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The Second-Order Asymptotic Properties of Asymmetric Least Squares Estimation

Author

Listed:
  • Tae-Hwy Lee

    (University of California)

  • Aman Ullah

    (University of California)

  • He Wang

    (University of California)

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.

Suggested Citation

  • Tae-Hwy Lee & Aman Ullah & He Wang, 2019. "The Second-Order Asymptotic Properties of Asymmetric Least Squares Estimation," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 201-233, September.
    • Handle: RePEc:spr:sankhb:v:81:y:2019:i:1:d:10.1007_s13571-019-00189-8
    DOI: 10.1007/s13571-019-00189-8
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    1. Kuan, Chung-Ming & Yeh, Jin-Huei & Hsu, Yu-Chin, 2009. "Assessing value at risk with CARE, the Conditional Autoregressive Expectile models," Journal of Econometrics, Elsevier, vol. 150(2), pages 261-270, June.
    2. Pagan,Adrian & Ullah,Aman, 1999. "Nonparametric Econometrics," Cambridge Books, Cambridge University Press, number 9780521355643, October.
    3. Bao, Yong & Ullah, Aman, 2007. "The second-order bias and mean squared error of estimators in time-series models," Journal of Econometrics, Elsevier, vol. 140(2), pages 650-669, October.
    4. Yao, Qiwei & Tong, Howell, 1996. "Asymmetric least squares regression estimation: a nonparametric approach," LSE Research Online Documents on Economics 19423, London School of Economics and Political Science, LSE Library.
    5. Rilstone, Paul & Srivastava, V. K. & Ullah, Aman, 1996. "The second-order bias and mean squared error of nonlinear estimators," Journal of Econometrics, Elsevier, vol. 75(2), pages 369-395, December.
    6. Aigner, D J & Amemiya, Takeshi & Poirier, Dale J, 1976. "On the Estimation of Production Frontiers: Maximum Likelihood Estimation of the Parameters of a Discontinuous Density Function," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 17(2), pages 377-396, June.
    7. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    8. Shangyu Xie & Yong Zhou & Alan T. K. Wan, 2014. "A Varying-Coefficient Expectile Model for Estimating Value at Risk," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(4), pages 576-592, October.
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    Cited by:

    1. Arnab Bhattacharjee & Tapabrata Maiti, 2019. "P. C. Mahalanobis in the Context of Current Econometrics Research," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 1-11, September.

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    More about this item

    Keywords

    Asymmetric least squares; Expectile; Delta function; Second-order bias; Monte Carlo.;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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