# Generalized Method of Moments: Introduction

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Generalized Method of Moments: Introduction
Amine Ouazad Ass. Professor of Economics

Outline Introduction: Moments and moment conditions
Generalized method of moments estimator Consistency and asymptotic normality Test for overidentifying restrictions: J stat Implementation (next session). Next session: leading example of application of GMM, dynamic panel data.

Moments Moment of a random variable is the expected value of a function of the random variable. The mean,the standard deviation, skewness, kurtosis are moments. A moment can be a function of multiple parameters. Insight: All of the estimation techniques we have seen so far rely on a moment condition.

Moment conditions Estimation of the mean:
m satisfies E(yi – m)=0 Estimation of the OLS coefficients: Coefficient b satisfies E(xi’(yi – xib))= 0 Estimation of the IV coefficients: Coefficient b satisfies E(zi’(yi – xib))= 0 Estimation of the ML parameters: Parameter q satisfies the score equation E(d ln L(yi;q) / dq ) = 0 As many moment conditions as there are parameters to estimate.

Method of moments The method of moments estimator of m is the estimator m that satisfies the empirical moment condition. (1/N) Si (yi-m) = 0 The method of moments estimator of b in the OLS is the b that satisfies the empirical moment condition. (1/N) Si xi’(yi-xib) = 0

Method of moments Similarly for IV and ML.
The method of moments estimator of the instrumental variable estimator of b is the vector b that satisfies: (1/N) Si zi’(yi-xib) = 0 . Empirical moment condition The method of moments estimator of the ML estimator of q is the vector q such that: (1/N) d ln L(yi;q) / dq = 0. The likelihood is maximized at that point.

Framework and estimator
iid observations yi,xi,zi. K parameters to estimate q = (q1,…,qK). L>=K moment conditions. Empirical moment conditions: GMM estimator of q minimizes the GMM criterion.

GMM Criterion GMM estimator minimizes: Or any criterion such as:
Where Wn is a symmetric positive (definite) matrix.

Assumption Convergence of the empirical moments. Identification
Asymptotic distribution of the empirical moments.

Convergence of the empirical moments
Satisfied for most cases: Mean, OLS, IV, ML. Some distributions don’t have means, e.g. Cauchy distribution. Hence parameters of a Cauchy cannot be estimated by the method of moments.

Identification Lack of identification if:
Fewer moment conditions than parameters. More moment conditions than parameters and at least two inconsistent equations. As many moment conditions as parameters and two equivalent equations.

Asymptotic distribution
Satisfied for means such as the OLS moment, the IV moment, and also for the score equation in ML (see session on maximum likelihood).

GMM estimator is CAN Same property as for OLS, IV, ML.
Variance-covariance matrix VGMM determined by the variance-covariance matrix of the moments.

Variance of GMM Variance of GMM estimator is:
Hansen (1982) shows that the matrix that provides an efficient GMM estimator is:

Two step GMM The matrix W is unknown (both for practical reasons, and because it depends on the unknown parameters). Estimate the parameter vector q using W=Identity matrix. Estimate the parameter vector q using W=estimate of the variance covariance matrix of the empirical moments.

Overidentifying restrictions
Examples: More instruments than endogenous variables. More than one moment for the Poisson distribution (parameterized by the mean only). More than 2 moments for the normal distribution (parameterized by the mean and s.d. only).

Testing for overidentifying restrictions
With more moments than parameters, if the moment conditions are all satisfied asymptotically, then Converges to 0 in probability, and has a c2 distribution. The number of degrees of freedom is the rank of the Var cov matrix.

Testing for overidentifying restrictions
With more conditions than parameters, this gives a test statistic and a p-value. Sometimes called the J Statistic.