using extreme value statistics to measure value at risk for daily electricity spot prices

8/22/2019 Using Extreme Value Statistics to Measure Value at Risk for Daily Electricity Spot Prices

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Using extreme value theory to measure value-at-risk for daily

electricity spot pricesB

Kam Fong Chan a,*, Philip Gray b,1

aDepartment of Accounting and Finance, Faculty of Business and Economics, The University of Auckland, Private Bag 92019,

Auckland, New Zealandb UQ Business School, The University of Queensland, St. Lucia 4072, Australia

Abstract

The recent deregulation in electricity markets worldwide has heightened the importance of risk management in energy

markets. Assessing Value-at-Risk (VaR) in electricity markets is arguably more difficult than in traditional financial markets

because the distinctive features of the former result in a highly unusual distribution of returns electricity returns are highly

volatile, display seasonalities in both their mean and volatility, exhibit leverage effects and clustering in volatility, and feature

extreme levels of skewness and kurtosis. With electricity applications in mind, this paper proposes a model that accommodates

autoregression and weekly seasonals in both the conditional mean and conditional volatility of returns, as well as leverage

effects via an EGARCH specification. In addition, extreme value theory (EVT) is adopted to explicitly model the tails of thereturn distribution. Compared to a number of other parametric models and simple historical simulation based approaches, the

proposed EVT-based model performs well in forecasting out-of-sample VaR. In addition, statistical tests show that the

proposed model provides appropriate interval coverage in both unconditional and, more importantly, conditional contexts.

Overall, the results are encouraging in suggesting that the proposed EVT-based model is a useful technique in forecasting VaR

in electricity markets.

D 2005 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

JEL classification: C14; C16; C53; G11

Keywords: Extreme value theory; Value-at-risk; Electricity; EGARCH; Conditional interval coverage

1. Introduction

The recent worldwide deregulation of wholesale

electricity markets has created opportunities and

incentives for market participants to trade electricity

spot prices and related derivatives. Trading in

electricity markets is challenging because spot prices

are highly volatile and exhibit occasional extreme

0169-2070/$ - see front matterD 2005 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.ijforecast.2005.10.002

B This paper is a revised version of Chapter Five of the first

authors Ph.D. thesis at The University of Queensland, Australia.

* Corresponding author. Tel.: +64 9 373 7599x85172.

E-mail addresses: [email protected] (K.F. Chan),

[email protected] (P. Gray).1 Tel.: +61 7 3365 6992.

International Journal of Forecasting 22 (2006) 283300

www.elsevier.com/locate/ijforecast


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price movements of magnitudes rarely seen in

markets for traditional financial assets.2 As a result,

energy industry participants often self-impose trading

limits to prevent extreme price fluctuations fromadversely affecting firm profitability and indeed the

operation of the entire industry. Firms also require

optimal trading limits to allocate capital to cover

potential losses should the trading limits be violated.

Obviously, over-capitalization implies idle capital

which compromises the firms profitability. On the

other hand, under-capitalization may cause financial

distress should the firm be unable to honour its

trading contracts.

One tool commonly used to establish optimal

trading limits is Value-at-Risk (VaR). In general,VaR measures the amount a firm can lose with a%

probability over a certain time horizon s. If, for

example, a=5% and s is one day, the VaR can be

interpreted as the maximum potential loss that will

occur for five days on average over each 100-day

period. An extensive discussion of VaR use in

traditional financial markets can be found in Dowd

(1998), Duffie and Pan (1997), Jorion (2000), Holton

(2003) and Manganelli and Engle (2004), whilst

energy VaR is detailed in Clewlow and Strickland

(2000) and Eydeland and Wolyniec (2003).

The conventional approaches to estimating VaR in

practice can be broadly classed as parametric and non-

parametric. Under the parametric approach, a specific

distribution for asset returns must be postulated, with

a Normal distribution being a common choice. In

contrast, non-parametric approaches make no assump-

tions regarding the return distribution. As an example,

the popular historical simulation method utilizes the

empirical distribution of returns to proxy for the likely

distribution of future returns. Both approaches are

widely employed in financial markets, where prices

seldom exhibit extreme movements. In electricitymarkets, however, the high volatility and occasional

price spikes result in an empirical distribution of

returns with a non-standard shape making it difficult

to specify a parametric form. As a result, parametric

approaches may not generate accurate VaR measures

in electricity markets. Similarly, the usefulness of non-

parametric approaches in electricity markets is largelyunknown.

One possible avenue for improving VaR estimates

in energy markets lies in extreme value theory (EVT),

which specifically models the extreme spot price

changes (i.e., the tails of the return distribution).

Focusing on extreme returns rather than the entire

distribution seems natural since, by definition, VaR

measures the economic impact of rare events. EVT

has already found numerous applications for VaR

estimation in financial markets.3 Longin (1996)

examines extreme movements in U.S. stock pricesand shows that the extreme returns obey a Frechet fat-

tailed distribution. Ho, Burridge, Cadle, and Theobald

(2000) and Gencay and Selcuk (2004) apply EVT to

emerging stock markets which have been affected by

a recent financial crisis. They report that EVT

dominates other parametric models in forecasting

VaR, especially for more extreme tail quantiles.

Gencay, Selcuk, and Ulugulyagci (2003) reach similar

conclusions for the Istanbul Stock Exchange Index

(ISE-100). Muller, Dacorogna, and Pictet (1998) and

Pictet, Dacorogna, and Muller (1998) compare the

EVT method with a time-varying GARCH model for

foreign exchange rates. Bali (2003) adopts the EVT

approach to derive VaR for U.S. Treasury yield

changes.

At present, applications of EVT to estimating VaR in

energy markets are sparse. Andrews and Thomas

(2002) combine historical simulation with a thresh-

old-based EVT model to fit the tails of the empirical

profit-and-loss distribution of electricity. They report

that the model fits the empirical tails better than the

Normal distribution. Rozario (2002) derives VaR for

Victorian half-hourly electricity returns using a thresh-old-based EVT model. While the model performs

well for moderate tails covering a =5 % to 1%,

it struggles when a is below 1%, a fact Rozario

attributes to the models failure to account for

clustering in the data.2 The extreme movements are attributable to several distinctive

features of electricity markets: (1) electricity cannot be stored

effectively through time and space; and (2) electricity prices have

inelastic demand curves and kinked supply curves (Cuaresma,

Hlouskova, Kossmeier, & Obersteiner, 2004; Knittel & Roberts,

2001; Escribano, Pena, & Villaplana, 2002).

3 Embrechts, Kluppelberg, and Mikosh (1997) and Reiss and

Thomas (2001) provide a comprehensive overview of EVT as a risk

management tool.

K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283300284


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It is important to note that EVT relies on an

assumption of i.i.d. observations. Clearly, this is not

true for electricity return series, and arguably financial

returns in general. One approach to this problem isprovided by McNeil and Frey (2000). Using a two-

stage approach, McNeil and Frey estimate a GARCH

model in stage one with a view to filtering the return

series to obtain (nearly) i.i.d. residuals. In stage two,

the EVT framework is applied to the standardized

residuals. The advantage of this GARCHEVT

combination lies in its ability to capture conditional

heteroscedasticity in the data through the GARCH

framework, while at the same time modelling the

extreme tail behaviour through the EVT method. As

such, the GARCH-EVT approach might be regardedas semi-parametric (Manganelli & Engle, 2004).

Bali and Neftci (2003) apply the GARCH-EVT

model to U.S. short-term interest rates and show that

the model yields more accurate estimates of VaR than

that obtained from a Student t-distributed GARCH

model. Fernandez (2005) and Bystrom (2004) also

find that the GARCH-EVT model performs better

than the parametric models in forecasting VaR for

various international stock markets. In an energy

application, Bystrom (2005) employs a GARCH-EVT

framework to NordPool hourly electricity returns. He

finds that the extreme GARCH-filtered residuals obey

a Frechet distribution. Furthermore, the GARCH-EVT

model produces more accurate estimates of extreme

tails than a pure GARCH model.

The objective of the current paper is to further

explore the usefulness of EVT in forecasting VaR in

electricity markets. There are several contributions.

First, the paper proposes a model that, when combined

with EVT, has the potential to generate more accurate

quantile estimates for electricity VaR. Based on daily

electricity returns, the model accommodates autore-

gression and weekly seasonals in both the conditionalmean and conditional volatility equations. Leverage

effects in conditional volatility are also modelled using

an Exponential GARCH (EGARCH) specification. In

forecasting VaR, EVT is applied to the standardized

residuals from this model. Clearly, the proposed

EGARCHEVT combination is a sophisticated ap-

proach to forecasting VaR. The second contribution,

therefore, is to compare the accuracy of VaR forecasts

under the proposed model with a number of conven-

tional approaches (both parametric and non-paramet-

ric). Tail quantiles are estimated under each competing

model and the frequency with which realized returns

violate these estimates provides an initial measure of

model success.While the use of violation frequencies is common in

assessing quantile estimators for VaR, the utility of

such an approach may be limited in electricity

applications where the true quantiles are likely to be

time varying. For example, a nave estimator con-

structed as the quantile of all historical returns will have

a perfect violation proportion on average. If, however,

the data series exhibit time-varying volatility (and

consequently, a time-varying return distribution), the

nave quantile estimator may struggle to differentiate

between periods of high volatility and periods ofrelative tranquility. As such, VaR violations from a

nave quantile estimator may well be clustered in time,

possibly during periods of turmoil when VaR forecasts

are most crucial.

The third contribution of this paper, therefore, is to

assess the VaR performance of a number of competing

models using formal statistical inference designed to

test both unconditional and conditional coverage of the

quantile estimators. Based on tests developed

by Christoffersen (1998), the findings shed new light

on the appropriateness of simple non-parametric

approaches to VaR estimation. Finally, the paper exa-

mines five electricity markets, each with defining

characteristics. Wolak (1997) notes that electricity

price behaviour is affected by how the electricity is

generated. This paper considers markets such as

Victoria, where electricity is primarily generated by

fossil fuel, and the NordPool, which utilizes hydro

generation. Indeed, the findings suggest that the

optimal approach to estimating VaR is very likely to

be a function of the characteristics of the underlying

power market. At the very least, the international

comparison allows an assessment of the generality ofour findings.

The remainder of the paper is structured as follows.

Section 2 describes the competing approaches used to

forecast VaR in this paper. A number of common

parametric and non-parametric models are included,

along with an EVT-based approach designed specif-

ically for electricity applications. Section 3 documents

the data employed in the study while Section 4

presents the results. Model estimates are presented in

Section 4.1, with particular emphasis given to the

K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283300 285


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implementation of the EVT framework. Section 4.2

documents the relative VaR performance of compet-

ing approaches, measured using (unconditional) vio-

lation frequencies. Section 4.3 extends the assessmentby conducting formal statistical tests of both uncon-

ditional and conditional coverage of the various

quantile estimators. Section 5 concludes the study.

2. Methods for estimating value-at-risk

This section presents the various approaches to

calculating VaR examined in this paper. Section 2.1

describes a simple non-parametric approach based on

the historical distribution of returns. Section 2.2outlines four parametric approaches based on an

autoregressive model for returns. Our proposed

model, termed AR-EGARCH-EVT, is detailed in

Section 2.3.

2.1. Historical simulation approach

Arguably, the most popular method of estimating

VaR is to utilise the empirical distribution of past

returns on the asset of interest. If, for example, one

requires the VaR for one day with an a= 5%

confidence level, one takes the 95% quantile from

the most-recent T observed daily returns. VaR for

longer horizons (for example, s days) can be similarly

obtained using the most-recent sample of non-over-

lapping s-day returns.4 Known as the Historical

Simulation (HS) approach, this simple method is

non-parametric in that it makes no arbitrary assump-

tions of the true distribution of returns. Of course, it

does assume that the past distribution is representative

of likely future returns. In this paper, the HS approach

serves as a nave benchmark against which more

sophisticated approaches are judged.

2.2. Parametric approaches

This paper considers four parametric approaches to

measuring VaR. First, we consider an autoregressive

(AR) model of returns with constant variance (here-

after denoted AR-ConVar). Since the data series are

sampled daily, an AR(7) model is proposed to capture

any weekly seasonality in electricity prices:

rt /0 X7j1

/jrtj et; 1

where rt= (StSt1) /St1 is the simple electricityreturn and St is the daily spot price. A distributional

assumption is made of the error term in the AR-

ConVar model; specifically, errors et are assumed to

be Normally distributed with zero mean and constant

variance (E(et2 /Xt1) =r

2). At any time t, the VaR

estimate from the AR-ConVar model is:

VaRq;t /0 X7

j1/jrtj F1 q rr; 2

where (/j= 1 , 2, . . .7, r ) are parameter estimates and

F1( q) is the q% quantile of the Normal distribution

function at an a% tail (i.e., q = 1a).The second parametric approach, a minor varia-

tion of the AR-ConVar model, combines the key

features of the autoregressive model and the histor-

ical simulation approach.5 The conditional mean is

again modelled using an AR(7) model. However,

rather than making a distributional assumption over

F1( q), the q% quantile for VaR is obtained by

bootstrapping the empirical distribution of residuals

from the fitted AR(7). Denoted AR-HS, this approach

is motivated by the likelihood that a historical

simulation from the empirical distribution of returns

is inappropriate in electricity VaR applications. Unlike

financial return series, which have near constant

mean, electricity returns have significant autocorrela-

tion in their conditional mean. The traditional HS

approach cannot capture these intertemporal charac-

teristics. In contrast, the proposed AR-HS method

accommodates the time-series properties through theautoregressive mean, while retaining the distribution-

free flavour by the use of bootstrapping. As such, the

AR-HS approach represents a more sensible imple-

mentation of the historical-simulation concept for

calculating VaR.6

4 Alternatively, s daily returns can be bootstrapped from the

empirical distribution and aggregated.

5 While we label this approach dparametricT, it is arguably a

hybrid semi-parametric approach.6 We are grateful to an anonymous referee for suggesting this

approach.



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Our third parametric approach specifically models

the serial correlation of both the conditional mean and

conditional volatility of returns. The mean return is

again modelled using an AR(7) but, rather thanconstant variance, the error term is assumed to follow

an EGARCH process:

E e2tjXt1 ht;

and ln ht b1 b2et1ffiffiffiffiffiffiffiffiht1

p

b3ln ht1

b4 et1

ffiffiffiffiffiffiffiffiht1p

E

et1

ffiffiffiffiffiffiffiffiht1p

b5

et7ffiffiffiffiffiffiffiffiht7

p

b6ln ht7 : 3

The adoption of an EGARCH formulation

for volatility is motivated by Knittel and Roberts

(2001), who argue that the convex nature of

the marginal costs of electricity generation causes

positive demand shocks to have a larger impact

on price changes than negative shocks. That is,

positive price shocks are conjectured to increase

volatility more than negative shocks, thus inducinga positive leverage effect.7 In addition to capturing

asymmetries (b4) , E q. ( 3) also accommodates

weekly seasonality in conditional volatility (b5 and

b6).

The corresponding VaR measure is calculated

in a similar fashion to Eq. (2), with the parameter

estimate of r replaced byffiffiffiffi

hhtp

from Eq. (3). Since

a number of distributional assumptions are common

in the GARCH literature, this study imposes

two distributions over the et error term: the Normal

distribution and the fat-tailed t-distribution with

m d eg re es o f f re e do m. T he f or me r m od el i s

termed AR-EGARCH-N, where the tail quantile

of F1( q) in its VaR model is also Normally

d is tr ib ut ed ; w hi ls t t he l at te r i s t er me d A R-

EGARCH-t, where the F1( q) quantile in its VaR

model is tm

-distributed.

2.3. The AR-EGARCH-EVT method

Following Bystrom (2005), this study adopts the

EVT approach of McNeil and Frey (2000) tomeasure VaR for electricity returns. McNeil and

Frey recognize that most financial return series

exhibit stochastic volatility and fat-tailed distribu-

tions. While the fat tails might be modelled directly

with EVT, the lack of i.i.d. returns is problematic.

McNeil and Freys solution is to first model the

conditional volatility using a GARCH approach.

The GARCH model serves to filter the return series

such that GARCH residuals are closer to i.i.d. than

the raw return series. Even so, GARCH residuals

have been shown to exhibit fat tails. In stage two,McNeil and Frey apply EVT to the GARCH

residuals. As such, the GARCHEVT combination

accommodates both time-varying volatility and fat-

tailed return distributions. We denote this approach

by AR-EGARCH-EVT.

The AR-EGARCH-EVT approach is implemented

as follows:

1. The AR-EGARCH model with a tm

-distribution

governing the et term (as described in Section 2.2)

is estimated from electricity returns. Maximum

likelihood estimation is employed over an in-

sample period (described shortly).

2. The residuals from the AR-EGARCH model are

standardized:

zzt rt /0

X7j1

/jrtj

( )ffiffiffiffiffiffiffiffiffi

hht1p

0BBBB@

1CCCCA 4

where T is the number of return observationsduring the in-sample estimation period.

3. EVT is applied to the standardized residuals zt to

model the tail quantile of F1( q) in deriving VaR.

In applying the EVT method, this paper adopts the

Peak Over Threshold (POT) EVT method.8 The POT

7 We are grateful to an anonymous referee for suggesting this

motivation for employing an EGARCH model.

8 For details on the POT EVT method, refer to McNeil and Frey

(2000) and Embrechts et al. (1997).



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method identifies extreme observations (that is, extreme

standardized residuals) that exceed a high threshold u

and specifically models these dexceedencesT separately

from non-extreme observations.Assume that the standardized residuals zt are

a sequence of i.i.d. random variables from an

unknown distribution function Fz. Let u denote a

high threshold beyond which observations of z are

considered exceedences (the choice of the threshold

u is discussed shortly). The magnitude of the exceed-

ence is given by yi =zi u, for i = 1, . . .Ny, where Nyis the total number of exceedences in the sample.

The distribution of y, for a given threshold u, is

given by:

Fu y Pr z u VyjzN u

Pr z u Vy;zN u Pr zNu

Fz y u Fz u 1 Fz u : 5

That is, Fu(y) is the probability that z exceeds the

threshold u by an amount no greater than y, given

that z exceeds u. Since z=y + u, re-arrange Eq. (5) to

obtain:

Fz z 1 Fz u Fu y Fz u ; zNu: 6

Balkema and de Haan (1974) and Pickands (1975)

show that, for a sufficiently high u, Fu(y) can be

approximated by the Generalized Pareto Distribution

(GPD), which is defined as:

Gn;m y 1 1 ny

m

1=nif n p 0

1 exp y=m if n 0;

8


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daily returns are quite large, the median returns are

close to zero.9 The high volatility of electricity

returns is evident in the standard deviation of

daily returns. Similarly, the positive skewness and

high kurtosis clearly illustrate the non-normality of

the distribution. LjungBox Q and Q2 statistics

indicate the presence of serial correlation at up to

7 lags, as well as potential time-varying volatility.These findings lend credence to the adoption of

the AR(7) and EGARCH models discussed in

Section 2.

Fig. 1 graphs spot prices, returns and QQ plots

for each power market. Together with Table 1,

Fig. 1 demonstrates the defining characteristics of

electricity markets: high volatility, occasional ex-

treme movements, volatility clustering and fat-

tailed distributions. These descriptive statistics and

plots further motivate the expl oration of the

alternative approaches to measuring VaR describedin Section 2.

4. Empirical results

This section presents the empirical findings of

the study. Section 4.1 reports the in-sample param-

eter estimates for all models proposed in Section 2.

In Section 4.2, an initial assessment is made of the

accuracy with which each model forecasts VaR,

Table 1

Descriptive statistics

Victoria NordPool Alberta Hayward PJM

Full sampleStart date 4 Jan 99 5 Jan 98 5 Jan 98 5 Jan 98 5 Apr 98

End date 31 Dec 04 31 Dec 04 31 Dec 04 31 Dec 03 31 Dec 03

No. of obs 2189 2553 2553 2184 2093

Mean 0.091 0.008 0.098 0.076 0.068

Median 0.014 0.003 0.011 0.005 0.015Std. dev. 1.100 0.141 0.575 0.775 0.560

Min 0.959 0.558 0.898 0.963 0.926Max 44.22 2.496 6.307 21.84 15.24

Skewness 30.52 3.742 4.007 17.01 12.91

Kurtosis 1191 50 31 398 301

Q(7) 196.4*** 236.7*** 130.4*** 141.7*** 329.2***

Q2(7) 46.64*** 111.4*** 67.16*** 59.86*** 58.92***

In-sample

Start date 4 Jan 99 5 Jan 98 5 Jan 98 5 Jan 98 5 Apr 98

End date 31 Dec 02 31 Dec 02 31 Dec 02 30 Apr 02 30 Apr 02

No. of obs 1459 1823 1823 1584 1493

Mean 0.104 0.012 0.094 0.062 0.052

Median 0.017 0.004 0.015 0.007 0.008Std. dev. 1.301 0.161 0.574 0.439 0.360

Min 0.959 0.557 0.897 0.963 0.925Max 44.22 2.496 6.306 21.84 15.24

Skewness 27.42 3.358 4.379 15.02 13.54

Kurtosis 911 40 35 304 292

Q(7) 146.3*** 165.9*** 113.0*** 99.22*** 217.2***

Q2(7) 38.25*** 74.66*** 73.83*** 36.45*** 36.18***

The table reports summary statistics for the daily simple net returns (rt) of five international power markets: Victoria, NordPool, Alberta,Hayward and PJM. The LjungBox Q(7) and Q2 (7) statistics test for serial correlation up to 7 lags for rt and rt

2 , respectively. *** Indicates

significance at the 1% level.

9 The non-zero mean return is directly attributable to the nature of

electricity returns. Extreme positive returns (sometimes exceeding

several hundred percent) occur semi-regularly. In contrast, the

minimum return is bounded from below at100%. As emphasizedby Bystrom (2005), this feature results in severe positive skewness

and non-zero mean returns, yet causes no major concerns as we

study the right tail of the distribution.



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Jan 99 Dec 00 Dec 02 Dec 040

200

400

600

800

1000

1200


0

10

20

30

40

50

-4 -2 0 2 4-2

0

2

4

6

8

10

Victoria prices Victoria returns Victoria QQ plot

Dec 98 Dec 00 Dec 02 Dec 040

200

400

600

800

1000

Dec 98 Dec 00 Dec 02 Dec 04-1

0

1

2

3

-4 -2 0 2 4-1

0

1

2

3NordPool prices NordPool returns NordPool QQ plot


100

200

300

400

500

600


0

2

4

6

8

-4 -2 0 2 4-2

0

2

4

6

8Alberta prices Alberta returns Alberta QQ plot


100

200

300

400

500


0

5

10

15

20

25

-4 -2 0 2 4

0

5

10

15

20

25

Hayward prices Hayward returns Hayward QQ plot

Apr 98 Dec 99 Dec 01 Dec 030

100

200

300

400

Apr 98 Dec 99 Dec 01 Dec 03

0

5

10

15

20

-4 -2 0 2 4

0

5

10

15

20PJM prices PJM returns PJM QQ plot

Fig. 1. Spot prices, returns and QQ plots. The figure shows summary plots for daily electricity data from five international power markets: Victoria,

NordPool, Alberta, Hayward and PJM. The left, middle,and right columns displayelectricity prices, returns and QQ plots for daily returnsrespectively.



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with the observed violation frequencies compared to

tail quantiles from each model. Section 4.3 exam-

ines VaR performance further by conducting statis-

tical tests of both the unconditional and conditional

interval coverage of each approach.

4.1. Model estimates

4.1.1. AR-ConVar model

Table 2 presents the ML estimates of the AR-

ConVar model (Eq. (1)). For each data series, the

period of estimation is the in-sample period indicated

in Table 1. The findings are quite similar across the

various power markets. Consider, for example, the

PJM market. The time-series properties of the return

series are evident, with statistically significant auto-

correlations at all seven lags.10 The first six lags

exhibit negative autocorrelation, while a day of the

week effect is confirmed by the positive estimate at

lag 7. These results support the deployment of an

AR(7) model for the return series. Taken together,

the AR estimates imply a long-term mean return of/0= 1

P7j1 /

j

0:052, which closely matches

the unconditional mean for PJM reported in Table

1. Similarly, the volatility estimate relating to AR

errors (r=0.332) approximates the unconditional in-

sample standard deviation.

4.1.2. AR-EGARCH model

Table 3 presents the ML estimates of the AR-

EGARCH-t model.11 The mean and conditional

volatility equations are given by Eqs. (1) and (3),

respectively, with a t-distribution governing the errors.

Estimates from the autoregressive mean equation have

changed little from Table 2. The estimates from the

conditional volatility equation are of particular interest.

The general tenor of the findings is as follows. There is

strong evidence of a first-order GARCH effect (b2 and

b3) in all markets except PJM. In addition, there

appears to be an asymmetric leverage effect (b4).12

Parameter estimates (b5 and b6) also suggest a weekly

seasonal effect in the conditional variance. Finally, ML

estimates of the parameterm suggest that returns have a

tail fatter than that implied by a Normal distribution. In

summary, the findings support the use of the AR-

EGARCH-t model as specified in Eqs. (1) and (3).

4.1.3. AR-EGARCH-EVT model

Since EVT relies on the assumption of i.i.d.

observations, Section 2.3 described a two-stageprocess designed to achieve (near) i.i.d. time-series.

First, the AR-EGARCH model is fitted and residuals

are standardized in an attempt to satisfy the i.i.d.

10 The vast majority of parameter estimates in Tables 2 and 3 are

significant at the 1% level and their p-values are not shown. p-

values are only explicitly shown when they are greater than 1%.

11 ML estimates for the AR-EGARCH-N model are qualitatively

similar and, to preserve space, are not reported. However, the VaR

performance of the AR-EGARCH-N model is reported in subse-

quent analysis.

Table 2

Parameter estimates for the AR-ConVar model


/0 0.079 0.018 0.109 0.077 0.092/1 0.119 0.026 (0.278) 0.144 0.077 0.185/2 0.101 0.113 0.127 0.158 0.254/3 0.126 0.023 (0.341) 0.009 (0.715) 0.071 0.167/4 0.085 0.164 0.051 (0.036) 0.044 (0.109) 0.102/5 0.094 0.154 0.018 (0.050) 0.036 (0.180) 0.138/6 0.043 (0.015) 0.119 0.001 (0.970) 0.013 (0.611) 0.106/7 0.222 0.122 0.168 0.167 0.184

r 0.422 0.153 0.555 0.424 0.332

R2 0.1015 0.0897 0.0639 0.0638 0.1526

The table reports maximum-likelihood estimates of the AR-ConVar model (Eq. (1)). For each data series, parameter estimates are based on the

in-sample period documented in Table 1. The majority of parameter estimates are statistically significant at better than the 1% level and their p-

value is not shown. p-values are shown in parentheses only when not significant at the 1% level.

12 Estimates of the leverage effect are comparable to those

reported by Knittel and Roberts (2001) and Duffie, Gray, and

Hoang (1998) where b4 is positive and statistically significant.



10/18

assumption. Second, EVT is applied to the standard-

ized residuals. Table 4 presents diagnostics for the

drawT and standardized AR-EGARCH residuals.The LjungBox Q and Q2 statistics provide an

indication of whether any serial correlation or hetero-

scedasticity is present in the data series. Panel A

strongly suggests that the raw AR-EGARCH residuals

are not i.i.d. as required by EVT. In contrast, the

standardized residuals in Panel B, whilst not perfectly

i.i.d., are better behaved. To a large extent, the filtering

procedure advocated by McNeil and Frey (2000) has

been effective in producing (near) i.i.d. residuals on

which EVT can be implemented. Table 4 Panel B does,

however, show that skewness and excess kurtosisremain in the standardized residuals. Similarly, QQ

plots (not presented) document heavy right tails. These

findings motivate the second stage of McNeil and

Freys (2000) EVT implementation, where the fat tails

of the standardized residuals are explicitly modelled.

To apply EVT, the threshold u is selected using

mean excess functions (MEF) and Hill plots.13 Table 5

13 Our approach to selecting u follows Gencay and Selcuk (2004)

closely and we do not present the MEFs and Hill plots here.

Table 3

Parameter estimates for the AR-EGARCH model


Estimates from the AR(7) mean (Eq. (1))/0 0.001 (0.907) 0.002 (0.153) 0.011 (0.092) 0.007 (0.129) 0.043/1 0.156 0.000 (0.990) 0.201 0.077 0.211/2 0.140 0.087 0.195 0.063 0.236/3 0.140 0.100 0.113 0.067 0.191/4 0.087 0.143 0.093 0.044 0.147/5 0.114 0.163 0.093 0.046 0.189/6 0.080 0.049 0.040 (0.020) 0.001 (0.970) 0.151/7 0.170 0.231 0.064 0.098 0.122

R2 0.1414 0.1414 0.1492 0.0596 0.1718

Estimates from the EGARCH conditional variance (Eq. (3))

b1 0.225 (0.051) 0.046 (0.338) 0.560 0.747 0.375b2

0.282

0.366

0.395

0.986

0.043 (0.416)

b3 0.057 (0.074) 0.377 0.835 0.879 0.039 (0.340)

b4 0.520 0.703 0.964 1.412 0.181

b5 0.223 0.208 0.110 0.023 (0.794) 0.232

b6 0.771 0.597 0.104 0.073 (0.047) 0.808

m 2.500 3.689 2.069 2.119 4.167

The table reports maximum-likelihood estimates of the AR-EGARCH model (Eqs. (1) and (3)), with a tm

-distribution governing the error terms.

For each data series, parameter estimates are based on the in-sample period documented in Table 1. The majority of parameter estimates are

statistically significant at better than the 1% level and theirp-value is not shown. p-values are shown in parentheses only when notsignificant at

the 1% level.

Table 4Summary statistics for AR-EGARCH residuals


Panel A: raw AR-EGARCH residuals

Median 0.014 0.001 0.013 0.015 0.004Mean 0.091 0.013 0.146 0.081 0.061

Std. dev. 0.423 0.155 0.563 0.427 0.333

Skewness 4.325 3.197 4.684 3.487 1.956

Kurtosis 28.29 46.20 36.92 21.67 10.18

Q(7) 7.189 114.6*** 75.40*** 21.79*** 8.521

Q2(7) 27.29*** 88.63*** 103.0*** 33.36*** 33.66***

Panel B: standardized AR-EGARCH residuals

Median 0.052 0.011 0.048 0.065 0.009Mean 0.354 0.078 0.444 0.212 0.138

Std. dev. 1.755 0.961 1.766 1.448 0.787

Skewness 4.880 4.281 4.095 3.230 2.348

Kurtosis 37.81 68.65 31.72 33.39 14.02

Q(7) 12.53* 34.00*** 11.14 20.72*** 8.860

Q2(7) 9.43 6.19 2.44 4.03 12.77*

The table reports summary statistics for the (in-sample) residuals

from the AR-EGARCH model, with a tm

-distribution governing the

error terms. Panels A and B report diagnostics for the drawT and

standardized residuals respectively. The latter are the basis of the

EVT estimation. * and *** indicate that the LjungBox Q and

Q2 statistics are significant at the 10% and 1% levels, respectively.



11/18

reports the threshold chosen in each market. In each

case, the resulting exceedences Ny total approximately

10% of the sample, which is consistent with percen-

tages reported by McNeil and Frey (2000).

Table 5 also reports ML estimates of the shape (n)

and scale (m) parameters, determined by fitting the GPD

Eq. (7) to the standardized residuals. Recall that values

of n N0 reflect heavy-tailed distributions. In each

power market, the n estimate is positive and statisti-

cally significantly different from zero, suggesting that

the right tail of the distribution of standardized

residuals is characterized by the Frechet distribution.14

Table 5 Panel B further documents the heavy tails of

the distribution by comparing the EVT tail quantiles

to those from a Normal distribution. EVT tail

quantiles Fz1( q) are obtained from Eq. (9) using

the Panel A reports ofT, u, Ny, n and m at the specified

end tail ofa%. In general, the tail quantiles from the

AR-EGARCH-EVT model are higher than those

under a Normal distribution. The fatness of the tail

is readily apparent, especially as we move to moreextreme quantiles (i.e., as a moves towards 0.5%).

Indeed, Gencay and Selcuk (2004) warn that using

quantile estimates from a Normal distribution when

the data is in fact fat tailed will cause VaR to be

underestimated.

4.2. Relative VaR performance of competing models

The primary goal of this paper is to assess the relative

ability of a number of alternate approaches to accurately

measure VaR in electricity markets. To do this, the full

data sample is divided into an in-sample period (on

which Section 4.1s model estimates are based) and an

out-of-sample period over which VaR performance is

measured. Measurement of VaR proceeds as follows.

On the first day of the out-of-sample period, the most-

recent T returns are used to estimate model parameters

for each parametric approach. The magnitude ofTis set

to be equal to the length of the in-sample period. That is,

T=1459 in Victoria, T=1823 in NordPool, and so on.

From the parameter estimates, the next-day VaR is

estimated using each method described in Section 2.

Should the realized next-day return exceed the

estimated VaR, this is labelled a dviolationT.15 Moving

to time t+ 1, the estimation procedure is rolled

forward one day and repeated. Note that the size of

the estimation window T is kept constant and simplyrolled forward one day at a time, thus ensuring that

model estimates are not based on stale data.16

The procedure differs slightly for the non-paramet-

ric (HS) and semi-parametric (AR-HS) approaches.

14 In a study of NordPool hourly prices, Bystrom (2005) also

finds that a Frechet distribution applies to the tail of the distribution

of standardized residuals.

Table 5

Parameter estimates for the AR-EGARCH-EVT model


Panel ATotal in-sample obs. T 1459 1823 1823 1584 1493

EVT threshold u 1.5 1.0 2.0 1.5 1.0

Number of exceedences Ny 151 185 167 187 149

% of exceedences in-sample Ny /T 10.35 10.15 9.16 11.81 9.98

GPD shape parameter n 0.552*** 0.305*** 0.332*** 0.305*** 0.344***

GPD scale parameter m 1.208*** 1.631*** 1.034*** 0.570*** 0.569***

Panel B Normal

q = 95% 1.645 2.582 1.459 3.289 2.375 1.444

q = 99% 2.326 7.265 2.935 7.498 5.194 2.996

q = 99.5% 2.576 10.97 3.832 10.05 6.956 3.978

Panel A reports in-sample ML estimates of the GPD distribution for the AR-EGARCH-EVT model. *** Denote significance at the 1% level.

Panel B presents EVT tail quantiles Fz1

( q) for the standardized residuals, along with tail quantiles from a Normal distribution.

15 Berkowitz (1999), Ho et. al. (2000), Bali and Neftci (2003),

Gencay and Selcuk (2004), Bystrom (2004, 2005) and Fernandez

(2005) adopt a similar procedure.16 Indeed, in relation to the EVT approach, plots (not reported)

show that rolling estimates n and m are clearly time varying. This

reinforces the necessity for using a rolling estimation window.



12/18

Under the AR-HS approach, parameters of the

autoregressive model are again estimated using a

rolling window of the most recentTobservations (this

ensures comparability with the plain-vanilla AR-

ConVar approach). However, tail quantiles are con-

structed by bootstrapping from the 500 most-recent

AR errors. Similarly, the na ve HS approach simply

bootstraps from the 500 most-recent raw returns.17

Table 6 documents the out-of-sample violation

ratios under each model for a range of quantiles. For

the 95% quantile (a = 5%), five violations are expected

every 100 days. Each model is evaluated by comparingthe actual and expected violation ratios and competing

models are ranked accordingly (rankings are shown in

parentheses). Consider, for example, the Victorian

market with a = 5%. The AR-EGARCH-t and AR-

GARCH-EVT models forecast right-tail quantiles

most accurately. The AR-EGARCH-N model (that is,

the autoregressive model with Normally distributed

errors) significantly underestimates the 95% quantile

resulting in an excessive number of violations; this is

to be expected when the actual returns have heavier

tails than assumed under a Normal distribution.

Curiously, the AR-ConVar model (which also assumes

Normal errors) overestimates the 95% quantile, while

the nave HS approach does surprisingly well. Moving

to the 99% quantile (a =1%), there is little consistency

in model rankings. The AR-HS and HS approaches

provide the most-accurate VaR forecasts, while the

AR-EGARCH-t and AR-EGARCH-EVT models un-derestimate and overestimate the 99% quantile respec-

tively. For the extreme quantile (a= 0.5%), rankings

approximate those reported fora=5%.

Examining the other power markets, little consis-

tency in model performance is evident. The HS

approach has superior performance for NordPool

and Alberta, irrespective of the quantile. The AR-

EGARCH-EVT model performs very well for Hay-

ward and PJM. While these inconsistent rankings are

not particularly encouraging for risk managers inter-

Table 6

Out-of-sample VaR violations


a= 5%HS 4.25 (3) 4.25 (1) 4.93 (1) 4.00 (2) 6.50 (2)

AR-ConVar 0.68 (6) 0.41 (5) 4.38 (4) 0.33 (6) 3.62 (1)

AR-HS 3.84 (4) 0.41 (5) 5.89 (6) 1.50 (5) 7.77 (5)

AR-EGARCH-N 6.99 (5) 3.70 (2) 4.65 (3) 4.83 (1) 7.83 (6)

AR-EGARCH-t 4.93 (1) 1.00 (4) 4.79 (2) 2.50 (4) 3.50 (2)

AR-EGARCH-EVT 4.65 (2) 2.19 (3) 4.11 (5) 3.50 (3) 6.67 (4)

a= 1%

HS 0.82 (2) 0.96 (1) 0.82 (1) 0.33 (2) 2.00 (5)

AR-ConVar 0.27 (5) 0.27 (3) 2.60 (5) 0 (4) 1.33 (2)

AR-HS 1.10 (1) 0 (4) 0.55 (4) 0 (4) 1.17 (1)

AR-EGARCH-N 3.56 (6) 1.23 (2) 3.01 (6) 2.17 (6) 4.83 (6)

AR-EGARCH-t 1.51 (4) 0 (4) 0.68 (3) 0.30 (3) 0.67 (2)

AR-EGARCH-EVT 0.69 (3) 0 (4) 0.69 (2) 0.5 (1) 1.50 (4)

a=0.5%

HS 0.41 (2) 0.69 (1) 0.55 (1) 0 (3) 1.00 (3)

AR-ConVar 0.27 (4) 0.27 (2) 2.60 (6) 0 (3) 1.17 (5)

AR-HS 0.14 (5) 0 (4) 0.27 (4) 0 (3) 0 (3)

AR-EGARCH-N 2.88 (6) 0.96 (3) 2.46 (5) 1.83 (6) 4.00 (6)

AR-EGARCH-t 0.68 (3) 0 (4) 0.14 (2) 0.17 (2) 0.17 (2)

AR-EGARCH-EVT 0.54 (1) 0 (4) 0.14 (2) 0.33 (1) 0.67 (1)

The table details the out-of-sample VaR violations for all competing models. A violation occurs if the realized empirical return exceeds the

predicted VaR on a particular day. The numbers in parentheses denote the ranking among the competing models for each quantile ata =5%, 1%

and 0.5%. All actual and expected violations are in percentage terms.

17 Manganelli and Engle (2004) note that it is common to utilize a

rolling window of between 6 and 24 months (i.e., between 180 and

730 observations) for HS approaches.



13/18

ested in forecasting VaR, a careful examination of the

violation ratios in conjunction with the Table 1

summary statistics is revealing. Consider the Victori-

an, Hayward and PJM markets, where the more-

sophisticated models like AR-EGARCH-EVT and

AR-EGARCH-t perform well. The summary statistics

for these markets reveal that electricity returns are

characterized by extremely high levels of skewness

and kurtosis, high variance and an extreme range.

Under such conditions, a sophisticated model like

EVT which explicitly models the tails of the return

distribution is better-equipped to produce accurate

VaR forecasts. In contrast, the NordPool and Alberta

summary statistics are notably differentthe skew-

ness and kurtosis statistics are an order of magnitude

lower than the other markets and the range of returns

is considerably narrower. The relative advantage of amore sophisticated VaR model is diminished in such

conditions and simpler models may suffice.18

In summary, the results in Table 6 extend the

findings of Bystrom (2005). Working with hourly

NordPool returns, Bystrom (2005) reports that VaR

performance under a GARCH-EVT framework is

superior to a number of competing parametric

approaches. The current findings (based on daily

returns) also show that the AR-EGARCH-EVT model

performs well, especially in markets where the

distributions of returns exhibit extreme moments.

However, the nave HS approach (not examined by

Bystrom) is also shown to perform well, particularly

in markets where the return distribution does not

display extreme skewness and kurtosis.

While the HS approach performs surprisingly well

in several energy markets, risk managers may

n o ne t he l es s b e ne f it f ro m a d op t in g t h e A R -

EGARCH-EVT model. A parametric model that

captures the time-series properties of both the mean

and volatility of returns, as well as explicitly

modelling the tails of the distribution, may offeradvantages during periods of market turmoil. To

illustrate, consider Fig. 2 which depicts the VaR

performance of the HS and AR-EGARCH-EVT

approaches during the out-of-sample period for the

Alberta market (a =5%). Although the HS model in

Table 6 has a marginally better violation ratio (4.93%)

than the AR-EGARCH-EVT model (4.11%), the latter

results in time-varying VaR forecasts that adapt

quickly to changing market conditions. During the

middle of 2003 and towards the end of 2004, the AR-

18 It is unclear why the distribution of electricity returns is so

different in these two markets. Arguably, differences might be

expected for NordPool where electricity is hydro-generated, yet

Alberta features traditional coal-fired generation.

Jan 03 Dec 03 Dec 04-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Fig. 2. Time-varying VaR forecasts and violations. The plot depicts the VaR forecasts from the HS (smooth, heavy red line) and AR-EGARCH-

EVT model (dashed, green line) for the Alberta market during the out-of-sample period ( a =5%). Daily returns are shown with the thin grey line.

Violations under the HS and AR-EGARCH-EVT models are displayed with triangles and circles respectively. (For interpretation of the

references to colour in this figure legend, the reader is referred to the web version of this article.)



14/18

EGARCH-EVT model produces more accurate and

robust VaR forecasts (dashed, green line). AR-

EGARCH-EVT dviolationsT (marked with circles)

are relatively evenly spaced throughout the out-of-sample period. In contrast, VaR forecasts under the

HS approach (smooth, heavy red line) are relatively

constant and persistent. As a result, HS violations

(marked by triangles) appear to be clustered during

periods of turmoil. This finding has obvious implica-

tionsa firm that forecasts VaR using the HS model

may experience a number of consecutive violations

during turbulent periods when accurate VaR measures

are needed most. In light of the possibility that true

quantiles are time varying, the following section

conducts formal statistical tests to assess the condi-tional coverage of various approaches to quantile

estimation.

4.3. Statistical analysis of model performance

The performance of competing approaches to VaR

measurement in Section 4.2 is based on an assessment

of the out-of-sample accuracy of estimated quantiles.

Specifically, the out-of-sample violation proportions

are compared to theoretical probabilities. Conducting

formal statistical inference on this unconditional

coverage is straightforward (see Berkowitz, 1999;

Christoffersen, 1998; McNeil & Frey, 2000).

Note, however, that evaluating quantile estimation

performance using unconditional coverage may be of

limited use if the true quantile is time varying. To

illustrate, consider a nave quantile estimator con-

structed as the quantile of all historical returns. On

average, the nave estimator will have a perfect

violation proportion (that is, correct unconditional

coverage). In any given period, however, the condi-

tional coverage may be incorrect. This scenario is

particularly relevant in financial time-series wherevolatility (and consequently, the return distribution)

varies over time. A nave quantile estimator may

entirely fail to differentiate between periods of high

volatility and periods of relative tranquility.19

Christoffersen (1998) clarifies the distinction be-

tween conditional and unconditional interval forecasts

and proposes statistical tests for each.20 Let LRccdenote a likelihood ratio test statistic examining

whether a quantile estimator has correct conditional

coverage. Christoffersen (1998) shows that LRcc canbe decomposed into a likelihood ratio test of correct

unconditional coverage (LRuc) and a likelihood ratio

test of independence (LRind). In brief, the test of

independence is concerned with the order in which

VaR violations occur observed violations should be

spread out over the sample rather than arriving in

clusters.

Table 7 reports statistical tests for conditional

coverage, unconditional coverage and independence.

In addition, the popular Binomial test of unconditional

coverage is also reported (see Fernandez, 2005;McNeil & Frey, 2000). As a quick reference guide,

the absence of dasterisksT in Table 7 indicates that the

difference between theoretical and empirical violation

ratios is not statistically significant. In addition, a

quantile estimator should be viewed with scepticism if

it passes the unconditional test but fails either or both

of the conditional and independence tests.

Almost immediately, we see examples of the issue

raised above. For example, with a=5%, the violation

ratio for Alberta passes the unconditional tests

(Binomial and LRuc), but fails the independence test,

and consequently the conditional coverage test. For

NordPool, the unconditional tests are passed, but the

independence test is failed. In contrast, the AR-

EGARCH-EVT approach (and arguably the AR-

EGARCH-t approach) demonstrates consistency be-

tween conditional and unconditional tests. In general,

the differences between theoretical and empirical

violation ratios from these models are not statistically

significant. Considering the results of Tables 6 and 7

as a whole, the only market where the AR-EGARCH-

EVT VaR forecast is not superior is the NordPool (at

any level of a). As noted in the previous section,NordPool features hydro-generation which seemingly

results in a return distribution with notably different

characteristics (as evidenced by the summary statistics

in Table 1). Table 7 suggests that the nave HS

approach is adequate in this market only.

19 We are grateful to an anonymous referee for articulating this

issue and suggesting tests of both unconditional and conditional

coverage.

20 Briefly, the tests can be implemented in a convenient likelihood

ratio framework and are distributed asymptotically chi-squared.

Readers are referred to Christoffersen (1998) for technical details on

the test statistics.



15/18

Table 7

Statistical tests of conditional and unconditional coverage


a= 5%HS Binomial test 0.93 0.93 0.09 1.12 1.68*

LRuc 0.92 0.92 0.01 1.35 2.61

LRind 10.43*** 2.75* 6.34** 0.95 5.42**

LRcc 11.35*** 3.67 6.35** 2.30 8.03**

AR-ConVar Binomial test 5.35*** 5.69*** 0.76 5.24*** 2.06*LRuc 44.54*** 53.61*** 0.61 46.52*** 4.85**

LRind 0.07 0.02 6.31** 0.01 1.24

LRcc 44.61*** 53.63*** 6.92** 46.54*** 6.09**

AR-HS Binomial test 1.44 5.69*** 1.10 3.93*** 3.00***LRuc 2.26 53.60*** 1.16 21.09*** 7.78***

LRind 12.75*** 0.02 5.68** 0.27 7.64***

LRcc 15.01*** 53.62*** 6.84** 21.36*** 15.42***

AR-EGARCH-N Binomial test 2.46***

1.61*

0.42

0.19 3.18***

LRuc 5.43*** 2.85* 0.18 0.04 8.71***

LRind 5.28*** 6.56** 6.21** 0.25 1.06

LRcc 10.71*** 9.41*** 6.39** 0.29 9.77***

AR-EGARCH-t Binomial test 0.08 6.03*** 0.25 2.81*** 1.69*LRuc 0.01 65.59*** 0.07 9.59*** 3.16*

LRind 2.35 0.00 0.35 0.77 1.52

LRcc 2.36 65.59*** 0.42 10.36*** 4.68*

AR-EGARCH-EVT Binomial test 0.42 3.48*** 1.10 1.68* 1.87*LRuc 0.18 15.21*** 1.29 3.16* 3.19*

LRind 2.96 0.72 0.05 0.09 1.52

LRcc 3.14 15.93*** 1.34 3.27 4.71*

a= 1%

HS Binomial test 0.48 0.11 0.11 1.64* 2.46**LRuc 0.25 0.01 0.25 3.63* 4.69**

LRind 4.44* 0.14 0.10 0.01 0.48

LRcc 4.69* 0.15 0.35 3.64 5.17*

AR-ConVar Binomial test 2.72*** 1.97* 4.35*** 2.46** 0.82LRuc 14.67*** 5.45** 13.13*** 12.06*** 0.61

LRind 0 0.01 0.42 0 0.22

LRcc 14.67*** 5.55* 13.55*** 12.06*** 0.83

AR-HS Binomial test 0.26 2.72*** 1.23 2.46** 0.41LRuc 0.07 14.67*** 1.80 12.06*** 0.16

LRind 3.26** 0 0.04 0 0.16

LRcc 3.33 14.67*** 1.84 12.06*** 0.32

AR-EGARCH-N Binomial test 6.96*** 0.63 5.47*** 2.87*** 9.44***

LRuc 29.14*** 0.37 19.44*** 6.19** 46.28***

LRind 0.01 0.22 1.37 1.18 0.14LRcc 29.15*** 0.59 20.81*** 7.37** 46.42***

AR-EGARCH-t Binomial test 1.37 2.72*** 0.86 1.64* 0.82LRuc 1.64 14.67*** 0.82 3.63* 0.76

LRind 0.34 0 0.07 0.01 0.05

LRcc 1.98 14.67*** 0.89 3.64 0.81

AR-EGARCH-EVT Binomial test 0.86 2.72** 0.86 1.23 1.23LRuc 0.82 14.67*** 0.82 1.86 1.31

LRind 0.07 0 0.07 0.03 0.27

LRcc 0.89 14.67*** 0.89 1.89 1.58

(continued on next page)



16/18

The statistical evidence favouring the use of VaR

forecasts based on the AR-EGARCH-EVT model

should come as no surprise. The AR(7) mean equation

accommodates the autoregression in returns. The

EGARCH component captures conditional volatility

clustering, asymmetric effects, and, in this case,

seasonality in volatility. The EVT component explic-

itly models the heavy tails of the standardized

residuals. Taken together, the features ensure that

quantile estimates from the AR-EGARCH-EVT model

at any given time reflect the most recent and relevantinformation.

5. Conclusion

The recent deregulation in electricity markets

worldwide has heightened the importance of risk

management in energy markets. This paper examines

a number of approaches to forecasting VaR for

electricity markets. Arguably, assessing VaR in

electricity markets is more difficult than in traditional

financial markets because the distinctive features of

the former result in a highly unusual distribution of

returns electricity returns are highly volatile, dis-

play seasonalities in both their mean and volatility,

exhibit leverage effects and clustering in volatility,

and feature extreme levels of skewness and kurtosis.

Accordingly, approaches to VaR measurement that

are common in financial markets may not necessarily

be appropriate in electricity markets.

In addition to popular parametric and non-para-metric approaches, this paper explores an approach to

VaR forecasting that incorporates extreme value

theory. The proposed model is specifically designed

for electricity applications. Given daily data series,

the model accommodates autoregression and weekly

seasonals in both the conditional mean and condi-

tional volatility equations. Leverage effects in condi-

tional volatility are modelled with an EGARCH

specification. Model residuals are standardized to

produce (near) i.i.d. observations, and EVT is applied

Table 7 (continued)


a=0.5%

HS Binomial test 0.34 0.71 0.18 1.74* 1.74*LRuc 0.89 0.45 0.03 6.02** 2.33

LRind 0.01 0.07 0.04 0 0.12

LRcc 0.90 0.52 0.07 6.02** 2.45

AR-ConVar Binomial test 0.87 0.87 8.06*** 1.74* 2.32**LRuc 7.31*** 0.89 32.32*** 6.02** 3.89**

LRind 0 0.01 0.43 0 0.16

LRcc 7.31** 0.90 32.75*** 6.02** 4.05

AR-HS Binomial test 1.39 1.92* 0.87 1.74* 1.74*LRuc 2.72 7.32*** 0.89 6.02** 6.02**

LRind 0.00 0 0.01 0 0

LRcc 2.72 7.32*** 0.90 6.02** 6.02**

AR-EGARCH-N Binomial test 9.10*** 1.76** 7.53*** 4.63*** 12.16***

LRuc 39.21*** 2.43 29.03*** 12.69*** 58.56***

LRind 0.23 0.14 0.91 1.73 2.00

LRcc 39.44*** 2.57 29.94*** 14.42*** 60.56***

AR-EGARCH-t Binomial test 0.71 1.92* 1.39 1.16 1.16LRuc 0.45 7.32*** 2.72* 1.81 1.81

LRind 0.07 0 0.00 0.00 0.00

LRcc 0.52 7.32*** 2.72 1.81 1.81

AR-EGARCH-EVT Binomial test 0.18 1.92* 1.39 0.58 0.58LRuc 0.03 7.32*** 2.72* 0.38 0

LRind 0.04 0 0.00 0.01 0.03

LRcc 0.07 7.32** 2.72 0.39 0.03

The table presents statistical tests of both conditional and unconditional coverage of the interval forecasts under each competing approach. *, **

and *** denote significance at the 10%, 5% and 1% level, respectively.



17/18

to the standardized residuals to forecast the tail

quantiles required for VaR.

The results support the deployment of the pro-

posed model. Autocorrelations exist at lags up to 7days, and conditional volatility displays leverage

effects. The two-step procedure of McNeil and Frey

(2000) produces standardized residuals that dbehaveT

significantly better than raw returns in terms of

independence, and thus better facilitate the EVT

implementation.

In terms of VaR performance, it is difficult to draw

consistent conclusions across the various methods,

quantile levels and energy markets. Of the parametric

models, the proposed AR-EGARCH-EVT method

arguably produces the most accurate forecasts of VaR.Somewhat surprisingly, the nave quantile estimator

based on historical simulation performs strongly in

several markets. Further examination suggests that the

distribution of returns in markets in which the HS

approach dominates may be differentthe distribu-

tion of returns in Nordpool and Alberta markets is

notably less skewed, has lower kurtosis, and exhibits

lower dispersion. In contrast, and as might be

expected, the sophisticated AR-EGARCH-EVT ap-

proach dominates in markets where the distribution of

returns is characterized by high skewness and

kurtosis, and high volatility.

The paper also examines VaR performance by

assessing the unconditional and conditional interval

coverage of the various approaches to forecasting

VaR. Assessing conditional coverage is important if

true tail quantiles are time varying. In such cases,

simple VaR approaches based on historical simula-

tion are likely to result in dclusteringT of VaR

violations, and this will occur during periods of

turmoil when accurate VaR forecasts are needed

most. The statistical tests of Christoffersen (1998)

suggest that the nave HS approach does indeed failto provide adequate conditional coverage. In con-

trast, the proposed AR-EGARCH-EVT approach

generates VaR forecasts that, by incorporating the

most recent market events, provide appropriate

conditional coverage. This finding is consistent

across nearly all energy markets examined. In

summary, the results of the paper support the

combination of the parametric AR-EGARCH model

with EVT for the purpose of estimating tail quantiles

and forecasting VaR.

Acknowledgements

We are grateful to two anonymous referees, Hans

Bystrom and seminar participants at the 2005AsianFA Conference for their helpful comments and

suggestions. The first author thanks the Department of

Education, Training and Youth Affairs (DETYA),

Australia, the University of Queensland and the

University of Auckland for the funding support. All

errors are our own.

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Kam Fong Chan is a PhD student at the UQ Business School, The

University of Queensland. He is a current lecturer in finance at The

University of Auckland. His research interests include financialeconometrics, testing asset pricing models and modelling jump-

diffusion and volatility processes. He has published in the

Multinational Finance Journal and Accounting & Finance.

Philip Gray is Associate Professor in Finance at the UQ Business

School at the University of Queensland. He completed a PhD at the

Australian Graduate School of Management in 2000. His research

interests include assessing return predictability, non-parametric

derivative pricing, and empirical testing of asset pricing models.

He has published in numerous scholarly journals including the

Journal of Business, Finance & Accounting, Journal of Futures

Markets, Journal of Finance, Economic Record, International

Review of Finance, Finance Research Letters, Accounting &

Finance and Journal of Banking and Finance.


using extreme value statistics to measure value at risk for daily electricity spot prices

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