The APLM can be considered as a modification of the AM by a parametric (linear) part or as a nontrivial extension of the linear model by additive components. The main motivation for this partial linear approach is that explanatory variables are often of mixed discrete-continuous structure. Apart from that, sometimes (economic) theory may guide us how some effects inside the regression have to be modeled. As a positive side effect we will see that parametric terms can be typically estimated more efficiently.
Consider now the APLM from (9.2) in the following way:
As known from Denby (1986) and Speckman (1988),
the parameter
can be estimated at
-rate.
The problematic part is thus the estimation of the additive part,
as the estimate of
lacks precision due to the curse of dimensionality.
When using backfitting for the additive components, the construction
of a smoother matrix is principally possible, however, eluding from
any asymptotic theory.
For this reason we consider a procedure based on
marginal integration, suggested first by
Fan et al. (1998). Their idea
is as follows: Assume
has
realizations being
. We can then calculate
the estimates
for each of the
subsamples
. Now we average over the
to obtain the final estimate
.
Note that this subsampling
affects only the pre-estimation, when
determining
at the points over which we have to integrate. To
estimate
,
we use a local linear expansion in the direction of interest
(cf. (8.16)), and minimize
![]() |
(9.8) |
![]() |
(9.9) |
Under the same assumptions as in Sections 8.2.1 and
8.2.2, complemented by regularity conditions for
and adjustments for the pdfs
, we obtain
the same asymptotic results for
as in Chapter
8. The mentioned adjustments are necessary as we
have to use conditional densities
,
, and
now;
for details we refer to Fan et al. (1998). For example, when
is discrete, we
set
.
Having estimated the nonparametric additive components ,
we turn to the estimation of the parameters
and
.
Note that
is now no longer the unconditional expectation of
,
but covers also the parametric linear part:
How does one to obtain such a -consistent
?
The solution is an ordinary LS regression of the partial
residuals
on
the
, i.e., minimizing
The condition on the bandwidth means that to obtain -consistent
estimator of
, we have to undersmooth
in the nonparametric part. A careful study of the proof reveals a
bias term of order
, which with
is faster
than
. Otherwise
would inherit the
bias from the
s as it is based on a
regression of the partial residuals
.
Unfortunately, in practice this procedure is hardly feasible if
has too many distinct realizations.
For that case, Fan et al. (1998) suggest a
modification that goes back to an idea of
Carroll et al. (1997). Their approach leads to much more
complicated asymptotic expressions, so that we only sketch the
algorithm.
The problem for (quasi-)continuous
is that we cannot longer work on subsamples.
Instead,
we simultaneously estimate the impact of
and
.
More precisely,
is estimated by
minimizing
![]() |
![]() |
The parametric coefficients
are estimated as
(female has children) and
(unemployment rate).
Figures 9.1 and 9.2
show the estimated curves
. The
approximate confidence intervals are constructed as
times
the (estimated) pointwise standard deviation of the curve
estimates. The displayed point clouds are the logarithms of the
distance to the partial residuals, i.e.
.
The plots show clear nonlinearities, a fact that has been confirmed in Härdle et al. (2001) when testing these additive functions for linearity. If the model is chosen correctly, the results quantify the extent to which each variable affects the female labor supply. For a comparison with a parametric model we show in Table 9.1 the results of a parametric least squares analysis.
Source | Sum of Squares | df | Mean square | ![]() |
Regression | 6526.3 | 10 | 652.6 | 9.24 |
Residual | 42101.1 | 596 | 70.6 | |
![]() |
![]() |
Variable | Coefficients | S.E. | ![]() |
![]() |
constant | 1.36 | 8.95 | 0.15 | 0.8797 |
CHILDREN | -2.63 | 1.09 | -2.41 | 0.0163 |
UNEMPLOYMENT | 0.48 | 0.22 | 2.13 | 0.0333 |
AGE | 1.63 | 0.43 | 3.75 | 0.0002 |
AGE![]() |
-0.021 | 0.0054 | -3.82 | 0.0001 |
WAGES | -1.07 | 0.18 | -6.11 | ![]() |
WAGES![]() |
0.017 | 0.0033 | 4.96 | ![]() |
PRESTIGE | 0.13 | 0.034 | 3.69 | 0.0002 |
EDUCATION | 0.66 | 0.19 | 3.58 | 0.0004 |
HOUSING | 0.0018 | 0.0012 | 1.56 | 0.1198 |
NETINCOME | -0.0016 | 0.0003 | -4.75 | ![]() |
As can be seen from the table, (squared AGE) and
(squared WAGE) have been added, and indeed their
presence is highly significant. Their introduction was motivated
by the nonparametric estimates
,
.
Clearly, the piecewise linear shapes of
and
are harder to model in a parametric model. Here, at least the
signs of the estimated parameters agree with the slopes of the
nonparametric estimate in the upper part. Both factors are highly
significant but the nonparametric analysis suggests that both
factors are less influential for young women and the low income
group.