Exchange Rate Prediction Euro vs NOK from financial and commodity information

Evaluation of Models for predicting the average monthlyEuro versus Norwegian krone exchange rate from financial

and commodity information

Raju RImal

Norwegian University of Life Sciences(NMBU)

April 22, 2015

Raju RImal (NMBU) Masters Thesis April 22, 2015 1 / 23

Table of Contents

1 The BIG picture2 Part I

Exchange rate determinationFactors affecting exchange rateForeign currency and ExchangerateBalance of Payment AccountRelevant Variables

3 Part IIStatistical Models

Linear ModelsMulticollinearity ProblemPCR and PLS regression ModelRidge RegressionCross-validation and Prediction

4 Part IIIComparison of ModelsComments on ModelComparisonDiscussions and Conclusions


The BIG picture

The BIG picture

1 Identify functional relationship of Exchange rate with financial andcommodity variables

2 Make prediction using different models

3 Compare the models


Part I

Identify functional relationship of Exchange ratewith financial and commodity variables


Part I Exchange rate determination

Exchange rate determination

Exchange Rate is a price of one currency in terms of another

Determined from the demand and supply of the currency in MoneyMarket (ForEx)


Part I Exchange rate determination

Exchange rate determination

Exchange Rate is a price of one currency in terms of anotherDetermined from the demand and supply of the currency in MoneyMarket (ForEx)

−1. 1. 2. 3. 4. 5. 6. 7. 8.−1.

1.

2.

3.

4.

5.

6.

7.

0

Demand of Currency

Supply of Currency

Quantity

Exchange

Rate

Equilibrium Point


Part I Factors affecting exchange rate

Factors affecting exchange rate

e = f (∆Inf,∆Int,∆Inc,∆Gc,∆Exp) (1)

∆Inf = Inflation differential betweentwo countries

∆Int = Interest Rate differential be-tween two countries

∆Inc = Income differential betweentwo countries

∆Gc = Government Control differen-tial between two countries

∆Exp = Expectation differential be-tween two countries

Consumer Price Index(CPI)Interest Rate (Norwayand Euro Zone)Loan Interest Rate










S0

D0

Valueof

EURO

per

NOK

Quantity of EURO

9.10

9.97

S1

D1

QEuro

Upward shift in Demand

of Euro due to inflation in

Norway

Downward shift in supply

of Euro purchasing NOK











Quantity of Euro(purchasing Norwegian Krone)

Price

ofEuro

(EUR/N

OK)

S0

S1

D0

D1

QEuro

NOK8.72

NOK9.10

Demand Shift

Supply Shift











Quantity of Euro(purchasing Norwegian Krone)

Price

ofEuro

(EUR/N

OK)

S0

D0

D1

Q◦(Euro)

NOK8.72

NOK9.10

Increased demand of for-eign goods due to in-creased income levels











Variable Selected:Consumer Price Index(CPI)Interest Rate (Norwayand Euro Zone)Loan Interest Rate


Part I Foreign currency and Exchange rate

Involvement of Foreign Currency and Exchange Rate

ExchangeRate Involve

during

Foreign InvestmentTrading of Goodsand Services

Travelling andmany otheractivities

Import Export

All these activities involve exchange of currency. These activities arerecorded as Balance of Payments account.


Part I Balance of Payment Account

Balance of Payment Account

Balance of Payment has two components - Current Account and CapitalAccount;

Current AccountPayments for merchandise and servicesFactor Income paymentsTransfer payments

Capital and Financial AccountDirect foreign investmentPortfolio investmentOther Capital InvestmentErrors, Omissions and Reserves





Current AccountPayments for merchandise and services

I Imports and Exports of Merchandise (tangibleproducts) and Services(tourism, consultingservice etc)

I The difference is referred as Balance of tradeI Import and Export of Good (Merchandise)

which are only availiable in Monthly formatare considered in this thesis

Factor Income paymentsTransfer payments





Current AccountPayments for merchandise and servicesFactor Income paymentsIncome as Interest and Dividents

I received by domestic investors on foreigninvestments

I payed to foreign investors on domesticinvestments

Transfer payments





Current AccountPayments for merchandise and servicesFactor Income paymentsTransfer paymentsRepresents aid, grands and gifts from one country toanother





Capital and Financial AccountDirect foreign investment

I Includes investment in fixed assets in foreigncountries

Portfolio investmentOther Capital InvestmentErrors, Omissions and Reserves





Capital and Financial AccountDirect foreign investmentPortfolio investment

I Includes long term transaction of long termfinancial assets such as bonds and stocks

Other Capital InvestmentErrors, Omissions and Reserves





Capital and Financial AccountDirect foreign investmentPortfolio investmentOther Capital Investment

I Includes short term financial assets such asmoney market securities

Errors, Omissions and Reserves






I Includes adjustment for negative balance incurrent account





Current AccountPayments for merchandise and servicesFactor Income paymentsTransfer payments


Variable SelectedImportOil Platform, Old Ship,New Ship, Excluding Oiland Ship PlatformExportCondense Fuel, Crude oil,Natural gas, Oil platform,Old and new ships,Excluding Ships and oilplatform


Part I Relevant Variables

Some relevant variables selected for analysis

Financial VariablesKey Policy Rate of Norway (KeyIntRate)Overnight lending rate (LoanIntRate)Money market interest rates of Euro Area(EuroIntRate)Consumer Price Index (CPI)

Price VariableEurope Brent Spot Price (OilSpotPrice)

Lagged VariablesFirst lag of Exchange Rate (ly.var)Second lag of Exchange Rate (l2y.var)First lag of Consumer Price Index (l.CPI)

Import VariablesOld Ship (ImpOldShip)New Ship (ImpNewShip)Oil Platform (ImpOilPlat)Excluding Ship and Oil Platform(ImpExShipOilPlat)

Export VariablesCrude Oil (ExpCrdOil)Natural Gas (ExpNatGas)Condensed Fuel (ExpCond)Old Ship (ExpOldShip)New Ship (ExpNewShip)Oil Platform (ExpOilPlat)Excluding Ship and Oil Platform(ExpExShipOilPlat)


Part II

Making prediction using different models


Part II Statistical Models

Models in use

Following models are used for prediction of Exchange Rate,Multiple Linear Model

Y = β0 + β1X1 + . . .+ βpXp

The OLS estimate of β is,

β̂ =(XtX

)−1 XtY

Ridge RegressionPrincipal Component RegressionPartial Least Square Regression



Models in use

Following models are used for prediction of Exchange Rate,Multiple Linear ModelRidge Regression

I Larger estimates due to multicollinearity is settled by using modifiedOLS estimate in case of Ridge Regression as,

β̂ridge =[λIp + XtX

]−1 XtY

I Here, ridge parameter λ is estimated by minimizing RMSEP throughcross validation

Principal Component RegressionPartial Least Square Regression



Models in use

Following models are used for prediction of Exchange Rate,Multiple Linear ModelRidge RegressionPrincipal Component Regression

I A new set of variables Z1, . . .Zk called principal components areconstructed from linear combination of predictor variables

I The variation present on predictor variables are accumulated on firstfew principal components

Partial Least Square Regression



Models in use

Following models are used for prediction of Exchange Rate,Multiple Linear ModelRidge RegressionPrincipal Component RegressionPartial Least Square Regression

I A new set of latent variables Z1, . . . ,Zk are constructed.I The variables tries to capture most information in predictor variable

that is useful for explaining response.


Part II Linear Models

Linear Models

Multiple Linear regression with full set of predictor variable results withfew significant variables (EuroIntRate, ly.var and l2y.var).Subset models are created from the full model using following criteria,

Minimum Mallow’s Cp and Maximum adjusted R2

Minimum AIC and BIC

Stepwise procedure (Forward and Backward) based on F-value



Linear Models

Multiple Linear regression with full set of predictor variable results withfew significant variables (EuroIntRate, ly.var and l2y.var).

Subset Model with criteria of

Minimum Mallow’s Cp and Maximum adjusted R2

1.2

0.04

0

0

−0.2

3 −0.0

3

1.1

R−Sq = 0.914 Adj R−Sq = 0.911

Sigma = 0.112 F = 264.8 (6,149)

0

5

10

15

(Int

erce

pt)

Eur

oInt

Rat

e

Exp

Crd

Oil

ImpO

ldS

hip

l2y.

var

Loan

IntR

ate

ly.v

ar

T−

Val

ue

0.65

0.06

0

0 0

0

0.01

−0.2

2−0

.03

1.08

0

R−Sq = 0.917 Adj R−Sq = 0.912

Sigma = 0.112 F = 160.9 (6,149)

0

5

10

15

(Int

erce

pt)

Eur

oInt

Rat

e

Exp

Crd

Oil

Exp

OilP

lat

ImpN

ewS

hip

ImpO

ldS

hip

l.CP

I

l2y.

var

Loan

IntR

ate

ly.v

ar

OilS

potP

rice

T−

Val

ue

Subset of linear model selected from criteria of minimum Mallow’s Cp (left) and maximumadjusted R2 (right)



Linear Models



Minimum AIC and BIC

0.65

0.06

0

0 0

0

0.01

−0.2

2−0

.03

1.08

0

R−Sq = 0.917 Adj R−Sq = 0.912

Sigma = 0.112 F = 160.9 (6,149)

0

5

10

15

(Int

erce

pt)

Eur

oInt

Rat

e

Exp

Crd

Oil

Exp

OilP

lat

ImpN

ewS

hip

ImpO

ldS

hip

l.CP

I

l2y.

var

Loan

IntR

ate

ly.v

ar

OilS

potP

rice

T−

Val

ue

0.67

0

−0.2

2

1.14

R−Sq = 0.91 Adj R−Sq = 0.909

Sigma = 0.114 F = 514.1 (6,149)

0

5

10

15

(Int

erce

pt)

ImpO

ldS

hip

l2y.

var

ly.v

ar

T−

Val

ue

Subset of linear model selected from criteria of minimum AIC (left) and BIC (right)



Linear Models



Stepwise procedure (Forward and Backward) based on F-value

0.67

0

−0.2

2

1.14

R−Sq = 0.91 Adj R−Sq = 0.909

Sigma = 0.114 F = 514.1 (6,149)

0

5

10

15

(Int

erce

pt)

ImpO

ldS

hip

l2y.

var

ly.v

ar

T−

Val

ue

1.2

0.04

0

0

−0.2

3 −0.0

3

1.1

R−Sq = 0.914 Adj R−Sq = 0.911

Sigma = 0.112 F = 264.8 (6,149)

0

5

10

15

(Int

erce

pt)

Eur

oInt

Rat

e

Exp

Crd

Oil

ImpO

ldS

hip

l2y.

var

Loan

IntR

ate

ly.v

ar

T−

Val

ue

Subset of linear model selected from F-test based criteria through forward selection procedure(left) and backward elimination procedure (right)



Linear Models


Following pairs of model are found equivalent as they constitute of sameset of variables,

Model selected from minimum AIC (aicMdl) and maximum AdjustedR2 (r2.model)Model selected from F-based backward elimination procedure(backward) and minimum Mallow’s Cp (cp.model)Model selected from minimum BIC (bicMdl) and F-based Forwardselected procedure (forward)


Part II Multicollinearity Problem

Multicollinearity Problem

Linear model with full set of predictor variable has seriousmulticollinearity problemsubset model selected from minimum AIC and Maximum Adjusted R2

criteria also have problems with multicollinearity.

0.0e+00

2.5e+08

5.0e+08

7.5e+08

1.0e+09

Key

IntR

ate

Loan

IntR

ate

Eur

oInt

Rat

eC

PI

OilS

potP

rice

ImpO

ldS

hip

ImpN

ewS

hip

ImpO

ilPla

tIm

pExS

hipO

ilPla

tE

xpC

rdO

ilE

xpN

atG

asE

xpC

ond

Exp

Old

Shi

pE

xpN

ewS

hip

Exp

OilP

lat

Exp

ExS

hipO

ilPla

tTr

Bal

TrB

alE

xShi

pOilP

lat

TrB

alM

land

ly.v

arl2

y.va

rl.C

PI

Variables

VIF

Linear Model

0.0

2.5

5.0

7.5

10.0

Loan

IntR

ate

Eur

oInt

Rat

e

OilS

potP

rice

ImpO

ldS

hip

ImpN

ewS

hip

Exp

Crd

Oil

Exp

OilP

lat

ly.v

ar

l2y.

var

l.CP

I

Variables

VIF

Model selected (criteria:AIC)

Using models such as PCR and PLS can solve this problem


Part II PCR and PLS regression Model

PCR and PLS Regression

0

25

50

75

100

0 5 10 15 20Components

Var

iatio

n E

xpla

ined

X PerEURO

Variation Explained by PCR Model

25

50

75

100

0 5 10 15 20Components

Var

iatio

n E

xpla

ined

X PerEURO

Variation Explained by PLS Model

More than 90 percent of variation present in Exchange Rate isexplained by 16 components of PCR model while PLS has explainedthat much of variation by 6 components.However, PCR model has captured most of the variation present inpredictor with fewer components than PLS model.


Part II Ridge Regression

Ridge Regression

Also known as shrinkagemethods as it shrinks theestimate that are enlarged byMulticollinearity.

The ridge parameter λ isestimated by minimizing theRoot mean square error(RMSECV) usingcross-validation technique.

Here, λ is found to be 0.005

0.1350

0.1375

0.1400

0.1425

0.0000 0.0025 0.0050 0.0075 0.0100λ

RM

SE

P

Setting up λ that minimize the RMSEP


Part II Cross-validation and Prediction

Cross-valudation and Prediction

All the models seemed to work fine with observations included, but how itbehave with new observation – here comes the role of cross-validation.

Jan 2000 – Dec 2012

Training dataset

Jan 2013 – Nov 2014Test dataset

Dataset is splitted into calibration set and test set as in figure aboveModels fitted with training set were analysed for its behaviour withnew observations through cross-validation with consecutive segment oflength 12

2000 2001 . . . 2012Training Set


Part III

Compare the models


Part III Comparison of Models

Comparison of Models

Linear Models are compared on the bases of their goodness of fit

Model AIC BIC R.Sq R.Sq.Adj Sigma F.valuelinear -207.178 -133.982 0.919 0.906 0.116 68.594cp.model -230.323 -205.925 0.914 0.911 0.112 264.849r2.model -227.995 -191.397 0.917 0.912 0.112 160.906aicMdl -227.995 -191.397 0.917 0.912 0.112 160.906bicMdl -229.234 -213.985 0.910 0.909 0.114 514.106forward -229.234 -213.985 0.910 0.909 0.114 514.106backward -230.323 -205.925 0.914 0.911 0.112 264.849

Selected Linear Models, Ridge Model, PCR model and PLS models arethen compared on the basis of predictability




Linear Models are compared on the bases of their goodness of fit

Model AIC BIC R.Sq R.Sq.Adj Sigma F.valuelinear -207.178 -133.982 0.919 0.906 0.116 68.594cp.model -230.323 -205.925 0.914 0.911 0.112 264.849r2.model -227.995 -191.397 0.917 0.912 0.112 160.906aicMdl -227.995 -191.397 0.917 0.912 0.112 160.906bicMdl -229.234 -213.985 0.910 0.909 0.114 514.106forward -229.234 -213.985 0.910 0.909 0.114 514.106backward -230.323 -205.925 0.914 0.911 0.112 264.849

As prediction is objective, aicMdl or r2.model can be selected since theyhave smallest residual standard error and explain the variation in exchangerate better than otherSelected Linear Models, Ridge Model, PCR model and PLS models are thencompared on the basis of predictability




Linear Models are compared on the bases of their goodness of fitSelected Linear Models, Ridge Model, PCR model and PLS models arethen compared on the basis of predictability

OO

O

O OO

RMSEP R2pred

0.10

0.11

0.12

0.13

0.14

0.84

0.87

0.90

Line

ar

AIC

Mod

el

BIC

Mod

el

Bac

kMod

el

Rid

ge

PC

R.C

omp1

5

PC

R.C

omp1

6

PC

R.C

omp1

7

PLS

.Com

p6

PLS

.Com

p7

PLS

.Com

p8

PLS

.Com

p9

Line

ar

AIC

Mod

el

BIC

Mod

el

Bac

kMod

el

Rid

ge

PC

R.C

omp1

5

PC

R.C

omp1

6

PC

R.C

omp1

7

PLS

.Com

p6

PLS

.Com

p7

PLS

.Com

p8

PLS

.Com

p9

Models

Val

ue (

RM

SE

P/ R

−sq

pre

d)

train test cv


Part III Comments on Model Comparison

Some Comments on model comparison

Figure alongside shows that,Linear Models predicts well for observationsincluded in the model

Ridge regression perform moderately buthas predicted closer than some linearmodels for new observationsPCR and PLS models have made moreaccurate prediction than other linear modelsboth in the case of cross-validation and testdatasetPLS model with 7 components has leastRMSEP while PCR model with 16components ave least RMSECV

OO

O

O O

O

RMSEP

R2pred

0.10

0.11

0.12

0.13

0.14

0.84

0.87

0.90

Line

ar

AIC

Mod

el

BIC

Mod

el

Bac

kMod

el

Rid

ge

PC

R.C

omp1

5

PC

R.C

omp1

6

PC

R.C

omp1

7

PLS

.Com

p6

PLS

.Com

p7

PLS

.Com

p8

PLS

.Com

p9

Models

Val

ue (

RM

SE

P/ R

−sq

pre

d)

train test cv




Figure alongside shows that,Linear Models predicts well for observationsincluded in the modelRidge regression perform moderately buthas predicted closer than some linearmodels for new observations

PCR and PLS models have made moreaccurate prediction than other linear modelsboth in the case of cross-validation and testdatasetPLS model with 7 components has leastRMSEP while PCR model with 16components ave least RMSECV

OO

O

O O

O

RMSEP

R2pred

0.10

0.11

0.12

0.13

0.14

0.84

0.87

0.90

Line

ar

AIC

Mod

el

BIC

Mod

el

Bac

kMod

el

Rid

ge

PC

R.C

omp1

5

PC

R.C

omp1

6

PC

R.C

omp1

7

PLS

.Com

p6

PLS

.Com

p7

PLS

.Com

p8

PLS

.Com

p9

Models

Val

ue (

RM

SE

P/ R

−sq

pre

d)

train test cv




Figure alongside shows that,Linear Models predicts well for observationsincluded in the modelRidge regression perform moderately buthas predicted closer than some linearmodels for new observationsPCR and PLS models have made moreaccurate prediction than other linear modelsboth in the case of cross-validation and testdataset

PLS model with 7 components has leastRMSEP while PCR model with 16components ave least RMSECV

OO

O

O O

O

RMSEP

R2pred

0.10

0.11

0.12

0.13

0.14

0.84

0.87

0.90

Line

ar

AIC

Mod

el

BIC

Mod

el

Bac

kMod

el

Rid

ge

PC

R.C

omp1

5

PC

R.C

omp1

6

PC

R.C

omp1

7

PLS

.Com

p6

PLS

.Com

p7

PLS

.Com

p8

PLS

.Com

p9

Models

Val

ue (

RM

SE

P/ R

−sq

pre

d)

train test cv




Figure alongside shows that,Linear Models predicts well for observationsincluded in the modelRidge regression perform moderately buthas predicted closer than some linearmodels for new observationsPCR and PLS models have made moreaccurate prediction than other linear modelsboth in the case of cross-validation and testdatasetPLS model with 7 components has leastRMSEP while PCR model with 16components ave least RMSECV

OO

O

O O

O

RMSEP

R2pred

0.10

0.11

0.12

0.13

0.14

0.84

0.87

0.90

Line

ar

AIC

Mod

el

BIC

Mod

el

Bac

kMod

el

Rid

ge

PC

R.C

omp1

5

PC

R.C

omp1

6

PC

R.C

omp1

7

PLS

.Com

p6

PLS

.Com

p7

PLS

.Com

p8

PLS

.Com

p9

Models

Val

ue (

RM

SE

P/ R

−sq

pre

d)

train test cv


Part III Discussions and Conclusions

Discussions and Conclusions

This thesis has attempted to make prediction in time series dataSome subset of linear model considered are free from multicollinearityIn case of multicollinearity problem, latent variable models like PLSand PCR can deal with the situationPLS and PCR models also outperformed in predicting newobservations that are not included in the modelAutocorrelation is inevitable in time series data, including laggeddependent variable in the model has corrected the problemResiduals obtained from selected model (pls.comp7) does not containany autocorrelation PACF plot

More practices are recommented to study the performance of latentvariable model in time-series data

Next



Partial Autocorrelation Function

Linear Ridge

PCR.16 PLS.7

−0.1

0.0

0.1

0.2

−0.1

0.0

0.1

0.2

0 5 10 15 20 0 5 10 15 20Var1

PAC

F

Partial Autocorrelation Function (PACF)



Acknoledgement

Thanks to my Supervisors,Ellen Sandberg and Trygve Almøy

and professorSolve Sæbø

for their guidance and encouragements


THANK YOU

Exchange Rate Prediction Euro vs NOK from financial and commodity information

Economy & Finance

Transcript of Exchange Rate Prediction Euro vs NOK from financial and commodity information