25 December 2008

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.


Brit said...

Sounds painful.

Peter Burnet said...

Are you sure you wouldn't rather argue about trains?

joe shropshire said...

So, we look up lepokurtotic and find that it is a fifteen-dollar word for "fat-tailed". But distributions with fat tails aren't a Christmas conundrum, they're more like a Christmas fact of life. What exactly are you puzzling over?

Duck said...

Read "The Black Swan". The normal distribution is a lie.

David said...

Well, lot's of things are normally distributed, particularly in biology. More things can be normalized so that statistics that assume a normal distribution of residuals are workable. Basically, anything that varies symmetrically around a mean is normal-ish.

But not everything is and when we try to use statistics that assume normal distribution on things that are not normally distributed, the answers are gigo.

Jolly Roger said...

The sub-prime mortgage and CDS implosion has an even more-basic explanation:

Past about 2005, EVERYONE working in the finance/real estate space KNEW that everyone was lying - the buyers were lying about their income, the appraisers were lying about the market value of the homes, the mortgage brokers were changing the app's to get them approvable, and the banks were approving KNOWN piles of steaming waste because they didn't expect to have to keep 'em.

But the money was too good for any major player to want to stop the cycle. Only gov't regulators would have any incentive to so do, and they'd long ago been bought or co-opted.

So here we are.