Transformation Models in High Dimensions

Abstract

Transformation models are a very important tool for applied statisticians and econometricians. In many applications, the dependent variable is transformed so that homogeneity or normal distribution of the error holds. In this article, we analyze transformation models in a high-dimensional setting, where the set of potential covariates is large. We propose an estimator for the transformation parameter and we show that it is asymptotically normally distributed using an orthogonalized moment condition where the nuisance functions depend on the target parameter. In a simulation study, we show that the proposed estimator works well in small samples. A common practice in labor economics is to transform wage with the log-function. In this study, we test if this transformation holds in American Community Survey (ACS) data from the United States.

Keywords:

• High-dimensional setting
• Lasso
• Post-selection inference
• Transformation models

Supplement Materials

The supplementary material includes additional technical material. The proofs are provided in Appendix A. In Appendix B, conditions for the uniform convergence rates of the lasso estimator are presented. Appendix C provides a theoretical result about inference on a target parameter in general Z-estimation problems with dependent and high-dimensional nuisance functions. Finally, simulation results and additional tables and figures are provided in Appendix D–F.

Acknowledgments

We thank the editor Prof. Fan, the associate editor, and two referees for very valuable comments. We also thank Victor Chernozhukov for insightful discussions.

Additional information

Funding

We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) — Project Number 43170191.