7.3.4 Uniformly Weighted Moving Average
Given out assumption that 
is a white noise, a standard approach for estimating 
is to treat the data {–αr, … , –1r, 0r} as a realization of a sample. We can then apply a statistical estimator for covariance matrices. For this purpose, we might use the sample estimator
[7.9]
where 
indicates a sample mean.
We already know that t | t–1
= 0 for all t, so a better estimator is
[7.10]
If 
is fixed from one value-at-risk analysis to the next, will always be estimated from the same number of historical data points. We call this the uniformly weighted moving average (UWMA) method for estimating .
7.3.5 Example: Aluminum Prices
Exhibit 7.4 indicates historical LME cash and forward prices for aluminum.
Let’s estimate a conditional mean vector 1 | 0μ and conditional covariance matrix 1 | 0Σ for key vector
[7.11]
For expositional convenience, we consider only the 10 data points of Exhibit 7.4. More data would be used in practice. Denote the data {–9r, –8r, … , 0r}. Assume t | t–1μ = t–1r for all t, so
[7.12]
Convert R to white noise with linear polynomial [7.4], where
[7.13]
and
[7.14]
Applying our linear polynomial to the data of Exhibit 7.4, we obtain white noise data {–8
, –7, … , 0}, which is indicated in Exhibit 7.5.
Estimator [7.10] becomes
[7.15]
Applying this to our white noise data, we estimate as
[7.16]
Applying [7.5], we estimate 1 | 0Σ as
[7.17]
Exercises
Reproduce the calculations of our aluminum example. Confirm that you obtain the same results.
Solution