[Paper Review] Probabilistic Forecasting in Day-Ahead Electricity Markets: Simulating Peak and Off-Peak Prices

Peru Muniain, Florian Ziel|arXiv (Cornell University)|Oct 19, 2018

Energy Load and Power Forecasting39 references44 citations

TL;DR

This paper proposes a bivariate mean-reverting jump diffusion (MRJD) model with correlated jump occurrences to improve probabilistic forecasting of German-Austrian day-ahead electricity peak and off-peak prices. By incorporating a bivariate Bernoulli jump process and evaluating forecasts using the energy score—sensitive to dependency structures—the model significantly outperforms univariate jump and independent jump models, especially in capturing joint price spikes and correlation dynamics.

ABSTRACT

In this paper we include dependency structures for electricity price forecasting and forecasting evaluation. We work with off-peak and peak time series from the German-Austrian day-ahead price, hence we analyze bivariate data. We first estimate the mean of the two time series, and then in a second step we estimate the residuals. The mean equation is estimated by OLS and elastic net and the residuals are estimated by maximum likelihood. Our contribution is to include a bivariate jump component on a mean reverting jump diffusion model in the residuals. The models' forecasts are evaluated using four different criteria, including the energy score to measure whether the correlation structure between the time series is properly included or not. In the results it is observed that the models with bivariate jumps provide better results with the energy score, which means that it is important to consider this structure in order to properly forecast correlated time series.

Motivation & Objective

• To improve probabilistic forecasting of peak and off-peak electricity prices in day-ahead markets.
• To incorporate complex dependency structures—specifically correlated jump occurrences—into multivariate electricity price models.
• To evaluate forecast performance using criteria sensitive to multivariate distributional properties, especially the energy score.
• To demonstrate that modeling cross-series dependencies enhances forecast accuracy for derivatives trading.
• To provide a framework that integrates mean reversion, seasonality, volatility clustering, and correlated spikes in a unified probabilistic model.

Proposed method

• Uses a two-step approach: first estimating the conditional mean via OLS and elastic net with seasonal and autoregressive components.
• Models residuals using a mean-reverting jump diffusion (MRJD) process with a bivariate jump component based on a correlated Bernoulli process.
• Incorporates time-varying volatility through a CCC-GARCH structure in the residual dynamics.
• Simulates future price paths using Monte Carlo methods based on the estimated bivariate MRJD model.
• Evaluates forecasts using four criteria: MAE, MSE, pinball score, and energy score (ES), with ES being sensitive to multivariate dependence.
• Applies the Diebold-Mariano test to assess statistical significance of performance differences between models.

Experimental results

Research questions

• RQ1 Does modeling correlated jump occurrences between peak and off-peak electricity prices improve forecast accuracy compared to independent or univariate jump models?
• RQ2 Can the energy score effectively detect improvements in multivariate forecast performance when dependency structures are correctly modeled?
• RQ3 How do different model specifications—especially those with bivariate jumps and CCC-GARCH—compare in out-of-sample forecasting performance?
• RQ4 To what extent do multivariate evaluation criteria like the energy score and pinball score outperform univariate metrics like MAE and MSE in capturing joint price behavior?
• RQ5 Is the inclusion of correlated jumps more beneficial for risk management (e.g., VaR) than for point forecasting?

Key findings

• The ARX-BiJ-µd model with bivariate jumps significantly outperforms other models in terms of the energy score, indicating superior capture of dependency structures.
• Models with bivariate jumps show slightly better performance on the pinball score, suggesting improved distributional forecasting, particularly for quantiles.
• Despite better in-sample fit, the ARX-BiJ-µd-GARCH model with CCC-GARCH structure underperforms its non-GARCH counterpart out-of-sample, indicating overfitting.
• The energy score is the only criterion that detects the benefit of correlated jumps, as it accounts for the full bivariate distribution and dependence.
• Observed price spikes occurring simultaneously in peak and off-peak markets are better captured by models with bivariate jumps, as evidenced by trajectory simulations.
• The ARX-BiJ-µd model produces more realistic forecast trajectories with wider spread and higher density around the median, compared to the more compact and less realistic ARX-enet model.