Modelling volatility using EViews - cmi.comesa.int
Transcript of Modelling volatility using EViews - cmi.comesa.int
Modelling and Forecasting Volatility in Financial Markets Using E-Views
User Guide
Modelling and Forecasting Volatility in Financial Markets Using E-Views
Prepared By
Dr Thomas Bwire
Senior Principal Economist
Bank of Uganda
Published By COMESA Monetary Institute (CMI)
First Published 2019 by COMESA Monetary Institute C/O Kenya School of Monetary Studies P.O. Box 65041 – 00618 Noordin Road Nairobi, KENYA Tel: +254 – 20 – 8646207 http://cmi.comesa.int Copyright © 2019, COMESA Monetary Institute (CMI)
All rights reserved. Except for fully acknowledged short citations for
purposes of research and teaching, no part of this publication may be
reproduced or transmitted in any form or by any means without prior
permission from COMESA.
Disclaimer
The views expressed herein are those of the author and do not in any way
represent the official position of COMESA, its Member States, or the
affiliated Institution of the Author.
Typesetting and Design Mercy W. Macharia
TABLE OF CONTENT
List of Figures ................................................................................................ vii
List of Tables ................................................................................................. viii
List of Acronyms ............................................................................................ ix
Preface .............................................................................................................. xi
Acknowledgements ....................................................................................... xii
1. INTRODUCTION TO MODELLING FINANCIAL MARKETS USING
E-VIEWS ............................................................................................ 1
1.1: EViews Software.................................................................................... 1
1.2: Getting Familiarity with EViews Window ......................................... 1
1.3: Getting Data into EViews .................................................................... 4
1.4: Viewing the Data ................................................................................... 9
2. FINANCIAL MARKETS VOLATILITY ..................................................... 13
2.1: Definition and Measurement of Volatility ....................................... 13
2.2: Demonstration of Empirical Properties of Financial Assets Return Using EViews .......................................................................... 15
3. MODELLING CONDITIONAL VOLATILITY ............................................. 27
3.1: Heteroskedasticity and Auto (Serial) correlation ............................ 27
3.1.1. Fitting autoregressive (integrated) moving average models in EViews ........ 29
3.2: Introduction to Modelling Conditional Volatility ........................... 41
3.2.1: The ARCH Effect ................................................................................. 45
3.2.2: The GARCH Model ............................................................................. 49
3.3: Models with Asymmetry ..................................................................... 53
3.3.1: The Threshold GARCH (TGARCH) model........................................ 53
3.3.2: The Exponential GARCH (EGARCH) Model .................................. 54
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3.4: Demonstrating the G(ARCH) and Related Models Estimations Using EViews Screens and Statistical Output Tables .................................................................................................... 55
3.4.1: Equation Estimation/Specification ......................................................... 56
3.4.2: Estimation Settings/ Method and Sample - obvious ................................ 59
4. MODELLING CONDITIONAL VOLATILITY IN A MULTIVARIATE
FRAMEWORK ................................................................................... 71
4.1: Introduction ......................................................................................... 71
4.2: The VECH Representation ................................................................ 72
4.3: The Diagonal VECH Representation ............................................... 74
4.4: The BEKK Representation ................................................................ 75
4.5: The Constant Conditional Correlation (CCC) Representation ..................................................................................... 76
4.6: Conditional Heteroskedasticity, Unit Roots and Cointegration ........................................................................................ 77
4.7: Demonstrating the Estimation of CPI and Exchange Rate Volatility within a Bi-variant GARCH(1,1) Framework using EViews screens and Statistical Output Tables ................................ 77
5. FORECASTING CONDITIONAL VOLATILITY AND FORECAST
PERFORMANCE EVALUATION ........................................................... 95
5.1: One-step-ahead Forecast .................................................................... 95
5.2: The j-step-ahead Forecasts ................................................................. 95
5.3: Forecast Evaluation ............................................................................. 96
5.4: Performing the Exchange Rate Forecast in a Univariate GARCH(1,1) Framework Using EViews Screens and Statistical Output Tables ..................................................................... 97
5.5: Performing the Forecast in a Multivariate GARCH(1,1) Framework Using EViews Screens and Statistical Output Tables .................................................................................................. 108
REFERENCES ................................................................................ 117
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List of Figures
Figure 1: Evolution of Ushs/USD nominal exchange rate ......................................... 17
Figure 2: Intraday % changes in nominal exchange rate .............................................. 19
Figure 3: Frequency distribution of intraday % changes of nominal
exchange rate ..................................................................................................... 21
Figure 4: Comparison of the Normal (solid line) and t– and GED
Distributions ...................................................................................................... 22
Figure 5: Correlogram ....................................................................................................... 24
Figure 6: Squares of log-returns ....................................................................................... 25
Figure 7: Inverse Roots of AR/MA Polynomial(s) ...................................................... 40
Figure 8: Autocorrelations decay: Exponential (left) or damped sine wave
(right) .................................................................................................................. 44
Figure 9: Normality of G(ARCH) errors........................................................................ 67
Figure 10: GARCH (1,1) Conditional variance-Measured .......................................... 68
Figure 11: Measured Conditional Covariance ................................................................ 90
Figure 12: In-the-Sample forecast performance ......................................................... 102
Figure 13: Actual and in-the-Sample forecast with confidence bands ................... 104
Figure 14: Depreciation forecast ................................................................................... 106
Figure 15: Nominal Exchange rate forecast ................................................................ 107
Figure 16: Historical and volatility forecast................................................................. 107
Figure 17: Inflation and Depreciation forecast .......................................................... 113
Figure 18: Inflation and Depreciation forecast .......................................................... 115
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List of Tables
Table 1: Residual serial correlation .................................................................................. 35
Table 2: Breusch-Godfrey Serial Correlation LM Test ................................................ 36
Table 3: ARMA(4,1) model Estimates ............................................................................ 38
Table 4: AR(1) model Estimates ...................................................................................... 39
Table 5: GARCH(1,1) with alternative restriction for ARCH-M in the
Variance equation ............................................................................................. 64
Table 6: Heteroskedasticity Test: ARCH ....................................................................... 66
Table 7: The General bi-variant mean model for Inflation ......................................... 81
Table 8: Estimates of the mean equation ....................................................................... 81
Table 9: EViews output for Diagonal BEKK Covariance specification ................... 85
Table 10: Diagonal BEKK estimates for bi-variant GARCH(1,1) ............................ 87
Table 11: E-Views output for Constant Conditional Correlation Covariance
specification ....................................................................................................... 91
Table 12: CCC estimates for bi-variant GARCH(1,1).................................................. 93
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List of Acronyms
ACF Autocorrelation function
AR Auto regressive
ARCH Autoregressive conditional heteroskedasticity
ARCH-M Autoregressive conditional heteroskedasticity in mean
ARMA Autoregressive moving average
BEKK Baba, Engel, Kraft and Kroner
BoU Bank of Uganda
C Constant
CCC Constant Conditional Correlation
CGARCH Component Generalized autoregressive conditional
heteroskedasticity
CMI COMESA Monetary Institute
COEF Coefficient
COMESA Common Market for Eastern and Southern Africa
CPI Consumer price index
D.G.P Data generating process
Docs Documents
D-W Durbin Watson
EGARCH Exponential Generalized autoregressive conditional
heteroskedasticity
EQN Equation
EViews Econometric Views
EXR Exchange rate
FIG Figure
GARCH Generalized autoregressive conditional heteroskedasticity
GED Generalized Error distribution
GENR Generate
I.I.D Identically and independently distributed
IGARCH Integrated Generalized autoregressive conditional
heteroskedasticity
IT Information technology
KSMS Kenya School of Monetary Studies
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LB Lower bound
LM Lagrange multiplier
LOG Natural logarithm
MA Moving Average
MAE Mean Absolute Error
MAPE Mean Absolute Percentage Error
MEFMI Macroeconomic and Financial Management Institute
M-L Maximum-Likelihood
MPC Monetary policy committee
MSE Mean Square Error
NA Not available
NYSE New York Stock Exchange
OLS Ordinary least squares
PACF Partial autocorrelation function
PARCH Power Autoregressive conditional heteroskedasticity
PDF Portable document format
PROC Procedure
QML Quasi-maximum likelihood
RESID Residual
RHS Right hand side
RMSE Root Mean Squared Error
SA Seasonally adjusted
SE. Standard error
S&P Standard and Poor’s
Std Dev Standard deviation
TGARCH Threshold Generalized autoregressive conditional
heteroskedasticity
UB Upper bound
U.K United Kingdom
VaR Value at risk
VECH Vector error conditional heteroskedasticity
xi |
Preface
The preparation of this User’s Guide followed a directive to COMESA
Monetary Institute (CMI) by the 22nd Meeting of the COMESA Committee
of Governors of Central Banks which was held in Bujumbura Burundi in
March 2017. Governors noted that financial markets volatility and the
associated volatility spillovers are a potential threat to financial stability and
can dampen prospects for economic growth.
The overall objective of the Guide is to serve as a knowledge product to
provide member central banks with analytical guide for modelling and
forecasting volatility in financial markets both in univariate and multivariate
frameworks. The guide provides step by step approach to central banks
modellers using EViews software to enable them adequately measure and
forecast both direct and spill over effects due to volatility in prices of
financial market assets. The guide is developed in such a manner that it
strikes a balance in theoretical and practical skills in modelling and
forecasting volatility in both in univariate and multivariate setting and lean
heavily on learning by doing approach.
It is hoped that the Guide will enable Central Bank modellers to undertake
rigorous and robust volatility analysis, which will enable decision makers to
undertake measures to mitigate the adverse effects of financial markets
uncertainty. It is also hoped that the Guide will be used by COMESA
member central banks as a reference material to train their staff.
Ibrahim A. Zeidy Chief Executive Officer
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Acknowledgements
The Author was grateful to the COMESA Monetary Institute (CMI), who
on behalf of the COMESA Committee of Governors provided the
opportunity to prepare the User’s Guide. He acknowledged technical and
logical support from the Director of CMI, Mr. Ibrahim Zeidy and the
Senior Economist, Dr. Lucas Njoroge. The Author also acknowledged the
assistance of CMI support staff as well as staff in the Member States Central
Banks for any information provided during the User’s Guide preparation.
The Author especially acknowledges comments from the participants of the
Validation Workshop held from 24–28 April, 2017 in Nairobi, Kenya that
provided the final inputs to the User’s Guide. The workshop was attended
by participants from the following COMESA member countries’ Central
Banks: Burundi, Djibouti, DR Congo, Egypt, Ethiopia, Kenya, Mauritius,
Sudan, Swaziland, Uganda, Zambia, and Zimbabwe.
Chapter 1
Introduction to Modelling Financial
Markets Using E-views
1.1: EViews Software
EViews – short form of Econometric Views is a user-friendly and powerful
statistical software package designed to provide sophisticated data analysis
and forecasting tools. The software comes with an extensive user guideline
which contains many useful examples/explanations/tutorials and is
extremely well-presented. Once you have gained basic familiarity with the
basic concepts and operations of the program, you should be able to
perform most operations without consulting the guideline. Moreover, the
program has a very extensive help menu (one of the entries in the Main
menu) which is simple to use and sufficient for most users, so one may not
again need to consult the guideline when conducting statistical analysis. This
introductory chapter aims to familiarise trainees with the basic fundamentals
of working with EViews pertinent to the purpose at hand, i.e. analysis of
financial markets. The detailed and in-depth exposition of EViews
fundamentals are given in the User’s Guide I which is in the drop-down menu
of the help menu in the earlier versions of EViews, or can be found in the
documents in PDF Docs file in the drop-down menu of the help menu in the
latest versions of EViews, notably EViews 9.5.All illustrations and output in
this guideline derive from EViews 8.0
1.2: Getting Familiarity with EViews Window
Launch the program (i.e. double click on EViews icon) - this assumes the
program has been properly installed on your computer (the installation is
usually done by authorized bank IT staff). This brings forth the EViews
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window as in the screen print below. We want to familiarize ourselves with
the title bar, main menu, command window, work space and status
line in there.
Title bar Main Menu
Command Window
Work space
Status line
At the very top of the main window, is the title bar, labelled EViews and is
generally darker, provided EViews is the active program in windows.
Immediately below the title bar is the main menu. If you move the cursor to
an entry in the main menu, say Object and click on the left mouse button, a
drop-down menu will appear. Some of the items in the drop-down menu, as
shown in the print screen below are listed in black while others are in gray.
Black items are executable while the gray items are not or simply unavailable.
Clicking on a black entry in the drop-down menu selects the highlighted item.
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Menu items
Greyed menu items not available
Darkened menu items are available
Drop down menu
Below the menu bar is the command window space in which EViews
commands such as log, first difference transformations are entered or typed.
Each of the commands is executed as soon as you hit the ENTER button
on the key board. The area in the middle of EViews window is the work
space and is where various objects that EViews creates are displayed. It is
actually analogous to a sheet of paper in an exercise book or work book on
your desk. To close EViews, select File/Exit from the main menu or click
on the x button in the upper right-hand corner of the EViews window. If
necessary, EViews will warn you and provide you with the opportunity to
save any unsaved work. EViews work file is saved in the same way as any
other computer-generated document. Select File in the main menu, then
Save As… in the drop- down menu and save the file. Subsequently, click on
Save through the File menu or as usual through the key board operations to
save any changes to the file. It is advisable that you save your work
continuously.
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1.3: Getting Data into EViews
For purposes of illustration, I have iteratively used both univariate and
multivariate data for Uganda with varied frequencies, depending on the
issues at hand. This has been provided in excel sheets in a folder I have
named, for convenience, COMESA_DATA_2017.
We will start with the data on one financial markets variable, driven not
only by economic fundamentals but also speculative market behaviour and
whose disruptive volatility is of great concern to monetary policy makers.
This is the nominal Ushs/USD exchange rate, observed at daily frequency
for the period 1 Jan 2005 to 3 Feb 2016. It is a very large dataset,
comprising some 4,051 observations, intentionally compiled to mimic
features of financial market variables which are observed at a much higher
frequency such as the sort of S&P composite index and /or NYSE
international 100 index. This data file in COMESA_DATA_2017 folder is
named Daily exr.xlsx. Note that this is not the sort of data you often use in
much of the macroeconomic time series analysis when undertaking
independent research.
Given the data, the first step in subjecting financial markets volatility
models (which we introduce later) to statistical analysis, is to read data into
an EViews work file. EViews obviously provides sophisticated tools for
reading in data from a variety of common data formats and sources, but
here I will demonstrate one of the easiest of the ways of doing so from an
Excel file.
Launch the EViews program to see the EViews window we’ve already
described above. On the Main menu, left click on File and in the drop-down
menu, point the cursor at NEW and navigate through to Work file. Click on
Work file and it’s in the Work file dialog box which pops (given in the
screen shot below) that the user uses to supply critical data information to
the software.
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Beginning with the work file structure type, choose from the drop-down menu,
the series type that you may be having at your disposal. Note that because
our data is dated and is at a regular frequency, we choose dated-regular
frequency (EViews default entry). Similarly, under Date specification,
choose from the drop-down menu, daily-7-day week. And supply the start
date and the end date (in the form: month, date, year e.g. 1/01/2005 for
start date and 2/03/2016 for end data). Although Work file name is
optional, here we provide, for convenience, COMESA_2017. These entries
are what we see in the EViews screen print above.
Click OK. A new window opens with two variables; C and resid. As we all
know, these two, namely constant (also known as intercept) and residual
appear as a given in a time series model and serve to capture important
information about the regressor or dependent variable. Holding everything
else constant, the minimum value that the dependent variable can take is
equal to the magnitude of the constant (be it positive or negative), subject
to its statistical significance and the model meeting standard statistical
criteria for evaluating the estimated results. Because it’s not claimed that the
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whole of economics or even the whole of economic theory can be
encompassed in a model, but a well devised model can bring out certain
features of interdependence among economic quantities that are not easily
comprehended without its help (Beach, 1958), the resid in a model captures,
among others, the unexplained component of the dependent variable.
The page (at the bottom of the window) is by default named untitled, but
can be renamed by right clicking on the name “untitled” and selecting
rename Work file page.
Constant & residual
Page we will need to baptize
Sample range & size
The window gives both the Range and Sample of the data, spanning
1/01/2005 to 2/03/2016, or some 4051 0bservations. It is at this point that
we load the data. To do so, interactively, open the excel data file Daily
exr.xlsx contained in COMESA_DATA_2017 folder. In the excel file, select
and copy the content in the column named EXR, the cell EXR inclusive,
ensuring you have copied both the series name and the series data. In the
open EViews window, check on the Quick command in the Main menu
and choose Empty group (edit series). A click on this produces a
spreadsheet similar to that in excel.
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The first column displays the sample range period, spanning from the start
to the end dates that you supplied earlier on. The first two rows are by
default empty. And unlike the second, the first is highlighted in blue (or)
colour. Either of these can be used to paste in the data, albeit with different
implications, which you might want to practically check. In the sheet, dates
only begin from the third row of the first column. Click on the second cell
in the second column, and while maintaining the cursor in there, make a
right hand click on your mouse and paste in the copied data. As you will see,
this simple way of importing the data also brings on board the series name,
exr, which then automatically becomes visible in the work space –
increasing the number of entries to three (c, exr, and resid). Once you are
sure the copy and paste command is complete, click on the x command
which appears at the top most right-hand side corner of the open excel-like
sheet window (containing the data you called in), and in the prompt
message, select yes. A window with three series names, c, exr, and resid
appears as in the print screen below.
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Series brought aboard.
Note that any mismatch between the actual series length and length
between the start and end date may prompt the software to reject the data
paste command. Also, in the event that during the copying, you omitted the
series initial, the software will assign the series a default name, usually
SER01 for the first series column, and so on in case of multiple series. You
can replace/rename the SER0i, i=1, 2,..,k with the actual series names or do
it more easily through the Genr (generate) command in the work space
window. Alternatively, highlight the variable/series you want to rename and
while keeping the cursor in there, right click on the mouse and choose
rename, then provide the name for the variable/series in question. Also,
take note that at times, actually quite often, the initial you give the series
may matter, and may be rejected by EViews especially if the initial is a
preserve of the software. In the event that this is so, try changing the initial
until it is acceptable. As mentioned earlier, there are many ways of
importing data into EViews, but the simplicity and complexity associated
with each of these ways is user specific. It is important that as a user, you
choose the easiest way – the above being simpler in my judgement.
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1.4: Viewing the Data
Make sure to verify that the data on exr is the actual data you copied in
from excel. To do this, click on exr to highlight the series and while
maintaining a cursor there, make a right-hand side click on your mouse and
select open. Data on exr will be displayed just as is shown in the EViews
print screen herewith.
Scroll up to the start of the data or down to the end of it
Data stops at 1/10/2016 instead of 2/03/2016
Missing observations
Always remember, whenever you enter data into EViews, to check it very
carefully so to ensure it is what you expect it to be and that it has not been
distorted in the process of calling it into EViews. Checking your data is
boring, but vital. In fact, if necessary, plot it because this gives you the
opportunity to get familiar with the data’s important characteristics -
features that make modelling even far more exciting. If you do not know
what the features of your data are, you are not going to be able to develop a
particularly good model of them. In financial markets modelling and time
series analysis in general, it is vital that you know your data.
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In this particular case, you will realize that our data stops at 1/10/2016 and
not 2/03/2016 as in excel for reasons we may well explore. About 24
observations, for the period 1/11/2016 all through to 2/03/2016 are
marked as NA or missing observations. We have to re-size our sample to
avoid the many NAs as is now. To do this, first, we close the open excel
like sheet by checking the x command at the top most right-hand side
corner of the sheet window. While in the Work space area, click on
Proc(second last icon at the extreme left-hand side corner of the work area
window) and choose Restructure/Resize Current Page.... In the new
window that opens, edit End date to a date consistent with that in the
EViews work file data, i.e. 1/10/2016, click OK, followed by Yes in the
prompt message to continue. This action downsizes the sample size by 24
NA observations to 4027 effective observations.
We might want to name this EViews page to be able to distinguish it as
we’re likely to have several of these. To do this, point the cursor in the sheet
given as Untitled and right click in it and in the drop-down menu, check
Rename Work file Page… In the new small window that pops up, under
name for page, let us write daily_exr and click OK. This effectively baptizes
the sheet in EViews working space.
Baptized work page
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Multiple series can also be read into EViews (by the same procedure above).
Similarly, viewing the same can also be done in one command. All you need
to do is to select the multiple series, but one at a time, while holding down
the Ctrl key on your key board. This highlights all series of interest. While
maintaining the cursor in the highlighted series, right click on your mouse
and choose open as a group. Data will be displayed for all the variables.
Examine the data to ensure it’s the actual data you intended to read into
EViews. The open window can then be closed using the x command.
You are now ready to undertake any data manipulation and analysis. In
what follows, we mimic practically the salient features of financial markets
volatility, starting with preliminaries of the definition and measurement of
volatility (in the tradition).
Chapter 2
Financial Markets Volatility
2.1: Definition and Measurement of Volatility
Risk, or in general uncertainty, associated with investment in financial assets,
is measured in terms of asset returns volatility, i.e. the degree to which financial
prices or returns fluctuate. Simply put, it is the time varying risk associated with
returns on an asset. However, the relationship between volatility and risk is
tenuous. Risk is more often associated with small or negative returns.
Volatility, on the other hand, makes no such distinction. Volatility is not an
observable metric, so its measurement is necessary.
In a simplistic world of finance practices, as in some circles of the academia
and policy arena, volatility is computed as a sample standard deviation of
returns associated with a financial asset – possibly motivated by the
simplicity with which it can be computed. In statistics, computation of a
sample standard deviation assumes a series of observations (N) on a
financial asset, with returns (Rt) on investing in it. The standard deviation
( ) of Rt is then derived as:
N
t
t RRN 1
21
(1)
Where R is the mean return and is a distribution free parameter which
depends on the dynamics of the underlying stochastic process and whether
or not the parameters are time varying.
Very often, when is used to measure volatility, the users usually assume
that the underlying stochastic process generating the returns follows a well
behaved normal distribution. In reality however, this has been shown not to
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always hold because in practice and as will be shown here, the underlying
stochastic process exhibits pronounced departures from the standard
Gaussian assumption. Besides, Eqn.1 only permits computation of historical
or realized volatility, i.e. the standard deviation of financial returns on an asset
computed over a window of a pre-specified number of past trading periods,
but the time frame over which it is computed has serious interpretational
implications. If it is computed over a very short horizon, realized volatility
will be too noisy while if the period is too long, it will not be so relevant for
today. Moreover, the statistical properties of a sample mean make it a very
inaccurate estimate of the true mean, especially in small samples because of
the influence of extremes, and as noted above, it does not always draw from
a normal distribution. Moreover, it is logically inconsistent to assume that
the variance is constant for a period such as one year ending today and at
the same time be constant for the year ending on the previous day but with
a different value – this raises the possibility that volatility is dynamic.
Importantly, asset holders are interested in the volatility of returns over the
holding period going forward and not over some historical past. This
forward-looking view of risk requires measures that are able to estimate
conditional volatility, i.e. expected volatility at some future time (say t+h) that is
deliberately informed by new information set available at time t (t ). It
means that tomorrow's volatility estimate depends on or is conditional on
certain new information available today. 1 Volatility is of great concern
because: valuation methods in finance and portfolio allocations in
Markowitz mean-variance framework depend on volatility and volatility
affects the spread between long-and short-term interest rates. It is also used
in the calculation of value at risk (VaR) of the financial position in risk
management. At policy level, financial market volatility and the associated
volatility spillovers are a potential threat to financial stability and prospects
1 The other form of volatility common in the literature, but not considered here is implied
volatility, which is volatility that delivers a no-arbitrage option pricing, and derives from Black-Scholes model. The value of a financial asset is priced on the basis of the assets own price, the exercise price, the time to maturity, the risk-free interest rate and the assets expected standard deviation. The approach however is criticized because the assumed geometric Brownian motion for the prices of the underlying asset may not hold in practice and the resultant implied volatility has been shown to be larger than that obtained by conditional volatility measures.
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for economy-wide growth. The downside is that volatility of an asset is not
observable, so it’s modelling is necessary. Although in literature, numerous
conditional volatility models have been suggested to capture the
characteristics of return for an asset, here we explore the Auto Regressive
Conditional Heteroskedasticity (ARCH) and their many extensions, in
particular, Generalized Auto Regressive Conditional Heteroskedasticity
(GARCH) type models.
2.2: Demonstration of Empirical Properties of Financial Assets
Return Using EViews
Financial markets literature identifies several salient features of financial
asset returns:
1. Returns on a financial asset evolve over-time in a continuous manner –
a feature we can demonstrate with ease using the daily nominal
Ushs/USD exchange rate series in EViews. To do this, we first
transform exr series to natural logarithms (this is done for reasons that
will become familiar in due course, if not familiar yet). To proceed,
make active the open EViews Daily_exr page above. In the command
window, type a command of the form genr lexr = log(exr), then press
the enter button on your key board. A new series, lexr is added onto
the list of variables as shown in the EViews screen print below.
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Supplied command
Natural logarithm of nominal exchange rate – a variable we’ve created
To proceed, double click on lexr to open the series and in the window,
click on View icon in there and chose and click on graph. By default,
EViews highlight Basic type under Option Pages and Line & Symbol
under graph type as in the screen here.
EViews generous highlights
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Click OK and you will see a graph given in Fig. 1. You could as well use
the Quick icon at the top- most window for the same results, except
that here, instead of opening the series data points, you have to
highlight the series name. All sorts of statistical manipulation of the
graph (s) are available in the graph window. It is still possible, whilst in
the graph window to retrieve the series spread sheet. Click on View and
in the drop-down menu, select Spread Sheet.
Figure 1: Evolution of Ushs/USD nominal exchange rate
It is a good idea to save such output. With the graph (s) on the screen,
right click on the mouse and choose copy to clipboard. You can then
paste the graph in an open word document. You could also, by the
same procedure, choose save graph to disk. It is also possible to keep
all results in the same work space. To do so, select Freeze and then
Name. This procedure allows you to appropriately assign a name to
your graph, e.g. Figure01.Note that the graph remains on the screen.
You can only navigate away from the graph window with the x
command and choosing yes in a small window that pops up.
Back to Fig. 1, the rather smooth curve shows what has happened to
the nominal exchange rate in Uganda over the last 11 years. We see a
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
18 |
shift in level, but a stable and somewhat appreciated exchange rate
before 2007 and before the financial crisis of 2008 and a steep sustained
depreciation and intermittent appreciation thereafter. At the
commencement of 2005, one USD was priced at about Ushs. 1,750 and
it cost about Ushs 3,500 in the first week of February 2016. The
exchange rate hit all time high of Ushs 3,700 per USD at the close of
September, 2015. This implies that one dollar invested in the foreign
exchange market at the commencement of 2005 had multiplied 1.99
times by the beginning of February 2016, while the shilling had
depreciated by 99.1% over the 11-yearperiod, amounting to an average
annual depreciation of roughly 10%.
2. Returns to financial assets meander in a fashion that contains periods
of high volatility followed by periods of lower volatility. In other
words, visually, periods of large movements in prices alternate with
periods during which prices hardly change, causing volatility clusters–
otherwise coined as volatility clustering. Because variations in returns over
time are not constant, the data generating process (stochastic process)
is said to be heteroskedastic. And it is this feature that gave birth to
conditional heteroscedasticity (Engel, 1982) of asset returns – a subject of
interest in financial assets modelling. Relatedly, is the existence of
asymmetric movement of the volatility, i.e. a large (small) change in
prices is more likely followed by a large (small) price change, i.e.
persistence in price movements.
Volatility clustering of financial asset returns can be demonstrated with
ease, and for some reason, it starts with transforming lexr series, in our
case, into intraday changes (or day-on-day depreciation). In the
command window, type genr dlexr = lexr-lexr(-1) and execute by
taping on the enter button on the key board. A new series, dlexr is
added onto the list of variables.
If we double check on dlexr, the series opens and repeating the
procedure for producing Fig. 1, we generate Fig. 2 for volatility
clustering.
................................................................................................................ Financial Markets Volatility
19 |
Figure 2: Intraday % changes in nominal exchange rate
While the picture of the level data in Fig. 1is appealing, it is the relative
price change over the holding period that really matters. Thus,
investors, portfolio and risk managers and monetary policy makers
focus attention on the relative intraday percentage change or
depreciation or returns as shown in Fig. 2. Clearly, the exchange rate
return series is centred around zero, with periods of large volatility,
followed by periods of relative tranquillity. We can note the world and
domestic events particularly in 2007, 2008-9, 2011/12 and more
recently 2014/15 when the size of depreciation and/or appreciation
dwarfs all the other changes. Volatile periods are hectic periods and
intuitively reflect heightened uncertainty. The usual suspect for such
uncertainty is uncertainty about the fundamentals in the economy. But
from an econometric point of view, uncertainty about fundamentals
only explains a moderate portion of the observed financial market
volatility. Comprehensive measurement of volatility, including its
forecast requires simple time series models beyond the fundamentals.
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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3. The other largely noticeable feature is the non-normal empirical
distribution of asset returns. Usually, the empirical density function has
a higher peak around its mean, but fatter tails than that of the
corresponding normal distribution or high excess kurtosis. The proof
of this is quite straight forward.
Under normality, the unconditional fourth moment or kurtosis is given
by
322
4
t
t
E
Ek
: 34 tE (2)
Where 12 tE
But as mentioned earlier, volatility is meaningful only if it’s deliberately
informed by new information set available at time t so considering
expectations, conditional fourth moment becomes
222
1
44
1
ttt
ttt
eEE
eEEk
(3)
Substituting 3 into 2, we get
2222
1
44
1
ttt
ttt
eEE
eEEk
(4)
So that 222
4
1
3
t
tk
for 12
1 tt eE and 34 tteE .
3322
4
t
tk
Showing, indeed that asset returns are not Gaussian normal. In
addition, the distribution of returns is typically negatively skewed,
largely because large negative movements in financial markets are not
usually matched by equally large positive movements.
To subject this notion to data, double click on dlexr, but because the
latest we have for this is a graph, we must get back the spread sheet
through the View icon, by checking spreadsheet, which should give
................................................................................................................ Financial Markets Volatility
21 |
us the data points. In the open spread sheet, click on View and choose
Descriptive Statistics & Tests, and navigate through to Histogram
and Stats. Check on Histogram and Stats as shown in the EViews
print screen. Check on Histogram and Stats to produce Fig. 3.
Road map
Figure 3: Frequency distribution of intraday % changes of nominal exchange rate
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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Indeed, kurtosis is excess at 30.7 and the distribution is negatively
skewed(Skewness of -0.92).Therefore, because there is a good reason
to believe that the asset return has a higher probability of a very large
loss or gain than indicated by the normal distribution; one might want
to estimate Maximum-Likelihood (M-L) using student’s t–distribution
and/or Generalized Error distribution (GED), which comparatively,
approximates a normal distribution as the sample size becomes
relatively large as shown in Figure 4.
Figure 4: Comparison of the Normal (solid line) and t– and GED Distributions
Den
sity
0.0
0.1
0.2
0.3
Den
sity
0.0
0.1
0.2
0.3
0.4
−4 −2 0 2 4 −4 −2 0 2 4
The t-distribution in particular places a greater likelihood on large
realizations than does the normal distribution. The good news is both
student’s t–distribution and GED tend to approximate a normal
distribution as t , and are explicitly given as options in EViews
estimations of the G(ARCH) models.
4. It has also been shown that the mean of log-return series is close to
zero (a feature clearly depicted in Fig. 2), which is intuitive from an
econometric perspective. If we assume the log-return on an asset at
time t to take the form:
)log()log( 1 ttt ppr , (5)
Where tp is the price level of the asset at time t. If
tr – essentially
differenced log of tp is stationary, then it follows that
tp itself is non-
stationary or is a random walk process around trend, something
synonymous with the plot in Fig.1. In time series econometrics, the
................................................................................................................ Financial Markets Volatility
23 |
data generating process (d.g.p) such as that for tp is said to contain a
unit root or unit roots, which is eliminated by transforming the series
into its differences. And if tp can become stationary upon first
differencing, it’s said to be integrated of order one or 1I .
It follows therefore that if 1~ Ip t, then )0(~log Ipr tt is
stationary or mean reverting, a behaviour akin to the plot in Fig. 2. In
time series, there are important distinctions between a stationary or
0I series and a non-stationary or 1I process. 0I Series fluctuate
around its mean (or is mean reverting) with a finite variance that does
not depend on time while its 1I variant wanders widely. 0I Series
has a limited memory of its past behaviour, while 1I series has
infinitely long memory i.e. innovations in the system permanently
affects the process. And as shown inFig.5, the autocorrelations of an
0I series decline rapidly to zero as the lag increases, while they decay
to zero very slowly in the case of 1I series.
Fig. 5 is produced from autocorrelation numbers over a 20-period
span. To do this, double click on lexr to open the spread sheet, and
under View, check Correlogram.... In the window that opens,
EViews by default checks in Level and chooses 36 lags to include.
Ireduced this to 20 for tractability. Click OK to see the following
output for Correlogram of LEXR. Repeating a similar process, but
checking first difference instead (which is dlexr) delivers Correlogram
of DLEXR.
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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Figure 5: Correlogram
Using non-stationary series in econometric analysis has been shown to
bear dire consequences. Non-stationary economic time series are
commonly characterized by strong trend components, i.e. a
deterministic and/or stochastic trend or some combination of the two.
As a consequence, many of these are said to contain a unit root or
simply non-stationary, that is, the variables in question may have a time
variant mean and/or non-constant variance. Working with such series
in their levels while analysing economic relationships has a high
likelihood of giving results that are economically meaningless –a
symptom that Granger and Newbold (1974) call spurious regression. This is
often characterised by significant t-statistic and a high explanatory
power, even when the regressors are economically unrelated to the
variable being explained. Moreover, no inference can be deduced from
such results since the least-square estimates are not consistent and the
customary tests of statistical inference, namely the ‘‘F’’ and ‘‘t’’ ratio test
statistics do not have the limiting distributions (Enders, 2010). Simply
put, non-stationary series may generate poor forecasts.
5. The other common feature of financial asset returns is that we do not
observe autocorrelation in levels, but we do with squares of log-
returns. As shown in Fig. 6, the autocorrelations of variance, and
particularly those of mean absolute deviation stay positive and
................................................................................................................ Financial Markets Volatility
25 |
significantly above zero for all lags. To demonstrate autocorrelation
with squares of log-returns, we transform dlexr to sq_dlexr using the
following command: genr dlexr_sq = dlexr^2, which we execute by
tapping enter button on the key board. And as before, graph sq_dlexr
to get Fig. 6.
Figure 6: Squares of log-returns
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Exercise 1
We have used daily frequency data on nominal exchange rate for Uganda in
the foregoing discussion to mimic the silent known features typical of
financial markets variables, the sort of S&P composite index. We have
provided similar data; in a separate folder we have named Exercise data. All
data is contained in a file given as Guideline Exercises data.xlsx, with two
sheets; daily_exr and multiple. This exercise requires you to use data on 5-
day week nominal exchange rate for Uganda, spanning September 01, 2005
to June 01, 2017 or some 2928 observations, given in daily_exr sheet of the
file to:
1. Demonstrate that returns on financial asset;
i) Evolve over-time in a continuous manner
ii) Meander in a fashion of clusters
iii) Has a higher peak around its mean, high excess kurtosis and that
returns are typically negatively skewed.
iv) Squares of log-returns display serial correlation; and
2. Demonstrate that that the autocorrelations of dlexr decay rapidly to
zero while those of lexr decay slowly as the lag increases.
Chapter 3
Modelling Conditional Volatility
3.1: Heteroskedasticity and Auto (Serial) Correlation
We have shown and also demonstrated that stochastic processes generating
financial asset returns among other features are heteroskedastic and that the
residuals are serially or auto correlated. It is important that we gain insights
into these twin econometric problems and importantly their implications on
regression coefficient. Only then shall we appreciate that pertinent
corrective modelling measure, of which autoregressive moving average
(ARMA) and the auto regressive conditional heteroskedasticity (ARCH)
models and related extensions thereof, must be undertaken. To underscore
these twin heteroskedasticity and serial correlation concepts in
macroeconomic and financial time series data, consider a standard but the
simplest stationary univariate model for a random stochastic variable, yt,
observed over a sequence of time t = 1... T:
lainedun
t
lained
tt yy
expexp
1
(5)
The equation implies that the variable, y, at time t is generated by its own
past behaviour, i.e. its own lags – the explained part and a disturbance (or
residual) term t -the unknown part. In econometrics,
t is assumed to be
governed by several standard classical or Gauss-Markov assumptions,
usually condensed as ),0.(~ 2 Nt, i.e. residuals are distributed as normal
with zero mean, and constant or time invariant variance – homoskedastic
residuals. This in principal is what econometricians call white noise process.
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An additional assumption about the behaviour of t is that
0 sttstt EC ; st . This means that the residuals in period t
are uncorrelated with those in period s, where period s is own past. This
behaviour is particularly important when we come to producing forecasts,
since it implies that the correlation structure between sty
and ty is the
same as that between ty and
sty , i.e. we can extrapolate backwards-looking
relationships into forward-looking relationships to yield forecasts of future
values of the series.
Providing, 0tE , 22 tE and 0sttE ; st hold, then
),0.(..~ 2 diit, i.e.
t are identically and independently distributed as
normal with zero mean and constant variance, These are the classical
Gauss-Markov simplifying assumptions that ensures a pure white noise
process. However, in contrast and in practice, residuals of financial and in
as many instances macroeconomic time series data exhibit a behaviour
which is such that:
a) 22 itE , i.e. residuals are heteroskedastic, which means
variance that is varying over time.
b) 0, 1 ttE , i.e. residuals are autocorrelated (a problem of serial
correlation) - a situation where there is interdependence in the
behaviour of residuals over time.
The concern is whether the d.g.p riddled with heteroskedastic and
autocorrelated residuals, i.e. departure from the standard Gauss-Markov
assumptions warrants a worry when modelling financial and
macroeconomic time series data. Certainly, it’s worrisome and this is why.
While developments in econometrics theory suggests that even with
heteroskedastic residuals, the popular ordinary least squares (OLS)
estimator, (in eqn. 5) is still consistent (i.e. we get the correct coefficient
values especially as the sample becomes reasonably large) and is reasonably
efficient (in terms of the variance of ), the real problem is that standard
errors (s.e ( )) are not correct. As a result, the t- and F-tests are no longer
valid and inference based on them could be awfully misleading. Residual
.......................................................................................................... Modelling Conditional Volatility
29 |
heteroskedastic behaviour is modelled, as we will show, by the
autoregressive conditional heteroskedasticity (ARCH) techniques and
related extensions thereof to obtain more efficient coefficient estimates.
Ensuring that the crucial assumption of time independence of residuals
holds is entirely an art craft of the modeller as it is purely a modelling aspect.
It requires that the modeller fits an appropriate ARMA process.
3.1.1. Fitting autoregressive (integrated) moving average models in
EViews
We will now describe how to test for and deal with serial correlation, i.e.
how to fit AR, MA and ARMA processes in practice. To see the
dependence structure of residuals in a random walk process, we will assume
for ease of exposition, an AR(1)process given as in (5) by,
ttt yy 1
Where ),0(~ 2 Nt, and additionally, the restriction that 1 - for mean
reversion to hold so that yt, is a stationary AR(1) process.
Showing the interdependence structure of it requires that the RHS is
expressed in terms of t only, a task easily achieved by backward recursive
substitution. Thus, given
ttt yy 1
tttt yy )( 12
2
2
1 tttt yy
)( 23
2
1 ttttt yy
3
3
2
2
1 ttttt yy
Repeating the back substitution 1j times, we obtain
jt
j
jt
j
tttt yy
)1(
1
2
2
1 ...
=
1
0
j
i
jt
j
it
i y , which still depends onjty .
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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However, given the restriction that 1 , then as j , 0 jt
j y ,
such that
0i
it
i
ty . (6)
This expression is an MA representation of the AR process.
As we show in detail under the build-up to conditional volatility, serial
correlation arises because residuals for periods not too far apart, such ast ,
1t and so on are related, but this dependence decay gradually to zero as
t either exponentially if 0 , or like a damped sine wave if 0 ,
providing 1 holds.
To rid the stochastic process of serial correlation, the process is sequentially
estimated, up to an order sufficient to remove any remaining serial
correlation. Usually, we want a model with as few lags as possible to get a
parsimonious model, but at the same time we want enough lags to remove
autocorrelation of the residuals. The stationarity condition for general
)( pAR processes is for the inverted roots of the lag polynomial to lie inside
the unit circle.
Unlike the AR(1) process, the simplest class of the moving average (MA)
process, the MA(1) process, given by eqn. 7 is stationary, i.e. has mean,
variance and auto covariance that does not depend on t.
1 ttty (7)
Where ),0(~ 2 Ntand 1
By definition, from eqn. 7,
)()( 1 ttt EyE
)()( 1 tt EE
0
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31 |
)()( 2
0 tt yEyVar
= }){( 2
1 ttE
= )2( 1
2
1
22
ttttE
= )1( 22
And indeed
1;0
1;),(
2
s
syyC sstt
Thus, for any value of , the mean, variance and auto covariance of the
MA(1) process simply does not depend on t, so the process is stationary.
The autocorrelation function for the MA(1) process is then
1;0
1;1 2
0 s
ss
s
Thus the MA(1) process is 1-dependent.
It follows thus that the specification of the ARMA (p,q) process is given by:
;...... 22112
2
1 componentqMA
qtqtt
componetpAR
tpt
p
ttty
),0(~ 2 Nt (8)
As with AR diagnostics, the estimated ARMA process is (covariance)
stationary if all the inverted AR and MA roots lie inside the unit circle. And
in addition, if the ARMA model is correctly specified, the autocorrelation
function (ACF) and partial autocorrelation function (PACF) of the residuals
from the model should be nearly white noise.
If in eq.5-7 the log of yt in levels is in non-stationary form or if it’s found to
contain a unit or unit roots, it would require to be differenced the
appropriate number of times, usually given as d times to induce stationarity.
The number of times that yt would be differenced until it becomes stationary
is its order of integration. In such a case we would instead state the process
in eq. 7 as autoregressive integrated moving average -ARIMA, incorporating
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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the order of integration explicitly in the modelling process, and so it
becomes ARIMA (p,d,q), with p - being the order of the AR process; d - the
order of integration and q - the order of the MA process. Lower order MA
models have in practice been found to be more useful in econometric
modelling than higher order MA models. Note that in all our usage in this
guideline, yt is assumed to take the definition we’re already very familiar with,
i.e. dlexr = lexr - lexr(-1), which by construction is already stationary so
the d.g.p is simply ARMA.
In the following, monthly frequency data on nominal exchange rate,
available in COMESA_2017 folder as Monthly_exr.xls excel file – some
241 observations spanning 1997m01 to 2017m01 is used. I will assume that
we can, as of now, easily call in this data to the EViews work file we have
used before. We shall name the page monthly_exr, using the procedure
described earlier on. We then transform exr series to natural logarithms (lexr)
and dlexr, i.e. m-o-m depreciation. This brings a screen with three series of
exr, lexr, dlexr and by default, c and resid. The data, despite being of a high
frequency nature is not adjusted for seasonal effects as this would contradict
the assumption of rational behaviour in financial markets, particularly
because the seasonality here is not regular (Bwire, Opolot and Anguyo
2013). The first task is to show that the residuals from a basic specification
of dlexr are serially correlated. We will use the screen print below to
illustrate how this is accomplished. Click on Quick icon at the top of the
command window, and in the drop down menu, select Estimate
Equation....., indeed as is shown by the arrows.
.......................................................................................................... Modelling Conditional Volatility
33 |
Road map
Choose Estimate Equation. This action returns an Equation Estimation
window, in which under Equation specification, we have to supply a
specification of the equation we want to estimate, in the order DLEXR
c(dependent variable followed by explanatory terms, with equation terms
separated by space). By default, both estimation method, and the sample
range are already provided, but can be changed if there is a reason to do so.
These entries are shown in the screen print below.
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Regression Specification
Various Estimation techniques: Scroll through for appropriate – here OLS
Sample range – freely adjustable
At this stage, click OK and the resulting results table is what is extracted as
Table 1 in the text. The results table has two parts: the first part shows the
dependent variable, method of estimation, date and time the estimation was
done, the sample and included number of obervations (after any
adjustment). It’s not often useful to report this upper part of the table and
here I do for exposition purposes only (and in this case only). The second
part is the estimated output –including all the available standard statistical
criteria for evaluating the estimated results.
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35 |
Table 1: Residual serial correlation
Dependent Variable: DLEXR
Method: Least Squares Date: 03/20/17 Time: 10:52
Sample (adjusted): 1997M02 2017M01
Included observations: 240 after adjustments Variable Coefficient Std. Error t-Statistic Prob.
C 0.005165 0.001492 3.462719 0.0006
R-squared 0.000000 Mean dependent var 0.005165
Adjusted R-squared 0.000000 S.D. dependent var 0.023109
S.E. of regression 0.023109 Akaike info criterion -4.693057
Sum squared resid 0.127629 Schwarz criterion -4.678554 Log likelihood 564.1668 Hannan-Quinn criter. -4.687213
Durbin-Watson stat 1.290424
An indicative way of detecting serially correlated residuals is the Durbin-
Watson (hereafter, D-W) statistic (usually appeared in the lower panel of the
regression output, as in Table1). Using a rule of tumb, a model is said to be
statistically free of serial correlation if 2WD , though in applied work,
judgement may be required especially that sometimes, you may have to
accept a model with moderate levels of serial correlation. In this application,
reading from Table 1, the D-W statistic is only about 1.29, which is too low
- raising the suspicion that the residuals could be serially correlated.
Beyond the D-W statistic, there are formal serial correlation tests embedded
in EViews routines – which we now demonstrate. To confirm the suspicion
of serial correlation detected by the D-W statistic described above, while in
the results window given in Table 1, click on View and navigate through to
Residual Diagnostics. Following the arrow, you see the two available test
checks for serial correlation namely Correlogram-Q-Statistics... and
Serial Correlation LM Test.... In both tests, the null hypothesis is that of
‘no serial correlation’. We will check out this, one at a time, beginning with
Correlogram-Q-Statistics..., and replace 20 for 36 lags in the Lag
Specification. This should yield output similar to those in Fig. 5 detailed
in the screen shot below.
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Clearly, the p-values of the Q-Stat at all lags are significantly different from
zero, suggesting we can safely reject the null of no serial correlation in the
residuals.
We will also implement Serial Correlation LM Test....in the same window
following the same procedure as above, though with 2 lags as default in Lag
Specification to yield the extract in Table 2.
Table 2: Breusch-Godfrey Serial Correlation LM Test
F-statistic 17.22048 Prob. F(2,237) 0.0000
Obs*R-squared 30.45166 Prob. Chi-Square(2) 0.0000
Consistent with the Q-Stat and D-W statistic evaluated earlier, the F-statistic
0000.0220.17237,2 F and Chi-Square-statistic 000.0452.3022 ,
both given in Table 2suggest we can safely reject the null of no serial
correlation in the residuals, as the corresponding p-values are statistically
different from zero.
This proof of existence of serially correlated residuals requires that we have
to incorporate AR lags in the basic specification, keeping in mind that we
want a model with as few AR parameters as possible to get a parsimonious
.......................................................................................................... Modelling Conditional Volatility
37 |
model, but at the same time we want enough AR parameters to remove any
remaining autocorrelation in the residuals.
An indicative way to identifying the ARMA structure is a visual inspection
of the autocorrelations and partial autocorrelations (Box-Jenkins, 1976) of
the DLEXR process, depicted in the screen shot above or in the right-hand
side panel of Fig. 5. Clearly, we see one extended bar at lag 1 for both the
autocorrelation and partial autocorrelation. Loosely speaking, the
autocorrelation is the AR term while the partial autocorrelation is the MA
term. Therefore, with one extended bar at lag 1 for both autocorrelation
and partial autocorrelation suggests one AR term (AR(1)) and one MA term
(MA(1)), which is ARMA(1,1) process. Note however that this easy to
identify scheme for ARMA structure is only feasible in a univariate case like
the one in our case and for a covariance stationary process. It is not possible
for 1n . Besides, this is a visual inspection scheme and may not necessarily
be robust to the formal identification procedure, described here-under.
The formal identification scheme involves fitting an ARMA model, with a
sufficient number of autoregressive parameters that ensures time
independence of the residuals, and as shown in section 3.1 that the MA
process is 1-dependent, one MA term. This is the long-tested general to specific
approach. Accordingly, here we began with k=4 AR parameters and for the
reason above, one MA component. Note however that for the sort of
frequency used in this illustration, i.e. monthly data, the generality of the
results may not be lost if the MA component is omitted altogether– the
reason being that a month’s period in financial markets is long enough for
the effect of previous month noise to have neutralized or even dissipated.
Such noise may however be very pronounced with higher frequency data
such as the daily observations. Nonetheless, for the benefit of doubt, the
MA(1) component is included. With this in mind, click on Quick icon at
the top of the command window, select Estimate Equation..... and suppy
a specification of the form DLEXR c ar(1)ar(2) ar(3) ar(4) ma(1).Noting that
by default, both estimation method, and the sample range are already
provided. Check OK to produce results extracted in Table 3:
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Table 3: ARMA(4,1) model Estimates
Variable Coefficient Std. Error t-Statistic Prob.
C 0.005141 0.001942 2.647304 0.0087
AR(1) -0.162363 0.416970 -0.389387 0.6974
AR(2) 0.158856 0.166143 0.956137 0.3400 AR(3) -0.008071 0.070508 -0.114474 0.9090
AR(4) -0.108533 0.068274 -1.589676 0.1133
MA(1) 0.534596 0.417367 1.280879 0.2015 R-squared 0.138359 Adjusted R-squared 0.119628
S.E. of regression 0.021794 Durbin-Watson stat 1.998234
Sum squared resid 0.109241 Inverted AR Roots .41+.37i .41-.37i -.50+.33i -.50-.33i
Inverted MA Roots -.53
Consistent with the prior that the MA component could be irrelevant for the
kind of data frequency considered in this illustration for financial data, the
MA(1) term, with a p-value of 0.2015 is indeed unimportant. Like the MA(1)
coefficient, all the AR parameters are apparently insignificant. Moreover,
implementing the model Residual Diagnostics following the procedures
described above the Q-static for some lags, notably the 11th – 15th are either
insignificant or borderline significantly different from zero. Nonetheless, all
other statistics seem to be parsimonious, i.e. the D-W statistic is 1.998,
which is very close to ranges where absence of serial correlation cannot be
safely rejected and the inverted root of the lag polynomials (inverted AR
and MA roots) are all inside the unit circle i.e. less than one in absolute
terms, which is consistent with the stationarity restriction imposed on and
parameters in the AR and MA equations, respectively, in section 3.1.
However, whilst the insignificance of the AR terms may be interpreted as
absence of evidence, it does not necessarily suggest that the d.g.p is devoid
of a dynamic structure – and could infact be a result of model
mispecification – in this case over parametization .As is the norm in
econometric modelling process, we have to drop regression terms
sequentially through general to specific approach, but with a particular
interest in the statistical performance of the resulting models. We dropped
.......................................................................................................... Modelling Conditional Volatility
39 |
first, the MA(1) term for the same reasons given above and then the AR
terms, one at a time beginning with AR(4) term, then AR(3), and eventually
AR(2). With the exception of AR(1) model structure, all higher order AR
terms in all the other specifications were insignificant and as in the general
case given in Table 3, the Q-static for the 11th – 15th lags are either
insignificant or borderline significantly different from zero – which in effect
suggests 1AR terms are not relevant. This reduces the appropriate d.g.p
in this class to AR(1) model, for which results are extracted in Table 4.As
can be seen, the AR(1) coefficient (0.354) is highly significant at the 1%
level of significance (p-value = 0.0000) and the D-W statistic (1.97) and the
inverted roots of the lag polynomial (0.35) are consistent with the absence
of serial correlation and the stationarity restriction imposed on parameter
in the AR equation, respectively in section 3.1.
Table 4: AR(1) model Estimates
Variable Coefficient Std. Error t-Statistic Prob.
DLEXR -- C 0.005268 0.002170 2.427636 0.0159
AR(1) 0.353713 0.060684 5.828735 0.0000 R-squared 0.125378 Adjusted R-squared 0.121687
S.E. of regression 0.021679 Durbin-Watson stat 1.969633
Inverted AR Roots .35 Notes: -- denotes the dependent variable
If you do the model Residual Diagnostics following the procedures
described above, you will see that consistent with the D-W statistic,
Breusch-Godfrey Serial Correlation LM Test yields 22 = 0.537 (0.765)
and Q-statistic probabilities are all above 10 percent, upholding the view
that we cannot safely reject the null hypothesis of no serial correlation in
AR(1) residuals.
Finally, we will demonstrate how to display the ARMA structure (inverted
AR roots). To do so, whilst in the estimated results window given in Table
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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4, select from the menu of this very estimated output, View/ARMA
Structure... and open the ARMA Diagnostic Views dialog box. We are
interested in inverse roots of the AR and/or MA characteristic polynomial,
one of the options on the left-hand side of the dialog box, which is already
selected by default. Check Graph on the right-hand side, to produce, as in
Figure 7, the roots in the complex plane. The horizontal axis is the real part
and the vertical axis is the imaginary part of the AR root.
Figure 7: Inverse Roots of AR/MA Polynomial(s)
Exercise 2
Exercise 2 makes use of the data used in exercise 1, i.e. data on 5-day week
nominal exchange rate for Uganda, spanning September 01, 2005 to June 01,
2017 or some 2928 observations. Based on the data:
1. Estimate a basic equationtty 0
, where dlexryt , and
demonstrate statistically that the model residuals are autocorrelated and
explain why testing for this kind of residuals behaviour is worthwhile
doing before a model can be used for statistical inference. .
2. Describe how the problem in (1) above can be dealt with in practice
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
AR
roo
ts
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41 |
3. Based on (2) above, fit an appropriate ARMA process for dlexr and
demonstrate statistically that this fitted model is free from serial
correlation
4. Demonstrate that the inverted roots of the lag-polynomial from the
fitted model lies inside the unit circle.
3.2: Introduction to Modelling Conditional Volatility
As mentioned earlier, uncertainty or risk plays an important role in financial
analysis and is usually measured with volatility. But as alluded to, volatility
of an asset is not observable, so it’s modelling is necessary. Based on the
constructed model, volatility can be both measured and predicted. Asset
holders are interested in the volatility of returns over the holding period,
not over some historical period. This forward-looking view of risk means
that it is important to be able to estimate and forecast the risk associated
with holding a particular asset. Although in literature, numerous volatility
models have been suggested to capture the characteristics of return for an
asset, here we explore the Auto Regressive Conditional Heteroskedasticity
(ARCH) and Generalized Auto Regressive Conditional Heteroskedasticity
(GARCH) type models.
As a starting point, consider the conventional AR(1) process in eqn. 5 given
as:
ttt yy 1 ;
Where t = 1, 2, ..., T, ),0.(..~ 2 diitand 1 so that
ty is a stationary
AR(1) process. If indeed ty is generated by this process, then it can be
shown that the mean and variance (covariance) of ty are constant or are
unconditional.
As we have shown, ttt yy 1
, behaved as above, in MA
representation is simply
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0i
it
i
ty .
Based on this, it is straight forward to show that:
0
)(i
it
i
t EyE
= )(0
i
it
i E
= 0.
)()( 2
tt yEyV
=
2
0i
it
iE
= 2
3
3
2
2
1 ... ttttE
=
...... 3
3
2
2
13
3
2
2
1 ttttttttE
= E ( ...2
3
62
2
42
1
22
tttt
{cross terms like jt
ji
where ji })
= 0...2624222
= 2 ( ...1 642 )
=
0
22
i
i
Since 1 , this infinite sum is convergent, so using the well-known
geometric series result that
0 1
1
i
i
rr when 1r
The proof of this is straight forward. Assume an infinite geometric seriesS ,
defined as
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43 |
1i
iarS Where 1r
...32 arararaS (a)
Multiplying through by r , becomes
...432 ararararrS (b)
Subtract (b) from (a) to give
....... 4322 arararararararSS
r
aS
aSr
1
1
So we have
2
2
0
22
01
)(
i
i
tyV . (9)
Showing that the mean and variance of the AR(1)process do not depend on
t provided 1 . This is the unconditional mean and variance ofty .
However, the covariance shows the residuals are serially correlated and the
proof of this is straight forward.
2
2
1,
s
sstt yyC (10)
It follows that the autocorrelation function for the AR(1) process will be
ss
s
0
(11)
Showing that the AR(1) process has non-zero autocorrelations at all lags,
but these decay to 0 as s because 1 (either exponentially if 0 ,
or like a damped sine wave if 0 ).To illustrate this, let us assume
20,...,2,1s and also assume that, 4.0 , i.e. positive but less than one
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and 4.0 , i.e. negative but less than one in absolute terms. If we
simulate these numbers in excel and plot them, we see indeed that at all lags,
the autocorrelations are non-zero and decay to 0 as s , but fashioned
differently, as shown in Figure 8.
Figure 8: Autocorrelations decay: Exponential (left) or damped sine wave (right)
The conditional mean of ty refers to the mean of
ty , conditional on
information set available at time it . Denoting this historical information
set available at time it asit , ,...2,1i , and assuming the same d.g.p
for ty given above, the conditional mean of
ty is given by
1| titt yyE (12)
And the conditional variance of ty is given by
2
2 ||
t
ittitt EyV
(13)
Consistent with the behaviour we have seen in Fig. 2, it is straight forward
from this expression that the conditional variance is time varying – a feature
seasoned econometricians call heteroskedasticity. An important question
though is how long t should be in calculating conditional volatility. As
mentioned earlier, if the period is too short, say one past year, it will be too
noisy and if it is too long, say past 30 years or more, such a long memory is
just not so relevant for today. The solution, according to Engel (2004, p.406,
in AER) is autoregressive conditional heteroskedasticity (ARCH) process, a process
which describes the forecast variance in terms of current observations. That
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45 |
is, the ARCH model, rather than use short or long sample standard deviations, takes
weighted averages of past squared forecast errors, a type of weighted variance which gives
more influence to recent information and less to the distant past, making the ARCH
model a simple generalization of the sample variance. We now turn to the ARCH
model in detail.
3.2.1: The ARCH Effect
Defined by Engel (1982), t for the ARCH model is:
21
2
110 ttt v (14)
Where 1,0..~ diivt , i.e. white noise process such that 12 v , tv
and 1t
are independent of each other, i.e. 0, 1 ttvC
, and 0,
1 are
constants and strictly positive, i.e. 00
, 10 1 .Unlike the
unconditional and conditional mean, both of which are zero, the unconditional
variance is non-zero and is given as:
1
02
0
2
1
0
2
11
2
2
110
2
110
1
2
2
2
0
2
1
1
t
t
tt
t
tt
tttt
E
E
EE
E
EvE
EEEE
Like the unconditional mean, the unconditional variance is unaffected by
the presence of the ARCH error process.
Given that 12 v , implies the conditional variance is influenced by the ARCH
error process defined in Eqn. 14, and in a manner consistent with the
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propositions of the asset pricing models. The risk premium of holding a
financial asset is the expected return on the asset over the holding period
and the variance of that return. Indeed, it can be shown that,
2
110
2
110
1
22 ||
t
titttitt vEhE
(15)
Thus, th , or conditional variance of
t is dependent on the realized value of
the past squared forecast errors, 2
1t . This implies that if the realized value
of 2
1t is large, h in period t will be large as well. Given this, th is said to
follow a first-order autoregressive, denoted ARCH(1). We just noted that in
an ARCH model, the unconditional and conditional expectations of t are
both zero. On the contrary, th is an autoregressive process resulting in
conditional heteroskedastic errors i.e. depends on the squared error term
from the last period ( 2
1t ). Each of the coefficients 0 and
1 are restricted
to be strictly positive to ensure the conditional variance satisfies certain
regularity conditions, including the non-negativity constraint and the
restriction on1 , i.e. 10 1 ensures stability of the process. It therefore
follows that since 1 and 2
1t cannot be negative, the minimum value that
th can take is0 . Thus, the volatility of {yt} is increasing in
0 and1 which
implies any unusually large shock in tv will be associated with a persistently
large variance in the {t } sequence, i.e. the larger
1 is, the larger the
persistence.
The ARCH(1) process in Eqn. 15 has been extended in several interesting
ways, including higher order ARCH processes – the Engel’s (1982) original
ARCH (q) model, to other univariate time series models, bivariate as well as
multivariate regression models and to systems of equations.
The AR(p)-ARCH(q) model form of Eqn. 5 is given as:
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47 |
p
i
titit yy1
;
(16)
Where 2
1
1
2
0
q
i
ititt v ; 1,0...~ diivt and 1
The ARCH(q) multiple regression model is given as:
k
i
titity1
0 x
(17)
where 2
1
1
2
0
q
i
ititt v ; 1,0...~ diivt and 1
Where itx is a vector of lagged exogenous explanatory variables, including
yt.
Equations (16) and (17) can be extended to include dummy variables in the
model for the conditional mean if there is a reason to capture particular
features of the market e.g. day-of-the- week’ effects.
For brevity, assuming daily data, and if we let 5,...,1i such that 1
(Monday), ..., and 5 (Friday), then the model adjusts to
k
i
t
l
ltlitit Dxy1
5
1
0 (18)
Where 2
1
1
2
0
q
i
ititt v ; 1,0...~ diivt, 1 and
0
1ltD ;
if daily feature is observed, zero otherwise, 00 and 0,...,, 21 q .
Generalizing the ARCH models to systems of equations (the multivariate
ARCH model) is a natural extension of the original specification.
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3.2.1.1 The ARCH-M model
The ARCH in mean (ARCH-M) is an extension to the basic ARCH
framework to allow the mean of ty to depend on its own th (Engle et al.,
1987). The model, relevant in financial applications is used to relate the
expected return on an asset to the expected asset risk. A risk-averse agent
will require compensation or risk premium for holding a long-term risky
asset. Since an asset’s riskiness is measured byth , it follows that the greater
th
of returns, the greater the risk premium necessary to induce the agent to
hold the long-term risky asset, i.e. the risk premium is an increasing
function of th of returns. Thus, the ARCH-M is of the form:
ttt
t
hy
; 0 (19)
Where: ty = excess return for holding a long-term asset relative to the risk
free one-period Treasury bond
t = risk premium necessary to induce the risk-averse agent to hold
the long- term risky asset rather than the risk free one-period Treasury
bond.
= is a measure of the risk-return trade-off
th = the ARCH(q) process, defined as:
q
i
itith1
2
0 , and
t = shock to the excess return on the long-term risky asset.
Note that holding a risky asset makes sense only if the tt yE 1.
Moreover, providing th is constant (i.e., if 0...21 q ) the
ARCH-M Model degenerates into the traditional case of a constant risk
premium.
Other variants of this ARCH-M model are:
ttt
t
hy
; 0 : (20)
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49 |
Where conditional standard deviation is used in place of conditional
variance, and
ttt
t
hy
log ; 0 (21)
Where the log of conditional variance is used in place of conditional
variance
Due to the large persistence in volatility, ARCH models typically require 5-8
lags of 2
t to adequately model th or fit the data. Moreover, to avoid
problems associated with a negative conditional variance it is necessary to
impose restrictions on the parameters of the model. Consequently, in
practice the estimation of ARCH models is not always straight forward.
Given this, Bollerslev (1986) extends the ARCH model to be an ARMA
process, i.e. allows for a more lag structure: the generalized ARCH
(GARCH) model.
3.2.2: The GARCH Model
Under the GARCH model, th is modelled as a function of lagged values of
2
t and th . In general, assuming a dgp for the sequence {yt} in (5) and
following Engel’s (1982) generalized definition of t for the ARCH model
as in (14), t for the GARCH (p, q) model is:
5.0
1 1
2
0
q
i
p
j
jtjititt hv
(22)
where 1,0...~ diivt ; 0q ; 0p ;
00 ;
0i , qi ,...,2,1 and
0j , pj ,...,2,1 .
The sufficient condition for covariance stationarity of the process t is for
q
i
p
j ji1 11 .
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It is straightforward to show from eqn. 22 that
The unconditional variance of t becomes:
q
i
p
j
ji
tV
1 1
02
1
And the conditional variance is:
q
i
p
i
jtjitit hh1 1
2
0 (23)
And since tv is a white noise process, the conditional and unconditional means
of t are equal to zero.
This generalized GARCH (p, q) model in eqn. 23 allows for ARMA
components in theth . It has been shown in applied settings that a GARCH
(p, q) model with low values of q and p provides a better fit to the data than
an ARCH (q) model with a high value of q.
In general, if we suppose in eqn.23that 0p and so are all values of j ,
and 1q , the GARCH (p, q) model collapses to an ARCH(1) or GARCH
(0,1) model. Thus, an ordinary ARCH model is a special case of a GARCH
specification in which there are no lagged forecast variances in the th
equation.
Key points to NOTE in GARCH Estimation
Estimating a GARCH process typically involves an estimation of two
interrelated equations:
a) tit
p
i
it yy
1
; pi ,...,1 Mean equation.
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51 |
Where
5.0
1 1
2
0
q
i
p
j
jtjititt hv
b)
q
i
p
i
jtjitit hh1 1
2
0 Variance equation.
1. The two equations are related in that th is the conditional variance of
t , hence t of the GARCH process is the conditional variance of the
mean equation. In fact 2
t is itself notth - and knowledge of this is
straightforward.
We know by definition that 5.0
ttt hv ttt hv22 .
Showing that 2
t is not th itself.
2. GARCH (1, 1) model is the most popular form of conditional volatility,
especially for financial data where volatility shocks are very persistent.
As such, eqn. 23 reduces to:
11
2
110 ttt hh (24)
i) As shown, th is a function of a constant,
0 , news about volatility
from the previous period, measured as the lag of 2
t from the
mean equation (the ARCH term) and the previous period’s
forecast variance: 1th (the GARCH term).
ii) If 01 , there is no volatility clustering
iii) If 01 , there is absence of a GARCH term in th
iv) GARCH (1, 1) process for t is weakly stationary if and only if
111 .
v) Volatility persistence is captured by 11 , i.e. the degree of
autoregressive decay of the squared residuals.
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vi) Conditional volatility increases with large values of both 1 and 1 ,
but in different ways. The response of thto new information is an
increasing function of the magnitude of 1 . And the larger the
value of 1 , the more is the autoregressive persistence of the th.
vii) The value of 1 must be strictly positive.
3.2.2.1: The GARCH-M model
Same as ARCH-M model above, except that th in GARCH (1,1) frame
work is:
11
2
110 ttt hh
So as shown before, the variance appears in the mean of the sequence {ty }
explicitly. The sequence {ty } is stationary as long as the variance process is
stationary.
3.2.2.2: The integrated GARCH model
In empirical estimations, 1 and
1 in GARCH (1,1) or
q
i
i
1
and
p
i
i
1
in
a general GARCH(q, p) model tend to sum to a value close to one.
IGARCH is the limit case, where the sum of these parameters equals exactly
one.
Once 111
p
i i
q
i i as is the case for the IGARCH, th is not
definite anymore, and while the sequence {ty } is no longer covariance
stationary, it remains strictly stationary because the unconditional density of
t does not change over time. The IGARCH (1, 1) model is written as:
11
2
110 1 ttt hh ; 10 1 . (25)
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53 |
This can be re-arranged to
1
2
1110 1 tttt hhh .
3.3: Models With Asymmetry
3.3.1: The Threshold GARCH (TGARCH) model
An interesting feature of asset prices is that ‘bad’ news tends to have a more
pronounced effect on volatility than does the ‘good’ news. TGARCH
models show how to allow the effects of good and bad news to have
different effects on conditional volatility (Glosten et al., 1993). In the model,
‘new information’ is measured by the size of the shockt , and the good
news is when 0it and bad news when 0it . The TGARCH process
is given by:
11
2
111
2
110 ttttt hdh (26)
where1td is a dummy variable that is equal to 1 if 01 t and is equal to
zero if 01 t . Thus, if
i) 01 t , 01 td , and the effect of an 1t shock on th is 2
11 t .
ii) 01 t , 11 td , and the effect of an 1t shock on th is 2
111 )( t .
a) If 01 , negative shocks will have larger effects on volatility
than positive shocks. This is the leverage effect, i.e. the tendency
for volatility to decline when returns rise and to rise when
returns fall (news impact curve, Enders, 156).
b) If 1 is statistically different from zero, the data contains a
threshold effect.
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3.3.2: The Exponential GARCH (EGARCH) Model
As opposed to the standard GARCH model which necessitates that all
estimated coefficients are positive, EGARCH does not require non
negativity constraint (Nelson, 1991), and is specified as:
11
1
11
1
110 lnln
t
t
t
t
tt h
hhh
. (27)
Note three interesting features about the EGARCH model:
i) Regardless of the magnitude of thln , the implied value of th
can never be negative so it is permissible for the coefficient to
be negative.
ii) The standardized value of 1t is used in the place of 2
1t ,
allowing for a more natural interpretation of the size and
persistence of shocks. 5.0
1
1
t
t
h
is a unit free measure.
iii) The model allows for leverage effects.
If
05.0
1
1
t
t
h
, the effect of the shock on thln is
11 ,
and
If
05.0
1
1
t
t
h
, the effect of the shock on thln is
11 .
(see Enders 2010: 157).
3. Others include the Power ARCH (PARCH) and the Component
GARCH (CGARCH) models.
Note that while we make reference to models with asymmetry: TGARCH
and EGARCH, PARCH and CGARCH, none of this is pursued in detail in
this guideline.
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55 |
3.4: Demonstrating the G(ARCH) and Related Models
Estimations Using EViews Screens and Statistical Output
Tables
The appropriate way to obtain a proper order of the GARCH process is to
estimate the mean and conditional variance equations simultaneously. As
such, GARCH processes are estimated by maximum-likelihood (M-L)
techniques so as to obtain estimates that are fully efficient. If the reader is
more interested in the exposition of the M-L technique, they will need to
refer to the many leading econometrics text books, including among others
Green (2003), Enders (2010), Harris and Sollis (2003) and Hamilton (1994).
EViews contains built in routines that estimate these models. Therefore, all
that the modeller needs is to specify the order of the process and the
assumption about the conditional distribution of t and the computer does
the rest. By default, G(ARCH) models in EViews are estimated by the
method of M-L under the assumption that the errors are conditionally
normally distributed.
In this section, we explore the estimation of a basic ARCH model, including
a wide range of specifications for the same available in EViews. For brevity,
we shall use ARCH and GARCH models interchangeably, but distinguish
where there is a possibility of confusion. We now turn to a step-by-step
practical estimation procedure, implemented using monthly_exr EViews
page.
As before, in the Main menu bar, click Quick/Estimate Equation....
Among the drop-down menu of Estimation settings is ARCH –
Autoregressive conditional heteroskedasticity as shown in the screen
print.
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Active page
Estimation method of choice - ARCH
Select ARCH – Autoregressive conditional heteroskedasticity – our
estimation method of choice for the G(ARCH) models. This selection
brings forth the ARCH specification dialog – where we have to supply the
mean and the variance specifications, the error distribution and the
estimation sample period.
3.4.1: Equation Estimation/Specification
Mean Equation
Enter, in the dependent variable edit box, the specification of the mean
equation in eqn.5 herein. Remember that at some point, we established
ARMA(1,0) as being appropriate for the d.g.p. Therefore, for now, we will
supply this very specification by entering dlexr c ar(1) in the mean equation
edit box.
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57 |
The upper right-hand side of the dialog provides for a specification of the
mean equation with an ARCH-M term. We will have to select the
appropriate item of the combo box by choosing from the drop-down menu,
alternative specifications of the ARCH-M model: the Std. Dev. (given in
eqn. 20), Variance(given in eqn. 19), or the Log(Var)(given in eqn. 21).
Getting the option that best fits the data is an iterative procedure, and so in
Table 5, we report results for all these options which we compare on the
basis of standard evaluation criteria.
Variance and distribution specification/ Model
We aim to estimate the standard GARCH model described in the text. So,
retain the default selection of GARCH/TARCH in the Model combo box.
The other entries in the drop-down menu; EGARCH, PARCH and
Component ARCH(1,1) correspond to more complicated variants of the
GARCH specification.
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Variance and distribution specification/ Order
Here we supply the number of ARCH and GARCH terms. As noted in eqn.
24, the default of one term for ARCH and one term for GARCH is by far
the most popular specification.
Variance and distribution specification/ Threshold order
Only if we’re to estimate an asymmetric model that we would have to
supply the number of asymmetry terms in the Threshold order edit field.
So, we shall maintain the default setting, with threshold order 0.
Variance and distribution specification/ Variance regressors
This edit box allows us to supply the list of variables we wish to include in
the variance equation specification. As you will discover, save for IGARCH
models, EViews always includes a constant as a variance regressor, so we
need not supply C here, but is an appropriate place for other exogenous
variables such as dummies providing these have been found to enter the
mean equation.
Variance and distribution specification/ Restrictions
As noted earlier, this option allows us to restrict the parameters of the
GARCH model, 1 and
1 , but in two ways:
1. One option is IGARCH, which restricts 1 and
1 to sum to one.
2. The other option is Variance Target, which restricts the constant
term, 0 , in such a way that: 110 1 th , i.e.
0 is a
function of the GARCH parameters and the unconditional variance.
For now, we shall leave the default setting of None unchanged, but
you might want to specify IGARCH at some point especially if we
want to see the difference it might make.
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Variance and distribution specification/ Error distribution
In this combo box, we have various options for the Error Distribution to
choose from. But as we have shown before, both student’s t–distribution
and Generalized Error distribution tend to approximate a normal
distribution as t . Little is gained by changing from the default -
Normal (Gaussian).
3.4.2: Estimation Settings/ Method and Sample - obvious
Equation Estimation/options
Next, click on the Option tab, where we may need to make additional input,
but if necessary.
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Back casting
Back casting is a method used in the computation of the2
0 , i.e. the initial
value used in initializing the MA estimation and 0h required for the
GARCH term. While computing 0h for the GARCH, EViews behind the
scene first uses the coefficient values to compute t of the ty equation,
and then computes an exponential smoothing estimator of the initial values,
which takes the form:
T
i
iT
iTT hh0
2122
00ˆ1ˆ
Where are residuals from the mean equation; 2h is the unconditional
variance estimate, computed as
T
i
t T1
2 , and the smoothing parameter
7.0 . Note though that from the Pre-sample variance drop-down list, we
have the option to choose from a number of weights in the range 0.1 – 1.0,
in increment of 0.1. Note that if 1 , then, 2
0 hh . Except if we have a
suitable reason, we will maintain 7.0 as in the default.
Coefficient covariance
This option of heteroskedasticity consistent covariance is used especially as
expected, if we suspect that the residuals are not conditionally normally
distributed, and is available as such, only if, in the previous screen, one
chose Normal (Gaussian) under the Error Distribution. The point is that
if the assumption of conditional normality does not hold, the ARCH
parameter estimates will still be consistent, provided the mean and variance
functions are properly specified. However, the estimates of the covariance
matrix will not be consistent, unless this option is specified, resulting in
incorrect standard errors. Therefore, given our knowledge of departure
from normality assumption, part of which we have seen earlier, we shall
check out this only box to compute the quasi-maximum likelihood (QML)
covariance and standard errors due to Bollerslev and Wooldridge (1992).
.......................................................................................................... Modelling Conditional Volatility
61 |
Derivatives/Select method to favour
Choose the method to favour accuracy (more function evaluation)
Iterative process/Starting coefficient values
By default, EViews supplies its own starting values using OLS regression
for the mean equation, but if we had a better knowledge of what is
appropriate, we would set starting values to various fractions of the OLS
starting values or specify the values by choosing the User Specified option
in the Option dialog. We now click OK to estimate the model. Estimates
of the various ARCH-M specifications, are extracted here as GARCH-M:
None; GARCH-M: std. dev.; GARCH-M: variance; and GARCH-M: log
var, and summarized for appropriate specification verification purposes and
tractability in Table 5.
GARCH-M: None
Variable Coefficient Std. Error z-Statistic Prob. C 0.005563 0.001714 3.245649 0.0012
AR(1) 0.365139 0.090046 4.055008 0.0001 Variance Equation C 5.92E-05 2.46E-05 2.411992 0.0159
RESID(-1)^2 0.323173 0.087123 3.709401 0.0002 GARCH(-1) 0.610927 0.075985 8.040092 0.0000 R-squared 0.125182 Mean dependent var 0.005234
Adjusted R-squared 0.121490 S.D. dependent var 0.023132 S.E. of regression 0.021682 Akaike info criterion -4.903957
Sum squared resid 0.111412 Schwarz criterion -4.831227
Log likelihood 591.0228 Hannan-Quinn criter. -4.874649
Durbin-Watson stat 1.991219 Inverted AR Roots .37
Notes: Dependent variable: DLEXR; Method: ML - ARCH (Marquardt) - Normal distribution; Included observations: 239 after adjustments.
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GARCH-M: STD.DEV. Variable Coefficient Std. Error z-Statistic Prob. @SQRT(GARCH) -0.383394 0.239740 -1.599211 0.1098 C 0.012035 0.004032 2.984840 0.0028
AR(1) 0.349790 0.083010 4.213853 0.0000 Variance Equation C 5.43E-05 2.19E-05 2.477799 0.0132 RESID(-1)^2 0.334234 0.086958 3.843615 0.0001
GARCH(-1) 0.612394 0.065935 9.287854 0.0000 R-squared 0.119554 Mean dependent var 0.005234 Adjusted R-squared 0.112093 S.D. dependent var 0.023132
S.E. of regression 0.021797 Akaike info criterion -4.906465
Sum squared resid 0.112129 Schwarz criterion -4.819190
Log likelihood 592.3226 Hannan-Quinn criter. -4.871296 Durbin-Watson stat 1.945799 Inverted AR Roots .35 Notes: Dependent variable: DLEXR; Method: ML - ARCH (Marquardt) - Normal
distribution; Included observations: 239 after adjustments.
GARCH-M: VARIANCE
Variable Coefficient Std. Error z-Statistic Prob. GARCH -6.082374 4.523817 -1.344523 0.1788
C 0.007515 0.001978 3.799458 0.0001
AR(1) 0.354549 0.083541 4.243996 0.0000 Variance Equation C 5.54E-05 2.29E-05 2.419761 0.0155
RESID(-1)^2 0.324512 0.086040 3.771629 0.0002
GARCH(-1) 0.616970 0.070153 8.794654 0.0000 R-squared 0.120146 Mean dependent var 0.005234
Adjusted R-squared 0.112690 S.D. dependent var 0.023132
S.E. of regression 0.021790 Akaike info criterion -4.902533
Sum squared resid 0.112054 Schwarz criterion -4.815257 Log likelihood 591.8526 Hannan-Quinn criter. -4.867363
Durbin-Watson stat 1.957874
Inverted AR Roots .35 Notes: Dependent variable: DLEXR; Method: ML - ARCH (Marquardt) - Normal distribution; Included observations: 239 after adjustments.
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GARCH-M: LOG (VAR) Variable Coefficient Std. Error z-Statistic Prob. LOG(GARCH) -0.002510 0.002082 -1.205376 0.2281
C -0.015342 0.017694 -0.867076 0.3859 AR(1) 0.354834 0.088316 4.017765 0.0001 Variance Equation C 5.45E-05 2.24E-05 2.436672 0.0148
RESID(-1)^2 0.345480 0.089545 3.858154 0.0001
GARCH(-1) 0.605935 0.069853 8.674459 0.0000 R-squared 0.124384 Mean dependent var 0.005234
Adjusted R-squared 0.116964 S.D. dependent var 0.023132
S.E. of regression 0.021737 Akaike info criterion -4.901086
Sum squared resid 0.111514 Schwarz criterion -4.813811 Log likelihood 591.6798 Hannan-Quinn criter. -4.865917
Durbin-Watson stat 1.971628 Inverted AR Roots .35
Notes: Dependent variable: DLEXR; Method: ML - ARCH (Marquardt) - Normal
distribution; Included observations: 239 after adjustments.
Based on the results in Table 5, the terms unique to variant ARCH-M
specifications are not significantly different from zero and are all wrongly
signed (in view of the restriction imposed on the coefficient in eqns. 19,
20 and 21). This signals that G(ARCH-M) variants of G(ARCH) are
inconsistent with the true d.g.p, and will not therefore be discussed beyond
this point. Given this, our subsequent interpretations of results in Table 5
draw solely from the pure G(ARCH) estimates.
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Table 5: GARCH(1,1) with alternative restriction for ARCH-M in the Variance equation
ARCH-M
Variable None Std. Dev Variance Log (Var)
Mean Equation
@SQRT(GARCH)
-0.383
GARCH
-6.082
LOG(GARCH)
-0.003
C 0.006** 0.012** 0.008* -0.015
AR(1) 0.365* 0.350* 0.355* 0.355* Variance Eqn.
C 0.000** 0.000** 0.000** 0.000*
RESID(-1)^2 0.323* 0.334* 0.325* 0.345*
GARCH(-1) 0.611* 0.612* 0.617* 0.606*
Adjusted R-squared 0.121 0.112 0.112 0.117
S.E. of regression 0.022 0.022 0.022 0.022
Sum squared resid 0.111 0.112 0.112 0.112
Durbin-Watson stat 1.991 1.946 1.958 1.972
Inverted AR Roots 0.37 0.35 0.35 0.35
Asterisks *, **, and *** represent 1 percent, 5 percent and 10 percent levels of significance, respectively.
Overall, the model seems plausible. All coefficients are both theoretically
plausible and statistically highly significant. The inverted AR Roots is 0.37,
which clearly lies inside the unit circle and the D-W statistic of 1.991 does
not point to serial correlation problems. The coefficient to the
autoregressive parameter of the mean equation, i.e. in eqn. 5 is 0.365,
which is less than one, implying the system is covariance stationary.
Moreover, the sum of conditional variance parameters, i.e. 11ˆˆ in eqn.
24 is 0.934, which is less than one, but very high. This implies 1,1GARCH
process for t is weakly stationary and depicts the high volatility persistency
inherent in exchange rate movements. The coefficient to RESID(-1)^2, or
1 in eqn. 24, is 0.323 and is significant at the conventional 1percent level.
This implies very strong evidence of volatility clustering and also that
availability of new information increases conditional volatility by a
magnitude of 0.32. Similarly, GARCH(-1) parameter, given as 1 in eqn. 24
.......................................................................................................... Modelling Conditional Volatility
65 |
is 0.611 and is also highly significant at the conventional 1 percent level,
which implies presence of a GARCH term and that there is autoregressive
persistence of conditional volatility.
In addition, the properties of the error term are reasonably good. To show
these properties, click on View within the results window reported in
column 1 of Table 5. In the drop-down menus, navigate through to Residual
Diagnostics. Here we find all available tests for the properties of the error
term: Correlogram – Q- statistics, Correlogram Squared Residuals,
Histogram – Normality Test and ARCHLM Test as shown in the screen
print below.
Beginning with the formal ARCH-LM Test (given in Table 6), both the F-
statistic 338.0922.0236,1 F and the Chi-Square statistic
336.0926.012 do not show any remaining ARCH effects.
To produce the above statistic results, given in Table 6, click on ARCH
LM Test..... This route, as shown in the screen print below, brings forth
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the Heteroskedasticity Tests window – in which EViews highlights ARCH
as default heteroskedasticity test and includes 1 lag.
Note that in the test, the null hypothesis is that of ‘no ARCH effects’. Click
OK to yield output extracted in Table 6.
Table 6: Heteroskedasticity Test: ARCH
F-statistic 0.9218 Prob. F(1,236) 0.338
Obs*R-squared 0.9260 Prob. Chi-Square(1) 0.336
Moreover, thei of the standardized residuals (
t
t
h
) and the standardized
squared residuals (t
t
h
2 ) do not show evidence of any remaining serial
correlation (check this in self- exercise following the same procedure for
results in Table 6).
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67 |
However, as shown in Figure 9 (produced through the same procedure
above), normality of the G(ARCH) errors is rejected, but this is an inherent
feature of the errors from regression models for financial data.
Figure 9: Normality of G(ARCH) errors
Finally, we will now demonstrate how to generate a graph of the measured
conditional variance from this modelling procedure. Whilst in the estimated
output window (for the preferred competing model specification, here
ARCH-M with None option), click on View/Garch Graph in the results
menu bar and select Conditional Variance as shown in the screen print.
Road map
0
5
10
15
20
25
30
35
40
-3 -2 -1 0 1 2 3 4 5
Series: Standardized Residuals
Sample 1997M03 2017M01
Observations 239
Mean 0.006712
Median -0.067399
Maximum 4.946636
Minimum -3.627497
Std. Dev. 1.001617
Skewness 0.601742
Kurtosis 6.238965
Jarque-Bera 118.8952
Probability 0.000000
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Click on Conditional Variance to generate a graph for the measured
conditional variance or exchange rate volatility, pasted in Figure 10.
Figure 10: GARCH (1,1) Conditional variance-Measured
It is also possible to generate the corresponding conditional variance series,
by checking Proc in the same window where the graph is appeared, and
selecting Make GARCH Variance Series…- a route shown in the screen
print.
.0000
.0005
.0010
.0015
.0020
.0025
.0030
.0035
.0040
1997
:03
1998
:01
1999
:01
2000
:01
2001
:01
2002
:01
2003
:01
2004
:01
2005
:01
2006
:01
2007
:01
2008
:01
2009
:01
2010
:01
2011
:01
2012
:01
2013
:01
2014
:01
2015
:01
2016
:01
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69 |
In the resulting window, EViews will assign Conditional Variance name to
this new series to be created, and if this is the first conditional variance that
is being estimated, it will be assigned garch01.Finally, click OK and a new
series, in this case, garch01 will be added to the list of variables in the series
window. It is then possible to open and extract the measured conditional
variance series-just like you can extract any other series. Going back to the
graph for measured conditional variance in Fig.10, you might want to
refresh yourselves with what could have caused the unusually high volatility
around 2008/09, 2011/12 and very lately around 2015 – based on the
Uganda country knowledge alluded to earlier. This is generally about
keeping abreast with developments that have influenced the data path being
dealt with.
Exercise 3
In demonstrating the estimation of the G(ARCH) process, we have used
nominal exchange rate data for Uganda observed on a monthly frequency.
In this exercise, we will use CPI data contained in the same folder and excel
file for exercises in multiple sheet. This data contains 143 observations,
spanning from July 2005 to May 2017. Use the data to:
1. Fit an appropriate mean equation or ARMA model.
2. Estimate an appropriate GARCH(1,1) variant model. Hint: consider
several competing ARCH-M assumptions.
Verify that both ARCH and GARCH terms enter the conditional variance and that these parameters are well behaved.
3. Estimate an appropriate IGARCH(1,1) model: Hint: consider several
competing ARCH-M assumptions.
4. Compare the results for the GARCH(1,1) and IGARCH (1,1) models
of choice, i.e. best models under the competing ARCH-M assumptions,
showing eventually, why the former would be preferred to the latter.
5. Generate and plot the series for measured conditional volatility from
the GARCH (1,1) and IGARCH (1,1) models. Assuming these results
is to be presented to the MPC of your central bank, which of the two
would you present and why?
Chapter 4
Modelling Conditional Volatility in a
Multivariate Framework
4.1: Introduction
Multivariate GARCH is a natural extension of the original G(ARCH)
specification and is founded on the basis that contemporaneous shocks to
variables can be correlated with each other, allowing for volatility spill-overs
(positive or negative). That is, volatility shocks to one variable might lead to
volatility of other related variables, but to magnitudes that only empirical
analysis can reveal. Multivariate GARCH models allow the variances and
Covariances to depend on the information set in a vector ARMA manner,
the type useful in multivariate financial models, which require the modelling
of both variances and Covariances (such as Capital asset pricing models
(CAPM) or dynamic hedging models).
Building from a univariate GARCH model, an n-variant model requires
allowing the conditional variance-covariance matrix of the n-dimensional
zero mean random variables t to depend on elements of information set
1t , that is:
tH,0~ Nt
To put it in context, assume an n-dimensional G(ARCH) model, which in a
compact form is represented as:
Ntttt xxx ,...,, 21x (28)
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Where tx = (N x 1). μx ittE / ; μ= (N x 1) & tittVar Hx / ;
tH = (N x N)
The diagonal elements of tH are the
ith terms and the off-diagonal elements
are the ijth terms -
ith and ijth are estimated simultaneously by M-L to capture
the interactions between volatility of the N-time series.
For ease of exposition, take a case of just two variables (N=2), tx1and
tx2,
and 1 qp . As in the univariate case, the GARCH (1, 1) errors for the
two error processes become:
5.0
1111 ttt hv
5.0
2222 ttt hv
Where th11 and th22 are the conditional variances of t1 and t2 , respectively
for 1varvar 21 tt vv as before.
To allow for the possibility that t1 and
t2 are correlated, the conditional
variance between t1 and
t2 becomesth12, and specifically,
tttt Eh 21112
Allowing in this framework the interaction of volatility terms with each
other, it is possible to construct various representations of the GARCH
models available in the literature, including VECH and diagonal VECH
(Engel and Kroner, 1995), diagonal BEKK (after Baba, Engel, Kraft and
Kroner, 1990), and constant correlation. In what follows, we discuss each of
these in detail.
4.2: The VECH Representation
Under the VECH representation of multivariate GARCH, conditional
variance of each variable, th11and
th22, depends on its own and the other
related variable’s past; the conditional covariance between the two variables,
th12; the lagged squared errors, 2
11 t and 2
12 t ; and the product of the lagged
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73 |
errors,1211 tt . To show this, define expressions for
th11,
th12and
th22as
follows:
2
1213121112
2
11111011 ttttth
122131121211111 ttt hhh
2
1223121122
2
11212012 ttttth
122231122211121 ttt hhh
2
1233121132
2
11313022 ttttth
122331123211131 ttt hhh
A compact matrix representation of these equations takes the form:
2
1,2
1,21,1
2
1,1
333231
232221
131211
30
20
10
,22
,12
,11
t
tt
t
t
t
t
h
h
h
1,22
1,12
1,11
333231
232221
131211
t
t
t
h
h
h
(29)
Showing a rich interaction between tx1 and tx2 .
There are several complications which make it very difficult to estimate the
VECH representation of multivariate GARCH models, despite the ease
with which it can be conceptualized.
The number of parameters necessary to estimate can be extremely large.
Even for a simple two variable GARCH (1,1) model in eqn. 29above, there
are 21 parameters, and the estimation can be quite complicated with more
variables added to the system and if the order of the GARCH process
increases. Moreover, as with univariate GARCH models, it is necessary to
impose restrictions on the parameters of the product of the lagged errors of
the model to ensure the non-negativity of the conditional variances of
individual series, which in practice can however be difficult to do.
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Overcoming these problems involves imposing restrictions on the general
model as in the diagonal VECH model.
4.3: The Diagonal VECH Representation
This is based on the assumption that each conditional variance is equivalent
to that of a univariate GARCH process and the conditional Covariance is
quite parsimonious. In the simplest case of N=2 and 1 qp , the
diagonal representation form of the VECH reduce the number of
parameters to be estimated to nine from 21, and compressed to:
2
1,2
1,21,1
2
1,1
22
12
11
20
12
10
,22
,12
,11
00
00
00
t
tt
t
t
t
t
h
h
h
1,22
1,12
1,11
22
12
11
00
00
00
t
t
t
h
h
h
(30)
Performing the matrix multiplication yields:
1,1111
2
1,11110,11 ttt hh
1,12221,21,11212,12 tttt hh
1,2222
2
1,22230,22 ttt hh
Showing that variances depend solely on past own squared residuals and past values of itself, and each element of the Covariance matrix,
th ,12
depends only on past values of itself and a product of the past values of
tt ,2,1 .
This easy way to estimate diagonal VECH model is handy, but setting all
0 ijij ji devoid the model of interactions among the variables –
the very essence of multivariate GARCH modelling. Indeed, as can be seen,
1,1 t shock affects th ,11and
th ,12but not
th ,22. Moreover, we require that
th ,12be positive definite for all values of
it in the sample space, a
restriction that can be difficult to check, let alone impose during estimation.
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4.4: The BEKK Representation
The BEKK (Baba, Engel, Kraft and Kroner, 1990) multivariate GARCH
model guarantees that that the conditional variances are positive, by forcing
all the parameters to enter the model via quadratic forms. It assumes the
following model fortH :
q
1i
p
1i
iitiiititi0t βHβαεεαξξH 0
The BEKK representation in the N=2 and 1 qp case becomes:
2212
1211
2212
1211
,22,12
,12,11
tt
tt
hh
hh
2212
21112
1,21,21,1
1,21,1
2
1,1
2
1,21,21,1
1,21,1
2
1,1
2221
1211
ttt
ttt
ttt
ttt
2212
2111
1,221,12
1,121,11
2221
1211
tt
tt
hh
hh
Performing the matrix multiplication yields:
1,22
2
121,1212111,11
2
11
2
1,2
2
121,21,12111
2
1,1
2
11
2
12
2
11,11
2
2
ttt
ttttt
hhh
h
2
1,222211,21,122111221
2
1,11211
122211,12
tttt
th
1,1222211,221,11221112211,121211 tttt hhhh
1,22
2
221,1222211,11
2
21
2
1,2
2
221,21,12221
2
1,1
2
21
2
22
2
12,22
2
2
ttt
ttttt
hhh
h
(31)
For N=2 and 1 qp , the BEKK, unlike the VECH, requires only 8
parameters (without the GARCH terms) but 11 parameters with the
GARCH terms inclusive to be estimated and allows for the interaction
effects that the diagonal VECH representation does not. As such the model
allows for the spill-over effects of shocks to the variance of one of the
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variables to the others. In principle, the BEKK representation improves on
both the VECH and diagonal multivariate GARCH representations.
However, despite its improvement over the VECH and diagonal
representations, in practice, the BEKK model has been shown to be quite
difficult to estimate. A large number of parameters in it are not globally
identified, and as such, convergence can be quite difficult to achieve.
Given this, the BEKK model in eqn. 31 is collapsed to a representation
similar to diagonal representation in eqn. 30, becoming diagonal BEKK,
where both ARCH and GARCH terms are diagonal matrices, while the
constant terms are an indefinite matrix. Unlike the diagonal VECH in
eqn.30, diagonal BEKK allows for the interaction of the two process
ARCH and GARCH terms in its conditional covariance. It takes the form:
1,11
2
11
2
1,1
2
11
2
12
2
11,11 ttt hh
1,221,112211
22
2211122211,12 1,21,1 ttt hhh
tt
(32)
1,22
2
22
2
1,2
2
22
2
22
2
12,22 ttt hh
This reduction of the original BEKK shown in eqn.32 reduces the
estimated variance-covariance parameters to 7 from the initial 11.
4.5: The Constant Conditional Correlation (CCC)
Representation
CCC representation of the multivariate GARCH model restricts the
conditional correlation coefficients to be equal to the correlation
coefficients between the variables, which are simply constants. Thus, as the
name suggests, the conditional correlation coefficients are constant over
time.
As such, in the simplest case of N=2 and 1 qp , the CCC model
assume:
1,1
2
1
2
1,1
2
1
2
2
2
1,1 ttt hh
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5.0
,2,112,12 ttt hhh (33)
1,2
2
2
2
1,2
2
2
2
2
2
1,22 ttt hh
The covariance equation entails only one parameter, and the variance terms
need not be diagonalized and the covariance terms are proportional to
5.0
,22,11 tthh .
4.6: Conditional Heteroskedasticity, Unit Roots and
Cointegration
Conditional heteroskedasticity is a common feature of many financial time
series, but as discussed earlier, an assumption of many time series models is
that the error terms are zero mean, homoskedastic, iid random variables.
This makes testing for a unit root and consequently, testing for
cointegrating relationships, in the presence of conditional heteroskedasticity
an important issue. However, developments in time series econometrics
have shown that, asymptotically, the Dickey Fuller (Dickey and Fuller, 1979)
tests are robust to the presence of conditional heteroskedasticity, such as
ARCH and GARCH (see e.g., among others, Phillips and Perron, 1988).
Consequently, in applied work on testing for unit roots or cointegration in
financial time series, it is extremely rare that potential difficulties caused by
presence of conditional heteroskedasticity are considered to be a problem.
4.7: Demonstrating the Estimation of CPI and Exchange Rate
Volatility Within a Bi-variant GARCH(1,1) Framework
Using EViews Screens and Statistical Output Tables
For illustration purposes, we will estimate a simple multivariate GARCH(1,1)
framework in both the BEKK and CCC representations involving the CPI
and Exchange rate i.e. N=2 and 1 qp , i.e. GARCH (1,1) over the
period July 2005 – January 2017, some 139 observations. The data set is
available in COMESA_DATA_2017 folder under monthly.xlsx excel file
as EXR_CPI excel sheet, and is given as Multivariate in the
demonstration EViews screen.
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The first step, as has been emphasized before, is to fit a mean equation, i.e.
a model in which the assumption of time independence of the residuals
holds. Because from economic theory, strong causation is more likely to run
from exchange rate to CPI inflation, the so-called exchange rate pass-
through, we will specify a mean equation in which CPI inflation is assumed
to depend on its lags and lagged exchange rate, while ensuring a sufficient
number of lags or autoregressive parameters that ensures time
independence of the residuals. But prior to the estimation of the mean
equation, it is important that the series, available at such a high frequency
like we have here (monthly) are tested for and adjusted for seasonality. This
is because economic analysis is focused on business cycles, so, performing
an analysis on variables with seasonality would be incorrectly characterizing
cyclical behaviour and the ensuing results would be spurious (Dejong and
Dave, 2007; Nyanzi and Bwire, 2017).
Unlike the dlexr, which we have argued before that if adjusted for
seasonality would contradict the assumption of rational behaviour in
financial markets, particularly because the seasonality here is not regular,
CPI series is particularly a suitable candidate for seasonality adjustment.
DLCPI series is thus tested for seasonal effects using the popular X12
method by the Census Bureau of Statistics, but no evidence of seasonality
was found. This procedure which could be useful in some other
applications is briefly described here-under.
Adjusting the DLCPI series for seasonality effects using the Census X12
method, like any other built in routine procedures in EViews, is an
automated procedure. The method is available only for monthly and
quarterly series for at least 3 full years – descriptions fitting our DLCPI
series at hand. To implement the procedure, highlight and open DLCPI
series in the active multivariate page. In the open series window (excel like
sheet), click Proc, and point the cursor on Seasonal Adjustment in the
drop-down menu of Proc and select Census X12... – a road map shown in the
print screen below.
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Selecting Census X12...brings forth the X12 Options screen, which is also
replicated in the screen print below. In the box for X11 Method – under
Seasonal Adjustment, a choice must be made between the EViews default
of Multiplicative and Additive – the most popular two adjustment methods
in applied time series. Multiplicative method applies when the series to be
adjusted is nonstationary and the series values are strictly non-negative.
Additive method applies when the series to be adjusted is in stationary form,
but in addition, the series values must be positive. DLCPI in our application
here is stationary, but with some negative entries. Given this, we check
Additive option, while accepting as an appropriate price to pay for the few
negative observations, which certainly will be lost in the process. With the
exception of changing the X11 method to favour, the rest of the default
options remain as given, noting that under Component Series to Save, the
Base name for the series to be adjusted is given as dlcpi and the Final
seasonally adjusted series will appear in the work space window with
extension_SA(where suffix SA is used to mean seasonally adjusted). This
box is by default checked.
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Click OK to implement the procedure. Based on the results (table is very
large to be reproduced here), all available tests for seasonality, including the
F-tests for seasonality; Nonparametric Test for the Presence of Seasonality
Assuming Stability; Moving Seasonality Test; and Test for the presence of
residual seasonality, all reveal no evidence of seasonality at conventional
levels of statistical significance.
With this formal proof of no seasonality effects in DLCP, we proceed to fit
the mean equation – with both dlexr and dlcpi unadjusted for seasonal
effects. We began with k=4 autoregressive parameters without MA
component. To proceed, in the active multivariate page of EViews window,
click on Quick at the top of the command window, and in the drop-down
menu, select Estimate Equation...by a mouse click. In the resulting
Equation Estimation window, we then supply in theEquation
specification, a specification of the form: dlcpi c dlcpi(-1 to -4) dlexr(-1 to -4), i.e.
in the order of dependent variable, constant and a list of lags explanatory
variables, a constant inclusive. Execution of this using the Least Squares
method generates results in Table 7.
............................................................. Modelling Conditional Volatility in a Multivariate Framework
81 |
Table 7: The General bi-variant mean model for Inflation
Variable Coefficient Std. Error t-Statistic Prob. C 0.002173 0.000830 2.616845 0.0100 DLCPI(-1) 0.117352 0.090634 1.294800 0.1978 DLCPI(-2) 0.267982 0.090318 2.967088 0.0036 DLCPI(-3) 0.190746 0.091682 2.080526 0.0395 DLCPI(-4) 0.039111 0.090599 0.431698 0.6667 DLEXR(-1) 0.056380 0.022538 2.501589 0.0137 DLEXR(-2) -0.006226 0.025167 -0.247397 0.8050 DLEXR(-3) 0.008312 0.024515 0.339058 0.7351 DLEXR(-4) -0.012151 0.022248 -0.546174 0.5859
S.E. of regression 0.006 Durbin-Watson stat 1.997
Breusch-Godfrey Serial Correlation LM Test:
F(2,123) = 0.928(0.398);
22 = 1.992(0.369)
Notes: 134 included observations
In the results, about four coefficients, c, dlcpi(-2), dlcpi(-3) and dlexr(-1) are
significantly different from zero. Asis the norm in econometric modelling
process, we have to drop sequentially through general to specific approach,
most insignificant lags, but first, those whose t-values are less or equal to
one. Following this procedure, the 2nd, 3rd and 4th lags of dlexr and the 4th lag
of dlcpi are dropped from the specification. Even after this reduction, the
1st lag to dlcpi still remains insignificant so subsequently, it is dropped in the
re-specification. This reduction process yields results in Table 8, which are
the most plausible estimate of the mean equation. Consistent with the
Durbin-Watson stat (1.77), formal Breusch-Godfrey Serial Correlation LM
Test does not point to any remaining autocorrelation in the model residuals.
Table 8: Estimates of the mean equation
Variable Coefficient Std. Error t-Statistic Prob.
C 0.002603 0.000753 3.456267 0.0007
DLCPI(-2) 0.302947 0.081914 3.698361 0.0003 DLCPI(-3) 0.232820 0.081549 2.854972 0.0050 DLEXR(-1) 0.059343 0.019757 3.003597 0.0032
S.E. of regression 0.005806 Durbin-Watson stat 1.770962
Breusch-Godfrey Serial Correlation LM Test:
F(2,129) = 1.059(0.350);
22 = 2.181(0.336)
Notes: p-values in parentheses; Included observations=135
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Therefore with the specification of the mean equation ably established as in
Table 8, we now turn to the implementation of multivariate GARCH(1,1)
framework in both the BEKK and CCC representations, but beginning with
the former.
To proceed, whilst in EViews work space, highlight the two variables in the
order dlcpidlexr. With the cursor in the highlighted area, right click and
choose Open as System…The screen that pops up, as shown in the screen
print below,is where we have to provide the system specification.
Like the VAR, each of dlcpi and dlexr are specified as potentially endogenous
or dependent variables. Under Regressors and AR() terms, enter in the
comb box for Equation Specific Coefficients, cdlcpi(-2 to -3) dlexr(-1),as
has indeed been determined beforehand. Leaving all the other entries in the
rest of the screen as default, click OK and as shown in the print screen
below, are the coefficients for the two mean equations. This is how the set-
up of the system coefficients for the mean equation will look like so it is
important this is familiar.
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83 |
In this very window, click on Estimate, which should lead you to the
System Estimation window, in which we provide information as follows:
Under Estimation method, choose from the drop-downlist ARCH as the
estimation method of choice (the default is OLS). You will then see that
under ARCH model specification/Model type, we have all the three
available specifications of multivariate GARCH, namely Diagonal VECH,
CCC and diagonal BEKK, all accessed through the drop-down menu(the
default is Diagonal VECH). We will implement diagonal BEKK and CCC,
but first BEKK. So, we select Diagonal BEKK as shown in the screen print.
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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All the other comb entries are obvious. Click Options to check the box for
Bollerslev-Wooldridge SE, if in the earlier versions of EViews or leave
defaults as they are if using latest EViews versions and click OK. The
resultant estimation output for both the mean and Variance Covariance
equations (Table 9), is reproduced here, but tabulated in a useable form in
Table 10.
We have earlier on shown that
DLCPI = C(1) + C(2)*DLCPI(-2) + C(3)*DLCPI(-3) + C(4)*DLEXR(-1)
DLEXR = C(5) + C(6)*DLCPI(-2) + C(7)*DLCPI(-3) + C(8)*DLEXR(-1)
So, the positioning and ordering of the coefficients C(1) to C(8), in Table
10given in the computer generated output in Table 9,should be quite
straight forward.
............................................................. Modelling Conditional Volatility in a Multivariate Framework
85 |
Table 9: EViews output for Diagonal BEKK Covariance specification
Coefficient Std. Error z-Statistic Prob. C(1) 0.002647 0.000771 3.434558 0.0006
C(2) 0.247868 0.093021 2.664659 0.0077
C(3) 0.196030 0.083824 2.338594 0.0194 C(4) 0.042519 0.013951 3.047669 0.0023
C(5) 0.005262 0.002081 2.528327 0.0115
C(6) -0.837201 0.304761 -2.747078 0.0060
C(7) 0.252033 0.239273 1.053332 0.2922 C(8) 0.464209 0.089039 5.213522 0.0000
Variance Equation Coefficients C(9) 1.38E-05 7.27E-06 1.899092 0.0576
C(10) 1.90E-06 4.82E-06 0.394011 0.6936
C(11) 4.15E-05 1.84E-05 2.261980 0.0237
C(12) 0.598470 0.168347 3.554985 0.0004 C(13) 0.535588 0.115586 4.633665 0.0000
C(14) 0.514698 0.266927 1.928237 0.0538
C(15) 0.832323 0.039751 20.93824 0.0000
Log likelihood 840.3703 Schwarz criterion -11.90490
Avg. log likelihood 3.112483 Hannan-Quinn criter. -12.09653
Akaike info criterion -12.22771
Equation: DLCPI = C(1) + C(2)*DLCPI(-2) + C(3)*DLCPI(-3) + C(4)*DLEXR(-1)
R-squared 0.232773 Mean dependent var 0.006129
Adjusted R-squared 0.215203 S.D. dependent var 0.006633
S.E. of regression 0.005876 Sum squared resid 0.004524
Durbin-Watson stat 1.702365
Equation: DLEXR = C(5) + C(6)*DLCPI(-2) + C(7)*DLCPI(-3) + C(8)*DLEXR(-1)
R-squared 0.208410 Mean dependent var 0.004929
Adjusted R-squared 0.190282 S.D. dependent var 0.025854
S.E. of regression 0.023264 Sum squared resid 0.070900
Durbin-Watson stat 1.921307 Covariance specification: Diagonal BEKK
GARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1 M is an indefinite matrix
A1 is a diagonal matrix
B1 is a diagonal matrix
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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Transformed Variance Coefficients Coefficient Std. Error z-Statistic Prob. M(1,1) 1.38E-05 7.27E-06 1.899092 0.0576
M(1,2) 1.90E-06 4.82E-06 0.394011 0.6936
M(2,2) 4.15E-05 1.84E-05 2.261980 0.0237 A1(1,1) 0.598470 0.168347 3.554985 0.0004
A1(2,2) 0.535588 0.115586 4.633665 0.0000
B1(1,1) 0.514698 0.266927 1.928237 0.0538
B1(2,2) 0.832323 0.039751 20.93824 0.0000
The variance equation coefficients, C(9) – C(15) and Transformed
Variance Coefficients: M(1,1), M(1,2), M(2,2), A1(1,1), A1(2,2), B1(1,1) and
B1(2,2) under the Covariance specification: Diagonal BEKK (appeared
in bold in Table 9) are the same. In other words, C(9) = M(1,1); C(10) =
M(1,2); C(11) = M(2,2); C(12) = A1(1,1); C(13) = A1(2,2); C(14) = B1(1,1)
and C(15)= B1(2,2). These correspond directly to the coefficients of the
system in eqn. 32. For brevity, mapping the system of the diagonal BEKK
model in eqn. 32 in light of the estimated output is as straight forward as:
1,11
932
2
11
2
1,1
932
2
11
932
2
12
2
11,11
)14()1,1(1
)12()1,1(1)9()1,1(
t
TABLEEQN
t
TABLEEQNTABLEEQN
t
hCB
CACMh
1,22
932
2
22
2
1,2
932
2
22
932
2
22
2
12,22
)15()2,2(1
)13()2,2(1)11()2,2(
t
TABLEEQN
t
TABLEEQNTABLEEQN
t
hCB
CACMh
22
932
2211
932
122211,12
1,21,1)2,2(1*)1,1(1
)10()2,1(
tt
TABLEEQN
TABLEEQN
t
AA
CMh
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87 |
1,221,11
932
2211 )2,2(1*)1,1(1
tt
TABLEEQN
hhBB
The coefficients to 2
1,2
2
1,1 tt and to1,221,11 tt hh , i.e. )2,2(1*)1,1(1 AA ; and
)2,2(1*)1,1(1 BB , respectively in the covariance equation, th ,12
are not
computer generated but are a simple product of the corresponding
coefficients in th ,11 and th ,22 , and are indicated as RESID1(-1)*RESID2(-1)
and Cov1_2(-1) in Table 10. All computer generated coefficients to the
ARCH, i.e. RESID(-1) and GARCH, i.e. GARCH(-1) terms are in quadratic
form as in eqn. 32.The computer-generated output in Table 9 is tabulated
in a useable form in Table 10.
Table 10: Diagonal BEKK estimates for bi-variant GARCH(1,1)
Estimated system mean Equations
Endogenous variable Cons DLCPI(-2) DLCPI(-3) DLEXR(-1) DLCPI
0.003* (3.434)
0.248* (2.665)
0.196** (2.339)
0.043* (3.048)
DLEXR
0.005** (2.528)
-0.837* (-2.747)
0.252 (1.053)
0.464* (5.214)
Variance and covariance Equations
Cons RESID1(-
1)^2 RESID2(-1)^2
GARCH1(-1)
GARCH2(-1)
RESID1(-1)* RESID2(-1)
COV1_2(-1)
GARCH1
0.000*** (1.899)
0.598* (3.555)
0.515*** (1.928)
COV1_2
0.000 (0.394)
0.321* (16.473)
0.428* (40.374)
GARCH2
0.000** (2.262)
0.536* (4.634)
0.832* (20.938)
Notes: Covariance specification assumption: Diagonal BEKK. ARCH and GARCH coefficients in the variance covariance equations are in quadratic form of the transformed variance coefficients while cross-terms are products of the transformed variance coefficients. In parentheses are z-statistics, but in product form for cross coefficient products, in the lower panel of the table. Robust standard errors and covariance are due to Bollerslev-Wooldridge, and the presample covariance assumption: backcast (parameter =0.7).ARCH and GARCH terms are diagonal matrices, while the constant terms are an indefinite matrix. Asterisks *, **, and *** represent 1 percent, 5 percent and 10 percent levels of significance, respectively.
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Overall, the estimates of the multivariate model seem plausible in both
theory and statistics sense. Each of the coefficients to the autoregressive
parameters (coefficients of the mean equations) is less than one in absolute
terms, which is consistent, in general, with the restriction on parameter in
eqn. 5. Therefore, the system is covariance stationary. In both CPI and
exchange rate depreciation mean equations, the coefficients to the second
lag of CPI, i.e. DLCPI(-2) and that on the lagged exchange rate depreciation,
i.e. DLEXR(-1) are significantly different from zero, and so, is the
coefficient to the 3rd lag of CPI, i.e. DLCPI(-3),in the CPI mean equation,
which is very intuitive.
In the CPI mean equation, significant coefficients on lagged CPI, i.e. 0.248
for DLCPI(-2) and 0.196 for DLCPI(-3)reflects inflation inertia or
persistence in inflation, i.e. prices are sticky downwards. The highly
significant coefficient of 0.043 on the first lag of exchange rate depreciation
(DLEXR(-1)) implies pass-through of exchange rate to domestic prices but
with a lag. This is consistent, in terms of direction and transmission, with
the exchange rate pass-through estimates for Uganda based on SVAR in a
wider context of key drivers of core inflation (Bwire, Opolot and Anguyo.,
2013, p. 41-68, in JSEM). Their estimate puts the pass-through at about 0.48
percentage points, with full impact being realized between 3 – 4 quarters.
These evidences are not surprising largely because small open economies,
without exception, experiences relatively high pass-through of exchange rate
depreciation to domestic inflation.
In the mean equation for exchange rate depreciation, the negative
significant coefficient of the second lag of CPI, i.e. -0.837 for DLCPI(-2)
implies that heightened domestic inflationary pressures depreciates the
exchange rate as this hurts the export sector while at the same time
promotes imports. A fall in exports reduces the inflow of foreign exchange
while an increase in imports puts pressure on foreign exchange demand. A
consequence of the resulting mismatch in supply of and demand for foreign
exchange, in a market driven environment, is heightened depreciation
pressures. The significant coefficient to the first lag of exchange rate
(DLEXR(-1))reflects persistence in the exchange rate movements.
............................................................. Modelling Conditional Volatility in a Multivariate Framework
89 |
The conditional variance parameters in the conditional variance equations
for inflation and exchange rate depreciation, i.e. 0.515 for GARCH1(-1) and
0.832 for GARCH2(-1) are positive and less than one, which is consistent
with the restrictions on j in Eqn. 22, and are quite reasonably high. This
depicts the high volatility persistency inherent in inflation dynamics and the
exchange rate movements. Decomposing 1,1GARCH process into past
squared forecast errors (RESID(-1))^2and past forecast variances
(GARCH(-1)), both RESID1(-1)^2 and RESID2(-1)^2 and GARCH(-1)
and GARCH(-2) are significant in the respective conditional variance
specifications. As shown by the asterisks, the past squared forecast errors
are significant at 1 percent, which in effect, reflect strong evidence of
volatility clustering in either case. The conditional volatility persistence
coefficients are 0.83 and 0.52 for exchange rate movements and inflation,
respectively, and are respectively significant at the 1 percent and 10 percent
levels of significance (see the asterisks). The persistence in exchange rate
volatility (o.83) is therefore more pronounced than that on inflation (0.52).
Importantly, the ARCH, i.e. RESID1(-1)*RESID2(-1) and GARCH, i.e.
COV1_2(-1) parameters of the conditional covariance equation are 0.321
and 0.428, respectively, and each of this is significantly different from zero
at the 1 percent level. This reflects indeed the spill over effects from shocks
in the system variables. One can therefore argue, in view of the mean
equation results, that volatility in exchange rates and that in inflation triggers
volatility in each other (correlation is both ways, i.e. lags of dlexr are
significant in the dlcpi equation and like- wise, lags of dlcpi are significant in
the dlexr equation).
Finally, we can follow the demonstration in the screen prints below to
generate a graphical exposition of the estimated variance covariance
structure. Whilst in the estimated output window, click on View and in the
drop-down menu, choose and click on Conditional Covariance… The
Conditional Variance window pops up as in the screen print. As can be
seen, we will be displaying Covariance (default in the drop-down menu) for two
variables, dlcpi and dlexr in graphs format over the sample period 2005m11
to 2017m01.
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The good news is that all these entries are already default and all we have to
do is to be in agreement, by clicking OK – to generate variance covariance
structure depicted in Figure 10.
Figure 11: Measured Conditional Covariance
.00000
.00005
.00010
.00015
.00020
.00025
.00030
05:1
1
06:0
1
07:0
1
08:0
1
09:0
1
10:0
1
11:0
1
12:0
1
13:0
1
14:0
1
15:0
1
16:0
1
Var(DLCPI)
-.0002
-.0001
.0000
.0001
.0002
.0003
.0004
05:1
1
06:0
1
07:0
1
08:0
1
09:0
1
10:0
1
11:0
1
12:0
1
13:0
1
14:0
1
15:0
1
16:0
1
Cov(DLCPI,DLEXR)
.000
.001
.002
.003
.004
05:1
1
06:0
1
07:0
1
08:0
1
09:0
1
10:0
1
11:0
1
12:0
1
13:0
1
14:0
1
15:0
1
16:0
1
Var(DLEXR)
............................................................. Modelling Conditional Volatility in a Multivariate Framework
91 |
Next, we estimate the same model but under the Covariance specification
assumption of Constant Conditional Correlation (CCC), following a similar
route as in Diagonal BEKK, the only departure being that we now choose
Constant Conditional Correlation in the drop-down menu of Model type in the
ARCH model specification shown in the screen print earlier. The resulting
EViews output is provided here in Table 11, which as with diagonal BEKK,
is organized in a usable form in Table 12.
Note that as with diagonal BEKK, it follows from the illustrations earlier
that
DLCPI = C(1) + C(2)*DLCPI(-2) + C(3)*DLCPI(-3) + C(4)*DLEXR(-1)
DLEXR = C(5) + C(6)*DLCPI(-2) + C(7)*DLCPI(-3) + C(8)*DLEXR(-1)
So that the positioning and ordering of the coefficients C(1) to C(8), in
Table 11, doesn’t have to be problematic and is as straight forward as
shown in Table 12.
Table 11: E-Views output for Constant Conditional Correlation Covariance specification
Coefficient Std. Error z-Statistic Prob. C(1) 0.003035 0.000762 3.982980 0.0001
C(2) 0.216693 0.097286 2.227385 0.0259
C(3) 0.158123 0.086229 1.833758 0.0667
C(4) 0.048551 0.014412 3.368877 0.0008 C(5) 0.003255 0.001916 1.698930 0.0893
C(6) -0.718685 0.278260 -2.582787 0.0098
C(7) 0.410830 0.244465 1.680525 0.0929
C(8) 0.493501 0.088435 5.580404 0.0000 Variance Equation Coefficients C(9) 9.41E-06 4.95E-06 1.901835 0.0572
C(10) 0.311793 0.180619 1.726253 0.0843
C(11) 0.413061 0.197150 2.095164 0.0362
C(12) 2.64E-05 1.67E-05 1.580114 0.1141
C(13) 0.449382 0.151510 2.966018 0.0030 C(14) 0.616743 0.067245 9.171604 0.0000
C(15) 0.107338 0.080460 1.334054 0.1822
Modelling and Forecasting Volatility in Financial Markets Using E-Views .........................
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Log likelihood 843.6630 Schwarz criterion -11.95368
Avg. log likelihood 3.124678 Hannan-Quinn criter. -12.14531
Akaike info criterion -12.27649
Equation: DLCPI = C(1) + C(2)*DLCPI(-2) + C(3)*DLCPI(-3) + C(4)*DLEXR(-1)
R-squared 0.223750 Mean dependent var 0.006129 Adjusted R-squared 0.205974 S.D. dependent var 0.006633
S.E. of regression 0.005911 Sum squared resid 0.004577
Durbin-Watson stat 1.678926
Equation: DLEXR = C(5) + C(6)*DLCPI(-2) + C(7)*DLCPI(-3) + C(8)*DLEXR(-1)
R-squared 0.205090 Mean dependent var 0.004929
Adjusted R-squared 0.186886 S.D. dependent var 0.025854
S.E. of regression 0.023313 Sum squared resid 0.071198 Durbin-Watson stat 1.965731
Covariance specification: Constant Conditional Correlation
GARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + B1(i)*GARCH(i)(-1)
COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j)) Transformed Variance Coefficients Coefficient Std. Error z-Statistic Prob. M(1) 9.41E-06 4.95E-06 1.901835 0.0572
A1(1) 0.311793 0.180619 1.726253 0.0843
B1(1) 0.413061 0.197150 2.095164 0.0362
M(2) 2.64E-05 1.67E-05 1.580114 0.1141
A1(2) 0.449382 0.151510 2.966018 0.0030 B1(2) 0.616743 0.067245 9.171604 0.0000
R(1,2) 0.107338 0.080460 1.334054 0.1822
The variance equation coefficients, C(9) – C(15) and Transformed
Variance Coefficients: M(1), A1(1), B1(1), M(2), A1(2), B1(2), R(1,2) under
the Constant Conditional Correlation are the same. In other words, C(9)
= M(1); C(10) = A1(1); C(11) = B1(1); C(12) = M(2); C(13) = A1(2); C(14)
= B1(2) and C(15)= R(1,2). These correspond directly to the coefficients of
the system in eqn. 33, which for brevity can be mapped as follows:
............................................................. Modelling Conditional Volatility in a Multivariate Framework
93 |
1,1
1133
2
1
2
1,1
1133
2
1
1133
2
2
2
1,1 )11()1(1)10()1(1)9()1(
t
TABLEEQN
t
TABLEEQNTABLEEQN
t hCBCACMh
1,2
1133
2
2
2
1,2
1133
2
2
1133
2
2
2
1,2 )14()2(1)13()2(1)12()2(
t
TABLEEQN
t
TABLEEQNTABLEEQN
t hCBCACMh
5.0
,2,1
1133
12,12 )15()2,1( tt
TABLEEQN
t hhCRh
Thus, an easy to follow form of computer generated output in Table 11 is
given in Table 12.
Table 12: CCC estimates for bi-variant GARCH(1,1)
Estimated system mean Equations
Endogenous variable Cons DLCPI(-2) DLCPI(-3) DLEXR(-1)
DLCPI 0.003* (3.983)
0.217** (2.227)
0.158*** (1.834)
0.049* (3.369)
DLEXR 0.003***
(1.698) -0.719* (-2.583)
0.411*** (1.681)
0.494* (5.580)
Variance and covariance Equations
Cons RESID1 (-1)^2
RESID2 (-1)^2
GARCH1 (-1)
GARCH2 (-1)
SQRT(GARCH1* GARCH2
GARCH1
0.000*** (1.902)
0.312*** (1.726)
0.413** (2.095)
COV1_2
0.107 (1.334)
GARCH2 0.000
(1.580) 0.449* (2.966)
0.617* (9.172)
Notes: Covariance specification assumption: Constant Conditional Correlation (CCC). In
parentheses are z-statistics. Robust standard errors and covariance are due to Bollerslev-
Wooldridge, and the presample covariance assumption: backcast (parameter =0.7).Asterisks *,
**, and *** represent 1 percent, 5 percent and 10 percent levels of significance, respectively.
These results are consistent with those estimated under the Covariance
specification assumption of diagonal BEKK, except that here, the spill over
effects are absent despite the strong causal effect of exchange rate
depreciation on CPI inflation and vice versa in the estimated system mean
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equations (upper window of the results table). I would be hard pressed
therefore to prefer this to the former, in the context of Uganda, and indeed
many COMESA small open economies, which are ideally vulnerable to both
domestic and external shocks.
Exercise 4
We have demonstrated the estimation of GARC(1,1) in a multivariate
environment, in particular, estimation of BEKK and CCC models. As with
exercise 3, the sheet named multiple in guideline exercises file, there are
three series, exr, cpi and 91-day T-bill, some 143 observations for each of
the variables for July 2005 to May 2017. In this exercise, consider a vector
of two variables: nominal exchange rate and the risk free 91-day T-bill rate
to perform the following tasks:
1. Use your economic intuition to fit a specification for a single equation
involving nominal exchange rate and 91-day T-bill
2. Based on (1), estimate a bi-variate GARC(1,1) model in
i). diagonal BEKK framework
ii). CCC representation framework
iii). Interpret the results and show how these compare across the two
competing analytical methodologies.
3. Plot conditional covariance structure arising from both the estimated
BEKK and CCC frameworks.
Chapter 5
Forecasting Conditional Volatility and
Forecast Performance Evaluation
Consider the GARCH (1,1) model:11
2
110 ttt hh
5.1: One-step-ahead Forecast
Simply update th by one period and since 2
t and th are known in period t ,
one–step-ahead forecast is:
ttt hh 1
2
101 .
5.2: The j-step-ahead Forecasts
First, we know that:
ttt hv22
Updating this by j periods gives:
jtjtjt hv 22
Taking the conditional expectations gives:
)( 22
jtjttjtt hvEE
The fact that 0,cov jtjt hv
and12 jttvE
, it follows that:
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jttjtt hEE 2 (34)
Update eq. 23 by j-periods and take the conditional expectation to obtain:
11
2
110 jttjttjtt hEEhE
Combine this with eq. 34 to get:
1110 )( jttjtt hEhE (35)
Using eq. 35 and given 1th , it is possible to obtain the j-steps-ahead forecast
of the conditional variance recursively.
t
jj
j
jtt hhE )(...1 11
1
11
2
11110
Providing 111 , the conditional forecast of jth will converge to the
long-run value
11
0
1
tEh
5.3: Forecast Evaluation
Once an appropriate model (one in which all ideal conditions hold) has
been chosen and all the diagnostic tests are done, the next step is to test for
accuracy of the forecasts. Several tests are used to determine the accuracy of
the forecasting model, including;
The Mean Absolute Error (MAE) =
n
t
tn 1
1
The Mean Square Error (MSE) =
n
t
tn 1
21 .
The lower the values of MAE and MSE, the better the forecasts
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Mean Absolute Percentage Error (MAPE) =
ht
Tt t
tt my
yy
1
ˆ100
Where ty and
ty are the actual and forecast values, respectively and m is
the number of observations. MAPE is used to compare the fits of different
forecasting and smoothing methods. Smaller values usually indicate a better
fitting model.
Root Mean Squared Error (RMSE) = 21
1
1
T
j
jtjt AFT
Where A represents actual values and F represents the forecasts. The RMSE
is representative of the size of the error because it is measured in the same
units as the data. A lower RMSE signifies better/more accurate forecasts
Theil’s Forecast Accuracy (U) =
5.0
1
2
5.0
1
2
n
i
i
n
i
ii
A
AP
Where P and A stand for a pair of predicted and observable values. If the
theil Coefficient is 0 then, P=A, therefore perfect forecasts. If the theil
coefficient is greater than 1, then the forecasts are not good. In other words,
the closer the theil coefficient is to zero, the better the forecasts.
5.4: Performing the Exchange Rate Forecast in a Univariate
GARCH(1,1) Framework Using EViews Screens and
Statistical Output Tables
The first thing to note is that the data from which GARCH(1,1) model in
column 2 of Table 5 derives contains observations spanning 1997M01 –
2017M01. This historical data is vital for performing in-sample forecast for
forecast evaluation purposes. To be able to perform out-of-sample forecast,
we will need to resize the sample to include the forecast horizon, which in
this exercise, shall be set at 9 months from the latest available observation.
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This then requires that the end date is set to 2017M10, a resizing which is
easily done from within the EViews work space window (this is
demonstrated later).
The first step to this task therefore, is to re-estimate the GARCH(1,1)
model in in column 2 of Table 5. As it is better having to forecast the
actual exchange rate series, we will this time re-estimate the GARCH(1,1)
model in column 2 of Table 5, butusing the following command for the
mean equation d(log(exr)) c ar(1) – where d(log(exr)) is the first difference of
the log of exr and c and ar(1) are the constant and the first autoregressive
parameters to be estimated. The rest of the provisions, including the
appropriate boxes in Options remain unchanged as described in the steps
leading to the results in Table 5 above. This alteration in the mean equation
command should appear as in the screen print here-under.
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With these entries, press the enter button on your key board to replicate the
exact results in column 2 of Table 5, and which for purposes of describing
the route to forecasting, the screen print is provided here-under.
I will now describe the route to performing in-sample forecast. Whilst in the
GARCH(1,1) output window (shown in the screen shot above), click on the
Forecast icon and as you will see in the Forecast dialog box (also
provided below) which pops up, EViews displays the necessary information
about the forecast.
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Ensure, as in default, the Series to forecast is EXR. The other alternative,
D(LOG(EXR)), would forecast month-on-month depreciation, but in log
form. Implicitly, the modeller would be required to make backward
calculations, to convert the resulting depreciation numbers to actual
exchange rate series. In addition to choosing the Series to forecast, i.e. EXR
(repeated for emphasis), we have to provide the following information:
Forecast name, which by default is given as exrf, where f is forecast. You
can however change this, but remember to make it different from that of
the dependent variable else the dependent variable is over written. Note that
the S.E. dialog box is given as optional and is indeed black. However,
because of the forecast uncertainty and the need as such to indicate 2
confidence bands, we will declare in this box, exrf_se.
Forecast sample: We want to perform, first, in-sample forecast to judge how
accurate our out-of-sample forecasts are likely to be. To produce
particularly appealing forecast, which is distinct from history, we will need
to adjust the data forecast range say to 2011M01 2017M01(which is part of
the series history) and is easily executed within the active output window.
Simply minimize (do not close) the current active GARCH(1,1) output
window and get back to the work space window. Double click on range in
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the series window and adjust the Start to 2011M01. Note that the resized
sample is well within our original sample data. Click OK (you will see this
resizing removes 168 historical observations – which you accept by
checking Yes). Here after, make active the output window again. With this,
the Forecast sample, adjusts to 2011m01 2017m01.
Forecasting Method, a choice has to be made between Dynamic and Static
forecast methods. Dynamic calculates dynamic, multi-step forecasts starting
from the first period in the forecast sample, and is available only when the
estimated equation contains dynamics, i.e. lagged terms of the dependent
variable. Static calculates a sequence of one-step ahead forecasts, using the actual
rather than forecasted values for lagged dependent variable. Both methods
perform in-sample forecasts, but the question of which of the two
outperforms the other is an empirical one. Because we will be performing
out-of-sample forecast, we will choose the default, Dynamic forecast, and
also select Coef uncertainty in S.E. calc, to allow for some degree of
forecast uncertainty, as having accurate forecasts are a wish but never a
reality in the modelling world.
Forecast standard error (S.E) measures the forecast variability, and are
majorly due to two sources of forecast uncertainty. First, is the innovation
uncertainty, which arises because the innovationst in the equation used to
forecast are unknown for the forecast period, but while performing
forecasts, EViews replaces this with t expectations. While it is true by
assumption that 0tE , we know that the individual values of t are
non-zero – which suggests that the larger the variation in the individual
residuals, the greater the overall error in the forecasts. Second, is coefficient
uncertainty, which arises because the estimated coefficient deviates from
the true coefficient in a random fashion. Coefficient uncertainty therefore
measures the precision with which the estimated coefficient measures the
true coefficient . Besides these two, an additional uncertainty is generated
when lags of the dependent variable are used as explanatory variables. These
uncertainties, collectively or individually, make point forecasts less realistic
and is a reason why in applied work, we report interval forecasts – informed
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by the S.E. Interval forecasts are given with 2 S.E bands, which provides
approximately 95 percent forecast interval. In other words, providing the
forecasting model is well specified and is evaluated as relatively accurate, the
forecast of the actual value of the dependent variable will fall inside these
bounds 95 percent of the time. This is much more nicely illustrated using
fan charts, but for technical reasons entirely associated with versions of
EViews prior to version 9.5, will not be produced in this guideline.
Output - shows us how to see the forecast output, as a graph and numerical
forecast evaluation. The latter is available only for in-sample forecasts.
With these specifications, click OK to display in-sample forecast results.
Figure 12: In-the-Sample forecast performance
The top graph is the forecast of exr (exrf) from the mean equation with 2
S.E bands. The second graph is the forecast of the conditional variance. As
can be seen, the Theil Inequality coefficient is 0.037, which is not only less
than one but also extremely low. The other ratios, namely RMSE (212.372),
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
2011 2012 2013 2014 2015 2016
EXRF ± 2 S.E.
Forecast: EXRF
Actual: EXR
Forecast sample: 2011M01 2017M01
Adjusted sample: 2011M03 2017M01
Included observations: 71
Root Mean Squared Error 212.3724
Mean Absolute Error 166.9608
Mean Abs. Percent Error 5.979584
Theil Inequality Coefficient 0.036861
Bias Proportion 0.035133
Variance Proportion 0.136396
Covariance Proportion 0.828471
.00060
.00065
.00070
.00075
.00080
.00085
.00090
2011 2012 2013 2014 2015 2016
Forecast of Variance
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MAE (166.961) and MAPE (5.979) may appear elevated, but relative to the
units of level EXR in which they are being measured, they are by all means
very low too. Indeed, as with Theil Inequality coefficient, the Bias (0.035)
and Variance (0.136) proportions are very close to zero and Covariance
proportion (0.828) is less than one. All these suggest relatively accurate
forecasts. In other words, the in-sample forecast performance seems
satisfactory, but this could be enhanced, especially by looking at forecast of
the same from other competing models. However, a key limitation of
ARIMA class of models is that they are known to have a very long memory
so that the resulting forecast, as is indeed the case herein, only seem sensible
over a very short horizon, while over the long term, they tend to revert to
the long-run average, provided in eqn. 1, 1 . The forecasts would be
explosive, i.e. not revert to the long-run steady state or average, if in the
estimated model used to generate forecasts 1 . This would be true if the
d.g.p used in the estimation of the model used for forecasting, is
nonstationary or contains a unit root or unit roots.
We now need to compare actual and forecast values of EXR, with
confidence bands around the forecast. To do this, first minimize (do not
close) the output window above and thereafter, perform several additional
important calculations. The first of this involve calculating the lower and
upper confidence bands around the forecast. Turning to the work space, we
see two additional series, exrf and exrf_se, in the series window. Note that
exrf_se series is only available for the forecast period of 2011M01 to
2017M01 and is NA elsewhere. We use the two series, exrf and exrf_se to
construct 2 confidence bands around the forecast, where (+) is the upper
band (ub) and (-) is the lower band (lb). To do this, execute in the command
window the following commands, but one at a time: genr
exrf_ub=exrf+1.96*exrf_se(where ub=upper band) and then genr
exrf_lb=exrf-1.96*exrf_se (where lb=lower band) and +/- 1.96 is the
statistical approximation of +/- 2 S.E. Once this is done, you will need
again to resize the historical EXR series, which is easily achieved through
the Range operation in the work space window following the procedure
described earlier. Note that in here, I do adjust the Start date back to
1997M01 and through the edit window, paste in the historical EXR series
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for the period 1997M01 to 2010M12. With this, open as a group and graph
exr (actual), exrf (the forecast), exrf_ub (upper band confidence interval) and
exrf_lb (lower band confidence interval), adjusting the sample say from
2008M01 to 2017M01 (which makes the graph even more appealing). This
is done by checking and adjusting the sample period in Sample icon of this
very graph window.
Figure 13: Actual and in-the-Sample forecast with confidence bands
This shows that the actual exchange rate, in 2011M03, which is exactly 3
months from the start of the forecast horizon, could fall in the range of Shs
2,253 – Shs 2,453 per USD, with the actual point forecast of Shs 2,353 per
USD, which is the same as the actual exchange rate corresponding to that
time. Extending this to 6 months into the forecast horizon, the forecast
reveals that the exchange rate could fall in the range of Shs 2,114 – Shs
2,666 per USD. The point forecast as of 2011M06 is Shs 2,390 per USD,
which is off the corresponding actual exchange rate of Shs 2,461 per USD,
but within the upper band. The quality of the forecast clearly deteriorates as
the forecast horizon increases – indeed as has already been discussed.
Next, if we can make active the output window for Fig. 12 (recall this was
minimized), we will choose Proc within this very active window and
navigate through the drop-down menu to Make GARCH Variance Series...,
1,000
2,000
3,000
4,000
5,000
6,000
2008
:01
2009
:01
2010
:01
2011
:01
2012
:01
2013
:01
2014
:01
2015
:01
2016
:01
EXREXRFEXRF_LB
EXRF_UB
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for which we will retain the default name as garch01. Click OKto complete
the procedure. After this series has been generated and now part of the
series in the variable window, double click on it (garch01) and you will see
that these forecasts cover the period for which exr series is observed and is
exactly as in Figure 10.
We now turn to performing out-of-sample forecast. Again, the first thing
we do is to ensure we have exactly the same model for the results in column
2 of Table 5. Minimize these results window and adjust the data Range, in
the work space window so we accommodate the out-of-sample forecast
period, here set at 9 months ahead of 2017M01. Double click on the Range
and adjust the End date to 2017M10 and click OK. EViews will show you
the number of observations; in this case 09 that have been added, and if in
agreement, click yes. If you recall the spread sheet for exr series, you will
see that the data points for the period 2017M02 – 2017M10 are all indicated
as NA. These are the out-of-sample data points that we have to forecast.
After adjusting for the data range, recall or make active the estimated output
window and check Forecast icon as before. Provide for exrf_se in the S.E
dialog box, as before, and edit the entry for Forecast sample to 2017M02
2017M10. Note that the forecast sample may or may not overlap with the
sample of observations used to estimate the equation. Edit the space for
GARCH (optional) with GARCH_forecast (here the forecasts will be
stored), as shown under.
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User supplied
Trimmed for out-of-sample
Check OK to display the following out-of-sample forecast results:
Figure 14: Depreciation forecast
This operation automatically updates exrf and conditional variance to
include the out-of-sample forecast and conditional variance/volatility. Both
of these are retrievable from the variable window. AS with Fig. 13, Fig. 14
plot exrf, exrf_lb and exrf_ub, where exrf_lb and exrf_ub are computed in
exactly the same way as before. Also note that exrf series is adjusted to
include historical data for the period 1997m01 – 2017m01 by EViews own
routines. In Fig. 15, we have intentionally adjusted the sample period for the
graph to cover 2016m01 to 2017m10.
2,800
3,200
3,600
4,000
4,400
4,800
M2 M3 M4 M5 M6 M7 M8 M9 M10
2017
EXRF ± 2 S.E.
.00025
.00030
.00035
.00040
.00045
.00050
.00055
02 03 04 05 06 07 08 09 10
2017
Forecast of Variance
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Figure 15: Nominal Exchange rate forecast
We see that exchange rate is forecast in the range of Shs 3471.43 – Shs
3780.226 at the beginning of the forecast period, but then the forecast
deteriorates to the range of Shs 3073.699 – Shs 4486.953 at the end of the
forecast horizon. The forecasts are off the actual exchange rate outturns for
March, April and May 2017 by Shs. 16.93, Shs -2.00 and Shs. -6.601,
respectively. Nonetheless, you must avoid as much as possible, mentioning
point forecasts (because very definitely, the outturn will be off the point
forecast). The graph below compares the measured and forecast conditional
volatility.
Figure 16: Historical and volatility forecast
3,000
3,200
3,400
3,600
3,800
4,000
4,200
4,400
4,600
I II III IV I II III IV
2016 2017
EXRFEXRF_LB
EXRF_UB
.0000
.0005
.0010
.0015
.0020
.0025
.0030
.0035
.0040
.0000
.0005
.0010
.0015
.0020
.0025
.0030
.0035
.0040
199
7:0
1
199
8:0
1
199
9:0
1
200
0:0
1
200
1:0
1
200
2:0
1
200
3:0
1
200
4:0
1
200
5:0
1
200
6:0
1
200
7:0
1
200
8:0
1
200
9:0
1
201
0:0
1
201
1:0
1
201
2:0
1
201
3:0
1
201
4:0
1
201
5:0
1
201
6:0
1
201
7:0
1
Historical conditional volatilityForecast conditional volatility(rhs)
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In both cases, compared to the historical depreciation and evolution of the
conditional variance, the future trajectory is less volatile, i.e. is relatively
stable. In fact, comparing history and forecast numbers, depreciation
averaged 0.86 percent in the 9 months to Jan 2017 (monthly average
depreciation of about 0.10 percent) and is forecast to average 0.54 percent
in the 9 months from February 2017 (average monthly depreciation of 0.06
percent). Nonetheless, the out-of-sample forecast over a relatively long
horizon converges to the unconditional variance, largely because of the
strong memory embedded in the AR structure.
5.5: Performing the Forecast in a Multivariate GARCH(1,1)
Framework Using EViews Screens and Statistical Output
Tables
As a starting point, we need to adjust the End date of the data points in the
Multivariate page of our EViews COMESA_2017 work file, following
pretty similar procedures described earlier, to 2017M10. Note that with this
resizing of the sample size, all these out-of-sample data points for the period
2017M02 – 2017M10 are indicated as NA and they are the data points that
we have to forecast. Next, we will need to recall the EViews window for the
BEKK estimated output in Table 9. Here, I reproduce the same following
the procedures detailed earlier.
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Output window
Whilst in the output window, click on Proc and select from the drop down
menuMake Model. You will see the following window given as Equations: 2
Baseline.
Equation 2
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Whilst in this window, click again at Proc and select from the drop-down
menuSolve Model…This gives us the following print screen.
This EViews screen has a number of dialog boxes in which we have to
provide user specific input. The good news though is that a great majority
of the input is common knowledge and as can be seen, EViews generously
fills out most of them and are given as default. Under Simulation type,
change from default of Deterministic to Stochastic- reason being that all
our entries in the mean equation are really stationary, i.e. in first difference
and lags of first difference. Deterministic applies if series are in level i.e.
non-stationary. Save for this, the default in the rest of the dialog boxes is
fine. The estimation allows for a dynamic structure of the model - the fact
that we have lags in the mean equation devoid it from being static and
allows for Std Dev in both the Baseline and Alternate Scenarios. Solution
sample should be the forecast horizon, here 2017m02 – 2017m10, which is
our 9 months out-of-sample forecast horizon. These have been adjusted for
in the screen shot above.
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The rest of the entries remain as in default, but briefly, under Solution
scenarios & output, we’re estimating an active model as Baseline,
providing for Std Dev – checked in the appropriate box. Check stochastic
option (in the menu of the window) and declare Baseline in the Page
Name dialog box and check Include coefficient uncertainty. Check also
Diagnostics (in the menu of the window) and check display detailed
messages including iteration count by block, and click OK to execute the
instructions, which then prints output of the form:
Ideally this shows whether the model has successfully solved or not and as
can be seen, here it is shown the model solve is complete with 1000
successful repetitions, 0 failure(s) – which is all good news in this round of
application – but note that in some applications, this is not always the case
as successive failures may be generated but consider this as a learning point
rather than frustration – it happens even to the most experienced seasoned
forecasters. Now turn to the series window, to see as highlighted, series
with extensions of _0m and _0s for all system variables being modelled.
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Check again Proc in this very window and in the drop-down menu, select
Make Graph…, a route shown in the print screen below:
This brings forth the following screen print:
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The only change we make in this is under Graph series, where in the dialog
box for Solution series, we choose from the drop-down menu, mean +/-2
Std. Deviation, replacing the default of Mean of Stochastic. Note that we
have set the Sample for graph to 2016M01 2017M10. Click OK to execute
this to give the graph in Fig.17.
Figure 17: Inflation and Depreciation forecast
The actual forecast numbers, if interested, can be retrieved either in a new
EViews page named Baseline (if you made provisions for it) or from the
workspace window, and is given with an extension _0 (if in a new EViews
page) or _0m if saved in the original work space window as in the screen
print below.
-.010
-.005
.000
.005
.010
.015
.020
III IV I II III IV
2016 2017
Actual
DLCPI (Baseline Mean)
DLCPI ± 2 S.E.
-.06
-.04
-.02
.00
.02
.04
.06
.08
III IV I II III IV
2016 2017
Actual
DLEXR (Baseline Mean)
DLEXR ± 2 S.E.
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_0m is the baseline mean forecast, both for in- and out of- sample forecast.
_0s is the baseline standard deviation, covering only the forecast horizon
and measures the extent to which the forecast deviates from the
unconditional variance. Using this, we might want to open as a group dlcpi,
dlcpi_0s, dlexr, and dlex_0s. Since dlcpi_0s and dlexr_0s only begin at 2017m02
and the level and baseline mean forecasts (_0m) are the same for the
corresponding periods, double click on Edit +/- (this allows us to edit the
open spread sheet), then copy the historical dlcpi/dlexr series and paste it
over the NA space in the columns for dlcpi_0s/dlexr_0s (2005m08 –
2017m01). After this, check again Edit +/- to effect the changes. This
should effectively enable us to plot and compare historical and predicted
values of dlcpi and dlexr as in Fig. 18.
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Figure 18: Inflation and Depreciation forecast
-.01
.00
.01
.02
.03
.04
2005:07
2006:01
2007:01
2008:01
2009:01
2010:01
2011:01
2012:01
2013:01
2014:01
2015:01
2016:01
2017:01
DLCPI (Baseline Mean)
DLCPI (Baseline S.D.)
-.12
-.08
-.04
.00
.04
.08
.12
2005:07
2006:01
2007:01
2008:01
2009:01
2010:01
2011:01
2012:01
2013:01
2014:01
2015:01
2016:01
2017:01
DLEXR (Baseline Mean)
DLEXR (Baseline S.D.)
Like in the univariate case, the future evolution of the depreciation and
inflation appear less volatile, at least when compared to history.
Depreciation averaged 0.86 percent in the 9 months to Jan 2017 (monthly
average depreciation of about 0.10 percent). Over the same 9 months to
October 2017, bi-variate period average nominal exchange rate depreciation
forecast is 0.4 percent, which is an average monthly depreciation of 0.04
percent. Unfortunately, in the framework, the accuracy of the forecast
cannot be evaluated and for some reason, purely related to the model at
hand, these are mean forecasts and not conditional covariance forecasts.
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Exercise 5
This exercise uses the models estimated in exercises 3 and 4.
1. Based on the appropriate model in exercise 3,
i). Perform and evaluate the in-sample forecast
ii). Perform 12-period ahead forecast of the measured conditional
volatility, and comment on the observed future trajectory of
inflation.
2. Based on the estimated a bi-variate GARC(1,1) model in exercise 4,
perform 18-month ahead forecast for exchange rate depreciation and
91-day T-bill in a diagonal BEKK framework
References
Baba, Y., R.F. Engel, D.F. Kraft and K.F. Kroner (1990), “Multivariate
Simultaneous generalizedARCH”, Mimeo, University of California at San
Diego, Department of Economics.
Beach, E.F. (1958), Economic Models: An Exposition, The Canadian Journal of
Economics and Political Science24 (3), 435 - 436
Bollerslev Tim (1986), Generalized Autoregressive Conditional Heteroskedasticity,
Journal ofEconometrics31: 307-27.
Bollerslev, Tim and Jeffrey M. Wooldridge (1992), Quasi- maximum Likelihood
Estimation andInferencein Dynamic Models with Time Varying
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