We examine the second order properties of various quantities of interest in the partially linear regression model. We obtain a stochastic expansion with remainder op(n -2/4), where ,u < 1/2, for the standardized semiparametric least squares estimator,a standard error estimator, and a studentized statistic. We use the second order expansions to correct the standard error estimates for second order effects, and to define a method of bandwidth choice. A Monte Carlo experiment provides favorable evidence on our method of bandwidth choice
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