A Simple Approach to Simultaneous Quantile Regression under Partial Homogeneity Constraints

Javier Alejo

doi:10.1515/jem-2025-0003

Article

A Simple Approach to Simultaneous Quantile Regression under Partial Homogeneity Constraints

Javier Alejo

Published/Copyright: January 19, 2026

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Journal of Econometric Methods

Abstract

Quantile regression is one of the most important methods to estimate heterogeneous effects on a variable of interest. In many applications there is a subset of covariates of interest, while the rest operate as controls in the regression equation. This work presents a straightforward empirical strategy for situations in which the control variables have a homogeneous effect on the conditional distribution. We develop the asymptotic theory for the proposed estimator and the corresponding inference procedures. An application using environmental pollution data illustrates the method by estimating the Environmental Kuznets Curve.

Keywords: quantile regression; ordinary least squares; efficiency

JEL Classification: C21; C14; C51

Corresponding author: Javier Alejo, IECON, Universidad de la República, Montevideo, Uruguay, E-mail: javier.alejo@fcea.edu.uy

Acknowledgments

The author would like to thank the anonymous referees and the editor for their helpful comments and suggestions, which substantially improved the manuscript. Any remaining errors are the author’s own.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: A Large Language Model (ChatGPT) was used solely for language editing and stylistic improvements.
Conflict of interest: The author states no conflict of interest.
Research funding: None declared.
Data availability: The data used in this study are publicly available from the World Development Indicators (WDI) database of the World Bank.

A Appendix

A.0 Joint Asymptotic Normality of OLS and QR

For this proof, we use the setup developed in Bera et al. (2014). The proposed estimator is implicitly defined in the following system of equations:

(5) ∑ i = 1 n x 1 i u i ( β ̂ 1 , β ̂ 2 ) = 0

(6) ∑ i = 1 n x 2 i u i ( β ̂ 1 , β ̂ 2 ) = 0

(7) ∑ i = 1 n x 2 i ψ τ ( v i ( β ̂ 1 , β ̂ 2 ( τ ) ) ) = 0

where u i ( β ̂ 1 , β ̂ 2 ) ≔ y i − x 1 i ′ β ̂ 1 − x 2 i ′ β ̂ 2 , v i ( β ̂ 1 , β ̂ 2 ( τ ) ) ≔ y i − x 1 i ′ β ̂ 1 − x 2 i ′ β ̂ 2 ( τ ) and ψ _τ(v)≔1(v > 0) + τ − 1. To simplify notation, we will simply write u _i and v _i.

Note that equations (5) and (6) are the first-order conditions (FOC) of OLS and represent the first stage of the proposed method, while (7) is the FOC of the QR problem when β ̂ 1 ( τ ) = β ̂ 1 , i.e. the second stage of the proposed method. In this way, the proposed estimator can be considered a Z-estimator (see van der Vaart (1998)). Therefore, since we are assuming that the standard conditions for both methods (OLS and QR) are valid, this system results in consistent estimators.

Let X ₁ and X ₂ be two matrices of dimensions (n × K ₁) and (n × K ₂), respectively. Let U and V denote the (n × 1) vectors containing u _i and v _i, respectively, for i = 1, …, n. Let K≔K ₁ + K ₂, and define X≔(X ₁, X ₂), a matrix of dimensions (n × K). Recall that β ≔ β 1 ′ , β 2 ′ ′ and β(τ):=(β ₁(τ)′, β ₂(τ)′)′ are the (K × 1) coefficient vectors of the model for the mean and the conditional quantile, respectively. For the estimators, let b ≔ β ̂ 1 ′ , β ̂ 2 ′ ′ and b ( τ ) ≔ β ̂ 1 ( τ ) ′ , β ̂ 2 ( τ ) ′ ′ denote the OLS and QR regression coefficients, respectively.

Assumption 1.

The sample X i ′ , Y i , i = 1 , … , n is independent but not necessarily identically distributed. The conditional distribution F Y i | X i ( y ) is absolutely continuous, with densities f _i(ξ) uniformly bounded away from 0 and ∞ at the point ξ τ ≔ F Y i | X i − 1 ( τ ) .

Assumption 2.

The parameters (β′, β(τ)′) are defined as the unique solution to the following system of equations:

(8) E X i Y i − X i ′ β = 0

(9) E X 2 i ψ τ ( Y i − X 1 i ′ β 1 − X 2 i ′ β 2 ( τ ) ) = 0

where ψ _τ(v) = τ − 1(v < 0), and 1(v < 0) is the indicator function that equals 1 if v < 0 and 0 otherwise.

Assumption 3.

Let X _i(j) denote the j-th element of the vector X _i:

E | Y i − X i ′ β 2 X i ( j ) X i ( h ) | 1 + δ 1 < ∞ for some δ ₁ > 0 for i = 1, …, n and h, j = 1, …, K;
E | X i ( j ) X i ( h ) | 1 + δ 2 < ∞ for some δ ₂ > 0 for i = 1, …, n and h, j = 1, …, K;
The matrices H n ( τ ) ≔ E n − 1 X ′ diag ( δ ( V ) ) X ) and V n Ω n ( τ ) Ω n ( τ ) τ ( 1 − τ ) J n are uniformly positive definite, where V _n≔ Var(n ^−1/2 X′U), Ω n ( τ ) ≔ E n − 1 X ′ diag ( ρ τ ( V ) ) X ) and J _n≔E(n ⁻¹ X′X). The matrices with “∼” refer to these same definitions using the submatrix X _a residualized with X _b, for a = 1, 2 and a ≠ b.

All these assumptions are standard in the OLS and QR literature. Assumption 2 is necessary for the model parameters to be correctly identified, and together with the other assumptions, it allows invoking the law of large numbers and the central limit theorem. See Bera et al. (2014) for more details on each specific assumption.

Lemma 1.

Under assumptions 1 to 3, the following holds:

n β ̂ − β β ̂ ( τ ) − β ( τ ) → d N ( 0 , D n ( τ ) )

for τ ∈ (0, 1) and D n ( τ ) ≔ J n − 1 V n J n − 1 J n − 1 Ω n ( τ ) H n − 1 H n − 1 Ω n ( τ ) J n − 1 τ ( 1 − τ ) H n − 1 Ω n ( τ ) H n − 1 .

Proof.

See the Supplemental Appendix in Bera et al (2014). □

Note that the matrix defined in Lemma 1 is partitioned according to whether the coefficients correspond to OLS or QR. However, another block partitioning is possible based on the partitioning of the covariates X :=(X ₁, X ₂), that is:

D n ( τ ) = D 11 ( τ ) D 12 ( τ ) D 21 ( τ ) D 22 ( τ )

It is also possible to partition D _n(τ) in a manner analogous to J _n, Ω_n, and H _n. The problem is that the exact form of each block in D _n(τ) is not known, particularly in the context where β ₁(τ) = β ₁. In sections A.1 and A.2, we derive each of these components.

A.1 Asymptotic Normality of β ̂ 2 ( τ )

We need the assumptions established in Section A.0 to prove this result. First, consider the following first-order asymptotic approximation of the system (5), (6), and (7):

(10) X 1 ′ U − X 1 ′ X 1 ( β ̂ 1 − β 1 ) − X 1 ′ X 2 ( β ̂ 2 − β 2 ) = 0

(11) X 2 ′ U − X 2 ′ X 1 ( β ̂ 1 − β 1 ) − X 2 ′ X 2 ( β ̂ 2 − β 2 ) = 0

(12) X 2 ′ ψ ( V ) − X 2 ′ diag ( δ ( V ) ) X 1 ( β ̂ 1 − β 1 ) − X 2 ′ diag ( δ ( V ) ) X 2 ( β ̂ 2 ( τ ) − β 2 ( τ ) ) = 0

From (10) and (11), it follows that ( β ̂ 1 − β 1 ) = X ̃ 1 ′ X ̃ 1 − 1 X ̃ 1 ′ U , where X ̃ 1 = M 2 X 1 and M 2 = I − X 2 X 2 ′ X 2 − 1 X 2 . Substituting this term into equation (12), it is possible to write:

n ( β ̂ 2 ( τ ) − β 2 ( τ ) ) = ( n − 1 X 2 ′ diag ( δ ( V ) ) X 2 ) − 1 ( n − 1 / 2 X 2 ′ ψ τ ( V ) ) − ( n − 1 X 2 ′ diag ( δ ( V ) ) X 2 ) − 1 ( n − 1 X 2 ′ diag ( δ ( V ) ) X 1 ) n − 1 X ̃ 1 ′ X ̃ 1 − 1 n − 1 / 2 X ̃ 1 ′ U

First term,

( n − 1 X 2 ′ diag ( δ ( V ) ) X 2 ) − 1 ( n − 1 / 2 X 2 ′ ψ τ ( V ) ) → d ( H 2,2 − 1 ( τ ) ) − 1 G n v ( τ )

where G n v ( τ ) = n − 1 / 2 X 2 ′ ψ τ ( V ) → d N ( 0 , τ ( 1 − τ ) J 2,2 ) , H 2,2 ( τ ) = E ( n − 1 X 2 ′ diag ( f ) X 2 ) and J 2,2 = E n − 1 X 2 ′ X 2 .

Second term,

( n − 1 X 2 ′ diag ( δ ( V ) ) X 2 ) − 1 ( n − 1 X 2 ′ diag ( δ ( V ) ) X 1 ) n − 1 X ̃ 1 ′ X ̃ 1 − 1 n − 1 / 2 X ̃ 1 ′ U → d ( H 2,2 ( τ ) ) − 1 H 2,1 ( τ ) J 1 ̃ , 1 ̃ − 1 G n u ( τ )

with G n u ( τ ) = n − 1 / 2 X ̃ 1 ′ U → d N ( 0 , V 1 ̃ , 1 ̃ ) , H 2,1 ( τ ) = E ( n − 1 X 2 ′ diag ( f ) X 1 ) , J 1 ̃ , 1 ̃ = E n − 1 X ̃ 1 ′ X ̃ 1 and V 1 ̃ , 1 ̃ = E ( n − 1 X ̃ 1 ′ diag ( U U ′ ) X ̃ 1 ) .

Then,

n ( β ̂ 2 ( τ ) − β 2 ( τ ) ) → d A ( τ ) G n v ( τ ) − B ( τ ) G n u ( τ )

with A ( τ ) = ( H 22 ( τ ) ) − 1 and B ( τ ) = ( H 22 ( τ ) ) − 1 H 21 ( τ ) J 1 ̃ , 1 ̃ − 1 .

Finally, considering that A C o v ( G n v ( τ ) , G n u ( τ ) ) = Ω 2 , 1 ̃ ( τ ) = E ( n − 1 X 2 ′ ρ τ ( V ) X ̃ 1 ) (see Bera et al. (2014)), the result of the proposition is reached.

Finally, the complete formula for the variance of β ̂ 2 ( τ ) is,

C ( τ ) = τ ( 1 − τ ) A ( τ ) J 2,2 A ( τ ) ′ + B ( τ ) V 1 ̃ , 1 ̃ B ( τ ) ′ − A ( τ ) Ω 2 , 1 ̃ B ( τ ) ′ − B ( τ ) ′ Ω 1 ̃ , 2 A ( τ ) .

□

A.2 Block Partition of D _n(τ)

We need the assumptions established in Section A.0 to prove this result. First, from OLS theory, we know that n ( β ̂ 1 − β 1 ) → d J 1 ̃ , 1 ̃ − 1 G n u , then.

D 11 = J 1 ̃ , 1 ̃ − 1 V 1,1 J 1 ̃ , 1 ̃ − 1

Secondly, n ( β ̂ 2 ( τ ) − β 2 ( τ ) ) → d A ( τ ) G n v ( τ ) − B ( τ ) G n u ( τ ) by Appendix A.1, then

D 22 ( τ ) = C ( τ )

Hence, the asymptotic covariance matrix is:

D 12 ( τ ) = A C o v β ̂ 2 ( τ ) , β ̂ 1 = E ( A ( τ ) G n v ( τ ) − B ( τ ) G n u ( τ ) ) G n u ′ J 1 ̃ , 1 ̃ − 1 = A ( τ ) Ω 2 , 1 ̃ ( τ ) J 1 ̃ , 1 ̃ − 1 − B ( τ ) V 1 ̃ , 1 ̃ J 1 ̃ , 1 ̃ − 1

Finally, the full set of components of D(τ) is as follows: D 11 = J 1 ̃ , 1 ̃ − 1 V 1,1 J 1 ̃ , 1 ̃ − 1 , D 12 ( τ ) = A ( τ ) Ω 2 , 1 ̃ J 1 ̃ , 1 ̃ − 1 − B ( τ ) W 1 ̃ , 1 ̃ J 1 ̃ , 1 ̃ − 1 = D 21 ( τ ) ′ and D ₂₂(τ) = C(τ). □

A.3 Asymptotic Normality of AQR

Formally, the proposed adaptive estimator is defined as.

β ̂ A ( τ ) = Λ n β ̂ PQR ( τ ) + ( 1 − Λ n ) β ̂ K B ( τ )

with

Λ n = 1 , if H 0 is accepted , 0 , if H 0 is rejected .

where H ₀: β ₁(τ) = β ₁ for all τ ∈ (0, 1).

Using Proposition 2 and standard results for QR, the asymptotic distribution of AQR is given by.

n ( β ̂ A ( τ ) − β ( τ ) ) → d Λ n n ( β ̂ PQR ( τ ) − β ( τ ) ) + ( 1 − Λ n ) n ( β ̂ K B ( τ ) − β ( τ ) ) .

Hence, the limit law is a mixture of two multivariate normal distributions and is therefore not strictly Gaussian. The variance depends on the pretest outcome: if H ₀ is not rejected, it equals D _n(τ); otherwise, it equals F(τ), the variance of Koenker and Bassett (1978). By the law of total variance,

Var [ β ̂ A ( τ ) ] = E Var ( β ̂ A ( τ ) | Λ n ) + Var E ( β ̂ A ( τ ) | Λ n ) = ( 1 − π ) Var [ β ̂ PQR ( τ ) ] + π Var [ β ̂ K B ( τ ) ] + … … + π ( 1 − π ) E ( β ̂ PQR ( τ ) − β ̂ Q R ( τ ) ) E ( β ̂ PQR ( τ ) − β ̂ Q R ( τ ) ) ′ ,

where π = lim_n→∞ P(Λ_n), which depends on the validity of H ₀. If the null is true, then π = α (the significance level), and by consistency of PQR and QR under H ₀, the third term vanishes asymptotically, yielding the mixed distribution stated in Proposition 3. If the null is false, the pretest is consistent, so π = 1, and the asymptotic distribution reduces to the standard QR theory.

□

B Additional Figures

Figure 2:

Partial homogeneity in the coefficients of X ₁.

Figure 3:

Environmental Kuznets Curve (EKC). Source: own calculations using the WDI database. Note: all other explanatory variables are held at their average values.

References

Allard, A., J. Takman, G. S. Uddin, and A. Ahmed. 2018. “The N-Shaped Environmental Kuznets Curve: An Empirical Evaluation Using a Panel Quantile Regression Approach.” Environmental Science and Pollution Research 25 (6): 5848–61, https://doi.org/10.1007/s11356-017-0907-0.Search in Google Scholar PubMed PubMed Central

Belloni, A., and V. Chernozhukov. 2011. “L1-Penalized Quantile Regression in High-Dimensional Sparse Models.” The Annals of Statistics 39 (1): 82–130.10.1214/10-AOS827Search in Google Scholar

Bera, A. K., A. F. Galvao, G. V. Montes-Rojas, and S. Y. Park. 2016. “Asymmetric Laplace Regression: Maximum Likelihood, Maximum Entropy and Quantile Regression.” Journal of Econometric Methods 5 (1): 79–101. https://doi.org/10.1515/jem-2014-0018.Search in Google Scholar

Bera, A., A. Galvao, and L. Wang. 2014. “On Testing the Equality of Mean and Quantile Effects.” Journal of Econometric Methods 3 (1): 47–62. https://doi.org/10.1515/jem-2012-0003.Search in Google Scholar

Breusch, and A. R. Pagan. 1979. “A Simple Test for Heteroscedasticity and Random Coefficient Variation.” Econometrica 47 (5): 1287–94. https://doi.org/10.2307/1911963.Search in Google Scholar

Canay, I. A. 2011. “A Simple Approach to Quantile Regression for Panel Data.” The Econometrics Journal 14 (3): 368–86, https://doi.org/10.1111/j.1368-423x.2011.00349.x.Search in Google Scholar

Chen, D., G. Li, and X. Feng. 2019. “Lack-Of-Fit Tests for Quantile Regression Models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 81 (3): 629–48. https://doi.org/10.1111/rssb.12321.Search in Google Scholar

Chernozhukov, V. 2005. “Extremal Quantile Regression.” Annals of Statistics 33 (2): 806–39. https://doi.org/10.1214/009053604000001165.Search in Google Scholar

Chernozhukov, V., and I. Fernandez-Val. 2011. “Inference for Extremal Conditional Quantile Models, with an Application to Market and Birthweight Risks.” Review of Economic Studies 78 (2): 559–89. https://doi.org/10.1093/restud/rdq020.Search in Google Scholar

de Castro, L., A. F. Galvao, D. M. Kaplan, and X. Liu. 2019. “Smoothed GMM for Quantile Models.” JoE 213 (1): 121–44, https://doi.org/10.1016/j.jeconom.2019.04.008.Search in Google Scholar

D’Haultfoeuille, X., A. Maurel, and Y. Zhang. 2018. “Extremal Quantile Regressions for Selection Models and the Black-White Wage Gap.” Journal of Econometrics 203 (1): 129–42. https://doi.org/10.1016/j.jeconom.2017.11.004.Search in Google Scholar

Efron, B. 1991. “Regression Percentiles Using Asymmetric Squared Error Loss.” Statistica Sinica 1 (1): 93–125.Search in Google Scholar

Flores, C. A., A. Flores-Lagunes, and D. Kapetanakis. 2014. “Lessons from Quantile Panel Estimation of the Environmental Kuznets Curve.” Econometric Reviews 33 (8): 815–53, https://doi.org/10.1080/07474938.2013.806148.Search in Google Scholar

Giles, J. A., and D. E. A. Giles. 1993. “Pre-Test Estimation and Testing in Econometrics: Recent Developments.” Journal of Economic Surveys 7 (2): 145–97. https://doi.org/10.1111/j.1467-6419.1993.tb00163.x.Search in Google Scholar

Goldfeld, S. M., and R. E. Quandt. 1965. “Some Tests for Homoscedasticity.” Journal of the American Statistical Association 60 (310): 539–47. https://doi.org/10.1080/01621459.1965.10480811.Search in Google Scholar

Huang, M. L., and A. Rat. 2017. “A Note on Quantile Regression.” Cogent Mathematics 4 (1): 1357237. https://doi.org/10.1080/23311835.2017.1357237.Search in Google Scholar

Huang, M. L., X. Xu, and D. Tashnev. 2015. “A Weighted Linear Quantile Regression.” Journal of Statistical Computation and Simulation 85 (13): 2596–618, https://doi.org/10.1080/00949655.2014.938240.Search in Google Scholar

Ike, G. N., O. Usman, and S. A. Sarkodie. 2020. “Testing the Role of Oil Production in the Environmental Kuznets Curve of Oil Producing Countries: New Insights from Method of Moments Quantile Regression.” Science of the Total Environment 711: 135208. https://doi.org/10.1016/j.scitotenv.2019.135208.Search in Google Scholar PubMed

Jiang, L., H. D. Bondell, and H. J. Wang. 2013. “Interquantile Shrinkage and Variable Selection in Quantile Regression.” Computational Statistics & Data Analysis 57 (1): 605–19.Search in Google Scholar

Kaplan, D., and Y. Sun. 2017. “Smoothed Estimating Equations for Instrumental Variables Quantile Regression.” Journal of Econometrics 200 (2): 262–83.10.1017/S0266466615000407Search in Google Scholar

Koenker, R. 1981. “A Note on Studentizing a Test for Heteroskedasticity.” Journal of Econometrics 17 (1): 107–12. https://doi.org/10.1016/0304-4076(81)90062-2.Search in Google Scholar

Koenker, R. 2004. “Quantile Regression for Longitudinal Data,” Journal of Multivariate Analysis 91 (1): 74–89, https://doi.org/10.1016/j.jmva.2004.05.006.Search in Google Scholar

Koenker, R. 2005. Quantile Regression. Cambridge: Cambridge University Press.10.1017/CBO9780511754098Search in Google Scholar

Koenker, R. 2017. “Quantile Regression: 40 Years on.” Annual Review of Economics 9 (1): 155–76, https://doi.org/10.1146/annurev-economics-063016-103651.Search in Google Scholar

Koenker, R., and G. W. Bassett. 1978. “Regression Quantiles.” Econometrica 46 (1): 33–49, https://doi.org/10.2307/1913643.Search in Google Scholar

Koenker, R., and G. W. Bassett. 1982. “Robust Tests for Heteroscedasticity Based on Regression Quantile.” Econometrica 50: 43–61. https://doi.org/10.2307/1912528.Search in Google Scholar

Koenker, R., and V. d’Orey. 1987. “Computing Regression Quantiles.” Applied Statistics 36 (4): 383–93. https://doi.org/10.2307/2347802.Search in Google Scholar

Koenker, R., and V. d’Orey. 1994. “Algorithm AS 229: Computing Regression Quantiles.” Applied Statistics 43 (3): 410–4. https://doi.org/10.2307/2986030.Search in Google Scholar

Koenker, R., and K. Hallock. 2001. “Quantile Regression.” Journal of Economic Perspectives 15 (4): 143–56. https://doi.org/10.1257/jep.15.4.143.Search in Google Scholar

Koenker, R., and K. Park. 1996. “An Interior Point Algorithm for Nonlinear Quantile Regression.” Journal of Econometrics 71 (1-2): 265–83. https://doi.org/10.1016/0304-4076(96)84507-6.Search in Google Scholar

Koenker, R., and Z. Xiao. 2002. “Inference on the Quantile Regression Process.” Econometrica 70 (4): 1583–612. https://doi.org/10.1111/1468-0262.00342.Search in Google Scholar

Lamarche, C. 2010. “Robust Penalized Quantile Regression Estimation for Panel Data.” Journal of Econometrics 157 (2): 396–408. https://doi.org/10.1016/j.jeconom.2010.03.042.Search in Google Scholar

Lamarche, C. 2019. Quantile Regression methods, 1–15. Springer International Publishing.10.1007/978-3-319-57365-6_42-1Search in Google Scholar

Leeb, H., and B. M. Pötscher. 2005. “Model Selection and Inference: Facts and Fiction.” Econometric Theory 21 (1): 21–59. https://doi.org/10.1017/s0266466605050036.Search in Google Scholar

Newey, W. K., and J. L. Powell. 1987. “Asymmetric Least Squares Estimation and Testing.” Econometrica 55 (4): 819–47. https://doi.org/10.2307/1911031.Search in Google Scholar

Özcan and Öztürk, 2019 Özcan, B., and I. Öztürk. 2019. Environmental Kuznets Curve (EKC): A Manual. London, United Kingdom: Academic Press.Search in Google Scholar

Romano, J. P., and M. Wolf. 2017. “Resurrecting Weighted Least Squares.” Journal of Econometrics 197 (1): 1–19. https://doi.org/10.1016/j.jeconom.2016.10.003.Search in Google Scholar

Tan, K. M., L. Wang, and W.-X. Zhou. 2022. “High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 84 (1): 189–208. https://doi.org/10.1111/rssb.12485.Search in Google Scholar

van der Vaart, A. 1998. Asymptotic Statistics. Cambridge: Cambridge University Press.10.1017/CBO9780511802256Search in Google Scholar

White, H. 1980. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.” Econometrica 48 (4): 817–38. https://doi.org/10.2307/1912934.Search in Google Scholar

Yaduma, N., M. Kortelainen, and A. Wossink. 2015. “The Environmental Kuznets Curve at Different Levels of Economic Development: a Counterfactual Quantile Regression Analysis for CO2 Emissions.” Journal of Environmental Economics and Policy 4 (3): 278–303, https://doi.org/10.1080/21606544.2015.1004118.Search in Google Scholar

Yi, C., and J. Huang. 2017. “Semismooth Newton Coordinate Descent Algorithm for elastic-net Penalized Quantile Regression.” Journal of Computational and Graphical Statistics 26 (3): 547–57. https://doi.org/10.1080/10618600.2016.1256816.Search in Google Scholar

Yu, K., and R. A. Moyeed. 2001. “Bayesian Quantile Regression.” Statistics and Probability Letters 54 (4): 437–47. https://doi.org/10.1016/s0167-7152(01)00124-9.Search in Google Scholar

Received: 2025-01-25

Accepted: 2025-12-27

Published Online: 2026-01-19

You are currently not able to access this content.

https://doi.org/10.1515/jem-2025-0003

Keywords for this article

quantile regression; ordinary least squares; efficiency