Testing Whether High VIX Predicts Subsequent S&P 500 Returns
Summary
The document asks how to assess whether unusually large weekly VIX gains predict positive S&P 500 returns over the following 45 days. The sample described contains 19 high-VIX episodes since 1990, with positive subsequent returns in 9 cases. It considers testing the selected returns against zero, comparing them with the unconditional historical average, or estimating a regression coefficient.
The response recommends treating the claim as a forecasting problem and benchmarking it against a simple unconditional-mean forecast or a passive 45-day S&P 500 strategy. It proposes a monthly regression with both the VIX level and an indicator for elevated VIX, then evaluating forecasts in rolling out-of-sample windows by comparing squared prediction errors. The small number of threshold events limits statistical power, and overlapping rolling return windows may complicate inference; the document does not specify adjustments for either issue. The suggested framework therefore offers a comparison design, not evidence that the proposed signal works.
Key ideas
- A small number of high-VIX episodes makes statistical significance difficult to establish.
- Frame the hypothesis as a forecast of future S&P 500 returns.
- A regression can include both the VIX level and an indicator for elevated VIX.
- Evaluate predictions out of sample against an unconditional-mean or passive-market benchmark.
- Overlapping forward-return windows can complicate inference and should be considered.
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Full text
# Evaluate the significance of the relationship among VIX and the S&P 500
# 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?
## Answer by phdstudent (score 3)
https://quant.stackexchange.com/a/39952
It is unlikely with so few observations that you will get statistical significance. Also you should benchmark your model against a "naive" model.
If your conjecture is that high VIX implies high expected returns and you want to use that for trading, I suggest that you run a Goyal and Welch (2008) type of model. They use regressions (you can easily use another model closer to yours).
My suggestion would be using monthly data to run the following regression:
$R_{t+1} = \alpha_i + \beta_1 VIX_t + \beta_2 1_{VIX > 0.35} + error$
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 $R_{OOS}^2$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.