Optimal instruments

A common statistical and econometric problem deals with conditional moment models, that satisfy regression relationships of the form ${\displaystyle E[Y\mid X]=\mu (X,\beta _{0})}$ where ${\displaystyle E[\varepsilon \mid X]=0}$. [How can you say "where" followed by an expression involving epsilon when the foregoing statement does not mention epsilon?] No additional constraints are imposed on the class of probability distributions that have generated the data. However, this means that we can use infinitely many moments. More specifically, the moments ${\displaystyle E[\varepsilon ^{*}X]=0}$, ${\displaystyle E[\varepsilon ^{*}X^{2}]=0}$, ${\displaystyle E[\varepsilon ^{*}X^{3}]=0,\ldots }$ are all consistent with the conditional moment restriction. A natural question to ask then, is whether an optimal set of instruments is available. Both econometricians[1][2] and statisticians[3] have extensively studied this subject.

Generally, it is assumed that ${\displaystyle (Y_{1},X_{1}),(Y_{2},X_{2}),\ldots ,(Y_{n},X_{n})}$ are iid. This model is pinned down by a finite-dimensional object ${\displaystyle \beta _{0}}$ and an infinite-dimensional object η(·) that characterizes the joint distribution of ${\displaystyle \varepsilon }$ and X. This infinite-dimensional object is often referred to as an infinite-dimensional nuisance parameter.[4] A popular choice for the conditional moment regression model is:

${\displaystyle E[YVX]=X\mathrm {B} _{0}}$ (1)

The mean zero assumption ${\displaystyle E[\varepsilon \mid X]=0}$ implies however, that we can estimate these models using an infinite set of unconditional moment restrictions, as ${\displaystyle E[\varepsilon \mid X]=0}$ implies ${\displaystyle E[h(X)(Y-X\beta _{0})]=0}$ for any function ${\displaystyle h(X)}$, reflecting the infinite-dimensional nature of the object η(·).[4] More specifically, this means that:

${\displaystyle E[X^{j}(Y-X\beta _{0})]=0}$ for all ${\displaystyle j=0,1,2,3,4,\ldots }$ (2)

An important question econometricians and statisticians therefore have focused on, is whether there is a finite set of optimal orthogonality conditions that provides asymptotically efficient estimators. Efficiency here means that adding additional moment conditions does not reduce the asymptotic variance.[5]

The answer to this question is generally that this finite set exists and have been proven for a wide range of estimators. Amemiya was one of the first to work on this problem and show the optimal number of instruments for nonlinear simultaneous equation models with homoskedastic and serially uncorrelated errors.[6] The form of the optimal instruments was characterized by Hansen,[7] and results for nonparametric estimation of optimal instruments are provided by Newey.[8] A result for nearest neighbor estimators was provided by Robinson.[9]