Eric Rasmusen’s Weblog

I am unhappy with Wordpress for various reasons, so I’m trying out Google’s Blogger. For a while I will be posting at http://www.rasmusen.org/t instead.

In doing statistics, when should we weight different observations differently?

Suppose I have 10 independent observations of $x$ and I want to estimate the population mean, $\mu$. Why should I use the unweighted sample mean rather than weighting the first observation .91 and each of the rest by .01?

Either way, I get an unbiased estimate, but the unweighted mean gives me lower variance of the estimator. If I use just observation 1 (a weight of 100% on it) then my estimator has the variance of the disturbance. If I use two observations, then a big positive disturbance on observation 1 might be cancelled out by a big negative on observation 2. Indeed, the worst case is that observation 2 also has a big positive disturbance, in which case I am no worse off by having it. I do not want to overweight any one observation, because I want mistakes to cancel out as evenly as possible.

All this is completely free of the distribution of the disturbance term. It doesn’t rely on the Central Limit Theorem, which says that as $n$ increases then the distribution of the estimator approaches the normal distribution (if I don’t use too much weighting, at least!).

If I knew that observation 1 had a smaller disturbance on average, then I *would* want to weight it more heavily. That’s heteroskedasticity.
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Jerusalem Post via National Review

Two weeks after Israel’s alleged bombing raid in Syria, which some
foreign reports said targeted North Korean nuclear material, the UN’s
nuclear watchdog elected Syria as deputy chairman of its General
Conference on Monday.

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The Frequentist view of probability is that a coin with a 50% probability of heads will turn up heads 50% of the time.

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Page 96 of David Cox’s 2006 Principles of Statistical
Inference has a very nice one-sentence summary of asymptotic theory:

[A]pproximations are derived on the basis that the amount of
information is large, errors of estimation are small, nonlinear
relations are locally linear and a central limit effect operates to
induce approximate normality of log likelihood derivatives.

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I received a rejection letter recently that puzzles me. My co- authors and I don’t really see how the criticisms below apply to our paper, apart from the single spacing and not citing any articles from that journal. We would welcome any comments. Don’t worry about being overly frank. We are especially interested in whether Empirical Finance has some customary style we are not following. Here is the paper’s abstract:

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Peggy Noonan’s “Now He Tells Us: If only they’d listened to Greenspan! And they might have, if only he’d spoken clearly” is good on how the former Fed chairman has saved his criticism of pork barrel spending for his book rather than making it when it might have actually stopped the spending:

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Has anyone pointed out that large landfills full of plastic bottles and disposable diapers are a solution to global warming?

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The Northern Rock bank run is interesting (background below from a newspape account). I’ll make three points: 1. Stupidly low deposit insurance limits caused a rational run, 2. The government should have made Northern Rock pay for the free full deposit guarantee it just got, and 3. It’s time to get out of savings accounts and bank stock shares if you’re British..

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Someone asked me about the past few years’ papers on exclusive dealing, so I did some thinking. I’ll lay it out in my own way.

Suppose we have 100 buyers, each with 1% of the market, one upstream incumbent seller who charges the monopoly price. Everybody knows that in a year a potential entrant seller will arise, and that he will need 15% of the market to achieve the necessary scale economies.

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“Intelligence in the Classroom: Half of all children are below average, and teachers can do only so much for them” by Charles Murray in the WSJ, is a good article. The subtitle says it all. In designing a school system, we need to think about children’s potential, not just what we’d like for them to be able to learn. Wherever we set a threshold, some children aren’t going to be able to make it. Read the rest of this entry »

p>Three Hierarchies makes the nice point that if you look closely, the Bible does not say that God directly created animals and plants, just that He created the earth, and that spontaneous generation– as opposed to God having created all living things directly— was long accepted by Christians and everyone else. It was only in the 1800’s that scientists showed that life comes from life, so that the origins of life became a puzzle: Read the rest of this entry »

Baylor University is clearly infringing on academic freedom when it tries to shut down Professor Robert Marks’s pro-intelligent-design website, located here with the disclaimers the professor put on in response to Baylor’s complaints that people might think Baylor officially approved of his research. I wish there were more publicity about this kind of thing. Even if a professor believes in astrology, if that’s his research, he should be allowed to pursue it, if not necessarily give pay raises as a result. Of course, a professor’s astrology page would *not* be shut down– it is precisely the plausbility of intelligent design that infuriates its opponents. They take its threat seriously. For an article in WORLD, see here.

I learned something this morning. When you’re trying to estimate the impact of a two-valued X on some two-valued Y with Y=1 being a rare event, you can get an unbiased estimate of the relative risk, Pr(Y=1|X=1)/Pr(Y=1|X=0), even if your sample is biased because you oversampled mainly Y=1 observations. This is not just restricted to logit estimation either. I learned this reading The analysis of case-control studies by NE Breslow and NE Day, but I have a writeup at http://www.rasmusen.org/x/2007/oddsratio.pdf that is much clearer.

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The London School of Economics has a great idea: a computerized seminar sign-up list. You choose the seminars that interest you, and get email notifications.

Peter Wilkensen preached on the Parable of the Sower at St. Ebbes today. Tho it was not his point, something new that struck me was that the wheat from the seed that falls among thorns does not die, as does that from the seed that falls on stony places. Rather, it is “unfruitful”. Presumably it lives, but so shaded from the sun by the distraction of the thorny weeds that it lives and dies naturally, but without fruit. This is quite different from the fate of the seed in the stones, but perhaps even sadder.

Here are the thorn texts from Matthew, followed by the text in full:

13:7 And some fell among thorns; and the thorns sprung up, and choked them:

13:22 He also that received seed among the thorns is he that heareth the word; and the care of this world, and the deceitfulness of riches, choke the word, and he becometh unfruitful.

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