# Using central limit theorem to test whether population average return is the same, before and after the recession

This is the task I have been asked to do. I've read up on what central limit theorem (clt) is, but I feel like I'm missing something.

The data I have is a matrix of monthly stock returns from 50 different companies from 1/1/2000 to 1/8/2014.

I've established I find the cross sectional average return before the recession (Rb), and the average return after the recession(Ra) and;

(Rb - Ra) is my X-bar in the clt, z-score formula.

I apologise for any inaccuracies, my knowledge is very little and I'm thankful for your patience

You cannot use the clt to test something, it is a theorem about convergence. You can only use a statistical test to test something which basis is in many cases the clt.

In this case you could e.g. use a so called t-test. In R you would e.g. type:

t.test(data.Rb,data.Ra)


to test whether the difference in the means is significant.

• Thanks for your response @vonjd! I'm using Excel, would the 't-test: two sample, assuming unequal variances' be ok with for me? As per this link – Harry Dec 30 '14 at 19:05
• @Harry: First a very warm welcome to Quant.SE and thank you for your question. Yes, this would work. If you found my answer helpful please upvote and accept it - Thank you :-) – vonjd Dec 30 '14 at 19:15
• Certainly! I think this website is going to be very helpful for my MSc Finance course :-) Edit: sorry for being a noob, I apparently need 15 votes to be able to vote up? – Harry Dec 30 '14 at 20:34
• @Harry, just looking through the link I provided, is it's way of hypothesis testing correct? Or should I only focus on P Values? – Harry Dec 31 '14 at 16:27
• @Harry Unfortunately I am not an expert concerning Excel :-( – vonjd Jan 1 '15 at 14:30

As vonjd mentioned, you could do a t-test. However, as stated in your comments if you believe the standard deviation for each group is different (maybe you should do a Levene's test), you shouldn't use a t-test for two means. You should consider a non-parametric test such as Mann-Whitney.

• Would a t-test be critically incorrect? It's just i've done it for 6 groups of data and written up about it. If it's seriously wrong, I would delete everything however I'd prefer to acknowledge in my conclusions that the Mann-Whitney is probably a better choice, providing the t-test still has some merit – Harry Jan 4 '15 at 1:39
• It depends on your data and which test you've used. If your test assumes equal variance, and your data is clearly not, maybe your conclusion is wrong. You really have to check with your data. – SmallChess Jan 4 '15 at 3:22
• I'm pretty sure that each sample has unequal variance (its average returns (and VaR figures) before and after the recession), but I've used the 't-test:with unequal variances' function in Excel – Harry Jan 4 '15 at 10:36
• Then you're fine. :-) – SmallChess Jan 4 '15 at 23:17