For the example in Section 12.5 .6 consider the estimator ^ that solves i =

For the example in Section 12.5 .6 consider the estimator $\hat{\alpha }$ that solves $\sum _{i=1}^N[y_i-$ $\frac{1}{S}\sum _{s=1}^S(\alpha +u_i^S)]=0$. Obtain analytical expressions for this estimator and its variance.

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12.5.6. Unobserved Heterogeneity Example We return to the example of Section 12.4.5. Then y; ~N[0+ui, 1], where u; has density g(u) given in (12.24). Since E[y; -0- u;] = 0, we can estimate by the method of moments estimator that solves (yi - E[u]) = 0, N (12.39) yielding MM = y - E[u]. Suppose that E[u] is unknown. Then we can instead use the MSM estimator MSM that solves 1 N N yi Yi-0. - u = 0, (12.40) where u are iid random draws from the extreme value distribution. The estimating equation (12.40) can be solved, yielding MSM = y-, (12.41)