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Unformatted text preview: Estimation for Box-Cox Transformation Model With Nonparametric restriction on the error term ∗ Yahong Zhou School of Economics, Shanghai University of Finance and Economics, Shanghai,200433 Abstract As well known, Box-Cox transformation model has been widely used in applied econometrics and statistics. Typically, under the restriction that the error term with normal distribution, estimation and inference procedures for the regression coefficients and transformation parameter under this model setting have been studied extensively. In this paper, we propose a simple semi-parametric estimation method for the Box-Cox transformation model with no specific paprametric assumption on the distribution of the error term. The proposed estimator is consistent and asymptotically normally distributed. Its covariance matrix can be in closed form which can be easily estimated. A small Monte Carlo experiment is done, which demonstrates good performance of our estimator. Keywords: Box-Cox transformation model, Semiparametric estimation, Rank condition, Smoothed kernel. AMS Subject Classification: 62G05, 62G20 . § 1 Introduction In this paper, we consider the estimation of the Box-Cox regression model with a linear structure of the form g ( α ,Y i ) = X ′ i β + u i (1.1) where g ( α,y ) = braceleftBigg y α − 1 α if α negationslash = 0 ln y if α = 0 for y > 0, α is the transformation coefficient, β is the vector of slope coefficients which correspond to the regressors, and u is the error term. When α = 1 , the left-hand side of (1.1) becomes the linear form, and the model reduces to the linear model; meanwhile if α = 0 , the left-hand side is ln Y i , it degenerates to be the log linear model. In (1.1), the conditional mean of the error term u should be assumed to be zero, otherwise the intercept in this model can not be identified, but its distribution remains unknown. * This paper is a revised version of one chapter of the author’s ph.D. dissertation at the Department of Economics of Hong Kong University of Science and Technology. Thanks to Prof. Songnian Chen for his guidance for this paper as well as seminar participants at HKUST and SHUFE for their valuable comments. And financial support is from research grants no.211-3-50 and no.211-3-70.. 1 For the estimation of the Box-Cox regression model, traditionally, the most common approach is us- ing the maximum likelihood method under the assumption that the error term u is normally distributed. However, as is well known, the normality assumption is not compatible with the transformation model. Furthermore, the error distribution is typically unknown and economic theory provides little guidance on this, any misspecification of the parametric distribution of the error term could lead inconsistency of the es- timates. Instead of making parametric specification for the error distribution, Amemiya and Powell (1981) proposed a nonlinear two-stage least squares (NL2SLS) estimator based on certain moment conditions.proposed a nonlinear two-stage least squares (NL2SLS) estimator based on certain moment conditions....
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