Suppose I want to build a linear regression to see if returns of one stock can predict returns of another. For example, let's say I want to see if the VIX return on day X is predictive of the S&P return on day (X + 30). How would I go about this?
The naive way would be to form pairs (VIX return on day 1, S&P return on day 31), (VIX return on day 2, S&P return on day 32), ..., (VIX return on day N, S&P return on day N + 30), and then run a standard linear regression. A t-test on the coefficients would then tell if the model has any real predictive power. But this seems wrong to me, since my points are autocorrelated, and I think the p-value from my t-test would underestimate the true p-value. (Though IIRC, the t-test would be asymptotically unbiased? Not sure.)
So what should I do? Some random thoughts I have are:
- Take a bunch of bootstrap samples on my pairs of points, and use these to estimate the empirical distribution of my coefficients and p-values. (What kind of bootstrap do I run? And should I be running the bootstrap on the coefficient of the model, or on the p-value?)
- Instead of taking data from consecutive days, only take data from every K days. For example, use (VIX return on day 1, S&P return on day 31), (VIX return on day 11, S&P return on day 41), etc. (It seems like this would make the dataset way too small, though.)
Are any of these thoughts valid? What are other suggestions?