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Day-ahead electricity price forecasting with high-dimensional structures: Univariate vs. multivariate modeling frameworks

Florian Ziel, Rafal Weron

arXiv:1805.06649v1stat.APq-fin.STstat.ML

TL;DR

The paper investigates whether univariate or multivariate structures are optimal for short-term electricity price forecasting through an extensive empirical comparison and lasso-based variable-selection analysis. Across 12 datasets, multivariate models have only a minor overall predictive edge and do not consistently outperform univariate models, while forecast combination can improve accuracy. The study consequently provides guidance on model design and selected explanatory variables.

  • Problem

    The paper addresses the unresolved question of whether short-term electricity price forecasting should use univariate or multivariate model structures.

  • Method

    The study conducts an extensive comparison of univariate and multivariate forecasting models across 12 datasets and analyzes variable selection in high-dimensional lasso-type models.

  • Results

    Multivariate models have a minor overall predictive edge but do not uniformly outperform univariate models across datasets, seasons, or hours; simple forecast averaging can improve accuracy.

  • Takeaways & Limitations

    Combining forecasts from both frameworks and incorporating important periodic structures can support better-performing electricity price forecasting designs.

  • Takeaways & Limitations

    The univariate specification tested does not improve from nonlinear effects, possibly because it represents periodic minimum and maximum effects with only two parameters.

Abstract

from arXiv · show

We conduct an extensive empirical study on short-term electricity price forecasting (EPF) to address the long-standing question if the optimal model structure for EPF is univariate or multivariate. We provide evidence that despite a minor edge in predictive performance overall, the multivariate modeling framework does not uniformly outperform the univariate one across all 12 considered datasets, seasons of the year or hours of the day, and at times is outperformed by the latter. This is an indication that combining advanced structures or the corresponding forecasts from both modeling approaches can bring a further improvement in forecasting accuracy. We show that this indeed can be the case, even for a simple averaging scheme involving only two models. Finally, we also analyze variable selection for the best performing high-dimensional lasso-type models, thus provide guidelines to structuring better performing forecasting model designs.

1. Introduction

The paper addresses an unresolved and insufficiently studied question in short-term electricity price forecasting: whether univariate or multivariate model structures are preferable. It evaluates this choice across hours, seasons, markets, and lagged price information.

  • Research question: The literature lacks consensus on representing the 24-hour price series with one univariate model or separate, potentially interdependent multivariate models.The choice concerns whether parameters are shared across load periods or estimated separately for each period.
  • Research gap: Existing comparisons are few and limited, with earlier studies reaching different conclusions about which framework forecasts better.One cited study favors separately modeled hours, while another reports that performance varies by time of day.
  • Research questions: The study aims to provide broad empirical evidence on framework superiority across datasets, hours, and seasons.It also asks which past spot-price values should enter EPF models.
  • Study design: The empirical design compares univariate and multivariate frameworks through ten forecasting models spanning eight model classes.The classes include benchmarks, similar-day methods, autoregressive structures, univariate AR models, and VAR models.

2. The univariate and multivariate modeling frameworks

The paper distinguishes multivariate frameworks that organize prices by day and hour from univariate frameworks that treat them as one hourly series. These choices differ in cross-hour dependence, variance assumptions, estimation, and forecasting schemes.

  • Multivariate framework: Day-ahead auctions disclose all 24 hourly prices together, motivating treatment of each day as a 24-dimensional multivariate vector.The multivariate perspective follows the market’s daily auction structure.
  • Univariate framework: A basic univariate structure estimates 24 separate same-hour models, which is simple and computationally light but omits cross-hour dependencies.Interdependencies can be added by including prices from other load periods as explanatory variables.
  • Fully multivariate framework: Fully multivariate models jointly estimate all load periods and impose a joint distribution on the 24-dimensional error vector.The fully multivariate specification is more general and nests sets of interrelated models.
  • Forecasting implications: Fully multivariate models can fit in-sample better without necessarily producing better forecasts.A two-step estimation procedure may ignore dependencies among individual specifications, although its forecasts need not be worse.
  • Forecasting schemes: Univariate forecasting produces 24-step-ahead predictions either recursively or directly for each load period.Recursive forecasts use earlier predicted prices, whereas direct forecasts avoid them but omit the most recent information.
  • Forecasting schemes: Recursive forecasting is vulnerable to accumulated propagated errors, while direct forecasting may lose useful recent information for late-night and early-morning hours.The two schemes therefore involve different forecast-risk trade-offs.
  • Forecasting implications: Changing representation has marginal forecasting impact for implicit innovation specifications because methods such as OLS are asymptotically consistent under both homoscedastic and heteroscedastic innovations.The paper illustrates alternative representations in an appendix.

3. Datasets and data preprocessing

The study uses 12 electricity price datasets and evaluates forecasting models with rolling calibration. It preprocesses diverse price series using an area hyperbolic sine transformation and robust median–MAD normalization.

  • Datasets: The empirical study covers 12 electricity spot-price datasets, with European day-ahead markets represented at 24 hourly load periods.The datasets include eleven European series and one GEFCom2014 dataset.
  • Datasets: The European datasets span six years, while GEFCom2014 covers three years.The table caption specifies the corresponding calendar ranges.
  • Evaluation design: Models are evaluated using a 730-day rolling calibration window, reestimating models and forecasting all 24 hours after each daily shift.The window is approximately two years long.
  • Variance stabilization: The area hyperbolic sine transformation is used as an automated alternative for datasets containing near-zero or negative prices.It reduces spike severity while retaining positive and negative price observations.
  • Variance stabilization: The transformation uses a 730-day calibration-sample median as its shift and a consistency-adjusted median absolute deviation as its scale.Models are calibrated on transformed prices and forecasts are inverse-transformed before evaluation.

4. Models

The paper evaluates a broad set of benchmark, autoregressive, multivariate, univariate, and lasso-based forecasting models using seasonal structures and transformed prices. Its designs emphasize comparisons between modeling frameworks and formal variable selection for high-dimensional specifications.

  • Model classes: The study considers 58 price-only models from eight model classes, while focusing on representative models in the main text.The models exclude stochastic weather, load, and renewable-generation forecasts, while retaining deterministic calendar dummies.
  • Model classes: The model classes include weekly hourly means, a similar-day naive benchmark, expert autoregressions, independent hourly AR models, VARs, and lasso-based AR structures.The VAR models are the only fully multivariate models in the study, while lasso shrinks redundant explanatory-variable coefficients to zero.
  • Seasonality and means: The forecasting designs use hour-of-day, day-of-week, and hour-of-week dummies to construct three time-varying means, alongside the full-sample mean.All four means are computed iteratively over each 730-day calibration window.
  • Benchmarks: The naive benchmark uses the price from a week earlier for Monday, Saturday, and Sunday, and yesterday’s price for other days.Two simpler similar-day variants were also considered but had worse predictive performance and were not reported.
  • Autoregressive models: Expert autoregressive models encode lagged prices, lagged daily minimum and maximum prices, day-of-week effects, and periodic effects through hour-specific coefficients.The lag structure reaches up to eight days, and coefficients are indexed by target hour, lag, hourly or aggregate daily value, and weekday.
  • Univariate autoregression: The ARHoW benchmark uses autoregressive parameters selected by minimizing AIC, with a potential memory of eight days, and previously outperformed sophisticated structures in one market.The model is estimated through the Yule-Walker equations.

5. Empirical results

Across 12 datasets, lasso models generally dominate, but neither modeling framework is uniformly superior across markets, seasons, or hours. Simple forecast averaging improves accuracy, while hourly tests reveal complementary strengths between the best univariate and multivariate models.

  • Overall performance: Lasso models dominate the competing structures, although no single lasso specification is best across all datasets.The multivariate 24lassoHQC DoW,p,nl and best univariate lasso forecasts are not significantly different in several markets.
  • Overall performance: 7 datasets select the multivariate 24lassoHQC DoW,p,nl model, while 5 select the univariate lassoHQC DoW,p model.
  • Overall performance: 1.90% m.p.d.f.b.MAE makes the simple univariate ARHoW model only slightly worse than sophisticated lasso structures.ARHoW even beats the overall best-performing 24lassoHQC DoW,p,nl model in EPEX.DE+AT and OTE.CZ.
  • Seasonal performance: Fall results give univariate models an edge over multivariate lasso models, whereas annual rankings generally persist in Spring, Summer, and Winter.The authors suggest that using all hourly prices from the past week may help univariate models adapt to year-end price and volatility increases.
  • Hourly performance: The best multivariate lasso model more often outperforms the univariate model in morning hours, while the univariate model more often wins in late evening and night hours.No universal daily pattern is observed, and the direction varies across markets.
  • Hourly performance: Hourly DM tests show that sophisticated lasso models are significantly better than competing structures only for some hours, not uniformly across the day.For BELPEX.BE, the univariate lasso is significantly better for seven hours and significantly worse for five.
  • Forecast combination: Arithmetic averaging of the two lasso forecasts improves accuracy over the ex-post selected lasso model for all 12 datasets.The improvement is significant at the 5% level for most markets; the 1.30% GEFCom2014 improvement is not significant, likely because its test period is shorter.

6. Conclusions and guidelines for energy forecasters

The study finds a minor overall predictive edge for multivariate EPF, but neither framework dominates across datasets, seasons, or hours. Forecast averaging and selected periodic, lagged, and lasso structures provide practical guidance for model design.

  • Framework comparison: The multivariate approach has a minor overall edge under MAE, RMSE, and m.p.d.f.b., but does not uniformly outperform univariate models.Univariate models sometimes perform better across datasets, seasons, and hours.
  • Forecast combination: A simple arithmetic average of the best multivariate and univariate lasso forecasts beats the better ex-post model on all 12 datasets.
  • Seasonal variation: Univariate models have an edge over multivariate lasso models in Fall, while annual model rankings persist in Spring, Summer, and Winter.The authors suggest season-specific models may further increase predictive efficiency.
  • Market variation: Across markets, forecasting accuracy varies, yet HQC-selected multivariate lasso or periodic-effect univariate lasso models consistently provide the best performance.The three multivariate lasso models are recommended generally because they never perform badly.
  • Variable selection: Variable-selection analysis highlights last-day same or neighboring hours, last-day midnight, and the same hour one week earlier.Periodic effects interacting daily dummies with last-day prices are especially important and are suggested for both expert and general designs.
  • Model assumptions: Converting between frameworks changes residual-variance assumptions: multivariate models allow hour-specific variance, whereas univariate estimation assumes equal variance across hours.Iterative reweighting can incorporate changing variance effects, especially in univariate approaches.

Appendix A. The set of models

The appendix defines additional C3–C8 models that were included in the empirical study but not discussed in the main comparison sections.

  • Appendix scope: Remaining models from classes C3–C8 are defined in the appendix and briefly evaluated in Appendix C.Their evaluation uses WMAE and m.p.d.f.b.

Appendix A.1. Expert models (class C3)

The expert-model appendix specifies variants that add day-of-week and periodic effects to asinh-transformed prices, with reduced forms omitting selected effects or using alternative demeaning.

  • Expert models: The generic expert models use the asinh-transformed price on day d and hour h.
  • Expert models: Collinearity permits dropping βh,14, βh,21, and βh,28, leaving effectively 25 parameters, or 18 when h = 24.The parameters are estimated using OLS.
  • Expert models: expertDoW,p,nl adds periodic effects to the expertDoW,nl specification.The model allows different structures, including previous-day midnight prices, for different days of the week.
  • Expert models: Variants remove periodic effects, nonlinear effects, or both from the full expertDoW,p,nl model.Restricting selected sums to Monday, Saturday, and Sunday yields another set of expert variants.
  • Expert models: The starred expert models replace the intercept with demeaning by the daily hourly mean, following the alternative formulation described by Uniejewski et al.

Appendix A.2. The second 24AR-type model (class C4)

The second 24AR-type model demeans the asinh-transformed price by the daily hourly mean and models each hour independently as an AR(ph) process.

  • 24AR-type model: The 24ARHoD specification uses daily-hourly-mean demeaning rather than weekly-hourly-mean demeaning.It is estimated analogously to 24ARHoW.

Appendix A.3. The second VAR-type model (class C5)

The VARHoD model is introduced as an alternative VARHoW specification, modeling 24 hourly values jointly and calibrating them through multivariate Yule-Walker equations.

  • VARHoD models the 24 hourly observations of each day as a vector centered by the calibration-sample mean vector.
  • The model is calibrated by solving multivariate Yule-Walker equations with maximum order pmax = 8.

Appendix A.4. Multivariate lasso models (class C6)

The multivariate lasso framework defines model families by restricting periodic and non-linear effects, while varying the information criterion used for estimation and tuning.

  • The framework defines 16 models from a general multivariate specification using restrictions on periodic and non-linear effects.
  • The four model groups comprise the full model, versions without periodic effects, versions without non-linear effects, and versions without both.
  • The superscript IC identifies AIC, HQC, BIC, or OLS as the information criterion used.
  • Lasso tuning selects λ by maximizing an in-sample generalized information criterion based on residual sum of squares, non-zero parameters, and residual variance.
  • AIC uses κ = 2, while BIC uses κ = log(n); OLS corresponds to κ = 0.
  • Two HQC-based multivariate models are included in the empirical study.

Appendix A.6. Univariate lasso models (class C8)

The univariate lasso framework constructs 16 model variants by combining restrictions on periodic and non-linear effects with four information-criterion choices.

  • The framework defines 16 univariate lasso models from a general autoregressive formula and restrictions on periodic and non-linear effects.
  • The four groups include the full model, models without non-linear effects, models without periodic effects, and models without both.
  • The superscript IC denotes AIC, HQC, BIC, or OLS, with estimation details referenced elsewhere.
  • Two HQC-based univariate models are analyzed in the empirical study.

Appendix B. Alternative representations

The appendix provides alternative representations of selected models to clarify autoregressive structures and relationships between multivariate and univariate frameworks.

  • The alternative representations are intended to clarify autoregressive structures and relationships between multivariate and univariate modeling frameworks.
  • The expertDoW,p,nl model is represented as a sparse 168-periodic autoregressive model with non-linear impact.
  • Its coefficients encode hour-of-day effects, lag-specific terms, periodic effects, and minimum and maximum values from the previous day.
  • The 24ARHoD model can be rewritten as a sparse 24-periodic AR model, while VARHoD can be rewritten as a 24-periodic AR model.
  • VARHoD may alternatively be expressed as a set of 24 single equations with coefficient restrictions defining the lag structure.
  • The univariate AR model can also be written as a special case of a 24-dimensional VAR process.

Appendix C. Model selection

Appendix C compares 58 models across major electricity markets and datasets, highlighting how model-selection criteria and weekly-seasonality terms affect forecasting performance. HQC generally favors the strongest lasso models, while multivariate advantages are small and calibration-sample limitations can substantially hurt complex models.

  • Results overview: Figures C.6 and C.7 summarize MAE results for all 58 models across four major markets and all 12 datasets.Figure C.6 reports full-period out-of-sample MAE; Figure C.7 reports mean percentage deviation from the best model.
  • Lasso model selection: HQC produces the best-performing lasso models on average for both multivariate and univariate specifications.The appendix contrasts HQC with previously tried AIC and BIC criteria.
  • Lasso model selection: HQC-selected multivariate lasso models outperform univariate lasso models on average, but the differences are small and vary across markets.For NP.DK1 and NP.DK2, lassoBIC_DoW is reported as good as lassoHQC_DoW,p.
  • Calibration sample size: Using the full model with the OLS criterion worsens performance, especially for multivariate lasso models.The appendix suggests that 730 observations may be too few for the multivariate, multi-parameter structure.
  • Weekly seasonality: Including dummies for every day of the week improves expert and multi-parameter models relative to counterparts without the DoW component.Examples include 24ARHoW over 24ARHoD, VARHoW over VARHoD, and ARDoW over AR.
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