# Efficiency Bounds for Semiparametric Regression

https://doi.org/0012-9682(199205)60:3<567:EBFSR>2.0.CO;2-2
p. 567-596

Gary Chamberlain

The paper derives efficiency bounds for conditional moment restrictions with a nonparametric component. The data form a random sample from a distribution $F$. There is a given function of the data and a parameter. The restriction is that a conditional expectation of this function is zero at some point in the parameter space. The parameter has two parts: a finite-dimensional component $\theta$ and a general function $h$, which is evaluated at a subset of the conditioning variables. An example is a regression function that is additive in parametric and nonparametric components, as arises in sample selection models. If $F$ is assumed to be multinomial with known (finite) support, then the problem becomes parametric, with the values of $h$ at the mass points forming part of the parameter vector. Then an efficiency bound for $\theta$ and for linear functionals of $h$ can be obtained from the Fisher information matrix, as in Chamberlain (1987). The bound depends only upon certain conditional moments and not upon the support of the distribution. A general $F$ satisfying the restrictions can be approximated by a multinomial distribution satisfying the restrictions, and so the explicit form of the multinomial bound applies in general. The efficiency bound for $\theta$ is extended to a more general model in which different components of $h$ may depend on different known functions of the variables. Although an explicit form for the bound is no longer available, a variational characterization of the bound is provided. The efficiency bound is applied to a random coefficients model for panel data, in which the conditional expectation of the random coefficients given covariates plays the role of the function $h$. An instrumental-variables estimator is set up within a finite-dimensional, method-of-moments framework. The bound provides guidance on the choice of instrumental variables.