Given the computational expense of estimating single index models, it is
desirable to know whether the distributional flexibility of these models
justifies the extra computational cost. This implies that the
performance of a specified parametric model must be compared
with that of an estimated single index model as given in (5.9).
To this end, Horowitz & Härdle (1994) designed a test
that considers the following hypotheses:
The main idea that inspires the test relies on the fact
that if the model under the null is true then a nonparametric
estimation of
gives a correct estimate of
. Thus, the specification of the
parametric model can be tested by comparing the
nonparametric estimate of
with the
parametric fit using the known link
.
The test statistic is defined as
Let us take a closer look at the intuition behind this test statistic.
The first
difference term in the sum measures the deviation of the estimated regression
from the true realization, that is it measures
.
If
holds, then this measure ought to be very small
on average. If, however, the parametric model under
the null fails to replicate the observed values
well, then
will increase. Obviously, we reject the hypothesis
that the data were generated by a parametric model if
becomes unplausibly large.
The second difference term measures the distance
between the regression values obtained under the null and
under the semiparametric alternative. Suppose the parametric
model captures the characteristics of the
data well so that
is small. Then even if
the semiparametric link deviates considerably from the parametric alternative
on average,
these deviations will be downweighted by the first difference term. Seen
differently, the small residuals of the parametric fit are blown up by
large differences in the parametric and semiparametric fits,
.
Thus, if
is true, the residuals should be small enough to accommodate
possible strong differences in the alternative fits.
Again, a small statistic will lead to maintaining the null hypothesis.
It can be shown that under and under some suitable
regularity conditions
is asymptotically
distributed as a
where
denotes the asymptotic sampling variance
of the statistic.
For related presentations of the topic we refer to Horowitz (1993), Horowitz (1998b) and Pagan & Ullah (1999, Chapter 7).
There is a large amount of literature that investigates the efficiency bound for estimators in semiparametric models. Let us mention Begun et al. (1983) and Cosslett (1987) as being two of the first references. Newey (1990) and Newey (1994) are more recent articles. The latter treats the variance in a very general and abstract way. A comprehensive resource for efficient estimation in semiparametric models is Bickel et al. (1993).
The idea of using parametric objective functions and substituting unknown components by nonparametric estimates has first been proposed in Cosslett (1983). The maximum score estimator of Manski (1985) and the maximum rank correlation estimator from Han (1987) are of the same type. Their resulting estimates are still very close to the parametric estimates. For that reason the SLS method of Ichimura (1993) may outperform them when the parametric model is misspecified.
The pseudo maximum likelihood version of the SLS was found independently by Weisberg & Welsh (1994). They present it as a straightforward generalization of the GLM algorithm and discuss numerical details. A different idea for adding nonparametric components to the the maximum likelihood function is given by Gallant & Nychka (1987). They use a Hermite series to expand the densities in the objective function.
The different methods presented in this chapter have been compared in a simulation study by Bonneu et al. (1993). They also include a study of Bonneu & Delecroix (1992) for a slightly modified pseudo likelihood estimator.
An alternative ADE method (without weight function) was proposed by Härdle & Stoker (1989). This estimator shares the asymptotic properties of the weighted ADE, but requires for practical computation a trimming factor to guarantee that the estimated density is bounded away from zero.