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A Smoothing-Splines-Like Estimator for Nonparametric Regression With a Linear Generalized Least Squares Interpretation
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In this paper a new estimator for nonparametric regression is suggested. It is a smoothing-splines-like estimator obtained by minimizing a two term criterion function. The first term of this criterion function is a sum of squares of residuals and the second term is a penalty function defined as a weighted sum of squared errors of local first order Taylor approximations. Under usual regularity conditions, almost sure uniform consistency is proved for both the function and its first derivatives. Instead of solving the optimization problem to explicitly obtain an element of the function space considered, a procedure is suggested to only estimate the values of the function and the first derivatives at the observation points. This is especially convenient in order to simplify the calculations and to keep the simplicity of the economic interpretation of the usual linear econometric specifications. This practical procedure also allows another interesting interpretation of the estimators as the generalized least squares estimators of a two regime expanded linear regression with a singular two components error covariance structure. The link with conventional linear regression theory is developed by showing that the estimators are unbiased if the true function is linear. Some Monte Carlo experiments are reported and the relative performance of the estimators is discussed. Finally, an application is presented, using aggregate data from OECD on primary energy requirements per unit of GDP and GDP per capita.
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Pinheiro, Maximiano Reis, A Smoothing-Splines-Like Estimator for Nonparametric Regression With a Linear Generalized Least Squares Interpretation (May, 1991). FEUNL Working Paper Series No. 170