Time Series Analysis by State Space Methods (Oxford Statistical Science Series)

Preface -11......Page 8
Contents -7......Page 12
1.2 Linear Gaussian model 1......Page 20
1.3 Non-Gaussian and nonlinear models 3......Page 22
1.5 Notation 4......Page 23
1.7 Website for the book 5......Page 24
I THE LINEAR GAUSSIAN STATE SPACE MODEL 7......Page 26
2.1 Introduction 9......Page 28
2.2.1 The Kalman Filter 11......Page 30
2.2.2 Illustration 12......Page 31
2.3 Forecast errors 13......Page 32
2.3.1 Cholesky decomposition 14......Page 33
2.3.2 Error recursions 15......Page 34
2.4.1 Smoothed state 16......Page 35
2.4.2 Smoothed state variance 17......Page 36
2.4.3 Illustration 18......Page 37
2.5 Disturbance smoothing 19......Page 38
2.5.2 Smoothed state disturbances 20......Page 39
2.5.3 Illustration 21......Page 40
2.6 Simulation 22......Page 41
2.7 Missing observations 23......Page 42
2.8 Forecasting 25......Page 44
2.9 Initialisalion 27......Page 46
2.10.1 Loglikelihood evaluation 30......Page 49
2.10.2 Concentration of loglikelihood 31......Page 50
2.11 Steady state 32......Page 51
2.12.1 Diagnostic tests for forecast errors 33......Page 52
2.12.3 Illustration 35......Page 54
2.13 Appendix: Lemma in multivariate normal regression 37......Page 56
3.1 Introduction 38......Page 57
3.2.1 Univariate models 39......Page 58
3.2.2 Multivariate models 44......Page 63
3.2.3 STAMP 45......Page 64
3.3 ARMA models and ARIMA models 46......Page 65
3.4 Exponential smoothing 49......Page 68
3.5 State space versus Box-Jenkins approaches 51......Page 70
3.8 Benchmarking 54......Page 73
3.9 Simultaneous modelling of series from different sources 56......Page 75
3.10.1 Local level model 57......Page 76
3.10.2 Local linear trend model 59......Page 78
3.11.1 Spline smoothing in discrete time 61......Page 80
3.11.2 Spline smoothing in continuous time 62......Page 81
4.1 Introduction 64......Page 83
4.2.1 Derivation of Kalman filter 65......Page 84
4.2.2 Kalman filter recursion 67......Page 86
4.2.4 State estimation errors and forecast errors 68......Page 87
4.3.1 Smoothed state vector 70......Page 89
4.3.2 Smoothed state variance matrix 72......Page 91
4.4.1 Smoothed disturbances 73......Page 92
4.4.3 Smoothed disturbance variance matrices 75......Page 94
4.4.4 Disturbance smoothing recursion 76......Page 95
4.5 Covariance matrices of smoothed estimators 77......Page 96
4.6.2 Filtering weights 81......Page 100
4.6.3 Smoothing weights 82......Page 101
4.7 Simulation smoothing 83......Page 102
4.7.1 Simulating observation disturbances 84......Page 103
4.7.2 Derivation of simulation smoother for observation disturbances 87......Page 106
4.7.3 Simulation smoothing recursion 89......Page 108
4.7.4 Simulating state disturbances 90......Page 109
4.7.5 Simulating state vectors 91......Page 110
4.8 Missing observations 92......Page 111
4.9 Forecasting 93......Page 112
4.10 Dimensionality of observational vector 94......Page 113
4.11 General matrix form for filtering and smoothing 95......Page 114
5.1 Introduction 99......Page 118
5.2.1 The basic recursions 101......Page 120
5.2.2 Transition to the usual Kalman filter 104......Page 123
5.2.3 A convenient representation 105......Page 124
5.3.1 Smoothed mean of state vector 106......Page 125
5.3.2 Smoothed variance of state vector 107......Page 126
5.4 Exact initial disturbance smoothing 109......Page 128
5.6.1 Structural time series models 110......Page 129
5.6.2 Stationary ARMA models 111......Page 130
5.6.3 Nonstationary ARTMA models 112......Page 131
5.6.4 Regression model with ARMA errors 114......Page 133
5.7.2 Augmented Kalman filter 115......Page 134
5.7.3 Filtering based on the augmented Kalman filter 116......Page 135
5.7.4 Illustration: the local linear trend model 118......Page 137
5.7.5 Comparisons of computational efficiency 119......Page 138
5.7.6 Smoothing based on the augmented Kalman filter 120......Page 139
6.2.1 Introduction 121......Page 140
6.2.3 Regression estimation by augmentation 122......Page 141
6.2.4 Least squares and recursive residuals 123......Page 142
6.3.1 Introduction 124......Page 143
6.3.2 Square root form of variance updating 125......Page 144
6.3.3 Givens rotations 126......Page 145
6.3.5 Square root filtering and initialisation 127......Page 146
6.4.1 Introduction 128......Page 147
6.4.2 Details of univariate treatment 129......Page 148
6.4.3 Correlation between observation equations 131......Page 150
6.4.4 Computational efficiency 132......Page 151
6.4.5 Illustration: vector splines 133......Page 152
6.6.1 Introduction 134......Page 153
6.6.2 The SsfPack function 135......Page 154
6.6.3 Illustration: spline smoothing 136......Page 155
7.2.1 Loglikelihood when initial conditions are known 138......Page 157
7.2.2 Diffuse loglikelihood 139......Page 158
7.2.3 Diffuse loglikelihood evaluated via augmented Kalman filter 140......Page 159
7.2.4 Likelihood when elements of initial state vector are fixed but unknown 141......Page 160
7.3.2 Numerical maximisation algorithms 142......Page 161
7.3.3 The score vector 144......Page 163
7.3.4 The EM algorithm 147......Page 166
7.3.5 Parameter estimation when dealing with diffuse initial conditions 149......Page 168
7.3.7 Effect of errors in parameter estimation 150......Page 169
7.5 Diagnostic checking 152......Page 171
8.2.2 Posterior analysis when parameter vector is unknown 155......Page 174
8.2.3 Non-informative priors 158......Page 177
8.3 Markov chain Monte Carlo methods 159......Page 178
9.2 Structural time series models 161......Page 180
9.3 Bivariate structural time series analysis 167......Page 186
9.4 Box-Jenkins analysis 169......Page 188
9.5 Spline smoothing 172......Page 191
9.6 Approximate methods for modelling volatility 175......Page 194
II NON-GAUSSIAN AND NONLINEAR STATE SPACE MODELS 177......Page 196
10.2 The general non-Gaussian model 179......Page 198
10.3 Exponential family models 180......Page 199
10.3.3 Binomial density 181......Page 200
10.3.5 Multinomial density 182......Page 201
10.4.1 /-Distribution 183......Page 202
10.5 Nonlinear models 184......Page 203
10.6.1 Stochastic volatility models 185......Page 204
10.6.2 General autoregressive conditional heteroscedasticity 187......Page 206
10.6.4 Trade frequencies: Poisson distribution 188......Page 207
11.1 Introduction 189......Page 208
11.2 Basic ideas of importance sampling 190......Page 209
11.3 Linear Gaussian approximating models 191......Page 210
11.4 Linearisation based on first two derivatives 193......Page 212
11.5 Linearisation based on the first derivative 195......Page 214
11.5.3 General error distribution 197......Page 216
11.6 Linearisation for non-Gaussian state components 198......Page 217
11.7 Linearisation for nonlinear models 199......Page 218
11.7.1 Multiplicative models 201......Page 220
11.8 Estimating the conditional mode 202......Page 221
11.9.2 Practical implementation of importance sampling 204......Page 223
11.9.3 Antithetic variables 205......Page 224
11.9.4 Diffuse initialisation 206......Page 225
11.9.5 Treatment of f-distribution without importance sampling 208......Page 227
11.9.6 Treatment of Gaussian mixture distributions without importance sampling 210......Page 229
12.2 Estimating conditional means and variances 212......Page 231
12.3 Estimating conditional densities and distribution functions 213......Page 232
12.4 Forecasting and estimating with missing observations 214......Page 233
12.5.2 Estimation of likelihood 215......Page 234
12.5.3 Maximisation of loglikelihood 216......Page 235
12.5.6 Mean square error matrix due to simulation 217......Page 236
12.5.8 Control variables 219......Page 238
13.2 Posterior analysis of functions of the state vector 222......Page 241
13.3 Computational aspects of Bayesian analysis 225......Page 244
13.4 Posterior analysis of parameter vector 226......Page 245
13.5 Markov chain Monte Carlo methods 228......Page 247
14.2 Poisson density: van drivers killed in Great Britain 230......Page 249
14.3 Heavy-tailed density: outlier in gas consumption in UK 233......Page 252
14.4 Volatility: pound/dollar daily exchange rates 236......Page 255
14.5 Binary density: Oxford-Cambridge boat race 237......Page 256
14.6 Non-Gaussian and nonlinear analysis using SsfPack 238......Page 257
References 241......Page 260
Author index 249......Page 268
Subject index 251......Page 270