Rafał Weron * FORECASTING WHOLESALE ELECTRICITY PRICES: A REVIEW OF TIME SERIES MODELS. 1. Introduction

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1 To appear as: R. Weron (008) Forecasting wolesale electricity prices: A review of time series models, in "Financial Markets: Principles of Modelling, Forecasting and Decision-Making", eds. W. Milo, P. Wdowiński, FindEcon Monograp Series, WUŁ, Łódź. Rafał Weron * FORECASTING WHOLESALE ELECTRICITY PRICES: A REVIEW OF TIME SERIES MODELS Abstract. In tis paper we assess te sort-term forecasting power of different time series models in te electricity spot market. We calibrate autoregression (AR) models, including specifications wit a fundamental (exogenous) variable system load, to California Power Excange (CalPX) system spot prices. Ten we evaluate teir point and interval forecasting performance in relatively calm and extremely volatile periods preceding te market cras in winter 000/001. In particular, we test wic innovations distributions/processes Gaussian, GARCH, eavy-tailed (NIG, -stable) or non-parametric lead to best predictions. Keywords: Electricity price forecasting; eavy tailed distribution; autoregression model; GARCH model; nonparametric noise; system load JEL Classification: C, C46, C53, Q40 1. Introduction In te last decades, wit deregulation of power markets and introduction of competition, electricity price forecasts ave become a fundamental input to an energy company s decisionmaking mecanism (Bunn 004; Weron 006). Extreme price volatility, wic can be even two orders of magnitude iger tan for oter commodities or financial instruments, as forced producers and wolesale consumers to edge not only against volume risk but also against price movements. Sort-term price forecasts (STPF) are of particular interest for participants of auction-type spot electricity markets wo are requested to express teir bids in terms of prices and quantities. In suc markets buy (sell) orders are accepted in order of increasing (decreasing) prices until total demand (supply) is met. Consequently, a market participant tat is able to forecast spot prices can adjust its own production and (to some extent) consumption scedule accordingly and ence maximize its profits. Tis paper is a continuation and a review of our earlier studies on STPF of California electricity prices wit time series models (Misiorek et al. 006; Weron 006; Weron and Misiorek 007). Consequently, as a bencmark and a starting point we coose te autoregressive time series specification tat as been found to perform well for pre-cras California power market data. We compare it not only wit autoregressive models allowing for eteroskedastic (GARCH) or eavy-tailed (NIG, α-stable) innovations, but also wit an autoregressive model calibrated witin a non-parametric framework, were te innovations density is estimated by te Parzen Rosenblatt kernel. We call te latter model semiparametric because it as a parametric autoregressive part and a non-parametric noise distribution. In a detailed empirical study we evaluate te point and interval forecasting * P.D., Assistant Professor, Hugo Steinaus Center, Institute of Matematics and Computer Science, Wrocław University of Tecnology, , Wrocław, Poland. 1

2 performance of all investigated models. In particular, te novel in electricity price forecasting literature semi-parametric approac seems to be promising.. Te data and te base model Like in our previous studies, an assumption is made tat only publicly available information is used to predict spot prices, i.e. generation constraints or line and production capacity limits are not considered. Te dataset used in tis analysis CA_ourly.dat is distributed wit te MFE Toolbox (Weron 006). It was constructed using data obtained from te UCEI institute ( and te California independent system operator (CAISO; oasis.caiso.com). Apart from ourly system prices quoted in te California Power Excange (CalPX), it includes two fundamental variables: system-wide loads and day-aead CAISO load forecasts for te California market, see Figure 1. Figure 1: Hourly system prices (top panel) and ourly system loads (bottom panel) in California for te period July 5, 1999 December 3, 000. Te canging price cap ( USD/MW) is clearly visible in te top panel. Te day-aead load forecasts of te system operator CAISO are indistinguisable from te actual loads at tis resolution; only te latter ave been plotted. Te logaritmic transformation was applied to price, p t log( P t ), and load, z t log( Z t ), data to attain a more stable variance. Furtermore, te mean price and te median load were removed to center te data around zero. Removing te mean load resulted in worse forecasts, peraps, due to te very distinct and regular asymmetric weekly structure wit te five weekday values lying in te ig-load region and te two weekend values in te low-load region. Te data from te period July 5, 1999 April, 000 was used only for

3 calibration. Suc a relatively long period of data was needed to acieve ig accuracy. For example, limiting te calibration period to data coming only from te year 000, like in Contreras et al. (003), led to a decrease in forecasting performance by up to 70%. Consequently, te period April 3 December 3, 000 was used for out-of-sample testing. Since in practice te market-clearing price forecasts for a given day are required on te day before, we used te following (adaptive) testing sceme. To calibrate te models and compute price forecasts for our 1 to 4 of a given day, data available to all procedures included price and system load istorical data up to our 4 of te previous day plus day-aead load predictions for te 4 ours of tat day. Te time of te year, te day of te week and te our of te day influence price patterns. Price forecasting models sould take tese time factors into account. However, since we are focused on sort-term (day-aead) forecasting te annual seasonal beavior does not play a major role (also te used adaptive testing sceme allows te analyzed autoregressive models to quickly adapt to te canging conditions trougout te year). Te ourly and weekly seasonal patterns were andled in two different ways. Since eac our displays a rater distinct price profile reflecting te daily variation of demand, costs and operational constraints te modeling was implemented separately across te ours, leading to 4 sets of parameters. Tis approac was also inspired by te extensive researc on demand forecasting, wic as generally favored te multi-model specification for sort-term predictions (Bunn 000; Weron 006). On te oter and, te weekly seasonal beavior wic is mostly due to variable intensity of business activities trougout te week was captured by a combination of (i) te autoregressive structure of te models and (ii) daily dummy variables. Te log-price was made dependent on te log-prices for te same our on te previous days, and te previous weeks, as well as te minimum of all prices on te previous day (te latter created te desired link between bidding and price signals from te entire day). Furtermore, tree dummy variables (for Monday, Saturday and Sunday) were considered to differentiate between te weekends, te first working day of te week and te remaining business days (tis particular coice of te dummies is a consequence of te significance of te dummy coefficients for particular days). Te electricity spot price is not only dependent on te weekly and daily business cycles but also on oter fundamental variables tat can significantly alter tis deterministic seasonal beavior. Recall, tat te equilibrium between demand and supply defines te spot price. Bot demand and supply are influenced by weater conditions, most notably air temperatures. In te sort-term orizon, te variable cost of power generation is essentially just te cost of te fuel, consequently, te fuel price is anoter influential exogenous factor. Oter factors like power plant availability (capacity) or grid traffic (for zonal and modal pricing) could also be considered. However, including all tese factors would make te model not only cumbersome but also sensitive to te quality of te inputs and conditional on teir availability at a given time. Instead we ave decided to use only publicly available, igfrequency (ourly) information. In te California market of te late 1990s tis includes system-wide loads and day-aead CAISO load forecasts. In particular, te latter are important as tey include te system operator s (and to some extent te market s) expectations regarding weater, demand, generation and power grid conditions prevailing at te our of delivery. Te knowledge of tese forecasts allows, in general, for more accurate spot price predictions. In te studied period (wit some deviations in te volatile weeks 11-35), te logaritms of loads (or load forecasts) and te log-prices were approximately linearly dependent (te Pearson correlation was positive, ρ > 0.6, and igly significant wit a p-value of approximately 0; null of no correlation). At lag 0 te CAISO day-aead load forecast for a given our was used, wile for larger lags te actual system load was used. Interestingly, te best models turned out to be te 3

4 ones wit only lag 0 dependence. Using te actual load at lag 0, in general, did not improve te forecasts eiter. Tis penomenon can be attributed to te fact tat te prices are an outcome of te bids, wic in turn are placed wit te knowledge of te CAISO load forecasts but not actual future loads. Extensive studies performed by Weron (006) led to te conclusion tat te best autoregressive model structure for te (log-)price p t, in terms of forecasting performance for te first week of te test period (April 3-9, 000), was given by: ( t 1 t 1 Mon Sat 3 Sun t B ) p z d D d D d D, (1) were te autoregressive part ) p p a p a p a p a mp, ( B t t 1 t 4 t 48 3 t t t mp was te minimum of te 4 ourly (log-)prices on te previous day, z t was te (log-)load forecast and D, D, D were te dummy variables (for Monday, Saturday and Sunday). In tis base Mon Sat Sun model, denoted in te text as ARX, te noise term t is i.i.d. Gaussian. Recall tat te model, as well as its extensions described in te following Section, were estimated using an adaptive sceme, i.e. instead of using a single model for te wole sample, for every day (and our) in te test period we calibrated te model (given its structure) to te previous values of prices and loads and obtained a forecasted value for tat day (and our). 3. Model extensions Te residuals obtained from te fitted ARX model seemed to exibit a non-constant variance. Indeed, wen tested wit te Lagrange multiplier ARCH test statistics (Engle 198) te eteroskedastic effects were significant at te 5% level. Following Weron (006) we calibrate an ARX-G model, were G stands for GARCH(1,1). It differs from te ARX model in tat te noise term t in eqn. (1) is not just iid (0, ) but is given by t t t wit t 0 1 t 1 1 t 1. It as been long known tat financial asset returns are not normally distributed. Rater, te empirical observations exibit excess kurtosis (Carr et al. 00; Racev and Mittnik 000). Bottazzi et al. (005) and Weron (006) ave sown tat electricity prices are also eavy-tailed. In particular, normal inverse Gaussian (NIG) and α-stable probability distributions provide a very good fit. Te pertinent question is weter models wit eavytailed innovations perform better in terms of forecasting accuracy tan teir Gaussian counterparts. Following Weron and Misiorek (007), we extend te basic model by allowing for a noise term t tat is governed by a eavy-tailed distribution: NIG or α-stable. Te resulting models are denoted by ARX-N and ARX-S, respectively. Recall, tat te NIG distribution is defined as a normal variance-mean mixture were te mixing distribution is te generalized inverse Gaussian law wit parameter λ= 0.5, i.e. it is conditionally Gaussian. Te probability density function of te NIG(α, β, δ, µ) distribution is given by: f NIG ( x) K ( 1 ( x ( x ) ) ) e ( x ), () 4

5 were δ > 0 and µ R are te usual scale and location parameters, wile α and β determine te sape, wit α being responsible for te steepness and β, β < α, for te skewness. Te normalizing constant K 1 (t) is te modified Bessel function of te tird kind wit index 1. Te tail beavior is often classified as semi-eavy, i.e. te tails are ligter tan tose of non- Gaussian stable laws, but muc eavier tan Gaussian (Weron 004). Stable laws also called α-stable, stable Paretian or Lévy stable require four parameters for complete description: te tail exponent (0,], wic determines te tail tickness, te skewness parameter β [ 1, 1] and te usual scale, σ > 0, and location, µ R, parameters. Wen α =, te Gaussian distribution results. Wen α <, te variance is infinite and te tails are asymptotically equivalent to a Pareto law, i.e. tey exibit a power-law decay of order x. In contrast, for α = te decay is exponential. From a practitioner s point of view te crucial drawback of te stable distribution is tat, wit te exception of tree special cases (α =, 1, 0.5), its density and distribution function do not ave closed form expressions. Tey ave to be evaluated numerically, eiter by approximating complicated integral formulas or by taking te Fourier transform of te caracteristic function (Weron 004). Heavy tailed laws provide a muc better model for electricity price returns tan te Gaussian distribution. Yet, a non-parametric kernel density estimator will generally yield a superior fit to any parametric distribution. Peraps, time series models would lead to more accurate predictions if no specific form for te distribution of innovations was assumed. To test tis conjecture we evaluate a semi-parametric model (denoted by ARX-NP; we call it semi-parametric because it as a parametric autoregressive part and a non-parametric noise distribution) for wic no specific form for te distribution of innovations t is assumed. Instead, to calibrate te parameters of te autoregression, we employ a non-parametric maximum likeliood (ML) routine. Suc ML estimators can be derived by extending te ML principle to a non-parametric framework, were te innovations density is estimated by te Parzen Rosenblatt kernel (Cao et al. 003, Hsie and Manski 1987). Tese non-parametric maximum likeliood estimators (NPMLE) generally perform well not only wen te error distribution is Gaussian (or any oter known parametric form), but also wen only regularity conditions are assumed about te error density. On te oter and, te deficiency of tese estimators wit respect to ordinary ML estimators, under normality, sould not be great if te non-parametric density estimator performs well. In tis study we use te smooted NPMLE proposed by Cao et al. (003). It (numerically) maximizes te likeliood L ) L(, ), were ( g, g, ( x) ( n 1 1) n i K x t ( ) (3) is te non-parametric density, K ( ) is te kernel and t ( ) are te model residuals for a given parameter vector. For te sake of simplicity we use te Gaussian kernel and = (wic rougly corresponds to te so-called rule of tumb bandwidt 1.06 ˆ n 1/, were ˆ is an estimator of te standard deviation of te error density). For more optimal bandwidt coices consult Jones et al. (1996). Finally, let us note tat nearly all computations are performed in Matlab 7.0. Te ARX model is calibrated using te armax.m function, wic minimizes te Final Prediction Error criterion (Weron 006). Te eavy-tailed and semi-parametric models are estimated by numerically maximizing te likeliood and te non-parametric likeliood function, respectively, wit te ARX models parameters as starting points of te unconstrained 5

6 simplex searc routine (fminsearc.m function). Only te ARX-G model is calibrated in SAS 9.0 (via ML), because Matlab s GARCH Toolbox yields significantly worse forecasts. 4. Empirical results To assess te prediction performance of te models, different statistical measures can be utilized. Te most widely used measures are tose based on absolute errors, i.e. absolute values of differences between te actual, P, and predicted, Pˆ, prices for a given our,. Te Mean Absolute Percentage Error (MAPE) is a typical example. However, wen applied to electricity prices, MAPE values could be misleading. In particular, wen electricity prices drop to zero, MAPE values become very large regardless of te actual absolute differences P Pˆ. Te reason for tis is te normalization by te current (close to zero, and ence very small) price P. Alternative normalizations ave been proposed in te literature. For instance, te absolute error P Pˆ can be normalized by te average price attained during te day: P P 4 1 given by (Conejo et al. 005; Weron 006):. Te resulting measure, also known as te Mean Daily Error, is MDE P ˆ P (4). P 4 Te Mean Weekly Error (MWE) corresponds to a situation wen te number 4 is replaced by 168 in (4). Bot errors are usually reported in percent, i.e. as MDE 100% or MWE 100%. Te forecast accuracy was cecked afterwards, once te true market prices were available. Te MWE errors for te wole test period (April 3 December 3, 000) and all models are given in Table 1. Furtermore, to distinguis te rater calm first 10 weeks of te test period from te more volatile weeks (see Fig. 1), in Table te summary statistics are displayed separately for te two periods. Tese statistics are based on te 35 Mean Weekly or 45 Mean Daily Errors. In particular, te number of weeks (days) a given model was best in terms of MWE (MDE), te mean MWE (MDE) and te mean deviation from te best model T in a given week (day). Te latter statistics is defined as 1 (E E, were i T t 1 i, t Best model, t ) ranges over all evaluated models (i.e. i = 5), T is te number of weeks (10, 5) or days (70, 175) in te sample and E is eiter MWE or MDE. Te obtained results suggest tat te semi-parametric specification ARX-NP is te best model. Of all te competitors it most often leads to te best point forecasts, bot in te calm and volatile periods and bot in terms of te weekly and daily measures (see rows labeled # best(mwe) and # best(mde) in Table ). Yet, it is not unanimously te best. Wile it as te lowest mean MWE in te calm period, in te latter 5 weeks, surprisingly, te ARX model beats it. Likewise, ARX-NP as te lowest mean MDE in te volatile period, but in te calm weeks bot eavy-tailed specifications sligtly overtake it. Te mean deviations from te best model lead to te same conclusions. Neverteless, ARX-NP can be considered te overall best model. 6

7 Table 1: Mean Weekly Errors (MWE; in percent) for all weeks of te test period. Best results in eac week are empasized in bold. Notice tat te results for te ARX and ARX-G metods in tis table were originally reported in Misiorek et al. (006) and are re-produced ere for comparison purpose. Week ARX ARX-G ARX-N ARX-S ARX-NP Table : Summary statistics for te Mean Weekly Errors (MWE; presented in Table 1) and te Mean Daily Errors (MDE). Te first number (before te slas) indicates performance during te first 10 weeks and te second during te latter 5 weeks. Best results in eac category are set in boldface. Statistics ARX ARX-G ARX-N ARX-S ARX-NP # best(mwe) 4/7 0/4 1/5 1/1 4/8 Mean(MWE) 13.36/ / / / /17.08 Mean dev. from best 0.69/ / / / /0.84 # best(mde) 17/35 9/43 15/5 5/18 4/54 Mean(MDE) 11.98/ / / / /17.64 Mean dev. from best 1.55/ / / / /.53 7

8 Table 3: Mean percent of exceedances of te 50%, 90% and 99% two-sided day-aead prediction intervals (PI) by te actual system price for te five considered models. Weeks 50% 90% 99% 50% 90% 99% ARX ARX-G ARX-N ARX-S ARX-NP At te oter end is te ARX-G model, wic is inferior to te remaining competitors in most categories. It fails spectacularly in terms of te mean errors and te mean deviation from te best model. However, it can lead to te best predictions from time to time. Finally, te eavy-tailed models beave similarly. Wile ARX-N more often yields te best forecasts, ARX-S performs sligtly better on average. Somewat surprisingly, it is te calm period and not te volatile one tat favors te eavy-tailed models relative to its competitors. Apart from point forecasts, we investigated te ability of te models to provide interval forecasts. For all considered models interval forecasts were determined analytically; for details on calculation of conditional prediction error variance and interval forecasts we refer to Hamilton (1994) and Weron (006). Afterwards, following Cristoffersen and Diebold (000) and Misiorek et al. (006), we evaluated te quality of te interval forecasts by comparing te nominal coverage of te models to te true coverage. Tus, for eac of te models we calculated prediction intervals (PIs) and determined te actual percentage of exceedances of te 50%, 90% and 99% two sided day-aead PIs of te models by te actual system price, see Table 3. If te model implied interval forecasts were accurate ten te percentage of exceedances sould be approximately 50%, 10% and 1%, respectively. Note tat in te calm period (first 10 weeks) 1680 ourly values were determined and compared to te system price for eac of te models, wile in te volatile period (weeks 11-35) 400 ourly values. Examining te exceedances of te 50% interval we note tat wile te Gaussian, GARCH and semi-parametric models yield too wide PIs, te eavy-tailed alternatives beave quite te opposite. In tis respect tey exibit a performance similar to te Markov regimeswitcing model analyzed in Misiorek et al. (006). Looking at te exceedances of te 90% interval we see all models performing alike and yielding too narrow PIs. Yet, te ARX PIs are sligtly better (wider) tan tose of te oter models. Finally, te exceedances of te 99% interval present a different picture. Te -stable innovations lead to te widest (even a bit too wide) and closest to te optimal PIs. Next in line is te ARX-N model, te oter tree trail far beind. All of tem yield too narrow PIs. In tis category, te ARX-N model beaves comparably to te nonlinear Tresold TARX model analyzed in Misiorek et al. (006). Overall, te interval forecasting results are muc less conclusive tan te point forecasting ones. Wile ARX, ARX-G and ARX-NP are better in 50% and 90% intervals, tey fail in 99% PIs, were te eavy-tailed models dominate. Among te tree models ARX, ARX-G and ARX-NP te latter model could be considered te best, as it leads to more accurate point forecasts and comparable interval forecasts. In te wole group, owever, te answer is not tat obvious. Te eavy tailed models could be preferred for risk management purposes since tey yield more accurate upper quantiles of te error distribution. 8

9 Teir beavior during te calm weeks is also comparable to tat of te ARX-NP model. Surprisingly, only during te volatile period tey perform below expectations. 5. Conclusions In tis paper we investigated te forecasting power of time series models for electricity spot prices. We expanded te standard autoregressive specification by allowing for eteroskedastic (GARCH), eavy-tailed (NIG, α-stable) and non-parametric innovations. Te models were tested on a time series of ourly system prices and loads from California. We evaluated te quality of te predictions bot in terms of te Mean Daily and Weekly Errors (for point forecasts) and in terms of te nominal coverage of te models to te true coverage (for interval predictions). Tere is no unanimous winner of te presented competition. Wile in terms of point forecasts te semi-parametric ARX-NP model generally yields te best performance, wen prediction intervals are considered te evidence is mixed. In particular, for risk management purposes, requiring accurate approximation of te upper quantiles, te eavy tailed models could be preferred. Altoug tis study adds an important voice to te discussion of electricity spot price forecasting, more researc including evaluation of te models on oter datasets is needed. Bibliograpy Bottazzi, G., Sapio, S., Secci, A. (005) Some statistical investigations on te nature and dynamics of electricity prices, Pysica A 355, Bunn, D.W. (000) Forecasting loads and prices in competitive power markets, Proceedings of te IEEE 88(), Bunn, D.W., ed. (004) Modelling Prices in Competitive Electricity Markets, Wiley. Cao, R., Hart, J.D., Saavedra, A. (003) Nonparametric maximum likeliood estimators for AR and MA time series, Journal of Statistical Computation and Simulation 73(5), Carr, P., Geman, H., Madan, D.B., Yor, M. (00) Te fine structure of asset returns: An empirical investigation, Journal of Business 75, Cristoffersen, P., Diebold, F.X. (000) How relevant is volatility forecasting for financial risk management, Review of Economics and Statistics 8, 1-. Conejo, A.J., Contreras, J., Espinola, R., Plazas, M.A. (005) Forecasting electricity prices for a day-aead poolbased electric energy market, International Journal of Forecasting 1(3), Contreras, J., Espinola, R., Nogales, F.J., Conejo, A.J. (003) ARIMA models to predict next-day electricity prices, IEEE Transactions on Power Systems 18(3), Engle, R.F. (198) Autoregressive conditional eteroscedasticity wit estimates of te variance of United Kingdom inflation, Econometrica 50, Jones, M.C., Marron, J.S., Seater, S.J. (1996) A brief survey of bandwidt selection for density estimation, Journal of te American Statistical Association 91, Hamilton, J. (1994) Time Series Analysis, Princeton University Press. Hsie, D. A., Manski, C. F. (1987) Monte Carlo evidence on adaptive maximum likeliood estimation of a regression, Annals of Statistics 15, Misiorek, A., Trück, S., Weron, R. (006) Point and Interval Forecasting of Spot Electricity Prices: Linear vs. Non-Linear Time Series Models, Studies in Nonlinear Dynamics & Econometrics, 10(3), Article. Racev, S., Mittnik, S. (000) Stable Paretian Models in Finance, Wiley. Weron, R. (004) Computationally intensive Value at Risk calculations, in Handbook of Computational Statistics: Concepts and Metods, eds. J.E. Gentle, W. Härdle, Y. Mori, Springer, Weron, R. (006) Modeling and Forecasting Electricity Loads and Prices: A Statistical Approac, Wiley. See also: ttp:// 9

10 Weron, R., Misiorek, A. (007) Heavy tails and electricity prices: Do time series models wit non-gaussian noise forecast better tan teir Gaussian counterparts?, Prace Naukowe Akademii Ekonomicznej we Wrocławiu Nr 1076, Te Polis Abstract (Streszczenie) Tytuł: Prognozowanie urtowyc cen energii elektrycznej: Przegląd modeli szeregów czasowyc Streszczenie: W pracy badamy efektywność krótkoterminowyc prognoz różnyc modeli szeregów czasowyc na spotowym rynku energii elektrycznej. Kalibrujemy modele autoregresji (AR), włączając w to modele ze zmienna zewnętrzną zapotrzebowaniem na energię, do danyc z giełdy kalifornijskiej CalPX. Następnie oceniamy efektywność prognoz punktowyc i przedziałowyc w stosunkowo spokojnyc, jak i bardzo zmiennyc okresac poprzedzającyc krac rynkowy w zimie 000/001. W szczególności testujemy jakie rozkłady/procesy innowacji gaussowskie, GARCH, gruboogonowe (NIG, -stabilne) czy nieparametryczne prowadzą do najlepszyc prognoz. 10