ECONOMETRIC MODELING AND
INFERENCE
JEAN-PIERRE FLORENS
University of Toulouse
V.ELAYOUDOM MARIMOUTOU
GREQAM, University of Aix-Marseille 2
ANNE P ˇEGUIN-FEISSOLLE
CNRS and GREQAM, France
Translated by Josef Perktold and Marine Carrasco
Foreword by James J. Heckman
Contents
Foreword page xvii
Preface xix
I Statistical Methods 1
1 Statistical Models 3
1.1 Introduction 3
1.2 Sample, Parameters, and Sampling Probability
Distributions 3
1.3 Independent and Identically Distributed Models 6
1.4 Dominated Models, Likelihood Function 8
1.5 Marginal and Conditional Models 10
2 Sequential Models and Asymptotics 17
2.1 Introduction 17
2.2 Sequential Stochastic Models and Asymptotics 17
2.3 Convergence in Probability and Almost Sure
Convergence V Law of Large Numbers 21
2.4 Convergence in Distribution and Central Limit Theorem 25
2.5 Noncausality and Exogeneity in Dynamic Models 27
2.5.1 Wiener-Granger Causality 28
2.5.2 Exogeneity 30
3 Estimation by Maximization and by the Method of Moments 33
3.1 Introduction 33
3.2 Estimation 33
3.3 Moment Conditions and Maximization 39
3.4 Estimation by the Method of Moments and Generalized
Moments 44
3.5 Asymptotic Properties of Estimators 48
ix
x Contents
4 Asymptotic Tests 61
4.1 Introduction 61
4.2 Tests and Asymptotic Tests 62
4.3 Wald Tests 65
4.4 Rao Test 69
4.5 Tests Based on the Comparison of Minima 73
4.6 Test Based on Maximum Likelihood Estimation 76
4.7 Hausman Tests 78
4.8 Encompassing Test 82
5 Nonparametric Methods 87
5.1 Introduction 87
5.2 Empirical Distribution and Empirical
Distribution Function 87
5.3 Density Estimation 91
5.3.1 Construction of the Kernel Estimator
of the Density 91
5.3.2 Small Sample Properties of the Kernel Estimator
and Choices of Window and Kernel 93
5.3.3 Asymptotic Properties 96
5.4 Semiparametric Methods 98
6 Simulation Methods 103
6.1 Introduction 103
6.2 Random Number Generators 103
6.2.1 Inversion of the Distribution Function 104
6.2.2 Rejection Method 105
6.2.3 Random Vector Generators 106
6.3 Utilization in Calculation Procedures 107
6.3.1 Monte Carlo Integration 107
6.3.2 Simulation-Based Method of Moments 109
6.4 Simulations and Small Sample Properties
of Estimators and Tests 116
6.5 Bootstrap and Distribution of the Moment Estimators
and of the Density 120
II Regression Models 127
7 Conditional Expectation 129
7.1 Introduction 129
7.2 Conditional Expectation 129
7.3 Linear Conditional Expectation 134
Contents xi
8 Univariate Regression 141
8.1 Introduction 141
8.2 Linear Regression 142
8.2.1 The Assumptions of the Linear Regression Model 142
8.2.2 Estimation by Ordinary Least Squares 144
8.2.3 Small Sample Properties 148
8.2.4 Finite Sample Distribution Under
the Normality Assumption 151
8.2.5 Analysis of Variance 156
8.2.6 Prediction 159
8.2.7 Asymptotic Properties 160
8.3 Nonlinear Parametric Regression 165
8.4 Misspecified Regression 169
8.4.1 Properties of the Least Squares Estimators 170
8.4.2 Comparing the True Regression
with Its Approximation 172
8.4.3 Specification Tests 174
9 Generalized Least Squares Method, Heteroskedasticity,
and Multivariate Regression 179
9.1 Introduction 179
9.2 Allowing for Nuisance Parameters in Moment Estimation 181
9.3 Heteroskedasticity 184
9.3.1 Estimation 185
9.3.2 Tests for Homoskedasticity 196
9.4 Multivariate Regression 199
10 Nonparametric Estimation of the Regression 213
10.1 Introduction 213
10.2 Estimation of the Regression Function by Kernel 214
10.2.1 Calculation of the Asymptotic Mean Integrated
Squared Error 216
10.2.2 Convergence of AMISE and Asymptotic
Normality 221
10.3 Estimating a Transformation of the Regression Function 223
10.4 Restrictions on the Regression Function 228
10.4.1 Index Models 228
10.4.2 Additive Models 231
11 Discrete Variables and Partially Observed Models 234
11.1 Introduction 234
11.2 Various Types of Models 235
xii Contents
11.2.1 Dichotomous Models 235
11.2.2 Multiple Choice Models 237
11.2.3 Censored Models 239
11.2.4 Disequilibrium Models 243
11.2.5 Sample Selection Models 244
11.3 Estimation 248
11.3.1 Nonparametric Estimation 248
11.3.2 Semiparametric Estimation by Maximum
Likelihood 250
11.3.3 Maximum Likelihood Estimation 251
III Dynamic Models 259
12 Stationary Dynamic Models 261
12.1 Introduction 261
12.2 Second Order Processes 262
12.3 Gaussian Processes 264
12.4 Spectral Representation and Autocovariance
Generating Function 265
12.5 Filtering and Forecasting 267
12.5.1 Filters 267
12.5.2 Linear Forecasting V General Remarks 270
12.5.3 Wold Decomposition 272
12.6 Stationary ARMA Processes 273
12.6.1 Introduction 273
12.6.2 Invertible ARMA Processes 274
12.6.3 Computing the Covariance Function of an
ARMA( p, q) Process 277
12.6.4 The Autocovariance Generating Function 278
12.6.5 The Partial Autocorrelation Function 280
12.7 Spectral Representation of an ARMA( p, q) Process 282
12.8 Estimation of ARMA Models 283
12.8.1 Estimation by the Yule-Walker Method 283
12.8.2 Box-Jenkins Method 286
12.9 Multivariate Processes 289
12.9.1 Some Definitions and General Observations 289
12.9.2 Underlying Univariate Representation of a
Multivariate Process 292
12.9.3 Covariance Function 294
12.10 Interpretation of a VAR( p) Model Under Its MA(≯) Form 294
12.10.1 Propagation of a Shock on a Component 294
12.10.2 Variance Decomposition of the Forecast Error 295
Contents xiii
12.11 Estimation of VAR( p) Models 296
12.11.1 Maximum Likelihood Estimation of  298
12.11.2 Maximum Likelihood Estimation of  300
12.11.3 Asymptotic Distribution of  and of  301
13 Nonstationary Processes and Cointegration 304
13.1 Introduction 304
13.2 Asymptotic Properties of Least Squares
Estimators of I (1) Processes 306
13.3 Analysis of Cointegration and Error Correction
Mechanism 325
13.3.1 Cointegration and MA Representation 326
13.3.2 Cointegration in a VAR Model in Levels 327
13.3.3 Triangular Representation 329
13.3.4 Estimation of a Cointegrating Vector 330
13.3.5 Maximum Likelihood Estimation of an Error
Correction Model Admitting a Cointegrating
Relation 335
13.3.6 Cointegration Test Based on the Canonical
Correlations: Johansenˇs Test 338
14 Models for Conditional Variance 341
14.1 Introduction 341
14.2 Various Types of ARCH Models 341
14.3 Estimation Method 346
14.4 Tests for Conditional Homoskedasticity 357
14.5 Some Specificities of ARCH-Type Models 361
14.5.1 Stationarity 361
14.5.2 Leptokurticity 362
14.5.3 Various Conditional Distributions 363
15 Nonlinear Dynamic Models 366
15.1 Introduction 366
15.2 Case Where the Conditional Expectation Is
Continuously Differentiable 367
15.2.1 Definitions 367
15.2.2 Conditional Moments and Marginal Moments in
the Homoskedastic Case: Optimal Instruments 368
15.2.3 Heteroskedasticity 372
15.2.4 Modifying of the Set of Conditioning Variables:
Kernel Estimation of the Asymptotic Variance 373
xiv Contents
15.3 Case Where the Conditional Expectation Is Not
Continuously Differentiable: Regime-Switching Models 376
15.3.1 Presentation of a Few Examples 377
15.3.2 Problem of Estimation 379
15.4 Linearity Test 383
15.4.1 All Parameters Are Identified Under H0 383
15.4.2 The Problem of the Nonidentification of Some
Parameters Under H0 387
IV Structural Modeling 393
16 Identification and Overidentification
in Structural Modeling 395
16.1 Introduction 395
16.2 Structural Model and Reduced Form 396
16.3 Identification: The Example of Simultaneous Equations 398
16.3.1 General Definitions 398
16.3.2 Linear i.i.d. Simultaneous Equations Models 401
16.3.3 Linear Dynamic Simultaneous Equations Models 407
16.4 Models from Game Theory 410
16.5 Overidentification 414
16.5.1 Overidentification in Simultaneous Equations
Models 417
16.5.2 Overidentification and Moment Conditions 418
16.5.3 Overidentification and Nonparametric Models 419
17 Simultaneity 421
17.1 Introduction 421
17.2 Simultaneity and Simultaneous Equations 422
17.3 Endogeneity, Exogeneity, and Dynamic Models 425
17.4 Simultaneity and Selection Bias 428
17.5 Instrumental Variables Estimation 431
17.5.1 Introduction 431
17.5.2 Estimation 433
17.5.3 Optimal Instruments 437
17.5.4 Nonparametric Approach and Endogenous
Variables 439
17.5.5 Test of Exogeneity 442
18 Models with Unobservable Variables 446
18.1 Introduction 446
18.2 Examples of Models with Unobservable Variables 448
Contents xv
18.2.1 Random-Effects Models and Random-Coefficient
Models 448
18.2.2 Duration Models with Unobserved Heterogeneity 450
18.2.3 Errors-in-Variables Models 453
18.2.4 Partially Observed Markov Models and State
Space Models 454
18.3 Comparison Between Structural Model and Reduced Form 456
18.3.1 Duration Models with Heterogeneity and Spurious
Dependence on the Duration 457
18.3.2 Errors-in-Variables Model and Transformation of
the Coefficients of the Linear Regression 459
18.3.3 Markov Models with Unobservable Variables and
Spurious Dynamics of the Model 460
18.4 Identification Problems 461
18.5 Estimation of Models with Unobservable Variables 462
18.5.1 Estimation Using a Statistic Independent of the
Unobservables 462
18.5.2 Maximum Likelihood Estimation: EM Algorithm
and Kalman Filter 464
18.5.3 Estimation by Integrated Moments 469
18.6 Counterfactuals and Treatment Effects 470
Bibliography 477
Index 493