For the MSM estimator in Section 12.5.3 suppose that the simulator is the frequency simulator.

(a) Show that ${\mathrm{V}}_{y,\mathbf{u}}\left[\hat{m}\left({\theta }_0\right)\right]=(1+1/S){\vee }_y\left[m\left({\theta }_0\right)\right]$.

(b) Hence show that the effect of simulation using the frequency simulator is to inflate the variance of the method of moments estimator by $(1+(1/S))$.

(c) How large is the efficiency loss for the standard errors if $S=10$ ?

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12.5.3. Method of Simulated Moments Suppose theory leads to a conditional moment condition E[m(yi, Xi, 0)|x;] = 0, (12.30) where m() is a scalar for simplicity. Let w; denote instruments, a function of x; and possibly 60, that satisfy E[wim(y, xi, 00)] = 0. (12.31)