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### Filling gaps in correlated bivariate data

Posted: October 7th, 2019, 10:07 am
Consider the data set below. The blue is the SP500 futures trading 24h/day. The other is the VIX which I only have values for 7h/day. Obviously both are somewhat inversely correlated.

What method would you recommend for filling the hidden/missing orange VIX points with 'most likely' values, given the observed correlation structure and boundary conditions? I've considered iterative mutlivariate singular spectrum analysis, but this is cumbersome.

Any help welcome.

### Re: Filling gaps in correlated bivariate data

Posted: October 9th, 2019, 11:22 pm
Just a thought to get started. You could first do a simple (zero-intercept) regression using the regular session (log)-returns, say at whatever frequency is convenient:

[$] R^{VIX}_t = \beta R^{SPX}_t + \epsilon_t[$].

Then, the predicted overnight returns, at the same frequency, could be

[$] R^{VIX}_t = \alpha + \beta R^{SPX}_t[$],

where [$]\alpha[$] was changed each night to make each full overnight VIX return come out correctly. Log returns would be easiest for this idea.

You could test the method by also using it to predict some known regular session VIX returns in the same way and see how it does. In other words, just pretend some of your regular session VIX returns were missing and try the same prediction method on those.

### Re: Filling gaps in correlated bivariate data

Posted: October 10th, 2019, 11:04 pm
There's a lot of real estate to fill in there. Might the correlated Brownian bridge technique be useful?

### Re: Filling gaps in correlated bivariate data

Posted: October 10th, 2019, 11:06 pm
SAS has functionality for this kind of stuff. E&Y has been applying it to fill in historical time series for FRTB clients.

### Re: Filling gaps in correlated bivariate data

Posted: November 30th, 2019, 4:29 pm
Thank you all. I've played with this quite a bit and I think I'll go with Alan's solution of regressed returns adjusted with a linear drift.