# Inference on the Quantile Regression Process

## Author Info

• Roger Koenker

(University of Illinois)

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## Abstract

Quantile regression is gradually evolving into a comprehensive approach to the statistical analysis of linear and nonlinear response models for conditional quantile functions. Just as classical linear regression methods based on minimizing sums of squared residuals enable one to estimate models for conditional mean functions, quantile regression methods based on minimizing asymmetrically weighted {\it absolute} residuals offer a mechanism for estimating models for the conditional median function, and the full Tests based on the quantile regression process can be formulated like the classical Kolmogorov-Smirnov and Cramer-von-Mises tests of goodness-of-fit employing the theory of Bessel processes as in Kiefer (1959). However, it is frequently desirable to formulate hypotheses involving unknown nuisance parameters, thereby jeopardizing the distribution free character of these tests. We characterize this situation as the Durbin problem'' since it was posed in Durbin (1973), for parametric empirical processes. In this paper we consider an approach to the Durbin problem involving a martingale transformation of the parametric empirical process suggested by Khmaladze (1981) and show that it can be adapted to a wide variety of inference problems involving the quantile regression process. In particular, we suggest new tests of the location shift and location-scale shift models that underlie much of classical econometric inference. The methods are illustrated in some limited Monte-Carlo experiments and with a reanalysis of data on unemployment durations from the Pennsylvania Reemployment Bonus Experiments. The Pennsylvania experiments, conducted in 1988-89, were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells.

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## Bibliographic Info

Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 0886.

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Handle: RePEc:ecm:wc2000:0886

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## References

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1. Hahn, Jinyong, 1995. "Bootstrapping Quantile Regression Estimators," Econometric Theory, Cambridge University Press, vol. 11(01), pages 105-121, February.
2. Donald W.K. Andrews, 1990. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Cowles Foundation Discussion Papers 943, Cowles Foundation for Research in Economics, Yale University.
3. Joel L. Horowitz, 1996. "Bootstrap Methods for Median Regression Models," Econometrics 9608004, EconWPA.
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## Citations

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Cited by:
1. Kostov, Philip & Patton, Myles & Moss, Joan E. & McErlean, Seamus, 2005. "Does Gibrat's Law Hold Amongst Dairy Farmers in Northern Ireland?," 2005 International Congress, August 23-27, 2005, Copenhagen, Denmark 24775, European Association of Agricultural Economists.
2. Santa-Clara, Pedro & Valkanov, Rossen, 2000. "Political Cycles and the Stock Market," University of California at Los Angeles, Anderson Graduate School of Management qt00n6f3ph, Anderson Graduate School of Management, UCLA.
3. Yu, Tiffany Hui-Kuang, 2011. "Heterogeneous effects of different factors on global ICT adoption," Journal of Business Research, Elsevier, vol. 64(11), pages 1169-1173.
4. Drescher, Larissa S. & Goddard, Ellen W., 2011. "Heterogeneous Demand for Food Diversity: A Quantile Regression Analysis," 51st Annual Conference, Halle, Germany, September 28-30, 2011 114484, German Association of Agricultural Economists (GEWISOLA).
5. Smith, Patricia K. & Bogin, Barry & Varela-Silva, Maria Ines & Loucky, James, 2003. "Economic and anthropological assessments of the health of children in Maya immigrant families in the US," Economics & Human Biology, Elsevier, vol. 1(2), pages 145-160, June.

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