Modeling Spot Price Dependence in Australian Electricity Markets with Applications to Risk Management

Transcription

1 Modeling Spot Price Dependence in Australian Electricity Markets with Applications to Risk Management This version: 22 November 2011 Katja Ignatieva a, Stefan Trück b a University of New South Wales, Sydney, Australia b Macquarie University Sydney, Australia Abstract We examine the dependence structure of electricity spot prices across regional markets in the Australian National Electricity Market (NEM). Our analysis is based on a GARCH approach to model the marginal price series in the considered regions in combination with copulae to capture the dependence structure between the different markets. We apply different copula models including Archimedean, elliptical and copula mixture models. We find a positive dependence structure between the prices for all considered markets, while the strongest dependence is usually exhibited between markets that are connected via interconnector transmission lines. Regarding the nature of dependence, among the considered Archimedean and elliptical copulae, the Student-t copula provides the best fit. On the other hand, the overall best results are obtained using mixture models due to their ability of also capturing asymmetric dependence in the tails of the distribution. We find significant tail dependence between Australian wholesale electricity prices, indicating that especially extreme observations like price spikes may happen jointly across regional markets. Examining the Value-at-Risk of stylized portfolios constructed from electricity spot contracts in different regional markets, we find that the Student-t and mixture copula models outperform the Gaussian copula in a backtesting study. Our results are important for risk management and hedging decisions of market participants, in particular for those operating in several regional markets simultaneously. Key words: Electricity markets, copula, dependence modeling, volatility Corresponding author. Address: Risk and Actuarial, Australian School of Business UNSW, Sydney NSW 2052 Australia. address: k.ignatieva@unsw.edu.au (Katja Ignatieva).

2 1 Introduction This paper examines the dependence structure of spot electricity prices across regional markets in the Australian National Electricity Market (NEM). The market operates as an interconnected grid comprising several regional networks providing supply of electricity to retailers and end-users. The NEM includes the states of New South Wales (NSW), Queensland (QLD), South Australia (SA), Victoria (VIC) and the Australian Capital Territory (ACT), while Tasmania (TAS) is connected to the network via an undersea inter-connector to Victoria. Within the national power grid, electricity can be transmitted between different regions via so-called interconnectors. The interconnectors may be of particular importance when the price of electricity in adjoining regions is low enough to displace local supply, but also when the energy demand in a particular region is higher than the amount of electricity that can be provided by local generators. Interestingly, as pointed out by Higgs (2009), the networks for each state are still usually characterized by a small number of market participants while sizeable differences in electricity prices can be observed. One of the major objectives in establishing a national market was to provide an efficient and integrated electricity market. However, the limitations of physical transfer capacity as well as congested interconnectors lead to sometimes significantly different price behavior in the considered regional markets. On the other hand, there have been various occasions when extreme price observations or price spikes happened jointly in several markets. Therefore, market participants are particularly interested in analyzing interconnections and dependencies between wholesale electricity prices in the considered regional markets. Our study is aimed to give a better understanding of the price dynamics in regional electricity spot markets. Hereby, we focus in particular on the dependence between regional prices and conduct a pioneer study on the use of copulae for capturing this dependence structure. Our study yields important insights with respect to joint price movements, spillover effects, extreme price outcomes and the impact of interconnection within the Australian electricity market. We complement and extend existing work by Worthington et al. (2005) and Higgs (2009) on Australian electricity markets by providing a deeper analysis of the actual dependence structure between observed spot prices. Our results are also of importance for the development of risk management and hedging strategies for market participants, in particular for those operating simultaneously in several of the considered regional markets. The study may also shed light into the Australian electricity market with respect to developing guidelines for market mechanisms or the construction of new interconnectors. So far only a limited number of studies, e.g., De Vany and Walls (1999); Worthington et al. (2005); Park et al. (2006); Haldrup and Nielsen (2006); Micola and Bunn (2007); Bollino and Polinori (2008); Dempster et al. (2008); Zachmann (2008); Le Pen (2008); Higgs (2009) have concentrated on the dependence or a multivariate analysis of different regional electricity markets. De Vany and Walls (1999) were the first to study the joint behavior of electricity spot prices in different U.S. markets. They find cointegration and some evidence for a pattern of nearly uniform prices despite a rather inefficient transmission network between the considered regional markets. Worthington et al. (2005) employ a multivariate GARCH model to investigate price and volatility spillovers in Australian electricity markets. This analysis is further extended by Higgs (2009) applying a family of constant and dynamic conditional correlation MGARCH models to examine the effects of cross-correlation volatility spillovers between regional Australian electricity markets. The author finds that the highest conditional correlations are evident between the well-connected markets indicating the presence of strong interdependence between these markets with weaker interdependence between the least interconnected markets. Park et al. (2006) investigate the connection between various U.S. spot markets using vector autoregression and so-called directed acyclic graph methods. The authors find that relationships among markets are not only dependent on transmission lines between regional markets but also affected by similar and dissimilar 2

3 institutional arrangements. Haldrup and Nielsen (2006) use Markov regime switching models in combination with long memory processes to model the dependence between pairs of regional electricity prices in the NordPool market. Micola and Bunn (2007) characterize the relationship between two network based oligopoly markets when local players share the interconnection s ownership on a natural gas pipeline. They suggest that the relationship between local price differentials and capacity utilization is increasing and convex. Bollino and Polinori (2008), focusing on different electricity markets in Italy, apply a model for contagion and identify significant contagion effects separately from interdependence of the markets. Applying Granger causality tests and cointegration analysis to Californian electricity markets, Dempster et al. (2008) find only a moderate degree of integration among these markets. Zachmann (2008) examines to which extent European electricity wholesale day-ahead prices converge towards arbitrage freeness. Using a Kalman filter to analyse Dutch-German and Danish-German cross-border capacity auction prices the author shows the absence of arbitrage opportunities between these markets when congestion costs are taken into account. Finally, Le Pen (2008) estimate a VAR-BEKK model and find evidence of return and volatility spillovers between the German, the Dutch and the British forward electricity market. Furthermore, investigating the impact of shocks on expected conditional volatility, the author suggests that a shock has a high positive impact only if its size is large compared to the current level of volatility. While a number of authors have examined the connection between different regional electricity markets, as mentioned above, none of these studies has focused on the use of copulae. Therefore, in our analysis, we aim to find a model which appropriately describes the price behavior of each of the regional electricity markets, followed by modeling the dependence structure among the markets using a multivariate copulae approach. Copulae allow to separate the study of univariate marginals, the regional electricity markets, from the study of a dependence structure. Using copulae is motivated by its ability to measure non-linear dependence between price series which arise for example when we deal with models which do not support the normality assumption. To our best knowledge so far no study in the literature has applied copulae to model the possibly non-linear dependence structure between different regional electricity markets. On the other hand, copulae have been extensively used in other financial markets when modeling the dependence between various assets in a portfolio, FX rates, or studying the dependencies between international stock markets, see e.g. Breymann et al. (2003); Cherubini et al. (2004); McNeil et al. (2005); Ignatieva and Platen (2010). Our study concentrates on Australian electricity markets that differ from other countries and continents in a sense that the market operates as a nationally interconnected grid, providing strong linkage between the regional markets. For the purpose of our analysis we, therefore, concentrate on five regions: NSW, QLD, SA, TAS and VIC. Four of these regions (except TAS) are the major regional markets in Australia that have also been considered in the previous literature, see e.g. Worthington et al. (2005); Higgs (2009). When dealing with electricity markets, one should take into account certain characteristics and stylized facts of the data. In particular, electricity is a non-storable commodity and spot prices generally exhibit mean reversion, seasonality, changes in volatility and price spikes. Therefore, prior to modeling the distribution of the prices, we need to employ an appropriate model to capture these characteristics. One could choose e.g. wavelets or linear step functions to remove seasonalities from the data, see Weron (2006), or employ e.g. GARCH type, regime-switching or jump-diffusion models to account for spikes and mean reversion, see Higgs and Worthington (2005); Geman and Roncoroni (2006); Bierbrauer et al. (2007); Janczura and Weron (2010). Further, similar to stock prices, electricity prices experience heavy tails and excess kurtosis which cannot be captured by the normal distribution. Therefore, some alternative distributions have to be investigated. In our analysis we also consider the class of Symmetric Generalized Hyperbolic distributions that are discussed e.g. in Platen and Rendek (2008); Wenbo and Kercheval (2008). Note that this class of distribu- 3

4 Maximal Dependence (NSW,QLD) on 31/07/2006(11:30) Minimal Dependence (NSW,QLD) on 30/04/2007(16:00) Fig. 1. Deseasonalized standardized log-prices for NSW and QLD electricity markets for July 2006 (left panel) and April 2007 (right panel). The left panel shows 1440 half-an-hourly observation corresponding to one month of data prior to 31/07/2006 when there is strong dependence between observed prices. The right panel shows 1440 half-an-hourly observation prior to 30/04/2007 when the dependence between prices is rather low. tions includes, for example, the Student-t, the Normal Inverse Gaussian (NIG) and the Hyperbolic (HYP) distribution. After capturing the marginal distributions for each regional market, we study the dependence between the markets using multivariate copulae. The usage of combining a model with time-varying volatility with a copula approach is also motivated by the fact that the dependence between regional electricity markets is not constant but may vary over time. To further motivate our point consider Figure 1 that provides a scatter-plot of half-hourly deseasonalized standardized log-prices for NSW and QLD electricity markets. The left panel shows one month of data corresponding to 1440 half-hourly observations in July 2006, whereas the right panel scatter-plots 1440 half-hourly observations in April One observes that the dependence between the NSW and the QLD electricity markets is not constant but may significantly vary through time. In addition, Figure 2 compares all five electricity markets in Australia (NSW, QLD, SA, TAS and VIC) on two different days. The left panel shows 48 half-hourly deseasonalized and standardized log-prices on 20/07/2006 where all five markets behave similarly over the day experiencing spikes at around 18:00-18:30. The right panel shows 48 half-hourly observations on 07/12/2009 when only the NSW electricity market experiences major spikes in the afternoon which are not present in the SA, VIC or TAS markets, while QLD shows only minor spikes. Copulae also allow to capture a time-varying dependence structure as described above in a more effective way. Therefore, we study the dependence between electricity markets in Australia in a static, as well as a time-varying setting. We consider different copula models which include the Gaussian, the Student-t and some mixture copulae, aiming to identify the best performing copula family. The remainder of the paper is organized as follows. Section 2 briefly reviews stylized facts of electricity markets, while Section 3 provides a brief overview on copula models. Section 4 presents the methods applied when estimating the marginals for each of the regional markets. Empirical results for the estimation of the marginals and the dependence structure between the markets are provided in Section 5. The following Section 6 describes results of a risk management application where several of the models were backtested in an out-of-sample forecasting study. Finally, Section 7 concludes. 4

5 Deseasonalized Log Price on 20/07/2006 Deseasonalized Log Price on 07/12/ NSW QLD SA TAS VIC NSW QLD SA TAS VIC Fig. 2. Deseasonalized standardized log-prices for NSW, QLD, SA, TAS and VIC on 20/07/2006 (left panel) and on 07/12/2009 (right panel). 2 Stylized Facts of Electricity Markets When modeling electricity spot prices, we have to take into account several stylized facts of these specific markets. Electricity is a non-storable good which causes demand and supply to be balanced on a knife-edge. Relatively small changes in load or generation can cause large changes in the spot price in a very short period of time what leads to electricity prices being far more volatile than other commodities. Next to extreme volatility, the strong relationship between demand and price also leads to typical features of electricity prices such as mean reversion, seasonality and short-lasting but often extreme price spikes. This section aims to summarize the specific characteristics of spot electricity markets. In the following we will provide a brief overview of the structure of the Australian National Electricity Market (NEM) as well as stylized facts of electricity markets in general. 2.1 The Australian electricity market Since the 1990s the Australian electricity market has experienced significant changes. Prior to 1997 the market consisted of vertically integrated businesses operating in each of the states and there was no connection between the individual states. The businesses were owned by the government and used to operate as monopolies. Overall, there were twenty-five electricity distributors being protected by the government from competition. To promote energy efficiency and reduce the costs of electricity production, in the late 1990s the Australian government commenced a structural reform which, among others, had the following objectives: the separation of transmission from electricity generation, the merge of twenty-five electricity distributors into a smaller number of distributors, and the functional separation of electricity distribution from the retail supply of electricity. Also retail competition was introduced through the electricity reform: state s electricity purchases could be made through a competitive retail market and customers were free to choose their retail supplier. As a wholesale market the National Electricity Market (NEM) in Australia began operating in December It is now an interconnected grid comprising several regional 5

6 networks which provide supply of electricity to retailers and end-users. The NEM includes the states of QLD, NSW, VIC, SA and the ACT. TAS is connected to the other NEM regions via an undersea inter-connector to VIC. The link between electricity producers and electricity consumers is established through a pool which is used to aggregate the output from all generators in order to meet the forecasted demand. The pool is managed by the Australian Energy Market Operator (AEMO) which follows the National Electricity Law and Rules and is in conjunction with market participants and regulatory agencies. Unlike many other markets, the Australian spot electricity market is not a day-ahead market but electricity is traded in a constrained real time spot market where prices are set each 5 minutes by AEMO. Therefore, generators submit offers every five minutes. The final price is determined every half-hour for each of the regions as an average over the 5-minute spot prices for each trading interval (Australian Energy Market Operator, 2010). Based on the half-hourly spot prices, also a daily average spot price for each regional market can be calculated. Note that in our empirical analysis we will concentrate on daily spot prices. 2.2 Mean Reversion Energy prices are in general regarded to be mean reverting (Schwartz, 1997). As pointed out by Weron (2006), spot electricity prices are perhaps the best example of antipersistent data. Note, however, that there is a critical difference between the form of mean reversion observed in electricity markets and most other financial markets. Interest rate markets, for instance, exhibit mean reversion in a weak form - the actual rate of reversion appears to be related to economic cycles and is therefore slow. In electricity markets, however, the rate of reversion is very strong, what can be explained by market fundamentals. When there is an increase in demand, generation assets with higher marginal costs will enter the market on the supply side, pushing prices higher. When demand returns to normal levels, these generation assets will be turned off again and prices will fall. This rational operating policy for the employment of generation assets supports the assumption of mean reversion in electricity spot prices. Further, other demand-driving factors such as weather and climate are cyclical as well. 2.3 Seasonality It is also well known that electricity demand exhibits seasonal fluctuations, see e.g. Pilipovic (1997) and Kaminski (1999). Due to the cyclical fluctuations in demand, also electricity prices tend to change accordingly. The seasonal component in electricity prices is more pronounced than in any other commodity market and several different seasonal patterns can be found in electricity prices during the course of a day, week and year. They mainly arise due to changing level of business activities or climate conditions, such as temperature or the number of daylight hours. Several techniques allow to remove seasonal components and trends from the data. These include, for example, differencing time series, moving average techniques, spectral decomposition, rolling volatility or wavelet decomposition. For an overview on suggested techniques for dealing with the seasonal components in electricty prices we refer to Weron (2006). Note that in this study wavelets are applied in order to remove a long-run, yearly, component from the univariate time series. After removing the long-run pattern, a moving average is applied in order to remove the short-run (weekly) component from the data. 6

7 2.4 Price Spikes In addition to mean reversion and strong seasonality on the annual, weekly and daily level, spot electricity prices exhibit infrequent, but large spikes. The spot price can increase tenfold during a single hour. Such extreme prices are usually the result of high load fluctuations caused e.g. by severe weather conditions but can also be due to generation outages or transmission failures. The observed price spikes are normally quite short-lived, and as soon as the weather phenomenon, outage or transmission failure is over, prices fall back to a normal level, see Kaminski (1999) and Weron et al. (2004). From a modeling point of view, price spikes are rather unpredictable discontinuities in the price process. In contrast, spikes are typically interpreted as the result of a sudden increase in demand. When demand reaches the limit of available capacity, electricity prices often exhibit positive price spikes. In periods when demand is reduced, electricity prices fall. Due to the operating cost or constraints of generators, who cannot adjust to the new demand level, also negative price spikes can occur (Fanone et al., 2011). From a modeling point of view, price spikes are short time intervals in which the price process exhibits a non-markovian behavior and prices increase or decrease significantly in a continuous way. The typical explanation for these phenomena is a highly non-linear supply-demand curve in combination with the non-storability of electricity. 2.5 Regional Markets Finally, it is important to note that spot and forward electricity prices may vary drastically from region to region, again due to non-storability and transmission constraints. In Australia, each state in the NEM initially developed its own generation, transmission and distribution network and is now linked to other states via interconnector transmission lines. Electricity can be transmitted between regions to meet energy demands that are higher than local generators can provide. However, the scheduling of generators to meet demand across the interconnected power system is significantly constrained by the physical transfer capacity of the interconnectors between the different regions (Australian Energy Market Operator, 2010). Especially during periods of peak demand, the interconnectors may become congested yielding reliability problems or only limited possibilities to transfer power between the regional markets. As a result, interstate trade represents only a small fraction of the total generation in the NEM leading to different regional prices. Given this limitation, one could argue that the regional markets are still relatively isolated and spot prices in each of the markets should behave quite differently. On the other hand, when demand in all markets is relatively high, one may expect extreme price observations like for example spikes, to happen jointly in different regional markets. Therefore, further investigation of the dependence structure between prices in regional Australian electricity markets is of significant importance for the market participants. 3 Copula Models The main objective of this paper is to investigate the dependence structure between electricity markets in Australia using copulae. Hereby, in the empirical analysis we will first perform a static copulae analysis and then proceed with a time-varying copulae approach. This section aims to briefly review the definition of copulae and also provides an overview of the parametric copula families used in the empirical analysis. Copulae are multivariate distribution functions which allow to connect d one-dimensional uniform-(0,1) marginals to the joint cumulative distribution. If F is a d-dimensional 7

8 distribution function with marginals F 1..., F d, then there exists a copula C with F (x 1,..., x d ) = C{F 1 (x 1 ),..., F d (x d )} (3.1) for every x 1,..., x d R. On the other hand, if C is a copula and F 1,..., F d are distribution functions, then the function F defined in (3.1) is a joint distribution function with marginals F 1,..., F d. Moreover, if F 1,..., F d are continuous, then C is unique. Thus, if X = (X 1,..., X d ) is a random vector with distribution X F X and continuous marginals X j F Xj (j = 1,... d), then the copula of X is the distribution function C X of u = (u 1,..., u d ) [0, 1] d, where u j = F Xj (x j ): C X (u 1,..., u d ) = F X {F 1 X 1 (u 1 ),..., F 1 X d (u d )}. (3.2) The above formulation with equations (3.1) and (3.2) being used to define copulae is known as a Sklar s theorem, see Joe (1997) for its proof. Note that the copula density c X which is used when estimating copulae, can be obtained by differentiating C X in (3.2): 1 f{fx c X (u 1,..., u d ) = 1 (u 1 ),..., FX 1 d (u d )} dj=1 f j {FX 1 (3.3) j (u j )} where f is the joint density of F X and f j is the density of F Xj. The density function of X is then given by with x j = F 1 X j (u j ). d f(x 1,..., x d ) = c X (u 1,..., u d ) f j (x j ) j=1 3.1 Correlation, Dependence and Tail Dependence The concept of correlation can still be considered as the most popular way to measure the degree of dependence among random variables in the literature. However, in contrast to the Pearson correlation coefficient r or Spearmann s ρ, which only measure linear dependence among random variables, copulae also allow to measure non-linear dependence between random variables (Embrechts et al., 2001b; Dias, 2004). This includes, for example, extreme dependence in the tails of the multivariate distribution. Thus, using copulae one can define the upper and the lower tail dependence coefficients (for the bivariate case) in the following way. For (U 1, U 2 ) denoting a pair of uniform variables on the unit square [0, 1] 2, the upper tail dependence coefficient λ u [0, 1] is defined as λ u = lim P (U C (u, u) 1 > u U 2 > u) = lim u 1 u 1 1 u. (3.4) Similarly, the lower tail dependence coefficient λ l [0, 1] is defined as λ l = lim u 0+ P (U 1 u U 2 u) = lim u 0+ C(u, u). (3.5) u If λ u falls into the interval (0, 1], then U 1 and U 2 are said to be asymptotically dependent in the upper tail, and if λ u = 0, then U 1 and U 2 are said to be asymptotically independent in the upper tail. Similarly, if λ l (0, 1] or λ l = 0, then U 1 and U 2 are said to be asymptotically dependent, or independent, respectively, in the lower tail. For properties of the lower and the upper tail dependence coefficients we refer to Embrechts et al. (2001a) and Embrechts et al. (2001b). Hu (2006) also reviews dependence and 8

9 tail dependence measures for mixture copula models. 3.2 Examples of Copulae Throughout the paper we will concentrate on two popular copula families: the elliptical copulae family, which includes the Gaussian copula and the Student-t copula, as well as the Archimedean copulae family, which has e.g. Frank, Gumbel and Clayton copulae as special cases. In addition, we will use a notion of a so-called survival copula C corresponding to a copula C from the Archimedean copulae family: F (x 1,..., x d ) = C {F 1 (x 1 ),..., F d (x d )} (3.6) where F (x 1,..., x d ) = P (X 1 > x 1,... X d > x d ). C For the bivariate case it can be defined as follows: C (u 1, u 2 ) = 1 u 1 u 2 + C(1 u 1, 1 u 2 ), (3.7) see Nelsen (1998). For the purpose of our study we will use the survival Gumbel and survival Clayton copulae Elliptical Copulae Elliptical copulae have a dependence structure generated by the elliptical distributions which include e.g. the normal and the Student-t distributions, see Lindskog et al. (2001), as well as the stable distribution class discussed in e.g. Rachev and Mittnik (2000) and Rachev and Han (2000). The Gaussian copula and Student-t copula will be presented in the following. The Gaussian copula generates the dependence structure of the multivariate normal distribution by combining normal marginals to form a multivariate normal distribution. If X j N(0, 1) and X = (X 1,..., X d ) N d (0, Ψ), where Ψ denotes a correlation matrix, an explicit expression for the Gaussian copula is given by = Φ 1 (u 1 ) C Ga Ψ (u 1,..., u d ) = F X {Φ 1 (u 1 ),..., Φ 1 (u d )} (3.8) Φ 1 (u d )... 2π d 2 Ψ 1 2 exp ( 1 ) 2 r Ψ 1 r dr 1... dr d. Defining ζ j = Φ 1 (u j ), ζ = (ζ 1,..., ζ d ), the density of the Gaussian copula can be written as c Ga Ψ (u 1,..., u d ) = Ψ 1 2 exp { 1 } 2 ζ (Ψ 1 I d )ζ. (3.9) The Student-t copula generates the dependence structure from the multivariate Studentt distribution. If X = (X 1,..., X d ) t d (ν, µ, Σ) has a multivariate Student-t distribution with ν degrees of freedom, mean vector µ and positive-definite dispersion or scatter matrix Σ, the Student-t copula is given by Cν,Ψ(u t 1,..., u d ) = t ν,ψ {t 1 ν (u 1 ),..., t 1 ν (u d )}, (3.10) where t 1 ν is the quantile function from the univariate t-distribution, Ψ is the correlation matrix associated with Σ 1. With ζ j = t 1 ν (u j ), j = 1,..., d the density of the Student-t 1 Since copula functions remain invariant under any series of strictly increasing transforma- 9

10 copula is given by c t ν,ψ(u 1,..., u d ) = Ψ 1 2 Γ( ν+d ) { Γ( ν 2 2 )} d 1 ( ν ζ Ψ 1 ζ ) ν+d 2 { Γ( ν+1 2 )} d dj=1 ( ν ζ2 j ) ν+1 2. (3.11) The Student-t copula allows to generate symmetric tail dependence. The tail dependence coefficients are defined by ) λ u = λ l = 2 ( t ν+1 (ν + 1)(1 ρ)/(1 + ρ), (3.12) where t ν denotes the Student-t distribution function, ν is the number of degrees of freedom, and ρ is the correlation coefficient. Modeling of dependency using elliptical distributions can be found in e.g. Hult and Lindskog (2001), Fang et al. (2002) and Frahm et al. (2003). Their application in finance and risk management are discussed, for instance, in Breymann et al. (2003), McNeil et al. (2005) and Dias and Embrechts (2008) Archimedean Copulae In our empirical study we will also apply the Gumbel and Clayton copula that belong to the family of Archimedean copulae. Both copulae also have a simple closed form and will be briefly reviewed below. Applications of Archimedean copulae to modeling portfolio credit risk have been studied e.g. in McNeil et al. (2005), Dias (2004) and Wu et al. (2006). The Clayton copula with the dependence parameter θ (0, ) is defined by 1/θ d C θ (u 1,..., u d ) = u θ j d + 1 j=1 (3.13) and density function (1/θ+d) d d c θ (u 1,..., u d ) = {1 + (j 1)θ}u (θ+1) j u θ j d + 1. (3.14) j=1 j=1 As the copula parameter θ tends to infinity, the dependence becomes maximal and as θ tends to zero, we have independence. The Clayton copula can mimic lower tail dependence but no upper tail dependence. The Gumbel copula with dependence parameter θ [1, ) is given by 1/θ d C θ (u 1,..., u d ) = exp ( log u j ) θ. (3.15) For θ > 1 this copula generates an upper tail dependence, while for θ = 1 it reduces to the product copula (i.e. independence): C θ (u 1,..., u d ) = d j=1 u j. Maximal dependence is achieved when θ tends to infinity. tions of X, such as e.g. standardization of the marginal distributions, see Nelsen (1998), the copula of a t d (ν, µ, Σ) distribution is identical to that of a t d (ν, 0, Ψ). j=1 10

11 3.2.3 Mixture Copulae In addition to the Archimedean and elliptical copulae discussed above we also consider some mixture models of Archimedean copulae as introduced in Joe (1993). Mixture copulae usually take the form of a convex combination of two or more copulae. Denoting C A and C B copulae with dependence parameters θ 1 and θ 2, respectively, the mixture model takes the following form: C X (u 1,..., u d, θ) = θ 3 C A X(u 1,..., u d, θ 1 ) + (1 θ 3 )C B X(u 1,..., u d, θ 2 ). (3.16) Empirical applications of mixture copulae can be found e.g. in Dias (2004) with application to FX rates or in Hu (2006) where mixture copulae are applied for modeling the dependence across international financial markets. In our empirical analysis we will consider four mixture models of copulae, namely the Clayton & survival Clayton, Clayton & Gumbel, survival Clayton & survival Gumbel and the Gumbel & survival Gumbel mixture copulae. 3.3 Copulae Estimation Generally, the maximum likelihood technique is applied to estimate parametric copulae. Assume that for a vector of random variables X = (X 1,..., X d ) with absolute continuous parametric univariate marginals F Xj (x j, δ j ), j = 1,..., d and a copula C X (θ), we aim to estimate parameters from the marginals δ 1,..., δ d as well as the copula dependence parameter θ. From Sklar s theorem the distribution of X is given by with a density where F X (x 1,..., x d ) = C{F X1 (x 1 ; δ 1 ),..., F Xd (x d ; δ d ); θ} (3.17) d f(x 1,..., x d ; δ 1,..., δ d, θ) = c{f X1 (x 1 ; δ 1 ),..., F Xd (x d ; δ d ); θ} f j (x j ; δ j ), (3.18) j=1 c(u 1,..., u d ) = d C(u 1,..., u d ) (3.19) u 1... u d is a copula density. For a sample of observations {x t } T t=1 where x t = (x 1,t,..., x d,t ), and a vector of parameters α = (δ 1,..., δ d, θ) R d+1 the likelihood function is given by T L(α; x 1,..., x T ) = f(x 1,t,..., x d,t ; δ 1,..., δ d, θ). (3.20) t=1 Combining (3.18) and (3.20), the corresponding log-likelihood function is given by l(α; x 1,..., x T ) = T ln [c{f X1 (x 1,t ; δ 1 ),..., F Xd (x d,t ; δ d ); θ}] + t=1 T t=1 j=1 d ln [f j (x j,t ; δ j )]. (3.21) Our objective is to maximize this log-likelihood numerically. The estimation can be performed using e.g. the exact maximum likelihood (EML), the inference for marginals (IFM) and the canonical maximum likelihood (CML) method. The exact maximum likelihood (EML) estimates copula dependence parameter α and 11

12 the marginal parameters δ 1,..., δ d in one step through α EML = arg max l(α). (3.22) α The estimates α EML = ( δ 1,..., δ d, θ) solve the first order condition ( l/ δ 1,..., l/ δ d, l/ θ) = 0. (3.23) The drawback of the EML method is that with an increasing scale of the problem the algorithm becomes computationally quite challenging. The inference for marginals (IFM) method is a sequential two-step maximum likelihood method, see e.g. McLeish and Small (1988) and Joe (1997): parameters from the marginals are estimated in the first step. Their estimates are then substituted into the copula to obtain the dependence parameter θ. For j = 1,..., d, the log-likelihood function for each of the marginal distributions is given by and the estimated marginal parameter is given by The pseudo log-likelihood function T l j (δ j ) = ln f j [x j,t ; δ j ] (3.24) t=1 ˆδ j = arg max l j (δ j ). (3.25) δ l(θ, ˆδ 1,..., ˆδ T d ) = ln [ c{f X1 (x 1,t ; ˆδ 1 ),..., F Xd (x d,t ; ˆδ d ); θ} ] (3.26) t=1 is maximized over θ to obtain the estimator ˆθ for the dependence parameter θ. The estimates ˆα IF M = ( ˆδ 1,..., ˆδ d, ˆθ) solve the first order condition ( l 1 / δ 1,..., l d / δ d, l/ θ) = 0. (3.27) Finally, the canonical maximum likelihood (CML) method maximizes the pseudo loglikelihood function with empirical marginal distributions T l(θ) = ln [ c{ ˆF X1 (x 1,t ),..., ˆF Xd (x d,t ); θ} ]. (3.28) t=1 Note that in contrast to the EML and IFM methods, here we do not have to make any assumptions about the parametric form of the marginal distributions. The empirical marginal cumulative distribution function is given by ˆF Xj (x) = 1 T + 1 T 1 Xj,t x, (3.29) t=1 see Genest and Rivest (2002). Using this method, the parameter can then be estimated in one step by using the estimate ˆθ CML = arg max l(θ). (3.30) θ 12

13 4 Estimation of the Marginals Prior to fitting marginal distributions, one should take into account certain characteristics and stylized facts of electricity markets. Electricity demand is subject to cyclical fluctuations and prices tend to change according to the hour, day, month and year. In order to account for a log-run (yearly) seasonality, we will apply a wavelet decomposition approach that is summarized below. A short-run (weekly) pattern is removed afterwards by using a moving average technique, see e.g. Brockwell and Davis (2002) and Weron (2006). Different distributions are then fitted to the deseasonalized logprices in order to find the best performing distribution family. Empirical studies have shown that electricity spot prices, similar to stock prices, exhibit heavy tails and excess kurtosis that cannot be modeled appropriately by the normal distribution. Therefore, we will also consider alternative distributions from the family of Symmetric Generalized Hyperbolic (SGH) distributions which includes the Student-t, the Normal Inverse Gaussian (NIG), the Hyperbolic (HYP) and the Variance Gamma (VG) distributions. 4.1 Wavelet Decomposition Wavelet decomposition is applied to find a long-run seasonal pattern and remove it from the data. Let P t denote the logarithmic spot price, which can be represented as a sum of two components: the predictable (seasonal component) f t and a stochastic component X t : P t = f t + X t. Further, we assume that f t can be decomposed into a yearly (long-term) seasonal trend T t and a short-term (weekly) periodic component s t : f t = T t + s t. In order to remove a long-term pattern T t from the data, we use a lowpass filtering approach which yields a linear smoother, that is, linear with respect to the coefficients of the series expansion. A wavelet family includes a father wavelet φ, which is used for the trend, or cycle, and a mother wavelet ψ which is responsible for the deviations from the trend. Any function, or in our case time series, f(t) can then be represented as a sequence of projections onto father and mother wavelets: f(t) = S J + D J + D J D 1 (4.1) with S J = s J,k φ J,k (t) and D j = d j,k ψ j,k (t). (4.2) k k The indices are k = 0, 1, 2,... and s = 2 j, j = 0, 1, 2,..., J where 2 J denotes the maximum scale sustainable by the number of data points. Coefficients s J,k, d J,k, d J 1,k,..., d 1,k measure the weight, or contribution of the corresponding wavelet function to the approximating sum, and the functions ( t 2 φ J,k (t) = 2 j/2 j ) ( k t 2 φ and ψ j,k (t) = 2 j/2 j ) k ψ (4.3) 2 j denote the approximating father and mother wavelets, respectively. After decomposing f(t) using signal decomposition in (4.1), the original signal can be obtained by inverting the procedure. The rough approximation of f(t) is given by S J. Higher level wavelets which increase the level of refinement of the signal, can be approximated by S J 1 = S J + D J. The refinement can be achieved by adding a mother wavelet D j of a lower scale j = J 1, J 2,... The smaller the number of mother wavelets being used, the less noisy will be the reconstructed series. After removing a long-term (yearly) component T t from the data, the weekly periodicity s t can be removed by applying a moving average, see e.g. Brockwell and Davis (2002). The deseasonalized log-price is then obtained by subtracting the resulting long- and 2 j 13

14 short-term seasonal patterns, i.e., P t T t s t. Finally, the deseasonalized log-prices can be shifted such that the minimum of the new process coincides with the minimum of P t, see e.g. Weron (2006). 4.2 Distribution for the Marginals Specifying the marginals and estimating their time-varying volatilities is an intermediate step prior to modeling the dependence structure via copulae. For the purpose of our analysis we concentrate only on symmetric representations 2 of the family of Symmetric Generalized Hyperbolic (SGH) distributions. The general form of the SGH density function is given by 1 1 ᾱ f X (x) = 1 + x2 2 (λ 1 2 ) K δσk λ (ᾱ) 2π (δσ) 2 λ ᾱ x2, (4.4) 2 (δσ) 2 where parameters α 0 if λ 0 and δ 0 if λ 0 and K λ ( ) denote a modified Bessel function of the third kind with index λ, see Abramowitz and Stegun (1972). The parameters λ and ᾱ are the shape parameters for the tails of the distribution. Varying λ and ᾱ allows to specify special cases of the SGH distribution. In particular, the Variance Gamma distribution is obtained by setting ᾱ = 0 and the shape parameter λ > 0, see Madan and Seneta (1990). Smaller values of λ indicate increasingly heavier tails. When λ the Variance Gamma density asymptotically approaches the Gaussian density. The Student-t distribution assumes ᾱ = 0 and λ < 0, see Praetz (1972). Note that the parameter σ is not the standard deviation of the random variable X, which is σ X = σ ν. When the number of degrees of freedom ν decreases, we observe an ν 2 increase in the tail heaviness of the density, which implies a larger probability of extreme values. Additionally, with an increase of the degrees of freedom ν, the Studentt density converges asymptotically to the Gaussian density. Finally, the Hyperbolic distribution is specified when λ = 1, see Eberlein and Keller (1995), and the Normal Inverse Gaussian distribution is obtained by setting λ = 0.5, see Barndorff-Nielsen (1995). Further details on the representation of the density functions can be found in e.g. Platen and Rendek (2008). In order to decide which distribution from the SGH family of distributions fits the data best, one could apply for example goodness-of-fit tests to the deseasonalized log-prices by using either the Kolmogorov-Simirnov (KS) distance or Anderson-Darling (AD) distance KS = sup F s (x) ˆF (x), (4.5) x R AD = sup x R F s (x) ˆF (x) ˆF (x)(1 ˆF (x)). (4.6) Hereby, F s (x) denotes an empirical sample distribution and ˆF (x) is the estimated distribution. In general, the drawback of the KS statistic is that it is more sensitive closer to the center of the distribution and fails to capture the tails. Since market participants are particularly interested in the risk of extreme electricity spot prices, in our empirical analysis we decided to use the AD distance to examine the fit of the estimated distribution to the price series, see e.g. Rachev and Mittnik (2000). The AD 2 The location of the distribution and the skewness are set equal to zero. 14

15 statistic captures both, the deviations around the median of the distribution, as well as the discrepancies in the tails. In addition to calculating the maximum of the distance as defined in (4.5) and (4.6), we will compute the following summary statistics for the AD distance to get more comprehensive results: the mean, the median, the standard deviation, the minimum and the range for all SGH distributions fitted to the data. 4.3 Model for the Marginals After specifying the best performing distributional family, we estimate parameters from the marginals, assuming that the conditional mean is constant and the volatilities are time-varying. Previous studies on the behavior of electricity spot prices in Australian markets report good results for univariate and multivariate GARCH type models for the time-varying volatility (Higgs and Worthington, 2005; Higgs, 2009). Therefore, for a sequence of i.i.d. random variables (u t ) t 0 with zero mean and unit variance, we assume that the deseasonalized log-price process (X t ) t 0 follows an AR(1) process X t = a 0 + a 1 X t 1 + σ t u t, (4.7) where innovations ε t = σ t u t have by definition mean zero and the conditional variance V ar(ε t F t ) = σt 2 is modeled via a GARCH(1,1) process σ 2 t = α 0 + α 1 σ 2 t 1 + β 1 ε 2 t 1, (4.8) where α 0 > 0, α 1 0, β 1 0, α 1 +β 1 < 1, and u t is independent of (X s ) s t. We estimate parameters via maximum likelihood for each marginal series assuming that the fitted residuals û t = ε t / σ t are approximately i.i.d. and follow one of the distributions from the SGH family which has been chosen based on the AD goodness-of-fit. 5 Empirical Results 5.1 Data In this study we use daily spot prices from five electricity markets in Australia: NSW, QLD, SA, TAS and VIC. For each market the sample of 1460 daily observations covers the time period from to Figures 3 and 4 show the system price (top left panel), the log-price together with the long-term seasonal component obtained using wavelet filtering (top right panel), the log-price after removing longand short-term seasonality pattern (bottom left panel), and the returns obtained by differencing the deseasonalized log-prices (bottom right panel) for the NSW and QLD electricity markets. 3 To remove the long-term seasonal component (thick line in the top right panel), we use the Daubechies wavelet with S 8 approximation, which roughly corresponds to annual (2 8 = 256 days) smoothing. Weekly periodicity s t is removed by applying the moving average technique as discussed in e.g. Brockwell and Davis (2002); Weron (2006). Figures 3 and 4 illustrate that Australian electricity prices reflect the characteristics and stylized facts discussed in Section 2, e.g., the price process is not smooth, prices 3 Note that the graphs for SA, TAS and VIC are not reported here but are available upon request to the authors. 15

16 2500 NSW 8 7 NSW Price [AUD/MWh] Log Price [AUD/MWh] daily [ ] daily [ ] Deseasonalized Log Price [AUD/MWh] NSW Deseasonalized Log Returns [AUD/MWh] NSW daily [ ] daily [ ] Fig. 3. The system price for NSW (top left), the log-price together with a long-term seasonal component obtained by wavelet filtering (top right), the log-price after removing long- and short-term seasonal pattern (bottom left), and the returns obtained by differencing the deseasonalized log-price (bottom right). The sample covers daily prices for the time period from to are mean-reverting and experience spikes due to non-linear supply-demand and nonstorability. From visual inspection one can also observe a similar pattern in the longterm component and the location of spikes for the states NSW and QLD (see the top right panel). When computing correlation coefficients 4 between the deseasonalized log-prices in Table 1, we observe a strong dependence between the pairs (NSW, QLD), (NSW, VIC) and (VIC, SA). Correlation between SA and TAS is relatively low, and these states also experience relatively low correlations with the remaining states. Slightly higher correlation between TAS and VIC might be due to the fact that TAS is connected to VIC via an undersea inter-connector. 5.2 Distribution of the Marginals In order to get a visual impression from the shape of the deseasonalized log-prices, we present a histogram for the pooled data taken for all states for the period from to The total number of observations corresponds to The histogram for the pooled data vs. the normal density is presented in the left panel, and 4 While Pearsons linear correlation, reported in the upper panel, depends on the distribution of the univariate marginals (i.e., keeping the dependence structure constant, different marginals might lead to different values for the joint distribution, see Dias (2004)), the other two rank correlations, Spearmann s ρ and Kendall s τ are independent of the univariate marginal distributions. 16

17 2500 QLD 8 7 QLD Price [AUD/MWh] Log Price [AUD/MWh] daily [ ] daily [ ] Deseasonalized Log Price [AUD/MWh] QLD Deseasonalized Log Returns [AUD/MWh] QLD daily [ ] daily [ ] Fig. 4. The system price for QLD (top left), the log-price together with a long-term seasonal component obtained by wavelet filtering (top right), the log-price after removing long- and short-term seasonal pattern (bottom left), and the returns obtained by differencing the deseasonalized log-price (bottom right). The sample covers daily prices for the time period from to Table 1 Pearson correlation r, Spearmann s ρ and Kendall s τ for the deseasonalized log-prices for NSW, QLD, SA, TAS and VIC from to Pearson correlation coefficient r NSW QLD SA TAS VIC NSW QLD SA TAS VIC Spearmann s ρ NSW QLD SA TAS VIC NSW QLD SA TAS VIC Kendall s τ NSW QLD SA TAS VIC NSW QLD SA TAS VIC vs. the Student-t density in the right panel of Figure 5. The figures suggest a much better fit of the Student-t density to the data compared to a relatively poor fit of the normal density. In the following, we fit univariate log-prices for NSW, QLD, SA, TAS and VIC using equations (4.7) and (4.8), assuming either normal, or SGH innovations which include the Student-t, the NIG, the HYP and the VG distributions discussed in Section

18 Normal density Empirical density Normal vs. Empirical density Student t density Empirical density Student t vs. Empirical density Fig. 5. Histogram for the pooled data vs. normal density (left panel) and Student-t density (right panel). Pooled data is taken for deseasonalized standardized log-prices for NSW, QLD, SA, TAS and VIC (daily data) from to We perform a goodness-of-fit test using the AD statistic defined in equation (4.6). We compute not only the maximal distance but also the moments and the range of the AD statistic. The results are summarized in Table 2. 5 Figure 6 shows the box-plots for the AD distance for the normal distribution as well as for all distributions from the SGH family for the states NSW and QLD. As we can observe from Table 2 and Figure 6, all distributions from the SGH family provide a considerably better fit than the normal distribution. The Student-t assumption on the marginals leads to the smallest AD max values, the smallest means and the smallest dispersions. We, therefore, conclude that it provides the best overall fit to the data. We now fit AR-GARCH(1,1) models (4.8) to each of the marginal price series, assuming Student-t innovations. The t-ar-garch(1,1)-fitted volatility at time t is computed using a moving window of deseasonalized log-prices, { ˆX s } t s=t n+1 of size n = scrolling in time for t = n,..., T. Average daily volatilities over the entire sample period vary from 17.35% for VIC to 20.50% for SA. The fitted volatilities are plotted in Figure 7 for the states NSW and QLD. 5.3 Copula Analysis In the following, we aim to fit parametric copulae assuming that the marginals follow the Student-t distribution. Hereby, we apply the inference for marginals method as discussed in Section 3.3. Thus, in a first step we specify the marginals using timevarying mean and volatilities estimated from equations (4.7) and (4.8), together with 5 We do not report the KS statistic since, as argued above, the AD statistics is preferred to the KS statistics due to its ability to captures both, the deviations around the median of the distribution, as well as the discrepancies in the tails. Therefore, we rely on the AD statistic when judging the goodness-of-fit of the considered models. 6 Moving window size n = 365 corresponds to one year of observations. Thus, the first estimate will correspond to