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Predictability of price movements in deregulated electricity markets OlgaY.Uritskaya QuantitativeDynamicsLLC,2205DarrowSt,SilverSpringMD,USA VadimM.Uritsky CatholicUniversityofAmerica,620MichiganAveNE,WashingtonDC,USA 5 1 0 Abstract 2 InthispaperweinvestigatepredictabilityofelectricitypricesintheCanadianprovincesofAlbertaandOntario, n a aswellasintheUSMid-Cmarket. Usingscale-dependentdetrendedfluctuationanalysis,spectralanalysis,andthe J probabilitydistributionanalysisweshowthatthestudiedmarketsexhibitstronglyanti-persistentpropertiessuggesting 3 thattheirdynamicscanbepredictedbasedonhistoricpricerecordsacrosstherangeoftimescalesfromonehourto 2 one month. For both Canadian markets, the price movements reveal three types of correlated behavior which can beusedforforecasting. Thediscoveredscenariosremainthesameondifferenttimescalesuptoonemonthaswell ] T as for on- and off- peak electricity data. These scenarios representsharp increases of prices and are not presentin S the Mid-C marketdue to its lower volatility. We arguethat extremeprice movementsin this market shouldfollow . the same tendency as the more volatile Canadian markets. The estimated values of the Pareto indices suggest that n the predictionof these events can be statistically stable. The results obtainedprovidenew relevantinformationfor i f managingfinancialrisksassociatedwiththedynamicsofelectricityderivativesovertimeframeexceedingoneday. - q [ Keywords: Deregulatedelectricitymarkets,efficientmarkethypothesis,detrendedfluctuationanalysis,financialforecasting 1 v 7 1 1. Introduction 1 8 Themodernelectricitymarketisnotonlyasystemforarrangingthepurchaseandsaleofelectricityusingsupply 0 anddemandtosettheprice,but,formostmajorgrids,isabasisforelectricityderivatives,suchaselectricityfutures . 5 and options, which are actively traded. The practical significance of this part of the market is increasing as is the 0 importanceoftherelatedscientificresearch[1,2,3]. Themarketsofelectricityderivativeshavedevelopedasaresult 5 oftheliberalizationandderegulationofelectricpowersystemsaroundtheworld.Deregulation,introducedinitiallyto 1 reduceandsimplifythecontrolofthebusinessinthisfield,hadafinalgoaltoreachfinancialefficiencyofelectricity : v markets[4, 5]. However,electricity is uniqueasit is a non-storablecommodity,andthe marketsremainextremely i X inefficient[6,7]. Electricity prices are not a result of long-termbut instant, usually on an hourly interval, balance of supply and r a demand. Moreover,asaconsequenceofthecomplexityofawholesaleelectricitymarket,itcanshowanextremely highpricevolatilityattimesofpeakdemandandsupplyshortages.Thispricespikesarehardtopredictandfinancial riskmanagementisstillahighpriorityforparticipantsinderegulatedelectricitymarketsduetothesubstantialprice andvolumerisksthatthemarketscanexhibit[8,9,10]. Theproblemofpredictabilityofelectricitypricesinderegulatedmarketshasbeenconsideredinmanyprevious studies(forinstance,[11,12,13,14]).Thevaluesofpricescanvarybyafactorof100overatimescaleofjustseveral hours. These dramatic changes tend to occur in a seemingly spontaneousfashion which is sometimes erroneously interpretedas a signature of a random uncorrelatedprocess (see for example [15]). A more detailed mathematical analysisrevealsnontrivialauto-correlationsinthesesuddenpricejumps[16,17,18,19]whichindicateapossibility ofpredictionofelectricitypricemovementsbasedontheinformationontheirhistoricevolution[7]. However,itisa widelyrecognizedfactthatpricefluctuationsinenergymarketsdisplayheavydistributiontails[16,17,20,21,22,23] PreprintsubmittedtoElsevier June1,2015 causingsubstantialdifficultiesinbuildingquantitativeforecastingmodelsofpricebehavior. Lessattentionhasbeen paidto theanalysisoftemporalpatternsunderlyingthe observedstatistical structureofelectricitymarketsandasa resultmodelingoftheirdynamicsisstillinitsinfancyandistypicallylimitedtoday-aheadmodels[2,24,25,26,27]. In this study, we take a few next steps toward answering fundamentalquestions related to the predictability of electricity prices. First of all, can deregulatedelectricity markets reach the state of efficiency with the Hurst index valueclosetootherwell-knownmarkets,orthisstateisnotreachableintheusualsense[28,29,30]?Thisiscrucially importantbecauseiftheelectricitymarketsareinherentlyinefficient,theforecastcanbebuiltatvarioustimescales. Inthiscontext,theinefficiencymeansthatpricehistoryisrelevanttothefuturepricechangesandcanbeusedfortheir forecasting[31]. TheproblemwithPareto-typestatisticsplaysaspecialpart,becausenotallheavydistributiontails canbeapproximatedbyasingleprobabilisticmodel. Theycanincludeseveraldynamicrangesdescribedbydistinct Paretoexponents. Ifsuchmarketsarepredictableinprinciple,theremightbeparticularpriceintervalsforwhichthe forecastisstatisticallystable,andtheseintervalsareimportanttoidentify. Another central question related to the predictability of electricity prices is how universal can be a model of electricity price behavior across different markets. In the present work, this question is addressed in frames of a quantitativeanalysisofelectricitypricesinthreeindependentmarketswithdifferentlevelsofliberalization–Alberta, Ontario(Canada)andMid-C(USA)markets. Dynamicalandstatisticalpropertiesofpricefluctuationsareinvestigatedusingseveralmethods.First,weevaluate correlationsinpricedynamicsacrossdifferenttimescalesusingthemethodofscale-dependentfractalexponent[7] obtained from detrended fluctuation analysis (DFA) [32, 33, 34, 35]. We also use the Fourier spectral analysis to identifycycliccomponentsintheelectricitypricedynamics,aswellastheParetoprobabilitydistributionanalysisfor testing the stability of statistical momentsof the studied data. Spectraland DFA analysisresultsshow no evidence ofinformationalefficiencyofelectricitypricefluctuationsatanytimescale. Allthreemarketsdemonstratedifferent levelsofinefficiencywhichcouldreflecttheirdifferentsizesandstructuraldiversification. Pricemovementsinthese markets are strongly anti-persistent [28]. Together with Pareto analysis results, this anti-persistence indicates that electricitypricemovementscanbepredictedbasedonhistoricpricerecords. Next,weverifythepossibilityofpriceforecastingusingphasediagramsrepresentingthecorrelationofprevious and current price increments. According to our results, the diagrams have a complex asymmetric shape revealing three basic scenarios of price movements. These scenarios remain the same for price movementsat different time scales,fromonehouruptoonemonth,andarefoundtorepresentstronglyvolatilemarketconditions.Basedonthese results,weshowthatpricefluctuationsinderegulatedelectricitymarketsarepredictablebytheirnature.Ourfindings layafoundationforfuturemathematicaldescriptionofmultiscaledynamicsinderegulatedelectricitymarkets. Theplanofthepaperisthefollowing.Section2containsadetaileddescriptionoftheanalyzeddatasets. Section 3describesmainresultsofourstatisticalanalysisdemonstratingthepossibilityofelectricitypriceforecasting. This possibilityisexploredfurtherinSections4. Section5providesabriefsummaryofourstudy. 2. Data Asanoutcomeoftheliberalizationpoliciespursuedinseveralcountriesfromthe80son,thesocalledday-ahead electricity market provides economists with a very challenging phenomenon. Electricity cannot be economically stored,whichimpliesthatdemandandsupplymustbecontinuouslybalanced,sothatthemarketpricemainlyreflects the demandand supply conditionsprevailingat the very momentit has to be delivered to final users. Then, rather complexmarketsystemshavebeensetup,withtheaimofreachingareasonabletrade-offbetweeneconomicefficiency and system reliability. These systems are built around a market operator, whose task is to manage uniform-price, sealed-bid,bilateralauctionsinordertoconstructaggregatedemandandsupplycurves,andtodetermineequilibrium pricesandquantities. Theknife-edgecharacterofsuchapricesettingmechanismisfatherlypushedtotheextreme byaverylowpriceelasticityofdemand,andbytechnicalconstraintswhichtimebytimeleadtonetworkcongestion (seee.g. [23,36]andreferencestherein). ThedatastudiedinthispaperconsistsofhourlyrealtimepoolelectricitypricesinAlberta,postedbytheAlberta ElectricSystemOperator(AESO),andOntario,postedbytheIndependentElectricitySystemOperator(IESO).The datacovertheperiodfromMay1,2002toJune6, 2009. InadditiontotheseCanadianmarkets,theMidColumbia (Mid-C) market has been considered during the time interval 1 July, 2001 to 31 Oct, 2006. For each of the three 2 Figure1: Timeseriesofhourlyelectricity pricesinAlberta(left), Ontario(center)andMid-C(right)markets. Fromtoptobottom: allhourly prices,on-peakprices,andoff-peakprices. AlbertaelectricitypricesdemonstratesignificantlyhigherfluctuationsthanthoseinOntariomarket, plottedonthesameverticalscale. FluctuationsofelectricitypricesinMid-ChavetwiceaslowamplitudeasthatinOntario,andabout5times smallerthaninAlberta. hourlydatasets, two secondarytime seriesconsistingof electricitypricesduringon-andoff-peakhourshavealso beenexamined.Figure1showsthetimeseriesunderstudy,includingtheoriginaldataandtheiron-andoff-subsets. Notethatallplotscontainnumerousspikeswithirregulartimingandamplitude. AlbertaandOntarioaretheonlytwoCanadianprovinceswherewholesaleelectricitymarketsarefullyderegulated [13,37].Alberta’smarketisdominatedbyfossilfuelgenerationandassuchfollowsmorecloselythepriceofnatural gas. Ontario’s generationinvolvesabout50% of nuclear and 25% of hydropower [38, 39] enabling a more stable price behavior [40]. The average level of volatility of electricity prices in Alberta is about twice as high as in the Ontariomarket. TheMidColumbiaelectricitymarketisnotasderegulatedasAlbertaandOntarioare[41]. Itisnotacentralized powermarket,butitisatradinghubwherepowerisbilaterallytradedamongutilitiesandmarketers.TheMid-Cprice hubisareferencepriceforthePacificNWregion,whichconsistsofWashington,Idaho,andOregon. Inthisregion, largeutilitiesowngenerationandserveloadunderregulatedrates. Thegenerationisprimarilyhydroandtheregion typically exportsto British Columbia and California [7, 42]. For these reasons, Mid-C prices are significantly less volatilethanthoseineitherCanadianmarket. 3. Statisticalsignaturesandpredictability 3.1. DFAandspectralsignatures Fortestingtheinformationalefficiencyofelectricitypricefluctuations,multiscalecorrelationsofpricedynamics were evaluated across different time scales. Two complementary approaches were used to achieve this goal – the scale-dependentDFAandtheFourierspectralanalysis. Theformerofthetwoapproacheshasbeenfirstintroducedin[7]. Incontrasttopreviousmethodsmanipulating with average scaling exponents characterizing broad scaling ranges, we investigated the distribution of local DFA exponentsover all time scales involved. This approach was shown to be the only suitable when the signals under study encompass qualitatively different types of behavior including random price movements, cycles, and spikes. UsingtheDFAasthebasealgorithmisjustifiedbythepresenceofmultiscaletrendsintheelectricitydata[29,43]. Thescale-dependentversionofthisalgorithmpresentedin[7]enablestheinvestigationofcomplextypesofnonlinear behavior of financial and economic indicators by providingdetailed informationon the distribution of correlations overdifferentscales,andisespeciallyusefulforquantitativeanalysisofmarketefficiency. 3 Figure2:DependenceofthedetrendedvariationF(eq. (2))andthelocalscale-dependentDFAslopeαonthetimescalenforallhourly,on-and off-peakelectricitypricesinAlberta(left),Ontario(center),andMid-C(right)markets.Thepresentedstatisticsrevealcomplexcorrelatedstructure ofpricemovementswithquasi-periodiccomponentsassociatedwithdailyandweeklycycles. Inallpresenteddatasetsthescale-dependentDFA exponentissignificantlybelowthelevel1.5definingthestateofinformationalefficiency,whichprovidesanopportunityofforecastingtheprices overwiderangesoftimescales. TheDFAtechniquewasappliedtothetime-integratedsignal k y(k)= (x(t)−hxi), (1) Xt=1 inwhichhxiistheaveragevalueofthehourlyelectricitypricexandk =1,...,N,whereN isthenumberofpointsin thestudiedtimeseries. Theintegratedsignaly(k)wasdevidedinto M = N/nnon-overlappingsubintervalsofequal length n rangingbetween 4 and 720 hours. The boxeswere indexedby m = 1,...,M and their starting times were labeledbyk . Foreverybox,theleastsquareregressionliney (k)representingthelocallineartrendinthatboxwas nm nm fittothedata. Usingthesefits, theintegratedseriesy(k)waslocallydetrendedandtherootmeansquarefluctuation oftheresultingdetrendedsignalwascalculated. ThedescribedcalculationwasrepeatedforeachoftheMboxesand theresultingvalueswereaveragedtoobtainthecharacteristicdependenceonthetimescale: 1 M 1knm+n F(n)= vutM n (y(k)−ynm(k))2 (2) Xm=1 kX=knm Forafractal(self-similar)financialtimeseriesx(t),thepower-lawrelationbetweentherootmeansquarefluctua- tionF andthetimescalenisexpected[44,43]: F(n)∼nα, (3) 4 inwhichαistheDFAscalingexponent[32,33]. Note that our definition of the DFA exponent differs from that used in the majority of other studies of price fluctuationsoperatingwithlogarithmicpricereturns,inwhichcaseαservesasaproxytotheHurstindexH.Because thetimeseriesoftheelectricitypricesarequitespiky(especiallyinAlberta),theDFAexponentsevaluatedusingthe standard return-based approach would be close to zero, making quantitative cross-market comparisons statistically unreliable. To improve the accuracy of our analysis, we apply the DFA method directly to the time series of the electricitypricesratherthantothepricereturns,whichisconsistentwiththeoriginalformulationofthemethod[33]. Sinceweomitthestepofcalculatingpricereturns,theresultingDFAexponentisgreaterthanthatderivedfromthe returnsbyone,andsoα=H+1. Keepingthisrelationshipinmind,efficientfinancialmarketsdescribedbyrandomBrownianwalkmodels[45,29] exhibitthevaluesofαwhicharecloseto1.5signalingtheabsenceofcorrelationsbetweenthepriceincrements[28]. Ithasbeenshown [7, 46] that deregulatedelectricity marketsdo notsatisfy this conditionandin that sense are not efficient. Inaddition,αvaluesofelectricitymarketstendtovarywithscalereflectingcomplexstructureoftheprice dynamicsinvolvingquasi-periodiccyclesandrandomvariations[7]. Fig. 2showsscale-dependentbehavioroftheDFAfunctionsF(n)characterizingstudiedelectricitymarkets. The local values of α exponentswere computedwithin narrown intervalsof exponentiallyincreasing width ensuringa uniformbinningonalogarithmicscale.ItcanbeseenthatCanadianmarketsobeytheanti-persistentconditionα<1.5 acrossalltemporalscales. TheMid-Cmarketshowsapersistentbehaviorcharacterizedbyα > 1.5atn < 12. This indicatesthatthe daily cycle in this marketdominatesrandomfluctuationsleadingto statistically significanttrends overtimespansshorterthanthehalf-dayinterval. Allthreemarketsexhibitanti-persistentregimesatn > 24andoperateinastronglyinefficientstate, confirming ourpreviousresultobtainedfordailyelectricityprices[7]. Table1providesaverageandminimumDFAindexvalues forthestudieddata,alongwithspectralanalysisresultsdiscussedbelow. Itcanbenoticedthatthedropofαassociatedwiththedailyperiodicityisshiftedtowardlargerrelativeton=24. Thisshiftmaybe caused byan interplaybetweenthe saturationof F(n)and a decreaseof the trendsat time scales greaterthatthecycleperiod.Qualitativelysimilar,albeitlesspronouncedshiftedsignatureisobservedneartheweekly periodicity. TheFourierspectralanalysiswasusedforamoreaccuratedescriptionofperiodiccomponentsofpricedynamics, and also as an independent test for its informational inefficiency. The power spectrum S(f) is obtained from the FouriertransformX ofthepricesignalx(t)definedinthecontinuouslimitas T T 1 X (f)= x(t)e−i2πftdt, S(f)= X X∗, (4) T Z0 T T T whereX∗ isthecomplexconjugateofX andT >>1/f isthelengthofthetimeintervaloftheanalysis. Themethod T T was implemented using the standard fast Fourier transform algorithm. For a self-similar signal the spectral power scaleswiththefrequency f asS(f)∼ f −β wheretheindexβisrelatedtotheDFAindexthroughβ=2α−1,sothat β=2isthecaseoftheefficientmarket(Fig. 3). TheobtainedFourierpowerspectraareconsistentwithresultsoftheDFAdiscussedabove(Fig. 3),andreveala stronglyinefficientbehaviorofthe electricitymarkets. Inaddition,spectraanalysisdemonstratesa relativelystable dailycycleaswellasalessstablebutevidentweeklyperiodicity.Theaverageβestimatesarestatisticallysignificantly belowthevalue2. Thevaluesα calculatedbasedonthespectralexponentsareinanapproximateagreementwith theor theaverageαvaluesobtaineddirectlyfromtheDFA(thefirstandthelastcolumnsofTable1). Bothmethodsclearlyshowthattherearestronganti-persistentcorrelationsbetweenthevaluesofelectricityprices atalltimescales forall-hourtime dataaswellas forthe subsetsoftheon-andoff-peakdata. Thisisevidentfrom theanalysisofthemaximumDFAindexvalues(secondcolumnitTable1)representingthelargestαacrosstheentire range of n scales for each market. In most of the data sets, [α(n)] < 1.5 meaning that at no scale the markets max becomeefficientorpersistent. TheonlyexceptionfromthistendencyistheMid-Cmarketatn < 12hoursshowing greaterthan1.5DFAindexvaluesforbothall-hourandon-peakprices. Thiseffectisabsentintheoff-peakMid-C datasuggestingthattheshort-terminformationalpersistencyofthismarketisafootprintofthehighdemandperiods. DespitethesignificantvariabilityoftheDFAexponentsacrosstemporalscales, theaverageαvaluesofall-hour pricesinAlbertaandMid-CmarketsshowninTable1areinanagreementwiththosedescribingdailypricesinthese 5 Figure3:FourierpowerspectraS(f)forall-hour,on-andoff-peakelectricitypricesinAlberta(left),Ontario(center),andMid-C(right)markets. Inalldatasetstheaveragespectralexponentβislowerthan2,whichconfirmstheconclusionoftheDFAanalysisregardingtheinefficientbehavior ofthestudiedpricedynamics.CanadianmarketsarecharacterizedbysignificantlysmallervaluesofβcomparedtotheAmericanMid-Cpool,and thereforehaveahigherdegreeofinefficiency. 6 Table1:AveragevaluesofDFAandspectralexponentsofhourlyelectricityprices. Timeseries <α(n)> [α(n)] β α =(β+1)/2 max theor Alberta,all-hoursprices 0.87±0.13 1.18±0.02 0.93±0.07 0.97±0.54 Alberta,on-peakprices 0.87±0.08 1.09±0.01 0.94±0.06 0.98±0.53 Alberta,off-peakprices 0.83±0.09 1.10±0.04 0.81±0.07 0.91±0.54 Ontario,all-hourprices 0.87±0.13 1.16±0.01 0.92±0.06 0.96±0.53 Ontario,on-peakprices 0.86±0.06 1.03±0.04 0.83±0.06 0.92±0.53 Ontario,off-peakprices 0.91±0.08 1.09±0.05 0.89±0.05 0.95±0.53 Mid-C,all-hourprices 1.17±0.19 1.69±0.04 1.95±0.04 1.48±0.52 Mid-C,on-peakprices 1.21±0.11 1.62±0.02 1.80±0.03 1.40±0.52 Mid-C,off-peakprices 1.22±0.05 1.33±0.04 1.63±0.06 1.32±0.53 marketsaccordingtoourprecedingstudy[7]: < α(n) >= 0.90±0.08forAlberta,< α(n) >= 0.90±0.08forMid- C. To complementthese earlierreportedfindings, we also conducteda DFA analysisof daily Ontariopriceswhich yielded< α(n) >= 0.93±0.08. Bycomparingthesenumberswiththe< α(n) >valuesinthetable,onecanseethat thediscrepancybetweenthetwosetsofDFAexponents(theonesobtainedfordailyandhourlyprices)iswithinthe statistical uncertainty of our measurements. The fact that the 24-houraggregationof the prices does not affect the averageDFA exponentindicatesthatthe latter isinsensitiveto the short-scalefluctuationsofthe demanddrivenby thedailysocioeconomicdynamics,includingthe24-hourcycleandintradaynonstationarities. Overall, the correlated price movements in all three markets present an opportunity of forecasting their future dynamics based on the historic behaviors over essentially all time scales involved. The Fourier spectral analysis confirmsthewellknownexistenceofdailycycleswhichcanbeusedasanauxiliaryfactorinthepriceforecasting. 3.2. Probabilitydistributions Toverifythestabilityofhighermomentsofelectricityprices,weinvestigatedtheParetoprobabilitydistribution ofhourly,on-andoff-electricity data(Fig. 4). The Paretoexponentsobtainedfromthis analysisprovidea critical pieceofinformationonwhetherthestatisticalpredictionofthepricescanyieldrobustresults. The probability distributions were estimated using the normalized discrete histograms N(x) with constant bin width chosen to be 5 currencyunits for Alberta and Ontario marketsand 1 unitfor the Mid-Cmarket. The rate of decay of the histogram tails was described by the Pareto index γ [47] defined by the power-law fit N(x) ∼ x−(γ+1) appliedtoaselectedpricerange. Asfollowsfromthedefinition,theParetoindexdescribestheasymptoticshapeof thecomplementarycumulativedistributionfunction.Thevaluesofγwerecomputedusingthestandardmean-square fittingalgorithmappliedonthelog-logscale. DependingonthenumericalvaluesoftheParetoindex,thefollowingcasesarepossible: ifγ ≤1,neithertheav- eragevalueofpricenoritsstandarddeviationcanbedetermined;if1<γ≤2,theaveragepricecanbecalculatedbut thestandarddeviationcannot;γ >2meansthatbothparameterscanbeevaluatedandtheresultsofpriceforecasting arepotentiallyrobust. ThevaluesoftheParetoexponentobtainedinourstudy(Fig. 4,Table2)indicatethatallthree marketspasstherequiredminimumlevelγ=2atleastforthepricerangex∈[1,200].Mid-Celectricitypricesdonot haveaheavy-taileddistribution,sothereisnoriskofunexpectedlyhighfluctuations,whereastheCanadianmarkets, especiallyAlberta,showpronouncedfattails. TheprobabilitydistributionsdescribingtheAlbertamarketapparently hastwocomponents–thePoissonian-likecentralportionwithawell-definedmaximum,andaheavypower-lowtail. InTable 2, this behavioris demonstratedbydividingthe distributionsintotwo intervals. Pareto indexesof Alberta electricity prices in the range x ∈ [200,1000]are lower than in the range x ∈ [1,200] and correspondto the case 1 < γ ≤ 2. Althoughthisdoesnotexcludethe possibility of forecastingthe averageprice, such predictionwill be statisticallyunstable,especiallyforpriceswhicharesignificantlyhigherthantheaverage. Ontarionumbersshowthe oppositetendency,withtherange x ∈ [200,1000]exhibitingafasterpowerlowdecaycomparetotheotherstudied price interval. Despite the presence of the fat tail, this market allows for robust statistical prediction even for the highestvaluesoftheprice. 7 Figure4:Normalizedprobabilityhistogramsofall-hour,on-andoff-peakelectricityprices.ThepresentedvaluesoftheParetoexponentγindicate apossibilityofrobuststatisticalpredictionofmeanpricevaluesforallthreemarketssincetheconditionγ>1ismet.OntarioandMid-Cmarkets arealsocharacterizedbystablestandarddeviationsduetoγ>2,whiletheAlbertamarketdoesnotsatisfythisrequirement. Table2:ValuesoftheParetoindexγfortworangesofelectricitypricesinthestudiedmarkets. Timeseries x∈[1,200] x∈[200,1000] Alberta,all-hourprices 2.072±0.053 1.318±0.068 Alberta,on-peakprices 2.360±0.084 1.273±0.070 Alberta,off-peakprices 2.394±0.084 1.670±0.082 Ontario,all-hourprices 3.016±0.064 3.508±0.045 Ontario,on-peakprices 3.177±0.067 4.106±0.128 Ontario,off-peakprices 2.964±0.075 3.095±0.115 Mid-C,all-hourprices 4.284±0.190 - Mid-C,on-peakprices 4.453±0.155 - Mid-C,off-peakprices 4.310±0.190 - 8 AllthreemarketsseemtobeinsensitivetothedemandlevelshowingcomparableParetoexponentsforall-hour, on-andoff-peakdatasetsforbothpriceranges,confirmingourearlierobservations[7]. Itremainstobeunderstood whythemarketdemandmakesnonoticeablecontributiontothestatisticalpropertiesoftheprocess. 4. Forecastingelectricitypricemovements Since anti-persistency implies a stochastic process with negative correlation between its increments, our next step is to investigate statistical interdependenceof previousand currentprice movements. To achieve this goal we introduceda new methodofmultiscale incrementssensitiveto such correlations. The multiscale increments∆x(n) i onthescalenarecomputedaccordingto 1 ti−1 ti+n ∆x = x(t)− x(t) (5) i n  t=tXi−n−1 Xt=ti  andrepresentarunningdifferenceofsubsequentaggregatedpricevaluesobtainedbyaveragingx(t)innon-overlapping binsof width n. The statistical relationship ∆x versus∆x between the currentand precedingincrementscarries i i−1 informationaboutcorrelatedpricedynamicsatagivenscaleandforagivenpricerange. Ifonaverage∆x ∼ ∆x , i i−1 thepricemovementsarepositivelycorrelatedandpersistenttrendsarepresent. Ontheotherhand,if∆x ∼ −∆x , i i−1 thecorrelationisnegativerevealinganti-persistentpricebehavior[29,7]. Fig. 5showsresultsoftheincrementanalysisofthethreeelectricitymarketsforafixedtimescalen=1. Thetop rowofpanelsdisplaysscatterplotsofsuccessivehourlypriceincrementswhichincludebothon-andoffpeakprices. The characteristic amplitude of price changes is considerably different for the studied markets, with the increment rangeinMid-Cmarketbeingabout10timessmallerthanthatinAberta. TheincrementplotsoftheCanadianmarketsexhibitacommonpattern. Datapointsinthefourthquadrantofthe coordinateplane definedby the conditions∆x < 0 and ∆x > 0 are aligned along a diagonalline with slope −1 i i−1 suggestingstronglyanti-correlatedbehaviorofpriceincrements. Mid-Cpricesalsoshowacorrelatedregressionpatternbutwithapositivesloperevealingpositivecorrelationsat thestudiedtimescale. DirectcomparisonofthisMid-CsignaturewithAlbertaandOntariopricesisnotjustifieddue toadrasticallysmallerrangeofpricefluctuationsintheMid-Cmarket. Becauseofitscompactshape,theregression plotofMid-CelectricitypricesfitswithinthecoreregionofAlbertaandOntariomarkets. Besidesthemaintendencyassociatedwiththeanti-correlation,Canadianmarketsshowtwoadditionalfeatures: alignment of the increment values along the positive vertical axis (∆x > 0, ∆x ≈ 0) and along the negative i i−1 horizontal axis (∆x ≈ 0, ∆x < 0). The first of these features reflects the tendency of the electricity price to i i−1 growabruptlystartingfromasteadyconditioncharacterizedbylowpricevariability. Thesecondfeaturereflectsthe oppositetendencyinwhichthepricedropsafteraprecedingpeakandreturnstoasteadycondition. The second and third rows of panels in Fig. 5 show scatterplots of price increments constructed for on-peak and off-peakprices. It can be seen that on- and off-peak prices in Alberta and Ontario demonstrate the same anti- correlation pattern as the one identified for the whole set of hourly prices in these markets. This pattern is more pronouncedin the on-peak price movementswhich is an expected result since the amplitude of the price jumps is muchhigherduringthepeaktimes. Surprisingly,neitheron-peaknoroff-peakpricemovementsintheMid-Cmarketshowpositivecorrelationsseen intheoriginalall-hourdata. Apossibleexplanationofthiseffectisthatthecorrelationspresentintheall-hourdata areproducedbytransitionsbetweenhigh-andlow-demandintervalsoftheMid-Cmarketratherthanbyitsdynamics duringtheseintervals. ThelastrowinFig. 5showsaggregatedscatterplotsconstructedbyaveraging∆x priceincrementsoverasetof i uniform∆x bins. Theseare constructedtorevealthe prevailingstatistical tendenciescontrollingthepricemove- i−1 mentsinthestudieddata,enablingtheirvisualandquantitativecomparisons. It can be seen that the Canadian markets are dominated by the anti-correlated price movements manifested in thefourthquadrantoftheaggregatedscatterplots. Theplotsexhibitanearlylineardecay,withtheslopecoefficient of about −0.5 for Alberta and −1 for Ontario market. This tendency is observed across the entire range of price incrementsforbothon-oroff-peakdatasets. Thelineardependenceofsmallpricemovementswasobstructedinthe originalscatterplotsbutisevidentintheaggregatedplots. 9 Figure 5: Regression diagrams showing current (∆xi) versus preceding (∆xi−1) hourly price increments for all-hour, on- and off- peak data. RegressionplotsforAlbertaandOntariomarketsaresimilarinshapeandrevealnegativecorrelationsofpricemovementsassociatedwiththespiky structureofthedata. TheMid-Cpriceincrementsexhibitpositivecorrelationsinall-hourdataonly. Bottompanelsdisplayaveragedversionsof thediagrams,with∆xivaluesaveragedover∆xi−1binsofthesamewidth. 10