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On multivariate quantile regression analysis

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Abstract

This paper investigates the estimation of parameters in a multivariate quantile regression model when the investigator wants to evaluate the associated distribution function. It proposes a new directional quantile estimator with the following properties: (1) it applies to an arbitrary number of random variables; (2) it is equivalent to estimating the distribution function allowing for non-convex distribution contours; (3) it satisfies nice equivariance properties; (4) it has desirable statistical properties (i.e., consistency and asymptotic normality); and (5) its implementation involves a modest computational burden: our proposed estimator can be obtained by solving parametric linear programming problems. As such, this paper expands the range of applications of quantile estimation for multivariate regression models.

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Notes

  1. For example, Hallin et al. (2010) proposed a directional quantile estimator involving hyperplanes that define halfspace Tukey-depth level sets.

  2. Note that multivariate unconditional quantiles are obtained as a special case when \(k=1\) and \(x=1\).

  3. An extreme point in a convex set is a point that cannot be expressed as a convex combination of any other points in the set.

  4. In turn, the solution of the dual problem (26b) evaluated at \(k^{*}\) can be used to conduct Lagrange multiplier/Rank tests on the parameters \(\beta \). See Koenker (2005, pp. 81–92).

  5. With (\(v_L,v_M)\) as bounds on simulated \(v_2 \), choosing \(y_2 =\frac{v_2-v_L }{v_M-v_L }\in [0,1]\) means that the distribution of \(Y_2 \) is a truncated version of the distribution of \(V_2 \), with truncation effects that vary with the simulated sample. As a result, we do not have a closed-form expression for the joint distribution of \((Y_1,Y_2)\). In this context, the quantile estimates reported below have the simple objective of illustrating that our proposed estimator is empirically tractable.

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Authors and Affiliations

  1. Agricultural and Applied Economics, Taylor Hall, University of Wisconsin, Madison, WI, 53706, USA

    Jean-Paul Chavas

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  1. Jean-Paul Chavas
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Correspondence to Jean-Paul Chavas.

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The author would like to thank two reviewers for useful comments on an earlier draft of the paper.

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Chavas, JP. On multivariate quantile regression analysis. Stat Methods Appl 27, 365–384 (2018). https://doi.org/10.1007/s10260-017-0407-x

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  • DOI: https://doi.org/10.1007/s10260-017-0407-x

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