -7.58 -1.7

3M 0.17 -0.97 1.31 0.19 -2.43 2.82 0.01 -3.48 3.56M 0.12 -1.06 1.29 0.18 -2.49 2.85 -0.02 -3.51 3.471Y 0.08 -1.14 1.3 0.21 -2.5 2.92 -0.01 -3.47 3.452Y 0.2 -1.04 1.44 0.41 -2.26 3.08 0.18 -3.12 3.473Y 0.44 -0.77 1.64 0.69 -1.86 3.25 0.46 -2.63 3.555Y 0.99 -0.1 2.08 1.28 -0.98 3.53 1.07 -1.59 3.737Y 1.47 0.49 2.44 1.75 -0.25 3.75 1.58 -0.74 3.89

10Y 1.97 1.12 2.81 2.24 0.53 3.96 2.1 0.14 4.0720Y 2.69 2.03 3.34 2.94 1.62 4.25 2.85 1.39 4.3230Y 2.94 2.34 3.53 3.18 2.01 4.35 3.11 1.82 4.4



Figure 4.13: IRF of NS-L-A and NS-L-B

xy$x

beta

1A

−0.

20.

00.

2

xy$x

beta

2A

−0.

20.

00.

2

xy$x

beta

3A

−0.

20.

00.

2

0 1 2 3 4 5 6 7 8 9 10

Orthogonal Impulse Response from beta1A

95 % Bootstrap CI, 100 runs

xy$x

beta

1A

−0.

20.

00.

2

xy$x

beta

2A

−0.

20.

00.

2

xy$x

beta

3A

−0.

20.

00.

2

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1A

0.0

0.4

0.8

xy$x

beta

2A

0.0

0.4

0.8

xy$x

beta

3A

0.0

0.4

0.8

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

−0.

20.

00.

2

xy$x

beta

2B

−0.

20.

00.

2

xy$x

beta

3B

−0.

20.

00.

2

0 1 2 3 4 5 6 7 8 9 10

Orthogonal Impulse Response from beta1B


xy$x

beta

1B

−0.

10.

10.

3

xy$x

beta

2B

−0.

10.

10.

3

xy$x

beta

3B

−0.

10.

10.

3

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

0.0

0.4

xy$x

beta

2B

0.0

0.4

xy$x

beta

3B

0.0

0.4

0 1 2 3 4 5 6 7 8 9 10




factors and four macroeconomic variables mentioned above: IPI, CPI, M1 and

USDI.

In the case of the macro-finance models NS-M-A and NS-M-B (which dif-

fer in the λ values as determined above), it comes out that the inclusion of

deterministic trend is unnecessary, or even biasing the model, leading to high

serial correlation of residuals. This may be considered as a signal, that the

4. Estimation 59

inclusion of the macroeconomic variables helps to explain the long-term move-

ments of the interest rates. The information criteria again indicate either 1 or

4 lags, and since the one-period-lagged model is after the testing obvious to

be poorly specified, the VAR(4) is again the resulting model, with the serial

correlation rather acceptable (p-value of the Breusch-Godfrey LM test is 0.047

in the case of NS-M-A, respectively 0.042 for the NS-M-B model). The Jarque-

Bera test rejects the normality of residuals in both cases, and the problem with

the heteroscedasticity still remains, although significantly weaker than for the

latent-factors-only model.

Estimated parameters of both models are shown in the Appendix A. In

the appendix, there are also included figures A.3, A.4, A.5 and A.6. Chosen

forecasts are displayed in the tables 4.12 and 4.13.

Table 4.12: NS-M-A Forecasts


β1 3.13 2.63 3.62 3.3 2.16 4.43 3.43 1.82 5.05β2 -2.64 -3.29 -1.98 -2.6 -4.05 -1.15 -2.53 -4.94 -0.13β3 -5.71 -7.16 -4.26 -5.5 -8.8 -2.2 -5.12 -9.08 -1.16

3M 0.26 -0.95 1.47 0.48 -2.24 3.21 0.71 -3.43 4.846M 0.1 -1.15 1.35 0.34 -2.48 3.16 0.58 -3.62 4.781Y -0.04 -1.33 1.25 0.2 -2.71 3.12 0.47 -3.76 4.72Y 0.06 -1.21 1.32 0.31 -2.56 3.18 0.59 -3.46 4.643Y 0.36 -0.83 1.55 0.61 -2.09 3.31 0.89 -2.88 4.665Y 1.02 0 2.05 1.26 -1.07 3.59 1.5 -1.73 4.737Y 1.51 0.61 2.42 1.73 -0.32 3.78 1.95 -0.9 4.8

10Y 1.97 1.18 2.76 2.17 0.38 3.97 2.37 -0.13 4.8720Y 2.55 1.91 3.19 2.73 1.27 4.2 2.9 0.84 4.9630Y 2.74 2.15 3.33 2.92 1.57 4.28 3.08 1.17 4.99



Finally, it is again useful to display the IRF. Since the causal direction

from the latent factors to the macroeconomic variables is omitted in the anal-

ysis, only the impact on latent factors is shown — separately impulses from

the latent factors themselves (Figure 4.14) and the impact of impulses of the

macroeconomic variables (Figure 4.15). Discussion of the results of the esti-

mates and the dynamics implied by them, as well as shapes and accuracy of

the predicted yield curves, are again postponed to the next Chapter 5 dealing

with the performance evaluation.

4. Estimation 60

Table 4.13: NS-M-B Forecasts


β1 3.4 2.92 3.88 3.56 2.49 4.63 3.67 2.14 5.2β2 -3.11 -3.74 -2.48 -3.09 -4.57 -1.6 -2.97 -5.47 -0.46β3 -4.97 -6.42 -3.51 -4.69 -7.84 -1.55 -4.34 -7.95 -0.73

3M 0.18 -0.97 1.34 0.38 -2.27 3.04 0.63 -3.47 4.736M 0.11 -1.08 1.31 0.33 -2.4 3.05 0.58 -3.55 4.721Y 0.05 -1.19 1.29 0.28 -2.53 3.1 0.55 -3.6 4.712Y 0.14 -1.13 1.4 0.39 -2.45 3.22 0.67 -3.39 4.723Y 0.36 -0.87 1.59 0.62 -2.13 3.37 0.89 -2.97 4.755Y 0.91 -0.2 2.02 1.15 -1.32 3.63 1.41 -2.02 4.847Y 1.39 0.39 2.38 1.62 -0.6 3.84 1.85 -1.21 4.91

10Y 1.9 1.03 2.76 2.11 0.18 4.04 2.31 -0.36 4.9820Y 2.63 1.95 3.31 2.82 1.31 4.33 2.98 0.87 5.0930Y 2.89 2.28 3.5 3.07 1.7 4.43 3.21 1.29 5.13



Figure 4.14: IRF of NS-M-A and NS-M-B: part 1

xy$x

beta

1A

−0.

20.

20.

4

xy$x

beta

2A

−0.

20.

20.

4

xy$x

beta

3A

−0.

20.

20.

4

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1A

−0.

10.

1

xy$x

beta

2A

−0.

10.

1

xy$x

beta

3A

−0.

10.

1

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1A

0.0

0.2

0.4

0.6

xy$x

beta

2A

0.0

0.2

0.4

0.6

xy$x

beta

3A

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

−0.

20.

00.

20.

4

xy$x

beta

2B

−0.

20.

00.

20.

4

xy$x

beta

3B

−0.

20.

00.

20.

4

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

−0.

10.

10.

2

xy$x

beta

2B

−0.

10.

10.

2

xy$x

beta

3B

−0.

10.

10.

2

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

0.0

0.2

0.4

0.6

xy$x

beta

2B

0.0

0.2

0.4

0.6

xy$x

beta

3B

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8 9 10




4.4 Affine Models

The estimation of the two affine models — latent-factors-only (henceforth noted

AF-L) model and macro-finance (AF-M) model — will follow the steps intro-

duced in the Section 3.4. The first three principal components have been al-

ready obtained in the Section 4.1, and based on them and the macroeconomic

4. Estimation 61

Figure 4.15: IRF of NS-M-A and NS-M-B: part 2

xy$x

beta

1A

−0.

20.

0

xy$x

beta

2A

−0.

20.

0

xy$x

beta

3A

−0.

20.

0

0 1 2 3 4 5 6 7 8 9 10

Orthogonal Impulse Response from IPI


xy$x

beta

1A

−0.

10.

10.

2

xy$x

beta

2A

−0.

10.

10.

2

xy$x

beta

3A

−0.

10.

10.

2

0 1 2 3 4 5 6 7 8 9 10

Orthogonal Impulse Response from CPI


xy$x

beta

1A

−0.

10.

00.

1

xy$x

beta

2A

−0.

10.

00.

1

xy$x

beta

3A

−0.

10.

00.

1

0 1 2 3 4 5 6 7 8 9 10

Orthogonal Impulse Response from M1


xy$x

beta

1A

−0.

20.

00.

1

xy$x

beta

2A

−0.

20.

00.

1

xy$x

beta

3A

−0.

20.

00.

1

0 1 2 3 4 5 6 7 8 9 10

Orthogonal Impulse Response from dollar.index


xy$x

beta

1B

−0.

20.

0

xy$x

beta

2B

−0.

20.

0

xy$x

beta

3B

−0.

20.

0

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

−0.

20.

00.

1

xy$x

beta

2B

−0.

20.

00.

1

xy$x

beta

3B

−0.

20.

00.

1

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

−0.

10.

00.

1

xy$x

beta

2B

−0.

10.

00.

1

xy$x

beta

3B

−0.

10.

00.

1

0 1 2 3 4 5 6 7 8 9 10



xy$x

beta

1B

−0.

20.

00.

1

xy$x

beta

2B

−0.

20.

00.

1

xy$x

beta

3B

−0.

20.

00.

1

0 1 2 3 4 5 6 7 8 9 10




variables described above as well, the models will be built10. The discussion

of the usefulness of the VAR models for non-stationary data, included in the

Section 4.3, is valid also for the affine models.

Latent-Factors-Only Model

For the latent-factors-only model, the first step is to estimate parameters of

VAR model, which the first three principal components are assumed to follow.

Using the information criteria, the proposed optimal lag is identical to the

Nelson Siegel model - AIC highlights four lags as the most appropriate, whereas

the other suggest only one lag. After estimating the VAR(1) model, the residual

diagnostics (LM-statistic) shows a problem with the residual serial correlation,

10With only one adjustment: since all the affine framework requires yields expressed inthe per one period (i.e. month) representation, the data have been transformed at the verybeginning of the analysis described in this section.

4. Estimation 62

similarly to the previous model. The problem has been resolved by a use of the

VAR(4) process (with a constant and without a trend).11

Before proceeding further, it is necessary to convert the VAR(4) model to

its VAR(1) representation, in order to insert the estimated parameters directly

into the Equation 3.33 without any modification of the model. This is done by

rewriting the Equation 3.35, enriching the original symbols with tildes in order

to keep the further notation simple and consistent:

Lt = γ0 + Γ1Lt−1 + Γ2Lt−2 + Γ3Lt−3 + Γ4Lt−4 + ΣLεt

Lt−1 = 0 + ILt−1 + 0Lt−2 + 0Lt−3 + 0Lt−4 + 0

Lt−2 = 0 + 0Lt−1 + ILt−2 + 0Lt−3 + 0Lt−4 + 0

Lt−3 = 0 + 0Lt−1 + 0Lt−2 + ILt−3 + 0Lt−4 + 0

which can be converted into a matrix form:Lt

Lt−1

Lt−2

Lt−3

=

γ0

0

0

0

+

Γ1 Γ2 Γ3 Γ4

I 0 0 0

0 I 0 0

0 0 I 0

Lt−1

Lt−2

Lt−3

Lt−4

+

ΣL 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

εt

0

0

0

Lt = γ0 + Γ1Lt−1 + ΣLεt (4.4)

This transformation allows to keep the key equations of the affine model un-

changed. However, the vector of the latent factors included into the model has

changed, enriched by the three lags. Consequently, the vectors dimension l,

used throughout the Section 3.4, is modified into 4l — similarly for matrices.

Matrices γ0, Γ1 and ΣL, built from estimated parameters of the VAR pro-

cess, are listed in the Appendix A. There is included also the Figure A.7

capturing the development of the actual and fitted values, as well as the ACF

and the PACF of the residuals. The impulse-responses resulting from the VAR

model are shown by the Figure 4.16 - implications will be discussed in the

Chapter 5, when comparing all the models.

The next step is to estimate the dynamics of the short rate (Equation 3.12).

The vector of the latent factors includes also the lagged values resulting from

the VAR(4) adjustment. Since the shortest maturity included in the sample

is three months, this will be assumed to represent the short rate. Using OLS,

11Similarly to the Section 4.3, the general assumptions imposed on the error term arenot fulfilled in terms of the normality and the homoscedasticity; however, following similarstudies, the models are used despite to this fact, emphasising they are studied mainly fromtheir forecasting ability point of view, rather than examining the exact values of parameters.

4. Estimation 63

Figure 4.16: IRF of AF-L

xy$x

PC

1

0.00

0.04

0.08

xy$x

PC

2

0.00

0.04

0.08

xy$x

PC

3

0.00

0.04

0.08

0 1 2 3 4 5 6 7 8 9 10

Orthogonal Impulse Response from PC1


xy$x

PC

1

−0.

020.

01

xy$x

PC

2

−0.

020.

01

xy$x

PC

3

−0.

020.

01

0 1 2 3 4 5 6 7 8 9 10



xy$x

PC

1

−0.

020.

000.

02

xy$x

PC

2

−0.

020.

000.

02

xy$x

PC

3

−0.

020.

000.

02

0 1 2 3 4 5 6 7 8 9 10




the estimates of the parameters a0 and a1 are obtained and included in the

Appendix A.

The last (and the most demanding) task is to find the optimal estimates of

λ0 and λ1 parameters values from the Equation 3.23. λ0 is in this case a 12-

dimensional vector, and λ1 is a 12×12-dimensional matrix of parameters, which

results in a high number (156) of parameters to be optimized jointly. However,

in the Equation 3.33 (which is crucial for the calculation of the optimal λ0

and λ1 values) the λ1 is pre-multiplied only by the ΣL matrix, which has zero

values for all rows or columns with the index higher than three: consequently,

without any impact on results, all rows of the λ1 matrix with the index higher

than three may be set to zero (similar argumentation holds for λ0 vector). This

reduces the number of the parameters to be estimates to only 39, which can be

considered as more eligible to enter the optimization procedure.

The procedure itself is built as follows: First, all the parameters are set to

their initial values equal to zero (following De Pooter et al. 2007). Then, since

the equation Equation 3.34 can be assumed to be differentiable with respect

to λ0 and λ1 (their elements), a gradient descent algorithm is used: in each

iteration, a partial derivative of the total difference between the observed and

model-implied yields (resulting from Equation 3.39) with respect to each of

the parameters to be optimized - relevant elements of λ0 and λ1 - is calculated

numerically, forming a gradient. Then, the new value of λ0 and λ1 is calculated

by subtracting the gradient from the previous values, multiplied by a speed

parameter, which is set as decreasing gradually to ensure both an acceptable

speed of the convergence and the best possible fit.12. Resulting λ0 and λ1 can

be found in Appendix A.

12The procedure was written manually in MATLAB. Final λ0 and λ1 were obtained atthe moment when a decrease of the RSS (Equation 3.40) between two iterations was smallerthan 10−6. This approach didn’t converge to the absolute minima of the function, and the

4. Estimation 64

Since the number of parameters of the model is restricted (and also because

of the imperfect convergence of the numerical solution), the fit of the observed

and model-implied values is, similarly to the Nelson-Siegel framework, not ex-

act. The time series are displayed in the Figure 4.17, with further discussion

again postponed to the next chapter.

Figure 4.17: Fitted and Observed Values - AF-L

03−1995 03−1998 03−2001 03−2004 03−2007 03−2010−1

0

1

2

3

4

5

6

7

8

9

10

time

annu

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tere

st r

ate

(%)

3M

original valuefitted by model

03−1995 03−1998 03−2001 03−2004 03−2007 03−2010−1

0

1

2

3

4

5

6

7

8

9

10

time

annu

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3Y


03−1995 03−1998 03−2001 03−2004 03−2007 03−2010−1

0

1

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3

4

5

6

7

8

9

10

time

annu

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10Y


03−1995 03−1998 03−2001 03−2004 03−2007 03−2010−1

0

1

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3

4

5

6

7

8

9

10

time

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30Y



Having all the parameters estimated, the forecasts can be produced. First,

the VAR model is iterated forward, obtaining forecasts of the latent state vari-

ables. Afterwards, utilizing estimates of C1(τ) and C2(τ), forecast for yields of

any maturity are obtained from the estimation of the Equation 3.34. Focusing

on the ten maturities included in the sample, the forecasting results summa-

rizes the Table 4.14. It is useful to remind, that the model uses monthly yields.

Consequently, the resulting yields are also calculated as per month - however, in

the table, the yields are already transformed to the per-annum representation

allowing to compare them with predictions resulting from the other models.

author admits, that using a different numerical procedure may produce better results - thistask is, however, left for a further research.

4. Estimation 65

Table 4.14: AF-L Forecasts


L1 -0.80 -0.91 -0.69 -0.79 -1.16 -0.43 -0.78 -1.36 -0.20L2 -0.12 -0.18 -0.06 -0.09 -0.24 0.05 -0.07 -0.26 0.12L3 -0.01 -0.04 0.02 -0.01 -0.06 0.03 -0.01 -0.06 0.04

3M 0.01 -0.33 0.35 -0.09 -1.27 1.09 -0.13 -2.04 1.796M -0.03 -0.41 0.36 -0.10 -1.30 1.09 -0.13 -2.06 1.811Y 0.05 -0.38 0.47 0.00 -1.22 1.21 -0.01 -1.98 1.962Y 0.39 -0.08 0.86 0.37 -0.92 1.66 0.38 -1.68 2.453Y 0.70 0.19 1.21 0.70 -0.67 2.07 0.73 -1.41 2.875Y 1.20 0.64 1.77 1.24 -0.23 2.71 1.30 -0.92 3.537Y 1.67 1.08 2.26 1.73 0.23 3.24 1.81 -0.42 4.05

10Y 2.35 1.76 2.94 2.43 0.95 3.91 2.53 0.38 4.6720Y 3.64 3.18 4.10 3.72 2.6 4.84 3.82 2.26 5.3730Y 4.20 3.88 4.52 4.26 3.49 5.04 4.33 3.27 5.40



Macro-Finance Model

Information criteria of the VAR model for the vector of state variables extended

by the macro-variables give the same results as in the previous case. However,

the model is not in this case specified well (it terms of the serial correlation

tests) for any reasonable lag, either with or without constant or even the time

trend. Following similar studies and keeping consistency thorough the thesis,

the model will be still estimated and commented on, utilizing the already men-

tioned flexibility of the VAR models. To reduce the number of parameters to

minimum, the VAR(1) model (again with a constant and without a time trend)

is henceforth estimated. Matrices ˆγ0,ˆΓ1 and ˆΣL, based on the estimated pa-

rameters of the VAR process, are included in the Appendix A, as well as the

Figure A.8 including the fitted values and the time series, ACF and PACF of

residuals. Moreover, Figure 4.18 includes the IRF of the model, with implica-

tions postponed to the next chapter.

The parameters of the short rate equation as well as ˆλ0 and ˆλ1 are obtained

in the very same way as in the case of the previous model. Fitted and observed

values are compared by the Figure 4.19. Moreover, the Table 4.15 includes a

sample of predictions, again for the 14-month horizon.

4. Estimation 66

Figure 4.18: IRF of AF-M

xy$x

PC

1

0.00

0.04

xy$x

PC

2

0.00

0.04

xy$x

PC

3

0.00

0.04

0 1 2 3 4 5 6 7 8 9 10



xy$x

PC

1

−0.

010.

010.

03

xy$x

PC

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010.

010.

03

xy$x

PC

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−0.

010.

010.

03

0 1 2 3 4 5 6 7 8 9 10



xy$x

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xy$x

PC

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01

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01

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01

0 1 2 3 4 5 6 7 8 9 10



xy$x

PC

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03−

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xy$xP

C2

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03−

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PC

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03−

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0 1 2 3 4 5 6 7 8 9 10



xy$x

PC

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01

xy$x

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xy$x

PC

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0 1 2 3 4 5 6 7 8 9 10




Table 4.15: AF-M Forecasts


L1 -0.80 -0.91 -0.69 -0.79 -1.10 -0.49 -0.77 -1.29 -0.26L2 -0.13 -0.19 -0.06 -0.10 -0.24 0.03 -0.07 -0.25 0.11L3 -0.01 -0.03 0.02 0.00 -0.04 0.04 0.00 -0.05 0.05

3M 0.09 -0.15 0.34 0.04 -0.75 0.82 -0.04 -1.54 1.476M 0.04 -0.13 0.22 -0.02 -0.72 0.69 -0.07 -1.53 1.381Y -0.05 -0.20 0.10 -0.10 -0.80 0.59 -0.12 -1.62 1.392Y -0.03 -0.26 0.19 -0.03 -0.89 0.83 0.05 -1.70 1.803Y 0.26 -0.06 0.59 0.32 -0.73 1.37 0.46 -1.53 2.465Y 1.08 0.59 1.58 1.21 -0.13 2.55 1.41 -0.91 3.737Y 1.79 1.21 2.38 1.93 0.48 3.38 2.14 -0.24 4.52

10Y 2.76 2.19 3.34 2.87 1.51 4.24 3.05 0.90 5.1920Y 4.35 4.01 4.69 4.40 3.6 5.2 4.48 3.25 5.7230Y 4.53 4.29 4.76 4.56 4.01 5.11 4.62 3.77 5.47



4. Estimation 67

Figure 4.19: Fitted and Observed Values - AF-M

12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−1

0

1

2

3

4

5

6

7

8

9

10

time

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12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−1

0

1

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6

7

8

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time

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12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−1

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12−1994 12−1997 12−2000 12−2003 12−2006 12−2009 12−2012−1

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Chapter 5

Performance Evaluation

5.1 In-Sample Characteristics

The first feature of the models to be evaluated is their in-sample performance.

In the thesis, it is measured as an accuracy of the model in terms of the sum

of squared differences between the actual and fitted values — the RSS. For

the Nelson-Siegel based models, the in-sample fit is the same for both latent-

factors-only and macro-finance models, depending only on the chosen value of

λ - it is not influenced by the estimated VAR process parameters. Contrary,

the fitted yields resulting from the affine models can be obtained only after

estimating the process for the state variables, utilizing parameters of the VAR

model, and hence is different for both approaches. Results for the models

includes the Table 5.1, together with the graphical illustration as captured by

figures 4.9, 4.11, 4.17 and 4.19.

Results clearly show that the fit is distinctively better for the Nelson-Siegel

framework. The main reason has been already mentioned — the basic Nelson-

Siegel approach is static (it is rather a statistic approach than a model), with

the term structure fitted for each period separately. However, the resulting

latent factors then enter an unrestricted VAR process, which may result in poor

predictive results, as will be tested below. Contrary, the affine models are more

complex, starting with imposing restrictions on the dynamics of the variables

— then, the model is fitted within boundaries of these restrictions, which is

both less flexible and more computationally demanding.1 On the other hand,

it is possible that the cross-equation restrictions may result in a more precise

1As noted in the previous chapter, the author admits that the estimation of parametersλ0 and λ1 has not converged to the optimal point, which might be an additional reason forthe poor fit of the affine models.

5. Performance Evaluation 69

Table 5.1: In-Sample Fit Results

AF-L AF-M NS-A NS-B

3M 3.51 3.27 2.86 2.476M 6.57 5.17 0.78 0.681Y 7.97 7.74 2.79 1.742Y 8.44 11.01 1.95 1.523Y 12.44 20.40 0.33 0.455Y 17.71 43.77 2.29 1.007Y 16.14 54.48 2.41 1.29

10Y 15.34 79.15 2.56 1.9020Y 66.22 151.50 5.49 4.1830Y 108.69 163.27 2.07 1.80

note: RSS calculated from in-sample periods 4-231 (Jan 1994 - Dec 2012).

First three months were omitted because of the lags included in some models.


estimation of the dynamics of the state variables, which would support the

predictive performance of the affine models.

Moreover, not surprisingly, the NS-B models fits the observed values better

than the NS-A model. This is resulting directly from the specification of the

λ parameter the models differ in - in case of NS-B, λ is estimated to ensure

an optimal fit, whereas in the other case, the λ is set in order to place the

”hump” into the most appropriate part of the yield curve. Focusing on the

affine models, the latent-factors-only model offers smaller RSS in case of longer

maturities (bigger than one year), whereas the short ones are fitted better by

the macro-finance model. In both affine cases, however, the fit is much worse

for the longest maturities than for the short ones.

To evaluate dynamic implications resulting from the estimated models, the

IRF of the VAR capturing development of the state variables will be used as a

both simple and representative tool. The responses are graphically captured by

the figures 4.13, 4.14 and 4.15 for the dynamic Nelson-Siegel models, and 4.16

and 4.18 for the affine approach. The most important findings are following:

NS-L-A and NS-L-B:

� An impulse from β1 might by related to structural changes in the be-

havior of economic subjects, changing either a required risk-premium

or a time preference of the subjects, often related to the expected

changes of the price level. This variable represents the level of the


term structure — a yield for an infinite maturity in this case. As ob-

vious from the figures, the impact of the impulse is only temporary

- a positive shock into this variable is gradually fading, which might

be considered to be related to the business cycle dynamics. More-

over, as the positive shock represents the growth of the longest end

of the yield curve, it is accompanied by a growth of the slope (i.e. a

decrease of β2). The curvature first increases, but after six periods

(a half of a year), its dynamics change and it decreases instead.

� A positive shock from β2 - a decrease of a slope - may be represented

by a monetary policy step, typically in a period of a monetary pol-

icy restriction. Despite the fact that the short rate jumps up, IRF

implies that it has no effect on the long rate, which is in line with

the reality preceding the Lehman-Brothers crisis — the unsuccessful

attempts of FED to increase the longer rates in order to decelerate

the unhealthy economic boom.

NS-M-A and NS-M-B:

� Impulses from βs are in case of the macro-finance models very similar

to the previous, however the shocks are slightly more persistent.

� The impact of the macro-variables is also in line with the macroeco-

nomic reality. A positive shock in the production, growth of the price

level, increase of the money supply as well as a depreciation of the

U.S.Dollar are all related to the well-performing economy2. Follow-

ing such impulses, β1 — the level — is gradually slightly decreasing,

i.e. the longest rates are shifted down, which is traditionally related

to a decrease of the risk premium in the periods of conjuncture.

On the other hand, these parts of the economic cycle are related to

the already mentioned attempts of the central banks to restrict the

boom to a reasonable level, which results in a growth of the short

rate and hence a decrease of the slope of the term structure (and

growth of the β2 factor) - which is exactly what the IRF show.

AF-L:

� In the case of the affine models, the level variable, represented by

the first principal component, is calculated as a sort-of-average of all

2Either as a cause or an evidence


rates (see the discussion of the eigenvectors in the Section 4.1). A

positive shock from this variable has at first an increasing impact on

the level, which then starts to diminish (very slowly) since the fifth

period. The related dynamics of the slope shows a slight decrease

of the slope, but only temporary. The interpretation is different as

compared to the previous models, but still interesting: a shock into

the level (typically caused by a growth of the risk premium preceding

a crisis) causes the whole yield curve to shift up, but slightly more

for the short rate (caused by simultaneous monetary steps tapering

the boom). Taking opposite direction, a negative shock to the yield

curve level3 is accompanied by an increase of the slope, i.e. a fall of

the short rates4. Obviously, such interpretation is in line with the

situation observed in the last ten years.

� Decrease of the slope, representing the impulse from the second prin-

cipal component, has again no significant effect on the level factor,

illustrating the inability of the central bank to regulate the major

part of the yield curve.

AF-M:

� The dynamics following impulses of the latent factors are similar

to the AF-L model, but (similarly to the Nelson-Siegel models) the

shocks are more persistent.

� The impact of the macroeconomic variables on the level in the affine

representation (i.e. position of the whole term structure) is reflecting

slightly different aspects as compared to the dynamic Nelson-Siegel

models, with the focus set to the changes of the risk premium and

a shift in expectations of the economic subjects. A positive shock

of the production results in an increase of the risk premium and an

upward shift of the yield curve.Contrary, an increase of the price

level is in this case a clear signal of the over-heated economy —

it is followed by a decrease of the expected inflation, as well as

supportive policy steps reacting on worsening economic conditions,

which results in a downward shift of the yield curve.

3Resulting for example from a large monetary expansion in terms of an intensive acquisi-tion of the debt instruments as observed in the post-Lehman period.

4As the short rate is the first instrument the central banks use to support the economy.


� A more difficult task is to interpret the growth of the rates after

an increase of the money supply. This growth is gradual, and could

be related to changes of the expected inflation, which increases the

nominal rates. Finally, the impact of a currency depreciation on

the term structure is in the case of the AF-M model positive, which

might by related to the international capital flows.

A discussion of the dynamics of the third latent factor — curvature — is

intentionally omitted, since this variable is usually considered as both unim-

portant from the dynamic point of view and not easy to be given a intuitive

interpretation.

Obviously, the relations of the macroeconomic variables and interest rates

are based on many transition channels. However, intuitively, it can be signalled

by the interpretation of the IRF, that the dynamics implied by the affine mod-

els is more related to the financial markets situation (including the price of

risk, the monetary authority behavior on the markets, changes in expectations

etc.), whereas the Nelson-Siegel based models, not utilizing any restrictions

imposed on the rates and the market price of risk, are explaining rather the

general macroeconomic dynamics and a relation of the interest rates of various

maturities to the business cycle. This difference may be considered as implied

by the different definition of the level factor — in case of the affine models,

it represents a position of the whole yield curve, whereas the level within the

Nelson-Siegel approach is considered as an infinite maturity yield.

Most importantly, using the interpretation offered above, the dynamics ex-

plained by the macro-variables is offering an exact representation of the macro-

finance relations, i.e. impact of the macro-variables on different parts of the

yield curve, which is what was aimed to obtain by constructing the macro-

finance models.

5.2 Predictive Performance

The predictive performance will be first evaluated only for the single forecasting

period, with a detailed discussion of the results. However, such comparison is

not sufficient for an overall evaluation of the forecasting abilities of the models

— for this reason, all the models will be multiply re-estimated for shorter

periods rolling forward through the whole sample, which allows to conduct a


more general comparison and examine the performance in various parts of the

business cycle.

Forecasting Results: Basic Estimation

The predictive performance of the models is at first evaluated for the forecasts

resulting for the estimations as described above. The accuracy of the forecasts

is measured similarly as in case of the in-sample performance: the used mea-

sure is the total squared difference between the observed and predicted values

(the total square error). Predictions are made for horizon 14 months, and are

graphically illustrated in the Appendix B. The comparison of models both

totally and according to the maturities, in terms of the total squared error, is

included in the Table 5.2.

Table 5.2: Predictions - Total Square Error

maturity RW NS-L-A NS-L-B NS-M-A NS-M-B AF-L AF-M

3M 0.01 0.62 0.17 3.31 2.27 0.28 0.036M 0.02 0.11 0.07 1.47 1.44 0.50 0.201Y 0.02 0.11 0.07 0.56 0.92 0.23 0.742Y 0.08 0.15 0.11 0.26 0.45 0.10 1.443Y 0.92 0.37 0.36 0.25 0.26 0.56 0.825Y 5.36 1.45 1.49 0.77 0.76 1.34 0.897Y 8.16 2.11 2.49 1.38 1.72 2.05 1.75

10Y 7.91 2.28 2.79 1.95 2.37 1.73 4.7920Y 7.26 3.11 2.57 3.47 2.60 5.91 23.2530Y 5.22 4.09 2.74 4.88 3.01 9.89 17.86

Total 34.96 14.4 12.86 18.3 15.8 22.59 51.77


Before commenting on the results, it is necessary to remind, that the fol-

lowing comparison of the models is valid only for the evaluated period; however

it is performed in order to be able to capture the differences into a large detail.

A more general comparison will be included in the next subsection. The most

important facts resulting from the predictions are following:

1. The random walk, i.e. predictions equal to the latest observation, is

performing very well for the shortest maturities — up to two years (in-

cluded). Especially for the shortest maturity, only AF-M is able to beat

its performance in some predictive horizons. Examining the predictive


quality into more detail, utilizing the Table 5.3 displaying the ranking of

the model for various combination of maturities and prediction horizons,

the model is the best predictor for either close horizon or short maturi-

ties. However, for longer maturities, almost all other models are able to

beat the naive predictions, especially for the horizon at least five months.

2. Focusing on the difference between the estimated models, similar tables

5.4 - 5.9 were created to offer a neat intuition about the performance.

Examining the difference between the Nelson-Siegel approach and affine

models, only for latent-factors-only models, the the NS-L-A and NS-L-B

models are better predictors for the short and medium horizon (across

all maturities5), whereas the AF-L model performs better for the longest

horizon. A reason for this lies in the fact, that the affine model implies

the term premium to be bigger than observed at the end of the in-sample

and the beginning of the out-of-sample period. However, since half of the

year 2013, the term premium is returning to the level expected by the

affine models.

3. Adding the macro-factors has a different impact on dynamic Nelson-Siegel

and affine models. In the first case, the macro-factors improve the per-

formance of the NS-L-A and NS-L-B models for the largest horizon, i.e.

extend the usefulness of these models — in some cases, it beats the AF-

L model performance. On the other hand, adding macro-factors to the

affine model leads to great a improvement of the model predictions for the

shortest maturity, as well as for the medium maturities (5Y-7Y), whereas

significantly decreases the performance quality for other maturities.

4. Finally, focusing purely on the Nelson-Siegel models, λB, based on the

optimal fit, provides better results for the longest maturities, whereas λA,

set according to the empirically observed curvature, performs better for

the medium maturities.

An important point of view offer also the forecasting intervals, calculated

on the 95% significance level, which are included in the Chapter 4 in the tables

displaying the samples of the predicted values. The widest prediction intervals

produced the models based on the dynamic Nelson-Siegel approach. On the

other hand, the affine models were particularly successful in this respect, with

the intervals much narrower than intervals calculated for the naive predictions.

5Except for the shortest maturities predicted well by the random walk.


Moreover, adding the macro-factors improved this result, as the AF-M model

shows the best performance in terms of the width of the forecasting intervals.

Consequently, although the affine models predict (by the point forecasts) neg-

ative values of the interest rates slightly more frequently as compared to the

other models, the forecasting intervals generally do not allow (on the 95% sig-

nificance level) the interest rates to be negative as often as the other models

do.

Table 5.3: Prediction Rankings - Random Walk

maturityhorizon 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 20Y 30Y

1 1 3 1 1 5 4 4 5 5 12 3 3 1 1 1 1 1 2 5 23 2 1 2 1 1 1 1 1 5 34 1 1 3 2 2 1 1 1 1 25 1 1 3 4 4 7 7 6 5 46 2 1 2 6 6 7 7 7 6 47 1 1 2 4 6 7 7 7 6 58 2 1 2 6 6 7 7 7 6 69 2 1 2 4 7 7 7 7 6 610 1 1 3 3 7 7 7 7 6 611 1 1 2 3 5 7 7 7 7 712 2 2 1 3 7 7 7 7 7 713 1 3 1 2 7 7 7 7 6 514 1 1 1 2 7 7 7 7 6 5


Table 5.4: Prediction Rankings - NS-L-A


1 6 4 5 5 2 6 5 1 3 42 4 2 6 4 6 6 5 5 3 43 5 3 5 2 6 7 6 5 2 44 5 4 1 4 6 7 6 5 4 45 5 4 1 3 5 6 5 4 2 26 5 3 1 1 5 1 2 2 2 37 5 4 1 2 5 1 2 2 3 38 5 4 1 4 2 1 2 3 3 49 5 4 1 1 2 1 2 3 4 410 5 3 1 1 2 3 3 2 4 311 5 3 3 2 2 4 3 4 4 412 4 1 3 5 4 5 5 5 6 613 5 1 4 4 3 5 5 5 5 414 4 2 4 4 4 5 5 5 5 4



Table 5.5: Prediction Rankings - NS-L-B


1 4 1 2 2 1 2 2 3 1 22 2 1 3 2 5 3 3 3 1 13 3 2 1 5 5 4 3 4 4 14 4 2 4 5 5 5 4 4 5 55 4 3 4 6 3 5 4 2 1 16 4 2 4 4 1 2 3 3 1 17 4 2 3 5 3 3 3 4 1 18 4 3 3 1 3 3 5 4 2 19 4 3 3 3 1 6 5 5 2 110 3 2 2 2 1 6 6 5 1 111 2 2 1 1 1 6 6 6 3 212 1 3 2 2 5 6 6 6 4 413 2 2 2 3 5 6 6 6 4 214 2 3 2 3 6 6 6 6 3 2


Table 5.6: Prediction Rankings - NS-M-A


1 7 5 6 6 3 3 3 2 4 52 7 4 4 5 4 5 4 4 4 53 7 6 3 4 3 5 4 3 3 54 7 7 2 1 3 4 3 3 2 15 7 7 2 2 1 3 2 1 4 56 7 7 3 5 4 3 5 4 4 57 7 7 5 1 1 2 4 5 5 68 7 7 4 2 5 2 3 5 5 59 7 7 5 5 4 2 3 4 5 510 7 7 6 5 5 5 2 3 5 411 7 7 5 5 6 2 2 3 5 612 7 7 6 4 1 1 2 3 5 513 7 6 6 5 2 1 2 2 3 314 7 6 6 5 2 1 2 2 2 3


Forecasting Results: Rolling Horizon

To compare the forecasting performance in general, it is necessary to extend

the forecasting over a certain period of time, which should be chosen in order to

include the economic conjuncture with a relatively stable yield curve, as well as

periods of an economics distress and the following stabilization. For this reason,

all the models are re-estimated on a 12-years samples from (Oct1993:Sept2005)

to (Mar2001:Feb2013), forming 90 estimations of the set of the six models.

Based on these estimates (plus the random walk producing the naive forecasts),

1-month, 6-months and 12-months forecasts are calculated and compared with

the reality. The models include the same number of lags as resulting from the

previous analysis, but all the parameters are re-estimated for each model and


Table 5.7: Prediction Rankings - NS-M-B


1 5 2 3 4 4 1 1 4 2 32 6 5 2 3 3 2 2 1 2 33 6 5 4 3 4 2 2 2 1 24 6 6 5 3 4 2 2 2 3 35 6 6 5 1 2 1 1 3 3 36 6 6 5 3 3 6 6 6 3 27 6 6 6 6 2 6 6 6 4 28 6 6 6 5 4 6 6 6 4 39 6 6 7 6 5 5 6 6 3 210 6 6 7 6 6 2 5 4 2 211 6 6 7 7 7 3 5 5 2 312 6 6 7 6 2 3 4 4 3 313 6 7 7 6 4 3 3 4 1 114 6 7 7 7 3 3 3 3 1 1


Table 5.8: Prediction Rankings - AF-L


1 3 7 4 3 7 7 6 6 6 62 5 7 5 6 7 7 6 6 6 63 4 7 6 6 7 6 5 6 6 64 3 5 6 6 7 6 5 6 6 65 3 5 6 5 6 4 3 5 6 66 3 5 6 2 2 4 4 1 5 67 3 5 4 3 4 4 5 1 2 48 3 5 5 3 1 5 4 2 1 29 3 5 4 2 3 4 4 1 1 310 4 5 4 4 3 4 4 1 3 511 4 5 4 4 4 5 4 2 1 112 5 5 4 1 3 4 3 2 1 113 4 5 3 1 1 4 4 1 2 614 5 5 3 1 1 4 4 1 4 6


period separately6.

From the first view, the results are not much encouraging. Comparing the

Root mean square error (RMSE) calculated over the whole 90 forecasts per each

maturity and forecasting horizon, the random walk7 performs best in almost

all cases, as displays the Table 5.10 showing the total aggregated RMSE over

all maturities (the results are, nevertheless, similar also for the single maturi-

ties). Regardless to this fact, it is obvious, that in case of the Nelson-Siegel

6For the affine models AF-L and AF-M, the number of iterations when numerically com-puting some of the parameters had to be restricted, due to the technical limitations (theauthor has used an ordinary laptop). To estimate these models as precisely as the mainestimation described above, it would require more than four months of a pure computationaltime.

7i.e. naive forecasts set to be equal to the latest observation


Table 5.9: Prediction Rankings - AF-M


1 2 6 7 7 6 5 7 7 7 72 1 6 7 7 2 4 7 7 7 73 1 4 7 7 2 3 7 7 7 74 2 3 7 7 1 3 7 7 7 75 2 2 7 7 7 2 6 7 7 76 1 4 7 7 7 5 1 5 7 77 2 3 7 7 7 5 1 3 7 78 1 2 7 7 7 4 1 1 7 79 1 2 6 7 6 3 1 2 7 710 2 4 5 7 4 1 1 6 7 711 3 4 6 6 3 1 1 1 6 512 3 4 5 7 6 2 1 1 2 213 3 4 5 7 6 2 1 3 7 714 3 4 5 6 5 2 1 4 7 7


framework, adding macro-factors leads to on average slightly worse results than

in case of the latent-factors-only models. Moreover, the models based on λB

produce generally more accurate forecasts than these using λA. In case of the

affine models, the macro-extended model AF-M performs generally better than

the AF-L form.

However, when examining the performance in a bigger detail, certain fea-

tures of the predictive dynamics can be identified:

� For the short rate (3M), the naive forecasts (random walk) systematically

outperform the other models only for the period beginning with the fall

of the short rates in 2008 - the situation is captured by the figures 5.1

to 5.3 illustrating the development of the RMSE calculated on a rolling

9-months window. It is not surprising, that at the period of extremely

low (and rather not volatile) short rates, the naive forecasts perform best

(see the Figure 4.3 for a context). However, for the years preceding this

period, the random walk was rarely the best performing model. For 1-

month prediction horizon, the affine models produced the most accurate

forecasts, whereas for longer horizons (6 and 12 months), also the NS-L-A

and NS-L-B were successful, when very closely followed the development

of the AF-L model. Regardless the horizon, the macro-extended Nelson-

Siegel models show the worst performance for the shortest maturity.

� Moving to the middle rates (for example 3Y), the situation is outlined

by figures 5.4-5.6. For these maturities, the performance of AF-L and

AF-M models becomes poor (especially for the shortest horizon), and the


Nelson-Siegel-based models are usually not able to beat the random walk

as well. The only exception is a period of growing rates during the years

2006 and 2007, when the AF-M model is performing the best.

� Finally, using the graphical illustration also for the longest yields (30Y)

in figures 5.7-5.9, the predictions of the longest rate resulting from all

the models are beating the naive forecasts in periods of changes on the

financial markets related to the change of the market price of risk, whereas

the random walk is performing best in the periods with a relatively stable

situation.

Table 5.10: Predictions - RMSE

modelhorizon RW NS-L-A NS-L-B NS-M-A NS-M-B AF-L AF-M

1 month 0.2510 0.2903 0.2780 0.3354 0.3250 1.0998 0.99976 month 0.7193 0.8451 0.8389 1.1378 1.1350 1.3941 1.280112 month 0.9878 1.2107 1.2032 1.8352 1.8340 1.7167 1.5694

note: RMSE calculated from out-sample periods September 2005 - February 2013.


Figure 5.1: 3M Yields Forecasting: One Month Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

period of prediction

RM

SE

: 9−

perio

ds w

indo

w

random walkNS−L−ANS−L−BNS−M−ANS−M−BAF−LAF−M


5.3 Comparison with Similar Studies

The implications of the forecasting performance evaluation can be well extended

by comparing the results with findings of similar studies, which were already


Figure 5.2: 3M Yields Forecasting: Six Months Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.5

1

1.5

2

2.5


RM

SE

: 9−

perio

ds w

indo

w



Figure 5.3: 3M Yields Forecasting: One Year Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.5

1

1.5

2

2.5

3

3.5

4

4.5


RM

SE

: 9−

perio

ds w

indo

w



mentioned throughout the thesis. Diebold & Li (2006) are dealing with the

dynamic Nelson-Siegel model — both an AR(1) version, auto-regressing the

latent factors separately, and a model as of the NS-L-A type. They anal-

yse U.S. data in the period from January 1985 to December 2000, including

yields of maturities up to ten years. The predictions are made for the period

starting at the beginning of 1994, and are based on the estimations resulting

from the whole sample before, i.e. the estimation sample is gradually increas-

ing. Authors find, that the AR(1) version is able to beat the random walk for

most maturities and forecasting horizons, whereas the VAR model is perform-

ing rather poorly. Authors explain this by the problem of a huge number of

the VAR parameters. Following this approach, Diebold et al. (2006) estimate


Figure 5.4: 3Y Yields Forecasting: One Month Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.5

1

1.5

2

2.5


RM

SE

: 9−

perio

ds w

indo

w



Figure 5.5: 3Y Yields Forecasting: Six Months Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.5

1

1.5

2

2.5


RM

SE

: 9−

perio

ds w

indo

w



the dynamic Nelson-Siegel model with the state variables vector enriched by

macroeconomic variables, forming a NS-M-A model. Authors do not examine

forecasting ability of the model, but conclude that the macroeconomic variables

have a strong impact on future yields.

Ang & Piazzesi (2003) use a much longer period — from June 1952 to De-

cember 2000, including maturities up to five years. Moreover, authors repre-

sent the macroeconomic situation by variables constructed as the first principal

components of two groups of macroeconomic variables, called inflation and real

activity. Based on the data, the models similar to AF-L and AF-M are created

(using a maximum likelihood method). Forecasting is made one step ahead,

and its results are following: for the maturity 3 months, the naive forecasts are


Figure 5.6: 3Y Yields Forecasting: One Year Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.5

1

1.5

2

2.5

3

3.5

4


RM

SE

: 9−

perio

ds w

indo

w



Figure 5.7: 30Y Yields Forecasting: One Month Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


RM

SE

: 9−

perio

ds w

indo

w



not beaten. For other maturities, AF-M offers the best results, and AF-L per-

forms better than the random walk as well. Author also use the AF-M model

with a lagged macro-variables, which however performs rather poorly.

Finally De Pooter et al. (2007) perform a comparative analysis of all men-

tioned models. They include 13 maturities up to 10 years, for the period

from January 1970 to December 2003, with the macro-variables based again

on the common factors extracted from a panel of macroeconomic data. Au-

thors conclude, that the models are generally able to outperform the random

walk, but adding the macro-factors is differently beneficial in different time

periods: in times of uncertainty, with volatile yield curves, they perform better

than the latent-factors-only models, whereas in the stable periods, they are


Figure 5.8: 30Y Yields Forecasting: Six Months Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


RM

SE

: 9−

perio

ds w

indo

w



Figure 5.9: 30Y Yields Forecasting: One Year Prediction Horizon

12−2005 12−2006 12−2007 12−2008 12−2009 12−2010 12−2011 12−20120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


RM

SE

: 9−

perio

ds w

indo

w



slightly weaker in terms of forecasting. Authors utilize this facts when building

a model based on a combination of the estimated models, which consequently

outperforms the single models.

Chapter 6

Conclusion

The thesis compares various approaches to the term structure of interest rates

modelling, focusing on the macro-finance models. The work is concerned with

capturing the real-world dynamics of the interest rates, with emphasis given to

the estimated models properties and a forecasting performance. Two widely-

used groups of models are first described: a dynamic Nelson-Siegel framework

and an affine class of models. Afterwards, the models are exactly specified

in several different ways, deriving both latent-factors-only and macro-finance

version for each of the groups of models. Moreover, the Nelson-Siegel based

models also vary by the two different ways to obtaining the fit-parameter λ.

A naıve model — the random walk — is used as a benchmark, as it produces

naive forecasts equal to the latest observation.

After the models are estimated on U.S. data, the estimation results show,

that the in-sample fit is better for the Nelson-Siegel based models as compared

to the affine models, mainly because of the parsimony of these models. How-

ever, it is shown that the models can be regarded as complementary, since the

latent factor representing the level of the yield curve is specified differently in

both cases — this is supported by an inspection of impulse responses result-

ing from the models, which are in line with both macroeconomic and financial

reality. An important implication from the impulse responses is also the fact,

that the explicit inclusion of the macroeconomic variables allows to examine

the impact of the macroeconomic variables on different parts of the yield curve.

The forecasts are performed in two ways — first, results of the estimated

models are used to create predictions for the most recent period, which is

then compared to the reality. It is shown, that the naive forecasts are not

outperformed for both shortest maturities and shortest prediction horizons —

6. Conclusion 85

except for the ability of the macro-extended affine model to beat the naive

forecasts in some cases. However, for longer maturities (above two years) and

a longer prediction horizon (more then four months), most of the estimated

models predict better than the naıve approach. Moreover, adding macro-factors

is rather beneficial for the affine models, whereas worsens results of the Nelson-

Siegel-based predictions. A certain complementarity of the models is again

present. Second, the models are re-estimated based on a rolling in-sample, with

forecasts resulting from them. Summing the results, the random walk performs

the best, followed by the Nelson-Siegel model, with the affine models performing

the worst. However, when focusing on the time-development of the forecasting

accuracy, the random walk is outperformed frequently, depending on the actual

conditions: For the shortest-maturity yields, the models are successful except

for the extremely low short rates as since the year 2009. Contrary, for the

longest maturities, the models are able to predict the reality well especially in

the times of changes on financial markets, with varying market price of risk.

As compared to similar studies, the biggest benefits of the thesis are fol-

lowing: First, maturities longer than 10 years are included, and the estimated

models are able to both capture their dynamics and produce reasonable fore-

casts. Second, it is shown that the dynamic Nelson-Siegel and affine models are

complementary into a significant extent, both of them performing differently

under various market and economic conditions; moreover, adding the macro-

factors is more beneficial for the affine models than for the Nelson-Siegel-based.

Third, it has been illustrated, that the fit-parameter of the dynamic Nelson-

Siegel model, based on an optimal in-sample fit, offers better predictive re-

sults as compared to the parameter value ensuring the optimal curvature of

the yield curve. Finally, the models were estimated on the most recent data,

which makes them stable in terms of structural changes and — since they try

to explain macroeconomic relations — can be regarded as resisting the Lucas

Critique. Consequently, the estimated relations of the variables, implying a

weak impact of the monetary policy steps on the longest maturities, can be

considered as generally valid.

In many cases, the analysis could be further extended. Different models

could be included into the analysis, for example a DSGE model with a term

structure specification, which would make the comparison more comprehen-

sive. Focusing on the included models, a different approach to the affine model

estimation could be used, particularly solving the difficulties related to the

numerical procedures.

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Appendix A

Estimated Parameters

NS-L-A

Figure A.1: NS-L-A estimation results

34

56

78

Diagram of fit and residuals for beta1A

0 50 100 150 200

−1.

00.

00.

5

0 2 4 6 8 10 12

0.0

0.8

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10.

2

Lag

PACF Residuals

−5

−3

−1

1


0 50 100 150 200

−1.

00.

00.

5

0 2 4 6 8 10 12

0.0

0.8

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals

−6

−2

02


0 50 100 150 200

−2

−1

01

2

0 2 4 6 8 10 12

−0.

20.

8

Lag

ACF Residuals

2 4 6 8 10 12

−0.

20.

1

Lag

PACF Residuals


A. Estimated Parameters II

Table A.1: Estimation results for equation beta1A:

Estimate Std. Error t value Pr(> |t|)

beta1A.l1 1.0139933 0.1073265 9.448 < 2e-16beta2A.l1 0.0015088 0.0791861 0.019 0.9848beta3A.l1 0.0133321 0.0226547 0.588 0.5568beta1A.l2 -0.2488234 0.1488980 -1.671 0.0962beta2A.l2 0.0755234 0.1116589 0.676 0.4995beta3A.l2 -0.0150110 0.0324617 -0.462 0.6443beta1A.l3 0.2426650 0.1491001 1.628 0.1051beta2A.l3 -0.1536011 0.1140515 -1.347 0.1795beta3A.l3 -0.0294117 0.0325206 -0.904 0.3668beta1A.l4 -0.1854988 0.1031092 -1.799 0.0734beta2A.l4 0.0665581 0.0749128 0.888 0.3753beta3A.l4 0.0272309 0.0249150 1.093 0.2756const 1.2808763 0.3087691 4.148 4.84e-05trend -0.0030140 0.0007073 -4.261 3.05e-05

Source: R-studio, author’s computations



beta1A.l1 -0.0361915 0.1484708 -0.244 0.8076beta2A.l1 0.9732143 0.1095425 8.884 2.73e-16beta3A.l1 0.0675338 0.0313395 2.155 0.0323beta1A.l2 0.4960026 0.2059790 2.408 0.0169beta2A.l2 0.1575704 0.1544639 1.020 0.3088beta3A.l2 0.0479330 0.0449060 1.067 0.2870beta1A.l3 -0.4426866 0.2062584 -2.146 0.0330beta2A.l3 -0.0194332 0.1577738 -0.123 0.9021beta3A.l3 -0.0751986 0.0449876 -1.672 0.0961beta1A.l4 0.0730462 0.1426367 0.512 0.6091beta2A.l4 -0.1729574 0.1036310 -1.669 0.0966beta3A.l4 0.0186514 0.0344663 0.541 0.5890const -0.7834363 0.4271375 -1.834 0.0680trend 0.0024783 0.0009785 2.533 0.0120


A. Estimated Parameters III



beta1A.l1 1.087963 0.325185 3.346 0.00097beta2A.l1 0.721926 0.239923 3.009 0.00294beta3A.l1 0.944119 0.068641 13.755 < 2e-16beta1A.l2 -1.224839 0.451141 -2.715 0.00717beta2A.l2 -0.895088 0.338311 -2.646 0.00876beta3A.l2 -0.146735 0.098354 -1.492 0.13721beta1A.l3 -0.016959 0.451753 -0.038 0.97009beta2A.l3 0.032708 0.345561 0.095 0.92468beta3A.l3 0.207122 0.098533 2.102 0.03672beta1A.l4 0.106531 0.312407 0.341 0.73344beta2A.l4 0.238280 0.226976 1.050 0.29500beta3A.l4 -0.173485 0.075489 -2.298 0.02252const 0.658438 0.935528 0.704 0.48232trend -0.004781 0.002143 -2.231 0.02674


A. Estimated Parameters IV

NS-L-B

Figure A.2: NS-L-B estimation results3

45

67

8

Diagram of fit and residuals for beta1B

0 50 100 150 200

−1.

0−

0.5

0.0

0.5

0 2 4 6 8 10 12

0.0

0.8

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals

−5

−3

−1


0 50 100 150 200

−1.

00.

00.

50 2 4 6 8 10 12

0.0

0.8

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals

−6

−2

02

4


0 50 100 150 200

−2

−1

01

2

0 2 4 6 8 10 12

0.0

0.8

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals


A. Estimated Parameters V

Table A.4: Estimation results for equation beta1B:


beta1B.l1 0.9758961 0.1056927 9.233 < 2e-16beta2B.l1 -0.0112432 0.0780282 -0.144 0.8856beta3B.l1 0.0178910 0.0216349 0.827 0.4092beta1B.l2 -0.2056409 0.1468627 -1.400 0.1629beta2B.l2 0.0903262 0.1128790 0.800 0.4245beta3B.l2 -0.0278579 0.0300483 -0.927 0.3549beta1B.l3 0.2326543 0.1470977 1.582 0.1152beta2B.l3 -0.1627541 0.1150401 -1.415 0.1586beta3B.l3 -0.0132403 0.0300067 -0.441 0.6595beta1B.l4 -0.1851969 0.1013710 -1.827 0.0691beta2B.l4 0.0677670 0.0754446 0.898 0.3701beta3B.l4 0.0229792 0.0231778 0.991 0.3226const 1.2980557 0.3085908 4.206 3.82e-05trend -0.0028650 0.0006945 -4.125 5.31e-05




beta1B.l1 0.059199 0.145493 0.407 0.6845beta2B.l1 1.058078 0.107411 9.851 <2e-16beta3B.l1 0.060664 0.029782 2.037 0.0429beta1B.l2 0.396035 0.202166 1.959 0.0514beta2B.l2 0.100679 0.155385 0.648 0.5177beta3B.l2 0.043377 0.041363 1.049 0.2955beta1B.l3 -0.441115 0.202489 -2.178 0.0305beta2B.l3 -0.040605 0.158360 -0.256 0.7979beta3B.l3 -0.074092 0.041306 -1.794 0.0743beta1B.l4 0.077945 0.139544 0.559 0.5770beta2B.l4 -0.152191 0.103854 -1.465 0.1443beta3B.l4 0.017502 0.031906 0.549 0.5839const -0.755286 0.424795 -1.778 0.0768trend 0.002067 0.000956 2.162 0.0318


A. Estimated Parameters VI



beta1B.l1 1.048395 0.330350 3.174 0.00173beta2B.l1 0.619486 0.243883 2.540 0.01179beta3B.l1 0.899242 0.067621 13.298 < 2e-16beta1B.l2 -1.315102 0.459030 -2.865 0.00459beta2B.l2 -0.891959 0.352811 -2.528 0.01219beta3B.l2 -0.127735 0.093918 -1.360 0.17525beta1B.l3 0.124540 0.459764 0.271 0.78675beta2B.l3 0.096676 0.359566 0.269 0.78829beta3B.l3 0.232782 0.093788 2.482 0.01384beta1B.l4 0.077828 0.316842 0.246 0.80620beta2B.l4 0.230451 0.235807 0.977 0.32954beta3B.l4 -0.199289 0.072444 -2.751 0.00645const 0.796934 0.964523 0.826 0.40959trend -0.005268 0.002171 -2.427 0.01606


A. Estimated Parameters VII

NS-M-A

Figure A.3: NS-M-A estimation results - latent variables3

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PACF Residuals


A. Estimated Parameters VIII

Figure A.4: NS-M-A estimation results - macroeconomic variables

−15

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Diagram of fit and residuals for IPI

0 50 100 150 200

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Diagram of fit and residuals for CPI

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Diagram of fit and residuals for M1

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Diagram of fit and residuals for dollar.index

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PACF Residuals


A. Estimated Parameters IX



beta1A.l1 1.065027 0.1157 9.205 <2e-16beta2A.l1 0.035073 0.087973 0.399 0.691beta3A.l1 0.036911 0.025172 1.466 0.144IPI.l1 -0.009958 0.021934 -0.454 0.65CPI.l1 -0.046912 0.052032 -0.902 0.368M1.l1 0.007334 0.016312 0.45 0.653dollar.index.l1 -0.003408 0.008401 -0.406 0.685beta1A.l2 -0.247524 0.159861 -1.548 0.123beta2A.l2 0.062193 0.120247 0.517 0.606beta3A.l2 -0.02725 0.035542 -0.767 0.444IPI.l2 0.013891 0.030797 0.451 0.652CPI.l2 0.041725 0.086345 0.483 0.629M1.l2 -0.024121 0.021408 -1.127 0.261dollar.index.l2 0.005681 0.012826 0.443 0.658beta1A.l3 0.280313 0.160372 1.748 0.082beta2A.l3 -0.112353 0.124705 -0.901 0.369beta3A.l3 -0.022411 0.035872 -0.625 0.533IPI.l3 -0.031735 0.030767 -1.031 0.304CPI.l3 -0.077255 0.085515 -0.903 0.367M1.l3 0.002024 0.021352 0.095 0.925dollar.index.l3 -0.01127 0.012811 -0.88 0.38beta1A.l4 -0.105135 0.114033 -0.922 0.358beta2A.l4 0.042888 0.085497 0.502 0.616beta3A.l4 0.002353 0.028272 0.083 0.934IPI.l4 0.023985 0.021924 1.094 0.275CPI.l4 0.042448 0.052751 0.805 0.422M1.l4 0.012842 0.015517 0.828 0.409dollar.index.l4 0.010244 0.008405 1.219 0.224const 0.181851 0.202665 0.897 0.371


A. Estimated Parameters X



beta1A.l1 -0.1425564 0.1541078 -0.925 0.3561beta2A.l1 0.836459 0.1171756 7.139 1.75e-12beta3A.l1 0.0373072 0.0335278 1.113 0.2672IPI.l1 0.0337623 0.0292157 1.156 0.2492CPI.l1 0.0069091 0.0693049 0.1 0.9207M1.l1 -0.0068317 0.0217275 -0.314 0.7535dollar.index.l1 0.0040184 0.0111895 0.359 0.7199beta1A.l2 0.4940774 0.2129275 2.32 0.0213beta2A.l2 0.175288 0.1601633 1.094 0.2751beta3A.l2 0.0695838 0.0473405 1.47 0.1432IPI.l2 -0.0191166 0.0410205 -0.466 0.6417CPI.l2 -0.0899581 0.115008 -0.782 0.435M1.l2 0.0187666 0.0285144 0.658 0.5112dollar.index.l2 -0.0224014 0.0170836 -1.311 0.1913beta1A.l3 -0.4416754 0.213608 -2.068 0.04beta2A.l3 -0.0395218 0.1661017 -0.238 0.8122beta3A.l3 -0.0654249 0.0477803 -1.369 0.1725IPI.l3 0.0119574 0.0409809 0.292 0.7708CPI.l3 0.1269091 0.1139026 1.114 0.2665M1.l3 -0.007893 0.0284401 -0.278 0.7817dollar.index.l3 0.0263026 0.0170635 1.541 0.1248beta1A.l4 -0.0002758 0.1518876 -0.002 0.9986beta2A.l4 -0.0927531 0.1138781 -0.814 0.4163beta3A.l4 0.0487715 0.0376567 1.295 0.1968IPI.l4 -0.0048517 0.0292016 -0.166 0.8682CPI.l4 -0.0230025 0.0702622 -0.327 0.7437M1.l4 0.0013997 0.0206676 0.068 0.9461dollar.index.l4 -0.0130001 0.011195 -1.161 0.2469const 0.3054586 0.2699415 1.132 0.2592


A. Estimated Parameters XI



beta1A.l1 1.25265 0.34049 3.679 0.000302beta2A.l1 0.80542 0.25889 3.111 0.00214beta3A.l1 0.98882 0.07408 13.348 <2e-16IPI.l1 -0.08226 0.06455 -1.274 0.204046CPI.l1 -0.18127 0.15313 -1.184 0.2379M1.l1 -0.02129 0.04801 -0.444 0.657853dollar.index.l1 -0.03766 0.02472 -1.523 0.12925beta1A.l2 -1.06897 0.47045 -2.272 0.024148beta2A.l2 -0.95125 0.35387 -2.688 0.007797beta3A.l2 -0.21673 0.1046 -2.072 0.039556IPI.l2 0.12619 0.09063 1.392 0.165375CPI.l2 0.20698 0.2541 0.815 0.416306M1.l2 0.06754 0.063 1.072 0.285018dollar.index.l2 0.07039 0.03775 1.865 0.063668beta1A.l3 -0.04039 0.47196 -0.086 0.931895beta2A.l3 0.17602 0.36699 0.48 0.63202beta3A.l3 0.26271 0.10557 2.489 0.013652IPI.l3 -0.11084 0.09055 -1.224 0.222366CPI.l3 0.01265 0.25166 0.05 0.959947M1.l3 -0.084 0.06284 -1.337 0.182849dollar.index.l3 -0.07183 0.0377 -1.905 0.058181beta1A.l4 0.11242 0.33559 0.335 0.737991beta2A.l4 0.13048 0.25161 0.519 0.60462beta3A.l4 -0.21199 0.0832 -2.548 0.011595IPI.l4 0.05798 0.06452 0.899 0.369927CPI.l4 0.02698 0.15524 0.174 0.862193M1.l4 0.04971 0.04566 1.089 0.277695dollar.index.l4 0.03374 0.02473 1.364 0.174036const -1.62638 0.59642 -2.727 0.006968


A. Estimated Parameters XII

Table A.10: Estimation results for equation IPI-A:


beta1A.l1 -0.413094 0.38005 -1.087 0.27838beta2A.l1 -0.269675 0.288971 -0.933 0.35184beta3A.l1 0.09593 0.082684 1.16 0.24736IPI.l1 0.932933 0.07205 12.948 <2e-16CPI.l1 0.184864 0.170915 1.082 0.28074M1.l1 -0.007979 0.053583 -0.149 0.88177dollar.index.l1 0.03897 0.027595 1.412 0.15946beta1A.l2 1.0092 0.525107 1.922 0.05605beta2A.l2 0.688533 0.394984 1.743 0.08285beta3A.l2 0.103409 0.116748 0.886 0.37683IPI.l2 0.204581 0.101162 2.022 0.04449CPI.l2 -0.344505 0.283625 -1.215 0.22595M1.l2 -0.043635 0.07032 -0.621 0.53563dollar.index.l2 -0.093862 0.04213 -2.228 0.02701beta1A.l3 -0.467053 0.526786 -0.887 0.37636beta2A.l3 -0.104953 0.409629 -0.256 0.79805beta3A.l3 -0.103243 0.117833 -0.876 0.38199IPI.l3 0.023824 0.101064 0.236 0.81388CPI.l3 0.245826 0.280899 0.875 0.38256M1.l3 -0.017421 0.070137 -0.248 0.80409dollar.index.l3 0.004411 0.042081 0.105 0.91662beta1A.l4 -0.049085 0.374575 -0.131 0.89587beta2A.l4 -0.295727 0.280839 -1.053 0.29362beta3A.l4 -0.094926 0.092867 -1.022 0.30794IPI.l4 -0.2113 0.072015 -2.934 0.00374CPI.l4 -0.227622 0.173276 -1.314 0.19049M1.l4 0.080779 0.050969 1.585 0.11459dollar.index.l4 0.053167 0.027608 1.926 0.05557const 0.03441 0.665712 0.052 0.95883


A. Estimated Parameters XIII

Table A.11: Estimation results for equation CPI-A:


beta1A.l1 0.157271 0.165636 0.949 0.34353beta2A.l1 -0.101837 0.125941 -0.809 0.41971beta3A.l1 -0.035127 0.036036 -0.975 0.33085IPI.l1 -0.005482 0.031401 -0.175 0.86158CPI.l1 1.285026 0.074489 17.251 <2e-16M1.l1 -0.016565 0.023353 -0.709 0.47895dollar.index.l1 -0.023474 0.012026 -1.952 0.05237beta1A.l2 -0.064059 0.228856 -0.28 0.77984beta2A.l2 0.222375 0.172144 1.292 0.19794beta3A.l2 0.074128 0.050882 1.457 0.14674IPI.l2 0.114941 0.044089 2.607 0.00983CPI.l2 -0.594818 0.123611 -4.812 2.96e-7M1.l2 0.019274 0.030647 0.629 0.53015dollar.index.l2 0.013332 0.018362 0.726 0.46865beta1A.l3 0.110232 0.229587 0.48 0.63166beta2A.l3 0.059775 0.178527 0.335 0.73811beta3A.l3 0.002229 0.051355 0.043 0.96542IPI.l3 -0.10798 0.044046 -2.452 0.01509CPI.l3 0.172639 0.122423 1.41 0.16006M1.l3 0.006854 0.030568 0.224 0.82281dollar.index.l3 -0.021671 0.01834 -1.182 0.23878beta1A.l4 -0.19743 0.16325 -1.209 0.22796beta2A.l4 -0.134652 0.122397 -1.1 0.27261beta3A.l4 -0.058922 0.040474 -1.456 0.14703IPI.l4 0.006446 0.031386 0.205 0.83749CPI.l4 0.035179 0.075518 0.466 0.64185M1.l4 -0.014738 0.022214 -0.663 0.50781dollar.index.l4 0.02104 0.012032 1.749 0.08191const 0.283294 0.290135 0.976 0.33005


A. Estimated Parameters XIV

Table A.12: Estimation results for equation M1-A:


beta1A.l1 -0.2321762 0.4890817 -0.475 0.6355beta2A.l1 -0.2153226 0.3718724 -0.579 0.5632beta3A.l1 -0.0382905 0.1064048 -0.36 0.7193IPI.l1 -0.0002715 0.0927199 -0.003 0.9977CPI.l1 -0.4742189 0.2199482 -2.156 0.0323M1.l1 0.8326987 0.0689552 12.076 <2e-16dollar.index.l1 -0.0348356 0.0355113 -0.981 0.3278beta1A.l2 -0.0555808 0.6757537 -0.082 0.9345beta2A.l2 -0.0012025 0.5082994 -0.002 0.9981beta3A.l2 -0.0496502 0.1502412 -0.33 0.7414IPI.l2 -0.1552327 0.1301839 -1.192 0.2345CPI.l2 0.3330265 0.3649932 0.912 0.3627M1.l2 -0.0106098 0.0904943 -0.117 0.9068dollar.index.l2 0.0685641 0.0542171 1.265 0.2075beta1A.l3 -0.0921122 0.6779135 -0.136 0.8921beta2A.l3 -0.0459945 0.5271458 -0.087 0.9306beta3A.l3 0.0843178 0.1516372 0.556 0.5788IPI.l3 -0.248727 0.1300583 -1.912 0.0573CPI.l3 0.6840449 0.3614849 1.892 0.0599M1.l3 0.1968976 0.0902586 2.181 0.0303dollar.index.l3 0.0118191 0.0541532 0.218 0.8275beta1A.l4 0.2639348 0.4820354 0.548 0.5846beta2A.l4 0.0620252 0.3614073 0.172 0.8639beta3A.l4 -0.1063405 0.1195087 -0.89 0.3746IPI.l4 0.4099631 0.0926752 4.424 0.000016CPI.l4 -0.4674619 0.2229863 -2.096 0.0373M1.l4 -0.1140857 0.0655914 -1.739 0.0835dollar.index.l4 -0.0288893 0.0355289 -0.813 0.4171const 0.102115 0.8566953 0.119 0.9052


A. Estimated Parameters XV

Table A.13: Estimation results for equation USDI-A:


beta1A.l1 0.750963 1.019285 0.737 0.462143beta2A.l1 0.494768 0.775011 0.638 0.52395beta3A.l1 0.331411 0.221756 1.494 0.136641IPI.l1 -0.124842 0.193236 -0.646 0.518987CPI.l1 0.06406 0.458389 0.14 0.888999M1.l1 0.103181 0.143708 0.718 0.473609dollar.index.l1 1.267837 0.074008 17.131 <2e-16beta1A.l2 -1.227661 1.408324 -0.872 0.384418beta2A.l2 -0.69714 1.059336 -0.658 0.511244beta3A.l2 0.053637 0.313115 0.171 0.864161IPI.l2 -0.447531 0.271313 -1.649 0.100632CPI.l2 1.178531 0.760675 1.549 0.122901M1.l2 -0.090343 0.188597 -0.479 0.632448dollar.index.l2 -0.412151 0.112993 -3.648 0.000338beta1A.l3 0.788821 1.412825 0.558 0.577251beta2A.l3 0.600461 1.098614 0.547 0.585295beta3A.l3 -0.552163 0.316024 -1.747 0.08215IPI.l3 0.927752 0.271052 3.423 0.000753CPI.l3 -1.658987 0.753363 -2.202 0.028812M1.l3 0.076378 0.188106 0.406 0.685153dollar.index.l3 0.075684 0.11286 0.671 0.503253beta1A.l4 -0.511221 1.0046 -0.509 0.611403beta2A.l4 -0.48123 0.753201 -0.639 0.523618beta3A.l4 0.261161 0.249066 1.049 0.295658IPI.l4 -0.375264 0.193143 -1.943 0.053441CPI.l4 0.917627 0.464721 1.975 0.049706M1.l4 -0.097845 0.136698 -0.716 0.474973dollar.index.l4 0.004225 0.074045 0.057 0.954553const -0.088452 1.785421 -0.05 0.960538


A. Estimated Parameters XVI

NS-M-B

Figure A.5: NS-M-B estimation results - latent variables3

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A. Estimated Parameters XVII

Figure A.6: NS-M-B estimation results - macroeconomic variables

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A. Estimated Parameters XVIII



beta1B.l1 1.029331 0.113764 9.048 <2e-16beta2B.l1 0.027114 0.086544 0.313 0.7544beta3B.l1 0.037094 0.023762 1.561 0.1201IPI.l1 -0.006783 0.021261 -0.319 0.75CPI.l1 -0.040469 0.050465 -0.802 0.4236M1.l1 0.008559 0.015894 0.539 0.5908dollar.index.l1 -0.001695 0.008164 -0.208 0.8358beta1B.l2 -0.216342 0.15742 -1.374 0.1709beta2B.l2 0.074599 0.121353 0.615 0.5394beta3B.l2 -0.035845 0.032578 -1.1 0.2725IPI.l2 0.010134 0.029844 0.34 0.7346CPI.l2 0.037715 0.083768 0.45 0.653M1.l2 -0.027528 0.020852 -1.32 0.1883dollar.index.l2 0.002447 0.012464 0.196 0.8446beta1B.l3 0.272887 0.157818 1.729 0.0853beta2B.l3 -0.128221 0.125575 -1.021 0.3085beta3B.l3 -0.010733 0.032663 -0.329 0.7428IPI.l3 -0.026621 0.029824 -0.893 0.3732CPI.l3 -0.082999 0.083002 -1 0.3185M1.l3 0.006986 0.020795 0.336 0.7373dollar.index.l3 -0.008111 0.012437 -0.652 0.515beta1B.l4 -0.103879 0.111849 -0.929 0.3542beta2B.l4 0.045187 0.085987 0.526 0.5998beta3B.l4 0.004301 0.025946 0.166 0.8685IPI.l4 0.020133 0.021333 0.944 0.3465CPI.l4 0.044712 0.051191 0.873 0.3835M1.l4 0.009774 0.015122 0.646 0.5188dollar.index.l4 0.008737 0.008158 1.071 0.2855const 0.24222 0.202204 1.198 0.2324


A. Estimated Parameters XIX



beta1B.l1 -0.0464334 0.1501457 -0.309 0.7575beta2B.l1 0.9154039 0.1142203 8.014 9.38e-15beta3B.l1 0.0454213 0.0313608 1.448 0.1491IPI.l1 0.0251214 0.0280605 0.895 0.3717CPI.l1 -0.0096574 0.0666037 -0.145 0.8849M1.l1 -0.0095624 0.0209765 -0.456 0.649dollar.index.l1 0.0005571 0.0107744 0.052 0.9588beta1B.l2 0.4067531 0.2077631 1.958 0.0517beta2B.l2 0.1158126 0.1601611 0.723 0.4705beta3B.l2 0.0583771 0.0429967 1.358 0.1761IPI.l2 -0.0082331 0.0393882 -0.209 0.8346CPI.l2 -0.0709828 0.110557 -0.642 0.5216M1.l2 0.0250092 0.0275201 0.909 0.3646dollar.index.l2 -0.0157955 0.0164506 -0.96 0.3381beta1B.l3 -0.4414666 0.2082875 -2.12 0.0353beta2B.l3 -0.0388578 0.1657334 -0.234 0.8149beta3B.l3 -0.0570123 0.043108 -1.323 0.1875IPI.l3 0.002513 0.0393622 0.064 0.9492CPI.l3 0.126593 0.1095457 1.156 0.2492M1.l3 -0.0158025 0.0274448 -0.576 0.5654dollar.index.l3 0.0197742 0.0164146 1.205 0.2298beta1B.l4 0.0100869 0.1476177 0.068 0.9456beta2B.l4 -0.0733414 0.1134857 -0.646 0.5189beta3B.l4 0.0356168 0.034243 1.04 0.2996IPI.l4 0.0024084 0.0281556 0.086 0.9319CPI.l4 -0.0205358 0.0675618 -0.304 0.7615M1.l4 0.0071122 0.0199574 0.356 0.7219dollar.index.l4 -0.0099796 0.0107669 -0.927 0.3551const 0.1913834 0.2668688 0.717 0.4741


A. Estimated Parameters XX



beta1B.l1 1.245875 0.346657 3.594 0.000411beta2B.l1 0.773333 0.263712 2.932 0.003759beta3B.l1 0.945147 0.072406 13.053 <2e-16IPI.l1 -0.082539 0.064786 -1.274 0.20415CPI.l1 -0.176723 0.153775 -1.149 0.251847M1.l1 -0.016769 0.048431 -0.346 0.729517dollar.index.l1 -0.038313 0.024876 -1.54 0.125117beta1B.l2 -1.131755 0.479685 -2.359 0.019279beta2B.l2 -0.952674 0.369781 -2.576 0.010714beta3B.l2 -0.191503 0.099271 -1.929 0.055149IPI.l2 0.117785 0.09094 1.295 0.196759CPI.l2 0.213387 0.255255 0.836 0.404175M1.l2 0.061082 0.063539 0.961 0.337556dollar.index.l2 0.075449 0.037981 1.986 0.048358beta1B.l3 0.066709 0.480895 0.139 0.889813beta2B.l3 0.235206 0.382646 0.615 0.53947beta3B.l3 0.269591 0.099528 2.709 0.007346IPI.l3 -0.115023 0.09088 -1.266 0.207121CPI.l3 0.001105 0.25292 0.004 0.996517M1.l3 -0.086563 0.063365 -1.366 0.173454dollar.index.l3 -0.07716 0.037898 -2.036 0.043083beta1B.l4 0.110261 0.340821 0.324 0.746647beta2B.l4 0.079209 0.262017 0.302 0.762737beta3B.l4 -0.238556 0.07906 -3.017 0.002885IPI.l4 0.059948 0.065006 0.922 0.357551CPI.l4 0.015405 0.155987 0.099 0.921429M1.l4 0.050408 0.046078 1.094 0.275294dollar.index.l4 0.036622 0.024859 1.473 0.142286const -1.752856 0.616148 -2.845 0.00491


A. Estimated Parameters XXI

Table A.17: Estimation results for equation IPI-B:


beta1B.l1 -0.418929 0.385221 -1.088 0.27814beta2B.l1 -0.245557 0.293049 -0.838 0.40308beta3B.l1 0.101998 0.080461 1.268 0.2064IPI.l1 0.934568 0.071993 12.981 <2e-16CPI.l1 0.181627 0.170882 1.063 0.28913M1.l1 -0.008724 0.053818 -0.162 0.87139dollar.index.l1 0.038624 0.027643 1.397 0.16391beta1B.l2 1.016896 0.533047 1.908 0.05788beta2B.l2 0.726865 0.410917 1.769 0.07845beta3B.l2 0.08772 0.110314 0.795 0.42746IPI.l2 0.202305 0.101056 2.002 0.04666CPI.l2 -0.341713 0.28365 -1.205 0.22976M1.l2 -0.043193 0.070607 -0.612 0.54141dollar.index.l2 -0.093697 0.042206 -2.22 0.02756beta1B.l3 -0.492061 0.534393 -0.921 0.35828beta2B.l3 -0.158372 0.425214 -0.372 0.70995beta3B.l3 -0.110017 0.1106 -0.995 0.32108IPI.l3 0.024861 0.100989 0.246 0.80581CPI.l3 0.24652 0.281056 0.877 0.38148M1.l3 -0.017873 0.070414 -0.254 0.79989dollar.index.l3 0.004178 0.042114 0.099 0.92108beta1B.l4 -0.034646 0.378735 -0.091 0.92721beta2B.l4 -0.309361 0.291165 -1.062 0.28931beta3B.l4 -0.071587 0.087856 -0.815 0.41615IPI.l4 -0.210699 0.072237 -2.917 0.00395CPI.l4 -0.228186 0.17334 -1.316 0.18956M1.l4 0.081265 0.051204 1.587 0.11409dollar.index.l4 0.053347 0.027624 1.931 0.05489const 0.076933 0.684692 0.112 0.91065


A. Estimated Parameters XXII

Table A.18: Estimation results for equation CPI-B:


beta1B.l1 0.160957 0.167895 0.959 0.33889beta2B.l1 -0.109462 0.127723 -0.857 0.39246beta3B.l1 -0.019793 0.035068 -0.564 0.57311IPI.l1 -0.005298 0.031378 -0.169 0.8661CPI.l1 1.285645 0.074477 17.262 <2e-16M1.l1 -0.01603 0.023456 -0.683 0.49514dollar.index.l1 -0.023339 0.012048 -1.937 0.05415beta1B.l2 -0.065198 0.232324 -0.281 0.77929beta2B.l2 0.239081 0.179095 1.335 0.18343beta3B.l2 0.049947 0.04808 1.039 0.30014IPI.l2 0.115645 0.044044 2.626 0.00932CPI.l2 -0.596178 0.123626 -4.822 0.00000282M1.l2 0.018982 0.030773 0.617 0.53806dollar.index.l2 0.013212 0.018395 0.718 0.47346beta1B.l3 0.113062 0.23291 0.485 0.62791beta2B.l3 0.06575 0.185326 0.355 0.72313beta3B.l3 0.002818 0.048204 0.058 0.95344IPI.l3 -0.109023 0.044015 -2.477 0.01409CPI.l3 0.173631 0.122496 1.417 0.15792M1.l3 0.006827 0.030689 0.222 0.82418dollar.index.l3 -0.021795 0.018355 -1.187 0.23649beta1B.l4 -0.20222 0.165068 -1.225 0.222beta2B.l4 -0.156037 0.126901 -1.23 0.22031beta3B.l4 -0.054729 0.038291 -1.429 0.1545IPI.l4 0.006337 0.031484 0.201 0.84068CPI.l4 0.035198 0.075549 0.466 0.6418M1.l4 -0.015076 0.022317 -0.676 0.50013dollar.index.l4 0.021144 0.01204 1.756 0.0806const 0.277749 0.298417 0.931 0.35312


A. Estimated Parameters XXIII

Table A.19: Estimation results for equation M1-B:


beta1B.l1 -0.231914 0.495459 -0.468 0.6402beta2B.l1 -0.228166 0.376911 -0.605 0.5456beta3B.l1 -0.029989 0.103486 -0.29 0.7723IPI.l1 0.001145 0.092596 0.012 0.9901CPI.l1 -0.473594 0.219783 -2.155 0.0324M1.l1 0.832793 0.069219 12.031 <2e-16dollar.index.l1 -0.035386 0.035554 -0.995 0.3208beta1B.l2 -0.063094 0.685588 -0.092 0.9268beta2B.l2 -0.02014 0.528509 -0.038 0.9696beta3B.l2 -0.048857 0.141883 -0.344 0.7309IPI.l2 -0.156067 0.129975 -1.201 0.2313CPI.l2 0.334151 0.364822 0.916 0.3608M1.l2 -0.01085 0.090812 -0.119 0.905dollar.index.l2 0.069348 0.054284 1.277 0.2029beta1B.l3 -0.115078 0.687319 -0.167 0.8672beta2B.l3 -0.031857 0.546897 -0.058 0.9536beta3B.l3 0.086229 0.14225 0.606 0.5451IPI.l3 -0.248483 0.129889 -1.913 0.0572CPI.l3 0.681508 0.361485 1.885 0.0609M1.l3 0.195726 0.090564 2.161 0.0319dollar.index.l3 0.011184 0.054166 0.206 0.8366beta1B.l4 0.292771 0.487117 0.601 0.5485beta2B.l4 0.044806 0.374487 0.12 0.9049beta3B.l4 -0.104017 0.112997 -0.921 0.3584IPI.l4 0.408756 0.09291 4.4 0.0000177CPI.l4 -0.466946 0.222944 -2.094 0.0375M1.l4 -0.113172 0.065857 -1.718 0.0873dollar.index.l4 -0.02855 0.035529 -0.804 0.4226const 0.101922 0.880629 0.116 0.908


A. Estimated Parameters XXIV

Table A.20: Estimation results for equation USDI-B:


beta1B.l1 0.815985 1.030442 0.792 0.429379beta2B.l1 0.611349 0.783888 0.78 0.436385beta3B.l1 0.303487 0.215227 1.41 0.160086IPI.l1 -0.12202 0.192578 -0.634 0.527063CPI.l1 0.069741 0.457098 0.153 0.87889M1.l1 0.106418 0.143961 0.739 0.46065dollar.index.l1 1.271509 0.073944 17.195 <2e-16beta1B.l2 -1.370061 1.425868 -0.961 0.337794beta2B.l2 -0.710775 1.099178 -0.647 0.518612beta3B.l2 0.095612 0.295084 0.324 0.746267IPI.l2 -0.453458 0.270319 -1.677 0.095024CPI.l2 1.15748 0.758747 1.526 0.128727M1.l2 -0.100156 0.188869 -0.53 0.596504dollar.index.l2 -0.418524 0.112899 -3.707 0.000272beta1B.l3 1.001184 1.429467 0.7 0.484506beta2B.l3 0.479771 1.13742 0.422 0.673624beta3B.l3 -0.617427 0.295848 -2.087 0.038171IPI.l3 0.932415 0.270141 3.452 0.000681CPI.l3 -1.638553 0.751807 -2.179 0.030476M1.l3 0.09345 0.188352 0.496 0.620342dollar.index.l3 0.080714 0.112653 0.716 0.474539beta1B.l4 -0.653906 1.013093 -0.645 0.519379beta2B.l4 -0.436836 0.778847 -0.561 0.575517beta3B.l4 0.315894 0.235008 1.344 0.180427IPI.l4 -0.37479 0.193231 -1.94 0.053849CPI.l4 0.914622 0.463673 1.973 0.049938M1.l4 -0.107827 0.136967 -0.787 0.432077dollar.index.l4 0.001533 0.073893 0.021 0.983466const -0.045705 1.831507 -0.025 0.980116


A. Estimated Parameters XXV

AF-L

Figure A.7: AF-L estimation results−

0.5

0.0

0.5

Diagram of fit and residuals for PC1

0 50 100 150 200−0.

20−

0.05

0.10

0 2 4 6 8 10 12

0.0

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals

−0.

20.

00.

2


0 50 100 150 200

−0.

050.

05

0 2 4 6 8 10 12

0.0

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals

−0.

060.

000.

04


0 50 100 150 200

−0.

040.

00

0 2 4 6 8 10 12

0.0

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals


A. Estimated Parameters XXVI

γ0

−0.0026

−0.0013

−0.0002

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

ΣL

0.0555 0 0 0 0 0 0 0 0 0 0 0

0.0175 0.0265 0 0 0 0 0 0 0 0 0 0

−0.0058 0.0003 0.0121 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

Γ1

1.2815 −0.0476 0.0928 −0.1755 −0.2808 −0.1977 −0.1201 0.5919 0.2866 0.0075 −0.2619 −0.1109

0.0386 1.0136 0.2357 −0.1399 −0.1524 −0.1443 0.0725 0.3273 0.1033 0.0258 −0.2409 0.0212

−0.0575 0.0345 0.7248 0.1025 −0.1121 0.0846 −0.0374 0.0652 0.229 −0.0055 0.0183 −0.2426

1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0

a0

3.0476

a1

0.3727

−0.4054

0.5231

−0.0266

0.0183

−0.0586

−0.0095

0.0194

−0.0067

0.0292

−0.0418

0.0450

λ0

−1.0043

−1.2787

−0.5715

0

0

0

0

0

0

0

0

0

λ1

0.3532 0.0222 0.0417 0.114 −0.0477 0.0457 −0.1126 −0.1145 0.0493 −0.3261 −0.1805 0.0522

0.093 −0.2903 −0.1043 0.0806 −0.2819 −0.1054 0.0677 −0.2719 −0.1067 0.0528 −0.2605 −0.1079

−0.0397 −0.1461 −0.1118 −0.08 −0.1473 −0.1117 −0.1197 −0.1481 −0.1116 −0.1587 −0.1489 −0.1116

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

A. Estimated Parameters XXVII

AF-M

ˆγ0

0.0085

0.0089

0.0043

0.4376

0.3108

0.0850

−1.6297

ΣM

0.0565 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0186 0.0260 0.0000 0.0000 0.0000 0.0000 0.0000

−0.0060 0.0007 0.0125 0.0000 0.0000 0.0000 0.0000

0.0214 −0.1639 −0.0381 0.8929 0.0000 0.0000 0.0000

0.0205 0.0159 0.0263 −0.0054 0.4085 0.0000 0.0000

0.0453 −0.1196 −0.1882 −0.2425 −0.1423 1.0948 0.0000

0.0100 −0.1351 −0.1522 0.2242 −0.7809 −0.2006 2.2204

ˆΓ1

1.0145 0.0051 −0.2648 0.0015 −0.0092 0.0016 −0.0008

0.0010 0.9391 0.3156 −0.0015 −0.0022 −0.0002 0.0002

−0.0005 −0.0011 0.7958 0.0004 −0.0018 −0.0002 0.0000

0.3155 1.2358 −4.4190 0.9934 −0.2053 0.0159 −0.0304

0.1741 −0.2373 0.6042 0.0104 0.8529 0.0030 −0.0196

−1.2385 1.1659 3.8795 −0.0269 0.1638 0.9008 0.0386

−0.3722 −0.4108 −14.8225 −0.0179 0.6828 −0.0178 0.9598

ˆa0

0.2588

ˆa1

0.3650

−0.4117

0.5325

−0.0004

−0.0011

−0.0003

0.0001

ˆλ0

−0.2855

−0.2014

−0.1737

−0.1783

−0.1790

−0.3137

0.1828

ˆλ1

0.0854 −0.0048 −0.0045 0.0204 −0.1800 0.0081 −0.0063

−0.0007 0.0003 −0.0003 −0.0370 −0.0473 −0.0450 0.0082

−0.0069 0.0007 0.0000 −0.0278 −0.0269 −0.0187 0.0011

−0.0025 0.0006 −0.0002 −0.0075 −0.0313 0.0368 −0.0164

0.0110 −0.0011 −0.0007 −0.0013 0.0000 −0.0060 0.0056

0.0433 −0.0016 −0.0029 −0.0315 −0.2191 −0.0401 0.0045

−0.0234 0.0019 0.0012 −0.0210 0.0249 −0.0088 −0.0022

A. Estimated Parameters XXVIII

Figure A.8: AF-M estimation results

−0.

50.

00.

5Diagram of fit and residuals for PC1

0 50 100 150 200−0.

20−

0.05

0.10

0 2 4 6 8 10 12

0.0

Lag

ACF Residuals

2 4 6 8 10 12

−0.

10

Lag

PACF Residuals

−0.

20.

00.

2


0 50 100 150 200−0.

100.

000.

10

0 2 4 6 8 10 12

−0.

2

Lag

ACF Residuals

2 4 6 8 10 12

−0.

2

Lag

PACF Residuals

−0.

060.

000.

04


0 50 100 150 200−0.

040.

000.

04

0 2 4 6 8 10 12

−0.

2

Lag

ACF Residuals

2 4 6 8 10 12

−0.

2

Lag

PACF Residuals

−15

−5

05


0 50 100 150 200

−4

−2

02

0 2 4 6 8 10 12

−0.

4

Lag

ACF Residuals

2 4 6 8 10 12

−0.

4

Lag

PACF Residuals

02

4


0 50 100 150 200

−2

−1

01

0 2 4 6 8 10 12

−0.

5

Lag

ACF Residuals

2 4 6 8 10 12

−0.

60.

4

Lag

PACF Residuals

−5

05

10


0 50 100 150 200

−4

02

4

0 2 4 6 8 10 12

−0.

4

Lag

ACF Residuals

2 4 6 8 10 12

−0.

4

Lag

PACF Residuals

−10

010

20


0 50 100 150 200

−6

−2

26

0 2 4 6 8 10 12

−0.

51.

0

Lag

ACF Residuals

2 4 6 8 10 12

−0.

4

Lag

PACF Residuals


Appendix B

Predictions

Figure B.1: Predictions for maturity 3M

2 4 6 8 10 12 14−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

prediction horizon

yiel

d (%

)

realityrandom walkNS−L−ANS−L−BNS−M−ANS−M−BAF−LAF−M


B. Predictions XXX

Figure B.2: Predictions for maturities 6M, 1Y and 2Y

2 4 6 8 10 12 14−0.4

−0.2

0

0.2

0.4

0.6

0.8

prediction horizon

yiel

d (%

)


2 4 6 8 10 12 14−0.4

−0.2

0

0.2

0.4

0.6

0.8

prediction horizon

yiel

d (%

)


2 4 6 8 10 12 14

−0.2

0

0.2

0.4

0.6

0.8

prediction horizon

yiel

d (%

)



B. Predictions XXXI

Figure B.3: Predictions for maturities 3Y, 5Y and 7Y

2 4 6 8 10 12 14

0

0.2

0.4

0.6

0.8

1

prediction horizon

yiel

d (%

)


2 4 6 8 10 12 140.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

prediction horizon

yiel

d (%

)


2 4 6 8 10 12 14

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

prediction horizon

yiel

d (%

)



B. Predictions XXXII

Figure B.4: Predictions for maturities 10Y, 20Y and 30Y

2 4 6 8 10 12 141.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

prediction horizon

yiel

d (%

)


2 4 6 8 10 12 14

2.5

3

3.5

4

4.5

prediction horizon

yiel

d (%

)


2 4 6 8 10 12 14

2.5

3

3.5

4

4.5

prediction horizon

yiel

d (%

)



University of Economics, Prague Faculty of Finance and ... · University of Economics, Prague...

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