One Possible Model of the VIX
Last week I posted about deconstructing the VIX, with an eye towards identifying the small part of VIX that corresponds to sentiment data not contained in or derivative of the price movement of the SPX. I proposed modeling the VIX based on some measure of price volatility and some measure price momentum, and using the error term as a new datum of sentiment.
For volatility, I am using the 45-day Average True Range (ATR) as my measure of volatility, dividing it by the Simple Moving Average (SMA) of the closing pricing over the same 45-day period. Of the several I tested, this one worked out very nicely, but I am by no means suggesting that it is the single best option. To represent a measure of fear/greed derived from price action alone, I am using the Rate Of Change (ROC) over the past 20 days. Other options for both were discussed last week.
Keep in mind, at this point I am not trying to predict the VIX at a future point in time (that would be another post). Right now, I’m attempting to look at price action to say what the VIX “should be” and then see if the “error term” or expected minus actual VIX tells me anything unique about sentiment.
For the data from 1/2/1990 to 2/22/2007, the regression to closing VIX looks like this:
The R-Squared of 79.8% is pretty convincing to me, suggesting that this simple model explains about 4/5th of the variation in the VIX over time. Given that I have over 4300 observations, the F-statistic is significant enough that Excel drops the decimal and returns a “zero.” For reference, you can use either HyperStat Online or the Aggie Stat 30x Notes if you want some help with the statistical terms.
The graph of actual versus predicted VIX looks like this:
When I define the error term as expected minus actual VIX, I can date match each error term to the VIX at that time, and (eventually) match that error term to some future change in the SPX or the VIX itself. First, I want to see how the error terms look for particular values of the VIX:
There is something very interesting about the error term – it is not normally distributed. Defined in either absolute (expected minus actual) or relative ((expected minus actual) divided by actual) terms, the error is notably larger when the actual VIX is small, and vice versa. This is something to keep in mind if/when we start using this error term to attempt prediction of future values for either VIX or SPX.
April 17th, 2007 at 6:27 am
Nice :)
maybe you should try to difference the data, i.e. test the changes, not actual values, because the data are not stationary (if you use actual VIX values), thus there might be spurious regression problems, variables are trendy, but not stacionary trendy, so this way you will always obtain high R^2, but it does not really say much about the regression. Try AR(1) process ( VIX(t) = a VIX(t-1) + e ) with the original VIX data, and with difference data ( ln ( VIX(t) / VIX(t-1) ) - it’s continual return, and you will see that in first case, you will get very nice >90% R^2, but the model is not valid, and useless, because there is unit root. In the second case, you will also get significant variable, but R^2 will be zero point zero nothing.
If you’re modelling time series data, stacionarity is crucial, because if there is unit root present, you will get vrong results
April 17th, 2007 at 8:22 am
Thanks for reading and thanks for commenting, Jozef! Interesting ideas.
It’s possible that testing the changes in the VIX over time might be indicative of sentiment extremes as well. Something to think about, as the purpose of the exercise is to find the unique or stepwise data contributed by VIX that is not included in the price movement of SPX. Doubtless there are dozens of ways to get at that …