Presentation on theme: "Generalized Method of Moments: Introduction"— Presentation transcript:
1Generalized Method of Moments: Introduction Amine OuazadAss. Professor of Economics
2Outline Introduction: Moments and moment conditions Generalized method of moments estimatorConsistency and asymptotic normalityTest for overidentifying restrictions: J statImplementation (next session).Next session: leading example of application of GMM, dynamic panel data.
3MomentsMoment 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.
4Moment conditions Estimation of the mean: m satisfies E(yi – m)=0Estimation of the OLS coefficients:Coefficient b satisfies E(xi’(yi – xib))= 0Estimation of the IV coefficients:Coefficient b satisfies E(zi’(yi – xib))= 0Estimation of the ML parameters:Parameter q satisfies the score equation E(d ln L(yi;q) / dq ) = 0As many moment conditions as there are parameters to estimate.
5Method of momentsThe method of moments estimator of m is the estimator m that satisfies the empirical moment condition.(1/N) Si (yi-m) = 0The 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
6Method 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 conditionThe 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.
7Framework 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.
8GMM Criterion GMM estimator minimizes: Or any criterion such as: Where Wn is a symmetric positive (definite) matrix.
9Assumption Convergence of the empirical moments. Identification Asymptotic distribution of the empirical moments.
10Convergence 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.
11Identification 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.
12Asymptotic 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).
13GMM estimator is CAN Same property as for OLS, IV, ML. Variance-covariance matrix VGMM determined by the variance-covariance matrix of the moments.
14Variance of GMM Variance of GMM estimator is: Hansen (1982) shows that the matrix that provides an efficient GMM estimator is:
15Two step GMMThe 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.
16Overidentifying 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).
17Testing for overidentifying restrictions With more moments than parameters, if the moment conditions are all satisfied asymptotically, thenConverges to 0 in probability, andhas a c2 distribution. The number of degrees of freedom is the rank of the Var cov matrix.
18Testing for overidentifying restrictions With more conditions than parameters, this gives a test statistic and a p-value.Sometimes called the J Statistic.