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 and instruments)
• \(θ\) is a \(k×1\) vector of unknown parameters
• \(f\) is a given vector valued function, \(f\left( w_i,θ \right)\) is \(k×1\) called elementary zero functions .
• Sample moments:

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

• \({\hat{θ}}_{MM}\) is called the method of moments estimator of \(θ\) if it solves

\[ \frac{1}{n}\sum_{i=1}^{n}{ f\left( w_i,{\hat{θ}}_{MM} \right) }=0\]

• Note that we have \(k\) equations in \(k\) unknowns.