Estimating Time-Varying Correlation from a Kalman Regression

Summary

The document asks whether a Gaussian Kalman filter applied to a regression with a changing slope can be used to estimate the changing correlation between two series. It sets out a linear observation model, first with an intercept and then without one, and notes that the regression slope is covariance divided by the predictor variance. Correlation can therefore be related to that slope if the relevant variances are also known over time.

The text does not give a complete estimation procedure or empirical evidence. A time-varying slope alone does not determine correlation: the calculation also requires suitable time-varying estimates of the two series’ variances, and the relationship depends on the regression assumptions and model specification. The document is best read as a question motivating how to combine state estimation with variance estimation, rather than as a validated trading method.

Key ideas

• A time-varying linear regression can represent a changing relationship between two series.
• The Kalman filter can estimate a latent regression slope over time under a specified state-space model.
• Correlation depends on the slope as well as the variances of both series.
• A changing slope by itself is insufficient to recover changing correlation.

Tags

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


# 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 $\rho_{y,x}(t)$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.