Modeling and Assessing Time-Varying Correlation
Summary
The document asks how to tell whether an estimated correlation between two assets is stable over time, and whether the relationship is meaningfully linear. It describes a rolling correlation estimate and an ad hoc stability rule based on the largest recent absolute change, but raises concerns that such thresholds may be unsatisfactory or overfit. The underlying setup assumes correlated geometric Brownian motion, even though observed rolling estimates vary through time.
The responses suggest Dynamic Conditional Correlation as a parametric extension of GARCH that models evolving correlations through conditional volatility estimates. Time-varying or conditional copulas are another option, with goodness-of-fit measures available for assessing a fitted dependence model. The parameters of these models can be examined when testing whether dependence is stable. A separate suggestion is to model the correlation series with a time trend and test whether its slope is zero, though this does not by itself capture richer dynamics. The exchange outlines candidate diagnostics rather than comparing them empirically, and provides no universal threshold for declaring a correlation unreliable.
Key ideas
- Rolling correlation estimates can vary substantially even when a model assumes constant correlation.
- A maximum-change band is an informal stability rule whose threshold is not established here.
- Dynamic Conditional Correlation models time-varying dependence using GARCH-related dynamics.
- Conditional copulas offer another way to model changing dependence and assess fit.
- A trend test can check for systematic change, but does not capture all forms of correlation dynamics.
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Full text
# Correlation: Test for linear dependence # 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... ## Answer by jlowin (score 3) https://quant.stackexchange.com/a/4114 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. ## Answer by user7056 (score 0) https://quant.stackexchange.com/a/4106 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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