Day-ahead electricity price forecasting with high-dimensional structures: Multi- vs. univariate modeling frameworks

Transcription

1 Day-ahead electricity price forecasting with high-dimensional structures: Multi- vs. univariate modeling frameworks Rafał Weron Department of Operations Research Wrocław University of Science and Technology, Poland Based on a working paper with Florian Ziel, available from RePEc: Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 1 / 41

2 Introduction Electricity markets and prices Markets for electricity in Europe Nord Pool (DK, EST, FIN, NOR, SWE) N2EX (UK) Belpex (BE) EPEX Spot (AT,CH, DE, FR) APX-ENDEX (NL) PolPX (PL) OTE (CZ) OKTE (SK) OPCOM (RO) OMIE (ES, PT) GME (IT) HUPX (HU) EXAA (AT) Borzen (SLO) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 2 / 41

3 Introduction Electricity markets and prices... in North America and Australia Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 3 / 41

4 Introduction Electricity markets and prices Electricity price time series Seasonality, mean-reversion and price spikes Daily POLPX spot price [PLN/MWh] Days [ ] Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 4 / 41

5 Introduction The spot price The electricity spot (day-ahead) price Day D Day D + 1 Day D + 2 Bidding for day D + 1 Bidding for day D hours of day D hours of day D + 2 Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 5 / 41

6 Introduction The spot price Supply and demand, renewables and negative prices Source: Ziel & Steinert (2016) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 6 / 41

7 Introduction The spot price Prices for different load periods Strongly correlated but seem to follow different data generating processes (DGPs) Load period 6 (2:30 3:00) Load period 36 (17:30 18:00) Log price Nov Mar Aug Jan 2013 Time Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 7 / 41

8 Introduction Electricity price forecasting (EPF) First read on electricity price forecasting (EPF) R.Hyndman: this paper alone is responsible for 0.7 of the current IF 2Y =2.642 ;-) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 8 / 41

9 Introduction Electricity price forecasting (EPF) Load and price forecasting classification (Hong & Fan, 2016, IJF; Weron, 2014, IJF) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 9 / 41

10 Introduction Electricity price forecasting (EPF) A taxonomy of (price) modeling approaches (Weron, 2014, IJF) Electricity price forecasting (EPF) and modeling approaches Multi-agent Fundamental Reduced-form Statistical Computational intelligence Nash- Cournot framework Parameter rich fundamental Jumpdiffusions Similar-day, exponential smoothing Feed-forward neural networks Supply function equilibrium Parsimonious structural Markov regimeswitching (Auto) regression models Recurrent neural networks Strategic productioncost Shrinkage, variable selection Fuzzy neural networks Agent-based Threshold AR Support vector machines GARCH-type Hybrid models, Forecast averaging Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 10 / 41

11 Introduction Electricity price forecasting (EPF) Two questions a forecaster has to ask 1 What technique shall I use for short-term EPF? (Auto)regression Neural net 2 Which modeling framework shall I use? Multivariate Univariate Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 11 / 41

12 Introduction Electricity price forecasting (EPF) Two most common EPF tools: Regression... Electricity price for day d and hour h: Y d,h = β h,1 + β h,2 Y d 1,h + β h,3 Y d 2,h + β h,4 Y d 7,h }{{} autoregressive terms + β h,5 Yd 1 min + β h,6 Yd 1 max }{{} non-linear effects + 7 β j=1 h,j+7d j +ε d,h, }{{} weekday dummies + β h,7 Y d 1,24 }{{} end-of-day effect Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 12 / 41

13 Introduction Electricity price forecasting (EPF)... and neural nets Input layer Y d 1,h Hidden layer Ouput layer Y d 2,h Y d 7,h Y min d 1 Y max d 1 Y d,h Y d 1,24 D 1 D 7. Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 13 / 41

14 Agenda Introduction Electricity markets and prices The spot price Electricity price forecasting (EPF) Modeling frameworks Multivariate Univariate Case study Datasets, variance stabilization 58 models Results Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 14 / 41

15 Multi- vs. univariate modeling frameworks Optimal representation of the price series No consensus in the existing literature on STPF Multivariate framework Implemented separately across the load periods, leading to 24 (48 or more) sets of parameters for each day More popular in the statistical EPF literature (Weron, 2014) Univariate framework One large model be implemented jointly for all load periods, yielding one large set of parameters More popular in the engineering EPF literature, dominated by neural network models (Aggarwal et al., 2009) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 15 / 41

16 Multi- vs. univariate modeling frameworks The multivariate framework Rationale for the multivariate framework 1 Load forecasting has favored the multi-model specification in the short-term 2 Each load period displays a rather distinct price profile 3 The day-ahead auction market structure 4 Simple to implement, only a small number of parameters is needed for each load period Log price Load period 6 (2:30 3:00) Load period 36 (17:30 18:00) Nov Mar Aug Jan 2013 Time Bidding for day D + 1 Day D Day D + 1 Day D + 2 Bidding for day D hours of day D hours of day D + 2 Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 16 / 41

17 Multi- vs. univariate modeling frameworks The multivariate framework Proponents of the multivariate approach Statistical time series models (Cuaresma et al., 2004; Misiorek et al., 2006; Weron, 2006; Zhou et al., 2006; Karakatsani & Bunn, 2008; Weron & Misiorek, 2008; Garcia-Martos et al., 2012b; Gaillard et al., 2016; Maciejowska et al., 2016; Nowotarski & Weron, 2016a; Hagfors et al., 2016; Ziel, 2016a) Neural network type models (Yamin et al., 2004; Amjady & Keynia, 2009a; Andalib & Atry, 2009; Dudek, 2016; Keles et al., 2016) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 17 / 41

18 Multi- vs. univariate modeling frameworks The multivariate framework Multivariate frameworks 1 A set of 24 univariate models: P d,1 = f 1 (P d 1,1, P d 2,1,...) + ε d,1 ˆP d,1,. P d,24 = f 24 (P d 1,24, P d 2,24,...) + ε d,24 ˆP d,24, no dependencies between the models for individual hours 2 A set of 24 interrelated models: P d,1 = f 1 (P d 1,1, P d 2,1,..., P d 1,24, P d 2,24,...) + ε d,1 ˆP d,1,.. P d,24 = f 24 (P d 1,1, P d 2,1,..., P d 1,24, P d 2,24,...) + ε d,24 ˆP d,24,. Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 18 / 41

19 Multi- vs. univariate modeling frameworks The multivariate framework A fully multivariate modeling framework 3 Treat the price series as panel data: P d,1. = f P d,24 P d 1,1 P d 2,1 ε d,1.,., P d 1,24 P d 2,24 ε d,24 ˆP d,1. ˆP d,24 Vector AutoRegression (Huisman et al., 2007; Panagiotelis & Smith, 2008; Haldrup et al., 2010; He et al., 2015) Works if the number of parameters is not too large Reduce dimensionality consider factor models (Garcia-Martos et al., 2012a; Wu et al., 2013; Maciejowska & Weron, 2015; Raviv et al., 2015) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 19 / 41

20 Multi- vs. univariate modeling frameworks The univariate framework Proponents of the univariate approach Statistical time series models (Nogales et al., 2002; Contreras et al., 2003; Conejo et al., 2005; Zareipour et al., 2006; Paraschiv et al., 2015; Ziel et al., 2015a) Neural network type models (Rodriguez & Anders, 2004; Amjady, 2006; Gareta et al., 2006; Amjady et al., 2010; Abedinia et al., 2015; Kim, 2015) Treat day-ahead electricity prices as one univariate, high-frequency time series: P t = f (P t 1, P t 2,...) + ε t Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 20 / 41

21 Multi- vs. univariate modeling frameworks Similarities and differences Multi- vs. univariate models In general, every multivariate model can be rewritten as univariate and vice versa (but the error structure changes) The real difference is the forecasting procedure used Multivariate framework prices for all load periods of the next day are predicted at once Univariate approach the forecasts are computed recursively, the price forecast for hour 1 is used as input (i.e., an explanatory variable) when making the prediction for hour 2, etc. Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 21 / 41

22 Multi- vs. univariate modeling frameworks Similarities and differences Multi- vs. univariate model comparisons Very few and very limited studies in the EPF literature: Cuaresma et al. (2004) apply variants of AR(1) and general ARMA processes and conclude that multivariate models present uniformly better forecasting properties than univariate Ziel (2016a) notes that, simple univariate models generally perform better for the first half of the day, whereas simple multivariate models are better in the second half of the day Our paper is the first through empirical study Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 22 / 41

23 Agenda Introduction Electricity markets and prices The spot price Electricity price forecasting (EPF) Modeling frameworks Multivariate Univariate Case study Datasets, variance stabilization 58 models Results Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 23 / 41

24 Case study What is this study about? We address three pertinent questions: 1 Which modeling framework multivariate or univariate is better for EPF? 2 If one of them is better, is it better across all hours, seasons of the year and markets? 3 How many and which past values of the spot price process should be used in EPF models? We use: 12 electricity spot price datasets 58 models from 8 model classes (C1-C8) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 24 / 41

25 Case study Datasets and variance stabilization Datasets Electricity market and region Acronym Unit Source BELPEX price for Belgium BELPEX.BE EUR/MWh belpex.be EPEX price for Switzerland EPEX.CH EUR/MWh epexspot.com EPEX price for Germany and Austria EPEX.DE+AT EUR/MWh epexspot.com EPEX price for France EPEX.FR EUR/MWh epexspot.com EXAA price for Germany and Austria EXAA.DE+AT EUR/MWh exaa.at GEFCom2014 competition data GEFCOM2014 USD/MWh Hong et al. (2016) Nord Pool price for West Denmark NP.DK1 EUR/MWh nordpoolspot.com Nord Pool price for East Denmark NP.DK2 EUR/MWh nordpoolspot.com Nord Pool System price NP.SYS EUR/MWh nordpoolspot.com OMIE price for Spain OMIE.ES EUR/MWh omie.es OMIE price for Portugal OMIE.PT EUR/MWh omie.es OTE price for the Czech Republic OTE.CZ EUR/MWh ote-cr.cz The GEFCom2014 dataset covers a 3 year period ( ) The remaining datasets a 6 year period from ( ) We use a 730-day (ca. 2-year) rolling calibration window Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 25 / 41

26 Case study Datasets and variance stabilization Variance stabilizing transformation (VST) We use the area hyperbolic sine transformation: asinh(x) = log ( x + x ) for standardized spot prices x = 1 b {P d,h a} with a = median and b = MAD The asinh transformation has been already used in the context of electricity prices by Schneider (2011, JEM)... but the article went unnoticed For a recent review and evaluation of different VSTs see Uniejewski, Weron & Ziel (2017, IEEE-TPWRS) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 26 / 41

27 Case study Datasets and variance stabilization Variance stabilization for EPEX.DE+AT Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 27 / 41

28 Case study 58 models from 8 classes 58 models: Benchmarks C1. the weekly mean of hourly frequency benchmark, which is a simple periodic function denoted by mean HoW C2. the so-called naive benchmark of Nogales et al. (2002), which belongs to the class of similar-day techniques naive Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 28 / 41

29 Case study 58 models from 8 classes 58 models: Experts C3. 16 parsimonious AR models within a multivariate framework, built on some prior knowledge of experts and following Uniejewski et al. (2016, Energies) and Ziel (2016) called expert models, estimated using OLS represented by expert DoW,nl Y d,h = β h,1 + β h,2 Y d 1,h + β h,3 Y d 2,h + β h,4 Y d 7,h }{{} autoregressive effects + β h,5 Y d 1,min + β h,6 Y d 1,max +β h,7 Y d 1,24 }{{} non-linear effects 7 + β h,7+j DoW j d,h + ε d,h j=1 }{{} weekday dummies Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 29 / 41

30 Case study 58 models from 8 classes 58 models: AR and VAR benchmarks C4. 2 AR specifications composed of sets of 24 independent models (for each hour of the day; max lag of 8 days) and estimated using Yule-Walker equations represented by 24AR HoW C5. 2 VAR models (max lag of 8 days) estimated using multivariate Yule-Walker equations represented by VAR HoW C7. 4 univariate AR models (max lag of 8 24 = 196 hours) estimated using Yule-Walker equations represented by AR HoW Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 30 / 41

31 Case study 58 models from 8 classes 58 models: Multi- and univariate LASSO models C6. 16 parameter-rich AR structures within a multivariate framework, estimated using the least absolute shrinkage and selection operator (LASSO), which shrinks to zero the coefficients of redundant explanatory variables represented by 24lasso HQC DoW,p,nl and 24lasso HQC DoW,nl C8. 16 univariate parameter-rich AR specifications estimated using the LASSO represented by lasso HQC HoW,p and lassohqc HoW Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 31 / 41

32 Case study LASSO Automated variable selection via the LASSO Consider a general regression: p ŷ i = β j x i,j + ε i j=1 How to select predictors x i,j? How to estimate β j s? LASSO: minimize the residual sum of squares (RSS) + a linear penalty function of the β j s: ˆβ = argmin β j { N ( p ) 2 n } y i β j x i,j + λ β j i=1 } j=1 {{ } j=1 }{{ } RSS penalty See Tibshirani (1996) and Uniejewski et al. (2016, Energies) Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 32 / 41

33 Case study LASSO How does LASSO work? 0.5 Lasso λ Blue area constraint region, i.e., β 1 + β 2 t Red ellipses contours of the least squares error function β the (unconstrained) OLS estimate Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 33 / 41

34 Case study LASSO Table 5: Mean occurrence (in %) of the multivariate lasso model parameters across all 12 datasets and the full out-ofsample test period. Columns represent the hours and rows the parameters of the 24lasso Byproduct: Variable significance across hours HQC DoW,p,nl model, see Eqn. (15) for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables. Continued in Table 6. Day (d 3) Day (d 2) Day (d 1) h φ h,1,1, φ h,1,2, φ h,1,3, φ h,1,4, φ h,1,5, φ h,1,6, φ h,1,7, φ h,1,8, φ h,1,9, φ h,1,10, φ h,1,11, φ h,1,12, φ h,1,13, φ h,1,14, φ h,1,15, φ h,1,16, φ h,1,17, φ h,1,18, φ h,1,19, φ h,1,20, φ h,1,21, φ h,1,22, φ h,1,23, φ h,1,24, φ h,2,1, φ h,2,2, φ h,2,3, φ h,2,4, φ h,2,5, φ h,2,6, φ h,2,7, φ h,2,8, φ h,2,9, φ h,2,10, φ h,2,11, φ h,2,12, φ h,2,13, φ h,2,14, φ h,2,15, φ h,2,16, φ h,2,17, φ h,2,18, φ h,2,19, φ h,2,20, φ h,2,21, φ h,2,22, φ h,2,23, φ h,2,24, φ h,3,1, φ h,3,2, φ h,3,3, φ h,3,4, φ h,3,5, φ h,3,6, φ h,3,7, φ h,3,8, φ h,3,9, φ h,3,10, φ h,3,11, φ h,3,12, φ h,3,13, φ h,3,14, φ h,3,15, φ h,3,16, φ h,3,17, φ h,3,18, φ h,3,19, φ h,3,20, φ h,3,21, φ h,3,22, φ h,3,23, φ h,3,24, Table 6: Mean occurrence (in %) of the multivariate lasso model parameters across all 12 datasets and the full out-ofsample test period. Columns represent the hours and rows the parameters of the 24lasso HQC DoW,p,nl model, see Eqn. (15) for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables. Continued in Table 7. Day (d 6) Day (d 5) Day (d 4) h φ h,4,1, φ h,4,2, φ h,4,3, φ h,4,4, φ h,4,5, φ h,4,6, φ h,4,7, φ h,4,8, φ h,4,9, φ h,4,10, φ h,4,11, φ h,4,12, φ h,4,13, φ h,4,14, φ h,4,15, φ h,4,16, φ h,4,17, φ h,4,18, φ h,4,19, φ h,4,20, φ h,4,21, φ h,4,22, φ h,4,23, φ h,4,24, φ h,5,1, φ h,5,2, φ h,5,3, φ h,5,4, φ h,5,5, φ h,5,6, φ h,5,7, φ h,5,8, φ h,5,9, φ h,5,10, φ h,5,11, φ h,5,12, φ h,5,13, φ h,5,14, φ h,5,15, φ h,5,16, φ h,5,17, φ h,5,18, φ h,5,19, φ h,5,20, φ h,5,21, φ h,5,22, φ h,5,23, φ h,5,24, φ h,6,1, φ h,6,2, φ h,6,3, φ h,6,4, φ h,6,5, φ h,6,6, φ h,6,7, φ h,6,8, φ h,6,9, φ h,6,10, φ h,6,11, φ h,6,12, φ h,6,13, φ h,6,14, φ h,6,15, φ h,6,16, φ h,6,17, φ h,6,18, φ h,6,19, φ h,6,20, φ h,6,21, φ h,6,22, φ h,6,23, φ h,6,24, Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 34 / 41

35 Case study LASSO Table Byproduct: 7: Mean occurrence (in %) of the multivariate lasso Variable model parameters across all 12 datasets significance and the full out-of- across hours cont. sample test period. Columns represent the hours and rows the parameters of the 24lasso HQC DoW,p,nl model, see Eqn. (15) for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables. Continued in Table 8. Day (d 8) Day (d 7) h φ h,7,1, φ h,7,2, φ h,7,3, φ h,7,4, φ h,7,5, Table 8: Mean occurrence (in %) of the multivariate lasso model parameters across all 12 datasets and the full out-ofsample test period. Columns represent the hours and rows the parameters of the 24lasso HQC DoW,p,nl model, see Eqn. (15) φ h,7,6, φ h,7,7, φ h,7,8, φ h,7,9,0 for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables φ h,7,10, φ h,7,11, h φ h,7,12, φ h,1,min, φ h,7,13, φ h,2,min, φ h,7,14, φ h,3,min, φ h,7,15, φ h,4,min, φ h,7,16, φ h,5,min, φ h,7,17, φ h,6,min, φ h,7,18, φ h,7,min, φ h,7,19, φ h,8,min, φ h,7,20, φ h,1,max, φ h,7,21, φ h,2,max, φ h,7,22, φ h,3,max, φ h,7,23, φ h,4,max, φ h,7,24, φ h,5,max, φ h,8,1, φ h,6,max, φ h,8,2, φ h,7,max, φ h,8,3, φ h,8,max, φ h,8,4, φ h,0,0, φ h,8,5, φ h,0,0, φ h,8,6, φ h,0,0, φ h,8,7, φ h,0,0, φ h,8,8, φ h,0,0, φ h,8,9, φ h,0,0, φ h,8,10, φ h,0,0, φ h,8,11, φ h,1,h, φ h,8,12, φ h,1,h, φ h,8,13, φ h,1,h, φ h,8,14, φ h,1,h, φ h,8,15, φ h,1,h, φ h,8,16, φ h,1,h, φ h,8,17, φ h,1,h, φ h,8,18, φ h,1,24, φ h,8,19, φ h,1,24, φ h,8,20, φ h,1,24, φ h,8,21, φ h,1,24, φ h,8,22, φ h,1,24, φ h,8,23, φ h,1,24, φ h,8,24, φ h,1,24, Daily minimums Daily maximums DoW dummies Periodic on Y,h Periodic on Y, Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 35 / 41

36 Case study Mean Absolute Errors (MAE) and m.p.d.f.b. MAE: 10 models, all datasets Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 36 / 41

37 Case study Mean Absolute Errors (MAE) and m.p.d.f.b. m.p.d.f.b.: All models, all datasets m.p.d.f.b. in % meanhow naive expert expertdow expertp expertdow,p expert * * expert DoW * expert p * expert DoW,p expertnl expertdow,nl expertp,nl expertdow,p,nl * expert nl * expert DoW,nl * expert p,nl * expert DoW,p,nl 24ARHoD 24ARHoW VARHoD VARHoW OLS 24lasso DoW AIC 24lasso DoW HQC 24lasso DoW BIC 24lasso DoW OLS 24lasso DoW,p AIC 24lasso DoW,p HQC 24lasso DoW,p BIC 24lasso DoW,p OLS 24lasso DoW,nl AIC 24lasso DoW,nl HQC 24lasso DoW,nl BIC 24lasso DoW,nl OLS 24lasso DoW,p,nl AIC 24lasso DoW,p,nl HQC 24lasso DoW,p,nl BIC 24lasso DoW,p,nl AR ARDoW ARHoW ARHoD OLS lasso HoW AIC lasso HoW HQC lasso HoW BIC lasso HoW OLS lasso HoW,p AIC lasso HoW,p HQC lasso HoW,p BIC lasso HoW,p OLS lasso HoW,nl AIC lasso HoW,nl HQC lasso HoW,nl BIC lasso HoW,nl OLS lasso HoW,p,nl AIC lasso HoW,p,nl HQC lasso HoW,p,nl BIC lasso HoW,p,nl Mean percentage deviation from the best model: m.p.d.f.b. i = j=1 ERR i,j ERR best model,j ERR best model,j 100% where ERR best model,j = min 1 i 58 ERR i,j and ERR = MAE, RMSE, etc. Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 37 / 41

38 Case study Seasonal and intra-day variations Seasonal variations Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 38 / 41

39 Case study Seasonal and intra-day variations Diebold-Mariano test: X,Y,d,h = ε X,d,h ε Y,d,h Ziel (2016a): simple univariate models perform better for the first half of the day, but... Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 39 / 41

40 Case study A surprising conclusion Forecast averaging? Where the multivariate loss differential series in the Diebold-Mariano (DM) test: X,Y,d = ε X,d 1 ε Y,d 1, (22) defines the differences of errors in the 1 -norm, i.e., ε X,d 1 = 24 i=1 ε X,d,h Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 40 / 41

41 Take home messages When top performing LASSO-based models are compared, the multivariate approach has a minor edge in predictive accuracy... but it does not uniformly outperform the univariate one across all datasets, seasons of the year or hours of the day A simple average of the two is yet better For unconstrained models, the univariate AR HoW outperforms: 24 independent models, i.e., 24AR HoW 24 interrelated models, i.e., expert DoW,nl fully multivariate VAR HoW Rafał Weron (Wrocław, PL) EPF: Multi- vs. univariate modeling , Uni Verona 41 / 41