As usual, I regret that I was not able to put out a technical post last Monday.
I was working on it… the fact is, right now I’m almost obsessed with it… but a couple of things conspired to keep me from finishing it last Sunday.
This particular regression post applies the 15 selection criteria (introduced in the most recent regression post) to all possible regressions that we could run on the Hald data, using the four given variables. There are, after all, only 16 such regressions. (Yes, that count assumes that every regression includes a constant term.)
It turns out that the selection criteria are relatively unambiguous: every one of the selection criteria selects one of two regressions. They weren’t unanimous, but there were only two candidates.
But I didn’t want to stop there… I wanted to learn more about the relative rankings of the regression. That turned out to be messy but straightforward.
Then I asked a different question. These selection criteria do not directly concern themselves with the t statistics of the coefficients. For my own purposes, I usually use far more relaxed standards (a t statistic greater than one in absolute value) than an economist would use (a t statistic greater than about two).
So I took my set of 16 regressions, and found the minimum absolute value of the t statistics in each one.
I was stunned by the results.
Please understand. What I have learned is that all of these selection criteria are a different way (from t statistics) of assessing the relative goodness of regressions. Nobody warned me about this. I knew it the way we know something we haven’t yet said explicitly; when we bump into it we say, “Oh, of course.”
What I found is that among the four regressions using exactly 3 variables, only one of those regressions has all of its t statistics significant.
It is also the regression which is ranked last among those four… by every single one of the 15 selection criteria.
The saving grace is that there are two 2-variable regressions with all t statistics significant, both of which unanimously outrank this particular 3-variable regression. The selection criteria have done their job.
Nevertheless, my early training in econometrics says that I wanted to know about this particular 3-variable regression. No, I didn’t fail to find the “best” regression – but I consider that I failed to find something of interest. That’s an interesting but not a devastating failure.
At the best of times, I would have wanted time to reconsider the post in the light of this discovery.
And it wasn’t the best of times… by the time I discovered this interesting regression, I also knew that I had to get up extremely early the next morning to show up on time for jury duty.
I bailed on writing the post… ran some errands… and relaxed that evening, allowing my mind to meditate on the discovery. (“To stew on” would be a better description. I was just letting ideas percolate to my conscious mind.)
As for jury duty, I was thrown off (ahem, “excused from”) the jury twice. I was the sixth of six alternates, i.e. not in the jury box. The plaintiff’s attorney used his second pre-emptory challenge (meaning, no justification or explanation required) to dismiss me, but the judge said he was limited to the 12 jurors actually in the jury box.
Well, a few more challenges later, by both attorneys, and I was in the jury box. And, to be honest, hoping that he still had another challenge left. (Come on. Even the judge said, “Try not to be too disappointed if we dismiss you. And don’t go high-fiving each other, either.”)
He did have at least one more. He used it on me as soon as I was seated in the box. The rest of the week was pretty normal.
It also turns out that the passage of time has made it clear that a reorganization of this next post would be a good thing, so it’s just as well that something stopped me from getting it out last Monday.
As for the kid (my inner child), he has moved from seismology, specifically the interior of the earth, to the interior of stars. I first saw the “stellar interior equations” as a sophomore, and I loved them. But that’s another story.