Generalized Method of Moments: Introduction Amine Ouazad Ass. Professor of Economics Outline 1. Introduction: Moments and moment conditions 2. Generalized method of moments estimator 3. Consistency and asymptotic normality 4. Test for overidentifying restrictions: J stat 5. 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). 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.