An Introduction to State - LISTINET

1 An Introduction to StateSpace Time Series AnalysisPractical EconometricsSeries EditorsJurgen Doornik and Bronwyn HallPractical econometrics is a series of books designed to provideaccessible and practical introductions to various topics in econo-metrics. From econometric techniques to econometric modellingapproaches, these short introductions are ideal for applied econo-mists, graduate students, and researchers looking for a non-technicaldiscussion on specific topics in Introduction to StateSpace Time Series AnalysisJacques J. F. CommandeurSiem Jan Koopman13 Great Clarendon Street, Oxford ox2 6 DPOxford University Press is a department of the University of furthers the University s objective of excellence in research, scholarship.

2 And education by publishing worldwide inOxford New YorkAuckland Cape Town Dar es Salaam Hong Kong KarachiKuala Lumpur Madrid Melbourne Mexico City NairobiNew Delhi Shanghai Taipei TorontoWith offices inArgentina Austria Brazil Chile Czech Republic France GreeceGuatemala Hungary Italy Japan Poland Portugal SingaporeSouth Korea Switzerland Thailand Turkey Ukraine VietnamOxford is a registered trademark of Oxford University Pressin the UK and in certain other countriesPublished in the United Statesby Oxford University Press Inc., New York Jacques Commandeur and Siem Jan Koopman 2007 The moral rights of the authors have been assertedDatabase right Oxford University Press (maker)First published 2007 All rights reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,without the prior permission in writing of Oxford University Press,or as expressly permitted by law, or under terms agreed with the appropriatereprographics rights organization.

3 Enquiries concerning reproductionoutside the scope of the above should be sent to the Rights Department,Oxford University Press, at the address aboveYou must not circulate this book in any other binding or coverand you must impose the same condition on any acquirerBritish Library Cataloguing in Publication DataData availableLibrary of Congress Cataloging in Publication DataData availableTypeset by SPI Publisher Services, Pondicherry, IndiaPrinted in Great Britainon acid-free paper byBiddles Ltd., King s Lynn, NorfolkISBN 978 0 19 922887 413579108642 PrefaceThis book provides an introductory treatment of State space methodsapplied to unobserved-component time series models which are alsoknown as structural time series models. The book started as a collectionof personal notes made by JJFC about what he discovered and understoodwhile studying State space methods for the first time.

4 When colleaguesand friends also found these notes useful and helpful, the idea came up tomake them publicly available. SJK started to cooperate with JJFC on thisbook project as part of the highly enjoyable joint projects for the SWOVI nstitute for Road Safety Research in Leidschendam, the (1989) and Durbin and Koopman (2001) treat the topic ofstate space methods at an advanced level suitable for postgraduate andadvanced graduate courses in time series analysis. Elementary time seriesbooks, on the other hand, provide only very limited space to the classof unobserved-component models. Most of the attention is given to theBox Jenkins approach to time series intended audience for this book is practitioners and researchersworking in areas other than statistics, but who use time series on a dailybasis in areas such as the social sciences, quantitative history, biology andmedicine.

5 This book offers a step-by-step approach to the analysis of thesalient features in time series such as the trend, seasonal and irregularcomponents. Practical problems such as forecasting and missing valuesare treated in some detail. The book may also serve as an accompanyingtextbook for a basic time series course in econometrics and statistics,typically at an undergraduate would like to acknowledge and thank the management and thecolleagues of the SWOV Institute for Road Safety Research for their mentaland financial contribution to this publication. The book is an importantcomponent of the SWOV Research Programme 2003 all SWOV colleagues, JJFC is especially indebted to FritsBijleveld, whose never abating and infectious enthusiasm for State spacevPrefacemethods was instrumental in stimulating JJFC to write this book.

6 Hewas always willing to answer any questions JJFC had, and is a genius inexploiting the enormous flexibility that State space methods have to authors are grateful to a referee for his positive remarks on an earlierdraft of the book. His many constructive comments have improved thebook considerably. Any mistakes and omissions remain the sole responsi-bility of the also wishes to thank members (some of them, former members)of the International Co-operation on Time Series Analysis (ICTSA): PeterChristens, Ruth Bergel, Joanna Zukowska, Filip Van den Bossche, GeertWets, Stefan Hoeglinger, Ward Vanlaar, Phillip Gould, Max Cameron,and Stewart Newstead, for their inspiring contributions to our in-depthdiscussions on time series analysis, and for their encouraging response toearlier drafts of the would like to thank his colleagues at the Department of Economet-rics, Vrije Universiteit Amsterdam, for giving him the opportunity to workon this usingtheMiKTeXsystem( ).

7 We thank Frits Bijleveld for his assistancein setting up the LATEX system. The Ox and SsfPack code for carryingout the analyses discussed in the book, as well as the data files,can be downloaded from and of FiguresxList of Tablesxiv1. Introduction12. The local level Deterministic Stochastic The local level model and Norwegian fatalities183. The local linear trend Deterministic level and Stochastic level and Stochastic level and deterministic The local linear trend model and Finnish fatalities284. The local level model with Deterministic level and Stochastic level and Stochastic level and deterministic The local level and seasonal model and UK inflation435. The local level model with explanatory Deterministic level and explanatory Stochastic level and explanatory variable526.

8 The local level model with intervention Deterministic level and intervention Stochastic level and intervention variable597. The UK seat belt and inflation Deterministic level and Stochastic level and Stochastic level and deterministic The UK inflation model70viiContents8. General treatment of univariate State space State space representation of univariate models Incorporating regression effects Confidence Filtering and Diagnostic Missing observations1039. Multivariate time series analysis State space representation of multivariate Multivariate trend model with regression Common levels and An illustration of multivariate State space analysis11310. State space and Box Jenkins methods for time series Stationary processes and related Stationary Random Moving average Autoregressive Autoregressive moving average Non-stationary ARIMA Unobserved components and State space versus ARIMA approaches13311.

9 State space modelling in TheSTAMP program State space representation inSsfPack Incorporating regression and intervention effects Estimation of a model inSsfPack Likelihood evaluation The score Numerical maximisation of likelihood The EM Some illustrations Prediction, filtering, and smoothing 15412. Further reading159 APPENDIX A. UK drivers KSI and petrol price162viiiContentsAPPENDIX B. Road traffic fatalities in Norway and Finland164 APPENDIX C. UK front and rear seat passengers KSI165 APPENDIX D. UK price changes167 Bibliography171 Index173ixList of Scatter plot of the log of the number of UK drivers KSI againsttime (in months), including regression Log of the number of UK drivers KSI plotted as a time Residuals of classical linear regression of the log of the numberof UK drivers KSI on Correlogram of random time Correlogram of classical regression Deterministic Irregular component for deterministic level Stochastic Irregular component for local level Stochastic level for Norwegian Irregular component for Norwegian Trend of stochastic linear trend Slope of stochastic linear trend Irregular component of stochastic linear trend Trend of stochastic level and deterministic slope Trend of deterministic level and stochastic slope model forFinnish fatalities (top), and stochastic slope component (bottom).

10 Irregular component for Finnish Log of number of UK drivers KSI with time lines for Combined deterministic level and Deterministic Deterministic Irregular component for deterministic level and seasonal Stochastic Stochastic of Stochastic seasonal for the year Irregular component for stochastic level and seasonal Stochastic level, seasonal and irregular in UK inflation Deterministic level and explanatory variable log petrol price . Conventional classical regression representation ofdeterministic level and explanatory variable log petrol price . Irregular component for deterministic level model withexplanatory variable log petrol price . Stochastic level and deterministic explanatory variable logpetrol price.