# Alternative ways to understand time-varying comovement between two time-series?

I have been looking into ways to better understand how the dependencies/correlations/etc between two time series can vary over time.

I first thought about using a Kalman/particle filter over a linear model to get a time-varying slope estimate. However, I'm worried that this will also pick up changing relative variances between the two time series and an increasing slope estimate doesn't actually mean a stronger relationship between the two time-series.

I have looked into time-varying quantile regression but am unconvinced that a changing slope parameter means much, even if it accounts for asymmetry over the various quantiles. The same thing goes for the new time-varying cointegration technology.

I have other reservations about DCC-GARCH because to the best of my knowledge it's a time-varying estimate of the Pearson estimator and is therefore not able to pick up non-linear dependencies (since it's essentially the time-varying square root of the $R^2$ of a linear regression). I'm concerned that the DCC-GARCH correlation estimate might decrease because linear dependencies are reducing, even if non-linear dependencies are increasing.

So what else is there and how can it help me to pick up the time-varying dependencies between two time series that accounts for both linear and non-linear relationships? Something like a time-varying Kendall tau or time-varying mutual information would be nice.

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"time-varying dependence between two time series". Do you mind elaborating what you mean with that? Differencing the time series? –  Matt Wolf Jan 16 '13 at 2:55
@Freddy Something that captures both the time-varying linear AND non-linear correlation/relationship between two time-series. I've edited the title to make it less confusing. Also, I have just learned that various 2006+ copula methods have been developed that can achieve what I want. –  Jase Jan 16 '13 at 3:49
thanks, interesting, I admit I lack a bit of knowledge when it comes to copulas. Do you have papers or any reference in that regards? –  Matt Wolf Jan 16 '13 at 4:09
@Freddy This one seems to be a good one: wisostat.uni-koeln.de/Institut/LSMosler/Manner/… –  Jase Jan 16 '13 at 4:21
@Freddy Apparently it also allows us to look at the time-varying correlation at the distributional extremes, sort of like a time-varying quantile correlation estimator (which actually has just been invented in the last 6 months, including standard error asymptotics, although nobody has referenced it yet). It's applied to good effect here: sciencedirect.com/science/article/pii/S1059056010001358 –  Jase Jan 16 '13 at 4:44