Root-N-consistent semiparametric regression.

*(English)*Zbl 0647.62100Summary: One type of semiparametric regression on an \({\mathcal R}\) \(p\times {\mathcal R}\) q-valued random variable (X,Z) is \(\beta 'X+\theta (Z)\), where \(\beta\) and \(\theta\) (Z) are an unknown slope coefficient vector and function, and X is neither wholly dependent on Z nor necessarily independent of it. Estimators of \(\beta\) based on incorrect parameterization of \(\theta\) are generally inconsistent, whereas consistent nonparametric estimators deviate from \(\beta\) by a larger probability order than \(N^{-}\), where N is sample size.

An estimator generalizing the ordinary least squares estimator of \(\beta\) is constructed by inserting nonparametric regression estimators in the nonlinear orthogonal projection on Z. Under regularity conditions \({\hat \beta}\) is shown to be \(N^{1/2}\)-consistent for \(\beta\) and asymptotically normal, and a consistent estimator of its limiting covariance matrix is given, affording statistical inference that is not only asymptotically valid but has nonzero asymptotic first-order efficiency relative to estimators based on a correctly parameterized \(\theta\).

We discuss the identification problem and \({\hat \beta}\)’s efficiency, and report results of a Monte Carlo study of finite-sample performance. While the paper focuses on the simplest interesting setting of multiple regression with independent observations, extensions to other econometric models are described, in particular seemingly unrelated and nonlinear regressions, simultaneous equations, distributed lags, and sample selectivity models.

An estimator generalizing the ordinary least squares estimator of \(\beta\) is constructed by inserting nonparametric regression estimators in the nonlinear orthogonal projection on Z. Under regularity conditions \({\hat \beta}\) is shown to be \(N^{1/2}\)-consistent for \(\beta\) and asymptotically normal, and a consistent estimator of its limiting covariance matrix is given, affording statistical inference that is not only asymptotically valid but has nonzero asymptotic first-order efficiency relative to estimators based on a correctly parameterized \(\theta\).

We discuss the identification problem and \({\hat \beta}\)’s efficiency, and report results of a Monte Carlo study of finite-sample performance. While the paper focuses on the simplest interesting setting of multiple regression with independent observations, extensions to other econometric models are described, in particular seemingly unrelated and nonlinear regressions, simultaneous equations, distributed lags, and sample selectivity models.

##### MSC:

62P20 | Applications of statistics to economics |

62G05 | Nonparametric estimation |

62J02 | General nonlinear regression |

62J05 | Linear regression; mixed models |