The behaviour ofday-ahead electricity pricesAnalysis of spot electricity prices using statistical, econometric, and econophysical methods
Zita MAROSSYCorvinus University of Budapestzita.marossy () uni-corvinus.hu
Workshop on Deregulated European Energy MarketCollegium BudapestSeptember 24-25, 2009
Zita Marossy: The behaviour of day-ahead electricity prices
3Workshop on Deregulated European Energy market
Topics covered Power exchanges, spot power prices „Stylized facts” of power price fluctuation Power price models
Time series models Distribution of spot prices
Own research results Detailed analysis of Hurst exponents Decomposition of multifractal feature of power prices Distribution of power prices: Fréchet distribution Deterministic regime switching model Intra-week seasonality filtering: GEV filter
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Power exchanges Actors:
– Power plants;– Power consumers;– Electricity trading companies.
Products:– Power supplied during a given time period
Organized markets Markets:
– Futures markets– Day-ahead (spot) markets– Balancing markets
Power price: P(t,T)
European power exchanges
Exchange CourntyEuropean Energy Exhange GermanyPowernext FranceAPX Power NL NetherlandsAPX Power UK UKEnergy Exchange Austria AustriaPrague Energy Exchange Czech
RepublicOpcom RumaniaPolish Power Exchange PolandNord Pool NorwayBorzen SloveniaItalian Power Exchange ItalyOMEL Madrid SpainBelpex Belgium
Source: RMR Áramár Portál. (March 30, 2009)
tHatáridős piac
Day-ahead (spot) piac
Kiegyenlítőpiac
T tHatáridős piac
Day-ahead (spot) piac
Kiegyenlítőpiac
T
Tt Day-ahead (spot) piac
Kiegyenlítőpiac
Határidős piac
Tt Day-ahead (spot) piac
Kiegyenlítőpiac
Határidős piac
Source: Geman [2005].
Futures markets
Futures markets
Day-ahead (spot) market
Day-ahead (spot) market
Balancing market
Balancing market
Zita Marossy: The behaviour of day-ahead electricity prices
5Workshop on Deregulated European Energy market
Market prices
Double auction for each hour of the next day Market price:
– Aggregated demand– Aggregated supply– Market clearing price– Transmission congestions:
• Nodal/Zonal prices
Source: Rules for the Operation of the Electricity Market, Borzen [2003].
Zita Marossy: The behaviour of day-ahead electricity prices
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Motivation for power price modelling
Future power prices are risky Power price forecasts help to
– determine the timing of buying/selling of power products– work out bidding strategies– price derivative products– manage risks
Therefore: the distribution of future prices are in the center of attention
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Spot time series
Hourly day-ahead prices One price for each hour Data:
– EEX hourly prices from June 16, 2000 to April 19, 2007
Time series of different products (apples & oranges)– Electricity can not be stored at reasonable cost
Stable correlation structure: existence of a data generating process
Daily prices: sum of 24 hourly prices for the given day („Phelix”: avg) Returns: hourly „log return”
0 1 2 3 4 5 6
x 104
0
500
1000
1500
2000
2500
Óra
Ár
(EU
R)
EEX árak
Zita Marossy: The behaviour of day-ahead electricity prices
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Modelling approaches
1. Stochastic model calibration, time series analysis Find a suitable model, calibrate, use it for forecasting
2. Fundamental models Driving factors of supply and demand are modelled Price behaviour is derived from market equilibrium
3. Agent-based models Description of market players’ actions (e.g. simulation)
4. Statistical models Directly investigate the distribution No prior knowledge about the driving factors & market players’ behaviour is needed
5. Artificial intelligence-based models E.g. neural networks, SVM Black box
Zita Marossy: The behaviour of day-ahead electricity prices
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Stylized facts 1/7 High prices (price spikes) in the time series
The volatility is extremely high (Weron[2006]):– T-note (<0.5%)– Equity (1-1.5%, risky: 4%)– Commodities (1.5-4%)– Electricity (50%)
The intensity of spikes changes in time, and it is higher in peak hours (Simonsen, Weron, Mo[2004]).
The price returns to the original level rapidly (Weron[2006]). Reason of spikes:
– „supply shocks” (electricity can not be stored) (Escribano, Pena, Villaplana[2002])– bidding strategies (Simonsen, Weron, Mo[2004])– long-term trends in the market factors (occurrence can be forecasted) (Zhao, Dong,
Li, Wong [2007])
Zita Marossy: The behaviour of day-ahead electricity prices
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Stylized facts 2/7
The time series exhibits seasonality.(Plot: EEX data)1. Annual
– Plot: 4-month MA-filtered data
2. Weekly3. Daily
– Plot: mean of hourly prices
0 500 1000 1500 2000 250012
14
16
18
20
22
24
26
28
Nap
Ár
(4 h
ónap
os m
ozgó
átla
g)
0 20 40 60 80 100 120 140 160 1800
10
20
30
40
50
60
70
Óra
Ár
Zita Marossy: The behaviour of day-ahead electricity prices
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Stylized facts 3/7
Stable autocorrelation structure with high autocorrelations
(Plot: EEX data) High autocorrelation
coefficients Slowly decreasing
autocorrelation function(persistency)
Periodicity (seasonality)
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Stylized facts 4/7 Volatility changes in time: heteroscedasticity
Hectic and calm periods GARCH-type models
– High shocks cause high volatility in the next period
– Volatility clustering– Stochastic (autoregressive) conditional
volatility
My arguments for deterministic conditional volatility:
– Volatility shows seasonal patters: it is higher in peak hours (Weron [2000]).
– Plot: Weron [2000] reproduced; hourly mean absolute percentage change (EEX data)
0 20 40 60 80 100 120 140 1600
5
10
15
20
25
30
35
40EEX
Óra
Átla
gos
absz
olút
vál
tozá
s
0 500 1000 1500 2000 2500-500
0
500Innovations
Inno
vatio
n
0 500 1000 1500 2000 25000
100
200Conditional Standard Deviations
Sta
ndar
d D
evia
tion
0 500 1000 1500 2000 25000
200
400Returns
Ret
urn
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Stylized facts 5/7
Price distributions have fat tails.
Heavier tails and higher kurtosis than Gaussian
– Plot: Q-Q plot of log EEX price versus Gaussian distribution
– Plot: histogram of EEX daily prices
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-2
0
2
4
6
8
Standard Normal Quantiles
Qua
ntile
s of
Inp
ut S
ampl
e
QQ Plot of Sample Data versus Standard Normal
0
200
400
600
800
1000
0 1000 2000 3000 4000 5000 6000 7000
Series: PR_D_EEXSample 6/16/2000 4/19/2007Observations 2499
Mean 773.0419Median 687.5300Maximum 7237.000Minimum 74.81000Std. Dev. 442.0329Skewness 3.951179Kurtosis 38.24476
Jarque-Bera 135845.7Probability 0.000000
Zita Marossy: The behaviour of day-ahead electricity prices
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Stylized facts 6/7 No consensus whether the price process has a unit root.
Eydeland, Wolyniec [2003]: Dickey-Fuller test (no unit root) Atkins, Chen [2002]: ADF (no unit root), KPSS (existence of u.r.)
Bosco, Parisio, Pelagatti, Baldi [2007]: traditional testing procedures can not be used
– additive outliers,– fat tails,– heteroscedasticity,– seasonality
Even robust tests disagree:– Escribano, Peña, Villaplana [2002] : no unit root (on outlier-filtered data)– Parisio, Pelagatti, Baldi [2007]: existence of unit root (weekly median prices)
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Stylized facts 7/7 Some authors argue that power prices are anti-persistent and mean reverting;
meanwhile others state that the price time series has long memory.Method: Hurst exponent (H) Mean reversion
– Weron, Przybyłowicz [2000] , Eydeland, Wolyniec [2003], Weron [2006], Norouzzadeh et al. [2007], Erzgräber et al. [2008],
Long memory– Carnero, Koopman, Ooms [2003] , Sapio [2004], Serletis, Andreadis [2004], Haldrup,
Nielsen [2006]
Large price changes behave differently: multifractality.Method: generalized Hurst exponent Multifractal property
– Resta [2004] , Norouzzadeh et al. [2007], Erzgräber et al. [2008] Monofractal property
– Serletis, Andreadis [2004] – other methodology
Zita Marossy: The behaviour of day-ahead electricity prices
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Reduced-form models
Geometric Brownian motion (GBM) GBM with mean reversion Stochastic volatility models:
– Constant Elasticity of Volatility (CEV)– Local volatility models– Hull-White model– Heston model
Jump diffusion Markov regime switching models
ydWydtdy dWdtydy
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Jump diffusion
Empirical findings: High mean reversion rate
– Positive jumps followed by a negative jump (Weron, Simonsen, Wilman [2004])– Mean reversion rate depends on jump size (Weron, Bierbrauer, Trück [2004])– „Regime jump model”: 3 regimes: normal, jump, return (Huisman, Mahieu
[2001])– „Signed jump model”: sign of a jump depends on the price (Geman, Roncoroni
[2006]) The intensity of jumps changes
– Intensity depends on the price (Eydeland, Geman [1999])– Non-homogeneous Poisson process with time-dependent jump intensity (Weron
[2008b])
: drift (usually: mean reversion): volatilityqt: jump (driven by e.g. a Poisson process)
),(),(),( tydqdWtydttydy ttttt
Zita Marossy: The behaviour of day-ahead electricity prices
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Regime-switching models 2 regimes with different price dynamics
Transition matrix: probability of changing regime
Weron[2006]: RS models do not outperform JD models with log prices Weron [2008a]: RS model provides better results than JD models with prices De Jong [2006] compares RS and JD models. Best fit: 2-state RS model. Haldrup, Nielsen [2006]: ARFIMA and RS models have similar forecasting power
UUU0
LLL
dWdt
dWdt
S
dS
2222
1111
1
1
qqQ
Zita Marossy: The behaviour of day-ahead electricity prices
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Time series models ARMA, ARIMA
– SARIMA• Seasonality + ARIMA
– ARFIMA• ARMA+fractional integration
– TAR (threshold AR)• Different price dynamics under and
above threshold– PAR (periodic AR)
• AR coefficients are different for each hour
GARCH– Stochastic volatility
Regime switching models– Different time series models in the
regimes
Exogenous variables:– (Forecasted) consumption– Seasonality variables– Weather– Coal, gas… prices– Capacities– …
Empirical findings:– Good fit for fractional models– RS models provide poor forecasting
performance
Zita Marossy: The behaviour of day-ahead electricity prices
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Modelling price spikes Price spikes are very important in risk management Definition varies:
mean + constant * standard deviationZhao, Dong, Li, Wong [2007]: constant depends on market, season, and time
Filtering:– „similar day”: mean of the hour– „limit”: threshold (T)– „damped”: T + Tlog10(P/T)
Adding to the model: jump diffusion, regime-switching models
Separate spike forecasting models– Zhao, Dong, Li, Wong [2007]:
„„An effective method of predicting the occurrence of spikes has not yet been observed in the literature so far.”
Zita Marossy: The behaviour of day-ahead electricity prices
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Own research results
I. Fractal feature„Detailed analysis of the fractal feature of day-ahead electricity prices”
II. Distribution of power prices„Extreme value theory discovers electricity price distribution”
III. Deterministic regime switching and filtering„Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”
Zita Marossy: The behaviour of day-ahead electricity prices
22Workshop on Deregulated European Energy market
Own research results
I. Fractal feature„Detailed analysis of the fractal feature of day-ahead electricity prices”
II. Distribution of power prices„Extreme value theory discovers electricity price distribution”
III. Deterministic regime switching and filtering„Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”
Zita Marossy: The behaviour of day-ahead electricity prices
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Persistency: Hurst exponent (H) H:
– A measure for self similar (self affine) processes– The increments b(t0,t) and r-Hb(t0,rt) r>0 are statistically indistinguishable– The process scales at a rate of H
0<H<1 1. For integrated processes (widely-used definition)
H = 0.5 the increments have no autocorrelation (e.g. Wiener-process)H > 0.5 persistent (the increments have a positive autocorrelation)H < 0.5 antipersistent (the increments have a negative autocorrelation)
2. For stationary processesH = 0.5 the process values have no autocorrelation (e.g. Gaussian white noise)H > 0.5 persistent (the process values have a positive autocorrelation)H < 0.5 antipersistent (the process values have a negative autocorrelation)
Zita Marossy: The behaviour of day-ahead electricity prices
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Persistency – example
fractional Wiener process (fractional Brownian motion)values and increments
(H = 0.25, 0.4, 0.5, 0.6, 0.75 )
Zita Marossy: The behaviour of day-ahead electricity prices
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Estimates on H in the case of EEX
Power prices have an H of 0.8-0.9 (1). Parentheses: „multiscaling” H = 1: pink noise
EEX
Price Log price Log return Price difference
R/S 0.88 0.77 0.26(0.77)* 0.30(0.71)*
Aggregated Variance 0.86 0.88 -0.03 -0.03
Differenced Variance 0.79 0.70 0.11 -0.02
Periodogram regression 0.83 1.06 0.22 -0.08
AWC 0.85 0.94 0.11 0.05
DFA2 0.84 0.87 0.06 0.08
hmod(2) 0.83 0.86 0.06 0.08
Zita Marossy: The behaviour of day-ahead electricity prices
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Multiscaling? MF-DFA(2) Data: EEX Tangents:0.76 ;0.11 ;0.03 Cut-off points:ln(44.7) ≈ 3.8
R/S method:101.5 ≈ 58
The cut-off point is difficult to explain
The log return (and the price increment) is not a self affine process
1 2 3 4 5 6 7 8 9 10 112
3
4
5
6
7
8
9
10
log(s)
log(
F(s
))
DFA2 (EEX ár)
1 2 3 4 5 6 7 8 9 10 11-3
-2
-1
0
1
2
3
4
5
6
7
log(s)
log(
F(s
))
DFA2 (EEX log ár)
1 2 3 4 5 6 7 8 9 10 11-2.5
-2
-1.5
-1
-0.5
0
log(m)
log(
F(s
))
DFA2 (EEX loghozam)
1 2 3 4 5 6 7 8 9 10 112
3
4
5
6
7
8
9
10
11
log(m)
log(
F(s
))
DFA2 (EEX szûrt ár)
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Multifractal feature Generalized Hurst exponent: h(q)
– Low q: persistency for small shocks– High q: persistency for large shocks
Sources of multifractality:– Fat tails– Correlations
Shuffling the time series helps to separate the two effects
Modified h(q).
-20 -15 -10 -5 0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
q
Hur
st
MFDFA (EEX ár)
h(q)h
mod(q)
hshuff led
(q)
-20 -15 -10 -5 0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6MFDFA (NordPool ár)
qH
urst
h(q)
hmod
(q)
hshuff led
(q)
5.0mod qhqhqh shuffled
Zita Marossy: The behaviour of day-ahead electricity prices
28Workshop on Deregulated European Energy market
Multifractality test Jiang, Zhou [2007]
H0: monofractalH1: multifractal
EEX: p = 0.36 monofractal NordPool: p = 0.00 multifractal
NordPool:h(q) for each hour of the week
– Upper plot: original h(q)s– Lower plot: modified h(q)s– p < 0.05 for 14 segments– p < 0.01 for 4 segments
The process is monofractal if the segments are separated. The different hours have different distributions:
– The distributions are mixed in the whole time series
-10 -5 0 5 100
0.5
1
1.5
2
q
h(q)
-10 -5 0 5 100
0.5
1
1.5
2
q
h mod
(q)
Zita Marossy: The behaviour of day-ahead electricity prices
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„There are no spikes”
The separate statistical modelling of price spikes is impossible as price spikes can not be distinguished in the price process.
a. Price spikes behave the same way regarding the correlations as prices at average level do.
b. Price spikes are high realizations of a fat tailed distribution. They constitute no separate regime, and they are not “outlier” from the price process. Giving them a separate name causes confusion in modelling.
Zita Marossy: The behaviour of day-ahead electricity prices
30Workshop on Deregulated European Energy market
Own research results
I. Fractal feature„Detailed analysis of the fractal feature of day-ahead electricity prices”
II. Distribution of power prices„Extreme value theory discovers electricity price distribution”
III. Deterministic regime switching and filtering„Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”
Zita Marossy: The behaviour of day-ahead electricity prices
31Workshop on Deregulated European Energy market
Distribuition of power prices
Weron [2006]:– Alfa-stable
– Hyperbolic distribution
– NIG (normal inverse Gaussian)
– Tests: on MA-filtered prices– Best fit (price difference, log prices): alfa-stable distribution
1
21
12
1
)( ln
zz
zizzi
tgz
zizzi
z
xx
H eK
xf22
)(2 221
22
22
221 )(22
x
xKexf x
NIG
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Generalized extreme value (GEV) distribution
3 parameters:– scale (k)
• Fréchet (k>0)• Weibull (k<0)• Gumbel (k=0)
– location ()– scale ()
k
k
xkxF
/1
,, 1exp
GEV pdfs
0
0,2
0,4
0,6
0,8
1
1,2
-5,0
0
-4,3
7
-3,7
4
-3,1
1
-2,4
8
-1,8
5
-1,2
2
-0,5
9
0,04
0,67
1,30
1,93
2,56
3,19
3,82
4,45
Fréchet
Gumbel
Weibull
0/1 xk
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GEV (Fréchet) fits the empirical dataData: EEX daily prices
cdf
Q-Q plot
Estimates
Statistical test
0 1000 2000 3000 4000 5000 6000 70000
0.5
1
1.5x 10
-3
Data
Den
sity
EEX (daily)
GEV (EEX)
0 1000 2000 3000 4000 5000 6000 70000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data
Cum
ulat
ive
prob
abili
ty
EEX (daily)
GEV (EEX)
Parameter Estimate (EEX)
k 0,12
586,81
258,38
Chi-squared statistics
p-value
APX 39,63 0,112
EEX 141,87 0,075
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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GEV provides better fit than LévyData: EEX daily prices. Marossy, Szenes [2008]
Difference in empirical and estimated cdfs
See Kolmogorov-Smirnov statistic
KS statistic:– Lévy: 0.0141, GEV: 0.0262, critical value: 0.068
Mean of the differences:– Lévy: 8.07*10-4, GEV: 7.18*10-4
GEV is better at the tails of the distribution
empFFD sup
0 100 200 300 400 500 600-0.03
-0.02
-0.01
0
0.01
0.02
0.03
price (EUR)
Fem
p-F
Differences in CDF
Lévy
GEV
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A theoretical model
Explaining why power prices have GEV distribution Background: extreme value theory
– Fisher-Tippett Theorem
Reason for Fréchet:– The price has to be an exponential function of the
quantity on the market supply curve– Empirical „supply stack”: exponential
Zita Marossy: The behaviour of day-ahead electricity prices
36Workshop on Deregulated European Energy market
Own research results
I. Fractal feature„Detailed analysis of the fractal feature of day-ahead electricity prices”
II. Distribution of power prices„Extreme value theory discovers electricity price distribution”
III. Deterministic regime switching and filtering„Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”
Zita Marossy: The behaviour of day-ahead electricity prices
37Workshop on Deregulated European Energy market
Distributions changing their shapes
The time series is divided into 168 segments The distributions differ not only in means but in shapes Plot: EEX data
0 10 20 30 40 50 60 700
50
100
150
200
250
300
350
400
450
500
mean
99th
per
cent
ile
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Estimated GEV parameters
For 168 segments of the time series Data: EEX 2 regimes:
– Different hours of weekbehave differently
– There are a few hourswith fatter tails
– These are more sensitiveto price spikes
Deterministic regime switchingExplains deterministic heteroscedasticity and changing spike
intensity
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1612
13
14
15
16
17
18
19
k
mu
GEV parameters
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39Workshop on Deregulated European Energy market
Changing distributions (EEX)
Upper plotVertical axes
left: mean of the hour;right: shape parameter k(dotted data)
Lower plotVetical axes
left: mean of the hour;right: regime (0 or 1 – normal or risky)(data with marker)
0 20 40 60 80 100 120 140 160 18020
25
30
35
óra0 20 40 60 80 100 120 140 160 180
0
0.05
0.1
0.15
0 20 40 60 80 100 120 140 160 18023
24
25
26
27
28
29
30
31
32
33
0 20 40 60 80 100 120 140 160 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
óra
Zita Marossy: The behaviour of day-ahead electricity prices
40Workshop on Deregulated European Energy market
Deterministic regime switching model in risk management Probability of exceeding a threshold tr (=cdf) Data: EEX
Line: theoretical probabilities.Dotted line: empirical probabilities (frequencies).
0 20 40 60 80 100 120 140 160 1800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
óra
p
tr=110
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Seasonality filtering (intra-week) Methods (Weron [2006]):
– Differencing (alters the correlation structure)– Median or average week (negative values)– Moving average– Spectral decomposition– Wavelet decomposition
Approaches:
Data = periodic component + stochastic part
Assume that distributions differ only in means.This is not true for the power prices.
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Suggested filter‘GEV filter’
Transformation:x original priceFln
-1 inverse of the lognormal cdfFi GEV cdf for hour iy filtered price
Properties:– If the prices have a GEV distribution, filtered prices have lognormal distribution– The transformation is always well-defined.– Risky distributions: heavy tails disappear (outlier filtering)– Time series models can be applied to filtered (log) prices– An inverse filter is defined accordingly.– Separate time series modelling and (outlier, seasonality, heteroscedasticity) filtering.
0 500 1000 1500 2000 2500 30000
0.5
1
1.5x 10
-3
x
a
GEV
lognormal
0 500 1000 1500 2000 2500 30000
0.5
1
1.5x 10
-3
x
b
GEV
lognormal
GEV with high expected value
2500 2550 2600 2650 2700 2750 2800 2850 2900 2950 30002
3
4
5
6
7
8
9
10x 10
-6
x
GEV
lognormal
xFFy i1
ln
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43Workshop on Deregulated European Energy market
Empirical results
Figures: periodogram of– ACF (orig prices)– ACF (filtered data)
Intraweekly filtering– successful
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Normalized Frequency ( rad/sample)
Pow
er/f
requ
ency
(dB
/rad
/sam
ple)
Power Spectral Density Estimate via Periodogram
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70
-60
-50
-40
-30
-20
-10
0
10
20
Normalized Frequency ( rad/sample)
Pow
er/f
requ
ency
(dB
/rad
/sam
ple)
Power Spectral Density Estimate via Periodogram
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44Workshop on Deregulated European Energy market
Price spikes and seasonality
Trück, Weron, Wolff [2007]– Price spikes influence the calculations during seasonality filtering.– With seasonality present, spikes are difficult to identify– The two filtering procedures are interconnected– Suggestion: iterative procedure
(seasonality -> spike - > seasonality)
My result: GEV filter– Filters fat tails and seasonality at the same time
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45Workshop on Deregulated European Energy market
Conclusions
Prices have long memory Price spikes constitute no separate regime
(monofractal property) Price spikes are high realizations of GEV (Fréchet)
distribution Deterministic regime switching causes time-
dependent jump intensity, heteroscedasticity and seasonality
It can be removed by the GEV filter
Zita Marossy: The behaviour of day-ahead electricity prices
46Workshop on Deregulated European Energy market
Thank you for your attention!
Questions zita.marossy () uni-corvinus.hu