Contents
1 Introduction and Motivations 1
1.1 Introduction 1
1.2 A Review of Nonlinear Estimation Methods 2
1.2.1 Parametric conditional models 2
1.2.2 Estimators defined by the optimization of a criterion function 3
1.2.3 Properties of optimization estimators 6
1.3 Potential Applications of Simulated Methods 7
1.3.1 Limited dependent variable models 8
1.3.2 Aggregation effect 10
1.3.3 Unobserved heterogeneity 11
1.3.4 Nonlinear dynamic models with unobservable factors 12
1.3.5 Specification resulting from the optimization of some expected
criterion 14
1.4 Simulation 15
1.4.1 Two kinds of simulation 15
1.4.2 How to simulate? 15
1.4.3 Partial path simulations 18
2 The Method of Simulated Moments (MSM) 19
2.1 Path Calibration or Moments Calibration 19
2.1.1 Path calibration 20
2.1.2 Moment calibration 20
2.2 The Generalized Method of Moments (GMM) 21
2.2.1 The static case 21
2.2.2 The dynamic case 22
2.3 The Method of Simulated Moments (MSM) 24
viii CONTENTS
2.3.1 Simulators 24
2.3.2 Definition of the MSM estimators 27
2.3.3 Asymptotic properties of the MSM 29
2.3.4 Optimal MSM 31
2.3.5 An extension of the MSM 34
Appendix 2A: Proofs of the Asymptotic Properties of the MSM Estimator 37
2A. 1 Consistency 37
2A.2 Asymptotic normality 38
3 Simulated Maximum Likelihood, Pseudo-Maximum Likelihood,
and Nonlinear Least Squares Methods 41
3.1 Simulated Maximum Likelihood Estimators (SML) 41
3.1.1 Estimator based on simulators of the conditional density
functions 42
3.1.2 Asymptotic properties 42
3.1.3 Study of the asymptotic bias 44
3.1.4 Conditioning 45
3.1.5 Estimators based on other simulators 48
3.2 Simulated Pseudo-Maximum Likelihood and Nonlinear Least
Squares Methods 50
3.2.1 Pseudo-maximum likelihood (PML) methods 50
3.2.2 Simulated PML approaches 55
3.3 Bias Corrections for Simulated Nonlinear Least Squares 56
3.3.1 Corrections based on the first order conditions 56
3.3.2 Corrections based on the objective function 57
Appendix 3A: The Metropolis-Hastings (MH) Algorithm 58
3A.1 Definition of the algorithm 58
3 A.2 Properties of the algorithm 59
4 Indirect Inference 61
4.1 The Principle 61
4.1.1 Instrumental model 61
4.1.2 Estimation based on the score 62
4.1.3 Extensions to other estimation methods 64
4.2 Properties of the Indirect Inference Estimators 66
4.2.1 The dimension of the auxiliary parameter 66
CONTENTS ix
4.2.2 Which moments to match? 67
4.2.3 Asymptotic properties 69
4.2.4 Some consistent, but less efficient, procedures 71
4.3 Examples 71
4.3.1 Estimation of a moving average parameter 71
4.3.2 Application to macroeconometrics 75
4.3.3 The efficient method of moment 76
4.4 Some Additional Properties of Indirect Inference Estimators 77
4.4.1 Second order expansion 77
4.4.2 Indirect information and indirect identification 82
Appendix 4A: Derivation of the Asymptotic Results 84
4A.1 Consistency of the estimators 85
4A.2 Asymptotic expansions 86
Appendix 4B: Indirect Information and Identification: Proofs 89
4B. 1 Computation of / I (0) 89
4B.2 Another expression of Ip(0) 91
5 Applications to Limited Dependent Variable Models 93
5.1 MSM and SML Applied to Qualitative Models 93
5.1.1 Discrete choice model 93
5.1.2 Simulated methods 94
5.1.3 Different simulators 96
5.2 Qualitative Models and Indirect Inference based on Multivariate
Logistic Models 100
5.2.1 Approximations of a multivariate normal distribution in a
neighbourhood of the no correlation hypothesis 100
5.2.2 The use of the approximations when correlation is present 102
5.3 Simulators for Limited Dependent Variable Models based on
Gaussian Latent Variables 103
5.3.1 Constrained and conditional moments of a multivariate
Gaussian distribution 103
5.3.2 Simulators for constrained moments 104
5.3.3 Simulators for conditional moments 107
5.4 Empirical Studies 112
5.4.1 Labour supply and wage equation 112
x CONTENTS
5.4.2 Test of the rational expectation hypothesis from business
survey data 113
Appendix 5 A: Some Monte Carlo Studies 115
6 Applications to Financial Series 119
6.1 Estimation of Stochastic Differential Equations from Discrete Observations
by Indirect Inference 119
6.1.1 The principle 119
6.1.2 Comparison between indirect inference and full maximum
likelihood methods 121
6.1.3 Specification of the volatility 125
6.2 Estimation of Stochastic Differential Equations from Moment
Conditions 133
6.2.1 Moment conditions deduced from the infinitesimal
operator 133
6.2.2 Method of simulated moments 137
6.3 Factor Models 138
6.3.1 Discrete time factor models 138
6.3.2 State space form and Kitagawa's filtering algorithm 139
6.3.3 An auxiliary model for applying indirect inference on factor
ARCH models 141
6.3.4 SML applied to a stochastic volatility model 142
Appendix 6A: Form of the Infinitesimal Operator 143
7 Applications to Switching Regime Models 145
7.1 Endogenously Switching Regime Models 145
7.1.1 Static disequilibrium models 145
7.1.2 Dynamic disequilibrium models 148
7.2 Exogenously Switching Regime Models 151
7.2.1 Markovian vs. non-Markovian models 151
7.2.2 A switching state space model and the partial Kalman
filter 152
7.2.3 Computation of the likelihood function 153
References 159
Index 173
Simulation-Based Econometric Methods.pdf
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