6.1 Estimating GPLMs
As mentioned above, a GPLM has the form
where
denotes the expected value of the dependent variable
given
,
which are vectors of explanatory variables.
The index
is linked to the dependent variable
via a known function
which is called the link function
in analogy to generalized linear models (GLM).
The parameter vector
and the function
need to be estimated.
Typically, generalized partial linear models are considered for
from an exponential family. We therefore assume for the variance
, i.e. a
dependence on the index
and on a dispersion parameter
.
6.1.1 Models
It is easy to see that GPLM covers a range of semiparametric models,
as for example:
- Partial linear regression
The model
with
implies
and
.
This gives a GPLM with identity link function
and variance function
.
- Generalized additive model (GAM) with
linear and nonparametric component
This is commonly written as
where
is assumed. By defining
we arrive at the above GPLM.
6.1.2 Semiparametric Likelihood
The estimation methods for the GPLM are based
on the idea that an estimate
can be found
for known
, and an estimate
can be
found for known
. The
gplm
quantlib implements
profile likelihood estimation and backfitting.
Details on the estimation procedure can be found in
Hastie and Tibshirani (1990),
Severini and Staniswalis (1994),
Härdle, Mammen, and Müller (1998),
Müller (1997).
The default numerical algorithm for likelihood maximization
is the Newton-Raphson iteration. Optionally, a Fisher scoring
can be chosen.
6.1.2.0.1 Profile Likelihood
Denote by
the individual log-likelihood or
(if the distribution of
does not belong to an exponential
family) quasi-likelihood function
The profile likelihood method considered in
Severini and Wong (1992) and Severini and Staniswalis (1994)
is based on the fact, that the conditional distribution
of
given
and
is parametric.
The essential method for estimation is to fix the
parameter
and to estimate the least favorable
nonparametric function in dependence of this
fixed
.
The resulting estimate for
is then used
to construct the profile likelihood
for
.
Suppose, we have observations
,
.
Denote the individual
log- or quasi-likelihood in
by
In the following,
and
denote the
derivatives of
with respect to
.
Abbreviate now
and
define
the smoother matrix with elements
 |
(6.1) |
and let
be the design matrix with rows
.
Denote further by
the identity matrix,
by
the vector and by
the diagonal matrix containing the first (
) and
second (
) derivatives of
,
respectively.
The Newton-Raphson estimation algorithm (see Severini and Staniswalis; 1994)
is then as follows.
The variable
is a sort of adjusted dependent
variable. From the formula for
it becomes clear,
that the parametric
part of the model is updated by a parametric method (with a
nonparametrically modified design matrix
).
Alternatively, the functions
can be
replaced by their expectations (w.r.t. to
) to obtain
a Fisher scoring type procedure.
6.1.2.0.2 Generalized Speckman Estimator
The profile likelihood estimator is particularly easy to derive
in case of a model with identity link and normally distributed
. Here,
and
. The latter yields the smoother matrix
with elements
 |
(6.2) |
Moreover, the update for
simplifies to
using the vector notation
,
.
The parametric component is determined by
with
and
.
These estimators for the partial linear model were
proposed by Speckman (1988).
Recall that each iteration step of a GLM is a
weighted least squares regression on an adjusted dependent variable
(McCullagh and Nelder; 1989).
Hence, in the partial linear model the weighted least squares
regression could be replaced by an partial linear fit on the
adjusted dependent variable
 |
(6.3) |
Again, denote
a vector and
a diagonal matrix containing the
first (
) and second (
)
derivatives of
,
respectively. Then, the Newton-Raphson type Speckman
estimator (see Müller; 1997)
for the GPLM can be written as:
The basic simplification of this approach consists in using the
smoothing matrix
with elements
 |
(6.4) |
instead of the matrix
from (6.1).
As before, a Fisher scoring type procedure is obtained by replacing
by their expectations.
6.1.2.0.3 Backfitting
The backfitting method was suggested as an iterative algorithm
to fit an additive model (Hastie and Tibshirani; 1990). The key idea is to
regress the additive components separately on partial residuals.
The ordinary partial linear model (with identity link function)
is a special case, consisting of only two additive functions.
Denote
the projection matrix
and
a smoother matrix.
Abbreviate
.
Then backfitting means to solve
For a GPLM, backfitting means now to perform an additive fit on
the adjusted dependent variable
which was defined in
(6.3), see Hastie and Tibshirani (1990).
We use again the kernel smoother matrix
from (6.4).
As for profile likelihood and Speckman estimation, we obtain a
Newton-Raphson or Fisher scoring type algorithm by using
or
, respectively.