Econometric Modeling A Likelihood Approach

378 Pages·2007·50.218 MB·English

by  David F. Hendry, Bent Nielsen

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
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In the United Kingdom: Princeton University Press
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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.
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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.