Fakulteta za Elektrotehniko

Jaroslava Hlouskova, Stephan Kossmeier, Michael Obersteiner, Alexander Schnabl

Short-Term Forecasting of Electricity Spot Prices

OSCOGENDiscussion Paper No. 4

November 2001

Contract No. ENK5-CT-2000-00094 Project co-funded by the European Community under the 5th Framework Programme (1998-2002)

Contract No. BBW 00.0627 Project co-funded by the Swiss Federal Agency for Education and Science

1

Contract No. CEU-BS-1/2001 Project co-funded by Termoelektrarna toplarna Ljubljana, d.o.o.

1 Introduction

This study presents an empirical analysis of electricity spot prices and was motivated by

Knittel and Roberts (2001). The data used here consists of hourly Leipzig Power Exchange

(LPX) electricity spot prices. The sample period begins on 1:00, June 16, 2000 (the opening

of the market) and ends on 24:00, October 15, 2001 for a total of 11 688 observations.

The behavior of electricity differs from the behavior of other commodity prices. One reason

for this is that electricity is a non-storable good, implying that inventories cannot be used to

arbitrage prices over the time.1 This inability to use arbitrage arguments for pricing securities

creates a need for accurate forecasts of electricity spot prices.

The behavior of electricity spot prices is characterized by several features:

• Mean reversion.

• Time of day effect - this is demonstrated in Figure 1 in the Appendix where average hourly

electricity spot prices measured in Euro per megawatt hour (Euro/MWh) for weekdays and

weekends are presented. As expected, prices are, on average, higher during the week when

demand is higher. The price begins to increase at 5:00 during the workday and continues to

increase until 12:00 when there is the first and biggest peak of the day. Then the price begins

to fall until 17:00 and after reaching its lowest point it starts to increase again until 19:00

when it reaches the second (and smaller) peak of the day. Prices begin to fall thereafter as the

workday ends and demand shifts to primarily residential usage.

• Weekend/weekday effect.

• Seasonal effects.

• Time varying volatility and volatility clustering.

• Extreme values.

1 Hydroelectric resources are arguably storable.

2

Table 1 presents summary statistics for the whole sample of hourly electricity prices and for

all 24 hours of daily electricity prices. The null hypothesis of normal distribution tested by

Jarque-Bera test statistic is not rejected at a 5% significance level for hours 1-5, 8. Thus,

models based on the normality assumption applied on the electricity prices of hours 1-5, 8

have bigger chance of accurately representing the data generating process than hours 6, 7, 9-

24. Note that the electricity prices for first peak hour 12 have the biggest mean and standard

deviation and the electricity prices for second peak hour 18 have the biggest skewness and

kurtosis.

2 Models and results

This section presents several different models of electricity spot prices. Each model is first

motivated and discussed in the context of the preliminary data analysis. Then the forecast

performance of the models are compared. The forecast horizon is 168 hours (one week) as

according to the frequency of the data only short-term forecasts are feasible. Out-of-sample

periods are of the length of 168, 169, 170 and 171 hours (of the same time period) and as

prices exhibit also the extreme values the out-of-sample periods are chosen at the end of the

sample data where no jump occurs (we refer later to this period as the first out-of-sample

period, see Figure 4) and the end of August and beginning of September where a jump

occurred (the second out-of-sample period, see Figure 5).

2.1 Model 1: Mean reverting process It has been well documented that an important property of energy spot prices is mean-

reversion (see Gibson and Schwartz (1990), Brennan (1991), etc.). This is usually modeled

using the Ohrnstein-Uhlenbeck process, which allows for autocorrelation in the series by

specifying prices as:

( ) )1.2(,)0(),()()( 0pptdWdttptdp =+−= σµκ

where p(t) is the electricity price at time t; µκ , and σ are unknown parameters and W(t) is

Wiener process. The intuition behind this specification is that deviations of the price from the

equilibrium level, tp−µ are corrected at rate κ and subject to random perturbation, tdWσ

(this assumes that 0>κ ).

3

Equation (2.1) is simply a first order autoregressive model. This can be seen by integrating

equation (2.1) to obtain:

( ) ∫ −−− +−+=t tstt sdWeepetp0

)(0 ).(1)( σµ κκκ

The exact discrete time version of this equation is

)2.2(,110 ttt pp ηβα ++= −

where and The error term, ( ) κκ βµα −− =−= ee 10 ,1 ∫−

−=t

t

tst sdWe

1

)( ).(ση κtη in (2.2) is

Gaussian white noise2 with variance Thus, the conditional mean is .2/)1( 222 κσσ κη

−−= e

11 −tt0 + βα and conditional variance is .2ησ

While these models capture some of the autocorrelation present in the price series, it suffers

from several serious shortcomings: (i) it ignores cycles present in the series (intraday,

weekend/weekday and seasonal); (ii) it assumes the error structure is independent across time;

(iii) it assumes that the volatility is constant over time; (iv) the normality assumption cannot

reproduce the extreme swings found in the data. All of these shortcomings appear in the

forecast results presented in Tables 3, 4 and Figure 8. Figure 8 presents the actual and

forecasted price series represented by the solid black and orange line, respectively. The two

dashed lines represent two standard errors around the forecasted value.

2.2 Model 2: Mean reverting process with time-varying mean The second model addresses the systematic variation found in electricity prices. We consider

the following extension to (2.1)

( ) )3.2(,)0(),()()()( 0pptdWdttpttdp =+−= σµκ

with

( ]( ) ( ) ( ) [ ] )4.2(11,11)(24

1∑∑ ∑ +∈+∈+−∈=

= monthtrendmonth

i daydayti tcmonthtcdaytciilctµ

where

2 A Gaussian white noise process is a sequence of independent, normally distributed random variables with zero mean and constant variance.

4

{ })5.2(

024,24,23,,124,24

=−∈−−

=t

ttt ktif

ktifktl

K

with k being the largest integer such that t Monday, Tuesday,

Wednesday, Thursday, Saturday, Sunday , month = January, February, March, April, May,

June, July, August, September, October, November, [t] is the largest integer not greater than t

and 1(.) denotes the indicator function. For instance

t { } =∈− daykt ,23,,1,024 K

( ]( ) ( ])6.2(

.,0,,1,1

,11 −∈

=−∈otherwise

iilifiil t

t

When modeling electricity prices for each hour separately the varying mean will have the

following form:

( ) ( ) [ ] )7.2(11)( ∑ ∑ +∈+∈=day month

trendmonthday tcmonthtcdaytctµ

where day= Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday and month is

the same as in (2.3). Equations (2.4) and (2.7) imply that )(tµ is a step function, constant

across any one hour. Integrating equation (2.3) yields

∫ ∫ −−− ++=t t

tstst sdWedssepetp0 0

)()(0 )()()( σκµ κκκ

∫ ∫− −

−−− ++−=t

t

t

t

tsts sdWedssetpe1 1

)()( ).()()1( σκµ κκκ

As one unit of time is an hour, )(tµ is constant over the interval Thus, the exact

discrete time version of (2.3) is

[ ).,1 tt −

)8.2(,11 tttt pp ηβα ++= −

where and is Gaussian white noise. The

only difference between equations (2.2) and (2.8) is in the intercept.

( ) κκ βµα −− =−= eett 1,1)( ∫ −

−=t

t

tst sdWe

1

)( )(ση κ

(2.8) can be viewed as an ARMA(1,0) model with the exogenous variables being the dummy

variables for an hour, a day of a week or a month of a year and can be written as

∑∑ ∑ +++++= −= month

ttttrendmonthtmonthi day

daytdayitit ptrenddddhp .11,

24

1,, ηβαααα

When modeling electricity prices for each hour separately (2.8) can be written as

∑ ∑ ++++= −day month

jtt

jt

jtrendmontht

jmonthdayt

jday

jt ptrendddp ,)(

1)(

1)(

,)(

,)()( ηβααα

for I.e., .24,,1K=j

5

∑ ∑ ∑=

+++=24

1,,, )9.2(

i day monthttrendmonthtmonthdaytdayitit trenddddh ααααα

when dealing with hourly data frequency and

∑ ∑∈ ∈

++=Weeki Monthi

tj

trenditj

iitj

ij

t trenddd )10.2()(,

)(,

)()( αααα

for when dealing with each hour separately. 24,,1K=j

2.3 Model 3: ARMA model with time-varying intercept ARMA3 models are more traditional time series approach to modeling electricity prices.

Working in a discrete time framework, price dynamics can be specified as

)11.2(tttp ηα +=

∑ ∑= =

−− ++=p

i

q

iitititit

1 1)12.2(εθεηρη

where tα is specified as in (2.9) (or as in (2.10) when dealing with each hour separately) and

{ }tε is Gaussian white noise with variance parameter when dealing with

hourly data frequency and when dealing with each hour separately. The

motivation for the lag structure in (2.12) follows from an examination of the correlogram.

.2σ 25,,1, K=qp

19,,1, K=qp4

2.4 Model 4: EGARCH model The data analysis reveals that electricity prices exhibit volatility clustering (see Figure 2). We

have also performed a Lagrange multiplier test for autoregressive conditional

heteroskedasticity (ARCH) in the residuals, which confirmed the presence of

heteroskedasticity in residuals.5 It is also observed and expected that positive price shocks

increase volatility more than negative shocks of the same magnitude. We refer to this as an

inverse leverage effect.6 The intuition behind this is that a positive shock to prices is an

3 Autoregressive moving average models are generalizations of the simple autoregressive model that deal with the serial correlation in the disturbance by introducing additional higher-order autoregressive terms and moving average terms which use lagged values of the forecast error to improve the current forecast. 4 The high correlation is usually present between the current price and the previous hour price and the current price and previous day price. All lag parameters in considered models are highly statistically significant; i.e., AR or MA terms with insignificant coefficients are discarded. 5 ARCH models were introduced by Engle (1982) and generalized autoregressive conditional heteroskedasticity (GARCH) models were introduced by Bollerslev (1986). 6 "Leverage effect", i.e., downward movements in the market are followed by higher volatilities than upward movements, is observed in the equity markets.

6

unexpected positive demand shock and since marginal costs are convex, positive demand

shocks have a larger impact on price changes relative to negative shocks. To account for these

phenomena we use the EGARCH(1,1) or Exponential GARCH model proposed by Nelson

(1991). The conditional variance specification is

( ) ( ) )13.2(loglog1

1

1

121

2

−

−

−

−− +++=

t

t

t

ttt σ

εγσετσδωσ

where is the one-period ahead forecast variance based on past information and is called

the conditional variance.

2tσ

( ,0

7 Assuming that the errors are conditionally normally distributed; i.e.,

then ARCH models are estimated by method of maximum likelihood. Note

that the left-hand side of (2.13) is the log of the conditional variance. This implies that the

inverse leverage effect is exponential, rather than quadratic and that forecasts of the

conditional variance are guaranteed to be nonnegative. The presence of the inverse leverage

effect can be tested by the hypothesis that

),~ 2tt N σε

;0>γ i.e., that the effect of positive shocks on the

variance of prices is amplified over negative shocks. In all our estimations the asymmetry

parameter γ is positive and significant and thus the positive shocks to prices amplify the

conditional variance of the process more than negative shocks. The conditional mean equation

is given by (2.11) and (2.12).

2.5 Jump-diffusion process

Another characteristic of the energy commodity prices is the presence of price spikes (see

Figure 2). Jump-diffusion models consider large variations of the underlying variable and thus

might be appropriate for modeling the electricity spot prices. The jump-diffusion models link

the price changes to arrival of information. There are two types of information: (i) the normal

news with smooth variation in prices (modeled by a mean-reversion continuous time process)

and (ii) the abnormal news with jumps in the prices (modeled with a Poisson discrete time

process). The distribution of time series which might follow the jump-diffusion process

usually presents a positive skewness8 and have fatter tails.9 LPX spot prices also exhibit

7 We refer to the forecasted variance from last period as the GARCH term, and to the volatility observed in the previous period as the ARCH term.

21−tσ 2

1−tε

8 Skewness is a measure of symmetry: it is 0 for symmetric distributions like the normal distribution. Positive skewness means distributions with long positive tails; i.e., skewed to the right. 9 Kurtosis shows how flat or peaked, and consequently fatter tailed, the distribution is. It is a measure of the distribution shape: the higher the peak in the distribution, the higher the kurtosis. Kurtosis helps to explain the likelihood of extreme events - fat tails suggest higher chances of prices being very high or very low.

7

positive skewness and fatter tails (see Figure 3). Most values are near the mean due to the

reversion force. Our price process is now specified by appending an additional term to the

equation (2.3), yielding

( ) )14.2(),()()()()( tdqtdWdttpttdp Φ++−= σµκ

where q(t) is a Poisson process with intensity Φ,λ is the jump size which is a random draw

from the normal distribution with mean jµ and standard deviation ;jσ i.e., .

We assume that Wiener process, W(t), Poisson process, q(t), and process describing jump

size, are mutually independent. We further assume that at most one jump can occur

between observations, which is the reasonable assumption given the periodicity of our data.

( ),~ 2jjN σµΦ

,Φ

λ is the probability of the jump to occur

−

=,1

,10)(

λλ

yprobabilitwithprobailitywith

tdq

i.e., with probability λ−

1−t

1 the electricity price is drawn at time t from a normal distribution

with mean 1+t pβα and variance and with probability 2ησ λ the electricity price is drawn

at time t from a normal distribution with mean jtt p µβα ++ −11 and variance .22jσση + 10 As

in this study we are focused more on the forecasting performance of price models than dealing

with distributional properties of prices, we do not perform simulations for forecasting the

prices with jumps but as the conditional mean should converge to

( )( ) ( ) jtpttt p jttp µλβαµβαλβαλ ++ −11=+− −111 +++ −11 we construct the forecasts for

"jump" model as

)15.2(ˆˆ 11 jttt pp µλβα ++= −

where is the forecasted value of the electricity price. tp̂

The jump parameters and jump intensity 2, jj σµ λ are estimated by a Recursive Filter

approach described in Clewlow and Strickland (2000) and are presented in Table 2 for both

the whole sample and each hour. Note that there are no jumps for hours 2-8.

Except for the conditional mean of Model 2 (M2) in (2.15) we use in our forecasting exercise

also Model 1 (M1), Model 3 (M3) and Model 4 (M4). To distinguish between models without

10 Note that while the mean may rise or fall when a jump occurs, the variance always increases.

8

jumps and models where jumps are included we introduce the following notation: (i) Mi.1 for

model Mi without jumps and (ii) Mi.2 for model Mi with jumps where i=1,2,3,4.11

2.6 Results

Before summarizing the forecasting results, we introduce the following notation for i=1,2,3,4

- models dealing with the whole sample based on hourly frequency (11 688 observations):

Mi.1a or Mi.2a

- models dealing with each hour separately (this approach was adopted from Ramanathan et.

al (1997)); i.e., we deal with 24 models based on samples of daily frequency (487observations

for each hour): Mi.1b or Mi.2b i=1,2,3,4.

Thus, we are dealing with 14 models.12 Forecasting ability across all 14 models is compared

by the following forecast error statistics:

( ) ( ) ,ˆ1

1 2∑+

=

−+

=hS

Sttt pp

hRMSEErrorSquareMeanRoot

( ) ,ˆ1

1 ∑+

=

−+

=hS

Sttt pp

hMAEErrorAbsoluteMean

where is forecasted and is the actual (observed) value of electricity price.tp̂ tp 13 In this

study we perform 168 hours (one week) ahead forecasts and thus h=167.

The out-of-sample performance analysis is done in the following way: a model is estimated

with the sample ranging from 1:00, June 16, 2000 to 21:00, October 8, 2001 for the first out-

of-sample period (to 21:00, August 26, 2001 for the second out-of-sample period.14 Next, one

to 168 steps (one week) ahead forecasts are performed; i.e., from 22:00, October 8, 2001 to

21:00, October 15, 2001 and the RMSE and MAE are calculated. In the next iteration the

estimation sample is expanded by one observation (i.e., from 1:00, June 16, 2000 to 22:00,

October 8, 2001) and then again 168 steps ahead forecasts are performed and the RMSE and

11 Note that we implicitly extended forecasting with jumps also to Models 3 and 4 by adding jµλ to the conditional mean of the original model estimated for the series without jumps. 12 Due to the fact that the volatility clustering appears relevantly only in the original series, the EGARCH model is estimated exclusively for the original data; i.e., we deal only with models M4.1a and M4.2a. 13 Note that the RMSE and the MAE differ in their loss function concerning the deviations of forecasts from actual values. While the MAE weights such derivations linearly, the RMSE does it in quadratic form. 14 For short of notation we will describe the out-of-sample performance analysis only for the first out-of-sample period.

9

MAE are calculated. This procedure was repeated four times. Tables 3 and 4 present the

averages of the RMSE and MAE for out-of-sample period increasing from 168 to 171 where

the out-of-sample period of 168 hours is the first out-of-sample period (Table 3) and the

second out-of-sample period (Table 4). For the case when the last out-of-sample period of 168

hours is the first out-of-sample period the best forecast performance for all four out-of-sample

periods has ARMA model with time varying mean when dealing with each hour separately

(M3.1b) measured by both RMSE and MAE (see Table 3).15 The forecasts over period 1:00,

October 9, 2001 to 24:00, October 15, 2001 are presented in Figure 7. The worse forecast

performance has M1.2a (see Table 3 and Figure 6). For the case when the last out-of-sample

period of 168 hours is the second out-of-sample period the best forecast performance

measured by RMSE for all four out-of-sample periods has ARMA model with time varying

mean when dealing with each hour separately (M3.1b).16 The best forecast performance

measured by MAE when the out-of-sample period is 168 hours has again M3.1b and for the

out-of-sample periods of 169-171 hours has the mean reverting process with time-varying

mean when dealing with each hour separately (M2.1b). This is presented in Table 4 and the

forecasts over period 1:00, August 27, 2001 to 24:00, September 9, 2001 are presented in

Figure 9. The worse forecast performance has M1.1a (see Table 4 and Figure 8).

The asymmetry parameter γ is positive and significant in both EGARCH models M4.1a and

M4.2a for both out-of-sample (ofs) periods.17

15 The second best model in terms of the MAE for all out-of-sample periods and in terms of the RMSE for the first two out-of-sample periods is ARMA model with jumps and time varying mean when dealing with each hour separately (M3.2b). See Figure 10. 16 The second best model in terms of both RMSE and MAE for all out-of-sample periods is ARMA model with jumps and time varying mean when dealing with each hour separately (M3.2b). See Figure 11. 17 104.0=γ in M4.1a for the 1st ofs, 305.0=γ in M4.2a for the 1st ofs, 501.0=γ in M4.1a for the 2nd ofs, 323.0=γ in M4.2a for the 2nd ofs.

10

3 Conclusions Mean reversion processes are unable to accommodate the persistence found in prices. Higher

order lags are needed to capture the autocorrelation in the prices. We also document the

inverse leverage effect using the EGARCH model where positive shocks increase price

volatility more than negative shocks.

11

4 Appendix

12

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour

Euro

/MW

h

WeekdayWeekend

Figure 1: Average hourly LPX electricity spot prices across the entire sample

13

0

20

40

60

80

100

120

140

160

180

200

1

380

759

1138

1517

1896

2275

2654

3033

3412

3791

4170

4549

4928

5307

5686

6065

6444

6823

7202

7581

7960

8339

8718

9097

9476

9855

1023

4

1061

3

1099

2

1137

1

Hours

Euro

/MW

h

Figure 2: LPX electricity spot price over the period June 16, 2000 - October 15, 2001

14

0

500

1000

1500

2000

2500

3000

3500

2.5 12.5

22.5

32.5

42.5

52.5

62.5

72.5

82.5

92.5

102.5

112.5

122.5

132.5

142.5

152.5

162.5

172.5

182.5

192.5

LPX Price

Figure 3: Histogram of LPX electricity spot price over the period June 16, 2000 - October 15,

2001 (Mean = 20.33, St. Dev. = 9.50, Skewness = 2.37, Kurtosis = 23.5)

15

0

5

10

15

20

25

30

35

40

45

50

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

Hours

LPX

Pric

e (E

uro/

MW

h)

Figure 4: LPX Price from Tuesday, 1:00, October 9, 2001 until Monday, 24:00, October 15,

2001 - first out-of-sample period

16

0

20

40

60

80

100

120

140

160

180

200

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

Hours

LPX

Pric

e (E

uro/

MW

h)

Figure 5: LPX Price from Monday, 1:00, August 27, 2001 until Sunday, 24:00, September 9,

2001 - second out-of-sample period

17

0

5

10

15

20

25

30

35

40

45

50

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

Hour

Euro

/MW

h LPX PriceForecastsForecasts + fseForecasts - fse

Figure 6: One-week ahead forecasts for the first out-of-sample period for mean reverting

process with jumps (Model 1.2a)

18

0

10

20

30

40

50

60

70

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

Hours

Euro

/MW

h

LPX PriceForecastsForecasts + fseForecasts - fse

Figure 7: One-week ahead forecasts for the first out-of-sample period for ARMA process with

time varying mean when modeling each hour separately (Model 3.1b)

19

0

20

40

60

80

100

120

140

160

180

2001 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

Hours

Euro

/MW

h LPX-priceForecastsForecasts + fseForecasts - fse

Figure 8: One-week ahead forecasts for the second out-of-sample period for mean-reverting

process (Model 1.1a)

20

0

20

40

60

80

100

120

140

160

180

2001 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

Hours

Euro

/MW

h LPX PriceForecastsForecasts + fseForecasts - fse

Figure 9: One-week ahead forecasts for the second out-of-sample period for ARMA process

with time varying mean when modeling each hour separately (Model 3.1b)

21

0

10

20

30

40

50

60

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

Hours

Euro

/MW

h

LPX PriceForecasts

Figure 10: One-week ahead forecasts for the first out-of-sample period for ARMA process

with time varying mean and jumps when modeling each hour separately (Model 3.2b)

22

0

20

40

60

80

100

120

140

160

180

200

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

Hours

Euro

/MW

h

LPX PriceForecasts

Figure 11: One-week ahead forecasts for the second out-of-sample period for ARMA process

with time varying mean and jumps when modeling each hour separately (Model 3.2b)

23

Table 1: Descriptive statistics for LPX electricity spot prices over period June 16, 2000 -

October 15, 2001

Hour Mean St. Dev. Skewness Kurtosis

1 15.01 4.04 0.077 2.9602 13.25 4.03 -0.038 2.5743 12.29 4.04 -0.004 2.6504 11.88 4.08 0.010 2.5705 12.18 4.23 -0.147 2.6206 13.19 4.38 -0.434 2.7857 15.62 5.43 -0.601 2.6948 20.43 8.11 -0.028 2.7349 23.52 8.89 0.348 3.50010 25.75 9.26 0.861 5.35911 28.30 10.01 1.105 5.62312 34.87 16.59 2.758 19.95013 28.34 10.07 3.526 36.81514 25.84 9.14 1.201 8.46415 23.31 8.19 0.783 4.68716 21.44 7.08 0.567 4.43517 20.37 6.51 0.603 4.24718 21.52 9.54 6.412 88.92019 22.25 7.69 1.329 6.31120 22.04 7.03 1.683 12.33321 20.90 5.53 1.360 9.29022 19.60 4.25 1.340 13.20023 19.29 3.65 0.641 7.14024 16.86 4.05 -0.135 3.388

Whole sample 20.33 9.50 2.370 23.500

24

Table 2: Mean of jumps ( jµ ), jump variances ( ) and jump probabilities (2

jσ λ ) for specific hours and the whole sample

Hour1 7.19 172.69 0.0082 no jumps3 no jumps4 no jumps5 no jumps6 no jumps7 no jumps8 no jumps9 29.60 19.72 0.01010 36.88 119.85 0.00811 39.74 54.90 0.01212 75.85 1258.28 0.01413 68.45 2128.27 0.00614 41.72 495.72 0.00615 29.04 40.44 0.01416 28.79 13.19 0.00817 23.79 7.45 0.01018 82.87 5859.03 0.00419 31.50 77.43 0.01020 29.03 200.77 0.01221 25.52 65.78 0.01022 17.92 75.60 0.01223 19.17 67.63 0.00424 7.93 182.66 0.008

Whole sample 36.41 17.30 0.018

jµ 2jσ λ

25

Table 3: Forecast performance tested on the first out-of-sample period where the forecast horizon is one week (168 hours) and the out-of-sample period increases from 168 till 171 hours

Model 168 169 170 171RMSE

M1.1a 7.625 7.637 7.653 7.685M1.2a 7.637 7.642 7.656 7.696M1.1b 4.638 4.854 4.879 4.931M1.2b 4.604 4.840 4.868 4.924M2.1a 4.261 4.273 4.292 4.324M2.2a 4.381 4.395 4.419 4.464M2.1b 3.876 4.114 4.407 4.608M2.2b 3.893 4.081 4.204 4.232M3.1a 3.796 3.803 3.812 3.822M3.2a 4.585 4.588 4.614 4.659M3.1b 3.325 3.554 3.739 3.808M3.2b 3.375 3.732 3.901 3.940M4.1a 5.788 - - -M4.2a 5.577 - - -

MAEM1.1a 6.162 6.183 6.205 6.242M1.2a 6.192 6.203 6.221 6.261M1.1b 3.362 3.578 3.624 3.672M1.2b 3.321 3.566 3.608 3.658M2.1a 3.285 3.299 3.319 3.353M2.2a 3.440 3.456 3.482 3.529M2.1b 2.966 3.121 3.434 3.666M2.2b 2.779 2.868 2.972 2.984M3.1a 2.953 2.962 2.976 2.987M3.2a 3.656 3.660 3.693 3.743M3.1b 2.383 2.510 2.637 2.674M3.2b 2.415 2.573 2.711 2.731M4.1a 4.487 - - -M4.2a 3.938 - - -

26

Table 4: Forecast performance tested on the second out-of-sample period where the forecast horizon is one week (168 hours) and the out-of-sample period increases from 168 till 171 hours

Model 168 169 170 171RMSE

M1.1a 20.803 20.797 20.797 20.796M1.2a 20.705 20.686 20.679 20.677M1.1b 18.718 18.531 18.458 18.411M1.2b 18.614 18.419 18.345 18.300M2.1a 17.777 17.777 17.777 17.782M2.2a 17.916 17.916 17.915 17.920M2.1b 15.886 15.838 15.796 15.780M2.2b 16.232 16.199 16.219 16.249M3.1a 16.836 16.834 16.833 16.836M3.2a 18.454 18.450 18.442 18.449M3.1b 15.012 14.943 14.930 14.927M3.2b 15.744 15.675 15.655 15.659M4.1a 18.775 - - -M4.2a 19.190 - - -

MAEM1.1a 9.322 9.334 9.353 9.353M1.2a 9.303 9.301 9.313 9.310M1.1b 8.274 7.939 7.907 7.830M1.2b 8.237 7.891 7.857 7.780M2.1a 6.770 6.768 6.765 6.776M2.2a 6.711 6.714 6.718 6.720M2.1b 5.308 5.139 5.052 5.037M2.2b 5.777 5.755 5.918 6.041M3.1a 6.750 6.744 6.741 6.752M3.2a 7.681 7.672 7.666 7.678M3.1b 5.194 5.166 5.364 5.491M3.2b 5.316 5.252 5.378 5.478M4.1a 8.257 - - -M4.2a 8.215 - - -

27

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References

[1] Brennan, M.J., (1991), The price of convenience and valuation of commodity contingent

claims.

[2] Bollerslev, T., (1986), Generalized autoregressive conditional heteroskedasticity, Journal

of Econometrics, 31, 307-327.

[3] Clewlow, L. and C. Strickland, (2000), Energy derivatives: Pricing and risk management,

Lacima Publications.

[4] Engle, R., F., (1982), Autoregressive conditional heteroskedasticity with estimates of the

variance of U.K. inflation, Econometrica, 50, 987-1008.

[5] Gibson, R., and E. Schwartz, (1990), Stochastic convenience yield and the pricing of oil

contingent claim, Journal of Finance, 45, 959-976.

[6] Knittel, C.R. and M. Roberts, (2001), An empirical examination of deregulated electricity

prices, mimeo.

[7] Nelson, D.B., (1991), Conditional heteroskedasticity in asset returns: A new approach,

Econometrica, 59, 347-370.

[8] Ramanathan, R., Engle, R., Granger, C.W.J., Vahid-Araghi, F. and C. Brace, (1997),

Short-run forecasts of electricity loads and peaks, International Journal of Forecasting, 13,

161-174.