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Quantile Vector Autoregression without Crossing

Tomohiro Ando, Tadao Hoshino, Ruey Tsay

Abstract

This paper considers estimation and model selection of quantile vector autoregression (QVAR). Conventional quantile regression often yields undesirable crossing quantile curves, violating the monotonicity of quantiles. To address this issue, we propose a simplex quantile vector autoregression (SQVAR) framework, which transforms the autoregressive (AR) structure of the original QVAR model into a simplex, ensuring that the estimated quantile curves remain monotonic across all quantile levels. In addition, we impose the smoothly clipped absolute deviation (SCAD) penalty on the SQVAR model to mitigate the explosive nature of the parameter space. We further develop a Bayesian information criterion (BIC)-based procedure for selecting the optimal penalty parameter and introduce new frameworks for impulse response analysis of QVAR models. Finally, we establish asymptotic properties of the proposed method, including the convergence rate and asymptotic normality of the estimator, the consistency of AR order selection, and the validity of the BIC-based penalty selection. For illustration, we apply the proposed method to U.S. financial market data, highlighting the usefulness of our SQVAR method.

Quantile Vector Autoregression without Crossing

Abstract

This paper considers estimation and model selection of quantile vector autoregression (QVAR). Conventional quantile regression often yields undesirable crossing quantile curves, violating the monotonicity of quantiles. To address this issue, we propose a simplex quantile vector autoregression (SQVAR) framework, which transforms the autoregressive (AR) structure of the original QVAR model into a simplex, ensuring that the estimated quantile curves remain monotonic across all quantile levels. In addition, we impose the smoothly clipped absolute deviation (SCAD) penalty on the SQVAR model to mitigate the explosive nature of the parameter space. We further develop a Bayesian information criterion (BIC)-based procedure for selecting the optimal penalty parameter and introduce new frameworks for impulse response analysis of QVAR models. Finally, we establish asymptotic properties of the proposed method, including the convergence rate and asymptotic normality of the estimator, the consistency of AR order selection, and the validity of the BIC-based penalty selection. For illustration, we apply the proposed method to U.S. financial market data, highlighting the usefulness of our SQVAR method.
Paper Structure (38 sections, 23 theorems, 220 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 38 sections, 23 theorems, 220 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Paper organization.
  3. Notation.
  4. Quantile VAR without Crossing
  5. The model
  6. Simplex QVAR representation
  7. Stationarity
  8. Estimation
  9. SCAD-penalized SQVAR estimator
  10. Regularization parameter selection
  11. Estimation of the joint innovation distribution
  12. Asymptotic Theory
  13. Asymptotic properties of the penalized SQVAR estimator
  14. Consistency of the BIC criterion
  15. Impulse Response Analysis
  16. ...and 23 more sections

Key Result

Lemma 2.1

Suppose that Assumption as:SMER holds. Then, we have where the elements of $\bm{\Phi}_i$ are all strictly increasing over $(0,1)$. In particular, the SQVAR coefficients $\bm{\Phi}_i(U_{it})$ and the original QVAR coefficients $\bm{\Theta}_i(U_{it})$ are related in the following manner: $\bm{\Theta}_i(U_{it}) = \bm V^{-1}\bm \Phi_i(U_{it})$, where $\bm

Figures (4)

  • Figure 6.1: Boxplots of RMSEs for SQVAR and standard QR methods
  • Figure 6.2: Daily returns of six U.S. asset portfolios formed by size and momentum from January 11, 1995 to June 30, 2025.
  • Figure 6.3: Estimated coefficient functions
  • Figure 6.4: Scenario-based analysis of the six portfolios formed be size and momentum.

Theorems & Definitions (44)

  • Remark 2.1
  • Definition 2.1: SMER
  • Lemma 2.1: SQVAR
  • Example 2.1
  • Lemma 2.2
  • Remark 3.1
  • Theorem 4.1: Rate of convergence
  • Theorem 4.2: Consistent model selection
  • Theorem 4.3: Asymptotic normality of the active coefficients
  • Remark 4.1: Choice of $H$
  • ...and 34 more