ARCH to EWMA

[arteja]

HI David,

I dont exactly understand how you say ARCH is a generalised form of EWMA. Assuming the w is 0, we basically have ARCH as a weighted sum of the spreads (or whatever those factors are). If the number of terms is infinite, and the weights are tending to zero, how does it stll go to the EWMA form? For the EWMA form the two weights have to add to 1, and I cant see how that is achieved by ARCH with the said conditions.

I missed EB#2, and that isnt in the member's section for download.

thanks,
Ravi

 

[David Harper CFA FRM]

Hi Ravi

(We don't have a recording of the EB #2. It was only our 2nd live and we messed up the live)

Agreed: if w = 0, we have a weighted sum of squared returns (if the weights are equal, then it's the plain old STDDEV or Moving Average)

The EWMA is an infinite series of weighted squared returns; e.g., if lambda = .94, then most recent weight (T-1) is 6%, then weight before that (T-2) is 6%*94%...each prior day's weight is lamba% of the nearer day's weight...and so on, until at some point it doesn't matter but technically infinite so you can see it's a form of the ARCH...

linda allen shows how this series convenienty reduces to the recursive form that you mention and that we work with (and that you are likely to be tested on. The test will either be: i. apply the recursive form, or ii. possible, give the weight applied T- n days priod). Our GARCH, too, in the three term format we are accustomed, itself is "an elegant reduction" of an infinite (exponentailly declining) series.

Thanks, David

 

[SURAJM]

I thought if you put alpha (the weights in ARCH) = (1 -lambda)* [ lambda ^ (n-1)] / (1 - lambda^n) it would turn to EWMA ??

 

[David Harper CFA FRM]

Surajm,

You are exactly correct, but now we are talking about 3 versions of EWMA
(all of which special cases of ARCH, but this is not particularly important for our purposes IMO)

The recursive EWMA -- i.e., =lambda*lag variance + (1-lambda)*lag return^2 --- is the solution to the infinite series where the weights decline by lambda.

Yours Surajm grosses them up a bit for a *finite* series, so that the finite series sums to 1.0. Yours is the "practical approximation" but technically is not exactly the same as the recursive.

you can see in action @ http://www.bionicturtle.com/premium/spreadsheet/2.b.5_volatilities_compared/

my sample here is ridiculously short n = 11. Finite series won't sum to 1.0. See bottom (row 40) for adjusted weights (these produce same as your formula Surajm). So this will be near to the recursive but mabye not exactly

For a realistic sample (e.g., n = 250 or 500) and a typical lamba (e.g., 0.90+) there will be no observable difference anyway between the two (infinite recursive of finite adjusted)

David