7.3.4 Uniformly Weighted Moving Average (UWMA)

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, … , –1r0r} 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


where indicates a sample mean.

We already know that t | t–1 = 0 for all t, so a better estimator is


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. 

Exhibit 7.4: Historical LME prices for cash, 3-month, 15-month, and 27-month aluminum. Source: LME.

Let’s estimate a conditional mean vector 1 | 0μ and conditional covariance matrix 1 | 0Σ for key vector


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


Convert R to white noise  with linear polynomial [7.4], where




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.

Exhibit 7.5: White noise data calculated from the data of Exhibit 7.4.

Estimator [7.10] becomes


Applying this to our white noise data, we estimate   as


Applying [7.5], we estimate 1 | 0Σ as



Reproduce the calculations of our aluminum example. Confirm that you obtain the same results.