^^trs^ ^ ^^orr^^ Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/consistencyasympOOnewe WS^W: Consistency and Asymptotic Normality of Nonparametric Projection Estimators Whitney K. Newey No. 584 Rev. July 1991 massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 -*-; ^ VO' ;?jf?K ; ''' U;>'<# • Consistency and Asymptotic Normality of Nonparametric Projection Estimators Whitney K. Newey No. 584 Rev. July 1991 M.J.T. LIBRARIES NOV 1 4 1991 RECEiVtu MIT Working Paper 584 Consistency and Asymptotic Normality of Nonparametric Projection Estimators Whitney K. Newey MIT Department of Economics March, 1991 Revised, July, 1991 Helpful comments were provided by Andreas Buja and financial support by the NSF and the Sloan Foundation. Abstract Least squares projections are a useful way of describing the relationship between random variables. These include conditional expectations and projections on additive functions. Sample least squares provides a convenient way of estimating such projections. This paper gives convergence rates and asymptotic normality results of least squares estimators of linear functionals of projections. General results are derived, and primitive regularity conditions given for power series and splines. Also, it is shown that mean-square continuity of a linear functional is necessary for v^-consistency and sufficient under conditions for asymptotic normality, and this result is applied to estimating the parameters of a finite dimensional component of a projection and to weighted average derivatives of projections. Keywords: Nonparametric regression, additive interactive models, partially linear models, average derivatives, polynomials, splines, convergence rates, asymptotic normality.