Analysis of Financial Time Series, Third Edition

10.1 Exponentially Weighted Estimate

Given the innovations , the (unconditional) covariance matrix of the innovation can be estimated by

where it is understood that the mean of is zero. This estimate assigns equal weight 1/(t − 1) to each term in the summation. To allow for a time-varying covariance matrix and to emphasize that recent innovations are more relevant, one can use the idea of exponential smoothing and estimate the covariance matrix of by

10.2

where 0 < λ < 1 and the weights (1 − λ)λj−1/(1 − λt−1) sum to one. For a sufficiently large t such that λt−1 ≈ 0, the prior equation can be rewritten as

Therefore, the covariance estimate in Eq. (10.2) is referred to as the exponentially weighted moving-average (EWMA) estimate of the covariance matrix.

Suppose that the return data are . For a given λ and initial estimate , can be computed recursively. If one assumes that follows a ...