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Semiparametric least squares estimation of multiple index models: Single equation estimation. (English) Zbl 0766.62065
Nonparametric and semiparametric methods in econometrics and statistics, Proc. 5th Int. Symp., Econ. Theory Econ., Durham/NC (USA) 1988, 3-49 (1991).
[For the entire collection see Zbl 0745.00029.]
The general framework of this contribution is the single equation multiple index model defined as follows:
1. $$y_ i=x_{0i}\alpha(\theta_ 0)+\varphi(x_{1i}\beta_ 1(\theta_ 0),\dots,x_{Mi}\beta_ M (\theta_ 0))+\varepsilon_ i$$, where $$i=1,\dots,n$$.
2. $$E(\varepsilon_ i\mid x_ i)=0$$, 3. $$\varphi$$ is not known, but $$M$$ is,
4. $$\alpha(\theta)$$, $$\beta_ 1(\theta),\dots,\beta_ M(\theta)$$ are all known functions of a basic parameter vector $$\theta$$.
Many econometric models are special cases of this model, among them simultaneous tobit models and disequilibrium models. Other models may be transformed to this form.
It is shown that all models of this form can be estimated by the semiparametric least squares method if identification conditions are met. Conditions giving consistency and asymptotic normality of estimators of the basic parameter vector are provided, as well as a consistent estimator of the asymptotic variance-covariance matrix. The extension to systems of equations and the possibility of cross equation parameter restrictions and exploitation of the covariance structure are left for future study.

##### MSC:
 62P20 Applications of statistics to economics 62G20 Asymptotic properties of nonparametric inference 62G99 Nonparametric inference