Parameterized expectations

We start with a set of equilibrium conditions on the following form. f(xt; dt; et) = 0 xt+1 = g(xt; dt; zt+1): where et = Et[h(dt+1; xt+1)] and zt is i.i.d. Under mild regularity conditions, there exists a function such that et = (xt): Suppose, for a moment, that we know the function . Then the decision function d solves the following trivial functional equation. f(x; d(x); (x)) = 0: The PEA algorithm proceeds, as the name suggests, by parameterizing the expectations. (x) (x; ): 1 If we know then we are in a position to simulate time series fxtg and fdtg. Once that is done, we can update so as to minimize, with respect to e the sum of squared deviations T−1∑ t=0 (xt; e) h(xt+1; dt+1))2 (1) where the square is interpreted elementwise. 2 Non-linear regression De ne yt = h(xt+1; dt+1): You already know what to do if is a linear function of , i.e. (x; ) = x′. Then all you do is choose so as to satisfy the orthogonality condition T∑ t=0 xt(yt x′t) = 0: To do this, you dene A = T∑ t=0 xt