Spline Quantile Regression with Cubic and Linear Smoothing Splines

Abstract

Spline quantile regression (SQR) is a method introduced recently by Li and Megiddo (2026) for linear quantile regression where the regression coefficients are treated as smooth functions of the quantile level. With the coefficients represented by cubic splines with fixed knots on a given set of quantiles, the SQR method produces an estimate for the functional coefficients by solving a penalized quantile regression problem. The $\ell_1$-norm of the second derivatives of the coefficients is employed as the penalty for regulating the roughness of the functional coefficients. This extends the SQR method by introducing additional pairings of the functional representation for the regression coefficients and the penalty for their roughness. The resulting cubic and linear SQR solutions are shown to be smoothing splines which are optimal in a functional space larger than the respective spline space with fixed knots. It is shown that the cubic SQR can be reformulated and solved as a quadratic program and the linear SQR as a linear program. A simulation study demonstrates that the SQR solutions not only offer a concise functional representation of the regression coefficients with distinct smoothness characteristics, but also provide a capability of producing more accurate estimates of the regression coefficients when the underlying functions are suitably smooth. Application of the SQR solutions is demonstrated by real-data examples, including a Granger causality analysis of stock market indices.

Figures (11)

• Figure 1: Total mean absolute error (MAE) of linear SQR estimates (left) and cubic SQR estimates (right) against smoothing parameter spar for the regression model (\ref{['lin']}) with $n=200$. Horizontal dotted line represents the error of QR estimates. Horizontal dashed line represents the best error achieved by the SQR estimator of Li and Megiddo (2026). Results are based on 2000 Monte Carlo runs.
• Figure 2: Total mean absolute error (MAE) of linear SQR estimates (left) and cubic SQR estimates (right) against smoothing parameter spar for the QAR model (\ref{['sim2']}) with (a) $n=200$ and (b) $n=500$. Horizontal dotted line represents the error of QR estimates. Horizontal dashed line and dash-dotted line represent the error of SQR estimates with spar selected by AIC and BIC, respectively. Results are based on 2000 Monte Carlo runs.
• Figure 3: Plot of AIC and BIC as functions of spar for a simulated data set from (\ref{['sim2']}). Circle identifies the global minimum.
• Figure 4: Total mean absolute error (MAE) of linear SQR estimates (left) and cubic SQR estimates (right) with different values of smoothing parameter spar from simulated data by (\ref{['sim']}) for (a) $n=200$ and (b) $n=500$ with MAE evaluated at $\EuScript{T} := \{ 0.04,0.06,\dots,0.96\}$. Solid line, estimates obtained with $\{ \tau_\ell \} := \{ 0.04,0.06,\dots,0.96\} = \EuScript{T}$. Dashed line, estimates obtained with $\{ \tau_\ell \} := \{ 0.04,0.08,\dots,0.96\} \subset \EuScript{T}$. Horizontal dotted line represents the error of QR estimates. Results are based on 2000 Monte Carlo runs.
• Figure 5: Cubic SQR estimates of the intercept (left) and the coefficient of household income (right) with a 90% bootstrap confidence band for the Engel food expenditure data. The smoothing parameter is selected by (a) AIC and (b) BIC. Dashed line, linear SQR. Circles, QR. Horizontal dashed lines and dotted lines show the ordinary least-squares estimate and a 90% confidence interval in the style of Koenker (2005, p. 302).

Theorems & Definitions (5)

• Lemma 1: de Boor 1963
• Lemma 2: Koenker et al. 1994; Pinkus 1988
• Theorem 1
• Theorem 2
• Theorem 3