# Evaluate the significance of the relationship among VIX and the S&P 500

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?

$R_{t+1} = \alpha_i + \beta_1 VIX_t + \beta_2 1_{VIX > 0.35} + error$
Then you have a proper model that you can test. Using rolling windows of the regression above then you can compare the realized return $\tilde{R}_{t+1}$ against the predicted return $\hat{R}_{t+1}$ from the model above and sum the squared differences which I call out-of-sample r-squared: $R^2_{OOS} =\sum_{t=n}^T (\hat{R}_{t+1} - \tilde{R}_{t+1})^2$ and compare it with a naive model of prediction of the SPX (such as the unconditional mean). You can do the same exercise using your strategy i.e. using all only the 19 time in history where VIX was high and using a 45 day return window for SPX. But you should benchmark it against a naive strategy of holding the SPX for 45 days and compute the $R_{OOS}^2$.