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Econometric Modeling A Likelihood Approach PDF

378 Pages·2007·50.218 MB·English
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Econometric Modeling Econometric Modeling A Likelihood Approach David F. Hendry Bent Nielsen PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright c 2007 by Princeton University Press Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Control Number 2006052859 ISBN-13: 978-0-691-13128-3 ISBN-10: 0-691-13128-7 ISBN-13: 978-0-691-13089-7 (pbk.) ISBN-10: 0-691-13089-2 (pbk.) British Library Cataloging-in-Publication Data is available This book has been composed in LATEX The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents Preface ix Data and software xi Chapter 1. The Bernoulli model 1 1.1 Sample and populationdistributions 1 1.2 Distribution functions anddensities 4 1.3 The Bernoulli model 6 1.4 Summaryand exercises 12 Chapter 2. Inference in the Bernoulli model 14 2.1 Expectation andvariance 14 2.2 Asymptotic theory 19 2.3 Inference 23 2.4 Summaryand exercises 26 Chapter 3. A first regressionmodel 28 3.1 The US censusdata 28 3.2 Continuous distributions 29 3.3 Regression model with an intercept 32 3.4 Inference 38 3.5 Summaryand exercises 42 Chapter 4. The logit model 47 4.1 Conditional distributions 47 4.2 The logit model 52 4.3 Inference 58 4.4 Mis-specification analysis 61 4.5 Summaryand exercises 63 Chapter 5. The two-variable regression model 66 5.1 Econometric model 66 5.2 Estimation 69 5.3 Structural interpretation 76 5.4 Correlations 78 5.5 Inference 81 vi CONTENTS 5.6 Summary andexercises 85 Chapter 6. The matrix algebra of two-variable regression 88 6.1 Introductory example 88 6.2 Matrix algebra 90 6.3 Matrix algebra inregression analysis 94 6.4 Summary andexercises 96 Chapter 7. The multiple regression model 98 7.1 The three-variable regressionmodel 98 7.2 Estimation 99 7.3 Partial correlations 104 7.4 Multiple correlations 107 7.5 Properties ofestimators 109 7.6 Inference 110 7.7 Summary andexercises 118 Chapter 8. The matrix algebra of multiple regression 121 8.1 More on inversion ofmatrices 121 8.2 Matrix algebra ofmultiple regressionanalysis 122 8.3 Numerical computation of regressionestimators 124 8.4 Summary andexercises 126 Chapter 9. Mis-specification analysis in cross sections 127 9.1 The cross-sectional regression model 127 9.2 Test for normality 128 9.3 Test for identical distribution 131 9.4 Test for functional form 134 9.5 Simultaneous applicationof mis-specification tests 135 9.6 Techniques for improving regressionmodels 136 9.7 Summary andexercises 138 Chapter 10. Strong exogeneity 140 10.1 Strong exogeneity 140 10.2 The bivariate normal distribution 142 10.3 The bivariate normal model 145 10.4 Inference with exogenous variables 150 10.5 Summary andexercises 151 Chapter 11. Empirical models and modeling 154 11.1 Aspects ofeconometric modeling 154 11.2 Empirical models 157 11.3 Interpreting regressionmodels 161 11.4 Congruence 166 11.5 Encompassing 169 11.6 Summary andexercises 173 CONTENTS vii Chapter 12. Autoregressions and stationarity 175 12.1 Time-series data 175 12.2 Describing temporaldependence 176 12.3 The first-order autoregressive model 178 12.4 The autoregressive likelihood 179 12.5 Estimation 180 12.6 Interpretation ofstationary autoregressions 181 12.7 Inference forstationary autoregressions 187 12.8 Summaryand exercises 188 Chapter 13. Mis-specification analysisin time series 190 13.1 The first-order autoregressive model 190 13.2 Testsfor both cross sections andtime series 190 13.3 Test for independence 192 13.4 Recursivegraphics 195 13.5 Example: finding a model for quantities of fish 197 13.6 Mis-specification encompassing 200 13.7 Summaryand exercises 201 Chapter 14. The vectorautoregressive model 203 14.1 The vector autoregressive model 203 14.2 A vectorautoregressive model for the fishmarket 205 14.3 Autoregressive distributed-lag models 213 14.4 Static solutions andequilibrium-correction forms 214 14.5 Summaryand exercises 215 Chapter 15. Identification of structuralmodels 217 15.1 Under-identified structural equations 217 15.2 Exactly-identified structural equations 222 15.3 Over-identified structural equations 227 15.4 Identificationfrom a conditionalmodel 231 15.5 Instrumental variables estimation 234 15.6 Summaryand exercises 237 Chapter 16. Non-stationary time series 240 16.1 Macroeconomictime-series data 240 16.2 First-order autoregressivemodel andits analysis 242 16.3 Empirical modelingof UK expenditure 243 16.4 Propertiesof unit-root processes 245 16.5 Inference aboutunit roots 248 16.6 Summaryand exercises 252 Chapter 17. Cointegration 254 17.1 Stylized example of cointegration 254 17.2 Cointegration analysis ofvector autoregressions 255 viii CONTENTS 17.3 A bivariatemodel for moneydemand 258 17.4 Single-equation analysis ofcointegration 267 17.5 Summary andexercises 268 Chapter 18. Monte Carlo simulation experiments 270 18.1 Monte Carlo simulation 270 18.2 Testingin cross-sectional regressions 273 18.3 Autoregressions 277 18.4 Testingfor cointegration 281 18.5 Summary andexercises 285 Chapter 19. Automatic model selection 286 19.1 The model 286 19.2 Model formulation and mis-specification testing 287 19.3 Removingirrelevant variables 288 19.4 Keeping variables that matter 290 19.5 A general-to-specific algorithm 292 19.6 Selection bias 293 19.7 Illustration usingUK money data 298 19.8 Summary andexercises 300 Chapter 20. Structural breaks 302 20.1 Congruence intime series 302 20.2 Structural breaks and co-breaking 304 20.3 Location shifts revisited 307 20.4 Rational expectations andthe Lucas critique 308 20.5 Empirical tests ofthe Lucas critique 311 20.6 Rational expectations andEuler equations 315 20.7 Summary andexercises 319 Chapter 21. Forecasting 323 21.1 Background 323 21.2 Forecastingin changingenvironments 326 21.3 Forecastingfrom an autoregression 327 21.4 A forecast-errortaxonomy 332 21.5 Illustration usingUK money data 337 21.6 Summary andexercises 340 Chapter 22. The way ahead 342 References 345 Author index 357 Subject index 359 Preface Thisbook provides alikelihood-based introduction toeconometrics. Inessence, the idea is to carefully investigate the sample variation in the data, then exploit thatinformation tolearnabout theunderlying economic mechanisms. Therelative likelihood reflects howwell different models ofthe data perform. We findthisis a useful approach for both expository andmethodological reasons. Thesubstantive contextofeconometrics iseconomics. Economic theory is con- cerned about how an economy might function and how agents in the economy behave, but not so muchabout a detailed description of the data variation thatwill be observed. Econometrics faces the methodological challenge that much of the observed economic data variability isdue tofactors outside of economics, such as wars, epidemics, innovations, and changing institutions. Consequently, we pursue amethodology where fairlygeneral econometric models areformulated tocapture thesamplevariation,withoneorseveraleconomic theories beingspecialcasesthat can be tested. This approach offers the possibility offalsifying that theory, by not imposingthe structure of an economic theory at the outset. More constructively, it ispossible toenhance the economic theory by also explainingimportant variations in the economy. An example is the theory of consumption smoothing. By look- ing at the variation in a quarterly time series, we will see that consumption is not smooth but in fact shows a large increase every Christmas, followed by a slump in the first quarter of the following year. It is then clear that a general theory of consumption would havetotake this seasonal fluctuation intoaccount. We will use a maximum likelihood approach to analyze distributional models for thedata. Themodelswe consider haveparameters thatcharacterize features of the data: to learn about these parameters, each is accorded its most likely value. Within such a framework, it is then relatively easy to explain why we use the var- ious econometric techniques proposed. The book is organized so as to develop empirical, as well as theoretical, econometric aspects in each chapter. Quite often the analysis ofdata sets is developed over a sequence of chapters. This continued dataanalysis serves asaconstant reminderthatinpracticeonehas tobe pragmatic, and there willbe many situations where one hasto make compromises. By devel- oping a likelihood approach, it is easier to realize where compromises are made anddecide on the best resolution.

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