# Testing predictability of a proposed predictor in case of multiple returns

Say I have a T daily observations for the last ten years on a new predictor $$x_t$$ which I think is a predictor of the expected weekly return on the stock market, $$r_{t,t+5} = r_{t+1}+...+r_{t+5}$$, where $$r_t$$ is log return for that period. In this case, how can I test the null hypothesis that $$x_t$$ has no predictability? The answer would be simple if we are concerned about only one return. But, here, I want to test the predictability for multiple returns which may be correlated with each other. Any idea how to test the hypothesis correctly? What's the correct test statistic and variance of it? Reference to a procedure or an academic paper is also welcome!

• You could run a regression where the dependent variable is $r$ and the independent variable is $x$. However, as you point out, the returns may be correlated, so OLS does not work. You would need to use statistical methods which allow for this. See quant.stackexchange.com/questions/740/… also quant.stackexchange.com/questions/35216/… Mar 12, 2023 at 9:20
• Yes, sorry, I should have said "the errors are autocorrelated". Mar 12, 2023 at 12:37