# Using variance-ratios as moment conditions in a GMM estimation

I have a few time-series, say ${X, Y, Z}$ at a given frequency, say, yearly.

Now I compute the yearly growth rates (ie log-differences):

$dx = log(X_{t}/X_{t-1}), dy = ..., dz = ...$

and I compute the variances $vx = var(dx), vy = var(dy), vz = var(dz)$.

Assume I re-do the same at various frequencies $n = 2, 3, 4, 5,...$:

$(d_n)x =log(X_{t}/X_{t-n})$

and I compute the variances of those $n$-years growth rates.

Then I compute the variance-ratios: $vr(n) = var((d_n)x) / (n var(dx) )$.

Assume that I have a model that provides these variance ratios: $vr(n, theta)$ for some parameter vector $\theta$.

How do I perform a GMM estimation of $\theta$?

I ask this because $dx$ and $(d_n)x$ are different time-series even if the latter is a transformation of the former.