Simulation-Based Econometric Methods

by Gouriéroux, Christian; Monfort, Alain

• ISBN13: 9780198774754
• ISBN10: 0198774753
• Format: Hardcover
• Copyright: 1997-04-10
• Publisher: Oxford University Press

Summary

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.

Table of Contents

1 Introduction and Motivations	1	(18)
1.1 Introduction	1	(1)
1.2 A Review of Nonlinear Estimation Methods	2	(5)
1.2.1 Parametric conditional models	2	(1)
1.2.2 Estimators defined by the optimization of criterion function	3	(3)
1.2.3 Properties of optimization estimators	6	(1)
1.3 Potential Applications of Simulated Methods	7	(8)
1.3.1 Limited dependent variable models	8	(2)
1.3.2 Aggregation effect	10	(1)
1.3.3 Unobserved heterogeneity	11	(1)
1.3.4 Nonlinear dynamic models with unobservable factors	12	(2)
1.3.5 Specification resulting from the optimization of some expected criterion	14	(1)
1.4 Simulation	15	(4)
1.4.1 Two kinds simulation	15	(1)
1.4.2 How to simulate?	15	(3)
1.4.3 Partial path simulations	18	(1)
2 The Method of Simulated Moments (MSM)	19	(22)
2.1 Path Calibration or Moments Calibration	19	(2)
2.1.1 Path calibration	20	(1)
2.1.2 Moment calibration	20	(1)
2.2 The Generalized Method of Moments (GMM)	21	(3)
2.2.1 The static case	21	(1)
2.2.2 The dynamic case	22	(2)
2.3 The Method of Simulated Moments (MSM)	24	(13)
2.3.1 Simulators	24	(3)
2.3.2 Definition of the MSM estimators	27	(2)
2.3.3 Asymptotic properties of the MSM	29	(2)
2.3.4 Optimal MSM	31	(3)
2.3.5 An Extension of the MSM	34	(3)
Appendix 2A: Proofs of the Asymptotic Properties of the MSM Estimator	37	(4)
2A.1 Consistency	37	(1)
2A.2 Asymptotic normality	38	(3)
3 Simulated Maximum Likelihood, Pseudo-Maximum Likelihood, and Nonlinear Least Squares Methods	41	(20)
3.1 Simulated Maximum Likelihood Estimators (SML)	41	(9)
3.1.1 Estimator based on simulators of the conditional density functions	42	(1)
3.1.2 Asymptotic properties	42	(2)
3.1.3 Study of the asymptotic bias	44	(1)
3.1.4 Conditioning	45	(3)
3.1.5 Estimators based on other simulators	48	(2)
3.2 Simulated Pseudo-Maximum Likelihood and Nonlinear Least Squares Methods	50	(6)
3.2.1 Pseudo-Maximum likelihood (PML) methods	50	(5)
3.2.2 Simulated PML approaches	55	(1)
3.3 Bias Corrections for Simulated Nonlinear Least Squares	56	(2)
3.3.1 Corrections based on the first order conditions	56	(1)
3.3.2 Corrections based on the objective function	57	(1)
Appendix 3A: The Metropolis-Hastings (MH) Algorithm	58	(1)
3A.1 Definition of the algorithm	58	(1)
3A.2 Properties of the algorithm	59	(2)
4 Indirect Inference	61	(32)
4.1 The Principle	61	(5)
4.1.1 Instrumental model	61	(1)
4.1.2 Estimation based on the score	62	(2)
4.1.3 Extensions to other estimation methods	64	(2)
4.2 Properties of the Indirect Inference Estimators	66	(5)
4.2.1 The dimension of the auxiliary parameter	66	(1)
4.2.2 Which moments to match?	67	(2)
4.2.3 Asymptotic properties	69	(2)
4.2.4 Some consistent, but less efficient, procedures	71	(1)
4.3 Examples	71	(6)
4.3.1 Estimation of moving average parameter	71	(4)
4.3.2 Application to macroeconometrics	75	(1)
4.3.3 The efficient method of moment	76	(1)
4.4 Some Additional Properties of Indirect Inference Estimators	77	(7)
4.4.1 Second order expansion	77	(5)
4.4.2 Indirect information and indirect identification	82	(2)
Appendix 4A: Derivation of the Asymptotic Results	84	(5)
4A.1 Consistency of the estimators	85	(1)
4A.2 Asymptotic expansions	86	(3)
Appendix 4B: Indirect Information and Indentification: Proofs	89	(4)
4B.1 Computation of II( )	89	(2)
4B.2 Another Expression of IP ( )	91	(2)
5 Applications to Limited Dependent Variable Models	93	(26)
5.1 MSM and SML Applied to Qualitative Models	93	(7)
5.1.1 Discrete choice model	93	(1)
5.1.2 Simulated methods	94	(2)
5.1.3 Different simulators	96	(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