# EWMA covariance matrix number of lags

When calculating an exponentially weighted covariance matrix for t observations, formula 10.2 here:

https://www.oreilly.com/library/view/analysis-of-financial/9781118017098/c10_level1_1.xhtml

Uses observations 1 (earliest) to t-1 (oldest-1). Why doesn't it use the oldest observation, t, like a simple moving average or exponential moving average?

It says "For a sufficiently large t such that λt−1 ≈ 0....", so if you have a smaller number of observations where λt−1 >0, you would use observations 1 to t in formula 10.2?

Thank you

• develarist: there's something weird going on with that formula for exponential smoothing. I don't see where the denominator on the outside, namely $(1-\lambda^{t-1})$ is coming from. If that term wasn't there and the numerator in the summation was $j+1$ instead of $j-1$, that would make sense to me because that formula then makes sense. But the denominator confuses me and that could be causing the indexing to be different. If anyone can see where the denominator arises, I'd be interested. – mlofton Sep 25 '20 at 6:58
• compare the formula to the source shown in my answer since it's the original, not oreilly. Besides the question was on the ewma covariance matrix, not univariate ewma which is what the op shows – develarist Sep 25 '20 at 7:51
• Hi: It doesn't really matter what the indices are in the summation as long as there is one $\lambda$ term that is raised to zero and one that is raised to $t$. How you get there is irrelevant. The question for me is the denominator. When I have more time, I'll check out the source you pointed to. But, as I said, how one does the indexing doesn't matter as long as $\lambda$ get raised to the powers from $0$ to $t$. – mlofton Sep 26 '20 at 14:20

Depends what smaller observations would mean. Page 81 of the RiskMetrics 1996 technical document where EWMA was introduced shows an example with only 22 observations for a commonly used $$\lambda$$ setting, so yes it decays fairly fast with lags farther in the past having an almost negligble contributing weight.
• I agree that the EWMA is quite well known as a smoother. But I just wanted to give a heads up to posterity that it shows up in a lot of other frameworks besides smoothing. All of the holt-winters machinery has its origins in exponential smoothing time series models. ( see Rob Hyndman's yellow book ). In Sargent's 1979 text, "Macro-economic Theory", there are a whole bunch of examples of how the ewma can arise as an optimal solution to various state space representations. It can also be viewed as an adaptive expectations model in econometrics where the $\beta$ parameter is set to 1.0. – mlofton Sep 25 '20 at 7:14