Thread by @jmwooldridge

Because of a recent post at Data Colada, I've been asked about my take on the various heterosk-robust standard errors. In the taxonomy of MacKinnon-White and Davidson-MacKinnon, there's HC0, HC1, HC2, HC3.

#metricstotheface

datacolada.org/99

HC0 was the original variance matrix estimator proposed in White (1980, Econometrica). HC1 = [n/(n-k)]*HC0 makes a simple df adjustment. Clearly, HC1 - HC0 is positive semi-definite (even PD).

HC2 divides the squared resids, u^(i)^2, by 1 - h(i,i) where the h(i,i) are diag elements from the "hat" or projection matrix. It can be shown that this produces n different unbiased estimators of sigma^2 under homoskedasticity.

HC3 uses [u^(i)^2]/{[1 - h(i,i)]^2} and is closely related to the jackknife estimator. Because 0 < h(i,i) < 1, it is easy to see that HC3 - HC2 is PSD.

All four estimators, when properly scaled by the sample size, are consistent for Avar[sqrt(n)*(beta^ - beta)].

Simulation studies show that HC2 and HC3 lead to better -- with small n, possibly much better -- confidence intervals than HC1.

The datacolada analysis shows that using HC1 [the Stata default with vce(robust)] and HC3 can be very different when sample sizes are not large.

The QJE study cited used HC1 when comparing with randomization inference in experiments. But HC3 turns out to work quite well even with pretty small n. It seems like a good idea to use HC2 or HC3. It won't make much difference if n is large.

In thinking about this, it is natural to wonder whether
HC2 - HC1 is PSD. To me, it's not obvious looking at formulas. If we know it's true then we would always have, in terms of standard errors,

HC0 <= HC1 <= HC2 <= HC3

As I said, the first and third are easy to show.

I can't stand not knowing about whether the middle inequality is true. So I did a simulation with n = 200, k = 3, and 1,000,000 replications. So, 3 million standard errors. And every time, HC1 <= HC2. Another proof by Stata!

I did some other simulations, too. So out of about 5 million chances, HC1 > HC2 never holds.

So, hotshots in linear algebra: Please provide a proof that HC2 - HC1 is PSD! I suspect it's out there but I've checked the obvious econometrics sources.

Oh, and I'm willing to share co-authorship of the Economics Letters paper. 😬🤓

Here's the summary of what I know:

HC0 <= HC1
HC0 <= HC2
HC2 <= HC3

Ambiguous: HC1 vs HC2 and HC1 vs HC3.

With k/n small, HC1 tends to be smaller than HC2.

In my shared Dropbox folder I added a folder, het_robust, with a Stata program that simulates data, computes HC1, HC2, and HC3, and reports the fraction of times HC1 > HC2, HC1 > HC3.