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.