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


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.


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

AMS (2000) subject classification

Primary 62F Secondary 62J 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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


  1. Aigner, D.J., Amemiya, T. and Poirier, D. (1976). On the estimation of production frontiers: Maximum likelihood estimation of the parameters of a discontinuous density function. Int. Econ. Rev. 17, 377–396.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bao, Y. and Ullah, A. (2007). The second-order bias and mean squared error of estimators in time-series models. J. Econometr. 140, 650–669.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Gelfand, I.M. and Shilov, G.E. (1964). Generalized Functions, 1. Academic Press, New York.Google Scholar
  4. Koenker, R. and Bassett, G.S. (1978). Regression quantiles. Econometrica 46, 33–50.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Kuan, C., Yeh, J. and Hsu, Y. (2009). Assessing value at risk with CARE, the conditional autoregressive expectile models. J. Econometr. 150, 261–270.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Newey, W.K. and Powell, J.L. (1987). Asymmetric least squares estimation and testing. Econometrica 55, 4, 819–847.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Pagan, A. and Ullah, A. (1999). Nonparametric econometrics. Cambridge University Press.Google Scholar
  8. Rilstone, P., Srivastava, V.K. and Ullah, A. (1996). The second-order bias and mean squared error of nonlinear estimators. J. Econometr. 75, 369–395.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Xie, S., Zhou, Y. and Wan, A. (2014). A varying-coefficient expectile model for estimating value at risk. J. Business Econ. Stat. 32:4, 576–592.MathSciNetCrossRefGoogle Scholar
  10. Yao, Q. and Tong, H. (1996). Asymmetric least squares regression estimation: A nonparametric approach. Nonparametr. Stat. 6, 273–292.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CaliforniaRiversideUSA

Personalised recommendations