5 semi-parametric efficiency of OLS

From your intro econometrics classes, you might recall the (celebrated?) result: the Gauss-Markov

theorem. The Gauss-Markov theorem says the following. If we have i.i.d data generated by the

model Yi = XiB + Ei, with (i) E[eilXi] = 0 and (ii) Elec'[X] = 02/ (homoskedasticity), then the

OLS estimator BOLS = (X'X) -1X'Y is BLUE - the best linear unbiased estimator (i.e. has smaller

variance than all other estimators in that class).

This is a slightly underwhelming result in so far as it relies on homoskedasticity (though this can be

relaxed) and a restricted class of estimators. Chamberlain (1987) provides a more general, asymp-

totic efficiency result: that OLS is semi-parametrically efficient in that only using the conditional

moment restriction Elei|Xi] = 0 (recall, we haven't specified how the errors are distributed), OLS

asymptotically has the lowest variance in the class of all regular estimators for B.

1. What do we know about asymptotic efficiency for maximum likelihood estimators? Consider

the data generating process Yi = XiB + Ei, with i.i.d data, and a sample size of n. Xi is a

single regressor / covariate. Is OLS asymptotically efficient if si ~ N(0, o2)?'

2. Now let the EilXi ~ Fax remain unspecified. But assume Yi and Xi are discrete; say Zi =

(Yi, Xi) take on one of J values: zj = (yj, ";), for j = 1, ..., J with P(Zi = zj) = Tj.

(a) Express the population coefficient B = E[X?]-E[X;Yi] in terms of Tj, xj, and yj.

(b) Set up the log-likelihood (i.e. the likelihood of observing { Zi}-1 = {(Yi, Xi) }=1). What

is the distribution of the data? What are the ML estimators 7,?

(c) Use the invariance property of maximum likelihood estimators to find BML, and show

that BML = BOLS. Conclude with a comment on the asymptotic efficiency of OLS.