Regularization Approach for Network Modeling of German Power Derivative Market
TThis is a post-peer-review, pre-copyedit version of an article published in Energy Economics. The finalauthenticated version is available online at: http://dx.doi.org/10.1016/j.eneco.2019.06.021
Regularization Approach for Network Modeling of GermanPower Derivative Market ∗ Shi Chen † , Wolfgang Karl H¨ardle ‡ , Brenda L´opez Cabrera § Abstract
In this paper we propose a regularization approach for network modeling ofGerman power derivative market. To deal with the large portfolio, we combinehigh-dimensional variable selection techniques with dynamic network analysis. Theestimated sparse interconnectedness of the full German power derivative market,clearly identify the significant channels of relevant potential risk spillovers. Our em-pirical findings show the importance of interdependence between different contracttypes, and identify the main risk contributors. We further observe strong pairwiseinterconnections between the neighboring contracts especially for the spot contractstrading in the peak hours, its implications for regulators and investors are also dis-cussed. The network analysis of the full German power derivative market helps usto complement a full picture of system risk, and have a better understanding of theGerman power market functioning and environment.
Keywords : regularization, energy risk transmission, connectedness, network, Germanpower derivative market
JEL : C1, Q41, Q47 ∗ Financial support from the IRTG 1792 ”High Dimensional Non Stationary Time Series”, as wellas the Czech Science Foundation under grant no. 19-28231X, the Yushan Scholar Program and theEuropean Union’s Horizon 2020 research and innovation program ”FIN-TECH: A Financial supervisionand Technology compliance training programme” under the grant agreement No 825215 (Topic: ICT-35-2018, Type of action: CSA), Humboldt-Universit¨at zu Berlin, is gratefully acknowledged. † Corresponding author, Chair of Statistics and Econometrics, Karlsruher Institut f¨ur Technologie,Bl¨ucherstr.17, 76185 Karlsruhe, Germany. Email: [email protected] ‡ School of Business and Economics, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, 10099Berlin, Germany; Singapore Management University, 50 Stamford Road, Singapore 178899 § School of Business and Economics, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, 10099Berlin, Germany a r X i v : . [ q -f i n . S T ] S e p Introduction
Affordable and reliable energy supply is essential for industrial growth. Achieving thesein times of growing demand, raw materials shortage and climate change poses challenges.Germany’s power system for the industry and the consumers is undergoing radical change,this transformation is being driven by the restructuring of electricity supply and byintense competition between suppliers (see BMWi (2016), Spiecker et al. (2014), Seifertet al. (2016), Grossi et al. (2017), Sinn (2017) and among others). However, the ongoingexpansion of renewable energy and the phase-out of nuclear energy for power generationwill change the composition of the electricity mix, which in return, will generate pricingsignals affecting the electricity trading (e.g. Benhmad and Percebois (2018), Ketterer(2014), Ballester and Furi´o (2015), Paraschiv et al. (2014)). As we know, electricity isthe commodity that should be supplied immediately. Unlike coal, oil, gas or other typicalcommodities, electricity cannot be stored. This results in the price of electricity beingvolatile and very depended on a secure supply. To hedge against the uncertainty arisen inthe market, we study the system-wide market risk of the whole German power derivativemarket. Therefore energy companies may invest in both electricity spot and derivativemarkets to diversity their existing portfolios. As electricity grids worldwide also beginrelying more heavily on renewable energy sources, analysis of German power market thusprovides useful insights for power generation companies and transmission organizationsacross the globe.However, the number of variables and relevant factors is typically huge. A properlydesigned subset selection has to be employed to identify the most informative powercontracts to representing energy market risk. The German power market is highly in-terconnected with a dense and wide range of electricity contracts, this motivates us tobuild up an ultra high dimensional network and investigate its sparse property. To bet-ter understand the interaction between power contracts, the iterated sure independencescreening (iterated-SIS, Fan et al. (2009)) method combined with regularization estima-tors are applied to estimate the sparse web of connections. Our network of interest isconstructed in the context of time series based on vector autoregression (VAR) models,the iterated-SIS method is important when building VAR models since the number ofparameters to estimate increases quadratically in the number of variables included. Toquantify the associations between individual power contract and energy exchange market,2he network we constructed is obtained from the forecast error variance decomposition(FEVD) based on VAR estimates in the framework of Koop et al. (1996) and Pesaranand Shin (1998). This kind of connectedness measure is also used by Diebold and Yil-maz (2009), Diebold and Yılmaz (2014) for conceptualizing and empirically measuringweighted, directed networks at a variety of levels. They proposed this variance decom-position networks as tools for quantifying and ranking the systemic risk of individualcomponent in a portfolio.In this paper we investigate the concept of connectedness in a realistic high dimen-sional framework, which is important for system-wide risk measurement and management.We aim to obtain a sparse network in which nodes represent power contracts and linksrepresent the magnitude of connectedness, local shocks and events can therefore be eas-ily amplified and turned into global events. While estimates of the network yield thequalitative links between power contracts, individual impact from specific contract canbe estimated and speculated accordingly. Hence the risk contribution from the marketcomponent can be identified, this will help us to learn more about the German powermarket functioning and environment. Following Diebold and Yılmaz (2014), Demireret al. (2018), Hautsch et al. (2014) and similar connectedness literature, the risk refers tothe uncertainty arisen in the system, and it measures the amount of future uncertaintycaused by other component in the system. For example, the market uncertainty may becaused by such as economic and financial uncertainties, or weather conditions, energyprices and regulation. While the systemic risk network yields qualitative information onrisk channels and roles of assets within the constructed portfolio, the risk is quantified byits contribution to the forecast error variance of other components in the whole system.Therefore we are able to capture the systemic risk of the component by summing up itstotal contribution. Understanding the risk transmission channels for investors is of greatimportance, for example, our results show that day-ahead spot power contracts that bid-ding between 9am and 13am are in the core of the German energy power market, the keyderivatives in connecting markets can be identified.The rest of the paper proceeds as follows. Section 2 reviews the relevant literatureand introduces the German energy market. In section 3, we describe in details how theregularization approach is applied to estimate the large portfolio and how the network isconstructed. Section 4 reports the data and discusses the model selection result. Section3 presents a static analysis of full-sample connectedness. Section 6 provides the empiricalresults of the dynamic network. Finally section 7 concludes.
As a tradable commodity electricity is relatively new, its dynamic properties have beenanalyzed with many different approaches, some recent contributions are Weron (2007),Geman and Roncoroni (2006), Bierbrauer et al. (2007) and Knittel and Roberts (2005).There is also a strand of literature that analyzes the multivariate behavior of electric-ity prices. For examples, Higgs (2009) examines the inter-relationships of wholesale spotelectricity prices across four Australasian markets by a multivariate GARCH model. Hen-riques and Sadorsky (2008) develops a four variable VAR model to explain the dependencestructure of a variety of energy equities, where they find that shocks to technology actu-ally have a larger impact on the stock prices of alternative energy companies than do oilprices. Castagneto-Gissey et al. (2014) studies the interactions of a representative sampleof 13 European (EU) electricity spot prices with dynamic Granger-causal networks.We take our starting point in the energy literature based on a vector regression frame-work, where the network of interest is constructed based on Diebold and Yilmaz (2009),Diebold and Yılmaz (2014). Most relevant studies explore the relationship between oiland energy equity prices in terms of volatility spillovers, they estimate the implied volatil-ity linkages across markets as source of future uncertainty. For example Du et al. (2011)conducts a Bayesian analysis to explain volatility spillovers among crude oil and variouseconomic factors. Arouri et al. (2012) investigates the volatility spillovers between oiland stock markets in Europe using VARDGARCH approach. Sadorsky (2012) applies amultivariate GARCH model to estimate the volatility spillovers between oil prices andthe stock prices of clean energy/technology companies. Joo and Park (2017) examinesthe time-varying causal relationship between the stock and crude oil price uncertaintiesusing a DCC GARCH-in-Mean specification. More empirical work are Diebold and Yil-maz (2012), Reboredo (2014), Maghyereh et al. (2016), Awartani et al. (2016), Zhang(2017), Apergis et al. (2017), and among others. There is also a fairly sizable literatureexploring the electricity market integration using vector regression framework, for exam-4le Worthington et al. (2005), Zachmann (2008), Bunn and Gianfreda (2010), Balaguer(2011), B¨ockers and Heimeshoff (2014), Castagneto-Gissey et al. (2014).However, relatively little research has focused on the systemic directional interactionbetween energy equities. A recent study by Lundgren et al. (2018) is the first to analyzethe connectedness network among different energy asset classes, their analysis examinesthe connectedness network among renewable energy stock, four investments, and un-certainties. Demirer et al. (2018) uses Lasso method to select, shrink and estimate ahigh-dimensional network. Other empirical work are with more focus on financial bank-ing contexts, like Yi et al. (2018) uses the VARX-L framework developed by Nicholsonet al. (2017) to conduct static and dynamic volatility spillovers among cryptocurrencies.More relevant work are Wang et al. (2018), Acharya et al. (2012), Hautsch et al. (2014),Giglio et al. (2016), Babus (2016), Brownlees and Engle (2016), Acharya et al. (2017)and among others.However, almost all existing energy literature are based only on moderate dimen-sions. This motivates us to examine the systemic risk transmission channels in a realistichigh dimensional framework. The main argument is the contracts trading in both spotand derivative markets share the same underlying contracts, and it is therefore natu-ral to consider a high dimensional portfolio. High-dimensional statistical problems arisefrom diverse fields of scientific research and technological development, including energymarkets. The traditional idea of best subset selection methods is computationally tooexpensive for many modern statistical applications. Variable selection techniques havetherefore been successfully developed in recent years and they indeed play a pivotal rolein contemporary statistical learning and techniques. Researchers have proposed variousregularized estimators with different penalty terms, a preeminent example being the leastabsolute selection and shrinkage operator (Lasso) of Tibshirani (1996). In recent years,Lasso has been extended to high-dimensional case, see Bickel et al. (2009). Other pop-ular methods contribute to the literature, such as smoothly clipped absolute deviation(SCAD) Fan and Li (2001), adaptive Lasso of Zou (2006), elastic net estimator of Zou andHastie (2005), Dantzig selector of Candes and Tao (2007). In an ultra high-dimensionalcase where the dimensionality of the model is allowed to grow exponentially in the samplesize, it is helpful to begin with screening to delete some significantly irrelevant variablesfrom the model. Fan and Lv (2008) introduce a method called sure independence screen-5ng for this goal. Even when the regularity conditions may not hold, Fan et al. (2009)extend the iterated-SIS method to work by iteratively performing feature selection torecruit a small number of features. Furthermore, the asymptotic properties of Lasso forhigh-dimensional time series have been considered by Loh and Wainwright (2011) and Wuet al. (2016). Kock and Callot (2015) establishes the high-dimensional VAR estimationwith focus on Lasso and adaptive Lasso. Basu et al. (2015) investigates the theoreticalproperties of regularized estimates in sparse high-dimensional time series models whenthe data are generated from a multivariate stationary Gaussian process.We address the above high dimensional problem by combining a regularization ap-proach with classic VAR model. In doing so, we contribute to the energy literature inseveral ways: First, we extend the current literature to investigate the sparse linkagebetween energy equities in a very large portfolio. Second, we identify the main riskcontributor to help investor diversity their existing portfolio rather than having largeholdings of individual electricity contract. Third, our study may further be extendedby including renewable energy assets and oil price across European energy market, thismay provide a better understanding of the overall energy market. In addition, our studycan also be applied to investigate electricity market integration by estimating the totalconnectedness for a wide range of components.
German electricity market
The German electricity market is Europe’s largest, withannual power consumption of around 530 TWh and a generation capacity of 184 GW.As a net energy exporter, the export capacity of Germany is expected to continue togrow as planned interconnections expand cross-border transmission capacity with severalneighboring countries. Germany has significant interconnection capacity with neighbor-ing EU member states as well. It is interconnected with Austria, Switzerland, the CzechRepublic, Denmark, France, Luxembourg, the Netherlands, Poland, and Sweden. Tomaintain stable and reliable supply of electricity, the so-called Transmission system op-erators (TSOs) keep control power available. Primary control, secondary control, andtertiary control reserve are procured by the respective TSOs within a non-discriminatorycontrol power market in accordance with the requirements of the Federal Cartel Office.Demand for control energy is created when the sum of power generated varies from the6ctual load caused by unforeseeable weather fluctuations in the case of renewable energies.Electricity is traded on the exchange and over the counter. Standardized productsare bought and sold in a transparent process on the exchange, which, for Germany, isthe European Energy Exchange EEX in Leipzig, the European Energy Exchange EPEXSPOT in Paris and the Energy Exchange Austria (EXAA) in Vienna. The EuropeanEnergy Exchange (EEX) is the leading energy exchange in Europe. It develops, operatesand connects secure, liquid and transparent markets for energy and commodity products.Contracts on power, coal and CO2 emission allowances as well as freight and agricul-tural products are traded or registered for clearing on EEX. EPEX SPOT, Powernext,Cleartrade Exchange (CLTX) and Gaspoint Nordic are also members of EEX Group.The German wholesale electricity market is broadly made up of three elements, a for-ward market, a day-ahead market and an intra-day market. These submarkets generatethe pricing signal which electricity production and consumption align to. The objective ofthis paper is to analyse the interaction of different future contracts traded in the forwardmarket, whether forward market is influenced by market power of spot prices traded inEPEX market.Figure 2.1: The distribution of European power derivatives in EEX market. Source: EEXwebsiteElectricity providers and electricity purchasers submit their bids in their nationalday-ahead market zones. The exchange price on the day-ahead market is determinedjointly for coupled markets. Electricity providers and electricity purchasers submit theirbids in their national day-ahead market zones. In an iterative process, the demand for7lectricity in the market zone is served by the lowest price offers of electricity from allthe market areas until the capacity of the connections between the market zones (cross-border inter-connectors) is fully utilized. As long as the cross-border inter-connectorshave sufficient capacity, this process aligns the prices in the coupled market areas. Onaccount of market coupling, the national power demand is covered by the internationaloffers with lowest prices. The upshot is that on the whole less capacity is required to meetthe demand. As shown in Figure 2.1, Phelix Future, as the product traded in Germany,is a financial derivatives contract settling against the average power spot market pricesof future delivery periods for the German/Austrian market area.
Phelix futures
Electricity supply deliveries in the forward market can be negotiatedup to seven years in advance, but for liquidity reasons typically only look out three years,and in fact one year ahead futures are traded at most. The Phelix Future is a financialderivatives contract referring to the average power spot market prices of future deliveryperiods of the German/Austrian market area.As the most liquid contract and benchmark for European power trading, the underly-ing of these future contracts is the Physical Electricity Index determined daily by EPEXSpot Exchange for base and peak load profiles. To be more specific, the Phelix Basecontract is average price of the hours 1 to 24 for electricity traded on spot market, whilethe Phelix Peak is the average price of the hours 9 to 20 for electricity traded on spotmarket. EEX offers continuous trading and trade registration of financially fulfilled Phe-lix Futures, with Day/Weekend Futures, Week Futures, Month Futures, Quarter Futuresand Year Futures available.The time series of Phelix day base and Phelix day peak prices are displayed in Figure2.2. Phelix day peak exhibit a larger volatility and more pronounced spikes than thePhelix day base. This is not surprising, since the Phelix day peak corresponds to hourswith high and variable demand. Both price series exhibit positive skewness and an excesskurtosis of about 1, implying a heavy-tailed unconditional distribution that is skewed tothe right.In addition, the Phelix market is also successfully connected to other European powermarkets. The products of Location Spread enables members to trade price differencesbetween markets, thus enabling participants to benefit from improved liquidity and tighter8igure 2.2: Phelix day base (black) and Phelix day peak (red) index from 2013-01-01 to2015-10-31. The red dotted line marks the end of the in-sample period.spreads, for instance, Phelix / French Power, Italian / Phelix Power, Phelix / NordicPower and Phelix / Swiss Power. For the empirical work of this paper, we use the PhelixFuture data to find price drivers and important variables in the big system we construct.The decision-making mechanism of energy companies will also be explored.
When there is a high-dimensional portfolio consisting of various power derivative con-tracts, standard methods are intractable and therefore require novel statistical methods.Here we are interested in addressing the following research questions, how all these con-tracts interact with each other? Which variables are crucial for the whole system? How-ever, due to the large number of variables in the system, some sparsity assumption mustbe imposed for the sake of an accurate estimate. The large dimensionality in our modelcomes from not only the varieties of power derivative products, but also the large lagorder in VAR model to avoid the correlation of error terms.The standard VAR( p ) model with lag order p is constructed according to L¨utkepohl92005), y t = ν + A y t − + A y t − + · · · + A p y t − p + u t = ν + ( A , A , . . . , A p ) (cid:0) y (cid:62) t − , y (cid:62) t − , . . . , y (cid:62) t − p (cid:1) (cid:62) + u t (3.1)where y t = ( y t , y t , . . . , y Kt ) (cid:62) is a ( K ×
1) random vector consisting K variables at time t , t = 1 , . . . , T . A i are unknown ( K × K ) coefficient matrices. ν is a ( K ×
1) vector ofintercept terms, u t = ( u t , u t , . . . , u Kt ) (cid:62) is a K -dimensional innovation process. Define Y = ( y , y , . . . , y T ) B = ( ν, A , A , . . . , A p ) Z t = (1 , y t , y t − p +1 ) (cid:62) Z = ( Z , Z , . . . , Z T − ) (3.2)For multivariate case, rewrite equation (3.1) as Y = BZ + U (3.3)where U = ( u , u , . . . , u T ). The compact form of (3.3) is vec ( Y ) = ( Z (cid:62) ⊗ I K ) vec ( B ) + vec ( U ) y = ( Z (cid:62) ⊗ I K ) β + u = X β + u (3.4)The dimension of model (3.4) to be estimated is pK and the number of observa-tions is KT . The ratio KpT could be large due to the reasons mentioned earlier, whichdeteriorates the accuracy of final estimate. Worse still, if
Kp > T , the model becomeshigh-dimensional with more unknown parameters than observations. Therefore we re-quire teniques other than traditional OLS method. That’s why most existing energyliterature are based only on moderate dimensions.Here we use variable selection technique, for example Lasso, to estimate the model.Besides, under normal assumption of error term, the upper bound of error in estimationis positively correlated in log( K p ) T , part of oracle inequality. the estimation results can befurther developed by adding one more step of sure independent screening (SIS) beforevariable selection step. Another advantage of SIS is that it could mitigate the problemcaused by multicollinearity, which is common in time series setting. The techniquesintroduced in the proceeding paragraph are of great importance in the sense that thetrue underlying model has a sparse representation.10 .2 Regularization estimator and iterated-SIS algorithm Regularization approach
Variable selection is an important tool for the linear re-gression analysis. A popular method is the Lasso estimator of Tibshirani (1996), whichcan be viewed to simultaneously perform model selection and parameter estimation. Re-lated literature includes bridge regression studied by Frank and Friedman (1993) andFu (1998), the least angle regression of Efron et al. (2004) and adaptive Lasso proposedby Zou (2006). Another remarkable example is a smoothly clipped absolute deviation(SCAD) penalty for variable selection proposed by Fan and Li (2001), they proved itsoracle properties.Let us start with consider model estimation and variable selection for equation (3.4), y = X β + u (3.5)The least square estimate is obtained via minimizing (cid:107) y − Xβ (cid:107) , where the ordinaryleast squares (OLS) gives nonzero estimates ω = X (cid:62) y to all coefficients. Normally bestsubset selection is implemented to select significant variables, but the traditional ideaof best subset selection methods is computationally too expensive for many statisticalapplications. Therefore the penalized least square with a penalty term that is separablewith respect to the estimated parameter ˆ β is considered here. In this paper we considertwo popular estimators, Lasso and SCAD.The Lasso is a regularization technique for simultaneous estimation and variable se-lection, with the estimator given by,ˆ β LASSO = arg min β (cid:107) y − Xβ (cid:107) + λ p (cid:88) j =1 | β j | = arg min β (cid:107) y − p (cid:88) j =1 x j β j (cid:107) + λ p (cid:88) j =1 | β j | (3.6)where λ is the tuning parameter. The second term in equation (3.6) is known as the (cid:96) -penalty. The idea behind Lasso is the coefficients shrink toward 0 as λ increases.When λ is sufficiently large, some of the estimated coefficients are exactly zero. Theestimation accuracy comes from the trade-off between estimation variance and the bias.Lasso is the penalized least square estimates with the (cid:96) penalty in the general leastsquares and likelihood settings. Furthermore, the (cid:96) penalty results in a ridge regressionand (cid:96) p penalty will lead to a bridge regression.11e proceed to the SCAD method. In the present context, the SCAD estimator isgiven by, ˆ β SCAD = sgn( ω )( | ω | − λ ) + when | ω | ≤ λ { ( a − ω − sgn( ω ) aλ } a − λ < | ω | ≤ aλω when | ω | > aλ (3.7)where a > p (cid:48) λ ( β ) = λ (cid:26) I ( β ≤ λ ) + ( aλ − β ) + ( a − λ I ( β > λ ) (cid:27) for a > β > Iterated-SIS algorithm for estimation
SIS method is proposed by Fan and Lv(2008) to select important variables in ultra high-dimensional linear models. The pro-posed two-stage procedure can perform better than other methods in the sense of sta-tistical learning problems. The SIS method is based on the concept of sure screening,is defined as the correlation learning which filters out the features that have weak cor-relation with the response. By sure screening, all the important variables survive aftervariable screening with probability tending to 1. Fan et al. (2009) improve iterated-SISto a general pseudo-likelihood framework by allowing feature deletion in the iterativeprocess. Fan et al. (2010) further extend the SIS model and consider an independentlearning by ranking the maximum marginal likelihood estimator or maximum marginallikelihood itself for generalized linear models. Here we combine the VAR( p ) model andSIS algorithm to find out the key elements in a big system. The basic idea of SIS isintroduced in the following.Let ω = ( ω , ω , . . . , ω p ) (cid:62) be a p-vector that is obtained by component-wise regression,i.e., ω = X (cid:62) y (3.9)where y is n vector of response and X is a n × p data matrix. ω is a vector of marginalcorrelations of predictors with the response of predictors with the response variable,rescaled by the standard deviation of the response.12hen there are more predictors than observation, LS (least square) estimator is noisy,that’s why ridge regression is considered. Let ω λ = ( ω λ , . . . , ω λp ) (cid:62) be a p − vector obtainedby ridge regression, i.e., ω λ = ( X (cid:62) X + λI p ) − X (cid:62) y (3.10)where λ > λ → ω λ → ˆ β LS and λ → ∞ , λω λ → ω . The component-wise regression is a specific case of ridge regressionwith λ = ∞ .The iterated-SIS algorithm applied for estimating the VAR( p ) model is,1. Apply SIS for initial screening, reduce the dimensionality to a relative large scaled;2. Apply a lower dimensional model selection method (such as lLassoasso, SCAD) tothe sets of variables selected by SIS;3. Apply SIS to the variables selected in the previous step;4. Repeat step 2 and 3 until the set of selected variables do not decrease. We construct our network using the fashionable directional connectedness measure pro-posed by Diebold and Yılmaz (2014). The connectedness is measured by cross-sectionalvariance decomposition, where the forecast error variance of variable is decomposed intoparts attributed to the various variables in the system.The interactions between the variables, i.e., the directional connectedness measure θ ij ( q ) is given by, θ ij ( q ) = σ − jj (cid:80) Q − q =0 (cid:16) e (cid:62) i ˆ B q Σ e j (cid:17) (cid:80) Q − q =0 (cid:16) e (cid:62) i ˆ B q Σ ˆ B (cid:62) q e i (cid:17) (3.11)where q is the lag order, e i is an pK × i -th elementand zeros elsewhere. Σ = E (cid:0) u t u (cid:62) t (cid:1) , is the covariance matrix of the non-orthogonalizedVAR( p ) in equation (3.1), with σ jj is the corresponding j -th diagonal element of Σ. ˆ B l are the coefficient matrices of (3.12). 13ith iterated-SIS algorithm to estimate the sparse VAR structure, we can acquire itsmoving average (MA) transformation, y t = ∞ (cid:88) i =0 B i u t − i (3.12)where the coefficient matrices B i obey B i = (cid:80) iyj =1 B i − j A j , with B = I K and A j = 0 for j > p . A j , j = 1 , , . . . , p is the coefficient matrices of VAR( p ) model.To measure the persistent effect of a shock on the behavior of a series, we aim toacquire the population connectedness table 3.1, according to Diebold and Yılmaz (2014). x x . . . x n From others x θ ( q ) θ ( q ) . . . θ n ( q ) (cid:80) nj =1 θ j ( q ) , j (cid:54) = 1 x θ ( q ) θ ( q ) . . . θ n ( q ) (cid:80) nj =1 θ j ( q ) , j (cid:54) = 2... ... ... ... ... x n θ n ( q ) θ n ( q ) . . . θ nn ( q ) (cid:80) nj =1 θ nj ( q ) , j (cid:54) = n To others (cid:80) ni =1 θ i ( q ) , i (cid:54) = 1 (cid:80) ni =1 θ i ( q ) , i (cid:54) = 2 . . . (cid:80) ni =1 θ in ( q ) , i (cid:54) = n n (cid:80) ni =1 ,j =1 θ ij ( q ) , i (cid:54) = j Table 3.1: Connectedness table of interest.The rightmost column gives the ”from” effect of total connectedness, and the bottomrow gives the ”to” effect. In particular, the directional connectedness ”from” and ”to”associated with the forecast error variation θ ij for specific power contract when the arisingshocks transmit from one asset to the others. These two connectedness estimators can beobtained by adding up the row or column elements, the pairwise directional connectednessfrom j to i is given by, C i ← j = θ ij ( q ) (3.13)The total directional connectedness ”from” C i ←· (others to i ), ”to” C ·← j ( j to others)and the corresponding net connectedness are defined as C i ←• = n (cid:88) j =1 θ ij ( q ) , i (cid:54) = jC •← j = n (cid:88) i =1 θ ij ( q ) , i (cid:54) = jC i = C to − C from = C •← i − C i ←• (3.14)The ”to” connectedness measures its total contribution to the forecast error variance ofother components in the system, and therefore captures the systemic risk of the compo-nent. 14 Data and Model Selection
As introduced in Section 2.2, EEX offers continuous trading data of Phelix Futures. Theavailable load profiles are base, peak and off-peak. The available products with differentmaturities have five kinds: Day/Weekend Futures, Week Futures, Month Futures, QuarterFutures and Year Futures. Nevertheless the products of Day/Weekend Futures and WeekFutures only have the off-peak load data, for all other contracts base and peak only. Herewe recall the underlying of the Phelix Futures data, the Phelix Base contract is averageprice of the hours 1 to 24 for electricity traded on spot market, while the Phelix Peak isthe average price of the hours 9 to 20 for electricity traded on spot market. Therefore weinvolve the products of spot prices as well. The contracts of spot prices are diversified inHours from 00-01h up to 23-24h, and in Blocks of Base Monthly, off-peak 01-08, off-peak21-24, Peak Monthly. The dataset we constructed is provided by Bloomberg, we have90 kinds of contracts in total. The time span is from 30.09.2010 to 31.07.2015. All thecontracts are listed on Table A.1 with detailed information in Table A.2. . − . C on t r a c t s P r i c e (a) Ribbon plot of prices over 90 contracts Time T y pe . . . . . . (b) Contour plot of log return Figure 4.1: Overview of datasetTo remove the redundant variable, we apply screening technique to select variablesusing the Phelix Futures consisting of different contracts and over different maturities.To implement the VAR model, first order difference of the data in Figure 4.1a is requiredto transform non-stationary data to stationary time series. The contour plot of theconstructed dataset is depicted in Figure 4.1b.15 i g u r e . : P a tt e r n o f i m pu t e dd a t a .
16n the market of Phelix Futures, final settlement at negative price is also possible.There are some missing values after transforming the original data to stationary timeseries by first order difference. To deal with the missing data, some quick fixes suchas mean-substitution may be fine in some cases. While such simple approaches usuallyintroduce bias into the data, for instance, applying mean substitution leaves the meanunchanged (which is desirable) but decreases variance, which may be undesirable. In ourpaper, we impute missing values with plausible values drawn from a distribution usingan approach proposed by Van Buuren and Oudshoorn (2000). The patterns of missingdata for the original dataset and imputation dataset are compared in the Figure 4.2. Thedistributions of the variables are shown as individual points, the imputed data for eachimputed dataset is shown in magenta while the density of the observed data is shownin blue. The distributions are expected to be similar based on the assumption. We canobserve that the shape of the magenta points (imputed) matches the shape of the blueones (observed). The matching shape tells us that the imputed values are indeed plausiblevalues.
The purpose of this section is to compare the performance of the regularization approachesand to select the best model used to construct connectedness measure. The estimationsteps are as follows,1. Given the lag order p , p is a constant.2. Recall iterated-SIS algorithm in 3.2, estimate VAR( p ) using either iterated-SIS-Lasso or iterated-SIS-SCAD.3. Select the best model with model selection criterion given by, IC ( p ) = log | ˆ H ( p ) | + ϕ ( K, p ) c T (4.1)where ϕ ( K, p ) is a penalty function. c T is a sequence indexed by the sample size T . The residual covariance matrix ˆ H ( p ) without a degrees of freedom correction isdefined as, ˆ H ( p ) = 1 T T (cid:88) t =1 u (cid:62) t u t (4.2)17ewrite equation (4.1) with different penalty functions, the three most commoninformation criteria are the Akaike (AIC), Schwarz-Bayesian (BIC) and Hannan-Quinn (HQ), AIC = log | ˆ H ( p ) | + 2 T pK (4.3) HQ = log | ˆ H ( p ) | + 2 log log TT pK (4.4) BIC = log | ˆ H ( p ) | + log TT pK (4.5)4. The selected model will be used to construct directional connectedness θ ij ( q ) definedin (3.11).The comparison of three information criteria is reported in Table 4.1. We observethat the VAR(2) estimated by iterated-SIS-Lasso performs best, with smallest IC valuesgiven by: AIC 4.5006, HQ 4.6426 and BIC 5.6076. The model with smaller IC values ismore likely to be the true model. Model AIC HQ BICiterated-SIS-Lasso, p = 1 4.5686 4.7249 5.7864iterated-SIS-Lasso, p = 2 4.5006 4.6426 5.6076iterated-SIS-Lasso, p = 3 6.9854 7.2315 10.0345iterated-SIS-SCAD, p = 1 4.5714 4.7277 5.7892iterated-SIS-SCAD, p = 2 6.1043 6.1043 9.5782iterated-SIS-SCAD, p = 3 7.2559 7.6820 10.5770Table 4.1: Model selection results according to the AIC, BIC and HQ criteria.We produce the coefficient paths for iterated-SIS-Lasso VAR estimation in Figure 4.3.Each curve corresponds to a variable. It shows the path of its coefficient against (cid:96) -normof the whole coefficient vector as λ varies. In addition, we partition the dataset into twosamples: we select the in-sample dataset as 30.09.2010-28.11.2014, and the out-of-sampledataset used to measure model forecast performance is from 31.12.2014 to 31.07.2015. Weroll each model through the out-of-sample dataset one observation at a time while eachtime forecasting the target variable one month ahead. By rolling window, the forecastmean squared errors (FMSE) for different models are calculated and compared in Table18
500 1500 2500 − L1 Norm C oe ff i c i en t s − Fraction Deviance Explained C oe ff i c i en t s Figure 4.3: Iterated-SIS-Lasso VAR(2) estimation results.4.2, VAR(2) with iterated-SIS-Lasso has the smallest FMSE of 0.067, this is consistentwith our previous finding. For the full sample dataset, we find that iterated-SIS-Lassooutperforms iterated-SIS-SCAD algorithm. Therefore the iterated-SIS-Lasso algorithmis selected for constructing the corresponding connectedness measure.Lag iterated-SIS-Lasso iterated-SIS-SCAD p = 1 0.0697 0.0697 p = 2 0.0670 0.0701 p = 3 0.0923 0.1413Table 4.2: FMSE of out-of-sample forecasting during 31.12.2014 - 31.07.201519 Static analysis of power market connectedness
GI1GI2GI3GI4 GI5GI6 GI7GT1 GT2GT3 GT4GT5 GT6GT7 HP1HP2HP3HP4HP5HP6 GJ1GJ2GJ3GJ4GJ5 GJ6 GJ7HI1 HI2HI3 HI4HI5 HI6HI7 NE1NE2NE3NE4NE5NE6 POA1POA2 POA3POA4 POA5POA6 POA7PDA1 PDA2PDA3 PDA4PDA5 PDA6PDA7 PBA1PBA2PBA3PBA4PBA5PBA6LPXBHR01LPXBHR02LPXBHR03 LPXBHR04LPXBHR05LPXBHR06 LPXBHR07LPXBHR08LPXBHR09LPXBHR10LPXBHR11LPXBHR12LPXBHR13 LPXBHR14LPXBHR15 LPXBHR16LPXBHR17LPXBHR18LPXBHR19LPXBHR20LPXBHR21LPXBHR22LPXBHR23LPXBHR24 LPXBHBLPXBHOP1LPXBHOP2 LPXBHPLPXBHRBLPXBHRP
GIGTHPGJHINEPOAPDAPBALPXBHRLPXBHxxGIGTHPGJHINEPOAPDAPBALPXBHRLPXBHxx
Figure 5.1: The graph for full-sample energy market network, across 11 different typeswith in total 90 contracts.The graph of our full-sample energy market network is depicted in Figure 5.1. We ob-serve the cluster phenomena in this graph, which motivates us to study the connectednessbetween contracts within and across 11 different types of energy contracts. In general, thecontracts that belong to the same type tend to appear inside the same cluster. We findout several pairs of strong connections between different types of contracts, for example,the upper-left area reveals that the LPXBHR-type and LPXBHxx-type are massivelyconnected. In addition, a cluster consisting of HP-type (Phelix Base Year Option), NE-type (Phelix Peak Year Future) and PBA-type (Phelix Off-Peak Year Future) indicatesthe closer relationship among these contracts, this implies the Year derivative contracts20re closer to each other while the month future and quarter future remain distinct.
GI GT HP GJ HI NE POA PDA PBA LPXBHR LPXBHxx FromGI 1.75 0.46 0.86 1.46 0.44 0.71 1.48 0.48 0.80 0.15 0.15 8.74GT 0.46 2.33 0.98 0.58 2.41 0.93 0.42 2.03 0.84 0.30 0.41 11.70HP 0.73 0.84 5.27 0.63 0.51 4.78 0.70 1.21 4.19 0.06 0.03 18.95GJ 1.46 0.58 0.73 1.80 0.63 0.64 1.09 0.51 0.65 0.13 0.13 8.36HI 0.44 2.41 0.60 0.63 2.65 0.63 0.37 1.92 0.49 0.34 0.48 10.96NE 0.61 0.79 4.78 0.55 0.54 5.09 0.55 1.05 3.34 0.08 0.12 17.52POA 1.48 0.42 0.81 1.09 0.37 0.64 1.60 0.48 0.79 0.19 0.18 8.04PDA 0.48 2.03 1.42 0.51 1.92 1.23 0.48 1.99 1.27 0.26 0.32 11.91PBA 0.68 0.72 4.19 0.56 0.42 3.34 0.67 1.09 3.88 0.20 0.12 15.88LPXBHR 0.80 1.24 0.50 0.67 1.35 0.50 0.90 1.12 1.01 7.86 9.81
LPXBHxx 0.13 0.35 0.03 0.11 0.41 0.12 0.16 0.28 0.13 2.70 3.86 8.28To 9.03 12.18
Table 5.1: The connectedness table for the aggregated network, including 11 types ofcontracts.
GI GT HPGJHINEPOAPDAPBALPXBHR LPXBHx
Figure 5.2: The graph for network across 11 different contract types.Given the cluster phenomena observed in the full sample connectedness graph, weaggregate the pairwise connectedness of the contracts that belong to the same type. Theaggregated network is then reported in Table 5.1, with in total 11 types of contracts(more details can be found in Table A.2). The off-diagonal elements represent the crosscontractType connectedness, while within contractType connectedness are on the diago-nal. 21ecall that for each component in the system, the “to” connectedness measures itstotal contribution to the forecast error variance of other components in the system, mak-ing possible to capture the systemic risk of the component. In this table, it is obviousthat the HP-type, NE-type and PBA-type are main contributors to systemic-wide risk.HP-type contracts have stronger links both from and to the other contract types. This ispotentially interesting because, although HP-type, NE-type are important for the wholemarket as shown in Figure 5.2, their net connectedness are negligible, with 4.52% and4.08% of the total market power contracts. The reason is that these two types of con-tracts (HP, NE) are mutually closely interconnected, they also have “from” impacts tothe system and thus offsetting their risk contributions. Moreover, the strongest pairwiseconnectedness is the impact of LPXBHxx-type on the LPXBHR-type, however the inverseimpact is not significant. We also observe very strong net impacts from LPXBHxx-typeon the LPXBHR-type contracts.
12 3 45 6789 101112 13 1415 1617 1819 2021 222324
Figure 5.3: The connectedness network graph based on different trading hours forLPXBHR-type spot contracts. The number in each node corresponds to the tradinghours ranging from 00-01 to 23-24h, e.g. 12 refers to the LPXBHR12 contract with tradinghours 11-12h. The LPXBHR-type power contracts that bidding from 09-13h and 16-20hare represented with darker purple color because of their higher “to”-connectedness.In terms of magnitude for different contract types reported in Table 5.1, the “net”22irectional connectedness is distributed rather tightly, in total 77.21% of LPXBHR-typeand LPXBHxx-type. We first investigate pairwise connectedness across 24 LPXBHR-typecontracts (full connectedness table is available in Table A.3).
Power Contract “from”-connectedness “to”-connectedness “net”-connectednessLPXBHR11.Index 12.48 -1.40LPXBHR10.Index 12.27 -1.45LPXBHR12.Index 11.83 -1.27LPXBHR13.Index 11.55 -1.27LPXBHR17.Index 11.46 -1.41LPXBHR19.Index 11.27 -1.23LPXBHR18.Index 11.06 -1.27LPXBHR20.Index 10.61 -0.88LPXBHR21.Index 9.51 9.31 -0.19LPXBHR22.Index 8.77 8.51 -0.26LPXBHR23.Index 7.80 8.48 0.68LPXBHR14.Index 7.48 8.16 0.68LPXBHR16.Index 7.02 7.16 0.14LPXBHR15.Index 6.51 7.15 0.64LPXBHR01.Index 5.99 6.99 0.99LPXBHR09.Index 5.64 6.54 0.90LPXBHR06.Index 5.60 6.42 0.83LPXBHR05.Index 5.35 6.30 0.94LPXBHR04.Index 5.28 6.19 0.91LPXBHR08.Index 5.13 5.65 0.52LPXBHR07.Index 5.04 4.98 -0.06LPXBHR02.Index 4.66 4.87 0.21LPXBHR03.Index 4.36 4.85 0.49LPXBHR24.Index 2.07 4.84 2.77
Table 5.2: Summary of “from”, “to” and “net” connectedness for all LPXBHR-typecontracts. The contracts are ranked by their system risk measure “to”- connectedness.Figure 5.3 visualizes the connectedness network according to different trading hours,the numbered nodes correspond to the trading hours from 00-01 to 23-24h. Some blocksof high connectedness are successfully detected and represented by darker purple color,i.e. the peak spot contracts with trading hours ranging from 09-13h and 16-20h. Wesummarize the “from”, “to” and “net” effects for the 24 LPXBHR-type contracts in de-scending order of importance, and rank the contacts by their system risk in Table 5.2.Our finding clearly shows that, the impacts from day-ahead spot power contracts thatbidding between 09:00 and 13:00 have highest “to”-connectedness values, and thereforestrongest impacts on the other day-ahead spot contracts. The peak spot contracts (trad-23ng hours 09-13h, 16-20h) have very large negative “net”-connectedness, revealing thatthey are main risk source in determining the electricity price.The pairwise directional impacts between LPXBHR-type and LPXBHxx-type areplotted in Figure 5.4, the colors of the nodes are the same as Figure 5.1 (the connected-ness table of the impacts from the six LPXBHxx-type contracts to the 24 LPXBHR-type spot contracts is available in the appendix, see Table A.4). We find a clusterof LPXBHR10, LPXBHR11, LPXBHR12, LPXBHR13, LPXBHRP, LPXBHRB andLPXBHB. The LPXBHB (Base hours 00:00 - 14:00) contract has significant impactson the spot contracts from hours 09 to 13, while the impacts from the LPXBHP (PeakHours 08:00 - 20:00) contract is negligible. In addition, both LPXBHRB (Baseload)and LPXBHRP (Peakload) contracts exhibit strong interconnectedness with the spotcontracts from hours 09 - 13, however only the LPXBHRP (Peakload) contract affectsthe spot prices trading from hours 16 - 18. Moreover, the graph exhibits strong mu-tual links between some of the spot contracts, for example, the pairs of LPXBHR10 andLPXBHR11, LPXBHR11 and LPXBHR12, LPXBHR12 and LPXBHR13, LPXBHR17and LPXBHR18 and among others.
LPXBHR01LPXBHR02LPXBHR03LPXBHR04LPXBHR05LPXBHR06LPXBHR07LPXBHR08LPXBHR09LPXBHR10LPXBHR11LPXBHR12LPXBHR13LPXBHR14LPXBHR15LPXBHR16LPXBHR17LPXBHR18LPXBHR19LPXBHR20LPXBHR21LPXBHR22LPXBHR23LPXBHR24LPXBHBLPXBHOP1LPXBHOP2LPXBHPLPXBHRBLPXBHRP
LPXBHR(1 − 24)LPXBHxxLPXBHR(1 − 24)LPXBHxx
Figure 5.4: The network graph for LPXBHR-type and LPXBHxx-type power contracts.The network graph in Figure 5.5 illustrates the pairwise directional connectedness24etween HP-type and NE-type contracts. Compared with Figure 5.1, we can see that,the links between these two types are very strong. NE-type contracts are the YearFutures with maturities up to six years, the underlying of these contracts is the averageprice of hours 9 to 20 for electricity traded on the spot market. HP-type contracts arethe European style options on the Phelix Base Future provided by EEX, the underlyingof Phelix Base is the average price of the hours 1 to 24 for electricity traded on the spotmarket. It is calculated for all calendar days of the year as the simple average of theAuction prices for the hours 1 to 24 in the market area Germany / Austria. This figureshows the similar connectedness pattern between HP1 and NE1 contracts.
HP1 HP2 HP3HP4HP5HP6NE1NE2NE3NE4NE5 NE6
HP−typeNE−typeHP−typeNE−type
Figure 5.5: The network graph for HP and NE-type power contracts.
We now study the dynamic network using rolling estimation. The number of observationsused in the rolling sample to compute prediction is 36 or correspondingly three years,and we examine dynamic evolution of the network for the following one year. In eachwindow, we repeat model selection and conduct iterated-SIS algorithm to obtain thesparse estimates. 25 ug Oct Dec Feb Apr Jun
Time t o t a l c onne c t edne ss Aug Oct Dec Feb Apr Jun
Time c r o ss / w i t h i n c on t r a c t T y pe c onne c t edne ss Figure 6.1: The time-varying network for the system-wide connectedness from August2014 to July 2015. The left panel is the time-varying full sample connectedness, it canbe decomposed into two parts: the cross contractType connectedness (upper curve in theright panel) and the within contractType connectedness(lower curve in the right panel).
We first calculate full sample system-wide connectedness for each window by summing upthe total directional connectedness whether “from” or “to”. In general, the full samplesystem-wide connectedness reflects the overall uncertainty arisen in the system. Thedynamic evolution path is plotted in the left panel of Figure 6.1, with the peak of thesystem-wide connectedness is December. In most regions of Germany, the coldest daysof the year are from around mid December to late January, which results in more workfor the heating systems. In particular, the amount of electricity used for the decorationsincrease dramatically over the holiday season in December. After December, the valueof the system-wide connectedness decreased to the lowest value of April, and it thenincreased gradually with increasing temperature until September.To measure the system-wide interaction, we further decompose the full sample system-26ide connectedness into two parts: cross contractType connectedness and within con-tractType connectedness as shown in the right panel of Figure 6.1. The cross contract-Type connectedness sums up the directional connectedness between the contracts comingfrom different contract types, while within contractType connectedness is the sum of di-rectional connectedness of all the contracts within the same contract type. The highervalues of cross contractType connectedness (upper curve) indicate that it is the maindriving force for system-wide connectedness. However the within-contractType connect-edness becomes less important, its values remain around 40 for the whole period.We further conduct a robustness risk by computing full sample system-wide connect-edness C h,t at different forecast horizons h = 8 , h = 9. Then the dynamic networks ofthe resulting time-varying C h ,t , C h ,t are compared with the above full sample system-wide connectedness C h ,t , h = 10 . To achieve this, we apply a Welch two sample t-test on H : C h ,t = C h ,t against H a : C h ,t (cid:54) = C h ,t . The p-value of the Welsh t-test is 0 . H : C h ,t = C h ,t against H a : C h ,t (cid:54) = C h ,t is 0 . To investigate the cross contractType connectedness, we aggregate the pairwise connect-edness of the contracts that belong to the same type. The resulting time-varying networksacross different contract types are summarized in Table 6.1. For each contract type, wereport the mean and standard deviation (s.d.), the numbers inside the parentheses rep-resent the 15% and 85% quantiles respectively. In general, the results are consistentwith the findings in section 5.1. The HP-type, NE-type and PBA-type contracts haverelatively higher values of both “to”-connectedness and “from”-connectedness, but theirnet impacts in the system are very low. Moreover, the LPXBHxx-type contracts havesignificant net impacts on the LPXBHR-type contracts, similar results are discussed insection 5.2.According to different trading hours, we further discuss how LPXBHxx-type contracts27 to”- connectedness “from”- connectedness “net”- connectednessmean s.d. mean s.d. mean s.d.GI 9.29 (9.02 - 9.68) 0.37 9.08 (8.85 - 9.33) 0.27 0.22 (0.06- 0.36) 0.16GT 12.40 (12.09 -12.57) 0.26 11.92 (11.80 - 12.05) 0.15 0.49 (0.24 - 0.69) 0.19HP 21.02 (20.44 - 21.52) 0.46 19.93 (19.43 - 20.41) 0.43 1.09 (1.03 - 1.15) 0.06GJ 8.49 (8.16 - 8.90) 0.32 8.31 (7.99 - 8.69) 0.32 0.18 (0.12 - 0.22) 0.06HI 11.92 (11.53 - 12.20) 0.34 11.16 (10.98 - 11.27) 0.18 0.75(0.52 - 0.92) 0.18NE 19.59 (19.24 - 19.92) 0.34 18.49 (18.07 - 18.89) 0.41 1.11 (1.02 - 1.20) 0.09POA 8.73 (8.28 - 9.29) 0.49 8.37 (8.15 - 8.70) 0.29 0.36 (0.13 - 0.61) 0.24PDA 12.35 (12.12 - 12.49) 0.20 12.15 (11.95 - 12.29) 0.19 0.20 (-0.03 - 0.40) 0.20PBA 18.87 (17.61 - 20.39) 1.35 17.60 (16.28 - 19.18) 1.39 1.27 (1.19 - 1.33) 0.10LPXBHR 15.16 (13.62 - 16.55) 1.46 29.92 (26.25 - 33.36) 3.33 -14.76 (-16.59 - -12.63) 1.89LPXBHxx 18.31 (16.74 - 19.86) 1.51 9.21 (8.43 - 9.85) 0.71 9.10 (8.30 - 9.88) 0.82
Table 6.1: The summary statistics of the connectedness measures for the aggregated net-work over one year, including 11 types of contracts. For each connectedness measure, wereport the corresponding mean(15% quantile - 85% quantile) and the standard deviation.may affect the LPXBHR-type contracts traded in the power derivative market. The meanand s.d. for the time-varying connectedness C LP XBHR i ← LP XBHxx j are reported in Table6.2. The spot contracts based on bid hours from 09-13 are closely related to the contractsof LPXBHB, LPXBHRB and LPXBHRP. We also find that the LPXBHRP affects thespot contracts from hours 16-19h most. To control for a relatively stable Germany spotelectricity price, the risk transmission channels among the contracts are not negligible.Investments in LPXBHB, LPXBHRB and LPXBHRP contracts help to limit the potentialrisk of loss when there are adverse movements of spot prices. This results may provideguide for policy maker, energy companies and investors.We also compute the averaged pairwise connectedness among the spot contracts basedon different trading hours, the interaction between the LPXBHR-type contracts is de-picted in Figure 6.2. We observe strong pairwise interconnection between the neighboringcontracts. For example, the LPXBHR12 has stronger connection with its nearest twoneighbors LPXBHR11 and LPXBHR13, but contract LPXBHR13 does not have stronglinkage with LPXBHR14. The most influential contracts are identified as the contractsbased on trading hours from 09-13 and 16-20. This is potentially interesting as it providespricing signals affecting the electricity trading.28 PXBHB LPXBHOP1 LPXBHOP2 LPXBHP LPXBHRB LPXBHRPLPXBHR01.Index 0.28 (0.01) 0.45 (0.01) 0.11 (0.01) 0.15 (0.01) 0.28 (0.01) 0.19 (0.01)LPXBHR02.Index 0.30 (0.01) 0.42 (0.02) 0.16 (0.01) 0.17 (0.01) 0.30 (0.01) 0.22 (0.01)LPXBHR03.Index 0.34 (0.02) 0.54 (0.02) 0.14 (0.01) 0.16 (0.01) 0.34 (0.02) 0.25 (0.02)LPXBHR04.Index 0.40 (0.02) 0.56 (0.02) 0.17 (0.01) 0.21 (0.01) 0.40 (0.02) 0.32 (0.01)LPXBHR05.Index 0.43 (0.02) 0.62 (0.02) 0.24 (0.01) 0.26 (0.02) 0.43 (0.02) 0.34 (0.01)LPXBHR06.Index 0.51 (0.02) 0.60 (0.03) 0.33 (0.01) 0.21 (0.01) 0.51 (0.02) 0.43 (0.02)LPXBHR07.Index 0.42 (0.01) 0.38 (0.01) 0.29 (0.01) 0.11 (0.01) 0.42 (0.01) 0.38 (0.01)LPXBHR08.Index 0.43 (0.01) 0.37 (0.01) 0.34 (0.01) 0.10 (0.01) 0.43 (0.01) 0.39 (0.01)LPXBHR09.Index 0.61 (0.02) 0.49 (0.02) 0.34 (0.01) 0.14 (0.01) 0.61 (0.02) 0.58 (0.02)LPXBHR10.Index (0.02) 0.68 (0.02) 0.47 (0.01) 0.38 (0.01) (0.02) (0.02)LPXBHR11.Index (0.02) 0.66 (0.02) 0.44 (0.01) 0.45 (0.01) (0.02) (0.02)LPXBHR12.Index (0.02) 0.60 (0.02) 0.40 (0.01) 0.50 (0.01) (0.02) (0.02)LPXBHR13.Index 0.78 (0.02) 0.53 (0.02) 0.34 (0.01) 0.51 (0.01) 0.78 (0.02) (0.02)LPXBHR14.Index 0.51 (0.01) 0.40 (0.01) 0.17 (0.01) 0.37 (0.01) 0.51 (0.01) 0.56 (0.02)LPXBHR15.Index 0.46 (0.01) 0.38 (0.01) 0.16 (0.01) 0.30 (0.01) 0.46 (0.01) 0.50 (0.02)LPXBHR16.Index 0.56 (0.02) 0.34 (0.01) 0.29 (0.01) 0.22 (0.01) 0.56 (0.02) 0.61 (0.02)LPXBHR17.Index 0.71 (0.02) 0.44 (0.01) 0.28 (0.01) 0.29 (0.01) 0.71 (0.02) (0.02)LPXBHR18.Index 0.71 (0.02) 0.44 (0.01) 0.32 (0.01) 0.28 (0.01) 0.71 (0.02) (0.02)LPXBHR19.Index 0.70 (0.02) 0.43 (0.01) 0.45 (0.01) 0.34 (0.01) 0.70 (0.02) 0.73 (0.02)LPXBHR20.Index 0.67 (0.02) 0.40 (0.01) 0.60 (0.02) 0.33 (0.01) 0.67 (0.02) 0.66 (0.02)LPXBHR21.Index 0.58 (0.02) 0.33 (0.01) 0.78 (0.02) 0.29 (0.01) 0.58 (0.02) 0.53 (0.01)LPXBHR22.Index 0.49 (0.01) 0.31 (0.01) (0.02) 0.18 (0.01) 0.49 (0.01) 0.39 (0.01)LPXBHR23.Index 0.43 (0.01) 0.22 (0.01) (0.02) 0.33 (0.01) 0.43 (0.01) 0.36 (0.01)LPXBHR24.Index 0.22 (0.01) 0.12 (0.01) 0.57 (0.02) 0.16 (0.01) 0.22 (0.01) 0.18 (0.01)
Table 6.2: The connectedness table reflects how the LPXBHxx-type contracts may affectspot contract according to different trading hours. In this table we report the mean ofthe time-varying networks, together with the s.d. inside the parentheses. Each elementmeasures the directional pairwise connectedness from the j th contract of LPXBHxx-typeto the i th LPXBHR-type, i.e. C LP XBHR i ← LP XBHxx j . The numbers larger than 0.8 aremarked with bold font. This paper combines regularization approach with dynamic network analysis in an ultrahigh-dimensional setting. We empirically analyze the sparse interconnectedness of thefull German power derivative market, clearly identify the significant channels of relevantpotential risk spillovers and thus complement the full picture of system risk. As we know,electricity is not storable and may be affected by various factors on the supply and demandside of the market, such as policy changes, weather conditions and external economicuncertainties. Nowadays Germany is transforming its power system towards renewable29
Figure 6.2: The averaged pairwise connectedness among the LPXBHR-type contractsover the rolling period. The numbered nodes correspond to the underlying trading hoursfor each contract. The colors are the same as Figure 5.3, the contracts with higherconnectedness values are represented with darker purple color.energy, analysis of German power derivative market thus provides useful insights forpower generation companies and transmission organizations across the globe.Our empirical findings suggest that the Phelix base year options and peak year fu-tures are the main contributors to the system risk. However these two types of contractsare mutually closely interconnected, they also have high “from” impacts received fromthe system and thus offsetting their risk contributions. In addition, the connectednessbetween different contract types are more significant compared with the connectednessamong contracts within the same type. Therefore it is important for policy makersand investors to take the interdependence between different contractTypes into account.Other interesting conclusions arise from the connectedness of spot and future contracts,when we examine each contract according to different trading hours. For spot contracts,we observe strong pairwise interconnections between the neighboring contracts especiallyfor the contracts trading in the peak hours, this provides investors pricing signals af-fecting the electricity trading. In addition, the monthly base future (BHB), monthlypeakload(BHRP)/baseload(BHRB) are identified as main driving force for the peak-hour30pot contracts. This evidence has implications for regulators to control for a relativelystable Germany spot electricity price. For energy companies and investors, it is impor-tant to diversity their existing portfolio rather than having large holdings of individualelectricity contract, for example, investments in LPXBHB, LPXBHRB and LPXBHRPcontracts help to limit the potential risk of loss when there are adverse movements ofspot prices. Another important characteristic of electricity is seasonality, this character-istic is reflected in our dynamic network analysis. We observe the dynamic evolution offull-sample system-wide connectedness in case of weather condition and external uncer-tainty, for example the full-sample system-wide connectedness increased gradually withincreasing temperature from April until September. In general, with the wide range ofpower derivative contracts trading in the German electricity market, we are able to iden-tify, estimated the risk contribution of individual power contract, this helps us to have abetter understanding of the German power market functioning and environment.
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Appendix
Symbol TypesGI GI1.Comdty GI2.Comdty GI3.Comdty GI4.ComdtyGI5.Comdty GI6.Comdty GI7.ComdtyGT GT1.Comdty GT2.Comdty GT3.Comdty GT4.ComdtyGT5.Comdty GT6.Comdty GT7.ComdtyHP HP1.Comdty HP2.Comdty HP3.Comdty HP4.ComdtyHP5.Comdty HP6.ComdtyGJ GJ1.Comdty GJ2.Comdty GJ3.Comdty GJ4.ComdtyGJ5.Comdty GJ6.Comdty GJ7.ComdtyHI HI1.Comdty HI2.Comdty HI3.Comdty HI4.ComdtyHI5.Comdty HI6.Comdty HI7.ComdtyNE NE1.Comdty NE2.Comdty NE3.Comdty NE4.ComdtyNE5.Comdty NE6.ComdtyPOA POA1.Comdty POA2.Comdty POA3.Comdty POA4.ComdtyPOA5.Comdty POA6.Comdty POA7.ComdtyPDA PDA1.Comdty PDA2.Comdty PDA3.Comdty PDA4.ComdtyPDA5.Comdty PDA6.Comdty PDA7.ComdtyPBA PBA1.Comdty PBA2.Comdty PBA3.Comdty PBA4.ComdtyPBA5.Comdty PBA6.ComdtyLPXBHR LPXBHR01.Index LPXBHR02.Index LPXBHR03.Index LPXBHR04.IndexLPXBHR05.Index LPXBHR06.Index LPXBHR07.Index LPXBHR08.IndexLPXBHR09.Index LPXBHR10.Index LPXBHR11.Index LPXBHR12.IndexLPXBHR13.Index LPXBHR14.Index LPXBHR15.Index LPXBHR16.IndexLPXBHR17.Index LPXBHR18.Index LPXBHR19.Index LPXBHR20.IndexLPXBHR21.Index LPXBHR22.Index LPXBHR23.Index LPXBHR24.IndexLPXBHxx LPXBHBMI.Index LPXBHOP1.Index LPXBHOP2.Index LPXBHPMI.IndexLPXBHRBS.Index LPXBHRPK.Index
Table A.1: Phelix Futures data traded at EEX.37 o. Symbol Description1 GI1.Comdty - GI7.Comdty Phelix Base Month Option, and the respectivenext six delivery months2 GT1.Comdty - GT7.Comdty Phelix Base Quarter Option, and the respectivenext six delivery quarters3 HP1.Comdty - HP6.Comdty Phelix Base Year Option, and the respective nextfive delivery years4 GJ1.Comdty - GJ7.Comdty Phelix Peak Month Future, and the respectivenext six delivery months5 HI1.Comdty - HI7.Comdty Phelix Peak Quarter Future, and the respectivenext six delivery quarters6 NE1.Comdty - NE6.Comdty Phelix Peak Year Future, and the respective nextfive delivery years7 POA1.Comdty - POA7.Comdty Phelix Off-Peak Month Future, and the respec-tive next six delivery months8 PDA1.Comdty - PDA7.Comdty Phelix Off-Peak Quarter Future, and the respec-tive next six delivery quarters9 PBA1.Comdty - PBA6.Comdty Phelix Off-Peak Year Future, and the respectivenext five delivery years10 LPXBHR01.Index - LPXBHR24.Index EEX Day-ahead Spot Market with Bid Type from00-01 to 23-24h, e.g. LPXBHR14.Index is EEXDay-ahead Spot price based on bid hours from 13-14.11 LPXBHRxx.Index EEX Day-ahead Spot Market with different BidTypes: LPXBHB.Index is Base Monthly 00-14h;LPXBHOP1.Index is Off Peak1 01-08h; LPXB-HOP2.Index is Off Peak2 21-24h; LPXBHP.Indexis Peak Monthly 08 - 20h; LPXBHRB.Index isBaseload; LPXBHRP.Index is Peakload.
Table A.2: Selected contracts from the file ”Products 2016” provided by European EnergyExchange EEX AG. 38
PXBHR01 LPXBHR02 LPXBHR03 LPXBHR04 LPXBHR05 LPXBHR06 LPXBHR07 LPXBHR08LPXBHR01 1.00 0.71 0.52 0.35 0.53 0.43 0.13 0.14LPXBHR02 0.42 0.59 0.47 0.36 0.31 0.18 0.06 0.07LPXBHR03 0.27 0.41 0.54 0.42 0.36 0.23 0.12 0.12LPXBHR04 0.19 0.32 0.41 0.53 0.38 0.25 0.21 0.15LPXBHR05 0.25 0.24 0.30 0.33 0.47 0.37 0.19 0.15LPXBHR06 0.18 0.12 0.18 0.18 0.34 0.41 0.22 0.23LPXBHR07 0.10 0.08 0.13 0.20 0.22 0.29 0.45 0.40LPXBHR08 0.10 0.08 0.13 0.14 0.18 0.30 0.45 0.54LPXBHR09 0.11 0.08 0.14 0.15 0.18 0.35 0.58 0.70LPXBHR10 0.25 0.23 0.23 0.33 0.49 0.66 0.46 0.49LPXBHR11 0.27 0.28 0.26 0.35 0.50 0.61 0.42 0.46LPXBHR12 0.26 0.29 0.23 0.30 0.42 0.48 0.28 0.31LPXBHR13 0.23 0.30 0.24 0.30 0.39 0.45 0.21 0.27LPXBHR14 0.16 0.15 0.17 0.23 0.26 0.30 0.13 0.13LPXBHR15 0.22 0.18 0.17 0.19 0.22 0.27 0.14 0.13LPXBHR16 0.07 0.06 0.10 0.13 0.19 0.34 0.36 0.41LPXBHR17 0.13 0.13 0.18 0.22 0.35 0.51 0.29 0.36LPXBHR18 0.13 0.11 0.15 0.20 0.32 0.47 0.24 0.27LPXBHR19 0.15 0.11 0.12 0.24 0.32 0.45 0.33 0.30LPXBHR20 0.10 0.07 0.08 0.17 0.24 0.38 0.32 0.28LPXBHR21 0.06 0.07 0.05 0.12 0.17 0.28 0.27 0.25LPXBHR22 0.11 0.13 0.11 0.12 0.19 0.31 0.23 0.26LPXBHR23 0.10 0.12 0.06 0.07 0.11 0.18 0.12 0.14LPXBHR24 0.00 0.00 0.01 0.01 0.01 0.00 0.00 0.00LPXBHR09 LPXBHR10 LPXBHR11 LPXBHR12 LPXBHR13 LPXBHR14 LPXBHR15 LPXBHR16LPXBHR01 0.16 0.25 0.27 0.26 0.23 0.13 0.08 0.03LPXBHR02 0.07 0.15 0.19 0.20 0.21 0.07 0.04 0.03LPXBHR03 0.10 0.13 0.14 0.13 0.14 0.07 0.05 0.05LPXBHR04 0.12 0.21 0.23 0.20 0.20 0.11 0.10 0.15LPXBHR05 0.12 0.24 0.25 0.22 0.21 0.11 0.08 0.11LPXBHR06 0.20 0.29 0.27 0.23 0.22 0.13 0.12 0.15LPXBHR07 0.38 0.27 0.26 0.20 0.16 0.09 0.09 0.16LPXBHR08 0.53 0.28 0.27 0.19 0.17 0.08 0.08 0.09LPXBHR09 0.73 0.33 0.31 0.22 0.19 0.10 0.11 0.09LPXBHR10 0.45 PXBHR23 0.14 0.40 0.41 0.44 0.39 0.22 0.17 0.27LPXBHR24 0.00 0.03 0.04 0.07 0.07 0.02 0.01 0.03LPXBHR17 LPXBHR18 LPXBHR19 LPXBHR20 LPXBHR21 LPXBHR22 LPXBHR23 LPXBHR24LPXBHR01 0.13 0.13 0.15 0.10 0.06 0.11 0.10 0.00LPXBHR02 0.08 0.07 0.08 0.07 0.14 0.17 0.22 0.38LPXBHR03 0.09 0.08 0.07 0.06 0.11 0.12 0.15 0.42LPXBHR04 0.16 0.16 0.18 0.15 0.17 0.16 0.18 0.36LPXBHR05 0.17 0.15 0.16 0.14 0.19 0.18 0.22 0.50LPXBHR06 0.22 0.20 0.21 0.19 0.25 0.24 0.27 0.55LPXBHR07 0.20 0.19 0.23 0.22 0.20 0.18 0.16 0.17LPXBHR08 0.20 0.16 0.18 0.18 0.21 0.20 0.17 0.20LPXBHR09 0.24 0.17 0.20 0.19 0.17 0.18 0.10 0.00LPXBHR10
Table A.3: Population connectedness table for LPXBHR contracts.40
PXBHB LPXBHOP1 LPXBHOP2 LPXBHP LPXBHRB LPXBHRPLPXBHR01 0.29 0.49 0.10 0.10 0.29 0.18LPXBHR02 0.30 0.48 0.12 0.14 0.30 0.18LPXBHR03 0.28 0.50 0.07 0.04 0.28 0.18LPXBHR04 0.38 0.54 0.10 0.12 0.38 0.27LPXBHR05 0.53 0.78 0.15 0.14 0.53 0.40LPXBHR06 0.66 0.75 0.24 0.08 0.66 0.55LPXBHR07 0.43 0.43 0.20 0.01 0.43 0.36LPXBHR08 0.44 0.40 0.20 0.01 0.44 0.39LPXBHR09 0.42 0.36 0.20 0.01 0.42 0.37LPXBHR10
LPXBHR11
LPXBHR12
LPXBHR13
LPXBHR14 0.49 0.29 0.22 0.33 0.49 0.56LPXBHR15 0.43 0.23 0.17 0.21 0.43 0.51LPXBHR16 0.52 0.29 0.31 0.14 0.52 0.59LPXBHR17 0.79 0.48 0.35 0.24 0.79
LPXBHR18 0.77 0.44 0.37 0.24 0.77
LPXBHR19 0.75 0.43 0.50 0.30 0.75
LPXBHR20 0.72 0.37 0.65 0.31 0.72 0.73LPXBHR21 0.65 0.33
Table A.4: The “To” impacts from the six LPXBHxx-type power contracts to the 24LPXBHR-type contracts. The numbers larger than 0 . ug Sep Oct Nov Dec 2015 Feb Mar Apr May Jun Jul time t o t a l c onne c t edne ss Figure A.1: The time-varying full sample system-wide connectedness estimated at differ-ent forecast horizons, from August 2014 to July 2015. The solid line is C h ,t , the dashedline is C h ,t , and the dotted line is C h ,t,t