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

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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.

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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: – satisfies E(y i – )=0 Estimation of the OLS coefficients: – Coefficient satisfies E(x i ’(y i – x i ))= 0 Estimation of the IV coefficients: – Coefficient satisfies E(z i ’(y i – x i ))= 0 Estimation of the ML parameters: – Parameter q satisfies the score equation E(d ln L(y i ; ) / d ) = 0 As many moment conditions as there are parameters to estimate.

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Method of moments The method of moments estimator of is the estimator m that satisfies the empirical moment condition. -(1/N) i (y i -m) = 0 -The method of moments estimator of in the OLS is the b that satisfies the empirical moment condition. -(1/N) i x i ’(y i -x i b) = 0

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

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Framework and estimator iid observations y i,x i,z i. K parameters to estimate = ( 1,…, K ). 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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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

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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). 1.Estimate the parameter vector using W=Identity matrix. 2.Estimate the parameter vector 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 2 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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