I have the weekly time series of returns for both VIX and S&P 500.
For the VIX I'm looking at 1 week return period (e.g. this is a 5 day return series rolling weekly)
For the S&P 500, instead, I'm looking at 45 Day return period (e.g. this is a 45 day return series rolling weekly)
What I'd like to evaluate is the following relationship:
I assume that every time VIX 5 day retruns was above 35%, then the S&P 500 had a following positive 45 day return.
My doubt is about how to test the significance of this relationship. Starting from the 1990 I found that 19 times in the history, the VIX was above 35% and the S&P 500 next to that performance was positive 9 times.
I'd like to test the significance of this relationship. I was wondering about:
test the average of those 9 positive return where the null hypothesis was that they were zero
test the average of those 9, against the average of all the history of the S&P 500 45 day series and look if the averages were different,
run a regression and test the beta was different than zero.
How do you suggest to proceede to evaluate the significance of that relationship?
