Walter Kramer Harald Sonnberger The Linear Regression Model Under Test Physica-Verlag Heidelberg Wien
Professor Dr. WALTER KRAMER, Fachbereich Wirtschaftswissenschaften, UniversWit Hannover, Wunstorfer Str. 14, D-3000 Hannover 91, FRG Dr. HARALD SONNBERGER, Leiter der Projektgruppe las-system, Institut fur Hohere Studien, Stumpergasse 56, A-1060 Wi en, Austria ISBN-13: 978-3-642-95878-6 e-isbn-13: 978-3-642-95876-2 DOl: 10.1007/978-3-642-95876-2 CIP-Kurztitelaufnahme der Deutschen Bibliothek Kramer. Walter: The linearregression model undertestiwalter Kramer; Harald Sonnberger. - Heidelberg; Wien: Physica-Verlag, 1986. ISBN-13: 978-3-642-95878-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use a fee is payable to 'Verwertungsgesellschaft Wort', Munich. by Physica-Verlag Heidelberg 1986 Soficover reprint of the hardcover 1st edition 1986 The usc of registered names, trademarks, etc. in this publication does not imply, even in the absence ofa specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover: Erich Kirchner, Heidelberg 712017130-543210
PREFACE This monograph grew out of joint work with various dedicated colleagues and students at the Vienna Institute for Advanced Studies. We would probably never have begun without the impetus of Johann Maurer, who for some time was the spiritus rector behind the Institute's macromodel of the Austrian economy. Manfred Deistler provided sustained stimulation for our research through many discussions in his econometric research seminar. Similar credits are due to Adrian Pagan, Roberto Mariano and Garry Phillips, the econometrics guest professors at the Institute in the 1982-1984 period, who through their lectures and advice have contributed greatly to our effort. Hans SchneeweiB offered helpful comments on an earlier version of the manuscript, and Benedikt Poetscher was always willing to lend a helping. hand when we had trouble with the mathematics of the tests. Needless to say that any errors are our own. Much of the programming for the tests and for the Monte Carlo experiments was done by Petr Havlik, Karl Kontrus and Raimund Alt. Without their assistance, our research project would have been impossible. Petr Havlik and Karl Kontrus in addition. read and criticized portions of the manuscript, and were of great help in reducing our error rate. Many of the more theoretical results in this monograph would never have come to light without the mathematical expertise of Werner Ploberger, who provided most of the statistical background of the chapter on testing for structural change.. We are also indebted to the people at the Institute's computer center (WSR), who, without blinking, watched us use up record amounts of CPU time orr their Univac 1100 machine. Beatrix Krones did an expert typing job, and Jorg Breitung and Ottmar von Holtz then edited the manuscript on the Hanover University text processing system. We thank all of them for their help and the effort they have put into our project. Finally we apologize for biting the hands that so generously fed us with data, in particular to John Rea, Dan Nichols and the editorial staff of the Journal of Money, Credit and Banking. Hanover and Vienna, May 1986 W. Kramer H. Sonnberger
CONTENTS page 1. INTRODUCTION 1 2. TECHNICAL PRELIMINARIES.. 3 a) The Linear Regression Model... 3 Notation and assumptions.... 3 Regression residuals.. 5 b) LR-, Wald- and LM-Tests. 7 Basic principles.... 8 A simple example...... 10 Estimating the information matrix..... 13 c) Miscellaneous Terminology.. 14 3. TESTING DISTURBANCES..... 17 a) Autocorrelation...... 17 The Durbin-Watson test... 17 The power of the Durbin-Watson test... 19 The DW inconclusive region... 24 Relationship to the LM principle..... 25 Dynamic regressions. 27 Simultaneous equations...... 30 Other alternatives to the DW test... 31 b) Heteroskedasticity.... 32 Comparing empirical variances... 32 A general LM test........ 35 Testing for arbitrary heteroskedasticity.. 38 c) Normality:..... 40 Comparing moments 40 Residuals vs. true disturbances.... 42 4. TESTING REGRESSORS.... 43 a) Structural Change........ 43 The Chow test and related methods 44 Testing subsets of the regression coefficients.. 45 Coefficient stability vs. stability of the variance. 46
VIII Shift point unknown................. 48 The CUSUM and CUSUM of squares tests............. 49 The CUSUM and CUSUM of squares tests with dummy exogenous variables.................... 54 Alternatives to the CUSUM and CUSUM of squares tests...... 56 The 'Fluctuation test.............. 59 Local power........................ 62 A Monte Carlo comparison of the CUSUM and Fluctuation tests... 64 Dynamic models... '................... 70 A modification of the CUSUM test............... 74 A modification of the CUSUM of squares test............ 75 b) Functional Form.................... 78 Ramsey's RESET................ 79 The Rainbow test............ 80 Convex or concave alternatives............ 81 Linear vs. log-linear functional form............. 83 Outliers....................... 84 c) General Misspecification................... 85 Hausman-type specification tests............ 85 Errors in variables................. 87 Computing the Hausman test..... '...... 88 Exogeneity.................................. 92 A Hausman test with trending data........... 95 Hausman tests versus classical tests............. 97 The differencing test.................. 101 First differences as an IV estimator............. 103 The information matrix test................. 105 5. UNIFYING THEMES.............. 111 a) Variable Transformation................... 111 Linear transformations.................... 111 Examples...................... 113 Relationship to IV estimation............... 115 b) Variable Addition.................. 116 Data transformation tests as RESETs............... 116 Hausman tests as RESETs............ 118 Power comparisons................ 119 Asymptotic independence.................. 120 6. DIAGNOSTIC CHECKING IN PRACTICE......... 123 a) Empirical Results...................... 123 Demand for money................ 124 Currency substitution.............. 128
Bond yield........................ 133 Growth of money supply............ 137 The Value of stocks............... 140 Wages.......... 142 Unemployment...... 144 Discussion......... 146 b) Issues in Multiple Testing.............. 147 Controlling the size............ 147 Ruger's test......... 148 Global vs. multiple significance level............... 150 Holm's procedure.............. 152 Robustness................... 154 Further reading..................... 155 APPENDIX...................... 157 a) Sample Data.............. 158 b) The IAS SYSTEM............ 168 REFERENCES................. 173 AUTHOR INDEX......................... 185 SUBJECT INDEX........... 188 IX