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Nonlinear Filters 2nd Edition Springer-Verlag Berlin Heidelberg GmbH Hisashi Tanizaki Nonlinear Filters Estimation and Applications Second Revised and Enlarged Edition With 18 Figures and 45 Tables Springer Associate Professor Hisashi Tanizaki Kobe University Faculty of Economics Rokkodai, Nadaku Kobe 657, Japan Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Tanlzakl, Hlsashl: Nonlinear filters : estimation and applications / Hisashi Tanizaki. - 2. rev. and enl. ed. - Berlin ; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan ; Paris ; Santa Clara; Singapore; Tokyo : Springer. 1996 ISBN 978-3-642-08253-5 ISBN 978-3-662-03223-7 (eBook) DOI 10.1007/978-3-662-03223-7 This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned, specifically the rights of translation, reprinting. reuse of illustrations. recitation, broadcasting, reproduction on microfilms or in other ways. and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993, 1996 Originally published by Springer-Verlag Berlin Heidelberg New York in 1996 Softcover reprint of the hardcover 2nd edition 1996 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN 10540183 42/2202-543210 -Printed on acid-free paper To My Wife Miyuki Preface Preface to Second Edition This book is a revision of Nonlinear Filters: Estimation and Applications, (Lecture Notes in Economics and Mathematical Systems, No.400), which was published from Springer-Verlag in 1993. Compared with the first edition, I have made a substantial revision in the second edition. First, titles in the following chapters, sections, terms and so on are changed as follows. The First Edition The Second Edition Chapter 3 Chapter 3 Nonlinear Filters based on Taylor ==> Traditional Nonlinear Filters Series Expansion Chapter 4 Chapter 4 Nonlinear Filters based on Density =* Density-Based Nonlinear Filters Approximation Section 4.3 Section 4.3 Numerical Density Approximation =* Numerical Integration Filter by Piecewise Linear Functions: Modified Kitagawa Estimator Section 4.4 Section 4.4 Simulation-based Density Estimator =* Importance Sampling Filter Chapter 5 Chapter 5 Comparison of Nonlinear Filters: =* Monte-Carlo Experiments Monte-Carlo Experiments Chapter 6 Chapter 6 An Application of Nonlinear ==> Application of Nonlinear Filters Filters: Estimation of Permanent Consumption Monograph =* Book VIII Preface Thus, the most appropriate title is taken or the title is made short. Second, new contents are briefly summarized as follows. • Section 2.3.3 Minimum Mean Square Linear Estimator As the third derivation method of the Kalman filter, I put this section. This section is not utilized for the proceeding chapters but it is added as a survey of the standard Kalman filter. • Appendix A2.2 Conditional Normal Distribution The derivation method under normality assumption is discussed in Sec tion 2.3.1. Lemmas and Proofs used in Section 2.3.1 are summarized in this appendix. • Section 4.5 Density-based Monte-Carlo Filter This nonlinear filter is one of the recent new nonlinear filters. • Section 4.6 Rejection Sampling Filter The rejection sampling filter is also one of the recent topics, where a recursive algorithm of random draws from filtering densities is derived. • Appendix A4.5 Density-Based Monte-Carlo Filter For the filtering estimates by the density-based Monte-Carlo filter, the asymptotic properties are discussed. • Appendix A4.6 Rejection Sampling Random number generation by rejection sampling is discussed in a gen eral form. • Appendix A5.1 On Initial Value of State-Variable In the Kalman filter algorithm, the filtering estimates are recursively obtained given the initial value. We analyze how the filtering estimates are sensitive to the initial value. • Appendix A5.3 On Random Draws by Importance Sampling Monte-Carlo integration with importance sampling is utilized to the non linear filter in Section 4.4, where random draws have to be generated from the importance density. We discuss about the random draws generated from the importance density. • Appendix A5.4 Rejection Sampling Using rejection sampling, random draws are generated by a computer. Precision of random draws are examined. • Chapter 7 Prediction and Smoothing The density-based filtering algorithms discussed in Chapter 4 are ex tended to prediction and smoothing. Moreover, the following chapters are substantially revised. • Chapter 5 Monte-Carlo Experiments I changed a functional form of the nonlinear measurement and transition equations in some simulation studies. Preface to Second Edition IX • Chapter 6 Application of Nonlinear Filters In the first edition, only the U.S. data are used. In the second edition, the same type of state-space model is estimated for Japan, U.S., U.K., France, Spain, Italy, Canada and Germany. I focus on estimation of the unknown parameters and estimation of a ratio of per capita permanent consumption relative to per capita total consumption for the above coun tries. Thus, the second edition is substantially changed compared with the first edition. Finally, I am grateful to the editor Dr. Werner A. Mueller, who gave me a chance to revise the monograph. March,1996 Author: Hisashi Tanizaki Associate Professor Faculty of Economics, Kobe University Rokkodai, Nadaku, Kobe 657, Japan E-mail: [email protected] Preface to First Edition Acknowledgements Originally, this monograph is a revision of my Ph.D. dissertation (" Nonlinear Filters: Estimation and Applications, " December, 1991) at the University of Pennsylvania. In writing the dissertation, many people took care of me in different ways. In particular, Professor Roberto S. Mariano was very patient as my main thesis advisor, and he gave me many useful suggestions and comments. I wish to thank him for recommending me this line of research and giving me the suggestions and comments. Also, I would like to acknowledge the support of NSF grant SES 9011917 in Professor Roberto S. Mariano's supervision of the thesis. Professor Marc Nerlove supported me financially during my staying in the U.S. in spite of my trouble with English. I would like to acknowledge the support of NSF grant SES 5-25056. Professor Francis X. Diebold has stimulated me through his studies. Although the dissertation was not directly related to their research, their work significantly affected me and will influence my future research. I am grateful to them for that. Moreover, I do not forget to thank the classmates in the Graduate Pro gram in Economics who had been encouraging me since I started studying at the University. Especially, I got many " invisible things " from Michael Blackman, Celia Chen, Haria Fornari and Til Schuermann (in alphabetical order) and asked them to correct my English. Also, I am grateful to Jun Sato and Shinichi Suda (the same year Japanese Ph.D. students in Economics) and Suminori Tokunaga (Japanese Ph.D. student in Regional Science) for helping me. I could not continue to study there without them. When I was depressed just after coming to the U.S., they encouraged me. Furthermore, I wish to thank Professors Mitsuo Saito, Kazuo Ogawa, Toshihisa Toyoda and Kazuhiro Ohtani. When I was in Japan before going to the U.S., I learned a lot of things under them at Kobe University. Especially, Professor Mistuo Saito, who was my main advisor in Japan, recommended that I continue to study econometrics at the University of Pennsylvania. And I am grateful to Shinichi Kitasaka, who was one of my classmates at Kobe University, for his suggestions and comments. Moreover, I have been stimulated by Shigeyuki Hamori's passion for research. He was one of my classmates at Kobe University, and he studied economics at Duke University at the same time that I was at the University of Pennsylvania. I am grateful to him for his valuable suggestions and advice. Also, I wish to say " Thanks" to my parents. They have not understood what I have done, but they supported me with regard to everything. Thus, I believe that I could finish the dissertation thanks to all the professors and all the friends who know me.