Generalized Method of Moments:Introduction

Amine OuazadAss. Professor of Economics

Outline

1. Introduction:Moments and moment conditions

2. Generalized method of moments estimator3. Consistency and asymptotic normality4. Test for overidentifying restrictions: J stat5. 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 theempirical 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.

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

Asymptotic distribution

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).

1. Estimate the parameter vector q using W=Identity matrix.

2. 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.