generalized method of moments: introduction
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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.
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