Quantile Regression

Scaling Conditional Autoencoders for Portfolio Optimization via Uncertainty-Aware Factor Selection ArXiv ID: 2511.17462 “View on arXiv” Authors: Ryan Engel, Yu Chen, Pawel Polak, Ioana Boier Abstract Conditional Autoencoders (CAEs) offer a flexible, interpretable approach for estimating latent asset-pricing factors from firm characteristics. However, existing studies usually limit the latent factor dimension to around K=5 due to concerns that larger K can degrade performance. To overcome this challenge, we propose a scalable framework that couples a high-dimensional CAE with an uncertainty-aware factor selection procedure. We employ three models for quantile prediction: zero-shot Chronos, a pretrained time-series foundation model (ZS-Chronos), gradient-boosted quantile regression trees using XGBoost and RAPIDS (Q-Boost), and an I.I.D bootstrap-based sample mean model (IID-BS). For each model, we rank factors by forecast uncertainty and retain the top-k most predictable factors for portfolio construction, where k denotes the selected subset of factors. This pruning strategy delivers substantial gains in risk-adjusted performance across all forecasting models. Furthermore, due to each model’s uncorrelated predictions, a performance-weighted ensemble consistently outperforms individual models with higher Sharpe, Sortino, and Omega ratios. ...

The Evolution of Probabilistic Price Forecasting Techniques: A Review of the Day-Ahead, Intra-Day, and Balancing Markets ArXiv ID: 2511.05523 “View on arXiv” Authors: Ciaran O’Connor, Mohamed Bahloul, Steven Prestwich, Andrea Visentin Abstract Electricity price forecasting has become a critical tool for decision-making in energy markets, particularly as the increasing penetration of renewable energy introduces greater volatility and uncertainty. Historically, research in this field has been dominated by point forecasting methods, which provide single-value predictions but fail to quantify uncertainty. However, as power markets evolve due to renewable integration, smart grids, and regulatory changes, the need for probabilistic forecasting has become more pronounced, offering a more comprehensive approach to risk assessment and market participation. This paper presents a review of probabilistic forecasting methods, tracing their evolution from Bayesian and distribution based approaches, through quantile regression techniques, to recent developments in conformal prediction. Particular emphasis is placed on advancements in probabilistic forecasting, including validity-focused methods which address key limitations in uncertainty estimation. Additionally, this review extends beyond the Day-Ahead Market to include the Intra-Day and Balancing Markets, where forecasting challenges are intensified by higher temporal granularity and real-time operational constraints. We examine state of the art methodologies, key evaluation metrics, and ongoing challenges, such as forecast validity, model selection, and the absence of standardised benchmarks, providing researchers and practitioners with a comprehensive and timely resource for navigating the complexities of modern electricity markets. ...

Quantile Predictions for Equity Premium using Penalized Quantile Regression with Consistent Variable Selection across Multiple Quantiles ArXiv ID: 2505.16019 “View on arXiv” 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. ...

Quantile deep learning models for multi-step ahead time series prediction ArXiv ID: 2411.15674 “View on arXiv” Authors: Unknown Abstract Uncertainty quantification is crucial in time series prediction, and quantile regression offers a valuable mechanism for uncertainty quantification which is useful for extreme value forecasting. Although deep learning models have been prominent in multi-step ahead prediction, the development and evaluation of quantile deep learning models have been limited. We present a novel quantile regression deep learning framework for multi-step time series prediction. In this way, we elevate the capabilities of deep learning models by incorporating quantile regression, thus providing a more nuanced understanding of predictive values. We provide an implementation of prominent deep learning models for multi-step ahead time series prediction and evaluate their performance under high volatility and extreme conditions. We include multivariate and univariate modelling, strategies and provide a comparison with conventional deep learning models from the literature. Our models are tested on two cryptocurrencies: Bitcoin and Ethereum, using daily close-price data and selected benchmark time series datasets. The results show that integrating a quantile loss function with deep learning provides additional predictions for selected quantiles without a loss in the prediction accuracy when compared to the literature. Our quantile model has the ability to handle volatility more effectively and provides additional information for decision-making and uncertainty quantification through the use of quantiles when compared to conventional deep learning models. ...

Generalized Distribution Prediction for Asset Returns ArXiv ID: 2410.23296 “View on arXiv” Authors: Unknown Abstract We present a novel approach for predicting the distribution of asset returns using a quantile-based method with Long Short-Term Memory (LSTM) networks. Our model is designed in two stages: the first focuses on predicting the quantiles of normalized asset returns using asset-specific features, while the second stage incorporates market data to adjust these predictions for broader economic conditions. This results in a generalized model that can be applied across various asset classes, including commodities, cryptocurrencies, as well as synthetic datasets. The predicted quantiles are then converted into full probability distributions through kernel density estimation, allowing for more precise return distribution predictions and inferencing. The LSTM model significantly outperforms a linear quantile regression baseline by 98% and a dense neural network model by over 50%, showcasing its ability to capture complex patterns in financial return distributions across both synthetic and real-world data. By using exclusively asset-class-neutral features, our model achieves robust, generalizable results. ...

Enhancing Causal Discovery in Financial Networks with Piecewise Quantile Regression ArXiv ID: 2408.12210 “View on arXiv” Authors: Unknown Abstract Financial networks can be constructed using statistical dependencies found within the price series of speculative assets. Across the various methods used to infer these networks, there is a general reliance on predictive modelling to capture cross-correlation effects. These methods usually model the flow of mean-response information, or the propagation of volatility and risk within the market. Such techniques, though insightful, don’t fully capture the broader distribution-level causality that is possible within speculative markets. This paper introduces a novel approach, combining quantile regression with a piecewise linear embedding scheme - allowing us to construct causality networks that identify the complex tail interactions inherent to financial markets. Applying this method to 260 cryptocurrency return series, we uncover significant tail-tail causal effects and substantial causal asymmetry. We identify a propensity for coins to be self-influencing, with comparatively sparse cross variable effects. Assessing all link types in conjunction, Bitcoin stands out as the primary influencer - a nuance that is missed in conventional linear mean-response analyses. Our findings introduce a comprehensive framework for modelling distributional causality, paving the way towards more holistic representations of causality in financial markets. ...

EX-DRL: Hedging Against Heavy Losses with EXtreme Distributional Reinforcement Learning ArXiv ID: 2408.12446 “View on arXiv” Authors: Unknown Abstract Recent advancements in Distributional Reinforcement Learning (DRL) for modeling loss distributions have shown promise in developing hedging strategies in derivatives markets. A common approach in DRL involves learning the quantiles of loss distributions at specified levels using Quantile Regression (QR). This method is particularly effective in option hedging due to its direct quantile-based risk assessment, such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). However, these risk measures depend on the accurate estimation of extreme quantiles in the loss distribution’s tail, which can be imprecise in QR-based DRL due to the rarity and extremity of tail data, as highlighted in the literature. To address this issue, we propose EXtreme DRL (EX-DRL), which enhances extreme quantile prediction by modeling the tail of the loss distribution with a Generalized Pareto Distribution (GPD). This method introduces supplementary data to mitigate the scarcity of extreme quantile observations, thereby improving estimation accuracy through QR. Comprehensive experiments on gamma hedging options demonstrate that EX-DRL improves existing QR-based models by providing more precise estimates of extreme quantiles, thereby improving the computation and reliability of risk metrics for complex financial risk management. ...

Postprocessing of point predictions for probabilistic forecasting of day-ahead electricity prices: The benefits of using isotonic distributional regression ArXiv ID: 2404.02270 “View on arXiv” Authors: Unknown Abstract Operational decisions relying on predictive distributions of electricity prices can result in significantly higher profits compared to those based solely on point forecasts. However, the majority of models developed in both academic and industrial settings provide only point predictions. To address this, we examine three postprocessing methods for converting point forecasts of day-ahead electricity prices into probabilistic ones: Quantile Regression Averaging, Conformal Prediction, and the recently introduced Isotonic Distributional Regression. We find that while the latter demonstrates the most varied behavior, it contributes the most to the ensemble of the three predictive distributions, as measured by Shapley values. Remarkably, the performance of the combination is superior to that of state-of-the-art Distributional Deep Neural Networks over two 4.5-year test periods from the German and Spanish power markets, spanning the COVID pandemic and the war in Ukraine. ...

Portfolio diversification with varying investor abilities ArXiv ID: 2311.06519 “View on arXiv” Authors: Unknown Abstract We introduce new mathematical methods to study the optimal portfolio size of investment portfolios over time, considering investors with varying skill levels. First, we explore the benefit of portfolio diversification on an annual basis for poor, average and strong investors defined by the 10th, 50th and 90th percentiles of risk-adjusted returns, respectively. Second, we conduct a thorough regression experiment examining quantiles of risk-adjusted returns as a function of portfolio size across investor ability, testing for trends and curvature within these functions. Finally, we study the optimal portfolio size for poor, average and strong investors in a continuously temporal manner using more than 20 years of data. We show that strong investors should hold concentrated portfolios, poor investors should hold diversified portfolios; average investors have a less obvious distribution with the optimal number varying materially over time. ...

Econometric Model Using Arbitrage Pricing Theory and Quantile Regression to Estimate the Risk Factors Driving Crude Oil Returns ArXiv ID: 2309.13096 “View on arXiv” Authors: Unknown Abstract This work adopts a novel approach to determine the risk and return of crude oil stocks by employing Arbitrage Pricing Theory (APT) and Quantile Regression (QR).The APT identifies the underlying risk factors likely to impact crude oil returns.Subsequently, QR estimates the relationship between the factors and the returns across different quantiles of the distribution. The West Texas Intermediate (WTI) crude oil price is used in this study as a benchmark for crude oil prices. WTI price fluctuations can have a significant impact on the performance of crude oil stocks and, subsequently, the global economy.To determine the proposed models stability, various statistical measures are used in this study.The results show that changes in WTI returns can have varying effects depending on market conditions and levels of volatility. The study highlights the impact of structural discontinuities on returns, which can be caused by changes in the global economy and the demand for crude oil.The inclusion of pandemic, geopolitical, and inflation-related explanatory variables add uniqueness to this study as it considers current global events that can affect crude oil returns.Findings show that the key factors that pose major risks to returns are industrial production, inflation, the global price of energy, the shape of the yield curve, and global economic policy uncertainty.This implies that while making investing decisions in WTI futures, investors should pay particular attention to these elements ...