Science is like a crossword puzzle

This is one of my favorite analogies, because it illustrates how doing science --- or, more broadly, learning about a subject --- is like putting up a scaffolding: every part supports every other part.

When you start filling in words in a difficult crossword puzzle, you're not very certain about your guesses.  They're often far apart from each other.  Some of them are more tentative than others.  But once you take them, tentatively, for granted, you can start guessing harder, longer words.

Sometimes you just can't make sense of a longer word's clue with the words you have down.  This is evidence that some of the words you've chosen are wrong.  They don't work.  But sometimes, harder clues just answer themselves --- this is evidence in favor of the words that you've already put down.

It supports itself.  Each new word is evidence to evaluate earlier words.  As you continue to fill in words, your confidence in the earlier words grows.  You're never 100% certain that you've got them right (even after you're done -- what if you have stumbled upon a perfectly consistent but wrong solution to the clues?) but you're close enough that it's not worthwhile to waste time qualifying "I got the crossword!" with "(except in the the improbable, but not quite zero-chance, case that I happened upon a self-consistent solution to all of the clues which is not correct)."

Science is the same way.  New discoveries don't happen independently of old discoveries; they're assisted and informed by old discoveries.  We have to re-evaluate those old discoveries in light of the new discoveries, the old theories in light of new theories, and the old observations in light of the new observations.

A crossword puzzle is not a perfect analogy.  Scientifically, many observations (the "clues" in the puzzle) rely on existing theory ("words").  Imagine a crossword puzzle with almost all of the clues embedded in the puzzle itself, and you're getting somewhere.  But it's good enough to illustrate the point that science isn't many independent observations all pointing at some abstract "truth" - it's a series of interlocking theories, each of which is evaluated in light of each other.

Test of MathJax

Test of MathJax:

Offset equation:
\[ \int_{\partial M} \omega = \int_M d\omega \]
$$ df(v) = \langle \nabla f, v\rangle  $$

Inline equations: \( \|f\|_q = \int_M \langle q,*q\rangle = \int_M |q|^2 \).

Addendum

If the Laffer curve is concave down, when taxes are decreased, revenue will still be below projections.

Things Mankiw Didn't Say

Saw this post on Greg Mankiw's blog just now. Here's a rephrase.

What Mankiw said: where we are, the Laffer curve is concave down.
What Bloomberg heard: where we are, the Laffer curve is decreasing.

Maybe I've just been tutoring too many calculus students recently, but this seemed a pretty humorous way to put the misunderstanding.

(Detail for math wonks: the Laffer curve in real life is certainly much more complicated than it is in a typical economics class, where it's derived from linear supply and demand curves. Mankiw says -- and I'd tend to usually agree -- that a marginal increase in taxes will result in less revenue than static models would tend to predict.

A static model is based on a first order approximation, i.e., extrapolating out on a tangent line. So Mankiw is claiming that the actual Laffer curve falls below the tangent line approximation, i.e., it is concave down.

Bloomberg misinterpreted Mankiw's claim, and thought he was repeating the nutcase claim that the current US tax rate is on the far side of the Laffer curve, i.e., that revenue would fall with an increase in taxes.

It all boils down to the difference between first and second derivatives.)

Neat

It strikes me as particularly beautiful that SO(3) is homeomorphic to RP3. That's all.

Most arrogant paragraph, this week edition

Source: http://econlog.econlib.org/archives/2010/11/not_who_you_thi.html

Here in the United States, one thing that strikes me about my most liberal friends is how conservative their thinking is at a personal level. For their own children, and in talking about specific other people, they passionately stress individual responsibility. It is only when discussing public policy that they favor collectivism. The tension between their personal views and their political opinions is fascinating to observe. I would not be surprised to find that my friends' attachment to liberal politics is tenuous, and that some major event could cause a rapid, widespread shift toward a more conservative position.

My only response:

Here in the United States, one thing that strikes me about my most conservative friends is how liberal their thinking is at a personal level. For their own children, and in talking about specific other people, they passionately stress generosity. It is only when discussing public policy that they favor individualism. The tension between their personal views and their political opinions is fascinating to observe. I would not be surprised to find that my friends' attachment to conservative politics is tenuous, and that some major event could cause a rapid, widespread shift toward a more liberal position.

Measures of unemployment closely track U3

This is pretty cool.

Everybody knows that the "official" unemployment rate isn't really an accurate measure of unemployment. Why? Well, it only counts people who have been looking for jobs in the last 15 weeks and haven't found them. That's such a restrictive measure of unemployment that one can't help but wonder whether the BLS is using "Lie with Statistics 101," which tells us (inter alia) to never cite a statistic that measures what you claim it's measuring.

The problem here is that "unemployment" is not a simple concept. Are people who aren't looking for jobs "unemployed"? What about people who look for jobs sometimes, but not always? People who are taking whatever they can get, instead of picking the best job for them? People who are between jobs? And so forth.

The BLS number represents essentially an arbitrary measure. It captures some, but not all, of what we mean when we say "unemployed" --- it's the best we can do with a single number. But in addition to the "unemployment rate", the BLS also tracks five other numbers of varying restrictiveness: U1, U2, U4, U5, and U6. The unemployment rate you hear in the news is called U3.

The BLS has been following U1 and U3 since 1948, U2 since 1967, and U4-6 since 1994. U6 is the most expansive measure of unemployment --- it measures not only who are unemployed individuals, but those who have taken part-time jobs for lack of anything better and those who aren't looking for jobs at all, but would start if the economy improved (the so-called 'marginally unattached').

Is it possible to figure out U4-6 for past recessions, so we can get a better picture of how past labor markets compare to today's labor markets?

Turns out, yes. Very definitely. I ran some regressions this afternoon on the BLS employment numbers and the five unemployment measures U1, U2, U4, U5, and U6 are very, very tightly correlated in the short-run with U3:

\[U_1 = (0.76 \pm 0.01)U_3 − (2.31 \pm 0.07)\]
\[U_2 = (0.78 \pm 0.01)U_3 − (1.47 \pm 0.05)\]
\[U_4 = (1.066 \pm 0.003)U_3 − (0.09 \pm 0.01)\]
\[U_5 = (1.099 \pm 0.005)U_3 − (0.43 \pm 0.03)\]
\[U_6 = (1.70 \pm 0.07)U_3 − (0.3 \pm 0.1)\]

Isn't that beautiful? This is from the 1994-2010 panel data. Look at how tight those spreads are! Apparently, over short (one-two decade) periods, U3 is able to predict upwards of 95% of the variation of U1-6.

This breaks down over the longer run, as you might expect. The linear correlation actually migrates slowly over time. I believe this reflects changing population demographics, such as the aging of the baby boomers, and technological change. So you can't take these numbers and extrapolate backwards with them too carelessly. However, it does mean that you can get a good idea of how past labor markets compare to this labor market just by comparing U3, and bearing in mind that the relationship between U3 and U1-6 across many decades isn't as tight as you might want it to be.