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?
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1Perhaps 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
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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
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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 $W\propto \ \Omega ^{{-1}}$.
Michael L.
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