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ARMA and GARCH-type Modeling Electricity Prices PDF

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MASTER’S THESIS ARMA and GARCH-type Modeling Electricity Prices YAN DONG Department of Mathematical Sciences CHALMERS UNIVERSITY OF TECHNOLOGY GÖTEBORG UNIVERSITY Göteborg Sweden 2012 Thesis for the Degree of Master of Science ARMA and GARCH-type Modeling Electricity Prices YAN DONG Department of Mathematical Sciences Chalmers University of Technology and Göteborg University SE-41296 Göteborg, Sweden Göteborg, May 2012 Matematisktcentrum Göteborg2012 Abstract In this master thesis statistical models for electricity prices during Jan- uary and February 2012 is developed. The time series is modeled based on autoregressive and moving average (ARMA) model and extreme value theory. ThespikesaresimulatedbyGeneralizedParetodistribution(GPD). Theinnovationprocessisanalyzedbyautoregressiveconditionallyheteroskedas- tic (GARCH) process and exponential GARCH (EGARCH) process. All theparametersareestimatedbymaximumlikelihoodmethod. Acknowledgments I would like to thank my supervisor Partik Albin for his encouragement and advice. I also learn a lot from him during the whole master period. He not only helps me in academic level, but also encourage me during my studyingoverseas. IalsowouldliketothankmyexaminorAnastassiaBaxe- vani. Ihaveobtainedbothstatisticalknowledgeandscientificmethodsfrom her course: Computer Intensive Statistical Methods. Lastly, I would like to thankRobinAxelsson,anoutstandingSwedishmasterstudentinChalmers. Contents 1 Introduction 4 2 Electricity Markets 5 2.1 TheNordPool . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 TheMainFeaturesofElectricityPrices . . . . . . . . . . . . . 5 3 TheoreticalBackground 6 3.1 ARMAModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 GARCHModel . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 ExponentialGARCHModel . . . . . . . . . . . . . . . . . . . 7 3.4 StationaryProcess . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.5 ExtremeValueTheory . . . . . . . . . . . . . . . . . . . . . . 8 3.5.1 GeneralizedParetoDistribution . . . . . . . . . . . . 8 3.5.2 PeakOverThreshold . . . . . . . . . . . . . . . . . . . 9 4 Methods 11 4.1 MaximumLikelihoodEstimationMethod . . . . . . . . . . . 11 4.2 EstimationforARMAModel . . . . . . . . . . . . . . . . . . 12 4.3 EstimationforGARCHModel . . . . . . . . . . . . . . . . . 14 4.4 AIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.5 AnalysisofResiduals . . . . . . . . . . . . . . . . . . . . . . . 16 5 Modeling 17 5.1 DataSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 FittheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2.1 RemoveExtremeValues . . . . . . . . . . . . . . . . . 20 5.2.2 RemovetheSeasonalComponents . . . . . . . . . . . 20 5.3 ARMAModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.4 ARMA-GARCHModel . . . . . . . . . . . . . . . . . . . . . 28 5.5 ARMA-EGARCH Model . . . . . . . . . . . . . . . . . . . . 32 5.6 ExtremeValueDistributionModel . . . . . . . . . . . . . . . 37 5.7 SimulationofGARCH-TypemodelandExtremeValueDis- tributionModel . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 Conclusions 41 3 1 Introduction Manystatisticalmodelsfortimeseriesareworkedthroughresearchingmean orvariance(volatility)oftheprocess. TheARMAmodelisusedtopresent the stationary time series based on autoregressive process and moving av- erage of noises. On the other hand, the autoregressive conditional het- eroskedasticity (ARCH) model is focus on time varying conditional vari- ance. Inpractice,however,highARCHorderhastobeselected. Bollerslev extended this model to Generalized ARCH (GARCH) model, which can solvethisproblem. GARCHmodeldescribevarianceatacertaintimewith bothpastvaluesandpastvariances. Mosttimeseriesissufficientlymodeled usingGARCH(1,1)thatonlyincludesthreeparameters. Electricitypricesaresignificantlyaffectedbythedemandandsupplyonthe market. Generally, high demand results in high price. Moreover, electricity pricesare affected byexternal factors suchas weather, pricesof fossil fuels, availabilityofnuclearpower,waterreservoirlevels,pricesofexhaustrights etc. Like most financial time series, electricity price has the characteristic ofvolatilityclustering. Mandelbrotquoted(1963): largechangestendtobe followed by large changes, and small changes by small changes. GARCH- type model successfully captures this property. Another main feature of electricity price is fat tails. Extreme values in electricity prices are better analyzedbyextremevaluetheory. Inthisproject,weaimtoexplorethepropertiesofelectricitypriceapplying ARMA , GARCH-type models and extreme value theory. We will mainly focus on modeling hourly electricity prices from Nord Pool in this winter, Jan-Feb 2012, hoping those external factors might be ignored. We pay at- tention to periodic components and extreme values. At first, in Section 4.2, the predictable periodicities of 24-hours and 168-hours and extreme values are removed. Then, in Section 4.3, we will fit the time series into ARMA model. In Sections 4.4 and 4.5, to analyze the volatility of data set, we em- ploy GARCH-type models. In this thesis, extreme value theory is also in- volved in Section 4.6. At last, in Section 4.7, we simulate time series using abovemodelsandGPdistribution. 4 2 Electricity Markets 2.1 The Nord Pool Nord Pool market was created in 1996 as a result of the establishment of common electricity market of Norway and Sweden. Nord Pool Sport runs the largest market for electrical energy in the world, measured in volume traded (TWh) and in market share. It operates in Norway, Denmark, Swe- den,FinlandandEstonia. Morethan70%ofthetotalconsumptionofelec- tricalenergyintheNordicmarketistradedthroughNordPoolSpot. Itwas the world’s first multinational exchange for trading electric power. Nord PoolSpotoffersbothday-aheadandintra-daymarkets,see [18]. 2.2 The Main Features of Electricity Prices The first characteristic is periodicity of different length. Electricity prices exhibit various seasonality over days, weeks and months. Weather condi- tions affect demand of electricity over months. In this paper we analyze electricity price on winter months. So, the periodic behaviors in daily and weekly are considered. And they explain periodicity components strongly, since the need for electricity is various during whole day and whole week. For example, the electricity demand is higher during daytime than at night. On the other hand, the electricity supply performs different ways between weekdaysandweekends. Secondly,presenceofspikesisdistinctinelectricityprice. Thisfeatureisre- latedtoinstantaneoussupplyanddemand,whichisquitedifferfromstocks. Anditshouldbetreated byappropriatemodel. The third one is stationary over short intervals. This means that electricity priceismeanreversionovershorterperiods. Itcanbeobservedthat,during winter,thepricefluctuatearoundastablelevel butnotfollowatrend. Last but not least, high volatility plays important role among features of electricity price. The volatility means the standard deviation of the hourly price. For electricity price, the volatility is not a constant but various from timetotime. 5 3 Theoretical Background 3.1 ARMA Model The general autoregressive and moving average (ARMA) statistical model isusedtodescribeatimeseriesthatevolvesovertime. Inthisprocessthere is a linear relationship between the values at a certain time point and past values,noiseaswell. According to [1], time series {X } is an ARMA(p,q) process if {X } is sta- t t tionaryandifforeveryt, X = φ X −...+φ X +ǫ +θ ǫ +...+θ ǫ , t 1 t−1 p t−p t 1 t−1 q t−q where{ǫ }isi.i.d. N(0,σ2)andthepolynomials t (1−φ ǫ−...−φ ǫp) 1 p and (1+θ ǫ+...+θ ǫq) 1 q havenocommonfactors. Theprocess{X }issaidtobeanARMA(p,q)processwithmeanµif{X −µ} t t is an ARMA(p,q) process. The time series {X } is said to be an autoregres- t siveprocessoforderp,andamoving-averageprocessoforderq. 3.2 GARCH Model According to [4], the generalized autoregressive conditional heteroskedas- ticit (GARCH) is a model that is used to estimate the volatility of an as- set. Itindicatesthatthepresentvolatilitydependsonpastobservationsand volatilities. ThetimeseriesX canbemodeledby t X = σ ǫ , t t t where {ǫ } is i.i.d. N(0,1) random variables. GARCH model is used to t estimatethevarianceσ2 q p σ 2 = ω + α X2 + β σ 2. t i t−i j t−j Xi=1 Xj=1 6

Description:
5.7 Simulation of GARCH-Type model and Extreme Value Dis-. tributionModel . exhibit various seasonality over days, weeks and months. Weather condi-.
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