International Journal of Forecasting 22 (2006) 283 – 300
www.elsevier.com/locate/ijforecast
Using extreme value theory to measure value-at-risk for daily
electricity spot pricesB
Kam Fong Chan a,*, Philip Gray b,1
a
Department of Accounting and Finance, Faculty of Business and Economics, The University of Auckland, Private Bag 92019,
Auckland, New Zealand
b
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 the
return 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
B
This paper is a revised version of Chapter Five of the first
author’s Ph.D. thesis at The University of Queensland, Australia.
* Corresponding author. Tel.: +64 9 373 7599x85172.
E-mail addresses: k.chan@auckland.ac.nz (K.F. Chan),
p.gray@business.uq.edu.au (P. Gray).
1
Tel.: +61 7 3365 6992.
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 matter D 2005 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.ijforecast.2005.10.002
284
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
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 from
adversely 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 firm’s 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 nonparametric. 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 assumptions 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 electricity
markets, however, the high volatility and occasional
price spikes result in an empirical distribution of
returns with a non-standard shape making it difficult
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).
to specify a parametric form. As a result, parametric
approaches may not generate accurate VaR measures
in electricity markets. Similarly, the usefulness of nonparametric approaches in electricity markets is largely
unknown.
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 prices
and shows that the extreme returns obey a Fréchet fattailed distribution. Ho, Burridge, Cadle, and Theobald
(2000) and Gençay and Selçuk (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.
Gençay, Selçuk, and Ulugülyaĝci (2003) reach similar
conclusions for the Istanbul Stock Exchange Index
(ISE-100). Müller, Dacorogna, and Pictet (1998) and
Pictet, Dacorogna, and Müller (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 threshold-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 threshold-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 model’s failure to account for
clustering in the data.
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) 283–300
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 is
provided by McNeil and Frey (2000). Using a twostage 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 GARCH–EVT
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 regarded
as 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 Byström (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, Byström (2005) employs a GARCH-EVT
framework to NordPool hourly electricity returns. He
finds that the extreme GARCH-filtered residuals obey
a Fréchet 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 autoregression and weekly seasonals in both the conditional
mean 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
EGARCH–EVT combination is a sophisticated approach to forecasting VaR. The second contribution,
therefore, is to compare the accuracy of VaR forecasts
under the proposed model with a number of conventional approaches (both parametric and non-paramet-
285
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 naı̈ve estimator constructed 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
naı̈ve quantile estimator may struggle to differentiate
between periods of high volatility and periods of
relative tranquility. As such, VaR violations from a
naı̈ve 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 examines 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 of
our 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 specifically 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
286
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
implementation of the EVT framework. Section 4.2
documents the relative VaR performance of competing approaches, measured using (unconditional) violation frequencies. Section 4.3 extends the assessment
by conducting formal statistical tests of both unconditional 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 2.2
outlines 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-overlapping s-day returns.4 Known as the Historical
Simulation (HS) approach, this simple method is
non-parametric in that it makes no arbitrary assumptions 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 naı̈ve 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 (hereafter denoted AR-ConVar). Since the data series are
4
Alternatively, s daily returns can be bootstrapped from the
empirical distribution and aggregated.
sampled daily, an AR(7) model is proposed to capture
any weekly seasonality in electricity prices:
rt ¼ / 0 þ
7
X
/j rtj þ et ;
ð1Þ
j¼1
where r t = (S t S t1) / S t1 is the simple electricity
return and S t is the daily spot price. A distributional
assumption is made of the error term in the ARConVar model; specifically, errors e t are assumed to
be Normally distributed with zero mean and constant
variance (E(e2t / X t1) = r 2). At any time t, the VaR
estimate from the AR-ConVar model is:
VaRq;t ¼ /̂ 0 þ
7
X
/̂j rtj þ F 1 ðqÞr̂
r;
ð2Þ
j¼1
where (/̂ j = 1, 2, . . .7, r̂ ) are parameter estimates and
F 1( q) is the q% quantile of the Normal distribution
function at an a% tail (i.e., q = 1 a).
The second parametric approach, a minor variation of the AR-ConVar model, combines the key
features of the autoregressive model and the historical simulation approach.5 The conditional mean is
again modelled using an AR(7) model. However,
rather than making a distributional assumption over
F 1( 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 autocorrelation in their conditional mean. The traditional HS
approach cannot capture these intertemporal characteristics. In contrast, the proposed AR-HS method
accommodates the time-series properties through the
autoregressive mean, while retaining the distributionfree flavour by the use of bootstrapping. As such, the
AR-HS approach represents a more sensible implementation of the historical-simulation concept for
calculating VaR.6
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.
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
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 than
constant variance, the error term is assumed to follow
an EGARCH process:
E e2t jXt1 ¼ ht ;
et1
and lnðht Þ ¼ b1 þ b2 pffiffiffiffiffiffiffiffi þ b3 lnðht1 Þ
ht1
et1 et1 þ b4 pffiffiffiffiffiffiffiffi E pffiffiffiffiffiffiffiffi ht1
ht1
et7
þ b5 pffiffiffiffiffiffiffiffi þ b6 lnðht7 Þ:
ht7
ð3Þ
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 inducing
a positive leverage effect.7 In addition to capturing
asymmetries (b 4 ), Eq. (3) also accommodates
weekly seasonality in conditional volatility (b 5 and
b 6).
The corresponding VaR measure is calculated
in a similar fashion to Eq.p(2),
ffiffiffiffi with the parameter
estimate of r̂ replaced by ĥ
h t from Eq. (3). Since
a number of distributional assumptions are common
in the GARCH literature, this study imposes
two distributions over the e t error term: the Normal
distribution and the fat-tailed t-distribution with
m degrees of freedom. The former model is
termed AR-EGARCH-N, where the tail quantile
of F 1( q) in its VaR model is also Normally
distributed; whilst the latter is termed AREGARCH-t, where the F 1( q) quantile in its VaR
model is t m -distributed.
7
We are grateful to an anonymous referee for suggesting this
motivation for employing an EGARCH model.
287
2.3. The AR-EGARCH-EVT method
Following Byström (2005), this study adopts the
EVT approach of McNeil and Frey (2000) to
measure VaR for electricity returns. McNeil and
Frey recognize that most financial return series
exhibit stochastic volatility and fat-tailed distributions. While the fat tails might be modelled directly
with EVT, the lack of i.i.d. returns is problematic.
McNeil and Frey’s 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 GARCH–EVT combination
accommodates both time-varying volatility and fattailed 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 t m -distribution
governing the e t term (as described in Section 2.2)
is estimated from electricity returns. Maximum
likelihood estimation is employed over an insample period (described shortly).
2. The residuals from the AR-EGARCH model are
standardized:
(
)1
0
7
X
/̂j rtj C
B rt /̂0 þ
C
B
j¼1
C
B
pffiffiffiffiffiffiffiffiffi
ð4Þ
ẑz t ¼ B
C
A
@
ĥh t1
where T is the number of return observations
during the in-sample estimation period.
3. EVT is applied to the standardized residuals ẑ t to
model the tail quantile of F 1( q) in deriving VaR.
In applying the EVT method, this paper adopts the
Peak Over Threshold (POT) EVT method.8 The POT
8
For details on the POT EVT method, refer to McNeil and Frey
(2000) and Embrechts et al. (1997).
288
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
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 z t are
a sequence of i.i.d. random variables from an
unknown distribution function F z . 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 exceedence is given by y i = z i u, for i = 1, . . .N y, where N y
is the total number of exceedences in the sample.
The distribution of y, for a given threshold u, is
given by:
and beta distributions. In most financial applications,
the data exhibit heavy-tails suggesting n N 0.
Given that F u ( y) can be approximated by Eq. (7),
and noting that F z (u) is determined by (T N y ) / T,
Eq. (6) simplifies to:
Fu ð yÞ ¼ Prð z u V yjz N uÞ
Eq. (9), together with the conditional mean (Eq.
(1)) and conditional variance (Eq. (3)), is termed AREGARCH-EVT. The VaR measure is defined as:
¼
Prð z u V y; z N uÞ
Prð zNuÞ
Fz ð y þ uÞ Fz ðuÞ
¼
:
1 Fz ðuÞ
ð5Þ
zNu:
ð6Þ
Balkema and de Haan (1974) and Pickands (1975)
show that, for a sufficiently high u, F u ( y) can be
approximated by the Generalized Pareto Distribution
(GPD), which is defined as:
8
<
ny 1=n
1
1
þ
Gn;m ð yÞ ¼
m
:
1 expð y=mÞ
ð8Þ
This tail estimator can be inverted for the purpose
of calculating VaR:
"
#
n
m
T
1
Fz ðqÞ ¼ u þ
a
1 :
ð9Þ
n
Ny
VaRq;t ¼ /̂0 þ
7
X
/̂j rtj þ Fz1 ðqÞ
qffiffiffiffi
ĥh t :
ð10Þ
j¼1
That is, F u ( 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 ð zÞ ¼ ½1 Fz ðuÞFu ð yÞ þ Fz ðuÞ;
Ny
nð z uÞ 1=n
1þ
F z ð zÞ ¼ 1 :
m
T
if n p 0
ð7Þ
if n ¼ 0;
where n and m N 0 are shape and scale parameters
respectively. Note that the GPD subsumes various
distributions. A value of n N 0 corresponds to the heavytailed distributions such as Pareto, Student t, Cauchy,
loggamma and Fréchet, whose tails decay like power
functions. A n = 0 corresponds to thin-tailed distributions such as Gumbel, normal, exponential, gamma and
lognormal, whose tails decay exponentially. A n b 0
corresponds to finite distributions such as the uniform
Finally, a reasonable threshold u must be chosen to
implement the POT method. Ideally, u should be set
sufficiently high so that the POT asymptotic theory
applies. However, if u is set too high, there will be too
few exceedences from which to estimate the parameters of the GPD. This paper follows the approach of
Gençay and Selçuk (2004), who determine a reasonable threshold u using a combination of two popular
techniques: the mean excess function (MEF) and the
Hill plots (Hill, 1975). Further details are provided in
Section 4.1’s discussion of results.
3. Data description
The data examined in this paper are daily
aggregated electricity spot prices from five international power markets: Victoria (Australia), NordPool
(Scandinavia), Alberta (Canada), Hayward (New
Zealand) and PJM (US). Table 1 reports descriptive
statistics for daily (simple) returns in all markets.
Statistics are shown for the full sample, as well as
the in-sample period used subsequently in model
estimation.
Full sample sizes range from 2093 days (PJM) to
2553 days (NordPool and Alberta). While the mean
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
289
Table 1
Descriptive statistics
Victoria
NordPool
Alberta
Hayward
PJM
Full sample
Start date
End date
No. of obs
Mean
Median
Std. dev.
Min
Max
Skewness
Kurtosis
Q(7)
Q 2(7)
4 Jan 99
31 Dec 04
2189
0.091
0.014
1.100
0.959
44.22
30.52
1191
196.4***
46.64***
5 Jan 98
31 Dec 04
2553
0.008
0.003
0.141
0.558
2.496
3.742
50
236.7***
111.4***
5 Jan 98
31 Dec 04
2553
0.098
0.011
0.575
0.898
6.307
4.007
31
130.4***
67.16***
5 Jan 98
31 Dec 03
2184
0.076
0.005
0.775
0.963
21.84
17.01
398
141.7***
59.86***
5 Apr 98
31 Dec 03
2093
0.068
0.015
0.560
0.926
15.24
12.91
301
329.2***
58.92***
In-sample
Start date
End date
No. of obs
Mean
Median
Std. dev.
Min
Max
Skewness
Kurtosis
Q(7)
Q 2(7)
4 Jan 99
31 Dec 02
1459
0.104
0.017
1.301
0.959
44.22
27.42
911
146.3***
38.25***
5 Jan 98
31 Dec 02
1823
0.012
0.004
0.161
0.557
2.496
3.358
40
165.9***
74.66***
5 Jan 98
31 Dec 02
1823
0.094
0.015
0.574
0.897
6.306
4.379
35
113.0***
73.83***
5 Jan 98
30 Apr 02
1584
0.062
0.007
0.439
0.963
21.84
15.02
304
99.22***
36.45***
5 Apr 98
30 Apr 02
1493
0.052
0.008
0.360
0.925
15.24
13.54
292
217.2***
36.18***
The table reports summary statistics for the daily simple net returns (r t ) of five international power markets: Victoria, NordPool, Alberta,
Hayward and PJM. The Ljung–Box Q(7) and Q 2 (7) statistics test for serial correlation up to 7 lags for r t and r2t , respectively. *** Indicates
significance at the 1% level.
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. Ljung–Box Q and Q 2 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 extreme movements, volatility clustering and fattailed distributions. These descriptive statistics and
plots further motivate the exploration of the
alternative approaches to measuring VaR described
in Section 2.
4. Empirical results
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 at 100%. As emphasized
by Byström (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.
This section presents the empirical findings of
the study. Section 4.1 reports the in-sample parameter 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,
290
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
Victoria prices
Victoria returns
Victoria QQ plot
1200
50
10
1000
40
8
800
6
30
600
4
20
400
2
10
200
0
Jan 99
Dec 00
Dec 02
Dec 04
0
0
Jan 99
Dec 00
Dec 02
Dec 04
-2
-4
1000
800
-2
0
2
4
NordPool QQ plot
NordPool returns
NordPool prices
3
3
2
2
1
1
0
0
600
400
200
0
-1
Dec 98
Dec 00
Dec 02
Dec 04
Dec 98
Dec 00
Dec 02
Dec 04
-1
-4
-2
600
500
0
2
4
2
4
Alberta QQ plot
Alberta returns
Alberta prices
8
8
6
6
400
4
4
300
200
2
2
0
100
0
0
Dec 98
Dec 00
Dec 02
Dec 04
Dec 98
Dec 00
Dec 02
Dec 04
-2
-4
Hayward returns
Hayward prices
25
25
400
20
20
300
15
15
10
10
5
5
100
0
0
Jan 98
Dec 99
Dec 01
Dec 03
Jan 98
PJM prices
0
Dec 99
Dec 01
Dec 03
-4
-2
PJM returns
400
300
200
0
Hayward QQ plot
500
200
-2
0
2
4
2
4
PJM QQ plot
20
20
15
15
10
10
5
5
0
0
100
0
Apr 98
Dec 99
Dec 01
Dec 03
Apr 98
Dec 99
Dec 01
Dec 03
-4
-2
0
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 display electricity prices, returns and QQ plots for daily returns respectively.
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
291
Table 2
Parameter estimates for the AR-ConVar model
/0
/1
/2
/3
/4
/5
/6
/7
r
R2
Victoria
NordPool
Alberta
Hayward
PJM
0.079
0.119
0.101
0.126
0.085
0.094
0.043 (0.015)
0.222
0.422
0.1015
0.018
0.026 (0.278)
0.113
0.023 (0.341)
0.164
0.154
0.119
0.122
0.153
0.0897
0.109
0.144
0.127
0.009 (0.715)
0.051 (0.036)
0.018 (0.050)
0.001 (0.970)
0.168
0.555
0.0639
0.077
0.077
0.158
0.071
0.044 (0.109)
0.036 (0.180)
0.013 (0.611)
0.167
0.424
0.0638
0.092
0.185
0.254
0.167
0.102
0.138
0.106
0.184
0.332
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 pvalue is not shown. p-values are shown in parentheses only when not significant at the 1% level.
with the observed violation frequencies compared to
tail quantiles from each model. Section 4.3 examines VaR performance further by conducting statistical 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 ARConVar 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 autocorrelations 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
P
/̂0 = 1 7j¼1 /ˆ j ¼ 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 insample standard deviation.
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. pvalues are only explicitly shown when they are greater than 1%.
4.1.2. AR-EGARCH model
Table 3 presents the ML estimates of the AREGARCH-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 (b 2 and
b 3) in all markets except PJM. In addition, there
appears to be an asymmetric leverage effect (b 4).12
Parameter estimates (b 5 and b 6) also suggest a weekly
seasonal effect in the conditional variance. Finally, ML
estimates of the parameter m suggest that returns have a
tail fatter than that implied by a Normal distribution. In
summary, the findings support the use of the AREGARCH-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-stage
process 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.
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 subsequent analysis.
12
Estimates of the leverage effect are comparable to those
reported by Knittel and Roberts (2001) and Duffie, Gray, and
Hoang (1998) where b̂4 is positive and statistically significant.
292
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
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)
/1
0.156
0.000 (0.990)
/2
0.140
0.087
0.140
0.100
/3
/4
0.087
0.143
/5
0.114
0.163
0.080
0.049
/6
/7
0.170
0.231
R2
0.1414
0.1414
0.011 (0.092)
0.201
0.195
0.113
0.093
0.093
0.040 (0.020)
0.064
0.1492
0.007 (0.129)
0.077
0.063
0.067
0.044
0.046
0.001 (0.970)
0.098
0.0596
0.043
0.211
0.236
0.191
0.147
0.189
0.151
0.122
0.1718
Estimates from the EGARCH conditional variance (Eq. (3))
b1
0.225 (0.051)
0.046 (0.338)
b2
0.282
0.366
b3
0.057 (0.074)
0.377
0.520
0.703
b4
b5
0.223
0.208
b6
0.771
0.597
m
2.500
3.689
0.560
0.395
0.835
0.964
0.110
0.104
2.069
0.747
0.986
0.879
1.412
0.023 (0.794)
0.073 (0.047)
2.119
0.375
0.043 (0.416)
0.039 (0.340)
0.181
0.232
0.808
4.167
The table reports maximum-likelihood estimates of the AR-EGARCH model (Eqs. (1) and (3)), with a t m -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 their p-value is not shown. p-values are shown in parentheses only when not significant at
the 1% level.
Table 4
Summary 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
0.004
Mean
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
Q 2(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
0.009
Mean
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
Q 2(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 t m -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 Ljung–Box Q and
Q 2 statistics are significant at the 10% and 1% levels, respectively.
assumption. Second, EVT is applied to the standardized residuals. Table 4 presents diagnostics for the
drawT and standardized AR-EGARCH residuals.
The Ljung–Box Q and Q 2 statistics provide an
indication of whether any serial correlation or heteroscedasticity 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 kurtosis
remain in the standardized residuals. Similarly, QQ
plots (not presented) document heavy right tails. These
findings motivate the second stage of McNeil and
Frey’s (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 Gençay and Selçuk (2004)
closely and we do not present the MEFs and Hill plots here.
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
293
Table 5
Parameter estimates for the AR-EGARCH-EVT model
Panel A
Total in-sample obs.
EVT threshold
Number of exceedences
% of exceedences in-sample
GPD shape parameter
GPD scale parameter
T
u
Ny
N y /T
n
m
Panel B
q = 95%
q = 99%
q = 99.5%
Normal
1.645
2.326
2.576
Victoria
NordPool
Alberta
Hayward
PJM
1459
1.5
151
10.35
0.552***
1.208***
1823
1.0
185
10.15
0.305***
1.631***
1823
2.0
167
9.16
0.332***
1.034***
1584
1.5
187
11.81
0.305***
0.570***
1493
1.0
149
9.98
0.344***
0.569***
2.582
7.265
10.97
1.459
2.935
3.832
3.289
7.498
10.05
2.375
5.194
6.956
1.444
2.996
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.
1
Panel B presents EVT tail quantiles F ( q) for the standardized residuals, along with tail quantiles from a Normal distribution.
z
reports the threshold chosen in each market. In each
case, the resulting exceedences N y total approximately
10% of the sample, which is consistent with percentages 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 N 0 reflect heavy-tailed distributions. In each
power market, the n estimate is positive and statistically significantly different from zero, suggesting that
the right tail of the distribution of standardized
residuals is characterized by the Fréchet 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 F z 1( q) are obtained from Eq. (9) using
the Panel A reports of T, u, N y, n and m at the specified
end tail of a%. 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 more
extreme quantiles (i.e., as a moves towards 0.5%).
Indeed, Gençay and Selçuk (2004) warn that using
quantile estimates from a Normal distribution when
the data is in fact fat tailed will cause VaR to be
underestimated.
14
In a study of NordPool hourly prices, Byström (2005) also
finds that a Fréchet distribution applies to the tail of the distribution
of standardized residuals.
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 4.1’s 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 mostrecent T returns are used to estimate model parameters
for each parametric approach. The magnitude of T is 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 simply
rolled 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-parametric (HS) and semi-parametric (AR-HS) approaches.
15
Berkowitz (1999), Ho et. al. (2000), Bali and Neftci (2003),
Gençay and Selçuk (2004), Byström (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.
294
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
Table 6
Out-of-sample VaR violations
Victoria
NordPool
Alberta
Hayward
PJM
a = 5%
HS
AR-ConVar
AR-HS
AR-EGARCH-N
AR-EGARCH-t
AR-EGARCH-EVT
4.25
0.68
3.84
6.99
4.93
4.65
(3)
(6)
(4)
(5)
(1)
(2)
4.25
0.41
0.41
3.70
1.00
2.19
(1)
(5)
(5)
(2)
(4)
(3)
4.93
4.38
5.89
4.65
4.79
4.11
(1)
(4)
(6)
(3)
(2)
(5)
4.00
0.33
1.50
4.83
2.50
3.50
(2)
(6)
(5)
(1)
(4)
(3)
6.50
3.62
7.77
7.83
3.50
6.67
(2)
(1)
(5)
(6)
(2)
(4)
a = 1%
HS
AR-ConVar
AR-HS
AR-EGARCH-N
AR-EGARCH-t
AR-EGARCH-EVT
0.82
0.27
1.10
3.56
1.51
0.69
(2)
(5)
(1)
(6)
(4)
(3)
0.96
0.27
0
1.23
0
0
(1)
(3)
(4)
(2)
(4)
(4)
0.82
2.60
0.55
3.01
0.68
0.69
(1)
(5)
(4)
(6)
(3)
(2)
0.33
0
0
2.17
0.30
0.5
(2)
(4)
(4)
(6)
(3)
(1)
2.00
1.33
1.17
4.83
0.67
1.50
(5)
(2)
(1)
(6)
(2)
(4)
a = 0.5%
HS
AR-ConVar
AR-HS
AR-EGARCH-N
AR-EGARCH-t
AR-EGARCH-EVT
0.41
0.27
0.14
2.88
0.68
0.54
(2)
(4)
(5)
(6)
(3)
(1)
0.69
0.27
0
0.96
0
0
(1)
(2)
(4)
(3)
(4)
(4)
0.55
2.60
0.27
2.46
0.14
0.14
(1)
(6)
(4)
(5)
(2)
(2)
0
0
0
1.83
0.17
0.33
(3)
(3)
(3)
(6)
(2)
(1)
1.00
1.17
0
4.00
0.17
0.67
(3)
(5)
(3)
(6)
(2)
(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 at a = 5%, 1%
and 0.5%. All actual and expected violations are in percentage terms.
Under the AR-HS approach, parameters of the
autoregressive model are again estimated using a
rolling window of the most recent T observations (this
ensures comparability with the plain-vanilla ARConVar approach). However, tail quantiles are constructed by bootstrapping from the 500 most-recent
AR errors. Similarly, the naı̈ve 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 comparing
the 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 ARGARCH-EVT models forecast right-tail quantiles
most accurately. The AR-EGARCH-N model (that is,
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.
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 naı̈ve 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 underestimate and overestimate the 99% quantile respectively. For the extreme quantile (a = 0.5%), rankings
approximate those reported for a = 5%.
Examining the other power markets, little consistency in model performance is evident. The HS
approach has superior performance for NordPool
and Alberta, irrespective of the quantile. The AREGARCH-EVT model performs very well for Hayward and PJM. While these inconsistent rankings are
not particularly encouraging for risk managers inter-
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
295
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
Jan 03
Dec 03
Dec 04
Fig. 2. Time-varying VaR forecasts and violations. The plot depicts the VaR forecasts from the HS (smooth, heavy red line) and AR-EGARCHEVT 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.)
ested in forecasting VaR, a careful examination of the
violation ratios in conjunction with the Table 1
summary statistics is revealing. Consider the Victorian, Hayward and PJM markets, where the moresophisticated 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 different—the skewness 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 a
more 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 Byström (2005). Working with hourly
NordPool returns, Byström (2005) reports that VaR
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.
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 naı̈ve HS approach (not examined by
Byström) 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
nonetheless benefit from adopting the AREGARCH-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 offer
advantages 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-
296
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
EGARCH-EVT model produces more accurate and
robust VaR forecasts (dashed, green line). AREGARCH-EVT dviolationsT (marked with circles)
are relatively evenly spaced throughout the out-ofsample 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 implications —a 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 conditional 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 naı̈ve quantile estimator constructed as the quantile of all historical returns. On
average, the naı̈ve estimator will have a perfect
violation proportion (that is, correct unconditional
coverage). In any given period, however, the conditional coverage may be incorrect. This scenario is
particularly relevant in financial time-series where
volatility (and consequently, the return distribution)
varies over time. A naı̈ve quantile estimator may
entirely fail to differentiate between periods of high
volatility and periods of relative tranquility.19
Christoffersen (1998) clarifies the distinction between conditional and unconditional interval forecasts
19
We are grateful to an anonymous referee for articulating this
issue and suggesting tests of both unconditional and conditional
coverage.
and proposes statistical tests for each.20 Let LRcc
denote a likelihood ratio test statistic examining
whether a quantile estimator has correct conditional
coverage. Christoffersen (1998) shows that LRcc can
be 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 AREGARCH-EVT approach (and arguably the AREGARCH-t approach) demonstrates consistency between 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-EGARCHEVT 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 naı̈ve HS
approach is adequate in this market only.
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.
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
297
Table 7
Statistical tests of conditional and unconditional coverage
a = 5%
HS
AR-ConVar
AR-HS
AR-EGARCH-N
AR-EGARCH-t
AR-EGARCH-EVT
a = 1%
HS
AR-ConVar
AR-HS
AR-EGARCH-N
AR-EGARCH-t
AR-EGARCH-EVT
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
test
test
test
test
test
test
test
test
test
test
test
test
Victoria
NordPool
Alberta
Hayward
PJM
0.93
0.92
10.43***
11.35***
5.35***
44.54***
0.07
44.61***
1.44
2.26
12.75***
15.01***
2.46***
5.43***
5.28***
10.71***
0.08
0.01
2.35
2.36
0.42
0.18
2.96
3.14
0.93
0.92
2.75*
3.67
5.69***
53.61***
0.02
53.63***
5.69***
53.60***
0.02
53.62***
1.61*
2.85*
6.56**
9.41***
6.03***
65.59***
0.00
65.59***
3.48***
15.21***
0.72
15.93***
0.09
0.01
6.34**
6.35**
0.76
0.61
6.31**
6.92**
1.10
1.16
5.68**
6.84**
0.42
0.18
6.21**
6.39**
0.25
0.07
0.35
0.42
1.10
1.29
0.05
1.34
1.12
1.35
0.95
2.30
5.24***
46.52***
0.01
46.54***
3.93***
21.09***
0.27
21.36***
0.19
0.04
0.25
0.29
2.81***
9.59***
0.77
10.36***
1.68*
3.16*
0.09
3.27
1.68*
2.61
5.42**
8.03**
2.06*
4.85**
1.24
6.09**
3.00***
7.78***
7.64***
15.42***
3.18***
8.71***
1.06
9.77***
1.69*
3.16*
1.52
4.68*
1.87*
3.19*
1.52
4.71*
0.48
0.25
4.44*
4.69*
2.72***
14.67***
0
14.67***
0.26
0.07
3.26**
3.33
6.96***
29.14***
0.01
29.15***
1.37
1.64
0.34
1.98
0.86
0.82
0.07
0.89
0.11
0.01
0.14
0.15
1.97*
5.45**
0.01
5.55*
2.72***
14.67***
0
14.67***
0.63
0.37
0.22
0.59
2.72***
14.67***
0
14.67***
2.72**
14.67***
0
14.67***
0.11
0.25
0.10
0.35
4.35***
13.13***
0.42
13.55***
1.23
1.80
0.04
1.84
5.47***
19.44***
1.37
20.81***
0.86
0.82
0.07
0.89
0.86
0.82
0.07
0.89
1.64*
3.63*
0.01
3.64
2.46**
12.06***
0
12.06***
2.46**
12.06***
0
12.06***
2.87***
6.19**
1.18
7.37**
1.64*
3.63*
0.01
3.64
1.23
1.86
0.03
1.89
2.46**
4.69**
0.48
5.17*
0.82
0.61
0.22
0.83
0.41
0.16
0.16
0.32
9.44***
46.28***
0.14
46.42***
0.82
0.76
0.05
0.81
1.23
1.31
0.27
1.58
(continued on next page)
298
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
Table 7 (continued)
a = 0.5%
HS
AR-ConVar
AR-HS
AR-EGARCH-N
AR-EGARCH-t
AR-EGARCH-EVT
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
Binomial
LRuc
LRind
LRcc
test
test
test
test
test
test
Victoria
NordPool
Alberta
Hayward
PJM
0.34
0.89
0.01
0.90
0.87
7.31***
0
7.31**
1.39
2.72
0.00
2.72
9.10***
39.21***
0.23
39.44***
0.71
0.45
0.07
0.52
0.18
0.03
0.04
0.07
0.71
0.45
0.07
0.52
0.87
0.89
0.01
0.90
1.92*
7.32***
0
7.32***
1.76**
2.43
0.14
2.57
1.92*
7.32***
0
7.32***
1.92*
7.32***
0
7.32**
0.18
0.03
0.04
0.07
8.06***
32.32***
0.43
32.75***
0.87
0.89
0.01
0.90
7.53***
29.03***
0.91
29.94***
1.39
2.72*
0.00
2.72
1.39
2.72*
0.00
2.72
1.74*
6.02**
0
6.02**
1.74*
6.02**
0
6.02**
1.74*
6.02**
0
6.02**
4.63***
12.69***
1.73
14.42***
1.16
1.81
0.00
1.81
0.58
0.38
0.01
0.39
1.74*
2.33
0.12
2.45
2.32**
3.89**
0.16
4.05
1.74*
6.02**
0
6.02**
12.16***
58.56***
2.00
60.56***
1.16
1.81
0.00
1.81
0.58
0
0.03
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.
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 explicitly 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 relevant
information.
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, display 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-parametric 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 conditional volatility equations. Leverage effects in conditional volatility are modelled with an EGARCH
specification. Model residuals are standardized to
produce (near) i.i.d. observations, and EVT is applied
K.F. Chan, P. Gray / International Journal of Forecasting 22 (2006) 283–300
to the standardized residuals to forecast the tail
quantiles required for VaR.
The results support the deployment of the proposed model. Autocorrelations exist at lags up to 7
days, 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 naı̈ve 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 different — the distribution 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 approach 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 simulation 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 naı̈ve HS approach does indeed fail
to provide adequate conditional coverage. In contrast, 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.
299
Acknowledgements
We are grateful to two anonymous referees, Hans
Byström and seminar participants at the 2005
AsianFA 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 financial
econometrics, testing asset pricing models and modelling jumpdiffusion 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.