## Time Series Analysis by State Space MethodsThis excellent text provides a comprehensive treatment of the state space approach to time series analysis. The distinguishing feature of state space time series models is that observations are regarded as made up of distinct components such as trend, seasonal, regression elements and disturbence terms, each of which is modelled separately. The techniques that emerge from this approach are very flexible and are capable of handling a much wider range of problems than the main analytical system currently in use for time series analysis, the Box-Jenkins ARIMA system. The book provides an excellent source for the development of practical courses on time series analysis. |

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### Contents

Chapter 1 Introduction | 1 |

13 NonGaussian and nonlinear models | 3 |

14 Prior knowledge | 4 |

16 Other books on state space methods | 5 |

The linear Gaussian state space model | 7 |

Chapter 2 Local level model | 9 |

22 Filtering | 11 |

23 Forecast errors | 13 |

Chapter 6 Further computational aspects | 121 |

63 Square root filter and smoother | 124 |

64 Univariate treatment of multivariate series | 128 |

65 Filtering and smoothing under linear restrictions | 134 |

Chapter 7 Maximum likelihood estimation | 138 |

73 Parameter estimation | 142 |

74 Goodness of fit | 152 |

Chapter 8 Bayesian analysis | 155 |

24 State smoothing | 16 |

25 Disturbance smoothing | 19 |

26 Simulation | 22 |

27 Missing observations | 23 |

28 Forecasting | 25 |

29 Initialisation | 27 |

210 Parameter estimation | 30 |

211 Steady state | 32 |

212 Diagnostic checking | 33 |

Lemma in multivariate normal regression | 37 |

Chapter 3 Linear Gaussian state space models | 38 |

32 Structural time series models | 39 |

33 ARMA models and ARIMA models | 46 |

34 Exponential smoothing | 49 |

35 State space versus BoxJenkins approaches | 51 |

36 Regression with timevarying coefficients | 54 |

39 Simultaneous modelling of series from different sources | 56 |

310 State space models in continuous time | 57 |

311 Spline smoothing | 61 |

Chapter 4 Filtering smoothing and forecasting | 64 |

42 Filtering | 65 |

43 State smoothing | 70 |

44 Disturbance smoothing | 73 |

45 Covariance matrices of smoothed estimators | 77 |

46 Weight functions | 81 |

47 Simulation smoothing | 83 |

48 Missing observations | 92 |

49 Forecasting | 93 |

410 Dimensionality of observational vector | 94 |

411 General matrix form for filtering and smoothing | 95 |

52 The exact initial Kalman filter | 101 |

53 Exact initial state smoothing | 106 |

54 Exact initial disturbance smoothing | 109 |

55 Exact initial simulation smoothing | 110 |

57 Augmented Kalman filter and smoother | 115 |

83 Markov chain Monte Carlo methods | 159 |

Chapter 9 Illustrations of the use of the linear Gaussian model | 161 |

93 Bivariate structural time series analysis | 167 |

94 BoxJenkins analysis | 169 |

95 Spline smoothing | 172 |

96 Approximate methods for modelling volatility | 175 |

NonGaussian and nonlinear state space models | 177 |

Chapter 10 NonGaussian and nonlinear state space models | 179 |

103 Exponential family models | 180 |

104 Heavytailed distributions | 183 |

105 Nonlinear models | 184 |

106 Financial models | 185 |

Chapter 11 Importance sampling | 189 |

112 Basic ideas of importance sampling | 190 |

113 Linear Gaussian approximating models | 191 |

114 Linearisation based on first two derivatives | 193 |

115 Linearisation based on the first derivative | 195 |

116 Linearisation for nonGaussian state components | 198 |

1 17 Linearisation for nonlinear models | 199 |

118 Estimating the conditional mode | 202 |

119 Computational aspects of importance sampling | 204 |

Chapter 12 Analysis from a classical standpoint | 212 |

123 Estimating conditional densities and distribution functions | 213 |

124 Forecasting and estimating with missing observations | 214 |

125 Parameter estimation | 215 |

Chapter 13 Analysis from a Bayesian standpoint | 222 |

133 Computational aspects of Bayesian analysis | 225 |

134 Posterior analysis of parameter vector | 226 |

135 Markov chain Monte Carlo methods | 228 |

Chapter 14 NonGaussian and nonlinear illustrations | 230 |

outlier in gas consumption in UK | 233 |

pounddollar daily exchange rates | 236 |

OxfordCambridge boat race | 237 |

146 NonGaussian and nonlinear analysis using SsfPack | 238 |

### Common terms and phrases

a,+i algorithms antithetic variables approach approximating model ARIMA models ARMA augmented Kalman filter Bayesian analysis Box-Jenkins calculated Chapter Cholesky decomposition coefficients components computed consider denote derived discussion disturbance smoothing disturbance vectors Durbin elements equation error variance exact initial Kalman example exponential family filtering and smoothing forecast errors function Gaussian state space given Harvey importance density importance sampling initial Kalman filter initial state vector initialisation Koopman level model linear Gaussian model linear trend model Linearisation lower triangular matrix maximising maximum likelihood estimate MCMC mean square error methods missing observations multivariate non-Gaussian and nonlinear non-informative prior observation vector obtain parameter estimation parameter vector Poisson distribution r-distribution random regression score vector seasonal series analysis series models Shephard simulation sample simulation smoother smoothed estimators smoothing recursion space model spline smoothing stochastic volatility structural time series techniques treatment univariate values variance matrix zero