A. Colin Cameron 743that they be fully efficient in that they attain semiparametric efficiency bounds (see
Chamberlain, 1987; Newey, 1990; Severini and Tripathi, 2001) that are extensions
of Cramer–Rao lower bounds or the Gauss–Markov theorem to semiparametric
models.
There are many semiparametric models, and for each model there can be several
different ways to obtain estimators. Partial linear and single index models are two
leading examples that are also the building blocks for more general models.
The partial linear model specifies the conditional mean to be the usual linear
regression function plus an unspecified nonlinear component, so:
E[yi|xi,zi]=xi′β+λ(zi), (14.17)where the scalar functionλ(·)is unspecified. An example is the estimation of a
demand function for electricity, wherezreflects time of day or weather indicators
such as temperature. A second example is a sample selection model whereλ(z)is
the expected value of a model error, conditional on the sample selection rule. In
applications interest may lie inβ,λ(z)or both. Various estimators for the partial
linear model have been proposed. The differencing method proposed by Robinson
(1988) estimatesβby OLS regression of(yi−m̂yi)on(xi−m̂xi), wherem̂yiand
m̂xiare predictions from nonparametric regression of, respectively,yandxonz.
Robinson used kernel estimates that may need to be oversmoothed. Other methods
that additionally estimateλ(z), at least for scalarz, include a generalization of the
cubic smoothing spline estimator and using a series approximation forλ(z).
The single index model specifies the conditional mean to be an unknown scalar
function of a linear combination of the regressors, with:
E[yi|xi]=g(x′iβ), (14.18)where the scalar functiong(·)is unspecified and the parametersβare then only
identified up to location and scale. An example is a binary choice model with Pr[y=
1 |x]=g(x′β)whereg(·)is unknown. The single index formulation is attractive as
the marginal effect of a change in thejth regressor isg′(x′iβ)βj, so that the ratio of
parameter estimates equals the ratio of marginal effects. Estimators for the single
index model include an average derivative estimator, a density weighted average
derivative estimator (see Powell, Stock and Stoker, 1989), and semiparametric least
squares.
Microeconometricians have focused on semiparametric estimation for limited
dependent variable models – binary choice with an unspecified function for the
probabilities, censored regression and sample selection. Nonparametric and semi-
parametric methods are also used in the treatment effects literature detailed in
section 14.5.1. The literature is vast. References include the applied study by Belle-
mare, Melenberg and van Soest (2002), and books by Pagan and Ullah (1999) and
Li and Racine (2007). The latter book is accompanied by many routines in R for
nonparametric and semiparametric regression.