Jurgen A. Doornik and David F. Hendry Empirical Econometric Modelling PcGiveTM 14 Volume I OxMetrics 7 Published by Timberlake Consultants Ltd www.timberlake.co.uk www.timberlake-consultancy.com www.oxmetrics.net, www.doornik.com Empirical Econometric Modelling – PcGiveTM 14: Volume I Copyright c 2013 Jurgen A Doornik and David F Hendry First published by Timberlake Consultants in 1998 Revised in 1999, 2001, 2006, 2009, 2013 All rights reserved. No part of this work, which is copyrighted, may be reproduced or used in any form or by any means { graphic, electronic, or mechanical, including photocopy, record taping, or information storage and retrieval systems { without the written consent of the Publisher, except in accordance with the provisions of the Copyright Designs and Patents Act 1988. Whilst the Publisher has taken all reasonable care in the preparation of this book, the Publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsability or liability for any errors or omissions from the book, or from the software, or the consequences thereof. British Library Cataloguing-in-Publication Data A catalogue record of this book is available from the British Library Library of Congress Cataloguing-in-Publication Data A catalogue record of this book is available from the Library of Congress Jurgen A Doornik and David F Hendry p. cm. { (Empirical Econometric Modelling { PcGiveTM 14: Vol I) ISBN ISBN 978-0-9571708-3-4 Published by Timberlake Consultants Ltd Unit B3, Broomsleigh Business Park 842 Greenwich Lane London SE26 5BN, UK Union, NJ 07083-7905, U.S.A. http://www.timberlake.co.uk http://www.timberlake-consultancy.com Trademark notice All Companies and products referred to in this book are either trademarks or regis- tered trademarks of their associated Companies. Contents Front matter iii Contentsv List of Figures xiii List of Tables xv Preface xvii I PcGive Prologue1 1 Introduction to PcGive3 1.1 The PcGive system.........................3 1.2 Single equation modelling.....................4 1.3 The special features of PcGive...................5 1.4 Documentation conventions....................9 1.5 Using PcGive documentation.................... 10 1.6 Citation............................... 11 1.7 World Wide Web.......................... 11 1.8 Some data sets........................... 11 II PcGive Tutorials 13 2 Tutorial on Cross-section Regression 15 2.1 Starting the modelling procedure.................. 15 2.2 Formulating a regression...................... 17 2.3 Cross-section regression estimation................ 18 2.3.1 Simple regression output................. 20 2.4 Regression graphics........................ 23 2.5 Testing restrictions and omitted variables............. 24 2.6 Multiple regression......................... 26 v vi CONTENTS 2.7 Formal tests............................. 28 2.8 Storing residuals in the database.................. 29 3 Tutorial on Descriptive Statistics and Unit Roots 30 3.1 Descriptive data analysis...................... 31 3.2 Autoregressive distributed lag................... 34 3.3 Unit-root tests............................ 36 4 Tutorial on Dynamic Modelling 40 4.1 Model formulation......................... 40 4.2 Model estimation.......................... 42 4.3 Model output............................ 43 4.3.1 Equation estimates.................... 43 4.3.2 Analysis of 1-step forecast statistics........... 44 4.4 Graphical evaluation........................ 45 4.5 Dynamic analysis.......................... 46 4.6 Mis-specification tests....................... 48 4.7 Specification tests.......................... 50 4.7.1 Exclusion, linear and general restrictions......... 50 4.7.2 Test for common factors................. 52 4.8 Options............................... 53 4.9 Further Output........................... 54 4.10 Forecasting............................. 55 5 Tutorial on Model Reduction 58 5.1 The problems of simple-to-general modelling........... 58 5.2 Formulating general models.................... 58 5.3 Analyzing general models..................... 59 5.4 Sequential simplification...................... 61 5.5 Encompassing tests......................... 65 5.6 Model revision........................... 66 6 Tutorial on Automatic Model Selection using Autometrics 67 6.1 Introduction............................. 67 6.2 Modelling CONS.......................... 67 6.3 DHSY revisited........................... 75 7 Tutorial on Estimation Methods 77 7.1 Recursive estimation........................ 77 7.2 Instrumental variables....................... 79 7.2.1 Structural estimates.................... 81 7.2.2 Reduced forms...................... 81 7.3 Autoregressive least squares (RALS)............... 82 7.3.1 Optimization....................... 84 CONTENTS vii 7.3.2 RALS model evaluation.................. 87 7.4 Non-linear least squares...................... 88 8 Tutorial on Batch Usage 93 8.1 Introduction............................. 93 8.2 Generating and running Batch code................ 93 8.3 Generating and running Ox code.................. 95 9 Non-linear Models 98 9.1 Introduction............................. 98 9.2 Non-linear modelling........................ 98 9.3 Maximizing a function....................... 99 9.4 Logit and probit estimation..................... 100 9.5 Tobit estimation........................... 105 9.6 ARMA estimation......................... 108 9.7 ARCH estimation.......................... 110 III The Econometrics of PcGive 113 10 An Overview 115 11 Learning Elementary Econometrics Using PcGive 118 11.1 Introduction............................. 118 11.2 Variation over time......................... 119 11.3 Variation across a variable..................... 120 11.4 Populations, samples and shapes of distributions......... 122 11.5 Correlation and scalar regression.................. 123 11.6 Interdependence.......................... 126 11.7 Time dependence.......................... 127 11.8 Dummy variables.......................... 129 11.9 Sample variability......................... 131 11.10 Collinearity............................. 131 11.11 Nonsense regressions........................ 133 12 Intermediate Econometrics 135 12.1 Introduction............................. 135 12.2 Linear dynamic equations..................... 136 12.2.1 Stationarity and non-stationarity............. 136 12.2.2 Lag polynomials..................... 136 12.2.2.1 Roots of lag polynomials............ 138 12.2.2.2 Long-run solutions............... 138 12.2.2.3 Common factors................ 139 12.3 Cointegration............................ 139 12.4 A typology of simple dynamic models............... 143 viii CONTENTS 12.4.1 Static regression...................... 145 12.4.2 Univariate autoregressive processes........... 145 12.4.3 Leading indicators.................... 146 12.4.4 Growth-rate models.................... 146 12.4.5 Distributed lags...................... 147 12.4.6 Partial adjustment..................... 147 12.4.7 Autoregressive errors or COMFAC models........ 148 12.4.8 Equilibrium-correction mechanisms........... 149 12.4.9 Dead-start models..................... 151 12.5 Interpreting linear models..................... 151 12.5.1 Interpretation 1: a regression equation.......... 151 12.5.2 Interpretation 2: a (linear) least-squares approximation. 152 12.5.3 Interpretation 3: an autonomous contingent plan..... 152 12.5.4 Interpretation 4: derived from a behavioural relationship 152 12.6 Multiple regression......................... 153 12.6.1 Estimating partial adjustment............... 154 12.6.2 Heteroscedastic-consistent standard errors........ 155 12.6.3 Specific-to-general.................... 156 12.6.4 General-to-specific (Gets)................. 159 12.6.5 Automatic model selection using Autometrics...... 160 12.6.6 Time series........................ 162 12.6.7 Equilibrium correction.................. 162 12.6.8 Non-linear least squares, COMFAC, and RALS..... 163 12.7 Econometrics concepts....................... 166 12.7.1 Innovations and white noise............... 166 12.7.2 Exogeneity........................ 167 12.7.3 Constancy and invariance................. 168 12.7.4 Congruent models..................... 170 12.7.5 Encompassing rival models................ 170 12.8 Instrumental variables....................... 172 12.9 Inference and diagnostic testing.................. 173 12.10 Model selection........................... 175 12.10.1 Three levels of knowledge................ 175 12.10.2 Modelling criteria..................... 176 12.10.3 Implicit model design................... 176 12.10.4 Explicit model design................... 177 13 Statistical Theory 179 13.1 Introduction............................. 179 13.2 Normal distribution......................... 179 13.3 The bivariate normal density.................... 180 13.3.1 Marginal and conditional normal distributions...... 180 13.3.2 Regression......................... 181 13.4 Multivariate normal......................... 182 CONTENTS ix 13.4.1 Multivariate normal density................ 182 13.4.2 Multiple regression.................... 183 13.4.3 Functions of normal variables: χ2, t and F distributions. 184 13.5 Likelihood............................. 185 13.6 Estimation............................. 186 13.6.1 The score and the Hessian................ 186 13.6.2 Maximum likelihood estimation............. 187 13.6.3 Efficiency and Fisher’s information............ 188 13.6.4 Cramer–Rao´ bound...................