An Introduction to State Space Time Series Analysis Book

List of Figures x

List of Tables xiv

1 Introduction 1

2 The local level model 9

2.1 Deterministic level 10

2.2 Stochastic level 15

2.3 The local level model and Norwegian fatalities 18

3 The local linear trend model 21

3.1 Deterministic level and slope 21

3.2 Stochastic level and slope 23

3.3 Stochastic level and deterministic slope 26

3.4 The local linear trend model and Finnish fatalities 28

4 The local level model with seasonal 32

4.1 Deterministic level and seasonal 34

4.2 Stochastic level and seasonal 38

4.3 Stochastic level and deterministic seasonal 42

4.4 The local level and seasonal model and UK inflation 43

5 The local level model with explanatory variable 47

5.1 Deterministic level and explanatory variable 48

5.2 Stochastic level and explanatory variable 52

6 The local level model with intervention variable 55

6.1 Deterministic level and intervention variable 56

6.2 Stochastic level and intervention variable 59

7 The UK seat belt and inflation models 62

7.1 Deterministic level and seasonal 63

7.2 Stochastic level and seasonal 64

7.3 Stochastic level and deterministic seasonal 67

7.4 The UK inflation model 70

8 General treatment of univariate state space models 73

8.1 State space representation of univariate models 73

8.2 Incorporating regression effects 78

8.3 Confidence intervals 81

8.4 Filtering and prediction 84

8.5 Diagnostic tests 90

8.6 Forecasting 96

8.7 Missing observations 103

9 Multivariate time series analysis 107

9.1 State space representation of multivariate models 107

9.2 Multivariate trend model with regression effects 108

9.3 Common levels and slopes 111

9.4 An illustration ofmultivariate state space analysis 113

10 State space and Box-Jenkins methods for time series analysis 122

10.1 Stationary processes and related concepts 122

10.1.1 Stationary process 122

10.1.2 Random process 123

10.1.3 Moving average process 125

10.1.4 Autoregressive process 126

10.1.5 Autoregressive moving average process 128

10.2 Non-stationary ARIMA models 129

10.3 Unobserved components and ARIMA 132

10.4 State space versus ARIMA approaches 133

11 State space modelling in practice 135

11.1 The STAMP program and Ssfpack 135

11.2 State space representation in SsfPack 136

11.3 Incorporating regression and intervention effects 139

11.4 Estimation of a model in SsfPack 142

11.4.1 Likelihood evaluation using SsfLikEx 144

11.4.2 The score vector 146

11.4.3 Numerical maximisation of likelihood in Ox 149

11.4.4 The EM algorithm 150

11.4.5 Some illustrations in Ox 151

11.5 Prediction, filtering, and smoothing 154

12 Conclusions 157

12.1 Further reading 159

Appendix A UK drivers KSI and petrol price 162

Appendix B Road traffic fatalities in Norway and Finland 164

Appendix C UK front and rear seat passengers KSI 165

Appendix D UK price changes 167

Bibliography 171

Index 173