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**Generalized Method of Moments: Introduction**

Amine Ouazad Ass. Professor of Economics

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**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.

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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.

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**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.

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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

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**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.

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**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.

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**GMM Criterion GMM estimator minimizes: Or any criterion such as:**

Where Wn is a symmetric positive (definite) matrix.

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**Assumption Convergence of the empirical moments. Identification**

Asymptotic distribution of the empirical moments.

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**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.

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**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.

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**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).

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**GMM estimator is CAN Same property as for OLS, IV, ML.**

Variance-covariance matrix VGMM determined by the variance-covariance matrix of the moments.

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**Variance of GMM Variance of GMM estimator is:**

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

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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.

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**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).

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**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.

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**Testing for overidentifying restrictions**

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

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