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The behaviour ofday-ahead electricity
pricesAnalysis of spot electricity prices using
statistical, econometric, and econophysical
methods

• Zita MAROSSY
• Corvinus University of Budapest
• zita.marossy () uni-corvinus.hu
• Workshop on Deregulated European Energy Market
• Collegium Budapest
• September 24-25, 2009

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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

• European power exchanges
• Exchange Cournty
• European Energy Exhange Germany
• Powernext France
• APX Power NL Netherlands
• APX Power UK UK
• Energy Exchange Austria Austria
• Prague Energy Exchange Czech Republic
• Opcom Rumania
• Polish Power Exchange Poland
• Nord Pool Norway
• Borzen Slovenia
• Italian Power Exchange Italy
• OMEL Madrid Spain
• Belpex Belgium
• Source RMR Áramár Portál. (March 30, 2009)

• 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)

Futures markets
Day-ahead (spot) market
Balancing market
Futures markets
Day-ahead (spot) market
Balancing market
Source Geman 2005.
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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.
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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

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Modelling approaches

• Stochastic model calibration, time series
  analysis
• Find a suitable model, calibrate, use it for
  forecasting
• Fundamental models
• Driving factors of supply and demand are modelled
• Price behaviour is derived from market
  equilibrium
• Agent-based models
• Description of market players actions (e.g.
  simulation)
• Statistical models
• Directly investigate the distribution
• No prior knowledge about the driving factors
  market players behaviour is needed
• Artificial intelligence-based models
• E.g. neural networks, SVM
• Black box

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Stylized facts 1/7

• High prices (price spikes) in the time series
• The volatility is extremely high (Weron2006)
• T-note (lt0.5)
• 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,
  Mo2004).
• The price returns to the original level rapidly
  (Weron2006).
• Reason of spikes
• supply shocks (electricity can not be stored)
  (Escribano, Pena, Villaplana2002)
• bidding strategies (Simonsen, Weron, Mo2004)
• long-term trends in the market factors
  (occurrence can be forecasted) (Zhao, Dong, Li,
  Wong 2007)

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Stylized facts 2/7

• The time series exhibits seasonality.
• (Plot EEX data)
• Annual
• Plot 4-month MA-filtered data
• Weekly
• Daily
• Plot mean of hourly 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)

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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

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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, Przybylowicz 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

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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

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Jump diffusion

• m drift (usually mean reversion)
• s volatility
• qt jump (driven by e.g. a Poisson process)

• 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)

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Regime-switching models

• 2 regimes with different price dynamics
• Transition matrix probability of changing regime
• Weron2006 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

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Time series models

• ARMA, ARIMA
• SARIMA
• Seasonality ARIMA
• ARFIMA
• ARMAfractional 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

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Modelling price spikes

• Price spikes are very important in risk
  management
• Definition varies
• mean constant standard deviation
• Zhao, 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.

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Own research results

• Fractal feature
• Detailed analysis of the fractal feature of
  day-ahead electricity prices
• Distribution of power prices
• Extreme value theory discovers electricity
  price distribution
• Deterministic regime switching and filtering
• Deterministic regime-switching, spike
  behaviour, and seasonality filtering of
  electricity spot prices

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Own research results

• Fractal feature
• Detailed analysis of the fractal feature of
  day-ahead electricity prices
• Distribution of power prices
• Extreme value theory discovers electricity
  price distribution
• Deterministic regime switching and filtering
• Deterministic regime-switching, spike
  behaviour, and seasonality filtering of
  electricity spot 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) rgt0 are
  statistically indistinguishable
• The process scales at a rate of H
• 0ltHlt1
• For integrated processes (widely-used definition)
• H 0.5 the increments have no autocorrelation
  (e.g. Wiener-process)
• H gt 0.5 persistent (the increments have a
  positive autocorrelation)
• H lt 0.5 antipersistent (the increments have a
  negative autocorrelation)
• For stationary processes
• H 0.5 the process values have no
  autocorrelation (e.g. Gaussian white noise)
• H gt 0.5 persistent (the process values have a
  positive autocorrelation)
• H lt 0.5 antipersistent (the process values have
  a negative autocorrelation)

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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 )

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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 EEX EEX 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

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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

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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).

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Multifractality test

• Jiang, Zhou 2007
• H0 monofractal
• H1 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 lt 0.05 for 14 segments
• p lt 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

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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.
• Price spikes behave the same way regarding the
  correlations as prices at average level do.
• 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.

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Own research results

• Fractal feature
• Detailed analysis of the fractal feature of
  day-ahead electricity prices
• Distribution of power prices
• Extreme value theory discovers electricity
  price distribution
• Deterministic regime switching and filtering
• Deterministic regime-switching, spike
  behaviour, and seasonality filtering of
  electricity spot prices

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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

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Generalized extreme value (GEV) distribution

• 3 parameters
• scale (k)
• Fréchet (kgt0)
• Weibull (klt0)
• Gumbel (k0)
• location (m)
• scale (s)

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GEV (Fréchet) fits the empirical dataData EEX
daily prices

• pdf
• cdf

• Q-Q plot
• Estimates
• Statistical test

Parameter Estimate (EEX)
k 0,12
m 586,81
s 258,38
Chi-squared statistics p-value
APX 39,63 0,112
EEX 141,87 0,075
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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 8.0710-4, GEV 7.1810-4
• GEV is better at the tails of the distribution

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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

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Own research results

• Fractal feature
• Detailed analysis of the fractal feature of
  day-ahead electricity prices
• Distribution of power prices
• Extreme value theory discovers electricity
  price distribution
• Deterministic regime switching and filtering
• Deterministic regime-switching, spike
  behaviour, and seasonality filtering of
  electricity spot prices

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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

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Estimated GEV parameters

• For 168 segments of the time series
• Data EEX
• 2 regimes
• Different hours of week
• behave differently
• There are a few hours
• with fatter tails
• These are more sensitive
• to price spikes
• Deterministic regime switching
• Explains deterministic heteroscedasticity and
  changing spike intensity

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Changing distributions (EEX)

• Upper plot
• Vertical axes
• left mean of the hour
• right shape parameter k
• (dotted data)
• Lower plot
• Vetical axes
• left mean of the hour
• right regime (0 or 1 normal or risky)
• (data with marker)

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Deterministic regime switching model in risk
management

• Probability of exceeding a threshold tr (cdf)
• Data EEX
• Line theoretical probabilities.
• Dotted line empirical probabilities
  (frequencies).

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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 filterGEV filter

• Transformation
• x original price
• Fln-1 inverse of the lognormal cdf
• Fi GEV cdf for hour i
• y 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.

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Empirical results

• Figures periodogram of
• ACF (orig prices)
• ACF (filtered data)
• Intraweekly filtering
• successful

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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 -gt spike - gt seasonality)
• My result GEV filter
• Filters fat tails and seasonality at the same time

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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

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Thank you for your attention!

• Questions
• zita.marossy () uni-corvinus.hu