# Time-varying correlation via state-space representation and Kalman filter

Let a linear time-varying mode like this one:

$y_{t}=\alpha_{t}+\beta_{t}x_{t}+\epsilon_{t}$.

You can also suppress the constant term to simplify this example:

$y_{t}=\beta_{t}x_{t}+\epsilon_{t}$.

Kalman filter allows you to get $\beta_{t}$ value under some hypothesis which there's no need to list here.

Knowing that, e.g. in the linear regression case,

$\beta_{y,x}=\dfrac{\sigma(y,x)}{\sigma^{2}(x)}$

and

$\rho_{y,x}=\dfrac{\sigma(y,x)}{\sigma(y)\sigma(x)}$,

it's very easy to see that

$\rho_{y,x}=\beta_{y,x}\dfrac{\sigma^{2}(x)}{\sigma(y)\sigma(x)}$.

My question: is there any way I can use the output of the Gaussian Kalman filter applied to this obs. equation to get the time-varying value of $\rho_{y,x}(t)$?

• You probably already know this, but as a side note consider time-varying correlations through (i) (AG-)DCC-GARCH, (ii) time-varying copulas and stochastic copulas, (iii) wavelet coherence. If you're using Kalman Filters because you want an "online" time-varying estimate of correlation, then (i) and (ii) aren't useful but (iii) with causal wavelets might be. If you want to just know what happened in the past, (ii) would be better because you can look at non-linear time-varying correlations such as a time-varying Kendall's Tau or Spearman's rho. – Jase Aug 3 '13 at 14:37
• @Jase, any good reference to start with time-varying copulas and stochastic copulas? Something easy to deal with... – Lisa Ann Aug 5 '13 at 6:52
• Stochastic copulas such as SCAR are complex. First, look at normal time-varying copulas in the 2006 paper by Andrew Patton. Code and paper references for the time-varying copulas can be found on public.econ.duke.edu/~ap172/code.html. – Jase Aug 5 '13 at 9:38