Christmas Conundrum

Trying to run a hierarchical linear model, I've discovered that my dependent variable, Return on Assets, is leptokurtotic. I can improve it somewhat by running a full model on the raw data, analyzing the residuals and dropping some outliers, but ROA is still leptokurtotic. A logit transform, which is supposed to reduce kurtosis, just makes things worse. It turns out that ROA, which is a very common dependent variable in Strategic Management, is almost always kurtotic because it violates the assumption of proportionality; that is, the relationship between net income and total assets is non-linear.

What is kurtosis? Here's a picture of leptokurtotic data plotted against a normal curve:



What makes it kurtotic is that it is "peakier" than the normal curve. Why should we care? Because the most common statistical methods assume that the residuals of the dependent variable (how far each point is from the predicted value) are normally distributed. If they aren't, then the estimate of how likely it is that a particular result found in a particular sample is chance rather than real is unreliable. Since that's all that statistics do, that's a problem. So, for example, if researchers assume that headaches are distributed normally in the population but they're not, then a treatment that seems effective might not be.

More to the point, some economists think that Long Term Capital Management failed because its models assumed that certain financial measures, like Return on Assets, were normally distributed when they're not. At least one economist has argued that the Black-Scholes option pricing model, which is related to the LTCM model is also wrong for the same reason. Since Black-Scholes and other, related, models are the basis for most modern finance, that implies that people pricing the risk of various financial instruments without much of a history might have assumed that Return on Assets, etc., are normally distributed. Because, contrary to this assumption, ROA is leptokurtotic, the models might have overestimated the expected return on investment, underestimated the risk and mispriced the instruments.

In other words, the fact that ROA is leptokurtotic is a possible explanation, and a more satisfying explanation than most out there, for the sub-prime mortgage and CDS implosion.