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Sankhya B

pp 1–33 | Cite as

The Second-Order Asymptotic Properties of Asymmetric Least Squares Estimation

  • Tae-Hwy Lee
  • Aman UllahEmail author
  • He Wang
Article
  • 6 Downloads

Abstract

The higher-order asymptotic properties provide better approximation of the bias for a class of estimators. The first-order asymptotic properties of the asymmetric least squares (ALS) estimator have been investigated by Newey and Powell (Econometrica55, 4, 819–847 1987). This paper develops the second-order asymptotic properties (bias and mean squared error) of the ALS estimator, extending the second-order asymptotic results for the symmetric least squares (LS) estimators of Rilstone et al. (J. Econometr.75, 369–395 1996). The LS gives the mean regression function while the ALS gives the “expectile” regression function, a generalization of the usual regression function. The second-order bias result enables an improved bias correction and thus an improved ALS estimation in finite sample. In particular, we show that the second-order bias is much larger as the asymmetry is stronger, and therefore the benefit of the second-order bias correction is greater when we are interested in extreme expectiles which are used as a risk measure in financial economics. The higher-order MSE result for the ALS estimation also enables us to better understand the sources of estimation uncertainty. The Monte Carlo simulation confirms the benefits of the second-order asymptotic theory and indicates that the second-order bias is larger at the extreme low and high expectiles.

Keywords

Asymmetric least squares Expectile Delta function Second-order bias Monte Carlo. 

AMS (2000) subject classification

Primary 62F Secondary 62J 

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Notes

Acknowledgements

We are thankful to the editors and two anonymous referees for many valuable comments that have helped improving the paper.

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Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CaliforniaRiversideUSA

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