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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