college football

For something (almost) completely different, I'd like to make some proposals regarding playoffs for the top division in college football.

My objection to an expansive playoff is primarily that the best team doesn't win every game, even insofar as there is such a thing as a best team; if #16 beats #1 in the first elimination game, I'm going to be wondering whether the winner of the tournament really is better than the team that lost that first match.  Of course, polls and selection committees aren't perfect, either, which is at least part of why playoff systems are popular; if there's a legitimate case to be made that #3 is likely to be better than #1 or #2, let's let them play each other and find out.^*

Part of the difficulty of comparing teams on the basis of their 12 games each is that there is so frequently little overlap among those teams' opponents.  Not only did Oklahoma State not play Alabama last year, but no team Oklahoma State played played Alabama last year.  There is an increasing tendency for teams to play almost all of their games in their own conference; this is sometimes hailed as a good thing, typically because the best teams tend not to play very good teams in their non-conference games — which makes it even harder to compare good teams in different conferences than if the occasional shared opponent were at least good. It would be better to have stronger schedules that weren't quite so incestuous.

On the other hand, in college basketball, some of the lower "mid-major" conference teams have complained that they have a hard time getting teams from the top conferences to play them precisely because strength-of-schedule is so emphasized; the top teams don't want to play a team that might hurt their strength-of-schedule metrics, so those teams never get a chance to develop the resumes that would make them look strong.  It would be nice if incentives induced teams to give the outsiders a chance to show what they have, but without creating too many rewards for beating up truly weak opponents.

Accordingly, here is my proposal:

1. A six-team tournament, comprising
   • the four teams believed by some reasonable process (e.g. a committee) to have the best claim to being the best team in the country
   • The next two teams selected by a process that emphasizes schedule strength and diversity
2. Teams 5 and 6 are selected according to the following ratings (probably modified for teams that play more than 12 games) that awards points to each team for each game between division I-A football teams:
   • For teams i and j, let n_ij be the smallest number of games required to link them; thus n_ii=0, n_ij=1 if i plays j, n_ij=2 if i did not play j (and i is not j) but they played a team in common, etc.
   • Team i gets 1/8^n_ij points for any game that team j wins
   Thus you get 1 point for a game you win, 1/8 for a game an opponent of yours wins, etc. — but you can only get credit for a game once

The rating system gives too much credit for strength of schedule, but only the two "incentive spots" in the playoffs are awarded on that basis; they won't be bad teams, and they won't get to the semifinals without beating a top 4 team, so they won't screw up the integrity of the playoffs, they just provide an inducement to athletic directors not to try to schedule too many weak teams.  On the other hand, if a mediocre team is being ignored by all the good teams, and is going to end up 8-4 against a weak schedule, then adding that team to your schedule is going to do you more good than adding a 6-6 team with a tougher schedule against teams that some of your other opponents are also playing; you move those 8 wins to a lower n_ij, as well as the wins of their opponents.  The general equilibrium effect, then, is actually to improve the ability of the committee to select the top four teams; when scheduling has been done knowing that, whatever else happens, a team with a good rating can snag one of these two incentive spots as a backup, that scheduling will result in more Oklahoma States scheduling decent common opponents with more Alabamas, increasing the likelihood that the teams that make the semifinals include the best one or two teams in the country.

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^*The race isn't always to the swift nor the battle to the strong, but that's usually the way to bet.

credit ratings and regulation

The ratings agencies were comically^* inept at rating credit tranches of, well, everything five years ago, and are probably not much better now.  Even when rating boring old corporate and sovereign bonds, they've typically been at best a lagging indicator of events; wherever there's a quasi-liquid market indication of whatever it is the credit ratings are supposed to mean, the market has typically priced in an increase in credit risk long before the ratings get around to doing downgrades.  It may be that they provide some value in the case of new issues; their ratings of bonds from companies that haven't previously issued bonds before may impound some research that would be costly for a lot of bond investors to replicate.  It seems mostly, however, as though 1) the credit ratings are inadequate, 2) the market knows it, and 3) the only reason the credit ratings carry any importance at this point is because of capital regulations — various banks and insurance companies are required to hold a certain amount of their assets in securities with various ratings, or else are required to maintain leverage ratios that depend on the securities' credit ratings.

Many large investment banks spent much of the middle of the last decade specifically constructing securities that they knew would get high credit ratings, but with as high a yield as possible, so that regulated entities would want to buy them. The very fact that they had high yields should have been a tip-off to regulators; the market is not always efficient, but there are certain ways in which it tends not to be too inefficient, and if a liquid marketable security, in times of low credit spreads and rising stock markets, has a high yield, it might be worth the regulators' wondering whether the market collectively knows something that a simplistic model has missed.

If we really think a fund of some sort should only, by regulation, invest in securities with some maximal leverage, perhaps it would make sense to impose (say) a 75% tax on gains above some maximum rate of return; perhaps 2 percentage points above ten or thirty year treasuries would be "investment grade".  This idea isn't quite just motivated by a general belief in a risk/reward tradeoff, though it's perhaps more closely related than should be made distinct: the idea is to get these investors in the mindset that, conditional on such-and-such a return, we're optimizing risk, rather than vice-versa.  This shifts the competition for pension-fund dollars from price-competition to quality-competition, in essentially the classical way in which this is done; by making it impossible for a creator of securities to compete by offering higher yields, I'm hoping they will be compelled to compete by offering more safety — safety as determined by what the market knows now, and not just information that is old enough to have finally been acknowledged by the ratings agencies.

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^* I have a sick sense of humor.

ceteris paribus and causality

One of economists' favorite latin phrases is "ceteris paribus", meaning "the rest unchanged"; the idea is that we're looking at the result of changing one thing while leaving everything else the same to isolate just the effect this one change has.  The moderately perceptive student notes that at some level this is impossible, or in any case is at variance with the rest of what is being taught.  For example, Y=MV, where Y is (nominal) income, M is the money supply, and V is the velocity of money.  If I double the money supply, either income has to increase, or velocity has to decrease, or both; it is not possible to double the money supply but leave everything else the same.

"Everything else the same" is approximately the idea behind the partial derivative from multivariate calculus.  If I have a function f(x,y), and I ask for ∂f/∂x, I'm asking for the change in f in response to a small change in x with y held constant.  Thermodynamics, in physics, is the only context in which I've seen the ceteris paribus problem formally attended to; for an ideal gas, for example, PV=nRT, and entropy can be written as a function of any two of P, V, and T; specific heat at constant volume C_V is T(∂S/∂T)_V, that is to say with volume constant and P changing as T changes, while C_P=T(∂S/∂T)_P is the corresponding expression for constant pressure (with V changing linearly with T). The two can be related; C_P-C_V=VT(α²/κ) where α=(1/V)(∂V/∂T)_P and κ=-(1/V)(∂V/∂ P)_T are the volume coefficient of expansion and the isothermal compressibility of the gas, respectively. This is related to the Slutsky equation from microeconomics, but the typical notation there serves to obscure rather than elucidate the fundamental point that partial derivatives can't hold "everything else" fixed.

Anyway, economists frequently look at statistical relationships between variables and seek to tease out causal relationships; if increases in investment tend to be followed by economic growth, for example, one might suppose that the growth is caused by the increase in investment, but it seems logical that investors would invest more when they expect economic growth than when they don't, so even the time-ordering in this case may run reverse to the causal relationship.^*  Of course, it's possible that both causal relationships exist, or even that neither does; perhaps both are being driven by something else.  The data do not — indeed, no set of data can — make a distinction between causal explanations.  In order to tease out causal effects econometricians will employ "instrumental variables", but even in that case instruments are useful at implying causation because there is an outside theoretical reason to believe that they bring in a causal relationship themselves; without outside theory, simply throwing more variables at the problem can not give causation.

At this point, then, it is worth asking what causality means.  Perhaps something causes us to expect an increase in investment that isn't connected to other changes we usually expect to affect GDP, so we predict an increase in GDP, or perhaps there's an increase in GDP forecasts that is not connected with other things we expect to drive investment, so we expect an increase in investment.  Presumably this means we believe something is happening that wasn't happening before — something we expect to change the data-generating process.  From that standpoint, causal interpretations of empirical data affect out-of-sample forecasts based on previous observations.

Colloquially, causality is often connected with intention.  If increases in investment cause later increases in GDP, perhaps we can increase investment, and that will increase GDP.  If this is the case, perhaps it matters how "we" — presumably action by the government or some other relatively unitary actor — go about increasing investment.  Formally, perhaps there is a set of choice variables available to "us", and the DGP is presumed to be a function of those choice variables; here, again, if we're looking to ascertain the distribution of variables effected by a set of choices for which we have no empirical data from empirical data we do have (for other values of our choice variables), this is essentially again out-of-sample prediction, and the extrapolation is again going to have to be theory-driven.

So this is how I'm currently thinking about causality, at least from an econometric point of view: Causality is simply about the ways in which I think an underlying data generating process is likely to change, and those ways in which it is not.  (I may think differently next month.)

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^* Indeed, Le Chatlier noted that near a locally stable equilibrium, effects tend to inhibit their causes; if both take place with a lag, you may see a diagonal time/variable cross-correlation of either sign at various lags, as one causal relationship outweighs the other.