# natalie fuller thesis

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School of Computing, Engineering and Mathematics

Can GDP Be Forecasted Using Statistical Models

Natalie Fuller

May 2014

Declaration

I declare that no part of the work in this report has been submitted in support of an application

for another degree or qualification at this or any other institute of learning.

Natalie Fuller

i

Acknowledgements

I would like to acknowledge my supervisor Alison Bruce for her help and encourangement with

this project.

ii

Abstract

This project compares alternative types of statistical modelling techniques to model and forecast

the rate of change of GDP in the UK. Statistical modelling has been carried out using publicly

available data measured quarterly from quarter 1 - 1970 through to quarter 3 - 2013. Within this

project the Box-Jenkins ARIMA, ARCH and GARCH modelling techniques are compared, and

the optimal models for each technique are decided. The Box-Jenkins ARIMA model is used to

forecast for GDP itself, whereas ARCH and GARCH models are employed to forecast the variance

in the series. Inflation is added as an explanatory variable to the modelling technique with the

best fit to the data thus creating a bivariate model. Forecasts are calculated for each modelling

technique, and the forecasting technique with the highest predictive performance is found to be

the bivariate AR(1)/GARCH(1,1) model.

Supervisor: Alison Bruce

iii

Contents

1 Introduction 1

1.1 General objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Specific objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Notation and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Review 4

3 Data Overview and Analysis 6

3.1 GDP - The expenditure approach . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Timeplot analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Univariate Box-Jenkins Modelling 9

4.1 ARIMA (p,d,q) modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 The ARIMA modelling process . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Check for non-stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . 12

iv

4.2.2 Model identification and selection . . . . . . . . . . . . . . . . . . . . . 14

4.2.3 Parameter testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.4 Residual analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.5 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Forecasting GDP by an ARMA(1,1) model . . . . . . . . . . . . . . . . . . . . . 20

5 Univariate GARCH Modelling 23

5.1 GARCH(p,q) modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.1.1 Check for heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1.2 Parameter testing and model identification . . . . . . . . . . . . . . . . . 26

5.1.3 Residual analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 AR(P)/GARCH(p,q) modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 Choosing the order of the autoregressive term . . . . . . . . . . . . . . . 32

5.2.2 Parameter testing and model identification . . . . . . . . . . . . . . . . . 37

5.2.3 Residual analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3 Forecasting GDP by a univariate GARCH model . . . . . . . . . . . . . . . . . . 39

6 Multivariate GARCH Modelling 42

6.1 Inflation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

v

6.2 Bivariate GARCH modelling Process . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2.1 Parameter testing and model identification . . . . . . . . . . . . . . . . . 45

6.2.2 Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.3 Forecasting GDP by a bivariate GARCH model . . . . . . . . . . . . . . . . . . 49

7 Results 51

7.1 Optimal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2 Bivariate AR(1)/GARCH(1,1) model analysis . . . . . . . . . . . . . . . . . . . 52

8 Conclusion 55

8.1 Suggestions to improve upon this investigation . . . . . . . . . . . . . . . . . . . 56

Bibliography 60

Appendix 61

A.1 UK Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.2 MSE Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.3 Parameter P values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.4 SAS code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

vi

Chapter 1

Introduction

Throughout recent years the UK economy has been through a tough time, experiencing World

Wars, political upheavals, and banking crises. During the 1970s the economy was suffering due

to political malice[1], but also thriving due to the Bank of England reducing the regulations on

mortgages[2]. Now we are in the early 21st century the status of the UK economy is rising. A

banking crisis hit the UK in 2008[3], and the stability of the economy has been improving since

then.

The overall status of the economy in the UK is measured by a quarterly figure called gross domestic

product (GDP). GDP figures are published approximately a month after the banking quarter end

causing a one month lag. This lag in data retrieval means that there is uncertainty about how

the economy is performing at this present moment, or where it could potentially be in the future.

There is a great demand for forecasts as when making important economic decisions, it is helpful

to know the current state of the economy.

NIESR (National Institute of Economic and Social Research) provide economic forecasts by using

expertise in both quantitative and qualitative methods [4]. Forecasting methods used by such

companies are undisclosed, therefore a review of the subject will be carried out to examine the

methods of other statisticians when forecasting GDP globally.

This project will focus on forecasting GDP using statistical models. A time series is a collection

of data points that have been measured sequentially throughout time. GDP can be described as

1

CHAPTER 1. INTRODUCTION 2

time series data as it has been measured throughout history at quarterly time points. A large

number of time series statistical models are available, and these can be used to calculate forecasts.

In this investigation a collection of univariate (a modelling process involving one variable) and

bivariate (a modelling process involving two variables), Box-Jenkins and heteroscedastic mod-

elling techniques are employed. These are techniques that have been widely used by statisticians.

The Box-Jenkins approach carried out is the autoregressive integrated moving average (ARIMA)

modelling process. The ARIMA model combines autoregressive (AR) and moving average (MA)

models to forecast the value of GDP in the future. The ARIMA model assumes that the variance

is constant over time, though it is suggested that for this data set the variance could be sporadic.

Consequently, the autoregressive conditional heteroscedasticity (ARCH) model and its extension

of the generalised autoregressive conditional heteroscedasticity (GARCH) model are used to ac-

commodate for the changes in variance. The variance is modelled to calculate a forecasted value

for the percentage change in GDP at the next time point.

Inflation is introduced into the modelling process as an explanatory variable to create a bivariate

model. The aim of bivariate modelling is to improve upon the univariate forecasting methods.

1.1 General objective

The general objective of this investigation is to determine whether the percentage change in GDP

from quarter to quarter can be forecasted using statistical models.

1.2 Specific objectives

To model GDP using a variety of statistical modelling techniques, and find a final modelthat fits the data for each technique.

Calculate a forecast value for each model for the percentage change in GDP from theprevious quarter in the UK for Q4-2013.

Determine the optimal model/modelling technique to forecast GDP in the UK.

Determine the level of accuracy of the model.

CHAPTER 1. INTRODUCTION 3

1.3 Notation and abbreviations

Notation

Greek letters are used to denote parameters.

Greek letters emphasised with a hat denote the estimate of a parameter. For example theestimated value of is .

Parameter tests are carried out with a general hypothesis of:H0: Parameter = 0.

H1: Parameter 6= 0.

A 95% confidence limit is assumed for each hypothesis test.

All modelling and forecasting is carried out using SAS 9.3 statistical software.

Abbreviations

ACF Autocorrelation Function

ADF Augme