7.3.8 Non-Positive Definite Covariance Matrices

7.3.8  Non-Positive Definite Covariance Matrices

Estimated by UWMA, EWMA or some other means, the matrix ^1|0Σ may fail to be positive definite. This typically occurs for one of two reasons:

1. Usually, the cause is ¹R having high dimensionality n, causing it to be multicollinear. Roundoff error in applying UWMA, EWMA or some other estimator causes the estimated matrix ^1|0Σ to have one or more eigenvalues that are zero or slightly negative.
2. Less common, the problem may be insufficient historical data for ¹R. If + 1 < n, any of estimators [7.10], [7.18], or [7.20] will produce a singular positive semidefinite matrix ^1|0Σ, assuming exact calculations. Roundoff error is unlikely to leave any eigenvalue precisely equal to zero, but it may cause one or more eigenvalues to be slightly negative.

We discuss covariance matrices that are not positive definite in Section 3.6. The Cholesky algorithm fails with such matrices, so they pose a problem for value-at-risk analyses that use a quadratic or Monte Carlo transformation procedure (both discussed in Chapter 10). There are two ways we might address non-positive definite covariance matrices

One way is to use a principal component remapping to replace an estimated covariance matrix that is not positive definite with a lower-dimensional covariance matrix that is. See Section 9.5. This approach recognizes that non-positive definite covariance matrices are usually a symptom of a larger problem of multicollinearity resulting from the use of too many key factors. The solution addresses the symptom by fixing the larger problem.

Alternatively, and less desirably, ^1|0Σ may be tweaked to make it positive definite. Various means have been proposed for accomplishing this. Some are quite elaborate. Since our goal is merely to correct for small roundoff error, whatever approach we adopt should alter ^1|0Σ very slightly. No approach should have a material impact on calculated value-at-risk as compared to the others. For this reason, it is reasonable to adopt a simple solution, such as replacing ^1|0Σ with

[7.21]

for some small ε > 0 and I the identity matrix. Generally, ε can be selected small enough to have no material effect on calculated value-at-risk but large enough to make covariance matrix [7.21] positive definite.

Exercises

7.3

Consider the covariance matrix

[7.22]

which has one slightly negative eigenvalue. Select a value ε and apply [7.21] to construct a positive definite approximation for the matrix.
Solution