8.1 Weak Stationarity and Cross-Correlation Matrices
Consider a k-dimensional time series . The series is weakly stationary if its first and second moments are time invariant. In particular, the mean vector and covariance matrix of a weakly stationary series are constant over time. Unless stated explicitly to the contrary, we assume that the return series of financial assets are weakly stationary.
For a weakly stationary time series , we define its mean vector and covariance matrix as
where the expectation is taken element by element over the joint distribution of
. The mean is a k-dimensional vector consisting of the unconditional expectations of the components of . The covariance matrix is a k × k matrix. The ith diagonal element of is the variance of rit, whereas the (i, j)th element ...