Concerning the recent exchange many of us had about @BruceEHansen's new Gauss-Markov Theorem, I now understand a lot more and can correct/clarify several things I wrote yesterday. I had a helpful email exchange with Bruce that confirmed my thinking.
#metricstotheface
#metricstotheface
A lot was written about the "linear plus quadratic" class of estimators as possible competitors to OLS. Here's something important to know: Bruce's result does not allow these estimators in the comparison group with OLS unless they are actually linear; no quadratic terms allowed.
If one looks at Theorem 5 concerning OLS, you'll see a distinction between F2 and F2^0. All estimators in the comparison group must be unbiased under the very large class of distributions, F2. This includes all distributions with finite second moments -- so unrestricted SIGMA.
Because the LPQ estimator uses Var(Y|X) = (s^2)*I (which holds under GM), these estimators are biased for many distributions in LPQ. In fact, I'm pretty sure that the unbiasedness condition for LPQ when SIGMA can be anything is that H(j) = 0 in GKTZ (1992)'s notation.
This closely relates to my discussion of asymptotics yesterday, and now I see there is no conflict. It's somewhat subtle. If we only use the conditional mean assumption in estimation and homosk happens to be true, OLS is efficient -- in a restricted class of estimators.
OLS uses the optimal IVs based on E(y|x) = x*b if Var(y|x) happens to be constant. Using only the conditional mean restrictions mean we allow for distributions that include heterosk. Estimators that use homosk to estimate beta are generally inconsistent.
If we are allowed to use the homosk assump to estimate beta then we can do better than OLS (at least under asymmetry).
The same thing is happening in Bruce's exact result. One can only use E(Y|X) = X*b in obtaining unbiased estimators because the class F2 is large.
The same thing is happening in Bruce's exact result. One can only use E(Y|X) = X*b in obtaining unbiased estimators because the class F2 is large.
Then, if it happens that Var(Y|X) is scalar, OLS is best unbiased in the class of estimators. But it's not BU if one uses Var(Y|X) = (s^2)*I in estimating beta. The Koopmann and GKTZ papers show that.
There remains an interesting question: Does the class of estimators unbiased under F2 include any nonlinear estimators? It seems it doesn't include any LPQ estimators; if that's the case, it makes me think they're all linear estimators.
Another way to understand this is that if F2 is replaced with F2^0 in Theorem 5 then the theorem is no longer true; there are LPQ estimators that outperform OLS. Using F2^0 is smaller than F2 and so expands the class of unbiased estimators in competition with OLS.
I'm relieved because it would be very odd for the exact result to be so at odds with the asymptotic result that I teach to first-year PhD students. Now I know they're entirely analogous.
For the asymptotic result, I don't know of any non-silly estimators that wouldn't have the form (linear in the y(i)),
Sum(i=1,...,n) g(x(i))'*y(i)
for some functions g().
Sum(i=1,...,n) g(x(i))'*y(i)
for some functions g().
The second “LPQ” should be “F2.”
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