Thread by @jmwooldridge

Not exactly. I like Bruce's approach in this paper and it yields nice insights. But in twitter and private exchanges last week, and what I've learned since, it seems that the class of estimators in play in Theorem 5 include only estimators that are linear in Y.

#metricstotheface

https://twitter.com/BorelliLuan/status/1494719603821449224

Theorem 5 is correct and neat, but leaves open the question of which estimators are in the class that is being compared with OLS. Remember, we cannot simply use phrases such as "OLS is BUE" without clearly defining the competing class of estimators. This is critical.

The class of distributions in F2 is so large -- only restricting the mean to be linear in X and assuming finite second moments -- that it's not surprising the class of unbiased estimators is "small." So small, it is estimators linear in Y.

Stephen Portnoy, @lihua_lei_stat, and I have independently come to this conclusion (I've had exchanges with both and with Bruce). So the statement in Theorem 5 concerning the class of estimators is the same as saying estimators linear in Y.

If we entertain estimators that are unbiased when the full GM assumptions are used -- so that Var(Y|X) = (s^2)*I -- then OLS is not best unbiased. Interestingly, when we add normality, by the C-R lower bound, shows OLS is best unbiased in a very large class of estimators.

This goes to show that efficiency is not a "monotonic" function as we relax restrictions on the class of estimators.
To show my summary is wrong, you need to find a nonlinear estimator that is unbiased for b for any distribution in F2. Portnoy has shown it's not possible.

@lihua_lei_stat and I have argued the same thing using results mentioned here:

https://twitter.com/lihua_lei_stat/status/1493291015129550849

https://twitter.com/jmwooldridge/status/1492990971218440197

Here's a more succinct way to state my conclusion: 1. OLS is BUE in the class of unbiased estimators E2, defined by the the distributions F2 in Hansen. 2. E2 includes only linear estimators.

But this is what we mean when we say OLS is BLUE.

In the original GM Theorem, the class of estimators is explicitly stated to be linear and unbiased. In Hansen Theorem 5, the class of estimators is implicitly linear and unbiased. But the conclusions about OLS are the same.

Here's another observation that may help. We must separate the class of estimators we're willing to entertain from the assumptions under which we claim efficiency. It's easy to confound them. It's desirable to have estimators unbiased under A1 only.

Such estimators are necessarily linear. Then, OLS is best in this class under Assumptions A1 and A2. If we want to consider estimators unbiased under A1, A2, and A3 then there is no ambiguity: OLS is best in this class under A1, A2, and A3.