Periodic seasonal reg-ARFIMA-GARCH models for daily electricity spot prices

TLDR: In this article, periodic extensions of dynamic long-memory regression models with autoregressive conditional heteroscedastic errors are considered for the analysis of daily electricity spot prices, and the parameters of the model with mean and variance specifications are estimated simultaneously by the method of approximate maximum likelihood.

Abstract: Novel periodic extensions of dynamic long-memory regression models with autoregressive conditional heteroscedastic errors are considered for the analysis of daily electricity spot prices. The parameters of the model with mean and variance specifications are estimated simultaneously by the method of approximate maximum likelihood. The methods are implemented for time series of 1,200–4,400 daily price observations in four European power markets. Apart from persistence, heteroscedasticity, and extreme observations in prices, a novel empirical finding is the importance of day-of-the-week periodicity in the autocovariance function of electricity spot prices. In particular, the very persistent daily log prices from the Nord Pool power exchange of Norway are effectively modeled by our framework, which is also extended with explanatory variables to capture supply-and-demand effects. The daily log prices of the other three electricity markets—EEX in Germany, Powernext in France, and APX in The Netherlands—are less...

Figures

Figure 1: Daily spot prices for the Nord Pool

Figure 3: Log daily spot prices for three new European electricity markets

Figure 2: Weekly water reservoir levels and daily consumption for the Nord Pool

Table 3: ML estimates daily log-prices NordPool Jan 18, ’93-April 10, ’05, Deterministic xt.

Table 4: Effect Water Reservoir Levels Nord Pool Jan 18, ’93-April 10, ’05

Table 5: Effect Reservoir Levels and Power Consumption Nord Pool, Feb 26, ’01-Apr 10, ’05.

Full text

TI 2005-091/4
Tinbergen Institute Discussion Paper
Periodic Seasonal Reg-ARFIMA-
GARCH Models for Daily Electricity
Spot Prices
Siem Jan Koopman
a
Marius Ooms
a
M. Angeles Carnero
b
a
Department of Econometrics, Vrije Universiteit Amsterdam, and Tinbergen Institute;
b
Dpt. Fundamentos del Análisis Económico, Alicante.

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Periodic seasonal Reg-ARFIMA-GARCH
models for daily electricity spot prices
September 16, 2005
Siem Jan Koopman, Marius Ooms
Free University Amsterdam, Department of Econometrics,
De Boel elaan 1105, NL-1081 HV Amsterdam
s.j.koopman@feweb.vu.nl and mooms@feweb.vu.nl
M. Angeles Carnero
University of Alicante,
Dpt. Fundamentos del An´alisis Econ´omico,
03071 Alicante
acarnero@merlin.fae.ua.es
1

Periodic seasonal Reg-ARFIMA- GARCH models
for daily electricity spot pr ices
Siem Jan Koopma n, Marius Oom s and M. Angeles Carne ro
Abstract
Novel periodic extensions of dynamic long memory regression models with autoregressive
conditional heteroskedastic errors are considered for the analysis of daily electricity spot
prices. T he parameters of the model with mean and variance s pecifications are estimated
simultaneously by the method of approximate maximum likelihood. The methods are
implemented for time series of 1, 200 to 4, 400 daily price observations. Apart from persis-
tence, heteroskedasticity and extreme observations in prices, a novel empirical finding is
the importance of day-of-the-week periodicity in the autocovariance function of electricity
spot prices. In particular, daily log prices from the Nord Pool power exchange of Nor-
way are modeled effectively by our framework, which is also extended with explanatory
variables. For the daily log prices of three European emerging electricity markets (EEX
in Germany, Powernext in France, APX in The Netherlands), which are less pers istent,
periodicity is also highly significant.
Keywords: Au toregressive fractionally integrated moving average model; Generalised
autoregressive conditional heteroskedasticity model; Long memory process; Periodic
autoregressive model; Volatility.
1

1 Introduction
Electricity supply has been the responsibility of public-private companies in many OECD coun-
tries unt il recently. It is anticipated that the private trading of electricity will intensify further
in future and eventually move towards fully privatised electricity markets. In such markets large
volumes of electricity power will be traded for the short and long term together with future
contr acts and options. Although similarities with financial markets exist with resp ect to its
operations, the price formation at electricity markets is more complex since it strongly depends
on the short- t erm characteristics of the energy supply function. The instantaneous nature of
electricity and the availability of different plant technologies lead to atypical supply functions.
On the other hand, electricity demand functions typically depend on weather variables, seasons
in the year, day-of-week effects and holidays. These characteristics of electricity supply and
demand functions determine the specific behaviour of electricity prices that is encountered in
empirical wo r k. The dynamic behaviour of prices is important for derivative pricing and real
option analysis. Therefore, the empirical time series modeling of electricity prices is important
for financial traders and investors.
Following the standard practice of modeling volatility in financial returns, we are interested
in the conditional mean and variance of price innovations. For many efficient financial and
commodity markets, log prices are assumed to behave as a random walk and price innovations
are simply obtained by taking first differences of log prices. The mean process of electricity
log prices can not simply be described by a random walk because of its specific characteristics,
see Escribano, Pe˜na, and Villaplana (2002) and Bunn and Karakatsani (2003) for reviews of
the salient features of electricity prices. The following characteristics are often considered:
(i) Seasona l i ty in prices is due to the strong dependence of electricity demand on weather
conditions but also on social and economic activities leading to different holiday and seasonal
effects; (ii) Mean-reversion in electricity prices exists since weather is a dominant factor and
influences equilibrium prices thro ugh changes in demand; (iii) Jumps and spike s can be due to
the difficulty in storing large quantities of electricity so that supply and demand shocks cannot
easily be smoot hed out; (iv) Volatility clustering is regarded as a typical feature in financial
markets where heavy trading takes place on underlying assets.
The literature on modeling and analyzing electricity prices is growing quickly, see the col-
lection of articles in Bunn (2004) where different linear and nonlinear time series techniques are
adopted in empirical work. Particular contributions of interest in the literature are by Lucia
and Schwartz (2002) and Knittel and Roberts (2005) who argue f or a mean-reversion model
with deterministic seasonal mean functions and apply it to daily prices from the Nord Pool elec-
tricity power exchange and to Californian hourly electricity prices, respectively. Escribano et al.
(2002) f ocus on volatility aspects using generalized autoregressive conditional heteroskedastic-
ity (GARCH) models with possibly a jump-diffusion intensity parameter for daily spot prices
2

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Periodic seasonal reg-arfima- garch models for daily electricity spot prices" ?

In this paper, Bunn and Karakatsani proposed a model for daily electricity spot prices in the Nord Pool power exchange of Norway. 

Q2. What is the effect of the persistence in the GARCH-parameter?

The strong persistence in the volatility, measured as α̂1 + β̂1 being close to unity, has a profound influence on the estimation of the autoregressive parameters. 

Q3. What are the largest inverse roots of the characteristic polynomial of the AR component?

The largest inverse roots of the characteristic polynomial of the AR component are 0.47, 0.36 and 0.20 for EEX, Powernext and APX, respectively. 

Q4. What is the reason for the high number of zeros in the empirical histogram?

The high number of zeros in the empirical histogram stems from holiday variables that occur once and put associating residuals to zero. 

Q5. What is the importance of periodicity in the autocovariance function of electricity spot prices?

Apart from persistence, heteroskedasticity and extreme observations in prices, a novel empirical finding is the importance of day-of-the-week periodicity in the autocovariance function of electricity spot prices. 

Q6. What is the largest inverse root of the characteristic polynomial?

The largest inverse root of the characteristic polynomial equals 0.95, see Boswijk and Franses (1996) for unit root tests in periodic AR models. 

Q7. Why are the periodic autocorrelations not presented here?

Due to space considerations, other statistics are not presented here but the autocorrelations remain periodic when nonstationarities due to other day-of-the-week effects and yearly weather cycles have been removed from the data by regression or by seasonally differencing. 

Q8. What are the main contributions of interest in the literature on electricity prices?

Particular contributions of interest in the literature are by Lucia and Schwartz (2002) and Knittel and Roberts (2005) who argue for a mean-reversion model with deterministic seasonal mean functions and apply it to daily prices from the Nord Pool electricity power exchange and to Californian hourly electricity prices, respectively. 

Q9. What are the main factors that influence the demand of electricity?

On the other hand, electricity demand functions typically depend on weather variables, seasons in the year, day-of-week effects and holidays. 

Q10. How is the ARFIMA-GARCH model used in the context of electricity prices?

The inclusion of regression effects in the variance specification of a non-seasonal and non-periodic AR-GARCH model in the context of modeling electricity prices is considered by Byström (2005). 

Q11. What is the exact Gaussian loglikelihood function of the standard ARFIMA?

The exact Gaussian loglikelihood function of the standard ARFIMA model (1) with µt = µ, σ2t = σ 2 and s = 1 is given bylogL(y;ψ) = −T 2 log 2πσ2 − 1 2 log |Vy| − 1 2σ2 (y − µ)′V −1y (y − µ), (6)where the parameter vector ψ collects all unknown coefficients of the model (1) andy = (y1, . . . , yT ) ′, σ2Vy = V ar(y),with the variances and autocovariances in Vy for an ARFIMA process computed by efficient methods such as the ones developed by Sowell (1992) and Doornik and Ooms (2003), who also discuss efficient methods for the computation of logL(y;ψ) for the ARFIMA model using Durbin-Levinson methods for the necessary Choleski decomposition of Vy. 

Q12. What is the generalisation towards an ARFIMA model with s > 1?

The generalisation towards an ARFIMA model with s > 1, periodic coefficients and seasonal lags, can in principle be implemented for the evaluation of the loglikelihood function, However, the computation of Vy is intricate and not practical for large T as no analytical expressions for Vy exist in the case of a periodic seasonal ARFIMA model. 

Q13. What is the way to estimate the mean and variance parameters?

Given the persistent changes in volatility, simultaneous estimation of mean and variance parameters is preferred above two-step methods. 

Q14. What is the significance of periodicity in the log prices of three European emerging markets?

For the daily log prices of three European emerging electricity markets (EEX in Germany, Powernext in France, APX in The Netherlands), which are less persistent, periodicity is also highly significant. 

Q15. what is the GARCH likelihood function for t-disturbances?

The GARCH likelihood function for t-disturbances is given byℓ∗(ψ) = {T − max(p, r)} log c(ν) − 1 2T∑t=max p,r+1[ log dt(ν) + (ν + 1) log{1 + dt(ν)−1 ε2t} ] , (9)wherec(ν) = Γ( ν 2 + 1 2 )Γ ( ν 2 ) , dt(ν) = (ν − 2) σ2t , ν > 2,with j = j(t) for t = max(p, r)+ 1, . . . , T . 

Q16. How is the time series corrected for the mean and the autocovariance features?

The time series is corrected for the mean and the autocovariance features using the appropriate recursive filter for which the initial observations are treated as fixed and known. 

Q17. What are the relevant and closely watched variables for the hydropower market of Nord Pool?

The two most relevant and closely watched variables for the hydropower market of Nord Pool are daily data on Norwegian power consumption and weekly measurements of the overall water reservoir levels in Norway.