The Linear Regression Model Under Test

Transcription

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2 Walter Kramer Harald Sonnberger The Linear Regression Model Under Test Physica-Verlag Heidelberg Wien

3 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: e-isbn-13: DOl: / CIP-Kurztitelaufnahme der Deutschen Bibliothek Kramer. Walter: The linearregression model undertestiwalter Kramer; Harald Sonnberger. - Heidelberg; Wien: Physica-Verlag, ISBN-13: 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

4 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 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

5 CONTENTS page 1. INTRODUCTION 1 2. TECHNICAL PRELIMINARIES.. 3 a) The Linear Regression Model... 3 Notation and assumptions Regression residuals.. 5 b) LR-, Wald- and LM-Tests. 7 Basic principles A simple example Estimating the information matrix c) Miscellaneous Terminology TESTING DISTURBANCES a) Autocorrelation The Durbin-Watson test The power of the Durbin-Watson test The DW inconclusive region Relationship to the LM principle Dynamic regressions. 27 Simultaneous equations Other alternatives to the DW test b) Heteroskedasticity Comparing empirical variances A general LM test Testing for arbitrary heteroskedasticity.. 38 c) Normality: Comparing moments 40 Residuals vs. true disturbances TESTING REGRESSORS a) Structural Change The Chow test and related methods 44 Testing subsets of the regression coefficients.. 45 Coefficient stability vs. stability of the variance. 46

6 VIII Shift point unknown The CUSUM and CUSUM of squares tests The CUSUM and CUSUM of squares tests with dummy exogenous variables Alternatives to the CUSUM and CUSUM of squares tests The 'Fluctuation test Local power A Monte Carlo comparison of the CUSUM and Fluctuation tests Dynamic models... ' A modification of the CUSUM test A modification of the CUSUM of squares test b) Functional Form Ramsey's RESET The Rainbow test Convex or concave alternatives Linear vs. log-linear functional form Outliers c) General Misspecification Hausman-type specification tests Errors in variables Computing the Hausman test..... ' Exogeneity A Hausman test with trending data Hausman tests versus classical tests The differencing test First differences as an IV estimator The information matrix test UNIFYING THEMES a) Variable Transformation Linear transformations Examples Relationship to IV estimation b) Variable Addition Data transformation tests as RESETs Hausman tests as RESETs Power comparisons Asymptotic independence DIAGNOSTIC CHECKING IN PRACTICE a) Empirical Results Demand for money Currency substitution

7 Bond yield Growth of money supply The Value of stocks Wages Unemployment Discussion b) Issues in Multiple Testing Controlling the size Ruger's test Global vs. multiple significance level Holm's procedure Robustness Further reading APPENDIX a) Sample Data b) The IAS SYSTEM REFERENCES AUTHOR INDEX SUBJECT INDEX IX