modeling long-term electricity forward prices

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 4, NOVEMBER 2009 1649

Modeling Long-Term Electricity Forward PricesMartin Povh and Stein-Erik Fleten

Abstract—In contrast to forwards and futures on storablecommodities, prices of long-term electricity forwards exhibit adynamics different to that of short-term and midterm prices. Wemodel long-term electricity forward prices through demand andsupply of electricity, adjusted with a risk premium. Long-termprices of electricity, oil, coal, natural gas, emission allowance,imported electricity, and aluminum are modeled with a vectorautoregressive model (VAR). For estimation, we use weeklyprices of far-maturity forwards relevant for the Nordic electricitymarket. Although electricity prices experienced a few substantialshocks during the period we analyzed, there is no evidence of astructural break. Cointegration analysis reveals two stationarylong-run relationships between all variables except the gas price,indicating that these variables move together over time. We findsome influence of the risk premium, however not on the long-termelectricity forwards at Nord Pool.

Index Terms—Cointegration, electricity prices, long-term for-ward prices, vector autoregressive (VAR) modeling.

I. INTRODUCTION

C OMMODITY forward markets are normally focused oncontracts with time to maturity up to 1.5 years. Since the

correlation between the short-term and long-term prices is highin many markets, long-term risks can be hedged with roll-overhedging using short-term and midterm forwards and futures.Unlike most commodities, electricity cannot be stored to anygreat extent. In an empirical analysis of forwards from NordPool, Koekebakker and Ollmar [1] show that the correlationbetween short-term and long-term electricity futures is lowand conclude that short-term contracts are not appropriate forhedging long-term exposures in electricity markets such aslong-term procurement costs and production revenues. Whilefar-maturity exposures can normally be hedged with short-termpositions, electricity companies can only properly hedge themwith long-term trading. Although the liquidity of long-termelectricity forwards is still often low, their maturities lie up tosix years in the future.

Long-term electricity forward prices also serve as importantinformation carriers, since they provide valuation signals forstrategic decisions like investments, mergers and acquisitions,and financing of new long-term generation assets. In recentyears, these decisions are also influenced by the environmentalpressure on the technology shift from traditional coal andnuclear to natural gas and renewable sources. Investment and

Manuscript received April 30, 2007; revised January 16, 2009. First publishedSeptember 22, 2009; current version published October 21, 2009. The work ofS.-E. Fleten was supported by the Research Council of Norway under project178374/S30. Paper no. TPWRS-00300-2007.

M. Povh is with the Faculty of Electrical Engineering, University of Ljubl-jana, Ljubljana SI-1000, Slovenia (e-mail: [email protected]).

S.-E. Fleten is with the Norwegian University of Science and TechnologyDepartment of Industrial Economics and Technology Management, TrondheimNO-7491, Norway (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRS.2009.2030285

disinvestment decisions are triggered by changes in the relativeeconomics of technologies, driven by changes in underlyingcommodity prices. The real options theory comprehensivelydescribed in [2] is appropriate for analyzing such decisions.The real option theory suggests using forward prices instead ofprojected future spot prices. The use of forward prices bypassesthe problem of risk adjustment of the discount interest rates,allowing the assets to be valued over time with a risk neutralpricing.

Since forward contracts are not traded far enough to be usedin real asset valuation, the forward prices beyond the tradedhorizon need to be forecasted. Extrapolation of quoted forwardprices might not give the best estimate, since it ignores the avail-able information about the long-term supply and demand. For-ward contracts towards the end of the term structure are oftenilliquid, reducing the trust in extrapolation. More sophisticatedmodels that focus on modeling long-term supply and demandand risk adjustment are therefore necessary to produce a betterestimate of forward prices beyond the term structure. Since suchmodels involve the understanding of what influence the pricesof traded far-maturity forwards, they might also prove useful inspeculative trading.

Long-term electricity prices are traditionally modeled withlong-term production-cost models [3], [4]. In a restructuredmarket, however, electricity prices do not necessarily equal pro-duction costs. Different extensions of production cost modelswere proposed to better reflect the real prices observed in thederegulated market. A hybrid approach, in which bottom-upmodels based on production cost variables are calibrated onmarket data, has gained increasing attention in recent years[5], [6]. Nonetheless, the literature on long-term electricityforwards is still scant, due to the lack of trusted long-termmarket data. Long-term forward prices are more often modeledas an extension of short-term forward-price modeling. Schwartz[7] uses models estimated on short-term oil futures and teststheir performance on the available long-term oil futures.The correlation between long-term electricity forward pricesand short- or midterm electricity forward prices, as shown inKoekebakker and Ollmar [1], is, however, low in many markets.This indicates that short-term models are unable to explain thedynamics of long-term electricity forward prices. An exampleof long-term electricity forward price modeling is provided in[8], which reports on a forward-price and volatility-forecastingmodel that combines risk adjustment and external long-termforecasting models.

In this paper, we focus on modeling the dynamic structureof long-term electricity forwards. To model these prices, wetry to identify the long-term information that influences the ex-pected long-term electricity supply, demand, and risk premium.We analyze the weekly prices of Nord Pool’s long-term elec-tricity forwards and how these are influenced by long-term for-ward prices of fuels, emission allowances, and imported elec-

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1650 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 4, NOVEMBER 2009

Fig. 1. Electricity spot price and long-term forward price dynamics from NordPool.

tricity. Due to possible endogeneity, we use vector autoregres-sive model, which do not require any ad-hoc assumptions onexogeneity.

This paper is organized as follows. Section II identifies thelong-term electricity forward price process. The use of the dataand univariate model representation is presented in Section III.Section IV starts with descriptive analysis of variables and mul-tivariate representation. This is followed by cointegration anal-ysis and estimation of vector error correction model. Section Vdraws the conclusions.

II. LONG-TERM FORWARD PRICE FORMATION

We define long-term electricity forward prices as prices ofelectricity forwards with a delivery period of one year and atime to maturity of more than one year year . Fig. 1presents an example of the price dynamics for a forward contractfrom the Nordic electricity exchange Nord Pool with delivery in2007. These contracts, though named forwards, correspond tothe definition of swaps, having a stream of cash flows that de-pends on the difference between the realized spot price and thefixed contract price. We will continue to denote them forwards.Fig. 1 demonstrates that the forward-price dynamics is differentfrom the spot-price dynamics when . As the delivery pe-riod closes , the forward-price dynamics becomes moresimilar to the spot-price dynamics. Long-term forward pricesand short-term forward prices (spot prices) are, therefore, gov-erned by somewhat different laws, which indicate the need tomodel them separately.

A. Setup

The nonstorability of electricity has important implicationson electricity trading and the valuation of forward contracts.While the cost-of-carry arbitrage is usually applied in valuationof commodity forwards, it cannot be used in case of electricityforwards, since electricity cannot be bought today at the spotprice and stored for subsequent sale at the forward price .As an alternative to the cost-of-carry arbitrage, one can use anequilibrium approach [9]

(1)

where the forward price is the (rational) expectation aboutthe spot price at delivery time, (or simply ) dis-counted with the risk-free interest rate and the risk premium .Due to the uncertainty of the expected spot price, , marketparticipants require a compensation for bearing the spot-pricerisk, i.e., they determine their own risk premium. When indi-vidual risk preferences are matched (e.g., on the exchange), theaggregated risk premium is obtained; this is also referred to as themarket price of risk. In (1), the forward price formation is there-fore an equilibrium process. If one is able to obtain an unbiasedestimate of the expected spot price , the supply and demandfor bearing the spot price risk determines the risk premium .

Transforming (1) to logs gives

(2)

Assuming constant risk-free interest rate and risk premiumand writing time to maturity as gives

(3)

where is the risk premium parameter defined as .In (3), the risk premium therefore depends only on time to ma-turity, which is a very common assumption in modeling fixedincome markets, foreign exchange markets, or commodity mar-kets. In case of electricity, an assumption that risk premium de-pends only on time to maturity is also often applied [10], despitesome empirical findings, which indicate that the risk premium inshort-term electricity forwards might be influenced by the prob-ability of price spikes (load seasonality) and the level of prices[11], [12]. However, some investigations, which also extend tofar-maturity contracts indicate that the magnitude and the vari-ability of the risk premium in far-maturity electricity forwardsis low [13], [14].

In (3), the forward prices are therefore mainly driven by theexpected spot prices, subject to information sets available tomarket participants. We assume that information sets in our caseinclude past information about as well as the informationthat influence the expected spot price , i.e., variables thatinfluence the expected supply and demand. We assume all par-ticipants (i.e., producers, buyers, and traders) have the same in-formation set.

B. Modeling the Long-Term Expected Spot Price

We define the long-term forwards as the contracts with de-livery period of one year, having a payoff that depends on therealized spot price over the delivery year. The long-term ex-pected spot price therefore represents the expected price of1 MW of annual base-load electricity. Expected electricity spotprices a few years into the future are influenced by the expectedsupply and demand at the delivery time . The supply and de-mand are however not observable variables. Instead we can usefundamental variables that influence the supply and demand toestimate their influence on the expected spot price. Electricitydemand can sufficiently be explained with weather, economicactivity, and demography, whereas the variables influencing thesupply can be grouped into five groups:

1) fuel prices (coal, natural gas, oil);2) water-reservoir level in hydro-rich systems;

POVH AND FLETEN: MODELING LONG-TERM ELECTRICITY FORWARD PRICES 1651

3) emission allowance prices ;4) supply capacity (market structure, available capacity);5) electricity prices in neighboring markets (in the case that a

significant share of electricity is imported or exported).When estimating the expected spot price in the long-term, weseek reliable information about the expected values of the fun-damental variables mentioned above. In the short term ,the fundamental variables can be predicted with high, thoughnot complete, accuracy. As the time to maturity increases, thevariance of these variables increases and their mean values areharder to predict. Still, there is a difference between variablesthat are considered stationary and integrated variables. With sta-tionary variables, the unconditional variance is bounded and theunconditional mean is based on historical average and expectedgrowth. Such are the hydro reservoir levels, supply capacity,and electricity demand, which can be predicted based on his-torical average value and expected long-term growth. Integratedvariables on the other hand have no unconditional distributionsince the shocks in these variables will persist and their uncon-ditional variance is therefore unbounded. In our case, these arefuel prices, emission allowance prices, and prices of electricityin neighboring markets. Fortunately the market offers securitiesto hedge their uncertain future evolution. Among these secu-rities, we use long-term forward prices of fuels, emission al-lowances, and electricity in neighboring markets to explain thedynamics of long-term electricity price.

Information on the forward prices of fuels, emission al-lowances, and imported electricity changes on a daily basis,since these forwards are usually traded each working day. In-formation on the long-term expected demand and the expectedsupply capacity changes only when new information on theunderlying factors (GDP, construction, and retirement plans)becomes available; this information changes less frequently(e.g., monthly, quarterly, or yearly). The problem with this datais also that it is not as reliable as market-based information.Unless it is published as a part of exchange information, marketparticipants need to estimate the expected demand and supplycapacity themselves. The expected spot price is, therefore, in-fluenced by high-resolution market-based information (forwardprices of fuels, emission allowances, and neighboring-marketelectricity) and by low-resolution estimated information (ex-pected demand and supply capacity).

Due to the different resolutions of both types of infor-mation, an estimation of the influence of these variables onelectricity forward prices is challenging. In this paper, weuse only high-resolution market-based information, whereaslow-resolution estimated information is the additional source ofuncertainty and it influences the variance structure of expectedlong-term electricity spot prices. Our model for the expectedlong-term spot price of electricity is therefore

(4)

where is the expected electricity spot price, is theforward price of fuel , is the forward price of the emis-sion allowance , and the forward price of electricity in aneighboring market .

TABLE ISAMPLE CONSTRUCTION

III. DATA AND DESCRIPTIVE ANALYSIS

We test the proposed model on the long-term electricity for-wards from the Nordic electricity exchange Nord Pool. NordPool is one of the oldest electricity exchanges, covering the areaof four Nordic countries: Norway, Sweden, Finland, and Den-mark. In 2005, most of the electricity in the Nordic electricitymarket was supplied by hydroelectric plants (54%), with the restcoming from nuclear (22%), renewable (8%), coal (6%), nat-ural gas (5%), imports (3%), oil (1%), and other sources (1%).In 2005, the Nord Pool financial market volume was 786 TWh,physical volume was 176 TWh, whereas the total production inthe market was 404 TWh. The market went through a numberof structural changes, the latest being the introduction of theEuropean emission trading scheme (ETS) in 2005. Since thischanged the overall price formation, we choose to analyze onlythe prices from the start of 2005 to the end of 2007. Our datasample is constructed in a way to include only prices of yearlycontracts with time to maturity between one year and two yearsas shown in Table I. For observation period 2005, ENOYR07 isused, and this contract is replaced with ENOYR08 with the startof 2006 and with ENOYR09 with the start of 2007. This way, weavoid the price shift when two consecutive contracts are rolledover. Since contracts with delivery period two and three yearsahead move very similarly, the difference between them is verysmall. For other variables, defined in the following of the paper,we use forward prices with the same observation and maturityperiod as electricity forwards.

The analysis of high-resolution financial data often involvesthe problem of nonsynchronous trading. The prices in our anal-ysis are quoted at different times, and due to the time mismatch,the integration between them is not clear. We use weekly res-olution instead of daily resolution, since the relative time mis-match is much lower in the case of weekly sampling. Althoughthe weekly sampling tends to smooth out the magnitude of pricejumps, the volatility structure should not be significantly dif-ferent to that when using the daily sampling. We use the closingprice from each Wednesday as the reference weekly price for allthe variables, giving the sample size of .

As shown in Fig. 2, there are no significant shifts at the time ofrollover. The sample, however, shows a significant price shockin April 2006 corresponding to observations 67 to 70. Beforethis shock, emission allowance prices were pushing elec-tricity prices up significantly; however, when the report on ac-tual emissions in EU was published in April 2006, the prices ofemission allowances dropped dramatically, which had a signif-icant effect on electricity prices. We will investigate this effectby testing whether this shock can be considered as a structuralbreak in the relationship between the variables.


Fig. 2. Estimation sample for long-term electricity forward price.

A. Fuel Prices

Fuel prices can be divided into two groups. In the first groupare the fuels that are not traded on an exchange, and so no trans-parent information about their prices exists. These fuels are ura-nium, biomass, water, wind, solar, and other renewable sources.Prices of these fuels are uncertain and they influence the vari-ance structure of electricity prices. In the second group are thefuels that are traded on an exchange, and at least some infor-mation about their long-term prices is available. These are oilderivatives, natural gas, and coal. Although their use in elec-tricity production in the Nordic market is small, they are oftenthe marginal source of production and can have a significant in-fluence on electricity prices. We model the fuel price withthe forward prices for coal , natural gas , and crudeoil , which also represents the price of all oil derivatives:

(5)

For the crude-oil price, we use the NYMEX WTI light sweetcrude oil data. Although the Brent crude oil data from the Inter-continental Exchange (ICE) might be a better choice for Nordiccountries, the availability of long-term oil prices is much betterat NYMEX. The long-term NYMEX WTI price represents aglobal price indicator of the world oil price in the long-term.

The steam-coal market cannot be characterized as a globalmarket like crude oil. The majority of coal is still traded overthe counter, mostly because coal is hard to standardize, due toits different energy values. Exchange forward trading with coalis still in its early stages. Instead, we use the TFS API2 indexas a reference for coal prices in the Nordic area. TFS API2 isa price index for coal delivered in Amsterdam, Rotterdam, andAntwerp harbor and should, therefore, also represent the coalprices in the Nordic area.

The natural gas consumed in the Nordic area comes mainlyfrom North Sea resources. Natural gas forwards of North Seagas is also traded on ICE. We use the ICE quarterly prices ofnatural gas forwards to construct the yearly forward prices fornatural gas.

B. Emission Allowance Prices

The price of emission allowances in our model include onlythe price of the European emission allowance (EUA)which were introduced by ETS in 2005 for carbon oxideemissions. The second part of (4) is therefore

(6)

where is the forward price of the EUA. We use the dataon EUA prices from Nord Pool. Since Nord Pool began tradingwith EUAs in March 2005, we use the Spectron EUA priceswhich precede that date. The difference between different EUAprices is negligible, since allowance can be purchased andused anywhere in Europe.

C. Neighboring-Market Price

The Nordic electricity market imports electricity from Russia,Germany, and Poland. We have no information on import pricesfrom Russia, so we use only the European Energy Exchange(EEX) long-term forward price as a reference price for the elec-tricity imported from Germany and Poland. The neighboring-market price is, therefore, the EEX long-term electricity forwardprice:

(7)

Combining (3)–(7) gives the following regression modeldescribing the long-term electricity forward prices from NordPool:

(8)

In (8), we include an error term , which represents the un-certainty in the expected spot price and the uncertainty in ex-planatory variables. We assume the error term follows anormal distribution.

IV. MULTIVARIATE MODEL

Model (8) defines the univariate relationship between a de-pendent variable on the left and explanatory variables on theright-hand side. The drawback of such a representation is thatexplanatory variables are assumed to be exogenous, which is anassumptionthatshouldbetestedrather thanassumedapriori.An-other drawback of the representation in (8) is that it fails to validlyestimate all the long-term relationships between the variables,particularly when variables are nonstationary and cointegrationbetween variables is present. In our model, we cannot assumeexogeneity or stationarity, since the prices of interdependentcommodities are often cointegrated and nonstationary. A modelthat overcomes the deficiencies of single equation models is avector autoregressive model (VAR). A VAR model assumes thatall variables are endogenous; hence, all variables are modeled asa function of own past values and past values of other endogenousvariables. We define a general Gaussian vector autoregressivemodel

(9)


TABLE IIDESCRIPTIVE ANALYSIS OF VARIABLES

rejects the null at 5% significance. rejects the null at 1% significance.

TABLE IIIVAR(2) DIAGNOSTIC TESTS


where is a vector of endogenous variables, vector of ex-ogenous variables, and intervention dummies to render theresiduals well-behaved. We use the logs of , , ,

, , and as endogenous variables, and we denotethem as , , , , , and , respectively. Time tomaturity is considered exogenous.

A. Sample Analysis

Table II gives descriptive analysis of variables in (9). Thevariables are not normally distributed; particularly the skewnessand excess kurtosis and are very high. Nonstationaritycannot be rejected in all cases except for and . Both sta-tionarity tests strongly reject the nonstationarity in first differ-ences (not presented here). Variables , , , and aretherefore integrated of order , while and may be oforder , although the results are not strongly significant.

B. VAR Setup

Based on stationarity test, we will assume that none of thevariables is and the system is therefore adequately modeledas .

The model (9) with lag length set to is estimated, andresults together with diagnostics are presented in Table III. Nointervention dummies are included at this point. All en-dogenous variables in VAR are significant. A significance test

on deterministic components show that the constant ismarginally significant, while time to maturity is not sig-nificant. The diagnostic tests involve -tests that there is noresidual autocorrelation ( , against 4th-order autoregression),that residuals are normally distributed , that there is no het-eroscedasticity , and that there is no autoregressive con-ditional heteroscedasticity ( , against 4th order). Misspec-

TABLE IVVAR(2) DIAGNOSTICS TESTS


ification tests reveal significant problems with all of these tests,particularly when vector tests are considered. Since VAR esti-mates are more sensitive to skewness than kurtosis, residualsskewness is also reported.

To overcome the undesired properties of residuals in Table III,we first focus on the structural specification of the model. In-creasing the lag length doesnothelp to removeresidual autocor-relation. Since residual autocorrelationalsosuggests an omissionof important variables that influence the dynamic structure of ourmodel, we analyze the price movement during this period andsearch for additional variables that might also be included in themodel. Among much nonquantifiable information, we find thataluminum prices also affected the prices of electricity in Scan-dinavia and Europe during this period. Aluminum prices rosesignificantly during this period, and this triggered some decisionsto postpone the decommissioning of some aluminum smelters,which could sell aluminum under increased long-term aluminumprices, with long-term electricity forwards as hedging instru-ments. The long-term aluminum prices therefore reflect changesin part of future electricity consumption and influence the de-mand for long-term electricity forwards. Descriptive analysis foraluminum forward price from London Metal Exchangealso show a non-normal distribution, while the values of ADFand Phillips-Peron test are 1.46 and 1.28indicating integration of order .

We also introduce a few dummies to the system to accountfor the shocks, which are known to induce erratic behaviorand nonlinear dynamics. First we introduce a blip dummy toset the residuals from 67 to 70, to zero, which correspond to

price shock in April 2006. A blip dummy of a typewith three lags is used for this purpose. Next we

add one transitory dummy of a typeto remove the effect of transitory shock in observations 27 and29. Additionally three blip dummies , , andare used to remove the largest outliers. The diagnostics of VARthat include these changes is presented in Table IV.

The results show that aluminum price and dummies help toimprove the properties of VAR. Most single equation and vectormisspecification tests are improved. There is a still slight au-tocorrelation present in and , but we will not pursuethis further, since we expect these two variables are weakly ex-ogenous and they do not have to be modeled themselves. The


Fig. 3. Recursive break-point Chow test.

vector tests on the other hand reject the autocorrelation and het-eroscedasticity in residuals. While strict normality is still notachieved, we managed to reduce the skewness, which is morecritical than kurtosis.

We test this specification for parameter constancy. In partic-ular, we are interested in the influence of price shock inApril 2006. The shock had a significant effect on the Nord Poolforward price as seen in Fig. 2 and also on the EEX forwardprice. To test whether this shock, or any other shock duringthis period, changed the overall structure of the data generatingprocess, we use the Chow test for structural break [15]. Fig. 3shows recursive break-point Chow test for each equation in thesystem and for the system as a whole. The 1% significance levelof the break-point test is never exceeded indicating that param-eters of individual equations and the system as a whole are con-stant throughout the sample. The price shock can thereforebe considered as a transitory shock, which can be removed withintervention dummies, rather than a structural break.

VAR in Table IV is also tested for stability by checking theroots of the companion matrix. All the roots lie inside the unitcircle with the moduli of the three largest roots being 0.981,0.981, and 0.948, respectively indicating that this representationof VAR is stable.

C. Cointegration Analysis

Since unit root testing indicates that first differences are ,we convert the model (9) to first difference model. This modelexplains only the short-run dynamics of the system, while thelong-run relationship between variables, which is important ifvariables are cointegrated, is lost. Cointegration between non-stationary variables can be captured with vector equilibriumerror correction model (VECM):

(10)

which has the same innovation process , since no restrictionshave been imposed by transformation from (9) to (10). In (10),

TABLE VCOINTEGRATION RANK TEST AND CHARACTERISTIC ROOTS


the right-hand side contains information about the short- andthe long-run adjustment to changes in . If containsvariables, then is , while must also befor to be a white noise process. Matrix can be decom-posed to , where represent the speed of adjustmentto disequilibrium and is the matrix of long-run coefficientssuch that represents up to stationary cointegratingrelationships, which ensure that converge to their long-runsteady-state solution. A note, however, is necessary that since

contains two variables that are possibly in levels, theyform a cointegrating relation by itself adding to the total numberof cointegrating relations. is unrestricted constant which ac-counts for a constant in the short-run model (trend in levels) anda constant in cointegration space.

To test for cointegration between variables, we employ Jo-hansen testing procedure [16] which concentrates on testingwhether the eigenvalues of the matrix in (10) are sig-nificantly different from 0. We test whether has a reducedrank , indicating that there are stationary cointe-grating relationships between nonstationary variables in VAR. If

, this would indicate that all variables are stationary, whilewould indicate no stable cointegrating relationships and

the VAR with first differences only would be adequate. To de-termine the rank , we use the trace test statistics

(11)

where is the sample size and are the estimated eigenvaluesof . The results of the cointegration rank test, presented inTable V, show that is strongly significant while andare on the borderline of significance with values 0.019 and0.074, respectively. It is hard to know exactly if they form astationary cointegrating vector or not. Since the choice of coin-tegration rank is crucial in modeling cointegrated systems, welook for additional indicators for determining , as specified in[17]. First we look at the moduli of the largest characteristicroots of the model, and see how they are changing for the hy-potheses in question, i.e., , 2, and 3. Table V again showindecisive results for and . It is hard to know ex-actly whether a moduli of 0.910 represent a unit root or not. Nextwe look at the significance of parameters of loading matrix .The -value of is 4.20 and for is 3.28 indicatingthat the second vector adds additional explanatory power to the


Fig. 4. Cointegrating vectors.

equation and third cointegrating vector to the equation.Finally we look at the graph of the first six cointegrating vec-tors presented in Fig. 4. The first two cointegrating vectors lookstationary and the last three are clearly not. For the third cointe-grating vector, it is hard to decide, so we test the third vector withADF test and Phillips-Perron test. Both of them reject station-arity with probability of 0.115 and 0.153, respectively. Based onthese findings, we choose cointegration rank . Althoughthe third vector might also help to explain the long-term rela-tionship in equation, it is not stationary.

To identify the two cointegration vectors, we test the restric-tions on estimated and . We first rotate the cointegrationspace by normalizing with respect to and . This way,beta is exactly identified and the significance of each variable incointegration space can be tested by putting additional restric-tions on parameters in . These tests based on standard LR testshow that in the second vector, only the parameter is signifi-cant, indicating that the second vector is exactly . This is con-sistent with the unit root test results showing being stationaryin levels. Gas price is insignificant in both cointegrating vectorsand therefore have no long-run explanatory power. Testing allrestrictions gives the following representation of , with stan-dard errors below.

The first cointegrating vector is interesting since it represents alinear combination of , , , , and . The NordPool price therefore increases approximately one to one withthe aluminum price and less than one to one with the coal price.On the other hand, the Nord Pool price falls almost one to onewith EEX price increase and falls by 0.13% if the emission al-lowance price rises by 1%. Since Nord Pool and EEX price arestrongly positively correlated, this means that if a positive shockoccurred in the EEX price in the last period, then similar posi-tive shock is also likely to have occurred in the Nord Pool price.The cointegrating vector would then pull the Nord Pool priceback down in the next period.

TABLE VIDIAGNOSTICS TEST OF VECM


In the second step, we test the restrictions on loading matrix, which is also known as the test for weak exogeneity. The test

involves testing the restrictions that particular row in the esti-mated loading matrix is insignificantly different from zero.The parameters of explain how the short-run model is ad-justed to the disequilibrium represented by cointegrating vec-tors . If the entire row in is zero, this indicates thatnone of the cointegrating vectors enters the equation associatedwith this row. Testing these restrictions additionally to restric-tions on shows that and are weakly exogenous inour model, while other variables should be treated as endoge-nous. The values of coefficients, reported below, show thatoil and gas price have very low and also insignificant speed ofadjustment to both cointegrating vectors, even though the weakexogeneity test for these two variables was rejected at 5% sig-nificance. The parameters for Nord Pool, EEX, and EUA priceare strongly significant. EUA price has the highest speed of ad-justment with about 39% of disequilibrium in the first vectorcorrected in one period:

Based on cointegration test and weak exogeneity test, weform a VECM as in (10). now includes endogenous vari-ables , , , , and , while includes two weaklyexogenous variables and and time to maturity .Estimation of (10) includes one lag of first differences of en-dogenous variables, the first lag of two cointegrating vectors,the first lag of weakly exogenous variables, eight dummy vari-ables, and a constant, giving 25, 10, 15, 40, and 5 parameters,respectively, a total of 95 parameters to estimate. We reduce themodel size with the standard -test. The reduced model showsthat is also insignificant in the short-run model so we com-pletely remove from the system. The 4-D model now in-cludes 40 parameters and the value of -test on reduction is

. The reduced model includes the first lag of, two cointegrating vectors, a constant, , and five dum-

mies only. The diagnostic tests presented in Table VI show thatthe main properties of residuals remained unchanged with stan-dard errors very close to values in Table IV. Since , ,and equation are removed from the system, vector auto-correlation and heteroscedasticity tests are improved. Normality


test, however, shows no improvement. Both cointegrating vec-tors are highly significant, confirming that the choice of coin-tegration rank is correct. is significant only in and

equation, indicating that the system shows some influ-ence of the risk premium; however, the significance is stronglyrejected in , which is in our interest.

V. CONCLUSION

We have analyzed the long-term electricity forward pricesusing a vector autoregressive model. The model is specifiedbased on variables that influence the expected long-term elec-tricity supply, demand, and risk premium. We use the long-termforward prices of oil, coal, natural gas, emission allowances, im-ported electricity, and aluminum prices to model the dynamicproperties of long-term electricity forward prices. The risk pre-mium is modeled as a function of time to maturity.

The model is estimated on weekly data from 2005 to 2007using variables relevant for the Nordic electricity market. Wespecify a 7-D VAR with two lags and few intervention dummies.The influence of the emission allowance price shock in April2006 is analyzed with a Chow breakpoint test, which showedno breaks in constant or trend. The variables in the model areall integrated of order , except oil and emission allowanceprice, which are but close to unit root. The system is testedfor cointegration using the Johansen cointegration test. The test,together with other indicators, indicates two stationary cointe-grating vectors, the first being a linear combination of nonsta-tionary variables. The Nord Pool price increases if aluminumprice and coal prices increase in the previous week and de-creases if the emission allowance price and the EEX price in-crease in the previous week. The second cointegrating vector isexactly the oil price, which is stationary in our case. Gas priceis found insignificant in both the short-and the long-run model.The model shows some influence of the risk premium; however,its influence on electricity forward prices from Nord Pool is notconfirmed. This indicates that the risk premium dynamics in thelong-term electricity forwards from Nord Pool is rather low andthat the risk premium could be considered as constant. Whilethese results hold for the Nordic electricity market, other mar-kets may have a different maturity level and their price dynamicsmay respond to other variables. Nevertheless, the general ap-proach could be used for analyzing other electricity markets.

ACKNOWLEDGMENT

The authors would like to thank Nord Pool ASA for the ac-cess to their FTP database and S. Westgaard, J. Wimschulte,S. Frydenberg, and N. van der Wijst for helpful comments.

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Martin Povh is pursuing the Ph.D. degree at the Faculty of Electrical Engi-neering, University of Ljubljana, Ljubljana, Slovenia.

He is a Research Assistant at the University of Ljubljana. His research isfocused on modeling electricity markets, electricity prices and their uncertainty,investment analyses under uncertainty, and electricity market operation.

Stein-Erik Fleten received the M.Sc. and Dr.Ing. degrees from the NorwegianUniversity of Science and Technology (NTNU), Trondheim, Norway, in 1995and 2000, respectively.

He is a Professor in the Department of Industrial Economics and TechnologyManagement at NTNU. His research interests include real options valuation,financial engineering in energy commodity markets, and stochastic program-ming. Applications typically involve restructured electricity markets.

modeling long-term electricity forward prices

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