Autocorrelation Inference Depends on Return Tails and Dependence
Summary
The post considers lag-one sample autocorrelation in daily stock log returns and asks when its scaled estimator is asymptotically standard normal under a model with normally distributed returns. It seeks a reference for the assumptions that justify this inference, especially the connection to central limit theory. The estimator and target limit are stated, but the question does not verify the formula or derive its distribution.
The response points readers to texts on econometric asymptotics and time-series statistical inference, ordering them roughly by complexity. Its main conceptual guidance is that asymptotic results depend on jointly controlling the existence of higher moments and the strength of serial dependence. This links tail behavior to the process's memory. The exchange gives no worked test, precise regularity conditions, or empirical evidence, so the references are starting points rather than a complete treatment.
Key ideas
- The post studies a lag-one sample autocorrelation estimator for daily stock log returns.
- A standard-normal asymptotic approximation for the scaled estimator requires assumptions that the post does not enumerate.
- Inference depends on both the finiteness of return moments and the dependence structure of the series.
- The suggested reading includes texts on econometric asymptotics and time-series inference.
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Full text
# Good reference on sample autocorrelation?
# Good reference on sample autocorrelation?
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
$$r = \frac{Cov(X_1,X_2)}{Var(X_1)}$$
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: https://math.stackexchange.com/questions/139408/good-reference-on-sample-autocorrelation
Statistics: https://stats.stackexchange.com/questions/27465/good-reference-on-sample-autocorrelation
## Answer by Ryogi (score 2)
https://quant.stackexchange.com/a/3391
Three good references are the
- Asymptotic theory for econometricians, H. White
- Stochastic Limit Theory, Davidson
- Asymptotic Theory of Statistical Inference for Time Series, Taniguchi and Kakizawa
They are roughly in order of complexity.
The crux of the matter is to balance the requirements of finiteness of higher moments of $X$ with its dependence structure, or, put it differently, to balance the thickness of the tails with the memory of the process.
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