Finite sample econometrics

241 Pages·2004·3.12 MB·English

by  Aman Ullah

ADVANCED TEXTS IN ECONOMETRICS
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Manuel Arellano Guido Imbens Grayham E.Mizon
Adrain Pagan Mark Watson
Advisory Editor
C. W. J. Granger
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Finite Sample Econometrics
AMAN ULLAH
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ISBN0-19-877447-8(hbk.)
ISBN0-19-877448-6(pbk.)
13579108642
Contents
Preface ix
1 Introduction 1
2 Finite Sample Moments 9
2.1 Introduction 9
2.2 Exact Moments: Normal Case 9
2.3 Exact Moments: Nonnormal Case 18
2.3.1 Binomial Distribution 18
2.3.2 Poisson Distribution 19
2.3.3 Gamma Distribution 19
2.3.4 Exponential Family 20
2.3.5K-Parameter Exponential Density 21
2.3.6 Mixtures of Distributions 22
2.3.7 Edgeworth Density or Gram–Charlier Density 23
2.4 Exact Moments: General Case 24
2.5 Approximations of Moments 26
2.5.1 Large Sample Approximations: Normal and Nonnormal 27
2.5.2 Small-σ Approximations: Normal and Nonnormal 36
2.5.3 Results for Non-i.i.d Cases 44
2.5.4 The Laplace Approximation: Normal and Nonnormal 45
2.6 Summary and Survey 48
3 Finite Sample Distributions 51
3.1 Introduction 51
3.2 Exact Distribution 51
3.2.1 Distribution of Ratio of Quadratic Forms 52
3.3 Approximations of the Distribution of Quadratic Forms 55
3.4 Limiting Distributions 56
3.5 Nonnormal Case 57
3.6 Large-n Edgeworth Expansion 57
vi CONTENTS
3.7 Small-σ Edgeworth Expansion of h(y) (Normal and Nonnormal) 67
3.8 Remarks on the Edgeworth Expansion 69
4 Regression Model 75
4.1 Introduction 75
4.2 Model Specification and Least Squares Estimation 75
4.3 Properties of Estimators 77
4.3.1 Coefficients Estimators 77
4.3.2 Residuals and Residual Sum of Squares 80
4.3.3R2 and Adjusted R2 83
4.3.4 The F-Ratio 86
4.3.5 Prediction 87
4.3.6 Exact Moments Under Nonnormal 89
4.3.7 Approximate Moments 90
4.3.8 Hypothesis Testing 94
4.3.9 Nonlinear Regression Models 96
5 Models with Nonscalar Covariance Matrix of Errors 97
5.1 Introduction 97
5.2 General Model with Nonscalar Covariance Matrix 97
5.2.1 Exact Moments 97
5.2.2 Approximate Distribution and Moments 100
5.2.3 Hypothesis Testing 104
5.3 Specialized Models 107
5.3.1 Heteroskedasticity 107
5.3.2 Heteroskedasticity Testing 112
5.3.3 Model with Autocorrelation 113
5.3.4 Seemingly Unrelated Regressions 116
5.3.5 Limited Dependent Variable Models 120
5.3.6 Panel Data Models 123
6 Dynamic Time Series Model 129
6.1 Introduction 129
6.2 Model and Least-Squares Estimator 129
6.3 Finite Sample Results for Dynamic Model 133
6.3.1 Review 133
6.3.2 Exact Results for AR(1) model 137
6.3.3 Approximate Methods 139
6.3.4 Probability Distributions 145
6.3.5 ARMAX model 147
6.3.6 Nonnormal Case 148
6.3.7 Cointegration Model 149
6.4 Conclusion 151
CONTENTS vii
7 Simultaneous Equations Model 153
7.1 Introduction 153
7.2 Simultaneous Equations Model 154
7.2.1 Model Specification 154
7.2.2 Moments of the Single Equation Estimators 155
7.2.3 Moments of the IV Estimators of β 162
7.2.4 General Case of m Endogenous Variables 162
7.2.5 Approximate Moments 164
7.2.6 Nonlinear Simultaneous Equations Model 164
7.2.7 Density Function of IV Estimator 166
7.2.8 Further Finite Sample Results 168
7.2.9 Summary of Results 170
7.3 Analysis of Weak Instruments 171
7.3.1 Effects on the Moments and Distribution 171
7.3.2 Issue of Optimal Instruments 175
Appendix AStatistical Methods 179
A.1 Moments and Cumulants 179
A.2 Gram–Charlier and Edgeworth Series 180
A.3 Asymptotic Expansion and Asymptotic Approximation 182
A.3.1 Asymptotic Expansion (Stochastic) 184
A.4 Moments of the Quadratic Forms Under Normality 185
A.5 Moments of Quadratic Forms Under Nonnormality 187
A.6 Moment of Quadratic Form of a Vector of Squared Nonnormal Random Variables 188
A.7 Moments of Quadratic Forms in Random Matrices 189
A.8 Distribution of Quadratic Forms 192
A.8.1 Density and Moments of a Noncentral Chi-square Variable 193
A.8.2 Moment Generating Function and Characteristic Function 194
A.8.3 Density Function Based on Characteristic Function 195
A.9 Hypergeometric Functions 196
A.9.1 Asymptotic Expansion 197
A.10 Order of Magnitudes (Small o and Large O) 197
References 199
Index 227
To my daughter, Sushana Ullah
Preface
Over the last five decades, significant advances in the estimation and inference of various econometric models have
taken place. This includes the classical linear model where the explanatory variables are nonstochastic (fixed) and the
error is normally distributed, and the non-classical models, where these classical assumptions are violated. These
modelsare frequentlyusedinappliedwork,suchas thesimultaneous equationmodels, modelswithheteroskedasticity
and/orserialcorrelation,limiteddependentvariablemodels,paneldatamodels,andalargeclassoftimeseriesmodels.
Many of these models may also be nonlinear, explanatory variables can be stochastic and errors follow nonnormal
distributions. Whilethe classical linear model is oftenestimated by the ordinary least squares (LS) or generalized least
squares (GLS) estimators, the non-classical models have largely used the maximum likelihood (ML), the method of
moments, the instrumental variable, and the extremum estimation techniques. Within this setup, establishing the
properties of estimators in the classical linear model are straightforward for samples of any size and they are well
presented in econometrics textbooks. For the non-classical models, however, textbooks have mostly presented large
sample theory results despitethe existing finite sample analytical results. One explanationof this may be the technical
difficulties in developing the existing finite sample results and the complexities of their expressions.
It is well known that the large sample theory properties may not imply the finite sample behavior of econometrics
estimators and test statistics. In fact, the use of asymptotic theory results for small or even moderately large samples
may give misleading results. The field of finite sample theory has been developing rapidly since the seminal
contributions of Sir R. A. Fisher. Its applications in improving the inference for finite samples, the issues of bias-
adjusted estimation, analyzing weak instruments, determining optimal instruments and bootstrapping have further
enhanced the importance of the large existing literature on the finite sample.
This bookisintendedtoprovidea somewhatfirstcomprehensiveandunifiedtreatmentoffinitesampletheoryandto
apply the basic tools of this to various estimators and test statistics used in various econometric models. Both time
series and cross section data models as well as panel data models are considered.