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Journal of Econometric Methods

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On Testing the Equality of Mean and Quantile Effects

Anil K. Bera
  • Corresponding author
  • Department of Economics, University of Illinois at Urbana-Champaign, 1407 W. Gregory Drive, Urbana, IL 61801, USA
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/ Antonio F. Galvao
  • Department of Economics, University of Iowa, W210 Pappajohn Business Building, 21 E. Market Street, Iowa City, IA 52242, USA
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/ Liang Wang
  • Department of Economics, University of Wisconsin-Milwaukee, Bolton Hall 821, 3210 N. Maryland Ave., Milwaukee, WI 53201, USA
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Published Online: 2013-11-02 | DOI: https://doi.org/10.1515/jem-2012-0003


This paper proposes tests for equality of the mean regression (MR) and quantile regression (QR) coefficients. The tests are based on the asymptotic joint distribution of the ordinary least squares and QR estimators. First, we formally derive the asymptotic joint distribution of these estimators. Second, we propose a Wald test for equality of the MR and QR coefficients considering a single fixed quantile, and also describe a more general test using multiple quantiles simultaneously. A very salient feature of these tests is that they produce asymptotically distribution-free nature of inference. In addition, we suggest a sup-type test for equality of the coefficients uniformly over a range of quantiles. For the estimation of the variance-covariance matrix, the use sample counterparts and bootstrap methods. An important attribute of the proposed tests is that they can be used as a heteroskedasticity test. Monte Carlo studies are conducted to evaluate the finite sample properties of the tests in terms of size and power. Finally, we briefly illustrate the implementation of the tests using Engel data.

This article offers supplementary material which is provided at the end of the article.

Keywords: least squares; quantile regression; testing; JEL Classification: C12; C21


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About the article

Corresponding author: Anil K. Bera, Department of Economics, University of Illinois at Urbana-Champaign, 1407 W. Gregory Drive, Urbana, IL 61801, USA, E-mail:

Published Online: 2013-11-02

Published in Print: 2014-01-01

See Manski (1991) for a general discussion on different forms of regression.

Wald tests designed for linear hypothesis were suggested by Koenker and Bassett (1982a,b), Koenker and Machado (1999) and more recently by Goh and Knight (2009). It is possible to formulate a wide variety of tests using variants of the proposed Wald test, from simple tests on a single quantile regression coefficient to joint tests involving many covariates and distinct quantiles at the same time. More recently, there is a large literature on goodness of fit for quantile regression models. See e.g., He and Zhu (2003), Whang (2006), and Escanciano and Velasco (2010).

Koenker and Xiao (2002) 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 a related work Chernozhukov and Fernandez-Val (2005) develop tests for quantile regression process based on subsampling. The test is not based on Khmaladzation.

Intuitively, the income has positive effect on the food expenditure, and also the economic theory conjectures that the food expenditure should be more volatile for the higher income households than for the lower income households; that is to say, there should be heteroskedasticity.

In Remarks (A.1) and (A.2) in Supplemental Appendix A1 we discuss the extensions of the results for the instrumental variables and nonlinear regression cases, respectively.

Here we show the results for only one quantile. The results for the general cases are simple extensions.

We plot size adjusted power function for the sup-Wald and KH tests. The empirical size of the sup-Wald tests for N(0,1), t(3), Exp(1), χ2(2), F(2,7), and F(7,7) are 0.030, 0.019, 0.039, 0.047, 0.036, and 0.036, respectively. The corresponding sizes for KH test are 0.136, 0.129, 0.172, 0.177, 0.200, and 0.142.

This is true in our case since the Engel data is a cross-sectional data set.

We use the heteroskedasticity-robust estimator here.

Citation Information: Journal of Econometric Methods, Volume 3, Issue 1, Pages 47–62, ISSN (Online) 2156-6674, ISSN (Print) 2194-6345, DOI: https://doi.org/10.1515/jem-2012-0003.

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