# How useful is the CLT in applications?

My lecturer just covered the Lindeberg-Lévy central limit theorem and the multivariate version, the Lindeberg-Feller CLT. I understood the basic concept and I can derive it, etc. But it would help my understanding a lot if someone could explain how all this used in real life applications of econometric analysis?

I've read some claims that the CLT is nice only on a piece of paper.

Some really cool industry applications or references would be appreciated.

The CLT certainly informs applications all the time, since we deal with distributions of averages or sums very frequently (including in cases that may not be always obvious; for example, $s_n^2$ - the sample variance with denominator $n$ - is an average, and so the ordinary sample variance is just a slightly rescaled average).

The CLT can tell you to expect an approach to normality with increasing sample size for a particular statistic, but not when, exactly, you can treat it as normal.

So while you know that normality should kick in eventually, to know if you're close enough at a particular sample size, you will need to check (say algebraically, or more often via simulation).

You may sometimes run into 'rules of thumb' that say "oh, n=30 is enough for the central limit theorem to kick in". Such rules are nonsense without specifying the exact circumstances (what the distribution is we're dealing with, and what properties we care about, and 'how close is close enough').

If you have an $X$ with a distribution like this:

Then sample means, $\bar X$ for $n=1000$ have a shape like this:

... which for some purposes might be just about okay to treat as normal (proportion within 2 s.d.s of the mean, say); for other purposes (probability of being more than 3 s.d.s above the mean, say), perhaps not.

Sometimes n=2 is plenty, sometimes n=1000 isn't enough.

Another example: the sample third and fourth moments are averages and so the CLT should apply. The Jarque-Bera test relies on that (plus Slutsky, I guess, for the denominator, along with asymptotic independence), in order to obtain a chi-square distribution for the sum of squares of standardized values. But as Bowman and Shenton had pointed out (5 years before!), this shouldn't be expected to work well until large sample sizes. Indeed my own simulations suggest that for normal data, bivariate normality of the skewness and kurtosis doesn't kick in well until the sample sizes are surprisingly large (at small and middling sample sizes, the contours of the joint distribution look more like a banana than a watermelon)

Increasingly often, however, sample sizes can be huge. I've helped with several real-data problems where the sample sizes were very large indeed (in the millions). In those situations, things the CLT suggests should approach the normal as $n$ approaches infinity are often extremely well approximated by normal distributions.

I wouldn't say the CLT is useless - it tells you what distribution to look for - but it doesn't do more than point to it as an eventual outcome; you still have to check whether it's a suitable approximation for your purposes at the sample size you have.

• Nicely put. As seen in other postings on this subject the CLT can be quite misleading and of questionable value in practice, not only because of the reasons above but because when the population variance has to be estimated and the distribution is skewed, the variance no longer applies in a certain sense, and bad things happen to the distribution of the $t$-statistic. – Frank Harrell Jan 3 '14 at 4:54
• @FrankHarrell Bad things do happen, yes. With the skewed distribution I drew at the top of my answer above (Gamma with shape 0.02), the distribution of the two-sample t statistic in medium-sized samples (say n=30) looks like the main cable on a suspension bridge - bimodal with two sharp peaks. – Glen_b Jan 3 '14 at 6:00
• @Glen_b just out of curiosity, could you tell what kind of analysis, variables or statistics you were working with? did the approximation prove reasonable? just to have an idea what fields could collect so much data and use CLT, since asymptotics always looked a bit academic for me. – mugen Dec 26 '14 at 23:17
• For example, one was a set of engineering measurements of some industrial process -- I wasn't given much detail of the exact thing being measured in that case, but I think they were a (very) large set of diameters or lengths. – Glen_b Dec 26 '14 at 23:35

Notwithstanding the problems with CLT described by Glen_b, if we are talking about econometric analysis, then practicaly all results are asymptotic (except when Bayesian analysis is used). Hence any application which is based on econometrics is based on CLT. For example Lars Hansen got the Nobel prize last year for work on generalized method of moments, which is based on CLT.

It might seem that such reliance on CLT is not a good thing, but on the other hand any econometric paper relying on asymptotics usually has a chapter with Monte Carlo simulations exploring the reliability of asymptotic results on small samples, and more often than not the results are not that bad.

• Thanks for the answer, and thanks for noticing that I bolded "Econometrics" in my question! It is indeed under an econometrics context that im learning the CLT. – Siddharth Gopi Jan 4 '14 at 10:57