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Evaluation of Models for predicting the

average monthly Euro versus Norwegian

krone exchange rate from financial and

commodity information

Raju Rimal

ADissertation

Presented to the Faculty

of Norwegian University of Life Sciences

in Candidacy for the Degree

of Masters of Bioinformatics and Applied Statistics

Recommended for Acceptance

by the Department of

IKBM

Supervisor: Ellen Sandberg and Trygve Almøy

Dec 2014

c

Copyright by Raju Rimal, 2014.

All rights reserved.

Abstract

Many multinational companies and policy makers carry out decisions by speculat-

ing exchange rate. Exchange rate is determined by the demand and supply of a

currency. It depends highly on variables like imports, exports, interest rates, oil

prices, inﬂation and even with its past values. Since these macroeconomic variables

are highly correlated with each other, latent variables or principal components can

solve the problem of multicollinearity. The application of latent variables and prin-

cipal components based methods such as Principal Component Regression (PCR)

and Partial Least Square (PLS) in time series data for prediction is uncommon.

Prediction of exchange rate of Norwegian Krone per Euro using Multiple linear re-

gression, Principal Component Regression (PCR) and Partial Least Square (PLS)

regression is performed in this dissertation.

Linear models and its subsets obtained using criteria such as minimum AIC or

BIC and maximum R2adj are compared on the basis of their goodness of ﬁt. The

selected model is then compared with models from principal component regression

and partial least square regression on the basis of predictability criteria of RMSEP

and R2predicted. The results have suggested the partial least square regression as

the best models among other. The residuals obtained from the models have no au-

tocorrelations so the application of this method has not only reduced the dimension

of data but also resolved the problem of multicollinearity and autocorrelations.

iii

Acknowledgements

IwouldliketoexpressmygratitudetomysupervisorsEllenSandbergandTrygve

Almøy for their guidance and invaluable suggestions. I am grateful to Prof. Solve

Sæbø for his helpful advice and instructions. I want to thank Statistic Norway

and Norges Bank for making valuable data easily available without which it is

impossible to complete my thesis.

Being a student of a developing country, I was unaware of programming, mod-

ern statistical methods and academic writing. With the encouragement of Prof.

Solve Sæbø and my supervisors, I have completed by thesis with extensive use of

R programming and modern statistical tools.

Finally, my special thanks also goes to my families and friends for their con-

tinuous support and encouragement.

iv

To my p arent s.

v

Contents

Abstract.................................... iii

Acknowledgements .............................. iv

ListofTables ................................. vii

ListofFigures.................................viii

1 Introduction 1

1.1 Methods opted for analysis . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Sourcesofdata ............................. 4

1.3 Objectiveofthesis ........................... 4

2 Data and Material 5

2.1 ForExMarket.............................. 5

2.2 The Norwegian krone (NOK) . . . . . . . . . . . . . . . . . . . . . 6

2.3 EURO .................................. 7

2.4 Factors inﬂuencing Exchange Rate . . . . . . . . . . . . . . . . . . 7

2.4.1 Inﬂation ............................. 9

2.4.2 InterestRate .......................... 11

2.4.3 IncomeLevels.......................... 14

2.4.4 Government Control . . . . . . . . . . . . . . . . . . . . . . 15

vi

2.4.5 Expectations .......................... 16

2.5 BalanceofPayment........................... 16

2.5.1 CurrentAccount ........................ 18

2.5.2 Capital and Financial Accounts . . . . . . . . . . . . . . . . 21

2.6 OilSpotPrice.............................. 23

2.7 Lagged response variable as predictor . . . . . . . . . . . . . . . . . 24

2.8 E↵ectofCrisisperiod.......................... 25

3 Models and Methods 26

3.1 AstatisticalModel ........................... 26

3.2 Linear Regression Model . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Least Square Estimation . . . . . . . . . . . . . . . . . . . . 27

3.2.2 Prediction............................ 29

3.3 Variableselection............................ 29

3.3.1 Criteria for variable selection . . . . . . . . . . . . . . . . . 30

3.3.2 Computational procedure for variable selection . . . . . . . . 31

3.4 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . 32

3.5 Principal Component Regression . . . . . . . . . . . . . . . . . . . . 35

3.6 Partial Least Square Regression . . . . . . . . . . . . . . . . . . . . 37

3.7 RidgeRegression ............................ 40

3.8 ComparisonCriteria .......................... 41

3.8.1 Goodnessofﬁt ......................... 41

3.8.2 Predictability .......................... 43

4 Data Analysis 47

4.1 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . 50

vii

4.2 Variable Selection Procedure . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Model selection using Mallows Cpand R2adjusted . . . . . 51

4.2.2 Model selection using AIC and BIC criteria . . . . . . . . . 51

4.2.3 Step wise procedures based on F-value . . . . . . . . . . . . 53

4.3 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . 55

4.4 Principal Component Regression . . . . . . . . . . . . . . . . . . . . 57

4.5 Partial Least Square Regression . . . . . . . . . . . . . . . . . . . . 58

4.6 RidgeRegression ............................ 59

4.7 CrossValidation............................. 60

4.8 Prediction on test Data . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.9 Comparison of Models . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.9.1 Goodnessofﬁt ......................... 63

4.9.2 Predictability .......................... 64

4.10 CoeﬃcientsEstimates ......................... 66

4.11 Autocorrelation and its resolution . . . . . . . . . . . . . . . . . . . 67

5 Discussions and Conclusion 68

5.1 Somediscussions ............................ 68

5.2 Conclusions ............................... 70

5.3 FurtherStudy.............................. 71

Bibliography 77

A Data Description 77

B R packages used 79

viii

C Some Relevent Plots 81

D Codes in Use 85

ix

List of Tables

2.2 Two components of Balance of Payments and their subdivision . . . 17

4.1 Summary Report of all the variables used in this report . . . . . . . 47

4.1 Summary Report of all the variables used in this report . . . . . . . 48

4.2 Variables signiﬁcant at ↵=0.05 while ﬁtting linear model . . . . . 50

4.3 Dispersion of data explained by principal components . . . . . . . . 56

4.3 Dispersion of data explained by principal components . . . . . . . . 57

4.4 Percentage of variation explained by PCR model in response and

predictor................................. 57

4.5 Percentage of variation Explained by PLS model in Response and

Predictor................................. 58

4.6 Summary statistic and information criteria for model comparison . 64

4.8 Validation result containing RMSEP and R2pred for training set,

cross-validation set and test set . . . . . . . . . . . . . . . . . . . . 66

4.9 Coeﬃcient Estimate for PLS and PCR model . . . . . . . . . . . . 67

x

List of Figures

2.1 Exchange rate of Norwegian Krone per Euro . . . . . . . . . . . . . 6

2.2 E↵ect of shifts on demand and supply of currencies on their Ex-

changerates............................... 8

2.3 E↵ect of inﬂation on Exchange Rate Equilibrium . . . . . . . . . . 10

2.4 Time Series plot of Consumer Price Index (CPI) . . . . . . . . . . . 11

2.5 E↵ect of interest rate change in Exchange Rate . . . . . . . . . . . 12

2.6 Market Rate inﬂuence on demand channel, exchange rate channel

and expectation channel . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Interest Rates from Norway and Eurozone and their comparision

with Exchange Rate showing a distinct inverse relationship . . . . . 13

2.8 E↵ect of change in relative income levels on exchange rate ceteris

paribus. ................................. 14

2.9 Current Account Balance prepared from quartely data from the year

1981to2014............................... 19

2.10 Time Series plot of major imports of Norway . . . . . . . . . . . . . 20

2.11 Time Series plot of major exports of Norway . . . . . . . . . . . . . 21

2.12 Time Series plot of variables related to capital account . . . . . . . 22

2.13 Time Series plot of oil spot price from Jan 2000 . . . . . . . . . . . 23

xi

2.14 Partial autocorrelation function for Exchange Rate of NOK per Euro 24

3.1 Model Error - Estimation Error and Prediction Error . . . . . . . . 43

3.2 Procedure adopted in the thesis . . . . . . . . . . . . . . . . . . . . 44

4.1 Correlation between response (Exchange Rate) and other predictor

variable ................................. 49

4.2 Number of variable against the criteria where the red dot corre-

sponds the number of variable to acheave the criteria, i . . . . . . . 52

4.3 Modelselectedby ........................... 52

4.4 Number of variable against the AIC vs BIC criteria . . . . . . . . . 53

4.5 Best subset model selected by AIC and BIC criteria . . . . . . . . . 54

4.6 Best subset model selected by F-test based criteria . . . . . . . . . 55

4.7 Variance Inﬂation Factor (VIF) of di↵erent models . . . . . . . . . 56

4.8 Variation Explained by PLS and PCR . . . . . . . . . . . . . . . . 59

4.9 RMSE and R2pred plots for di↵erent ridge regression paramter . . 60

4.10 RMSEP plot for PCR and PLS . . . . . . . . . . . . . . . . . . . . 61

4.11 Comparision of Model on the ground of calibration model, cross-

validation models and prediction model on the basis of RMSEP and 65

4.12 Coﬃcients estimates for predictor variables . . . . . . . . . . . . . . 67

C.1 Diagnostic plot for the subset of linear model selected from mini-

mum .................................. 81

C.2 Scatter loading plot of PLS with its ﬁrst and second components . . 82

C.3 Scoreplot of ﬁrst three component of PLS regression . . . . . . . . . 82

C.4 Residuals obtained after ﬁtting the model . . . . . . . . . . . . . . 83

xii

C.5 Partial Autocorrelation Function (PACF) of Residuals obtained af-

terﬁttingthemodel .......................... 83

C.6 Prediction made on trained and test dataset using di↵erent models 84

xiii

Abbreviations and Symbols

Abbreviations and their full forms used in this Thesis

Abbreviation FullForm

PC Principal Components

PCA Principal Component Analysis

PLS Partial Least Square

PCR Princiapal Component Regression

AIC Akaike Information Criterion

BIC Bayesian Information Criterion

Cp Mallows’sCp

VIF Variance Inﬂation Factor

RMSE Root-Mean-Square Error

RMSEP Root-Mean-Square Errorof Prediction

RMSECV Root-Mean-Square Errorof Cross-validation

R2pred PredictedR-squared

VAR Vector Autoregression

ARIMA Autoregressive Integrated Moving Average

ADL Autoregressive Distributed Lag

NOK Norwegian Krone

USD United State Dollor

Symbols and their meaning used in this Thesis

Symbols Meaning

Bold Symbols like,

X,YMatrices and Vectors

Sigma (S) Popularion (Sample) variance-covariance matrix

R2adj Adjusted coeﬃcient of determination

CVadj RMSECV adjusted for bias

cp.model Subset of linear model selected with minimum Mallow’s Cp

Criteria

r2.model Subset of linear model selected with maximum R2adjusted

Criteria

aic.model Subset of linear model selected with minimum AIC Criteria

bic.model Subset of linear model selected with minimum BIC Criteria

forward.model Subset of linear model selected based on F-test Criteria

using forward selection procedure

backward.model Subset of linear model selected based on F-test Criteria

using backward elimination procedure

train Training Dataset (From Jan 2000 to Dec 2012)

test Test Dataset (From Jan 2013 to Nov 2014)

Ridge Regression Parameter

Q2R2predicted

PerEURO Exchange Rate of Norweian Krone Per Euro (Response

Variable)

Chapter 1

Introduction

Apart from having distinct role in money market, exchange rate has inﬂuence

in almost all the sectors of economics and ﬁnance. Understanding its dynamics

enables multinational companies to make decision on their investment and assist

bureaucrats to update the monetary and ﬁscal policies. Di↵erent models are used

to understand the dynamics of exchange rate, however the use of latent variable in

the models is unconventional. Multicollinearity which is also a common problem

in economic researches, models based on principal components (latent variables)

such as Principal Component Regression(PCR) and Partial Least Square(PLS)

regression can resolve the problem. Although autocorrelation is a major problem

in time-series, inclusion of the past values of dependent variable in the model can

solve the problem in many situations. In this dissertation the exchange rate of

Norwegian Krone vs Euro is predicted from the classical linear regression models,

its subsets derived from various criteria, PCR and PLS models. The models are

compared on the basis of their performance. Under proper model speciﬁcation

1

and wise selection of required components, Principal Component Regression and

Partial Least Square regression can forecast better than the linear models.

Tradi ng has s tar t ed fr om the ver y be ginn ing of hu man ci vili zati on. Peo ple

used to trade with goods at the time but with advancement of development people

started using gold, silver and ﬁnally money. The process is not restricted within

a country. Some countries are powerful and some are not so as their currencies.

Currency of another country becomes essential to buy things from that country.

Here comes the role of exchange rate. Buying powerful currencies requires large

sum of weak currencies.

Any international trade is conducted through more than one currencies. Par-

ticipants in the international trade require to exchange their currency which is

performed by foreign exchange market. “The foreign exchange market (ForEx)

is the mechanism that brings together buyers and sellers of di↵erent currencies”

(Appleyard, Field, and Cobb, 2014).

As any other commodity, exchange rate is also determined from its demand and

supply in money market. All those economic activities that exist between countries

create demand and supply of the currencies which consequently determine the

exchange rate. The economic activities between countries are recorded as balance

of payment account. Thus the balance of payment account captures all the demand

and supply of foreign currency (Fang and Kwong, 1991). When the domestic

demand for foreign currency exceeds the foreign demand of domestic currency i.e.

a deﬁcit in the balance of payment, the domestic currency depreciate (Balance of

Payments Deﬁcits and Surpluses).

Foreign currencies are involved in various activities such as, (a) imports and ex-

ports of goods and services, (b) interest and dividends payed to foreign investment

2

in domestic market, (c) interest and dividends earned from investments made on

foreign market, (d) all the currencies that enter into and leave from a country as

income and expenditure.

Three factors a↵ecting exchange rate are considered in this thesis. Primarily,

total monthly imports and exports of goods are considered. Ships, oil platform,

chemicals and food stu↵s are major imports of Norway. Petroleum products,

machinery, equipment, chemicals and ﬁshes are the major exports. Since the

economy of Norway highly depend on petroleum products, apart from imports and

exports, the second component considered is the spot oil price. Third factor is the

ﬁnancial variables such as interest rate and consumer price index are considered.

In interest rate - (a) key interest rate of Norway, (b) Loan interest rate (c) key

interest rate of euro area are taken into account as factors a↵ecting interest rate.

1.1 Methods opted for analysis

Univariate time series analysis is very common in Econometric where Autoregres-

sive (AR), Moving Average (MA) and Autoregressive integrated Moving average

(ARIMA) are used. However, dealing a time series data with many predictor

variables using latent variables and principal components methods is unconven-

tional. This thesis aims to analysis a time series with ﬁnancial and commodity

data, as predictor, using statistical regression methods such as - Multiple Linear

Regression, Ridge Regression, Principal Component Regression (PCR) and Partial

Least Square (PLS) Regression. Apart from these, a subset models which selected

from the Multiparty Linear Regression using various criteria are also used. An

application of PCR and PLS on time series data makes this thesis distinct.

3

1.2 Sources of data

Data related to balance of payment such as import, export and trade balance used

here are obtained from Statistics Norway. Consumer price index is also obtained

from the same source. Interest rate variable related to Norway are obtained from

Norges’ Bank and the key interest rate for euro zone is obtained from Euro Bank

while the oil spot price is obtained from US Energy information system. The

average monthly spot price for Brent oil was on Dollar per Barrel unit which was

converted into NOK using NOK per USD exchange rate for that month.

1.3 Objective of thesis

There are three main objective of this thesis-

1. To anal yze th e rel a tio nshi p of for eign e xchan ge rat e with t he ﬁna ncia l (pri ce,

indices and exchange rate) and commodity (imports, exports and trade bal-

ance) information

2. Prediction of out-of-sample observations (Exchange Rate) using various mod-

els

3. Comparison of the Models considered on the basis of goodness of their ﬁt

and their predictive ability

4

Chapter 2

Data and Material

Prediction of dynamics of Exchange Rate through Economic and Financial indi-

cators is the main aim of this thesis. From these two broad categories, only those

factors were considered which are believed to be useful to understand the exchange

rate dynamics.

2.1 ForEx Market

Foreign Exchange(Fx) Market is the most traded and liquid ﬁnancial market where

individuals, ﬁrms and banks buy and sell foreign currencies. Forex market consti-

tute of monetary counters connected electronically which are in constant contact

forming a single international ﬁnancial market. The market remains open 24 hr a

day for ﬁve working days of a week (Introduction to the Forex Market ).

Currencies are exchanged for activities like trade, tourism and investments in

another countries. For instance, a person visiting France needs euro since euro is

accepted in France. On returning back from the visit (s)he might want to exchange

5

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9.5

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Date

PerEURO

Fig 2.1: Exchange rate of Norwegian Krone per Euro

back those Euros to Norwegian Krone. This transaction is a↵ected by the exchange

rate of Norwegian Krone per Euro. The exchange rate of NOK per Euro over time

is plotted in ﬁgure-2.1.

Exchange rate can be set according to di↵erent macroeconomic variables, such

as interest rate, price index, balance of payment etc. Such exchange rate deter-

mined by ForEx market transaction is called Floating exchange rate. Some country

ﬁx exchange rate while others pegged with other currency. Norway has a ﬂoating

exchange rate.

2.2 The Norwegian krone (NOK)

After introduction of Krone in April 1875 (Brief History Of Norges Bank 2014-

11), Norway was pushed to join the Scandinavian Monetary Union established on

6

1873 (Norwegian Kroner 2014/12). Although the Union was formally abolished

on 1972, Norway decided to keep the names of its currencies. In December 1982,

due to heavy speculation, Norges Bank (Central Bank of Norway) decided to ﬁx

Norwegian Krone which later ﬂoated on 1992 (Brief History Of Norges Bank 2014-

11).

2.3 EURO

Euro, the oﬃcial currency in the Eurozone, was introduced as a virtual currency

in 1999 and later as physical in 2002. It is the single currency shared by 191of the

European Union’s Member States of Euro Area. Although European Central Bank

(ECB) manages Euro, the ﬁscal policy (public revenue and expenditure) are in the

hands of individual national authorities. The single currency market throughout

the euro zone not only makes traveling across the countries easier but also helps the

member country to keep their economy sound and stable. This situation removes

currency exchange cost, smooth international trade and consequently gives them

more powerful voice in the world. A stable economy and larger area protects

euro zone from external economic ﬂuctuations, instability in currency market and

unpredictable rise in oil prices.(The euro 2015)

2.4 Factors inﬂuencing Exchange Rate

The demand of any currency relative to its supply determines its price, just like

any other commodity. For each possible price of a Norwegian Krone, there is

1https://www.ecb.europa.eu/euro/intro/html/index.en.html

7

a corresponding demand and supply to be exchanged with euro in the money

market. When demand of krone equals its supply, the price it exhibit at some

speciﬁc time is called its equilibrium exchange rate. Factors like inﬂation, interest

rates, expectation and government policy a↵ects the demand for any currency. But

the supply is mostly in control of the central bank. In a ﬂoating exchange rate

regime, the shift in demand (ﬁg-2.2a) and supply(ﬁg-2.2b) function determines

equilibrium exchange rate of any currency.

Q0

e0

ed0

e1

Q1

Supply Function

Demand Function

Shift in Demand Function

Deadweight

Loss

(a) Demand Shift and Exchange Rate Equilibrium

Q0

e0

e1

Q0

Q1

Supply Function

Demand Function

Shift in Supply Function

Deadweight

Loss

(b) Supply Shift and Exchange Rate Equilibrium

Fig 2.2: E↵ect of shifts on demand and supply of currencies on their Exchange rates

In case of demand shift, with constant currency supply, the exchange rate

will suddenly rise to edcreating dead weight loss (also known as excess burden

or allocative ineﬃciency2) which consequently pushes the supply from Q0to Q1

creating a new equilibrium exchange rate at e1. In the similar fashion, if the

market is over ﬂooded with currency, shifting the supply function and creating

dead weight loss, the exchange rate is pressed from e0to create a new equilibrium

at e1. In both the situation, the quantity supplied although being increased, the

ﬁrst one leads to a rise in exchange rate while the other leads to its fall.

2http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Deadweight_loss.html

8

Madura (2012, p. 103) suggested an equation consisting those macroeconomic

factors that can a↵ect the demand and supply of any currency and consequently

the exchange rate as,

e=f(INF,INT,INC,GC,EXP) (2.1)

where,

e: percentage change in spot exchange rate

INF: change in inﬂation di↵erential between two countries (currencies)

INT: change in interest rate di↵erential between two countries

INC: change in the income level di↵erential between two countries

GC: change in government control

EXP: change in currency value expectations

2.4.1 Inﬂation

Inﬂation is the steady rise in overall price level, i.e. a decrease in the value of

currency. In other words, more amount of money is needed to buy same goods than

previous. Relative change in inﬂation has e↵ect on exchange rate. For instance, an

abrupt rise in the inﬂation in Norway relative to the Eurozone, Norwegian products

becomes relatively expensive in terms of Norwegian Currency. On one hand, this

would increase the demands for Eurozone goods, and consequently the demand

for euro increases in the short run. On the other hand, expensive Norwegian

goods becomes less attractive in Eurozone and therefore reduce the supply of

euro purchasing Norwegian kroner. In ﬁgure -2.3, the demand function of Euro

9

shift upward due to inﬂation of NOK, i.e. Eurozone goods are more attractive

than Norwegian goods and the downward shift on supply function occurs as the

customers are less interested in Norwegian products. As a result the value of Euro

per NOK increases from 9.10 to 9.97, i.e Norwegian Krone deprecates against the

Euro (Madura, 2012, p. 104).

S0

D0

Value of EURO per NOK

Quantity of EURO

9.10

9.97

S1

D1

QEuro

Upward shift in De-

mand of Euro due to

inﬂation in Norway

Downward shift in sup-

ply of Euro purchasing

NOK

Source: Madura, 2012

Fig 2.3: E↵ect of inﬂation on Exchange Rate Equilibrium

Statistics Norway prepares and publishes the oﬃcial ﬁgures for inﬂation, the

consumer price index (CPI) with base year at 1998. Since the real value of money is

constantly declining, high inﬂation means that storing money is expensive. while

low and stable inﬂation contributes to an eﬃcient distribution of resources in

a market economy (FAQ: Monetary Policy, Inﬂation and Interest Rates 2007).

Since this is an important factor that can inﬂuence exchange rate, data for CPI is

10

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138

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Date

CPI

Source: Norges Bank

Fig 2.4: Time Series plot of Consumer Price Index (CPI)

obtained for this thesis from Norges bank. The time-series plot for CPI in ﬁgure-??

shows an steady increment over the time.

2.4.2 Interest Rate

Since Interest rate has impact on inﬂation and currency values, by manipulating it,

central banks exert inﬂuence over both inﬂation and exchange rates. For example,

a sudden increase in interest rate in Norway relative to Eurozone could have in-

crease on investment of Eurozone in Norway with interest-bearing securities. The

Eurozone investors wants to invest more in Norway which increases the demand

for NOK in Eurozone. Due to stronger incentives, Norwegians also increase their

domestic investment, as a result, the supply of NOK in currency market will re-

11

duce. The increase in Demand of NOK and decrease in its supply results a shift

in exchange rate to lower level. The process is illustrated in ﬁgure - 2.5.

Quantity of Euro

(purchasing Norwegian Krone)

Price of Euro (EUR/NOK)

S0

S1

D0

D1

QEuro

NOK

8.72

NOK

9.10

Demand Shift

Supply Shift

Source: Madura, 2012

Fig 2.5: E↵ect of Interest Rate change on Exchange Rate includes (a) Demand Shift:

Due to increased interest rate in Norway, demand of Norwegian Krone increases

creating a demand shift in demand function and (b) Supply Shift: The supply

of Krone decrease as Norwegian increase their domestic investment creating a

shortage of NOK in market.

The inﬂuence of market interest rate ﬂows through multiple channel such as

demand channel, exchange Rate channel and expectation Channel as shown in

ﬁgure-2.6 (E↵ect of Interest Rate Changes 2004).

According to Madura (2012), change in interest rate in third country can also

a↵ect the exchange rates between NOK and Euro. For instance, the sudden in-

crease of interest rate in US would shift the European investment from Norway to

12

Exchange

Rate

Imported

Price Inﬂation

Norge’s Key

Interest Rate Market Rates Inﬂation

Expectations Inﬂation

Consumption

Investment

Labor Market

Wages

Margins

Source: E↵ect of Interest Rate Changes 2004

Fig 2.6: Market Rate inﬂuence on demand channel, exchange rate channel and expec-

tation channel

US which consequently reduce the demand of NOK resulting a downward pressure

on its exchange rate with Euro.

Norwegian Key Interest Rate

Norweign Loan Interst Rate

Eurozone Key Interest Rate

0.0

2.5

5.0

7.5

2000

2005

2010

2015

2000

2005

2010

2015

2000

2005

2010

2015

Date

value

Eurozone Key Interest Rate

Exchange Rate of NOK per Euro

Norwegian Key Interest Rate

Norweign Loan Interst Rate

Fig 2.7: Interest Rates from Norway and Eurozone and their comparision with Exchange

Rate showing a distinct inverse relationship

Since the interest rate is a key factor inﬂuencing exchange rate, the key interest

rate of Norway and Eurozone along with the loan interest rate of Norway is consid-

13

ered in this thesis. The time series plot of these variables are in ﬁgure - 2.7. Due

to simultaneous act of other variables, the plot does not exhibit any discrete rela-

tionship. However, the model ﬁtted by the data collected suggest some in-depth

understanding of this relationship which is analysed and presented in chapter-4.

2.4.3 Income Levels

The rise in real income level increases the consumption level. Relative income

levels of a country is another factor which can a↵ect the demand of imported

goods which consequently a↵ect exchange rate (Madura, 2012). For instance, if

the income levels of people of euro zone rises, other factor being constant, the

demand for foreign goods in euro zone may increase which can shift the demand

function outward and subsequently increase the exchange rate (ﬁgure-2.8).

Quantity of Euro

(purchasing Norwegian Krone)

Price of Euro (EUR/NOK)

S0

D0

D1

Q(Euro)

NOK

8.72

NOK

9.10

Increased demand of

foreign goods due to in-

creased income levels

Source: Madura, 2012

Fig 2.8: E↵ect of change in relative income levels on exchange rate ceteris paribus.

14

The example considered above is on the assumption of ceteris paribus, which in

reality is not usual. The change in exchange rate due to income levels is also guided

through the e↵ect of income levels on interest rates and inﬂation. The increased

income levels increase the consumption cause the economy to overheat. Central

banks could increase interest rates to prevent overheating and increased inﬂation.

Thus the relative change in income levels can a↵ect exchange rates directly and

indirectly (Madura, 2012, p. 106).

2.4.4 Government Control

Government Control is the fourth factor Madura (2012) has considered that can

inﬂuence foreign exchange rate. Government can inﬂuence exchange rate in many

ways including, (a) imposing foreign exchange barriers, (b) imposing foreign trade

barriers, (c) intervening (buying and selling currencies) in the foreign exchange

markets, and (d) a↵ecting macro variables such as inﬂation, interest rates, and

income levels. Norges Bank could force the currency to depreciate by ﬂooding the

market with NOK (i.e increasing supply) if Norway wants to boost its exports.

Similarly, the bank could used their foreign currency reserve to purchase NOK to

rise its value. Such direct interventions make considerable impact on the exchange

rate. As a indirect intervention, the government can inﬂuencing the underlying

macroeconomic factors like inﬂation, interest rate and income level (Madura, 2012,

p. 107).

15

2.4.5 Expectations

Response to new information in foreign exchange market is similar to other ﬁ-

nancial market. The current expectation for the future value is reﬂected in the

exchange rate changes. Like in stock market, when a company publishes its pros-

perous ﬁnancial statement, the stock price suddenly rises; the forex market also

exhibit similar performance. For example, a news of increasing inﬂation in Norway

cause currency traders to sell Norwegian Krone expecting a decrease in its future

value. This expectation is immediately seen as a downward pressure on Norwe-

gian Krone. The similar e↵ect is obtained when speculator expects the currency

to depreciate (Madura, 2012, p. 107).

A person of one country need the currency of another country for various

purposes such as trade of goods and services, foreign investment and travelling.

The actual ﬂow of currency from one country to another is in these forms of

activities. The transaction of trade in terms of goods and services between speciﬁc

countries is kept recorded as a form of balance of payment which can even have

signal of possible shifts in exchange rate.

2.5 Balance of Payment

Although international trade is possessed in various forms, the transaction of mul-

tiple currency is common in each of them. A country keeps these transactions

with other countries as a form of Balance of Payments account. A balance of

payment account maintains a systematic records of these transactions conducted

at some speciﬁc time period between a home country and others (those countries

with which the transactions are made). A balance of payment account of a country

16

exhibit the size of its economic activities with rest of the world (Appleyard, Field,

and Cobb, 2014, p. 462).

Since Balance of Payment is a bookkeeping system for inter countries economic

activities, the items with payments inward to the home country are credited while

payments outward from the home country are debited. Exports, inﬂow of foreign

investment, interest and dividends obtained from the investment made on foreign

country by the home country are considered as credited items as they increase

the inward ﬂow of currency. Similarly, Imports, investment made on foreign coun-

tries, interest and dividends paid to foreign countries for their investment in home

country are the items to be debited (Appleyard, Field, and Cobb, 2014, p. 465).

Table 2.2: Two components of Balance of Payments and their subdivision

Balance of Payment

Current Account Capital Account

•Payments for Merchandise and Ser-

vices

•Factor Income Payments

•Transfe r Payment s

•Examples of Payment Entries

•Actual Current Account Balance

•Direct Foreign Investment

•Portfolio Investment

•Other Capital Investment

•Errors and Omissions and Reserves

Source: Madura, 2012

Balance of payment can be classiﬁed into two broad categories - (a) Current

Account and (b) Capital Account. The items that lies in these subcategories are

illustrated in table-2.2.

17

2.5.1 Current Account

Current account measures net imports and exports of a country. Imports and

exports are divided into three sub categories - (a) Tra de of go od s, (b) Tr ade of

services and (c) Income which includes the interest and dividend payed to inter-

national ﬁrms operating within home country and interest and dividends earned

from domestically owned ﬁrms abroad (Krugman and Obstfeld, 2006).

The current account balance is the di↵erence between export and import.

When export of a country exceed its import, there is current account surplus

and when import exceed export there is a current account deﬁcit.

Current Account = Total Exports Tota l Imp ort s (2.2 )

Above equation can also be expressed as a form of income and expenditure like

in equation-2.3 which is the di↵erence between Total National Income and Total

Domestic consumption (Krugman and Obstfeld, 2006).

Current Account Balance = Y

|{z}

GNP

(C + I + G)

| {z }

Total Domestic

Consumption

(2.3)

where,

C=Consumption

I=Investment

G=GovernmentPurchases

Current account incorporates a wide range of international transactions so

there is a vital role of exchange rate in each of those transaction. This thesis has

considered the monthly data for imports and exports of goods which is available

18

0

50

100

1980Q1 1985Q1 1990Q1 1995Q1 2000Q1 2005Q1 2010Q1 2015Q1

Date (in quarters)

Value (in NOK thousand)

BalanceGoods BalanceServices CurrentAccountBalance

Fig 2.9: Current Account Balance prepared from quartely data from the year 1981 to

2014

from Statistics Norway. In Norway, current balance is highly inﬂuence by the

balance in goods. Figure-2.9 shows that the balance in services in Norway is

decreasing while the balance in Goods has boost up after around 1998. Further,

the balance in services plotted in the same ﬁgure from the quarterly data exhibit

a seasonal trend which is usual in Norway.

Imports

Machinery & equipment, chemicals, metals and food stu↵saremajorimportsof

Norway. Sweden (13.6%), Germany (12.4%), China (9.3%), Denmark (6.3%), UK

(6.1%) and US (5.4%) are major import partners 3. The monthly imports of new

ships (ImpNewShip), oil platform (ImpOilPlat), old ships (ImpOldShip) and all

3https://www.cia.gov/library/publications/the-world-factbook/geos/no.html

19

ImpExShipOilPlat ImpNewShip

ImpOilPlat ImpOldShip

200

300

400

500

0

10

20

30

0

25

50

75

0

20

40

60

80

2000 2005 2010 2015 2000 2005 2010 2015

Date (Monthly)

Value (NOK hundreds)

Fig 2.10: Time Series plot of major imports of Norway

other items excluding ship and oil platform (ImpExShipOilPlat)areconsidered

as predictor variable in data analysis. The time-series plot for these variables are

presented in ﬁgure-2.10

Exports

Norway is richly endowed with natural resources - petroleum, hydro-power, ﬁsh,

forests, and minerals but the economy is highly dependent on the petroleum sec-

tor 3. Petroleum products, machinery and equipment, metals, chemicals, ships and

ﬁshes are major exports of Norway 3. The monthly time series for the Export of

condensed fuel (ExpCond), crude oil (ExpCrdOil), natural gas (ExpNatGas), new

ships (ExpNewShip), oil platform (ExpOilPlat), old ships (ExpOldShip) and all

other exports excluding ships and oil platforms (ExpExShipOilPlat)arepresented

in ﬁgure-2.11.

20

ExpCond ExpCrdOil

ExpExShipOilPlat ExpNatGas

ExpNewShip ExpOilPlat

ExpOldShip

0

5

10

15

20

200

300

400

500

600

700

800

900

100

200

0

5

10

15

20

0

10

20

30

0

5

10

15

20

2000 2005 2010 2015

Date (Monthly)

Value (NOK hundreds)

Fig 2.11: Time Series plot of major exports of Norway

2.5.2 Capital and Financial Accounts

The following text of capital and ﬁnancial accounts are adapted from International

ﬁnancial management by Madura (2012). A capital account includes transaction

of inter-country transfer of ﬁnancial assets due to immigration and non-ﬁnancial

assets such as buying and selling of patents and trademarks. These transaction

are relatively minor in comparison to the items of ﬁnancial accounts. The key

elements of ﬁnancial account are,

•Direct Foreign Investment includes investment in ﬁxed assets in foreign

countries.

21

CapitalTransferAbroad AcqPatentLeisense NetLending

−50

−40

−30

−20

−10

0

10

−2.5

0.0

2.5

5.0

0

400

800

1200

1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010

Date (Monthly)

Value (NOK hundreds)

Fig 2.12: Time Series plot of variables related to capital account

•Portfolio Investment includes transaction of long term ﬁnancial assets

such as bonds and stocks.

•Other Capital Investment includes short term ﬁnancial assets such as

money market securities.

•Errors, Omissions and Reserves includes adjustment for negative bal-

ance in current account.

Due to unavailability of monthly data for capital accounts, this thesis has not

included the data in the analysis. The time series plot from quarterly totals for

the variables related to capital account are plotted in the ﬁgure-2.12. The ﬁgure

shows that the economy of Norway has drastically heated after the year around

1998.

22

OilSpotPrice

0.5

1.0

1.5

2.0

2000 2005 2010 2015

Date (Monthly)

Value (NOK hundreds)

Fig 2.13: Time Series plot of oil spot price from Jan 2000

2.6 Oil Spot Price

After the discovery of oil in the North Sea in late 1969, economy of Norway has

transformed completely (Norway The rich cousin 2013). Since the economy of

Norway is highly depended on its petroleum products, oil spot price also has

inﬂuence on foreign exchange rate of Norway. However, Ferraro, Rogo↵,andRossi

(2012) argued that the predictive ability of exchange rate from oil price is more

e↵ective at a daily frequency and is hardly visible at monthly frequencies. Oil spot

price is also considered as predictive variable in this thesis. The heavy ﬂuctuation

in the oil spot price shown in time series plot (ﬁg-2.13) is due to the ﬁnancial crisis

of 2007-2009.

23

2.7 Lagged response variable as predictor

Exchange rate, being a time-series variable, contains autocorrelation which can be

checked out (soften) by including the lagged variables of the response as predictor.

Further, the correlation of response (PerEURO) with its ﬁrst lag and second lag

are 0.94 and 0.86 respectively. In addition, two spikes which are signiﬁcant in the

partial autocorrelation function as plotted in ﬁgure-2.14 also indicate for the use of

auto-regressive terms in the model. This thesis has included the ﬁrst and second

lag of response variable as a predictor.

−0.25

0.00

0.25

0.50

0.75

0 5 10 15 20

Lag

Partial.ACF

Fig 2.14: Partial autocorrelation function for Exchange Rate of NOK per Euro. The red

dashed line denotes the 95% level of signiﬁcance.

24

2.8 E↵ect of Crisis period

Financial crisis unleashed in the United State in summer 2007. The crisis extended

towards Europe which has created a series of diﬃcult situations in the ﬁnancial

market. Inter bank interest rate rose dramatically, stock market plunged and

banks incurred serious funding problem with losses on their head (The Financial

Market in Norway 2008: Risk outlook 2009).

Norway has been a↵ected by the crisis through various channels. Sharp fall

in commodity price, devaluation of companies and low international demand has

direct impact in exchange rate of NOK. The data during those period has high

inﬂuence in the statistical model using in this thesis. The inﬂuence of crisis is

visible in the plots of Appendix-C.

25

Chapter 3

Models and Methods

3.1 A statistical Model

A statistical model describes the relationship between a cause and its e↵ect. Let a

vector ycontains nnumber of responses and Xbe a n⇥pmatrix whose columns

are predictor variables and each of them have nobservations. These variables in

Xcan a↵ect yso, the relationship between Xand ycan be written in a functional

form as,

y=f(X)+✏(3.1)

where, ✏is a vector of unknown errors usually referred as ‘white noise’ when

dealing with time-series data which is assumed to have zero mean, constant vari-

ance and no autocorrelation.

26

3.2 Linear Regression Model

The linear regression model with a single response (Y=yt1,y

t2,...,y

tp)andp

predictor variable X1,X

2,...,X

phas form,

Y

Response

=0+1Xt1+2Xt2+...+pXtp

Mean Response explained by predictors only

+✏

Error Term

(3.2)

The model - 3.2 is linear function of p+1 unknown parameters ,1,2,...,p

which is generally referred as regression coeﬃcients. In matrix notation, equation-

(3.2) becomes,

Y

n⇥1=X

n⇥(p+1)

(p+1)⇥1

+✏

n⇥1(3.3)

3.2.1 Least Square Estimation

The estimate of the unknown parameter vector in (3.3) is obtained by minimizing

the sum of square of residuals, The sum of square of residuals is,

✏t✏=(YX)t(YX) (3.4)

On minimizing equation - 3.4, we get the OLS estimate of as,

ˆ

OLS =(XtX)1XtY(3.5)

For ordinary least square estimation, following basic assumptions (Wooldridge,

2012) are required,

1. Linear in parameter

27

2. Absence of Multicollinearity

3. No correlation between Error terms and predictor variable, mathematically,

E(✏i|X)=0,t =1,2,...,n

The equation implies that the error term at time tshould be uncorrelated

with each explanatory variable in every time period

4. Homoskedastic Error terms, i.e,

var(✏t|X)=var(✏t)=2I

5. No serial correlation (autocorrelation) in error terms, i.e,

corr(✏t,✏s)=0,8t6=s

For Hypothesis testing and inference using tand Ftest, an additional assumption

of normality is needed, i.e

✏t⇠N(0,2I)

Under the assumption from 1 to 5, the OLS estimate obtained from equation-3.5

is best linear unbiased estimator (BLUE) of .

28

3.2.2 Prediction

Using ˆ

obtained in equation-3.5, following two matrices can be obtained,

Predicted Values: ˆ

Y=Xˆ

=X(XtX)1XtY(3.6a)

Residuals:ˆ

✏=Yˆ

Y=[IX(XtX)1Xt]Y(3.6b)

Here equation-3.6a gives predicted values of Ywhich on subtracting from

observed value give the predicted error terms as is presented in equation-3.6b.

Equation-3.6a can also be written as,

ˆ

Y=Xˆ

=HY (3.7)

Here, His called Hat matrix and is the orthogonal projection of yonto the

space spanned by the columns of X.

3.3 Variable selection

Although including many variables in the model can add information, they are

also the source of unnecessary noise. In addition, many variables in a model is

also the cause of multicollinearity. So, a model that is simple yet contain useful

information is always desirable. Variable selection is intended for selecting best

subset of predictor variables. Some of the criteria for variable selection as described

in Applied linear regression by Weisberg (2005) are discussed below:

29

3.3.1 Criteria for variable selection

Suppose Xsis selected set of variable which gives the predicted output of,

ˆ

Y=E(Y|Xsxs)=0

sxs(3.8)

If Xsmisses important variables, the residual sum of squares of ﬁtted model in

equation-3.8 will be larger than the full model. Lack of ﬁt for selecting the set Xs

is measured by its Error sum of square.

Model statistic Approach

When a model is ﬁtted, various statistics such as R2,R2-adj, F-statistic are

obtained which measures the quality of that model. Based on these statistic,

a model is selected as better than others.

Information Criteria

Another common criterion, which balances the size of the residual sum of

squares with the number of parameters in the model (Johnson and Wich-

ern, 2007, p. 386), for selecting subset of predictor variable is AIC (Akaike

Information Criterion). It is given as,

AIC = nlog(RSSs/n)+k (3.9)

where, RSS=Residual Sum of Square, n=number of observation and

k=Number of variables included in the model

A model with smaller value of AIC obtained from equation-3.9 is better

better than other with larger AIC. An alternative to AIC is its Bayesian

30

analogue, also known as Schwarz or Bayesian information criteria. Bayesian

Information Criteria provides balance between model complexity and lack of

ﬁt. Smaller value of BIC is better.

BIC = nlog(RSSs/n)+klog(n) (3.10)

A third criterion that balances the complexity and lack of ﬁt of a model is

Mallows Cp(Mallows, 1973), where the subscript pis the number of variables

in the candidate model. The formula for this statistic is given in equation-

3.11,

Mallows Cp=RSS

ˆ2+2kn (3.11)

Where, ˆ2is from the full model. A plot of Cpvs kfor each subset of

predictors indicate models that predict the responses well. Better models

usually lie near the 45line of the plot.

3.3.2 Computational procedure for variable selection

When a model is large, ﬁtting all possible subsets is not feasible. Furnival and

Wilson (1974) suggested several algorithm to calculate residual sum of square of

all possible regression called leap and bound technique which has been widely

implemented in statistical software. However, this method is not appropriate for

criteria based on model statistic where step wise methods can be used. methods

has three basic variation (Weisberg, 2005, p. 221).

Forward selection procedure

Model is started without any variable and in each step a variable is added and

31

the model is ﬁtted. The variable is left in the model if the subset minimizes

the criterion of interest . Similar process is repeated for other predictor

variables.

Backward elimination procedure

This process is like the reverse of Forward selection procedure. In this pro-

cess, the model is ﬁtted with all the predictor variable and variables are

removed one at a time except those that are forced to be in the model. The

model is examined against the considered criteria. Usually, the term with

smallest t-value is removed since this gives rise to the residual sum of square.

Stepwise procedure

This combines both Forward selection procedure and Backward elimination

procedure. In each step, a predictor variable is either deleted or added so

that resulting model minimizes the criterion function of interest.

3.4 Principal Component Analysis

The purpose of PCA is to express the information in X=(X1,X

2,...,X

p)bya

less number of variables Z=(Z1,Z

2,...,Z

q); q<pcalled principal components

of X(Martens and Naes, 1992). These principal components are orthogonal and

linearly uncorrelated. Since they are computed from the linear combinations of

Xvariables, the variation in Xvariables are compressed in ﬁrst few principal

components. In other words, the ﬁrst principal components is the direction along

which the Xvariables have the largest variance (Massart, 1998). In this situation,

the multicollinearity in Xis not a problem any more.

32

The principal components can be performed on Covariance or Correlation ma-

trix. If the variables are of same units and their variances do not di↵er much, a

covariance matrix can be used. However the population correlation matrix is un-

known, its estimate can be used. In this thesis, sample correlation matrix is used

to compute sample principal components. Construction of principal components

requires following steps,

1. Estimate the correlation matrix Aof Xas,

corr(X) = (diag(⌃))1

2⌃(diag(⌃))1

2(3.12)

Using sample observation, equation-3.12 can be estimated as,

A= corr(X) = (diag(S))1

2S(diag(S))1

2(3.13)

Where Sis the sample estimate of covariance matrix ⌃,

S=Eh(XE[X]) (XE[X])Ti(3.14)

2. Calculate eigenvalue and eigenvector of the correlation matrix obtained in

equation-3.13. An eigenvalue ⇤of a square matrix Aof rank pis a diagonal

matrix of order pwhich satisﬁes,

AE =E⇤(3.15)

where,

33

⇤= diag(1,2,...,p) (3.16)

In PCA these eigenvalues are arranged in descending order, i.e. 1

2... p. For each eigenvalues there is an eigenvector. Let E=

(v1,v2,...,vp) be the matrix of eigenvector so that the correlation matrix

Acan be decomposed and expressed as,

A=E⇤E1=E⇤ET(3.17)

Equivalently, |AiIn|E= 0 which can only be realized if AiInis

singular, i.e.,

|AiIn|= 0 (3.18)

Equation-3.18 is called the characteristic equation where, Ais the correla-

tion matrix obtained from equation-3.13. The root of the equation is called

eigenvalues (Seber, 2008) and the vector Eiis called eigenvector correspond-

ing to the eigenvalue i. The eigenvector obtained from equation-3.15 are

then normalized, i.e. ||Ei||2=1.

3. Since, the variation explained in data are accumulated in ﬁrst few principal

components, only keigenvalues are considered. The corresponding eigenvec-

tors of those eigenvalues is called projection matrix. The projection matrix

is,

34

P=✓ET

1ET

2... ET

k◆T

(3.19)

The projection matrix in equation-3.19 projects the data matrix into lower

dimensional subspace Zi. i.e.,

Z=PX (3.20)

The column vectors of matrix Zobtained from 3.20 are the orthogonal pro-

jections of data matrix Xinto kdimensional subspace. These components

are the linear combination of the rows of matrix Xsuch that the most vari-

ance is explained by the ﬁrst column vector of Zand second one has less

variance than the ﬁrst one and so on. Here,

var(Zi)=iand

cov(ZiZj)=0fori6=j

3.5 Principal Component Regression

The components of Principal Component Analysis (PCA) accumulate the varia-

tion in predictor variables on ﬁrst few components. A linear regression ﬁtted with

only those components can give a similar results as the full linear model. How-

ever, Jolli↵e(1982)inhispaper“Anoteontheuseofprincipalcomponentsin

regression”, has given many examples taken from di↵erent papers of various ﬁelds

where the components with low variance are also included in regression equation

35

in order to explain most variation in the response variable. Following are the steps

to perform Principal Component Regression. These steps are based on the paper

“A comparison of partial least squares regression with other prediction methods”

by Yeniay and Goktas, 2002.

1. First principal components are obtained for Xas explained in section-3.4.

The PCs obtained are orthogonal to each other.

2. Suppose mPC which are supposed to inﬂuence the response are taken and

a regression model is ﬁtted as,

Y=Zm↵m+✏(3.21)

3. Here, ↵m=ZT

mZm1ZT

mYare the coeﬃcients obtained from OLS methods.

Using this alpha, one can obtain the estimate of as,

ˆ

PCR =PPTXTXP1PTXTY(3.22)

Here, Pis a projection matrix deﬁned in equation-3.19.

Since, PCR includes only mcomponents, the estimate obtained are biased. ;The

number of components mcan be chosen by cross-validation the prediction mean

squared error (RMSEP). If all the components are included in the model, estimates

obtained from PCR, i.e. PCR are identical to the estimates of OLS (OLS).

36

3.6 Partial Least Square Regression

Partial Least Square Regression (PLS) is relatively new method and it can be

used for both univariate and multivariate regression. It constructs a new set of

variables called latent variable (or factor or components) from the linear combi-

nation of predictor variables X1,X

2,...,X

n(Garthwaite, 1994) as in the case of

principal components, however PCR construct components (factors) maximizing

the variation of data matrix(X) while PLS construct them using the variation

in both Xand Y(Yeniay and Goktas, 2002). The intention of PLS is to create

latent variables (components) that capture most of the information in the Xvari-

ables that is useful for predicting Y1,Y

2,...,Y

p, while reducing the dimension of

the regression problem by using fewer components than the number of X-variables

(Garthwaite, 1994). Partial least square regression can be performed using follow-

ing steps. These steps are adapted from the paper “PLS-regression: a basic tool of

chemometrics” from Wold, Sj¨ostr¨om, and Eriksson (2001). The Xand Ymatrices

are column centered for the ease of computation.

1. PLS estimates the latent variables also called X-scores denoted by

ta,(a=1,2,...,A), where Ais the number of Components a model

has considered. These X-scores are used to predict both X and Y, i.e. both

X and Y are assumed to be modeled by the same latent variable. The

X-scores are estimated as linear combination of original variables with the

coeﬃcients W(wka) as in equation-3.23, i.e,

tia =

p

X

k=1

W⇤

kaXik (T=XW⇤)(3.23)

37

Where, W⇤is a vector of weights w⇤

aof X. It is obtained as in equation-3.24

below as a normalized coeﬃcients obtained on regressing Xon a column of

Y.

W⇤=Xty(i)

kXty(i)k(3.24)

Here, y(i)is any column of response matrix Y.

2. The x-scores (T) are used to summarize Xas in the equation-3.25. Since

the summary of Xexplained most of the variations, the residuals (E)are

small.

Xik =X

a

tiaPak +eik ;(X=TP0+E)(3.25)

A similar setup can be used to have the summary for Y-matrix as in equation-

3.26,

Yim =X

a

uiaqam +gim ;(Y=UQ0+G)(3.26)

where, U=YQand Q=TtY

3. The X-scores (T) are also good predictor of Y, i.e.,

yim =X

a

qmatia +fim (Y=TCt+F)(3.27)

Here, Fis the deviation between the observed and modeled response.

38

4. Coeﬃcients Estimates:

Equation(3.27) can also be written as,

yim =X

a

qma X

k

w⇤

kaxik +fim

=X

k

bmkxik +fim

In matrix notation this can be written as,

Y=XW⇤Ct+F=XB +F(3.28)

Thus, the estimates of PLS coeﬃcients are obtained as,

ˆ

bmk =X

a

qmaw⇤

ka (3.29)

i.e., BPLS =W⇤Ct(3.30)

Above process is repeated for each components (a), the matrix Xand Yare

“deﬂated” by subtracting their best summaries (TPtfor Xand QCtfor Y).

The Residuals obtained are used as new Xand Yin the computation process

for new component. However, the deﬂation of Yis not necessary since the result

is equivalent with or without the deﬂation (Wold, Sj¨ostr¨om, and Eriksson, 2001,

p. 5).

Various algorithm exist to perform PLS regression among which NIPLS and

SIMPLS are in fashion. This thesis has opted NIPLS (Nonlinear Iterative Partial

Least Square) regression which is performed by oscores method of pls package

in R. In the algorithm, the ﬁrst weight vector (w1)istheﬁrsteigenvectorofthe

39

combined variance-covariance matrix XtYYtXand the following weight vectors

are computed using the deﬂated version. Similarly, the ﬁrst score vector (t1)is

computed as the ﬁrst eigenvector of XXtYYtand the following x-scores uses the

deﬂated version of the matrices.

3.7 Ridge Regression

When the minimum eigenvalue of XtXmatrix is very much smaller than unity (i.e.

min << 1), the least square estimate obtained from equation-3.5 are larger than

average (Marquardt and Snee, 1975). Estimates based on [XtX+Ip],0

rather than XtXcan solve these problems. A.E. Hoel ﬁrst suggests that to control

instability of the least square estimate, on the condition above, can be;

ˆ

⇤

ridge =⇥XtX+I⇤1XtY;0

=WXtY(3.31)

The analysis build around equation-3.31 is called “ridge equation”. The relation-

ship of ridge estimate with ordinary least square is,

ridge =hIp+XtX1i1ˆ

OLS

=Zˆ

OLS (3.32)

40

Here, as !0,ˆ

ridge =ˆ

OLS and !1,ˆ

ridge = 0 Further, the hat matrix for

Ridge regression is given as,

Hridge =XXtX+I1Xt(3.33)

All the theory behind Ridge Regression described above are cited from “Ridge

regression: Biased estimation for nonorthogonal problems” by Hoerl and Kennard

(1970).

3.8 Comparison Criteria

After ﬁtting models with various methods, it becomes necessary to test their valid-

ity for their results to be trusted. Models react di↵erently for the new information

during prediction as the quality of model highly depends on their estimates. Since

the purpose of this thesis is to compare di↵erent models, the basis for their com-

parison are set as their (a) Goodness of ﬁt and (b) Predictability.

3.8.1 Goodness of ﬁt

A model is assumed to follow some hypothetical state of being ideal. Setting up this

state as null hypothesis (H), in many situations, the test of goodness of ﬁt for a

model construct an alternative hypothesis simply stating that the model gives little

or no information about the distribution of the data. However in other situation,

such as testing for no e↵ect of some speciﬁc variable in the model, rejection of

Hindicate that the variable is useful in the model (D’Agostino, 1986, p. 1). A

goodness of ﬁt for a model depends on many aspects such as,

41

Residual obtained after the model ﬁt

Residuals obtained from the ﬁtted model are assumed to be random and

normal considering that no useful information are still content on them.

Outlier

Outliers can distort the analysis toward unintentional direction creating false

estimates. Models without such outliers are considered better.

Variance explained by the model

The variance explained by the model is generally measured by R2or R2adj

in linear models. More the variation contained in the data is explained by

the model, better the model is considered. In the case of PLS and PCR, the

residuals contains very little information left on the ignored components.

Relative value of Information Criteria such as AIC and BIC

AIC (Akaike information criterion) and BIC (Bayesian information criterion

or Schwarz criterion) measures relative quality of models. Although, it is not

an absolute measure of the model quality, it helps to select a better model

among others. AIC is deﬁned as in equation - 3.34 which is free from the

ambiguities present in the conventional hypothesis testing system (Akaike,

1974).

AIC = (2) log(L)+2(k) (3.34)

where, L= maximum likelihood and k= number of independently adjusted

parameters within the model For least square case, above formula resembles

to equation - 3.9 (Hu, 2007).

42

3.8.2 Predictability

Prediction is highly inﬂuenced by the model in used. So, prediction strongly

depends on the estimates of a model. False and unstable estimate makes the

prediction poor and unreliable. On one side, providing more information (variable)

can well train the model resulting more precise prediction. On the other hand,

over-ﬁtting, which attempts to explain idiosyncrasies in the data, leads to model

complexity reducing the predictive power of a model. In the case of PLS and PCR,

adding more components results in including noise in the model.

123456789

1

2

3

4

5

6

0

Under ﬁtting Over ﬁtting

Error of prediction

Model Error

Estimation Error

Complexity of Calibration Model

Fig 3.1: Model Error - Estimation Error and Prediction Error

The relationship between the model complexity and the prediction error is

presented in ﬁgure-3.1 with the case of under-ﬁtting and over-ﬁtting of a model.

Furthermore, a model exhibits an external validity if it closely predicts the

observations that were not used to ﬁt the model parameters (Lattin, Carroll, and

Green, 2003, p. 72). An over-ﬁtted model fails to perform well for those obser-

43

vation that are not included during model parameter estimation. The dataset in

this thesis is divided into two parts. The ﬁrst part includes the observations from

Jan 2000 to December 2012 and the second one includes observation onward till

November 2014. A cross-validation approach is utilized on the ﬁrst set of observa-

tion to train the model. The model is used to predict the exchange rate of NOK

per Euro from the predictors of the second set of observations. Figure - 3.2 shows

the procedure adopted for prediction in this thesis.

Tra i n i n g D a taset

(Jan 2000 - Dec 2012)

Test Dataset

(Jan 2013 - Nov 2014)

Dataset

Y

train X

train

Y

test X

test

Calibrated

Model

•Linear Model

•PCR

•PLS

•Ridge

Model Compari-

sion Criteria

•Goodness of Fit

•Predictability

Best Model

Cross

Validation

Tes t d a t a

for Prediction

Tes t S t a t i stic

AIC, BIC, R2,

R2pred, RMSEP etc

Fig 3.2: Procedure adopted in the thesis for model comparison. A cross-validation tech-

nique is used to validate the trained dataset. The trained model is used to

predict the test response from with prediction errors are obtained.

Cross-Validation

There are various cross-validation techniques among which two are described be-

low;

K-Fold Cross-validation:

The dataset are split into kequal parts. For each i=1,2,...,k, a model

44

is ﬁtted leaving out the ith portion. A prediction error is calculated for this

model. The process is repeated for all i. The prediction error for K-fold

cross validation is obtained by averaging the prediction error of each of the

model ﬁtted.

Leave-one-out cross validation:

This is a special case of kfold cross-validation where k=n(number of

observation), i.e, each time one observation is removed and the model is

ﬁtted.

Prediction Error

Prediction of a model becomes precise if the error is minimum. Models can be

compared according to their predictability. Understanding of di↵erent measures

of prediction error is necessary to acknowledge their predictability and eventually

perform model comparison.

Root Mean Square Error (RMSE)

RMSE is the measure of how well the model ﬁt the data.

RMSE = v

u

u

t1

n

n

X

i=1

(yi