# Correlation: Test for linear dependence

Setting the scene: Assume a multivariate GBM with correlation matrix $\Sigma$. Further, one want to estimate the correlation between two of the assets. Assume one has a suitable estimator of the correlation, the exact estimator might not be directly relevant, so assume e.g. standard, 60 days moving average.

Identifying the issue: If one look at the time series of the correlations, it is obvious that a correlation is not constant over time, as postulated by gbm. Is there any good measure of "correlation volatility", i.e. a measure that says when a correlation seems stable? Alternatively, a measure that will quickly identify that the correlation is unrelieable?

Attempt: Ive tried to look at the maximum (absolute) changes of the close the last 60 days, and then created a band around todays estimation equal to $(p_t + max, p_t-max)$, where p is the correlation estimate and max is the max absolute movement. Then I have said that if this spread is higher than some given value (say 0.15), then the correlation is "unstable".

I have also tried different variation as looking at the maximum return, highest vs. lowest value etc. I am a bit vary to over-model this, so I hesistate to start giving correlation coefficient a probability distribution etc...

At the same time, I find my approach a bit unsatisfactory, and wondering if there is any "well designed" tests to see whether two variables satisfy such a linear dependence that correlation is...

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I have started to be a bit vary of just looking at the correlation coefficients, as they are simply a number and could be stable although the returns do not follow a lognormal (or any other assumed distribution) walk with correlation p. I guess one would have to look at the distribution as a whole, considering whether the observed data would be sampled from e.g. a bivariat normal distribution. – AdAbsurdum Sep 12 '12 at 13:35

Are you trying only to identify unstable correlations, or are you trying to incorporate time-varying correlation into your model?

If the latter, you may want to check out Engle's Dynamic Conditional Correlation, which is an extent of GARCH modeling. In simplest terms, DCC models time-varying correlation parametrically via the GARCH residuals.

Another, slightly more exotic option is to consider a time-varying or conditional copula, such as those introduced by Andrew Patton. See for example his paper Modelling Asymmetric Exchange Rate Dependence. Patton has also introduced goodness-of-fit measures for copulas.

Note that by inspecting the parameters of such models, you could evaluate the hypothesis that correlations are stable.

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Thanks. I think the second paper maybe can solve some of my issues (restated in a comment above). I will have a look at it. – AdAbsurdum Sep 12 '12 at 13:37

If you have time seria you could postulate a linear equation for them:

x(t)=a*t+b giving a=x(t+1)-x(t)

There are lots of methods to deal with the hypothesis that a is zero or not, including low sample statistics to boost your confidence (levels) in the final result. In addition, by b-estimation you could see the "volatility" of your correlations.

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