Correlation Cointegration
In a previous post I looked at ways of modeling the relationship between the CBOE VIX Index and the Year 1 and Year 2 CBOE Correlation Indices:
The question was put to me whether the VIX and correlation indices might be cointegrated.
Let's begin by looking at the pattern of correlation between the three indices:
If you recall from my previous post, we were able to fit a linear regression model with the Year 1 and Year 2 Correlation Indices that accounts for around 50% in the variation in the VIX index. While the model certainly has its shortcomings, as explained in the post, it will serve the purpose of demonstrating that the three series are cointegrated. A standard test rejects the null hypothesis of a unit root in the residuals of the linear model, confirming that the three series are cointegrated, order 1.
Vector Autoregression
We can attempt to take the modeling a little further by fitting a VAR model. We begin by splitting the data into an in-sample period from Jan 2007 to Dec 2015 and an out-of-sample test period from Jan 2016 to Aug 2017. We then fit a vector autoregression model to the in-sample data:
When we examine how the model performs on the out-of-sample data, we find that it fails to pick up on much of the variation in the series - the forecasts are fairly flat and provide quite poor predictions of the trends in the three series over the period from 2016-2017:
Conclusion
The VIX and Correlation Indices are not only highly correlated, but also cointegrated, in the sense that a linear combination of the series is stationary.
One can fit a weakly stationary VAR process model to the three series, but the fit is quite poor and forecasts from the model on't appear to add much value. It is conceivable that a more comprehensive model involving longer lags would improve forecasting performance.
Asset Management. Venture Capital.
7 年Happy to see Wolfram language put to work in this space. Could pct change in level of VIX on various timeframes be more correlated with 1w, 2w, 4w, forward changes on Correlation index? Can this analysis be used to help time capital allocation to L/S equity fund?
Chief Advisor at Central Bank of the Republic of Türkiye
7 年Nice work, thanks for sharing. There is nothing wrong with the model or the forecasts it generates. Actually, the model does exactly what it is supposed to do: focus on the "mean" not the volatility. Cointegration, if found, can be a powerful dynamic to exploit. It basically tells you not to worry about the short term fluctuations as they will eventually revert to an equilibrium (hence the flat lines). You can, of course, add some noise to these out-of-sample forecasts through Monte Carlo simulations and/or generate the confidence bands to emphasize the uncertainty surrounding them. Other than that, one cannot expect a satisfying long-term out-of-sample forecasting performance using high(er) frequency data (i.e. daily). It is really very difficult to do so, unless you capture a predictable trend component, which may require a completely different modeling framework. Despite all these caveats, I think VAR or VEC can be used more often in financial time series.
What is the meaning of stationary ?