the rules of matrix multiplication, it is easy to verify that the daily covariance matrix of asset returns can be computed as SQ{d)=R{d)'R(d)/T. However, as we know from our earlier discussion, in order to compute the monthly covariance matrix we must also estimate the covariances between returns observed on different days. This can easily be done in matrix form by introducing a new matrix R_k{d) which contains zeros in the first k rows, and the first T - k rows of R(d) in its last T - k rows. Again, one can verify that the matrix product Sk(d) = R(d)'R_k(d)/T provides sample estimates of the daily covariances between returns observed k days apart. If daily returns display a serial correlation of order q, the monthly covariance matrix estimator can be written as: S(m) = p.S0(d) + ^(p-k)[sk(d) + Sk(dy] (16.3') The more technically inclined reader will note that this estimator has the desirable feature of generating a monthly covariance matrix that is guaranteed to be positive semidefinite. Loosely speaking, this is the matrix equivalent of requiring that an estimator of the variance should be non-negative. In practice, this property guarantees that whenever it is used to estimate the risk of a portfolio, this estimator will generate a non-negative value.7 Weighting the Observations A common criticism of the estimator discussed in the previous section is that it assigns the same weight to each observation, no matter when the observation occurred. Obviously, this would not be a problem if daily returns were iid, because in that case all returns would be drawn from the same distribution. However, when the iid assumption becomes questionable, it might be desirable to associate a larger weight with recent observations. In the discussion that follows, we propose a simple way of incorporating this feature within the estimation framework developed so far. An intuitive weighting scheme assigns a weight of 1 to the most recent observation and discounts previous observations at a prespecified rate 5. Formally, if wt is the weight assigned to the observation at time t, then the sequence of weights can be computed in a recursive fashion from wt_x = (1 - )wf Intuitively, the larger the rate 5, the faster the decay process, or equivalently, the larger the relative weight assigned to recent observations. 7De Santis and Tavel (1999) provide a more technical discussion of this estimator. They show that the same estimator would be obtained by estimating a daily covariance matrix using the serial correlation correction proposed by Newey and West (1987), and then scaling the daily covariance matrix by the number of trading days in one month. They also show that this estimator provides a formal justification for the common practice of adjusting for serial correlation by averaging returns over several days (so-called overlapping).