## Forecasting, Structural Time Series Models and the Kalman FilterThis book provides a synthesis of concepts and materials that ordinarily appear separately in time series and econometrics literature, presenting a comprehensive review of both theoretical and applied concepts. Perhaps the most novel feature of the book is its use of Kalman filtering together with econometric and time series methodology. From a technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. This technique was originally developed in control engineering but is becoming increasingly important in economics and operations research. The book is primarily concerned with modeling economic and social time series and with addressing the special problems that the treatment of such series pose. |

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### Contents

Introduction | 1 |

12 Explanatory variables and intervention analysis | 4 |

13 Multivariate models | 7 |

14 Statistical treatment | 10 |

15 Modelling methodology | 11 |

16 Forecasting | 14 |

17 Computer software | 15 |

Univariate time series models | 17 |

65 Timevarying and nonlinear models | 341 |

66 Nonnormality count data and qualitative observations | 348 |

Exercises | 362 |

Explanatory variables | 365 |

72 Estimation in the frequency domain | 376 |

73 Estimation of models with explanatory variables and structural time series components | 381 |

74 Tests and measures of goodness of fit | 385 |

75 Model selection strategy and applications | 390 |

21 Introduction | 18 |

22 Ad hoc forecasting procedures | 23 |

23 The structure of time series models | 31 |

24 Stochastic properties | 49 |

25 ARIMA models and the reduced form | 65 |

26 ARIMA modelling | 75 |

27 Applications | 81 |

Exercises | 99 |

State space models and the Kalman filter | 100 |

32 The Kalman filter | 104 |

33 Properties of timeinvariant models | 113 |

34 Maximum likelihood estimation and the prediction error decomposition | 125 |

35 Prediction | 147 |

36 Smoothing | 149 |

37 Nonlinearity and nonnormality | 155 |

Appendix Properties of the multivariate normal distribution | 165 |

Exercises | 166 |

Estimation prediction and smoothing for univariate structural time series models | 168 |

42 Estimation in the time domain | 180 |

43 Estimation in the frequency domain | 191 |

44 Identifiability | 205 |

45 Properties of estimators | 209 |

46 Prediction | 222 |

47 Estimation of components | 226 |

Exercises | 232 |

Testing and model selection | 234 |

52 Lagrange multiplier tests | 239 |

53 Tests of specification for structural models | 248 |

54 Diagnostics | 256 |

55 Goodness of fit | 263 |

56 Postsample predictive testing and model evaluation | 270 |

57 Strategy for model selection | 273 |

Exercises | 281 |

Extensions of the univariate model | 283 |

62 Seasonally and seasonal adjustment | 300 |

63 Different timing intervals for the model and observations | 309 |

64 Data irregularities | 326 |

76 Intervention analysis | 397 |

77 Timevarying parameters | 408 |

78 Instrumental variables | 411 |

79 Count data | 418 |

Exercises | 422 |

Multivariate models | 423 |

82 Seemingly unrelated time series equations | 429 |

83 Homogeneous systems | 435 |

84 Testing and model selection | 442 |

85 Dynamic factor analysis | 449 |

86 Intervention analysis with control groups | 456 |

87 Missing observations delayed observations and contemporaneous aggregation | 463 |

88 Vector autoregressive models | 468 |

89 Simultaneous equation models | 474 |

Exercises | 477 |

Continuous time | 479 |

91 Introduction | 480 |

92 Stock variables | 486 |

93 Flow variables | 492 |

94 Multivariate models | 501 |

Appendix 1 Principal structural time series components and models | 510 |

Appendix 2 Data sets | 512 |

B US Real Gross National Product GNP Annual data 191070 | 515 |

C Purses snatched in Hyde Park area of Chicago | 516 |

D Rainfall in Fortaleza northeast Brazil | 517 |

E International airline passengers OOOs | 518 |

F Deaths and serious injuries in road accidents Great Britain Monthly data January 1969December 1984 | 519 |

G Tractors in Spain Annual data 195176 | 523 |

H Goals scored by England against Scotland in international football matches | 524 |

I Employment OOOs and output 1980100 in UK manufacturing seasonally adjusted Quarterly data 1963Q11983Q3 | 525 |

J Mink and muskrat furs sold by Hudsons Bay Company Annual data 18481909 | 526 |

Selected answers to exercises | 527 |

529 | |

543 | |

547 | |

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Forecasting, Structural Time Series Models and the Kalman Filter Andrew C. Harvey Limited preview - 1990 |

### Common terms and phrases

algorithm ARIMA ARIMA model asymptotic autocovariance autoregressive coefficients computed conditional constructed correlogram corresponding covariance matrix cycle defined denote deterministic diffuse prior disturbance term elements equal to zero evaluated EWMA example exogenous explanatory variables exponential expression forecast function formulation frequency domain generalised given gives Harvey hyperparameters information matrix intervention Kalman filter lag operator likelihood function linear trend model LM test logarithms measurement equation missing observations ML estimator MSE matrix multivariate normal distribution noise model non-stationary normally distributed null hypothesis obtained periodogram polynomial post-sample procedure properties random walk recursion reduced form regression residuals result sample seasonal adjustment seasonal component series models slope space form space model starting values stationary stationary process stochastic process structural time series test statistic time-invariant transition equation trend and seasonal trend component uncorrelated univariate univariate model unknown parameters unobserved components variance vector vector autoregressive walk plus noise yields