I found it helpful to look at a variety of covariance ellipses for tame distributions so that I would have a better feel for those I come across in the wild. Following is a quick tour that I hope will show the effect of changing the parameters of the distributions of the series in question on the covariance ellipses:
- For uncorrelated series, what is the effect of changing the standard deviations of the series?
- For series with identical standard deviations, what is the effect of changing the correlation coefficients?
- For series with different standard deviations, what is the effect of changing the correlation coefficients?
For an introduction to covariance ellipses, refer to my previous post: Covariance Ellipses
Varying Standard Deviation of Uncorrelated Series
Here is the covariance ellipse for two uncorrelated series each with standard deviation of 20%. Theoretically we should see a circle (a special case of an ellipse) with semi-axes parallel to the x and y axes. I say “theoretically” because for these randomly generated series, the variances of the distributions are not quite equal and the covariance is not exactly zero. So the ellipse is stretched just slightly in the direction of the distribution with the higher variance, and rotated slightly.
When we generate series with markedly different variances, even though the covariance still is not exactly zero the small effect that rotates the axes of the ellipse away from the axes of the plot is overwhelmed. So we see the ellipse stretched in the direction of the distribution with the largest standard deviation. The axes of the ellipse are theoretically parallel to the axes of the plots: The maximum variance portfolio of two uncorrelated series is made up only of the series with the largest variance. The minimum variance portfolio contains only the series with the smaller variance.In these cases, the minor effect of the non-zero covariance makes itself felt in the orientation of the ellipse semi-axes. The eigen vectors of the covariance matrix will be flipped one way or the other depending on the slight deviation from zero in the measured covariance. Notice how, in the second chart the measured correlation is just slightly negative and the semi-axes are -ve and + ve respectively. In the third chart where the correlation comes out just +ve, both the semi-axes are negative.
A final point: if I had plotted correlation ellipse rather than covariance ellipses, all the charts would be the same aside from random variations.
This is another of those “aha” situations. Intuitively, I always expect to see varying correlation cause the ellipse to rotate. Of course, it doesn’t rotate at all it causes the ellipse to change shape. Notice that the axes of all the three ellipses in the following chart have the same orientation of 45 degrees. The eigen vectors have coefficients of in all positions.
As the magnitude of the correlation coefficient increases the ellipse changes from a perfect circle (ρ = 0) elongating in the direction dictated by the sign of ρ and getting narrower perpendicular to this direction until it ultimately ends up as a straight line.
Different Variance, Different Correlation
Finally we see some rotation in the covariance ellipses. IF the distributions have different variances (which, in the wild, they mostly will) then when the magnitude of the correlation coefficient increases, the ellipse BOTH stretches AND rotates away from the axis of the series with the larger variance. The final resting angle of the ellipse (compressed to a line) that represents the fully correlated series is dictated by the relationship of the variances of the two series.
This brief synopsis gives me enough understanding to be able to look at changes in covariance ellipses over time, or as a result of partitioning data either by time frame or some other criteria, and make a quick qualitative interpretation of what has happened to the correlation relationships.
Edit: LaTeX issues fixed