Time series analysis by state space methods / J. Durbin and S.J. Koopman.

Format:

Book

Author/Creator:

Durbin, J. (James), 1923-2012.

Contributor:

Koopman, S. J. (Siem Jan)

Alumni and Friends Memorial Book Fund.

Series:

Oxford statistical science series ; 24.

Oxford statistical science series ; 24

Language:

English

Subjects (All):

Time-series analysis.

State-space methods.

Physical Description:

xvii, 253 pages : illustrations ; 25 cm.

Place of Publication:

Oxford ; New York : Oxford University Press, 2001.

Contents:

1.1 Basic ideas of state space analysis 1

1.2 Linear Gaussian model 1

1.3 Non-Gaussian and nonlinear models 3

1.4 Prior knowledge 4

1.6 Other books on state space methods 5

1.7 Website for the book 5

I The Linear Gaussian State Space Model

2 Local level model 9

2.2 Filtering 11

2.2.1 The Kalman Filter 11

2.3 Forecast errors 13

2.3.1 Cholesky decomposition 14

2.3.2 Error recursions 15

2.4 State smoothing 16

2.4.1 Smoothed state 16

2.4.2 Smoothed state variance 17

2.5 Disturbance smoothing 19

2.5.1 Smoothed observation disturbances 20

2.5.2 Smoothed state disturbances 20

2.5.4 Cholesky decomposition and smoothing 22

2.6 Simulation 22

2.7 Missing observations 23

2.8 Forecasting 25

2.9 Initialisation 27

2.10 Parameter estimation 30

2.10.1 Loglikelihood evaluation 30

2.10.2 Concentration of loglikelihood 31

2.11 Steady state 32

2.12 Diagnostic checking 33

2.12.1 Diagnostic tests for forecast errors 33

2.12.2 Detection of outliers and structural breaks 35

2.13 Appendix: Lemma in multivariate normal regression 36

3 Linear Gaussian state space models 38

3.2 Structural time series models 39

3.2.1 Univariate models 39

3.2.2 Multivariate models 44

3.2.3 Stamp 45

3.3 ARMA models and ARIMA models 46

3.4 Exponential smoothing 49

3.5 State space versus Box-Jenkins approaches 51

3.6 Regression with time-varying coefficients 54

3.7 Regression with ARMA errors 54

3.8 Benchmarking 54

3.9 Simultaneous modelling of series from different sources 56

3.10 State space models in continuous time 57

3.10.1 Local level model 57

3.10.2 Local linear trend model 59

3.11 Spline smoothing 61

3.11.1 Spline smoothing in discrete time 61

3.11.2 Spline smoothing in continuous time 62

4 Filtering, smoothing and forecasting 64

4.2 Filtering 65

4.2.1 Derivation of Kalman filter 65

4.2.2 Kalman filter recursion 67

4.2.3 Steady state 68

4.2.4 State estimation errors and forecast errors 68

4.3 State smoothing 70

4.3.1 Smoothed state vector 70

4.3.2 Smoothed state variance matrix 72

4.3.3 State smoothing recursion 73

4.4 Disturbance smoothing 73

4.4.1 Smoothed disturbances 73

4.4.2 Fast state smoothing 75

4.4.3 Smoothed disturbance variance matrices 75

4.4.4 Disturbance smoothing recursion 76

4.5 Covariance matrices of smoothed estimators 77

4.6 Weight functions 81

4.6.2 Filtering weights 81

4.6.3 Smoothing weights 82

4.7 Simulation smoothing 83

4.7.1 Simulating observation disturbances 84

4.7.2 Derivation of simulation smoother for observation disturbances 87

4.7.3 Simulation smoothing recursion 89

4.7.4 Simulating state disturbances 90

4.7.5 Simulating state vectors 91

4.7.6 Simulating multiple samples 92

4.8 Missing observations 92

4.9 Forecasting 93

4.10 Dimensionality of observational vector 94

4.11 General matrix form for filtering and smoothing 95

5 Initialisation of filter and smoother 99

5.2 The exact initial Kalman filter 101

5.2.1 The basic recursions 101

5.2.2 Transition to the usual Kalman filter 104

5.2.3 A convenient representation 105

5.3 Exact initial state smoothing 106

5.3.1 Smoothed mean of state vector 106

5.3.2 Smoothed variance of state vector 107

5.4 Exact initial disturbance smoothing 109

5.5 Exact initial simulation smoothing 110

5.6 Examples of initial conditions for some models 110

5.6.1 Structural time series models 110

5.6.2 Stationary ARMA models 111

5.6.3 Nonstationary ARIMA models 112

5.6.4 Regression model with ARMA errors 114

5.6.5 Spline smoothing 115

5.7 Augmented Kalman filter and smoother 115

5.7.2 Augmented Kalman filter 115

5.7.3 Filtering based on the augmented Kalman filter 116

5.7.4 Illustration: the local linear trend model 118

5.7.5 Comparisons of computational efficiency 119

5.7.6 Smoothing based on the augmented Kalman filter 120

6 Further computational aspects 121

6.2 Regression estimation 121

6.2.2 Inclusion of coefficient vector in state vector 122

6.2.3 Regression estimation by augmentation 122

6.2.4 Least squares and recursive residuals 123

6.3 Square root filter and smoother 124

6.3.2 Square root form of variance updating 125

6.3.3 Givens rotations 126

6.3.4 Square root smoothing 127

6.3.5 Square root filtering and initialisation 127

6.3.6 Ilustration: local linear trend model 128

6.4 Univariate treatment of multivariate series 128

6.4.2 Details of univariate treatment 129

6.4.3 Correlation between observation equations 131

6.4.4 Computational efficiency 132

6.4.5 Illustration: vector splines 133

6.5 Filtering and smoothing under linear restrictions 134

6.6 The algorithms of SsfPack 134

6.6.2 The SsfPack function 135

6.6.3 Illustration: spline smoothing 136

7 Maximum likelihood estimation 138

7.2 Likelihood evaluation 138

7.2.1 Loglikelihood when initial conditions are known 138

7.2.2 Diffuse loglikelihood 139

7.2.3 Diffuse loglikelihood evaluated via augmented Kalman filter 140

7.2.4 Likelihood when elements of initial state vector are fixed but unknown 141

7.3 Parameter estimation 142

7.3.2 Numerical maximisation algorithms 142

7.3.3 The score vector 144

7.3.4 The EM algorithm 147

7.3.5 Parameter estimation when dealing with diffuse initial conditions 149

7.3.6 Large sample distribution of maximum likelihood estimates 150

7.3.7 Effect of errors in parameter estimation 150

7.4 Goodness of fit 152

7.5 Diagnostic checking 152

8 Bayesian analysis 155

8.2 Posterior analysis of state vector 155

8.2.1 Posterior analysis conditional on parameter vector 155

8.2.2 Posterior analysis when parameter vector is unknown 155

8.2.3 Non-informative priors 158

8.3 Markov chain Monte Carlo methods 159

9 Illustrations of the use of the linear Gaussian model 161

9.2 Structural time series models 161

9.3 Bivariate structural time series analysis 167

9.4 Box-Jenkins analysis 169

9.5 Spline smoothing 172

9.6 Approximate methods for modelling volatility 175

II Non-Gaussian And Nonlinear State Space Models

10 Non-Gaussian and nonlinear state space models 179

10.2 The general non-Gaussian model 179

10.3 Exponential family models 180

10.3.1 Poisson density 181

10.3.2 Binary density 181

10.3.3 Binomial density 181

10.3.4 Negative binomial density 182

10.3.5 Multinomial density 182

10.4 Heavy-tailed distributions 183

10.4.1 t-Distribution 183

10.4.2 Mixture of normals 184

10.4.3 General error distribution 184

10.5 Nonlinear models 184

10.6 Financial models 185

10.6.1 Stochastic volatility models 185

10.6.2 General autoregressive conditional heteroscedasticity 187

10.6.3 Durations: exponential distribution 188

10.6.4 Trade frequencies: Poisson distribution 188

11 Importance sampling 189

11.2 Basic ideas of importance sampling 190

11.3 Linear Gaussian approximating models 191

11.4 Linearisation based on first two derivatives 193

11.4.1 Exponentional family models 195

11.4.2 Stochastic volatility model 195

11.5 Linearisation based on the first derivative 195

11.5.1 t-distribution 197

11.5.2 Mixture of normals 197

11.5.3 General error distribution 197

11.6 Linearisation for non-Gaussian state components 198

11.6.1 t-distribution for state errors 199

11.7 Linearisation for nonlinear models 199

11.7.1 Multiplicative models 201

11.8 Estimating the conditional mode 202

11.9 Computational aspects of importance sampling 204

11.9.2 Practical implementation of importance sampling 204

11.9.3 Antithetic variables 205

11.9.4 Diffuse initialisation 206

11.9.5 Treatment of t-distribution without importance sampling 208

11.9.6 Treatment of Gaussian mixture

distributions without importance sampling 210

12 Analysis from a classical standpoint 212

12.2 Estimating conditional means and variances 212

12.3 Estimating conditional densities and distribution functions 213

12.4 Forecasting and estimating with missing observations 214

12.5 Parameter estimation 215

12.5.2 Estimation of likelihood 215

12.5.3 Maximisation of loglikelihood 216

12.5.4 Variance matrix of maximum likelihood estimate 217

12.5.5 Effect of errors in parameter estimation 217

12.5.6 Mean square error matrix due to simulation 217

12.5.7 Estimation when the state disturbances are Gaussian 219

12.5.8 Control variables 219

13 Analysis from a Bayesian standpoint 222

13.2 Posterior analysis of functions of the state vector 222

13.3 Computational aspects of Bayesian analysis 225

13.4 Posterior analysis of parameter vector 226

13.5 Markov chain Monte Carlo methods 228

14 Non-Gaussian and nonlinear illustrations 230

14.2 Poisson density: van drivers killed in Great Britain 230

14.3 Heavy-tailed density: outlier in gas consumption in UK 233

14.4 Volatility: pound/dollar daily exchange rates 236

14.5 Binary density: Oxford-Cambridge boat race 237

14.6 Non-Gaussian and nonlinear analysis using SsfPack 238.

Notes:

Includes bibliographical references (pages [241]-247) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0198523548

OCLC:

45636928