Exponential weighted moving skewness/kurtosis

There are well-known on-line formulas for computing exponentially weighted moving averages and standard deviations of a process (xn)n=0,1,2,. For the mean,

μn=(1α)μn1+αxn

and for the variance

σ2n=(1α)σ2n1+α(xnμn1)(xnμn)

from which you can compute the standard deviation.

Are there similar formulas for on-line computation of exponential weighted third- and fourth-central moments? My intuition is that they should take the form

M3,n=(1α)M3,n1+αf(xn,μn,μn1,Sn,Sn1)

and

M4,n=(1α)M4,n1+αf(xn,μn,μn1,Sn,Sn1,M3,n,M3,n1)

from which you could compute the skewness γn=M3,n/σ3n and the kurtosis kn=M4,n/σ4n but I’ve not been able to find simple, closed-form expression for the functions f and g.


Edit: Some more information. The updating formula for moving variance is a special case of the formula for the exponential weighted moving covariance, which can be computed via

Cn(x,y)=(1α)Cn1(x,y)+α(xnˉxn)(ynˉyn1)

where ˉxn and ˉyn are the exponential moving means of x and y. The asymmetry between x and y is illusory, and disappears when you notice that yˉyn=(1α)(yˉyn1).

Formulas like this can be computed by writing the central moment as an expectation En(), where weights in the expectation are understood to be exponential, and using the fact that for any function f(x) we have

En(f(x))=αf(xn)+(1α)En1(f(x))

It’s easy to derive the updating formulas for the mean and variance using this relation, but it’s proving to be more tricky for the third and fourth central moments.

Answer

The formulas are straightforward but they are not as simple as intimated in the question.

Let Y be the previous EWMA and let X=xn, which is presumed independent of Y. By definition, the new weighted average is Z=αX+(1α)Y for a constant value α. For notational convenience, set β=1α. Let F denote the CDF of a random variable and ϕ denote its moment generating function, so that

ϕX(t)=EF[exp(tX)]=Rexp(tx)dFX(x).

With Kendall and Stuart, let μk(Z) denote the non-central moment of order k for the random variable Z; that is, μk(Z)=E[Zk]. The skewness and kurtosis are expressible in terms of the μk for k=1,2,3,4; for example, the skewness is defined as μ3/μ3/22 where

μ3=μ33μ2μ1+2μ13 and μ2=μ2μ12

are the third and second central moments, respectively.

By standard elementary results,

1+μ1(Z)t+12!μ2(Z)t2+13!μ3(Z)t3+14!μ4(Z)t4+O(t5)=ϕZ(t)=ϕαX(t)ϕβY(t)=ϕX(αt)ϕY(βt)=(1+μ1(X)αt+12!μ2(X)α2t2+)(1+μ1(Y)βt+12!μ2(Y)β2t2+).

To obtain the desired non-central moments, multiply the latter power series through fourth order in t and equate the result term-by-term with the terms in ϕZ(t).

Attribution
Source : Link , Question Author : Chris Taylor , Answer Author : whuber

Leave a Comment