long term impacts and empirics

There has been an accumulation of evidence in the past several years that labor markets respond to prices slowly and, at least at the first incidence, through flows rather than stocks; an exogenous increase in wages causes employers to pull back in hiring and increase layoffs, not by a lot (in terms of rate) but for a long time. A lot of previous studies had missed these effects because they looked at employment levels and included "corrections" for trend — when in fact the relevant "trend" was the very signal for which they were looking.

Labor is perhaps the strongest example, but there are a lot of markets in which we participate in which we establish, in some meaningful sense, relationships; if there are five supermarkets nearby, you may almost always go to one or two of them, such that you wouldn't respond to a sale at one of the others.  In other contexts, too, long-term behavior may obey very different rules than short-term behavior, but a lot of the most popular empirical techniques right now involve the use of changes shortly after other changes to infer causal relationships; long-term interrelationships are a lot harder to tease out causally.  (A change that happens in many places two or three months after legislation is announced, passed, or goes into effect is easy to attribute as an effect of the legislation; changes over the course of ensuing years are harder to distinguish from underlying trends that may, in fact, have motivated the legislation in the first place.)  When long-term and short-term impacts differ, I think there's a consensus that the long-term impacts are, from a policy standpoint, generally more important — the long-term is longer than the short-term, after all — but I worry that a lot our empirical studies now are trumpeting results about short-term relationships because teasing out causal directions can be done more convincingly.  Certainly there is something to be said for doing the possible rather than the impossible, but I think more studies of long-term behavior for which the "identification" [technical word] is less compelling should be encouraged, even if they might be harder to interpret in a crystal clear way.

A simple identity for Bayesian updating

For random variables A and X, consider the relationship

E{XA}=E{X}E{A}+ ρ_XAσ_Xσ_A

which, up to a bit of arithmetic, is basically the definition of correlation. If A is a binary variable, though, we can do more with this; among other things, in this case σ_A²=E{A}(1-E{A}). Conflating the variable A a bit with the "event" A=1, and doing a bit of algebra, we get

The effect of the arrival of new information on the expected value of a variable is proportional to the square root of the odds ratio. Among other things, it can't be more than σ_X times the square root of the odds ratio, though this bound, which (obviously?) is reached when X is a linear function of A and therefore is a binary variable, can be more directly derived in that context.