Years ago, shortly after I joined Deutsche Bank, my boss asked me to calculate some correlations for a note he was writing. When I placed the results on his desk, I noticed he looked a little surprised.
“Were you expecting something else,” I asked?
“Yes,” he replied. “I was expecting the correlations to be a bit greater.”
I smiled and said, “give me a minute.”
The next set of numbers I placed on his desk showed a set of correlations that were larger than the first set of figures. In response to his raised eyebrow, I explained that the second set of numbers was produced using a weekly frequency rather than a daily frequency, so that some of the high-frequency noise that otherwise would reduce the reported correlations had been effectively averaged out of the data.
I’ve long since forgotten the data series we were analyzing. And that’s a shame, because otherwise I’d love to go back and check a competing hypothesis – namely, that the variables interacted with one another so that the actual correlations, irrespective of any high-frequency noise, really did depend on the frequency with which the data was observed.
To see how this works, consider the stochastic differential equation of a vector-valued random variable, x(t), which follows a mean-zero, multivariate Ornstein-Uhlenbeck process
dx(t) = -A x(t) dt + S dw(t)
where A is a square transition matrix and S is a square scatter matrix that multiplies dw(t), the instantaneous change in a multivariate Wiener process, w(t).
The correlations between the changes in the elements of x(t) are the result of both the scatter matrix, S, and of the transition matrix, A. In particular, if any of the elements of A are non-zero – eg, if there are any attractions between variables – then the correlations between transitions will depend on the time between successive observations.
An example of this is shown in the figure below, which shows the correlation between two variables when the scatter matrix, S, is the identity matrix and when the transition matrix, A, captures mutual attraction between the two variables.
In fact, even the sign of the correlation can depend on the sampling frequency, as shown in the next graph. In particular, in this example, the scatter matrix imparts a positive correlation to the data, while the transition matrix specifies a strong repulsive force between the two variables. When the time between successive observations is small, the scatter matrix is the main determinant of correlation, with the result that the correlation is positive. But as the time between successive observations increases, the repulsive forces introduced via the transition matrix come to dominate, with the result that the correlation becomes strongly negative.
The bottom line is that even simple interactions between variables can have a profound effect on their behavior. And if the models we use to characterize the dynamics of these variables are incapable of capturing these dynamics, the consequences can be equally profound. For example, imagine the implications for a risk manager who concludes, on the basis of daily data, that the correlation between two variables is 0.7, when the correlation over a semi-annual horizon is actually -0.7.
This last example is somewhat contrived, and we shouldn’t expect to come across examples like this frequently in most markets. But the point remains that simple interactions between variables can and often do matter. Checking for such interactions isn’t difficult, and we all should be in the habit of conducting these sorts of checks as a matter of course when analyzing financial data…particularly when the boss has a strong prior regarding the correlations he’s asked us to calculate.
-Doug Huggins
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