# Interpretation and Semiparametric Efficiency in Quantile Regression under Misspecification

## Author

Listed:
• Ying-Ying Lee

() (Department of Economics, University of Oxford, Manor Road Building, Manor Road, Oxford OX1 3UQ, UK)

## Abstract

Allowing for misspecification in the linear conditional quantile function, this paper provides a new interpretation and the semiparametric efficiency bound for the quantile regression parameter β ( τ ) in Koenker and Bassett (1978). The first result on interpretation shows that under a mean-squared loss function, the probability limit of the Koenker–Bassett estimator minimizes a weighted distribution approximation error, defined as $$F_{Y}(X'\beta(\tau)|X) - \tau$$, i.e., the deviation of the conditional distribution function, evaluated at the linear quantile approximation, from the quantile level. The second result implies that the Koenker–Bassett estimator semiparametrically efficiently estimates the quantile regression parameter that produces parsimonious descriptive statistics for the conditional distribution. Therefore, quantile regression shares the attractive features of ordinary least squares: interpretability and semiparametric efficiency under misspecification.

## Suggested Citation

• Ying-Ying Lee, 2015. "Interpretation and Semiparametric Efficiency in Quantile Regression under Misspecification," Econometrics, MDPI, Open Access Journal, vol. 4(1), pages 1-14, December.
• Handle: RePEc:gam:jecnmx:v:4:y:2015:i:1:p:2-:d:61252
as

File URL: http://www.mdpi.com/2225-1146/4/1/2/pdf

File URL: http://www.mdpi.com/2225-1146/4/1/2/

## References listed on IDEAS

as
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Full references (including those not matched with items on IDEAS)

### Keywords

semiparametric efficiency bounds; misspecification; conditional quantile function; conditional distribution function; best linear approximation;

### JEL classification:

• B23 - Schools of Economic Thought and Methodology - - History of Economic Thought since 1925 - - - Econometrics; Quantitative and Mathematical Studies
• C - Mathematical and Quantitative Methods
• C00 - Mathematical and Quantitative Methods - - General - - - General
• C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
• C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
• C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
• C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
• C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics
• C5 - Mathematical and Quantitative Methods - - Econometric Modeling
• C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs

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