Paper: arXiv 2505.16019

Authors: Shaobo Li, Ben Sherwood

Abstract

This paper considers equity premium prediction, for which mean regression can be problematic due to heteroscedasticity and heavy-tails of the error. We show advantages of quantile predictions using a novel penalized quantile regression that offers a model for a full spectrum analysis on the equity premium distribution. To enhance model interpretability and address the well-known issue of crossing quantile predictions in quantile regression, we propose a model that enforces the selection of a common set of variables across all quantiles. Such a selection consistency is achieved by simultaneously estimating all quantiles with a group penalty that ensures sparsity pattern is the same for all quantiles. Consistency results are provided that allow the number of predictors to increase with the sample size. A Huberized quantile loss function and an augmented data approach are implemented for computational efficiency. Simulation studies show the effectiveness of the proposed approach. Empirical results show that the proposed method outperforms several benchmark methods. Moreover, we find some important predictors reverse their relationship to the excess return from lower to upper quantiles, potentially offering interesting insights to the domain experts. Our proposed method can be applied to other fields.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.2/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced statistical theory including selection consistency, group LASSO penalties, and Huberized quantile loss, indicating high mathematical density. It also provides simulation studies and empirical backtesting with benchmarks and OOS evaluation, demonstrating significant implementation and data-driven rigor.

Research Flowchart

  flowchart TD
  Goal["Research Goal<br>Equity Premium Prediction<br>vs. Traditional Mean Regression"] --> Inputs["Data Inputs<br>Equity Premium Data<br>Predictor Variables"]
  
  Inputs --> Method["Key Methodology<br>Novel Penalized Quantile Regression<br>Huberized Quantile Loss"]
  
  Method --> Process1["Computational Process<br>Simultaneous Estimation<br>Group Penalty for Cross-Quantile Consistency"]
  
  Process1 --> Process2["Computational Process<br>Augmented Data Approach<br>for Efficiency"]
  
  Process2 --> Results["Outcomes<br>Enhanced Interpretability<br>No Crossing Quantiles<br>Outperforms Benchmarks"]
  
  Results --> Insights["Key Findings<br>Variable Relationships Reverse<br>from Lower to Upper Quantiles"]

Cited References (185)

  1. Noncrossing quantile regression curve estimation (2010)
  2. Solution of an Overdetermined System of Equations in the $\ell_1$ Norm. (1974)
  3. L1-Penalized quantile regression in high-dimensional sparse models (2011)
  4. Optimal minimal neural interpretation of spectra (1992)
  5. grpreg: Regularization Paths for Regression Models with Grouped Covariates 3.1-2 (2017)
  6. hrqglas: Group Variable Selection for Quantile and Robust Mean Regression (2021)
  7. Constructive Approximation (2005)
  8. Statistical Inference (2002)
and 177 more
  1. Splines as linear combinations of B-splines. (1976)
  2. On additive conditional quantiles with high-dimensional covariates (2003)
  3. grppenalty: Concave 1-norm and 2-norm group penalty in linear and logistic regression (2014)
  4. Variable selection via nonconcave penalized likelihood and its oracle properties (2001)
  5. Nonconcave penalized likelihood with NP-dimensionality (2011)
  6. Nonconcave penalized likelihood with a diverging number of parameters (2004)
  7. Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models (2006)
  8. faraway: Functions and Datasets for Books by Julian Faraway 1.0.7 (2016) — link
  9. Multivariate adaptive regression splines (1991)
  10. ADMM for high-dimensional sparse penalized quantile regression (\noop{3001}in press)
  11. On Parameters of Increasing Dimensions (2000)
  12. Convergence rate of B-spline estimators of nonparametric conditional quantile functions (1994)
  13. Bivariate tensor-product B-splines in a partly linear model (1996)
  14. Estimation in semiparametric model for longitudinal data with unspecified dependence structure (2002)
  15. Quantile-adaptive model-free nonlinear feature screening for high-dimensional heterogeneous data (2013)
  16. Nonparametric estimation of additive quantile regression model (2005)
  17. Projection estimation in multiple regression with application to functional ANOVA models (1998)
  18. Functional ANOVA models for generalized regression (1998)
  19. A selective review of group selection in high-dimensional models (2012)
  20. Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors (2015)
  21. Variable Selection in nonparametric additive models (2010)
  22. Robust Estimation of a Location Parameter (1964)
  23. Robust Regression: Asymptotics, Conjectures and Monte Carlo (1973)
  24. Concave 1-norm group selection (2015)
  25. Combining Logisitc Regression Analysis and Association Rule Mining via MLR Algorithm (2016)
  26. Group Lasso for high dimensional sparse quantile regression models (2012)
  27. Quantile regression with varying coefficients (2007)
  28. Smoothly clipped absolute deviation on high dimensions (2008)
  29. Limiting distributions for $L_1$ regression estimators under general conditions (1998)
  30. Quantile Regression (2005)
  31. Regression Quantiles (1978)
  32. Computing Regression Quantiles (1987)
  33. A Remark on Algorithm AS 229: Computing Dual Regression Quantiles and Regression Rank Score (1994)
  34. Large sample properties of the SCAD-penalized maximum likelihood estimation on high dimensions (2012)
  35. Variable selection for nonparametric quantile regression via smoothing spline analysis of variance (2013)
  36. Component selection and smoothing in multivariate nonparametric regression (2006)
  37. Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space (2018)
  38. Practical variable selection for generalized additive models (2011)
  39. A coordinate descent algorithm for computing penalized smooth quantile regression (2017)
  40. High-dimensional additive modeling (2009)
  41. A Unified Framework for High-Dimensional Analysis fo $M$-Estimators with Decomposable Regualrizers (2012)
  42. An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression (2015)
  43. On the subspaces of $L^p$ ($p>2)$ spanned by sequences of independent random variables. (1970)
  44. On the span in $L^p$ of sequences of independent random variables (II). (1972)
  45. Spline Functions: Basic Theory (1981)
  46. Variable selection for additive partial linear quantile regression with missing covariates (2016)
  47. rqPen: Penalized Quantile Regression 2.0 (2017) — link
  48. Partially linear additive quantile regression in ultra-high dimension (2016)
  49. Optimal global rates of convergence for nonparametric regression (1982)
  50. Additive regression and other nonparametric models (1985)
  51. The dimensionality reduction principle for generalized additive models (1986)
  52. Nonparametric quantile estimation (2006)
  53. Convex analysis approach to d.c. programming: Theory, algorithms and applications (1997)
  54. Asymptotic Statistics (1998)
  55. Weak Convergence and Empirical Processes: With Applications to Statistics (1996)
  56. Probability in Banach Spaces (1991)
  57. Quantile regression in partially linear varying coefficient models (2009)
  58. Quantile regression of analyzing heterogeneity in ultra-high dimension (2012)
  59. Estimation and model selection in generalized additive partial linear models for correlated data with diverging number of covariates (2014)
  60. Spline-backfitted kernel smoothing of nonlinear additive autoregression model (2007)
  61. Group SCAD regression analysis for microarray time course gene expression data (2007)
  62. Quantile regression methods for reference growth charts (2006)
  63. On M-Processes and M-Estimation (1989)
  64. Variable selection in quantile regression (2009)
  65. Consistent variable selection in additive models (2009)
  66. Additive coefficient modeling via polynomial spline (2006)
  67. Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression (2017)
  68. A parallel algorithm for large-scale nonconvex penalized quantile regression (\noop{3001}in press)
  69. Nearly unbiased variable selection under minimax concave penalty (2010)
  70. Component selection and smoothing for nonparametric regression in exponential families (2006)
  71. Variable selection and model building via likelihood basis pursuit (2004)
  72. Variable selection in additive quantile regression using nonconcave penalty (2016)
  73. On model selection consistency of Lasso (2006)
  74. The adaptive Lasso and its oracle properties (2006)
  75. One-step sparse estimates in nonconcave penalized likelihood models (2008)
  76. Regularized Multivariate Regression for Identifying Master Predictors with Application to Integrative Genomics Study of Breast Cancer (2010)
  77. A Multivariate Regression Approach to Association Analysis of a Quantitative Trait Network (2009)
  78. Automatic Response Category Combination in Multinomial Logistic Regression (2017)
  79. The Cluster Elastic Net for High-Dimensional Regression With Unknown Variable Grouping (2014)
  80. Regularization and variable selection via the Elastic Net (2005)
  81. Predicting Multivariate Responses in Multiple Linear Regression (1997)
  82. Multivariate Sparse Group Lasso for Multivariate Multiple Linear Regression with Arbitrary Group Sparsity (2015)
  83. Sparse Multivariate Regression with Covariance Estimation (2010)
  84. Regularized Matrix Regression (2014)
  85. Regularized Paths for Genearlized Linear Models via Coordinate Descent (2008)
  86. Properties and Refinements of the Fused Lasso (2009)
  87. A Path Algorithm for the Fused Lasso Signal Approximator (2010)
  88. Pathwise coordinate optimization (2007)
  89. Asymptotics for minimisers of convex processes (1993)
  90. An efficient algorithm for a class of fused lasso problems (2010)
  91. Sparsity and smoothness via the fused lasso (2005)
  92. On $\ell_2$ Error Bounds of the Elastic Net when $p»n$ (2017)
  93. Regression Shrinkage and Selection via the Lasso (1996)
  94. Ridge Regression: Biased Estimation for Nonorthogonal Problems (1970)
  95. Correlated variables in regression: Clustering and sparse estimation (2013)
  96. Model selection and estimation in regression with grouped variables (2005)
  97. SPReM: Sparse Projection Regression Model for High-Dimensional Linear Regression (2015)
  98. Foundations for Envelope Models and Methods (2015)
  99. Envelope Models for Parsimonious and Efficient Multivariate Linear Regression (with discussion) (2010)
  100. Indirect multivariate response linear regression (2016)
  101. Simultaneous multiple response regression and inverse covariance matrix estimation via penalized Gaussian maximum likelihood (2012)
  102. Simultaneous Analysis of Lasso and Dantzig Selector (2009)
  103. Lasso-Type Recovery of Sparse Representations for High-Dimensional Data (2009)
  104. The Dantzig selector: statistical estimation when $p$ is much larger than $n$ (2007)
  105. On the conditions used to prove oracle results for the Lasso (2009)
  106. Transcriptome alterations in maternal and fetal cells induced by tobacco smoke (2011)
  107. DNA methylation differences at growth related genes correlate with birth weight: a molecular signature linked to developmental origins of adult disease? (2012)
  108. The relationship between the weight of the placenta and birth weight of the neonate in a Nigerian Hospital (2012)
  109. Relationship of fetal and placental weight in human beings: fetal/placental weight ratios at various gestational ages and birth weight distributions (1978)
  110. Fetal growth is directly related to maternal anthropometry and placental volume (2004)
  111. New Semiparametric Method for Predicting High-Cost Patients (\noop{3001}in press)
  112. A new perspective on robust $M$-estimation: Finite sample theory and applications to dependence-adjusted multiple testing (2018)
  113. Adaptive Huber Regression (2020)
  114. Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions (2017)
  115. Asymmetric Least Squares Estimation and Testing (1987)
  116. The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators (1997)
  117. Simultaneous estimation and variable selection in median regression using Lasso-type penalty (2010)
  118. Regularization of Case-Specific Parameters for Robustness and Efficiency (2012)
  119. The group lasso for logistic regression (2008)
  120. The benefit of group sparsity (2010)
  121. A group LASSO-based method for robustly inferring gene regulatory networks from multiple time-course datasets (2014)
  122. Adaptive group LASSO selection in quantile models (2019)
  123. A fast unified algorithm for solving group-lasso penalize learning problems (2015)
  124. Convex Optimization in R (2014)
  125. Oracle inequalities and optimal inference under group sparsity (2011)
  126. Regularization Paths for Generalized Linear Models via Coordinate Descent (2010)
  127. A tutorial on MM algorithms (2004)
  128. Strong rules for discarding predictors in lasso-type problems (2012)
  129. The MM alternative to EM (2010)
  130. hqreg: Regularization Paths for Lasso or Elastic-Net Penalized Huber Loss Regression and Quantile Regression (2017)
  131. AmesHousing: The Ames Iowa Housing Data (2020)
  132. Ames, I owa: Alternative to the B oston housing data as an end of semester regression project (2011)
  133. Recovering Low-Rank Matrices From Few Coefficients in Any Basis (2011)
  134. Quantile regression (2005)
  135. Private Optimal Inventory Policy Learning for Feature-Based Newsvendor with Unknown Demand (0)
  136. The Big Data Newsvendor: Practical Insights from Machine Learning (2019)
  137. Tail-Heaviness, Asymmetry, and Profitability Forecasting by Quantile Regression (2021)
  138. A Quantile Regression Approach to Generating Prediction Intervals (1999)
  139. Selecting the Best Alternative Based on Its Quantile (2021)
  140. A Multilevel Simulation Optimization Approach for Quantile Functions (2022)
  141. Quantile regression metamodeling: Toward improved responsiveness in the high-tech electronics manufacturing industry (2018)
  142. Quantile regression feature selection and estimation with grouped variables using Huber approximation (2022)
  143. Conditional value-at-risk: Aspects of modeling and estimation (2001)
  144. Predictive regressions with time-varying coefficients (2012)
  145. Forecasting tail risk measures for financial time series: An extreme value approach with covariates (2023)
  146. On LASSO for predictive regression (2022)
  147. Goodness of Fit and Related Inference Processes for Quantile Regression (1999)
  148. Semismooth Newton coordinate descent algorithm for elastic-net penalized Huber loss regression and quantile regression (2017)
  149. A fast unified algorithm for solving group-lasso penalize learning problems (2015)
  150. Regularization and variable selection via the elastic net (2005)
  151. Globally adaptive quantile regression with ultra-high dimensional data (2015)
  152. The dividend-price ratio and expectations of future dividends and discount factors (1988)
  153. Predicting excess stock returns out of sample: Can anything beat the historical average? (2008)
  154. A new robust inference for predictive quantile regression (2023)
  155. Dividend yields and expected stock returns (1988)
  156. On the out-of-sample predictability of stock market returns (2006)
  157. Predictive quantile regression with persistent covariates: IVX-QR approach (2016)
  158. Limit theory for cointegrated systems with moderately integrated and moderately explosive regressors (2009)
  159. A quantile regression approach to equity premium prediction (2014)
  160. A comprehensive 2022 look at the empirical performance of equity premium prediction (2024)
  161. Out-of-sample equity premium prediction: Combination forecasts and links to the real economy (2010)
  162. A comprehensive look at the empirical performance of equity premium prediction (2008)
  163. Exploratory Quantile Regression With Many Covariates: An Application to Adverse Birth Outcomes (2011)
  164. Regularized simultaneous model selection in multiple quantiles regression (2008)
  165. DANTZIG-TYPE PENALIZATION FOR MULTIPLE QUANTILE REGRESSION WITH HIGH DIMENSIONAL COVARIATES (2017)
  166. Interquantile Shrinkage in Regression Models (2013)
  167. Shrinkage estimation of varying covariate effects based on quantile regression (2014)
  168. Regularized quantile regression under heterogeneous sparsity with application to quantitative genetic traits (2016)
  169. Heteroscedasticity identification and variable selection via multiple quantile regression (2024)
  170. Penalized regression across multiple quantiles under random censoring (2015) — link
  171. Simultaneous multiple non-crossing quantile regression estimation using kernel constraints (2011)
  172. A multi-quantile regression time series model with interquantile lipschitz regularization for wind power probabilistic forecasting (2022) — link
  173. Quantile Curves without Crossing (1997) — link
  174. Quantile regression feature selection and estimation with grouped variables using Huber approximation (2022)
  175. Group penalized quantile regression (2022)
  176. Nonparametric estimation and inference on conditional quantile processes (2015)
  177. High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization (2021)

0 of 185 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.