## Monday, October 29, 2012

### Central Limit Theorems

When we first encounter asymptotic (large sample) theory in econometrics, one of the most important results that we learn about is the Central Limit Theorem.  Loosely speaking we learn that if we aggregate together enough values that are sampled randomly from the same distribution, with a finite mean and variance, then this aggregate starts to behave as if it is normally distributed.

However, too few courses make it clear that this "classical" central limit theorem is just one of several such results. The one that assumes independently and identically distributed values is actually the Lindeberg-Lévy Central Limit Theorem. There are other, related, results that deal with less restrictive situations.

For example, there's the Lindeberg-Feller Central Limit Theorem, and Liapounov's Theorem.

John Cook has a nice (older) blog post on the history of the central limit theorems. He also has a very readable discussion (here) of the theorems themselves, and the rate at which normality is approached, John's pieces should be recommended reading for students of econometrics!

© 2012, David E. Giles

#### 5 comments:

1. I've come to this post just after the trimmed PCE mean inflation introduced by the Dallas Fed http://www.dallasfed.org/research/pce/2012/pce1209.cfm. Instead of trimming would it be helpful to use the CLT and compile a price index for the inflation estimates from independent (low correlation) components with the mean normally distributed. Also the uncertainty (standard deviation) would have sense in statistical terms.

1. Ivan - thanks for the interesting comment. I'll take a look at the Fed paper you mentioned. Sounds interesting.

2. Invesco Ltd. has a financial product consisting of several independent ingredients (stocks, bonds, commodities, emergent markets, etc.) which best illustrates the CLT performance and properties. Because of the CLT they are able to estimate the uncertainty of the mean and provide some leverage depending on the current uncertainty. The CLT is extremely useful.

2. But never forget, in the real world, especially in risk management, the assumptions of finite mean and variance are wrong, and there are no central limits. Most power-law distributions don't have central limits. It can be conjectured that if you enumerate all the generating functions for distributions, the non-limitable ones vastly outnumber the limitable ones. Nor is the assumption of stability/stationarity generally valid. For example, consider prices of goods or securities in a monetary system in which inflation is possible. It gets even worse: Quasi-hurricane Sandy, currently devastating the Eastern US, is a perfect example of an assumption-destroying event. No statistical model of the stock market would predict the market to be completely closed for two days or more!

1. George - thanks for the comment. I understand your point, but don't forget that there are also central limit theorems for the case of an infinite variance - see the second link to John Cook in my post.

Not all real-world situations involve and infinite mean or variance, of course. Yes, there is non-stationarity out there - we all now that too.

I have other posts that discuss "extreme value analysis" - e.g., http://davegiles.blogspot.ca/2012/04/extremes-generalized-pareto.html

http://davegiles.blogspot.ca/2012/04/modelling-extremes.html