# 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

I think you fail to understand Multivariate Garch model such as DCC models since they do take into account non linearity. They are interested in jointly modeling the time series behavior of multiple conditional variance processes.

Each couple of series has its own particular conditional correlation process evolving trough time in a non-linear way. In fact they are devoted to this understanding of co-movement. I recommend you to have a look to this excellent survey:

Bauwens, L., Laurent, S., & Rombouts, J. V. K. (2006). Multivariate GARCH models: a survey. Journal of Applied Econometrics, 21(1), 79–109. http://doi.org/10.1002/jae.842

ex : the conditional correlation of serie A and B may be modeled as autoregressive process while each of one having its own different conditional variance process. The conditional correlation of serie A and C is a different autoregressive process. Both processes have square terms allowing no-linear relationship. Also, They can have explanatory variable and feedback to the conditional mean process (Arch-in- mean models). Don't see them as simple "Pearson estimator" processes.

However another interesting class of models you can have a look are Unobserved Components Models (which can be estimated via state space models). They are recents and you can model the joint behavior of multiple series. A good starting point is the following book :

An Introduction to State Space Time Series Analysis by Jacques J.F. Commandeur,Siem Jan Koopman (2007).

Finally, you can also have a look to multivariate stochastic volatility models, which you can also estimate with state space models and kalman filter.

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