Generalized method of moments estimator

Summary

• True moments:

\[E\left( f\left( w_i,θ \right) \right)=0\]

• where
• \(w_i\) is all data for individual \(i\) (dependent variable, explanatory variable, instruments)
• \(θ\) is a \(k×1\) vector of unknown parameters
• \(f\left( w_i,θ \right)\) is now \(r×1\) .
• Corresponding sample moments:

\[s\left( θ \right)= \frac{1}{n}\sum_{i=1}^{n}{ f\left( w_i,θ \right) }\]

• If \(r=k\) then the solution to \(s\left( θ \right)=0\) will give us the MM estimator of \(θ\) .
• If \(r≠k\) then the method of moments estimator will not, in general, exist ( \(r\) equations and \(k\) unknowns)
• If \(r<k\) then there is no general procedure of estimating \(θ\) (too little information).
• If \(r≥k\) then we define the quadratic form (scalar)

\[Q\left( θ \right)=s{\left( θ \right)}'Ws\left( θ \right)\]

• where \(W\) is an arbitrary \(r×r\) symmetric positive definite matrix called the weighting matrix .
• The generalized method of moments estimator,   \({\hat{θ}}_{GMM}\) for a given \(W\) is defined as

\[{\hat{θ}}_{GMM}=arg \min_{θ} Q\left( θ \right)\]

• Result: \({\hat{θ}}_{GMM}\) is a consistent estimator of \(θ\) (under regularity conditions)
• The asymptotic variance of \({\hat{θ}}_{GMM}\) will depend on \(W\) .
• We define

\[W^{opt}={\left( E\left( f\left( w_i,θ \right)f{\left( w_i,θ \right)}' \right) \right)}^{-1}\]

• as the optimal weighting matrix.
• The asymptotic variance of \({\hat{θ}}_{GMM}\) is minimized when \(W=W^{opt}\) .
• \(W^{opt}\) is, in general, unknown. However, it can often be estimated.