Take the 2-minute tour ×
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

I'm not a statistician but I'm writing my thesis on mathematical finance and I think it would be neat to have a short section about independence of stock returns. I need to get better understanding about some assumptions (see below) and have a good book to cite.

I have a model for stock prices $S$ in which the daily ($t_i - t_{i-1}=1$) log-returns

$$X_n = \ln\left(\frac{S(t_n)}{S(t_{n-1})}\right), \ \ n=1,...,N$$

are normally distributed with mean $\mu-\sigma^2/2$ and variance $\sigma^2$. The autocorrelation function with lag 1 is

$$r = \frac{\mathrm{Cov}(X_1,X_2)}{\mathrm{Var}(X_1)}$$

which I estimate by

$$\hat{r} = \frac{(n+1)\sum_{i=1}^{n-1} \bigl(X_i - \bar{X} \bigr)\bigl(X_{i+1} - \bar{X} \bigr)}{n \sum_{i=1}^{n}\bigl(X_i - \bar{X} \bigr)^2} $$

where

$$\bar{X} = \frac{1}{n}\sum_{i=1}^N X_i$$

Now I understand that under some some assumptions it holds that

$$\lim_{n \rightarrow \infty} \sqrt{n}\hat{r} \in N(0,1)$$

I would be very glad if someone could point me towards a good book which I can cite in my thesis and read about these assumptions (I guess it has something to do with the central limit theorem).

Thank you in advance!

Crossposting at:

Mathematics: http://math.stackexchange.com/questions/139408/good-reference-on-sample-autocorrelation

Quantative Finance: http://quant.stackexchange.com/questions/3390/good-reference-on-sample-autocorrelation

share|improve this question
3  
Please don't simultaneously cross post. –  Andy W May 1 '12 at 15:16

3 Answers 3

Van der Vaart's time series lecture notes are excellent, and chapter 5 has exactly the sort of theorem you're looking for.

share|improve this answer
    
Really excellent notes! Very witty too: "Many financial time series exhibit an exponential trend, not always in the right direction for the owners of the corresponding assets." –  Dimitriy V. Masterov May 1 '12 at 19:46

The assumptions are :we have i.i.d. observations (Xi, Yi), i = 1, . . . , n, for (X, Y ). Note that the the first 'i' means independent thus daily prices would be inapplicable due to their auto-correlative structure unless filtered by an ARIMA model with any necessary deterministic variables such as Pulses, Level Shifts , Seasonal Pulses and/or Local Time trends to render the process "identically distributed". You might want to look at some of my comments in Correlating volume timeseries . Pay close attention to the Yule paper (1926) that is cited as there continues to be a total oversell of the cross-correlation when dealing with non-Gaussian data, ignoring the critical assumptions that you wisely ask for. Your question is dead-on and deserves attention. You might also review a recent Masters Thesis at GSU Interval Estimation for the Correlation Coefficient by Aekyung Jung ajung1@student.gsu.edu at http://digitalarchive.gsu.edu/cgi/viewcontent.cgi?article=1109&context=math_theses. Hope this helps .

share|improve this answer

It might be worth checking out:

Analysis of Financial Time Series (2010) by Ruey S. Tsay

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.