# Selection of weighting matrix in GMM estimation

As stated in the title. Are there any assumptions or restrictions behind in selection of weighting matrix in doing the estimation? Does it exist a form which is suitable in most cases?

• Perhaps it would be better if you specified some alternative weighting matrices which you are considering to give people something concrete to work with. – mdewey Nov 22 '16 at 14:55
• Given the existence of an upvoted answer, I don't think this is too unclear to be answerable. – gung - Reinstate Monica Nov 22 '16 at 15:26

in general you do not need much assumptions in order to define a proper weighting matrix. The weighting matrix $W$ must be positive (semi)definite as a minimum condition. However, the power of GMM comes from its efficiency. But first things first. If you choose some weighting matrix $W$ such that $${\hat {W}}_{T}{\xrightarrow {p}}W\text{ } (1),$$ where W is positive semi-definite and, $${\displaystyle \,W\operatorname {E} [\,g(Y_{t},\theta )\,]=0}\text{ } (2)$$ iff $\theta=\theta_0$, then the GMME (GMM estimator that is the solution to $\operatorname {E}[\,g(Y_{t},\theta _{0})\,]=0$) is consistent (among other asusmptions not related to $W$).
But the important thing is the efficiency. It can be shown, that the GMME is asymptotically normal and efficient if $W\propto \ \Omega ^{{-1}}$, where $\Omega =\operatorname {E}[g(Y_{t} ,\theta _{0})g(Y_{t},\theta _{0})^{\mathsf {T}}]$. Hence, you can choose any $W$ that fulfills (1) and (2) if you are interested in consistency, but for asymptotic efficiency you need to use $W\propto \ \Omega ^{{-1}}$.