A single index model (SIM) summarizes the effects of the
explanatory variables
within
a single variable called the
index.
As stated at the
beginning of Part
,
the SIM is one possibility for generalizing the GLM or for
restricting the multidimensional regression
to overcome
the curse
of dimensionality and the lack of interpretability.
For more examples of motivating the SIM see Ichimura (1993). Among
others, this reference mentions duration, truncated regression (Tobit)
and errors-in-variables modeling.
As already indicated, the estimation of a single index model
Before we proceed to the estimation problem we first have to clarify identification of the model. Next we will turn your attention to estimation methods, introducing iterative approaches as semiparametric least squares and pseudo maximum likelihood and a the non-iterative technique based on (weighted) average derivative estimation. Whereas the former methods work for both discrete and continuous explanatory variables, the latter needs to be modified in presence of discrete variables.
The possibility of estimating the link function
nonparametrically suggests using the SIM for a model
check. Thus, at the end of this chapter we will also present a test to
compare parametric models against semiparametric alternatives
based on the verification of the link specification.