Comparative Performance of ARIMA and ARCH/GARCH Models on Time Series of Traffic ...

111 Pages·2016·2 MB·English

The Islamic University of Gaza
Deanship of Postgraduate Studies
Faculty of Science
Department of Mathematics

Comparative Performance of ARIMA and
ARCH/GARCH Models on Time Series of
Traffic Accidents in Gaza Strip
ثداوحلل ةينمزلا لسلاسلا ىلع شراق / شرآ جذامنو اميرآ جذامنل نراقملا ءادلأا
ةزغ عاطق يف ةيرورملا
Submitted By

Raja’ M. Al-Ajez

Supervised By

Dr. Bisher M. Iqelan

A Thesis Submitted In Partial Fulfillment Of Requirements For The
Degree Of M.SC. Of Mathematics

January, 2016

مِيحِرَّلا نِ ٰـَحْْ رَّلا هِـَّللا مِسْ ِب

نْ َأوَ يَّ دَِلاوَ ىٰ َلعَوَ يَّ َلعَ تَ مْ عَـْنَأ تَِّلا كَ َتمَعِْن رَكُ شْ َأ نْ َأ نِعْزِوَْأ ب ِ رَ"
" يَ ِلِِ اصَّ لا كَ دِاَبعِ فِ كَ ِتَحْْ رَِب نِلْخِ دَْأوَ ُ هاضَ رْـَت اًلِِ اصَ لَ مَعْ َأ

﴾١٩﴿ةيآ لمنلا ةروس
Dedication

To my country, Palestine.

To my family, my relatives and friends.
Contents
Acknowledgments vii
List of Abbreviations ix
Abstract x
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Research Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 Organization of The Research . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Fundamental Concepts and Definitions 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 The Backshift Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
i
2.6 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Autocorrelation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7.1 Autocorrelation Function . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7.2 Partial Autocorrelation Function . . . . . . . . . . . . . . . . . . . 17
2.8 Unit Root Tests for Stationarity . . . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Tests for Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.10 Model Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.10.1 The Akaike Information Criterion(AIC) . . . . . . . . . . . . . . . . 22
2.10.2 Bayesian Information Criterion (BIC) . . . . . . . . . . . . . . . . 22
2.10.3 The Corrected Akaike Information Criterion (AICc) . . . . . . . . . 23
2.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Seasonal ARIMA Models 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 General Linear Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Models for Stationary Time Series . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Autoregressive Models (AR) . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Moving Average Models (MA) . . . . . . . . . . . . . . . . . . . . . 27
3.3.3 Autoregressive Moving Average Models(ARMA) . . . . . . . . . . . 28
3.4 Models for Non-stationary Time Series . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Autoregrresive Integrated Moving Average ARIMA(p,d,q) Models . 30
3.4.2 Seasonal ARIMA Models (SARIMA) . . . . . . . . . . . . . . . . . 30
3.5 Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Identification of ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6.1 MA and AR Models . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6.2 ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Estimation of ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7.1 AR and MA Models . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7.2 ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ii
3.8 Diagnostic of ARIMA Models . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.9 Forecasting with ARIMA Models . . . . . . . . . . . . . . . . . . . . . . . 40
3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Conditional Heteroscedasticity:ARCH/GARCH Models 42
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 ARCH(q) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 ARCH(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.2 Weakness of the ARCH Models . . . . . . . . . . . . . . . . . . . . 45
4.3 GARCH(p,q) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 GARCH(1,1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Some Extentions of the GARCH Models . . . . . . . . . . . . . . . . . . . 51
4.5.1 Integrated GARCH Model . . . . . . . . . . . . . . . . . . . . . . . 51
4.5.2 ARIMA - GARCH Model . . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.7 Identifying an ARCH/GARCH Models . . . . . . . . . . . . . . . . . . . . 54
4.8 Estimation of GARCH (p,q) Models . . . . . . . . . . . . . . . . . . . . . . 55
4.9 Model Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.10 Testing for ARCH Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.11 Forecasting with GARCH(p,q) Models . . . . . . . . . . . . . . . . . . . . 58
4.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Case Study 60
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Data Description and Basic Statistics . . . . . . . . . . . . . . . . . . . . . 60
5.3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Fitting of SARIMA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.1 Preparing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.2 Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.3 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
iii
5.4.4 Parameter Estimation of SARIMA(3,1,2)(1,0,1) Model . . . . . 67
12
5.4.5 Diagnostic Checking for SARIMA(3,1,2)(1,0,1) Model . . . . . . 67
12
5.4.6 Forecasting with SARIMA(3,1,2)(1,0,1) . . . . . . . . . . . . . . 68
12
5.5 Fitting of GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5.1 Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5.2 Testing for ARCH Effects . . . . . . . . . . . . . . . . . . . . . . . 72
5.6 Model Selection and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.7 Parameter Estimation of ARMA(1,1)-GARCH(1,1) . . . . . . . . . . . . . 73
5.8 Diagnostic Checking of ARMA(1,1)-GARCH(1,1) Model . . . . . . . . . . 74
5.9 Forecasting with ARMA(1,1)-GARCH(1,1) Model . . . . . . . . . . . . . . 79
5.10 Comparison between Best ARIMA and Best GARCH Models . . . . . . . . 82
5.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Conclusions and Recommendations 84
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Refrences 86
iv
List of Figures
2.1 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1 Time plot and ACF and PACF of monthly Traffic Accidents in Gaza Strip 62
5.2 First difference for Traffic Accidents time series and ACF ,PACF . . . . . . 64
5.3 Residuals analysis for the SARIMA(3,1,2)(1,0,1) fit to Traffic Accidents
12
data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Sarima Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5 TimeseriesplotofstandardizedresidualsofARMA(1,1)-GARCH(1,1)pro-
cess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.6 Time series plot of standared deviation of ARMA(1,1)-GARCH(1,1) process 76
5.7 QQ plot of ARMA(1,1)-GARCH(1,1) model . . . . . . . . . . . . . . . . . 77
5.8 ACF plot of standardized Residuals . . . . . . . . . . . . . . . . . . . . . . 78
5.9 ACF plot of Squared standardized Residuals . . . . . . . . . . . . . . . . . 79
v