6. Single Index Models

A single index model (SIM) summarizes the effects of the explanatory variables $X_{1},\ldots,X_{d}$ 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 $E(Y\vert{\boldsymbol{X}})$ 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

$$E(Y\vert{\boldsymbol{X}})=m({\boldsymbol{X}})=g\left\{ v_{\boldsymbol{\beta}}({\boldsymbol{X}})\right\}$$ (6.1)

is carried out in two steps. First we estimate the coefficients vector ${\boldsymbol{\beta}}$, then using the index values for our observations we estimate $g$ by ordinary univariate nonparametric regression of $Y$ on $v_{\widehat{\boldsymbol{\beta}}}({\boldsymbol{X}})$.

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 $g$ 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.