Moment estimation for censored quantile regression

Abstract

In influential articles Powell (Journal of Econometrics 25(3):303–325, 1984; Journal of Econometrics 32(1):143–155, 1986) proposed optimization-based censored least absolute deviations estimator (CLAD) and general censored quantile regression estimator (CQR). It has been recognized, however, that this optimization-based estimator may perform poorly in finite samples (e.g., Khan and Powell, Journal of Econometrics 103(1–2):73–110, 2001; Fitzenberger, Handbook of Statistics. Elsevier, 1996; Fitzenberger and Winker, Computational Statistics & Data Analysis 52(1):88–108, 2007; Koenker, Journal of Statistical Software 27(6), 2008). In this paper we propose a moment-based censored quantile regression estimator (MCQR). While both the CQR and MCQR estimators have the same large sample properties, our simulation results suggest certain advantage of our moment-based estimator (MCQR). In addition, the empirical likelihood methods for the uncensored model (e.g., Whang 2006; Otsu, Journal of Econometrics 142(1):508–538, 2008) can readily be adapted to the censored model within our method of moment estimation framework. When both censoring and endogeneity are present, we develop an instrumental variable censored quantile regression estimator (IVCQR) by combining the insights of Chernozhukov and Hansen’s (Journal of Econometrics 132(2):491–525, 2006; Journal of Econometrics 142(1):379–398, 2008) instrumental variables quantile regression estimator (IVQR) and the MCQR. Simulation results indicate that the IVCQR estimator performs well.

Keywords:

• Quantile regression
• censoring
• moment condition

JEL CLASSIFICATION:

• C14
• C21
• C24
• C26

Acknowledgements

We would like to thank the associate editor and two anonymous referees for their very helpful comments that have greatly improved the presentation of the paper.

Notes

1 Here for two positive definite matrixes A and B, $A\ge B$ means that $A-B$ is a positive semi-definite matrix.

2 This essentially becomes the problem of quantile regression with exogenous variables since the control variable can be considered as a kind of exogenous regressor.

3 Although Chernozhukov and Hansen (2006, 2008) did provide some high level primitive conditions for identification, though they can be difficult to interpret.

4 As for the exogenous case, once the exact singular cases are ruled out, these constraints are practically irrelevant.