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Shrinkage for Categorical Regressors

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Document pages: 46 pages

Abstract: This paper introduces a flexible regularization approach that reduces pointestimation risk of group means stemming from e.g. categorical regressors,(quasi-)experimental data or panel data models. The loss function is penalizedby adding weighted squared l2-norm differences between group locationparameters and informative first-stage estimates. Under quadratic loss, thepenalized estimation problem has a simple interpretable closed-form solutionthat nests methods established in the literature on ridge regression,discretized support smoothing kernels and model averaging methods. We deriverisk-optimal penalty parameters and propose a plug-in approach for estimation.The large sample properties are analyzed in an asymptotic local to zeroframework by introducing a class of sequences for close and distant systems oflocations that is sufficient for describing a large range of data generatingprocesses. We provide the asymptotic distributions of the shrinkage estimatorsunder different penalization schemes. The proposed plug-in estimator uniformlydominates the ordinary least squares in terms of asymptotic risk if the numberof groups is larger than three. Monte Carlo simulations reveal robustimprovements over standard methods in finite samples. Real data examples ofestimating time trends in a panel and a difference-in-differences studyillustrate potential applications.

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