Simulation-Based Econometric Methods

, by Gouriéroux, Christian; Monfort, Alain

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ISBN: 9780198774754 | 0198774753
Cover: Hardcover
Copyright: 4/10/1997

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This book introduces a new generation of statistical econometrics. After linear models leading to analytical expressions for estimators, and non-linear models using numerical optimization algorithms, the availability of high- speed computing has enabled econometricians to consider econometric models without simple analytical expressions. The previous difficulties presented by the presence of integrals of large dimensions in the probability density functions or in the moments can be circumvented by a simulation-based approach. After a brief survey of classical parametric and semi-parametric non-linear estimation methods and a description of problems in which criterion functions contain integrals, the authors present a general form of the model where it is possible to simulate the observations. They then move to calibration problems and the simulated analogue of the method of moments, before considering simulated versions of maximum likelihood, pseudo-maximum likelihood, or non-linear least squares. The general principle of indirect inference is presented and is then applied to limited dependent variable models and to financial series.

1 Introduction and Motivations

(18)

1.1 Introduction

(1)

1.2 A Review of Nonlinear Estimation Methods

(5)

1.2.1 Parametric conditional models

(1)

1.2.2 Estimators defined by the optimization of criterion function

(3)

1.2.3 Properties of optimization estimators

(1)

1.3 Potential Applications of Simulated Methods

(8)

1.3.1 Limited dependent variable models

(2)

1.3.2 Aggregation effect

(1)

1.3.3 Unobserved heterogeneity

(1)

1.3.4 Nonlinear dynamic models with unobservable factors

(2)

1.3.5 Specification resulting from the optimization of some expected criterion

(1)

1.4 Simulation

(4)

1.4.1 Two kinds simulation

(1)

1.4.2 How to simulate?

(3)

1.4.3 Partial path simulations

(1)

2 The Method of Simulated Moments (MSM)

(22)

2.1 Path Calibration or Moments Calibration

(2)

2.1.1 Path calibration

(1)

2.1.2 Moment calibration

(1)

2.2 The Generalized Method of Moments (GMM)

(3)

2.2.1 The static case

(1)

2.2.2 The dynamic case

(2)

2.3 The Method of Simulated Moments (MSM)

(13)

2.3.1 Simulators

(3)

2.3.2 Definition of the MSM estimators

(2)

2.3.3 Asymptotic properties of the MSM

(2)

2.3.4 Optimal MSM

(3)

2.3.5 An Extension of the MSM

(3)

Appendix 2A: Proofs of the Asymptotic Properties of the MSM Estimator

(4)

2A.1 Consistency

(1)

2A.2 Asymptotic normality

(3)

3 Simulated Maximum Likelihood, Pseudo-Maximum Likelihood, and Nonlinear Least Squares Methods

(20)

3.1 Simulated Maximum Likelihood Estimators (SML)

(9)

3.1.1 Estimator based on simulators of the conditional density functions

(1)

3.1.2 Asymptotic properties

(2)

3.1.3 Study of the asymptotic bias

(1)

3.1.4 Conditioning

(3)

3.1.5 Estimators based on other simulators

(2)

3.2 Simulated Pseudo-Maximum Likelihood and Nonlinear Least Squares Methods

(6)

3.2.1 Pseudo-Maximum likelihood (PML) methods

(5)

3.2.2 Simulated PML approaches

(1)

3.3 Bias Corrections for Simulated Nonlinear Least Squares

(2)

3.3.1 Corrections based on the first order conditions

(1)

3.3.2 Corrections based on the objective function

(1)

Appendix 3A: The Metropolis-Hastings (MH) Algorithm

(1)

3A.1 Definition of the algorithm

(1)

3A.2 Properties of the algorithm

(2)

4 Indirect Inference

(32)

4.1 The Principle

(5)

4.1.1 Instrumental model

(1)

4.1.2 Estimation based on the score

(2)

4.1.3 Extensions to other estimation methods

(2)

4.2 Properties of the Indirect Inference Estimators

(5)

4.2.1 The dimension of the auxiliary parameter

(1)

4.2.2 Which moments to match?

(2)

4.2.3 Asymptotic properties

(2)

4.2.4 Some consistent, but less efficient, procedures

(1)

4.3 Examples

(6)

4.3.1 Estimation of moving average parameter

(4)

4.3.2 Application to macroeconometrics

(1)

4.3.3 The efficient method of moment

(1)

4.4 Some Additional Properties of Indirect Inference Estimators

(7)

4.4.1 Second order expansion

(5)

4.4.2 Indirect information and indirect identification

(2)

Appendix 4A: Derivation of the Asymptotic Results

(5)

4A.1 Consistency of the estimators

(1)

4A.2 Asymptotic expansions

(3)

Appendix 4B: Indirect Information and Indentification: Proofs

(4)

4B.1 Computation of II( )

(2)

4B.2 Another Expression of IP ( )

(2)

5 Applications to Limited Dependent Variable Models

(26)

5.1 MSM and SML Applied to Qualitative Models

(7)

5.1.1 Discrete choice model

(1)

5.1.2 Simulated methods

(2)

5.1.3 Different simulators

(4)

5.2 Qualitative Models and Indirect Inference based on Multivariate Logistic Models

100

(3)

5.2.1 Approximations of a multivariate normal distribution in a neighbourhood of the no correlation hypothesis

100

(2)

5.2.2 The use of the approximations when correlation is present

102

(1)

5.3 Simulators for Limited Dependent Variable Models based on Gaussian Latent Variables

103

(9)

5.3.1 Constrained and conditional moments of multivariate Gaussian distribution

103

(1)

5.3.2 Simulators for constrained moments

104

(3)

5.3.3 Simulators for conditional moments

107

(5)

5.4 Empirical Studies

112

(3)

5.4.1 Labour supply and wage equation

112

(1)

5.4.2 Test of the rational expectation hypothesis from business survey data

113

(2)

Appendix 5A: Some Monte Carlo Studies

115

(4)

6 Applications to Financial Series

119

(26)

6.1 Estimation of Stochastic Differential Equations from Discrete Observations by Indirect Inference

119

(14)

6.1.1 The Principle

119

(2)

6.1.2 Comparison between indirect inference and full maximum likelihood methods

121

(4)

6.1.3 Specification of the volatility

125

(8)

6.2 Estimation of Stochastic Differential Equations from Moment Conditions

133

(5)

6.2.1 Moment conditions deduced from the infinitesimal operator

133

(4)

6.2.2 Method of simulated moments

137

(1)

6.3 Factor Models

138

(5)

6.3.1 Discrete time factor models

138

(1)

6.3.2 State space form and Kitagawa's filtering algorithm

139

(2)

6.3.3 An auxiliary model for applying indirect inference on factor ARCH models

141

(1)

6.3.4 SML applied to a stochastic volatility model

142

(1)

Appendix 6A: Form of the Infinitesimal Operator

143

(2)

7 Applications to Switching Regime Models

145

(14)

7.1 Endogenously Switching Regime Models

145

(6)

7.1.1 Static disequilibrium models

145

(3)

7.1.2 Dynamic disequilibrium models

148

(3)

7.2 Exogenously Switching Regime Models

151

(8)

7.2.1 Markovian Vs. non-Markovian models

151

(1)

7.2.2 A switching state space model and the partial Kalman filter

152

(1)

7.2.3 Computation of the likelihood function

153

(6)

References

159

(14)

Index

173

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Simulation-Based Econometric Methods

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Table of Contents

Supplemental Materials