Thread by @jmwooldridge

One of the remarkable features of Bruce's result, and why I never could have discovered it, is that the "asymptotic" analog doesn't seem to hold. Suppose we assume random sampling and in the population specify

A1. E(y|x) = x*b0
A2. Var(y|x) = (s0)^2

#metricstotheface

https://twitter.com/CavaliereGiu/status/1492454733948362753

Also assume rank E(x'x) = k so no perfect collinearity in the population. Then OLS is asymptotically efficient among estimators that only use A1 for consistency. But OLS is not asymp effic among estimators that use A1 and A2 for consistency.

A2 adds many extra moment conditions that, generally, are useful for estimating b0 -- for example, if D(y|x) is asymmetric with third central moment depending on x. So there are GMM estimators more asymp efficient than OLS under A1 and A2.

With asymp analysis, we see a clear tradeoff between robustness and effic.

Perhaps my choice of asymptotic analogy isn't a good one, or the optimal IVs for b0 collapse to OLS (harder for me to believe). In any case, I still have more to learn ....

BTW, one would never have used the expanded class of nonlinear in Y estimators because they're generally biased (and inconsistent) if we only impose E(e|X) = 0. But it's very neat to know that they're inefficient relative to OLS even with using stronger assumps for unbiasedness.