Additive models have been proven to be very useful as they
naturally generalize the linear regression model and allow for an
interpretation of marginal changes, i.e. for the effect of one variable
on the mean function
when holding all others constant. This kind of
model structure is widely used in both theoretical economics and
in econometric data analysis. The standard text of
Deaton & Muellbauer (1980) provides many examples in microeconomics
for which the additive structure provides interpretability. In
econometrics, additive structures have a desirable statistical
form and yield many well known economic results. For instance,
an additive structure allows us to aggregate inputs into
indices; elasticities or rates of substitutions
can be derived directly. The separability of the input
parameters is consistent with decentralization
in decision making or optimization by stages.
In summary, additive models can easily be interpreted.
Additive models are also interesting from a statistical point of
view. They allow for a componentwise analysis and combine flexible
nonparametric modeling of multidimensional inputs with a statistical
precision that is typical of a one-dimensional explanatory variable.
Let be the dependent variable and
the
-dimensional vector of explanatory variables.
Consider the estimation of the general nonparametric regression
function
.
Stone (1985) showed that the optimal convergence rate for
estimating
is
with
an
index of smoothness of
. Note how high values of
lead to a
slow rate of convergence. An additive structure for
is a
regression function of the form
Additive models of the form (8.1) were first considered in the context of input-output analysis by Leontief (1947a) who called them additive separable models. In the statistical literature, additive regression models were introduced in the early eighties, and promoted largely by the work of Buja et al. (1989) and Hastie & Tibshirani (1990). They proposed the iterative backfitting procedure to estimate the additive components. This method is presented in detail in Section 8.1.
More recently, Tjøstheim & Auestad (1994b) and
Linton & Nielsen (1995) introduced a non-iterative method for
estimating marginal effects. Note that marginal
effects coincide with the
additive components
if the true regression function
is indeed additive. The idea of this method is to
first estimate a multidimensional functional of
and then use
marginal integration to obtain the marginal effects. Under
additive structure the marginal integration estimator yields the functions
up to a constant. This estimator will be introduced
starting from Section 8.2.
A comparison of backfitting and marginal integration is given
in Section 8.3.