using extreme value statistics to measure value at risk for daily electricity spot prices
TRANSCRIPT
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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.
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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
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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
Jan 99 Dec 00 Dec 02 Dec 04
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
Dec 98 Dec 00 Dec 02 Dec 040
100
200
300
400
500
600
Dec 98 Dec 00 Dec 02 Dec 04
0
2
4
6
8
-4 -2 0 2 4-2
0
2
4
6
8Alberta prices Alberta returns Alberta QQ plot
Jan 98 Dec 99 Dec 01 Dec 030
100
200
300
400
500
Jan 98 Dec 99 Dec 01 Dec 03
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
Victoria NordPool Alberta Hayward PJM
/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.
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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
Victoria NordPool Alberta Hayward PJM
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
Victoria NordPool Alberta Hayward PJM
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.
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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
Victoria NordPool Alberta Hayward PJM
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.
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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
Victoria NordPool Alberta Hayward PJM
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.
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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.)
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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.
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Table 7
Statistical tests of conditional and unconditional coverage
Victoria NordPool Alberta Hayward PJM
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)
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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)
Victoria NordPool Alberta Hayward PJM
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.
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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.
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283300300