Forecasting the
day-ahead Dutch Electricity
futures price
Abstract
In this paper we forecast the Dutch electricity futures price for the next day. Using data from 2014 up to and including 2016 an ARCH/GARCH model is specified including exogenous regressors. These exogenous regressors are tested on stationarity and cointegration. The found cointegration relationships are included in the forecasting model. Together with the use of a combination of rolling windows and the adaptive LASSO as a variable selection method a model is constructed. This model is compared to an AR(1) model and it is seen that the former yields higher forecasting accuracy.
Kasper de Harder (10274391) 31-07-2017
Supervisor: dr. M.J.G. (Maurice) Bun Second reader: prof. dr. H.P. (Peter) Boswijk Faculty of Economics
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Index
1 Introduction ... 2 2 Theoretical framework ... 4 2.1 Electricity market ... 4 2.1.1 Base load ... 42.1.2 Dutch power production ... 4
2.1.3 The Dutch electricity grid... 6
2.1.4 Exchange rate and EUA... 7
2.1.5 Electricity futures ... 8
2.1.6 Traditional futures pricing ... 8
2.2 Econometric methodology ... 9
2.2.1 Stationarity ... 9
2.2.2 Cointegration ... 11
2.2.3 Structural breaks ... 13
2.3 Model construction and testing ... 14
2.3.1 Model selection ... 14
2.3.2 Variable selection ... 15
2.3.3 Out of sample testing ... 17
3 Data description and analysis ... 18
4 Methodology ... 21 4.1 Extension ... 23 4.2 Expectations ... 23 5 Results ... 24 5.1 Day ahead ... 24 5.2 Week ahead ... 26 5.3 Cointegration... 27 6 Conclusion ... 28 References ... 30 Appendix A ... 32 Appendix B ... 32
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1 Introduction
Electricity, one of the most important commodities in the world. It is used in vast amounts in every industry worldwide. Electricity can be bought and sold on (certain) markets and so the price is depending on supply and demand. This is similar to other commodities (e.g. oil) but there are two important unique characteristics. The first one is that oil is a natural resource, extracted from the earth, electricity on the other hand is not. Although it is present in nature in the form of lightning, all the electricity that we use is generated. Because of this the supply depends on the factors involved in the generation of electricity. The second and probably biggest difference between electricity and other commodities is the storability. Unlike oil, coal and even gas, electricity can’t be stored. Directly after it has been generated it has to be used somewhere. Methods to store it in a different form of energy1 are not available on large scale with the current technology.
The resources from which electricity can be generated can be divided into two categories. On one hand there are renewable energy sources which can be replaced and will not run out. On the other hand are the non-renewable energy sources, which cannot be replaced once they are all used. Both have their drawbacks and advantages. Non-renewable energy sources are traditionally the most used form for the production of electricity. In the Netherlands 87,6% of the electricity that is produced comes from this category (CBS, 2016). It includes power from nuclear reaction and fossil fuels (primarily coal, gas and oil). Advantages with these kind of energy sources is the relative cheap way of production and the independency of the weather to generate electricity. On the other side we have the environmental harm, carbon dioxide which is emitted during the burning of fossil fuels and the radioactive waste from nuclear fission. The other disadvantage is that non-renewable sources, as said before, are limited and therefore one day will be depleted. These disadvantages make that renewable energy is seen as the future. Electricity generated from renewable energy causes almost no harm2,3 to the environment and are from a source which can’t be depleted. But there are also some drawbacks to the usage of these energy sources. First are the higher costs compared to non-renewable energy and second is the
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Chemical energy as with a battery or kinetic energy as with a reservoir lake 2
Biomass is arguably as carbon dioxide neutral as fossil fuels
3 The construction of power plants do have an impact on the environment (e.g. reservoir lakes for hydro plants)
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dependency on uncontrollable factors like the weather for the production of electricity4.
The second unique aspect of electricity, the non-storability, forces supply and demand to be in equilibrium. When this is not maintained black outs can be the result. To keep the equilibrium the use of non-renewable energy sources for electricity production is preferred as they are not dependent on factors like the weather. As a result, when a company would plan an increase in production and thus electricity usage for the next month, it is dependent on the price of generation in the next month.
The dependency of electricity on renewable energy sources and the
non-storability cause uncertainty about the price in the future. This uncertainty can cause risks for companies which they want to avoid. One solution is given by a futures contract, it is an agreement for the delivery of a certain good in the future for a certain price per unit. The contract gives certainty about the future price of electricity but the price of the contract does change day to day.
Forecasting the next day price of the electricity futures contract is the main focus of this paper. Earlier work focuses on forecasting models with and without exogenous variables (Contreras, Espinola, Nogales, & Conejo, 2003; Conejo, Plazas, Espinola, & Molina, 2005; Cuaresma, Hlouskova, Kossmeier, & Obersteiner, 2004) or cointegration relationships (de Jong & Schneider, 2009; Bower, 2002; Zachmann, 2008) of energy commodities. The combination of these with exogenous variables and techniques like the adaptive LASSO distinguishes this paper from previous literature. Next to this our focus is on the Dutch power market in contrast to majority of the literature about European power markets which mostly included the German, UK and Nordic markets. By focussing on the Dutch market, prices from leading markets like Germany and the Nordics can be used as explanatory variables.
The remainder of this paper is organized as follows. In section two existing literature on the various aspects of this research are discussed. Section three is about the data used. In section four the construction of the forecasting models is discussed. Section five include the results of comparing s of these forecasting models. Our concluding remarks will complete this paper.
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2 Theoretical framework
The theoretical framework on which this research is based will include an overview and explanation about the electricity market, factors influencing the electricity price and the electricity futures. Afterwards the mathematical background of the applied techniques will be discussed.
2.1
Electricity market
2.1.1 Base load
As explained before electricity can’t be stored on large scale like other commodities. This causes that demand and supply have to be in equilibrium5 at all times. To make sure this is the case electricity generation should be adjusted to the demand. Although it may sound straightforward this takes more planning than one may think. The main reason for this is the start-up6 time of a power plant, which can take from less than an hour7 up to a whole day8 (or more). When demand fluctuates, supply should be adjusted quickly. Power plants with relative short start-up time are most suitable help maintaining the equilibrium of demand and supply. Power stations with relative longest start-up time on the other hand, are preferably always “on” to provide a certain minimum amount of electricity. This minimum level of supply should be equal to the minimum level of demand. The term used for the minimum amount of
demanded electricity over 24 hours is base load. All the demand above this base line is called peak load and this fluctuates over the course of a day. Because base load is relatively constant and not flexible it is cheaper than peak load. Throughout this paper the focus is primarily on the price of base load.
2.1.2 Dutch power production
In 1886 the Netherlands got its first power station consisting of a steam engine with two dynamo’s together producing 15 kW. Now, more than 130 years later, the Netherlands have an electricity production capacity of 31,5 million kW (CBS, 2015). In
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Small deviations are accepted
6 The speed at which the power output can be increased and minimum power output is also of importance but when start up time is low, output adjusting time and minimum power output are usually also low
7 Gas turbines and wind turbines 8
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1998 the power supply and the management of the Dutch electrical grid9 has been split up (before there was one company supplying and maintaining the grid). Since that liberalisation TenneT10 is the company responsible for the maintenance and management of the grid in the Netherlands. The other branch, the supply of the electricity, is taken care of by commercial energy companies like Essent, Nuon and Eneco. They are responsible for the production and delivery of power to consumers. It is interesting to know that this production is centralized production, produced at power plants, and accountable for around 62% of the total power generation in the last few years. The other 38% is generated decentral, mostly by the agricultural,
horticultural11 and chemical industry. The vast majority of this locally produced electricity is used by the industry itself.
In Figure 1 the total amount of electricity produced in the Netherlands per source can be seen. The figure shows a steady increase until the peak in 2010 from where the total production started to decrease again. Besides the economic downfall this could also be explained by the growing international electricity trade (CBS, 2015). Looking at the energy sources separately we notice a decreasing share of fossil fuels in the total power production and a constant absolute amount of nuclear and renewable energy. But although the conventional fossil fuels as a whole are slowly losing ground they still make up for approximately 80% of the Dutch power production in 2014.
Figure 1. Total electricity production in the Netherlands between 2006 and 2014 split up in different sources (Breakdown of Electricity Generation by Energy Source, 2016)
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The interconnected network for delivering electricity from suppliers to consumers 10
TenneT actually bought the high voltage grid piece by piece until it owned the whole grid by 2010
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Hortus means garden in Latin
0 20000 40000 60000 80000 100000 120000 2006 2007 2008 2009 2010 2011 2012 2013 2014 G W h
Electricity production in the Netherlands
Other Biomass and Waste Wind Nuclear Oil
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The large share of fossil fuels suggests that price changes in these commodities could cause for price changes in the same direction for electricity (correlation). The different prices could also exhibit some certain constant balance or co-movement in the long term, this is called cointegration. This concept will be explained later in this paper. But the presence of cointegration between the electricity market and various other energy markets (mostly oil, gas and coal) have been found by many researches in other European countries. Bunn and Fezzi (2007) found that the natural gas and European Allowances (EUA) spot price were cointegrated with the electricity price in the UK. Asche et al. (2006) couldn’t reject the null hypothesis of cointegration of the natural gas, electricity and crude oil spot prices in the UK. The Dutch spot market on which gas is traded (TTF) is cointegrated with the power spot market but on a “forward time scale” (de Jong & Schneider, 2009). Frydenberg et al. (2014) find cointegration
between front future electricity prices in Europe (UK, Germany and the Nordic region) and oil, gas and coal future prices.
2.1.3 The Dutch electricity grid
The high voltage electricity grid of the Netherlands is currently12 connected to that of Belgium, England, Norway and Germany. The connection capacities vary between the countries. On three places the Dutch grid is connected to that of Germany which is the connection with the biggest capacity and as a result the most import and export of electricity is happening with the eastern neighbours. On average the Netherlands have a positive import balance with Germany and Norway and a negative balance with England and Belgium (CBS, 2015). The amount of power imported in 2012-2015 was around 25% of the total consumption in those years (CBS, 2016). These international connections have caused significant cointegration between the electricity prices in the Netherlands and the prices in Germany, England & Wales and the Nordic market13 already in 2001 (Bower, 2002). Other recent studies have found comparable
cointegrated European electricity markets while looking at prices from 2002 to 2006 (Zachmann, 2008) and electricity spot prices in the years 2009 to 2011 (Houllier & Menezes, 2013).
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See http://www.tennet.eu/our-grid/international-connections/about-international-connections/ for information about future planned international connections
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Since 2015 a new procedure for the capacity calculation of the interconnectors14, the flow based market coupling (FB), is active in the CWE (Central West Europe) region. This new procedure enables the optimization of the transmission capacity of the grid. FB resulted in an increase in import and export capacities (TenneT, 2016). The predecessor used fixed values for interconnection capacity between the countries in the CWE region. This may sound technical but because of the increased import and export capacities the relation between Dutch electricity prices and that of the other countries in the CWE region is expected to change.
2.1.4 Exchange rate and EUA
Next to the fuels which are used for electricity generation and the connection of the grid with other big markets there are at least two more apparent influential factors for the Dutch energy price.
The first one is the currency exchange rates which affects all international trading when multiple currencies are involved. As oil is globally traded in US dollars, the value of oil in Euros is both depending on the oil price and on the exchange rate Euro/USD. As the same holds for coal, also traded in dollars, the exchange rate very likely has a connection with the Dutch electricity base load price.
The other one is the EUA price, this is the price tag that has been put on the emission of one ton of CO2. The amount of EUA’s is limited to restrict the amount of carbon
dioxide emitted by companies in Europe. After the economic crisis in 2009 there were more of these allowances than needed which caused a drop in price and made them rather ineffective. Last few years the European economy has been growing again and the price of the EUA permits have started to rise and therefore will possibly affect the electricity price.
As a large part of the electricity generation in the Netherlands is done by coal and gas, we could argue that the two are substitutes. In this case, following basic economic theory, a price increase in gas leads to a higher demand of coal and so the influence of its price increases and vice versa. But as the use of coal for power production produces more carbon dioxide than that of gas, the price of the EUA’s starts to play a more
14 Electricity interconnectors are the physical link which allow the transfer of electricity across borders
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important role. So as the coal/gas price ratio decreases, the EUA price has an increasing influence on the electricity price (Seebregts & Volkers, 2005)
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2.1.5 Electricity futures
The electricity futures on which this paper focuses are traded on the ICE ENDEX. This is the term market for electricity and gas in the Netherlands for monthly, quarterly and yearly futures contracts. As mentioned before a futures contract is an agreement to buy or sell a certain good at a specified price and date in the future. In this paper we focus on the yearly contracts in which the agreement is about the price per unit of electricity that has to be paid in one of the coming years15. The price of these contracts are set on the market, depending on future expectations which influence demand and supply. In Figure 2 the price for this future (year ahead) is shown for the last twelve years.
Figure 2. Price level of the Dutch base load electricity futures from January 2004 until October 2016
2.1.6 Traditional futures pricing
Economic theory gives us the so called spot-future parity and suggests that no-arbitrage price for a futures contract is equal to:
In this equation F0 stands for the price of the futures contract at time t=0, on the right
side we have the spot price at t=0 (S0) multiplied by an exponential term including the
15 Until five years into the future these yearly futures are traded where a contract duration is from the beginning of January to the end of December of the specified year
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risk free interest rate (r), storage cost (u), convenience yield (y), dividend yield (q) and time until delivery date in years (t). This can be rewritten as:
This simply states that the price of a futures contract should be equal to the spot price corrected for the cost (c) and the benefit (b) of carrying the underlying asset until the delivery date. This parity can be applied to storable assets only and that doesn’t hold for asset of interest. As electricity can’t be properly traded, traditional valuation approaches (like the one showed here) can’t be used (Pirrong, 2005). To avoid this problem we approach the price forecasting by looking at other explanatory variables and their movements.
2.2
Econometric methodology
In the previous section the factors that could influence the futures price of electricity were discussed. This section will focus on the mathematical properties and relations of the electricity futures price and the variables (expected to be) affecting it. Based on these the appropriate forecasting method can be constructed.
2.2.1 Stationarity
When predicting time series one of the main concerns is stationarity16. When
regressing multiple non-stationary time series on each other it can lead to a spurious regression (Granger & Newbold, 1974). This happens when the series are not causally related but do exhibit a similar trend. As our research consists of several different energy commodities, a similar trend among them could very well be possible. To avoid spurious regressions the series should be de-trended when they are found to have one (non-stationary). A process Yt is said to be stationary if the first and second moments
are time invariant:
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The first two conditions are stating that the expected value is the same for every t and the variance is constant and finite on every moment in time, respectively. The third condition is saying that the autocovariance only depends on the decay on time and not in the time itself. If the assumption of stationarity is violated forecasts can be
unreliable.
To check for stationarity of a time series a unit root17 test is applied. Arguably the most used one is the Augmented Dickey-Fuller18 (ADF) unit root test (Dickey & Fuller, 1981). It has the null hypothesis of a unit root present in the series against the alternative that there is no unit root present. So if the null hypothesis is rejected it can be concluded that the time series is stationary. There are a few disadvantages about this test; the first is that under the null hypothesis of the variable being non stationary, the t-test statistic does not follow the usual t-distribution. This causes the problem that the usual critical values for the statistic are invalid. To solve this critical values have to be
simulated. Dickey and Fuller simulated these critical values with the use of the limiting distribution. Later MacKinnon (1990) simulated these critical values for finite samples and he simulated more accurate values in 2010.
The second problem with the ADF test is the choice of the lag length of the dependent variable to correct for autocorrelation. Information criteria (will be discussed later) are mostly used to choose the right length, but Ng & Perron (1995) argue that the use of a t-test is preferable over an information criterion. The reason they give for this is that the t test would have more robust size properties as it tends to prefer more lags in the model19. On the other hand the power of the test decreases as the lag length increases. The intuitive reason for this is that, as the lag length tends to infinity, the lagged terms become asymptotically colinear (Paparoditis & Politis, 2016). So the trade-off must be made between a good size (more lagged terms) and a good power (less lagged terms). A solution to avoid choosing a particular lag length is the use of the Phillips-Perron (PP) test (Phillips & Perron, 1988). This is a non-parametric test which correct for any serial correlation and heteroskedasticity by directly modifying the test statistic.
The third problem arises when the series is close to containing a unit root, but doesn’t have one. In this case the ADF test can fail to reject the null hypothesis of being
17 If a time series has a unit root it is by definition non stationary 18
This test is extended version of the original Dickey-Fuller test, in which there is corrected for autocorrelation
19 Although an overparameterized model tends to have low power, this diminishes as sample size increases
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stationary. To overcome this problem a test can be used which has the null hypothesis of stationarity like the KPSS test (Kwiatkowski, Phillips, Schmidt, & Shin, 1992).
When on the basis of these tests the time series is found to be non-stationary, it should be transformed to make it stationary. Taking differences can be done, this works for establishing stationarity for most economic time series. The amount of differences that have to be taken in order to make a times series stationary is called the order of integration. In the case of taking the first difference, we are effectively predicting the change in price rather than the absolute price.
2.2.2 Cointegration
Different than correlation20, two (or more) variables can be cointegrated. In the case of two variables we speak about cointegration when a linear combination of those two can be constructed which is stationary. As mentioned before cointegration has been found in literature between the electricity price and other price series. Presence of it in our data can be valuable as it holds additional predictive information. An intuitive explanation of cointegration can be as follows: “A drunk and a puppy wander around in the park, both walks could be seen as independent random walks. There would be no relationship to discover between the two walks as they are both random and
unrelated. But what if the dog belongs to the drunk? The dog and its owner will probably stay within a certain distance from each other, once you see one of the two the other one is unlikely to be far away. If this is right, then the distance between the two paths is stationary, and the walks of the drunk and her dog are said to be
cointegrated.” (Murray, 1994).
Probably the most intuitive way to test for cointegration is by using the two-step procedure recommended by Engle and Granger (1987). In the case of two time series (Xt and Yt which are both I(1)) we first estimate the long-run (equilibrium) equation :
The OLS residuals from this are a measure of the distance from the long run equilibrium.
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When we want to test for cointegration of Yt and Xt we can do so by testing whether ût
is stationary. This can be done using the stationarity tests discussed before. The problems of this approach has been discussed extensively in literature and different solutions have been proposed over the years. Bo Sjö (2008) points out three main problems. Firstly, the shortcomings of the ADF test and the critical choice of the number of lags in the augmentation. Secondly the assumption of one cointegration vector on which the test is based on and the third is the test assumption of the presence of a common factor in the dynamics of the system. Sjö suggests testing cointegration in a single equation Error Correction Model (ECM) and points out the superiority of Johansen’s test (Johansen, 1991). One of the problems of the Engle and Granger approach is that, since ût are estimates, new critical values need to be
tabulated. To solve this problem the improved21 MacKinnon (199022) critical values for cointegration test should be used. Next to using the ADF test on the residuals the Phillips-Perron unit root (known as the Phillips-Ouliaris cointegration test) can also be conducted. Another problem we can encounter is the choice for the dependent variable in the first stage regression, we need the regressors in this regression to be at least weak exogenous.
The Johansen test, mentioned before, has a big advantage over the Engle-Granger method in the way that permits more than one cointegration relationship. So different combinations of non-stationary variables which are stationary can be found at once. In short this test constructs a Vector Error Correction Model (VECM) after which it
calculates the rank of the matrix. Johansen originally derived two tests, the
eigenvalue test and the trace test, both for calculating the rank of the specific matrix. It has been found that the trace test is better since it appears to be more robust to skewness and excess kurtosis. Cointegration vectors found through the Johansen test are less intuitive as they will consist of all the variables used in the test. Another important difference between the Engle and Granger procedure and the Johansen test is that the latter is subject to asymptotic properties and thus has a lower power for finite samples.
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MacKinnon implements a much larger set of simulations for the tabulation of critical values and estimates response surfaces for the simulation results
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When the presence of such long run equilibriums is confirmed, any off equilibrium cointegration vector is expected to be corrected in the next period. Effectively increasing our forecast accuracy.
2.2.3 Structural breaks
The influence of certain variables on the electricity futures price can change, a possible example of this can be the new procedure used for the capacity calculation of the interconnectors (mentioned in 1.1.3). If as a result more German power is exported to the Netherlands, the German power price will probably have an increased effect. Such a change is called a structural change or break. Abrupt structural changes can possibly be present, but as Hansen (2001) points out, “it may seem unlikely that a structural break could be immediate and might seem more reasonable to allow a structural change to take a period of time to take effect.” A smooth structural change can be a more plausible expectation in various situations.
The Chow test (Chow, 1960) was the first to test for a structural break with a known break date. Since then a vast amount of work has been done on issues related to structural change. The Quandt Likelihood Ratio test (QLR) (Quandt, 1960) is an
extension to the Chow test. The QLR statistic is the maximum of all Chow statistics over a range of break dates.
When sufficient proof of the presence of a structural break is found the coefficient estimates of a regression using the whole sample period will be biased. Using an estimation window which include some pre-break data could be advantageous because of the important trade-off between bias and forecast error variance (Pesaran &
Timmermann, 2007). But Pesaran and Timmerman also notice that in practice it is hard to time the break or multiple breaks which in turn makes it hard to choose the optimal window length to fully exploit the bias-variance trade-off. As a solution to overcome this difficulty they suggest the use of multiple estimation windows and combine the forecasts off these different estimated models.
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2.3 Model construction and testing
2.3.1 Model selection
A popular and simple model used in forecasting economic time series is the Auto-Regressive-Integrated-Moving-Average (ARIMA23) model. This type of univariate model tends to capture and use historic values to make future predictions for stationary time series. It has been used for forecast many different commodities including the
forecasting of oil futures prices (Moshiri & Foroutan, 2006; Sadorsky, 2002) and the day ahead electricity price (Contreras et all, 2003; Conejo et all, 2005). An important reason for the widespread use the ARIMA model is the relative simplicity of setting it up and the ability to predicits multiple steps ahead.
Economic theory often suggests other variables that could help to forecast the variable of interest. When finding that certain variables X causes Y, lagged values of these stationary variables X can be added to the model to form an Autoregressive Distributed Lag model (ADL). Although extra valuable information could in this way be used in the model for forecasting, the amount of steps ahead which can be predicted is limited to the available explanatory variables.
The levels of the regressors could be cointegrated, these relationships can be exploited in an Error Correction model (ECM). This mechanism enables us to model the short-run dynamics between the cointegrated series. The idea is that the long run equilibrium between two or more time series can be estimated, based on this a disequilibrium from one period would be corrected in the next. This close relationship between cointegration and error correcting models was first suggested by Granger (1981) and later extended by Engle and Granger (1987). LeSage (1990) finds that the ECM model produces forecasts with much lower errors than any of the alternative VAR models. While the before mentioned models can be used to forecast the mean of the
dependent variable, they assume constant variance. But when examining the residuals of so called mean models, interesting things can sometimes be found. In economic time series it is not uncommon that there are periods of high and periods of low variance. The Auto Regressive Conditional Hetereoskedasticity (ARCH) and its
“Generalized” (GARCH) version can be useful in this situation. The former, introduced
23 Integrated stands for the order of integration of the dependent variable, if we are using a stationary variable this will be equal to 0 and the model boils down to an ARMA model
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by Robert Engle (1982), uses the plausible autocorrelation of the errors variance to give a better variance forecast. The GARCH model (Bollerslev, 1986) is an extension which allows for more flexible lag structure. This variance modelling has shown useful, particularly in financial applications, for analyzing and forecasting variance. But the ARCH and GARCH models have some disadvantages, the ‘standard’ version cannot model asymmetries. This can be a drawback as it could be argued that negative shocks have a larger effect than positive ones for many financial time series (also called the leverage effect). A test that can be used to check if this asymetric effect is present is the sign bias test (Engle & Ng, 1993). If true, asymetric GARCH models, like the popular Exponential GARCH (EGARCH), have been suggested to overcome this problem. Other then this, the ARCH and GARCH coefficients are not quite intuitive and hence are mostly only used for forecasting purposes and not for analysis. Estimation, mostly done with the use of maximum likelihood (because of the nonlinearity), can get
compuational intensive compared to the previous linear models.
2.3.2 Variable selection
Including past prices of different commodities is expected to contribute to the predictive power of our model. But the more lags we add the larger the amount of explanatory variables. To avoid overfitting we need to make a selection of the available variables. Below we discuss some approaches for this variable selection.
2.3.2.1 T and F tests
When Ordinary Least Squares (OLS) is used and the assumptions are met, the estimated standard errors of the estimator are valid. The t-test and F-test can in this case be used to check if, respectively, an individual or several variables together have a significant effect. The model can be improved by adding and/or deleting terms while taking the previous two statistics into account. This is known as forward and backward selection respectively. This method doesn’t check for all possible parameter
combinations and it only tells us if a variable is seemingly significant not if one model fits better than another.
2.3.2.2 Information criteria
To compare models objectively to each other Information Criteria (IC’s) have been created. First introduced by Akaike (1974) it has been used extensively in researches
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when model selecting was important. Since Akaike’s information criteria (AIC24) many variations have been made on the original formula. With some exceptions the general formula for such a criterion is made up of a log likelihood which measures the fit of the model on the data and a penalty term for the number of regressors. This penalty term differs per IC and has a strong influence on the complexity of the preferred model. The two most used are the AIC and the BIC (or SIC25) both shown below;
The latter has a stronger penalty (when the number of data points, n, is bigger than 7) on additional variables added to the estimated model. In practice the AIC tends to favour over fitted models (Hurvich & Tsai, 1989). On the other hand, the BIC is based on the idea that the “real” model is in the selection of models, which is unlikely in most cases. This lead to the BIC sometimes choosing under fitted models.
2.3.2.3 Adaptive LASSO
The last two techniques needed the models first to be calculated in order to select the most favourable one. This, like stated before, can be computational unattractive to look at all possible models if the amount of parameters is large. A selection technique which avoids this problem is the adaptive Least Absolute Shrinkage and Selection Operator (adaptive LASSO). This technique simultaneous selects and estimates the explanatory variables. It was introduced by Zou (2006) as an improvement of the original LASSO (Tibshirani, 1996). In short LASSO estimates the parameters of the model by least squares under the restriction that the sum of the absolute values of the coefficients is smaller than some tuning parameter t (this is an L1 norm penalty). The addition of Zou’s adaptive LASSO are the adaptive weights used for penalising different coefficients in the L1 penalty. This algorithm satisfies the oracle properties which, in linear regression, can be formulated as; An oracle estimator must be consistent in parameter estimation and variable selection. But to obtain that consistency a trade-off is made with regards to the bias of the estimates.
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Actually Akaike intended the abbreviation to mean “An Information Criterion”
25 BIC was developed by Akaike at the same time Schwarz developed the SIC, these criteria are totally equivalent
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2.3.3 Out of sample testing
When the available data is split into a training set and a test set, the chosen model can be trained and tested on “new” data. This can be used for determining the quality of the forecasts and with that performance of the model. In such a way by testing a few models, model selection based on out of sample testing can be done. However there are a few drawbacks, as this does make the model better at predicting already known values, it does not necessarily improve the future forecasts. Secondly, which measure should be used for the performance of the out of sample tests?
The mean absolute percentage error (MAPE) has become a commonly used measure. But the measure possesses some disadvantages as it is undefined if the real value is zero and it also puts a heavier penalty on negative error than positive ones. Another popular measure is the root mean squared forecast error (RMSFE), this absolute measure doesn’t suffer from the same points as the MAPE. Nevertheless it has some shortcomings, with its main drawback being scale dependent and besides this it is sensitive to outliers. Hyndman and Koehler (2006) propose that the mean absolute scaled error (MASE) should be used for comparing forecast accuracy across multiple time series. The method scales the forecast errors by the in sample mean absolute error obtained using the naïve forecasting method.
Next to measuring the accuracy of the point forecast we measure the accuracy of the forecast direction (upward or downward). The Mean Directional Accuracy (MDA) is used for this, it consists of the sum of the cases in which the forecasted direction was correct divided by the amount of forecasts. The simplicity makes it interpretable and therefore it is used as a secondary measure in this paper.
Measuring the accuracy of the prediction interval is much less intuitive. The accuracy of the predicted sigma (which is used to create the prediction interval) can’t be measured based on one forecast because it is the deviation from the point estimate. In other words, if the forecast error is very close to sigma that doesn’t necessarily mean the sigma forecast is good. In order to say something about it we will look at the percentage of times when the real value lies within the prediction interval. Another interesting result can be obtained after analysing the prediction interval, whenever zero is not included (when predicting the change in the dependent variable) we
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conclude that the change is significant26 different from zero. Comparing this to the amount of times this was true we can say something about how reliable those more extreme predictions are.
3 Data description and analysis
The data used in this paper to estimate the forecasting model is published on the ICE Endex. It consists of the historical data of the Dutch, German and Norwegian electricity futures prices of yearly contracts. Furthermore yearly futures prices of natural gas and coal are used as well as December month future exchange rate and futures prices of oil and EUA (a more detailed list of these variables is given in Appendix A). The choice for the monthly futures of the latter two is made based on the trading volume27. The prices of all of these contracts are published daily on working days, the sparse few days on which a few of these were not published have been linearly interpolated28. The data ranges from 02-01-2014 till 09-12-2016 and consists of 782 data points. This dataset consists of the year ahead29 contract prices.
In Figure 3 the electricity futures price of Holland, Germany and the Nordics is shown. A downward trend until 2016 can be seen after which it has started to climb again. This increase in price hasn’t been witnessed since the crisis of 2008. As expected of a year contract, no clear signs of seasonality or any other clear repetitive movement can be discovered.
26
The significance is depending on the chosen prediction interval
27 Trading volume for yearly futures of oil, EUA and FX are very low making the price of those less representative
28
3% of the data points are created this way
29 The contract which matures first (next year or next December in case of the monthly contracts)
19
Figure 3. The price series of the electricity futures of Holland, Germany and the Nordics
In Figures 4 and 5 the Dutch electricity future is displayed together with the fossil fuel30 prices and two other related series31 respectively. In all three figures we see similar movements in all the series and as shown in Table 1 all are non-stationary series. This combination hints towards possible cointegration relationships.
Figure 4. Price levels compared to the base p rice (31-12-2013) of Electricity, Gas, Coal and Oil futures
30
Dutch gas futures price, coal and oil futures prices are from the global market. More about these price series in Appendix A
31 The European price for EUA and the FX futures are from an American market. More about these price series in Appendix A
20
Figure 5. Price levels and exchange rate compared to the base price (31-12-2013) of Electricity, EUA and USD/EUR exchange rate futures
Stationarity tests ADF32 PP33 KPSS34
EBL.Cal -1.14 -1.13 9.15*** Gas.Cal -1.24 -1.13 10.47*** Coal.Cal -1.46 -1.48 8.81*** Oil.Dec -1.16 -0.98 10.12*** EUA.Dec -1.95 -1.92 1.95*** G.EBL.Cal -1.22 -1.33 9.20*** N.EBL.Cal -1.32 -1.45 8.86*** FX.Dec -1.16 -1.03 9.10***
Table 1. Statistics for three different stationarity tests. Significance level: 0.10*, 0.05** and 0. 01***
Because of the non-stationarity first differences are taken of all the series. These differenced series are stationary (Appendix B).
Table 2 shows the correlation coefficients of Dutch electricity base load with certain lags of the other series. We observe relative high correlation between electricity and some of the series at lag zero. But the correlation coefficients are quickly decreasing when higher lags of the explanatory values are taken. For the first lag most are still significant but in the third column it can be seen that electricity is only correlated significantly with second lags of EUA and gas. A theory to explain the quickly decreasing correlation coefficients is that most information about price changes are incorporated
32
Augmented Dickey Fuller with H0: non-stationary 33 Phillips-Perron with H
0: non-stationary 34
21
into the electricity price relative fast (relative to the frequency of our data, which is daily).
Pearson correlation dEBL.Cal dEBL.Cal (t+1) dEBL.Cal (t+2)
dCoal.Cal 0.49*** 0.16*** 0.05 dGas.Cal 0.65*** 0.26*** 0.07* dOil.Dec 0.10*** 0.20*** -0.01 dG.EBL.Cal 0.72*** 0.12*** 0.03 dN.EBL.Cal 0.47*** 0.14*** 0.02 dEUA.Dec 0.28*** 0.10*** -0.08** dFX.Dec 0.00 0.00 -0.04
Table 2. Pearson correlation coefficients between different commodities and Dutch EBL on the same day and with one and two day lags of the
commodities. Significance level: 0.10*, 0.05** and 0.01***
4 Methodology
Based on the literature review we will construct a model in a few steps which will then be used to forecast. Every day a model with exogenous variables and possible
cointegration vectors together with GARCH(1,1)35 errors is estimated. Although the general structure of the model is the same every day, the available training data does grow36. As a result the model will be slightly different every day, this is clarified by the step by step construction of the model given below.
Step 1
The last n available data points are selected to be used as training data. The number n depends on the length of the window used.
Step 2
Using this data we search and test for possible cointegration relationships, between the independent and the explanatory variables. The Engle and Granger (1987) method and the Johansen test are used to find these. When found the deviations from the long run equilibria are stored. These error correction vectors are then treated as variables in the next step.
35
The order is chosen based on the (partial) autocorrelation functions and the basic GARCH model is used based on absence of leverage effect shown by the sign bias test
36
22
Step 3
The group of available variables now consists of the differenced economic variables (and their lags) (X) and the error correction vectors (Z). The variable selection is done with two different methods;
1. Using OLS a linear regression is constructed including the whole group of variables. The least significant variable is deleted and OLS is applied again using the remaining regressors. Based on the AIC and the BIC the best model is chosen using this backward elimination technique
2. Using the adaptive LASSO on the full group of stationary variables a model is constructed. As it shrinks some of the coefficients to zero we are left with a selection of variables to work with.
Step 4
Incorporating the selected variables the model will have the following general form:
This model will then be estimated using the specified training data.
Step 5
Now that the coefficients of our forecasting model are estimated we can make a day ahead point and standard error prediction.
Step 6
The above five steps are repeated for different window lengths, the forecasts are then combined (the average is taken). So we end up with a point prediction of the change in price and a prediction interval. This interval is created by adding and subtracting one predicted standard error from the point forecast.
Combining these forecasts should, according to theory, make the predictions more robust against structural breaks (which the QLR test hints towards in our data). In the end 300 day ahead predictions are made, to measure the accuracy of point forecasts the MASE is used. Performance of the GARCH component is based on the
23
prediction interval and the fraction of times it was correctly specified. Preferable around 68% of the times this should be the case, similar to the normal distribution. Furthermore prediction intervals not including zero are of interest as we are predicting the change in price. The measures described are then used to compare our forecasting model to a baseline model which is a “simple” AR(1)-GARCH(1,1) model.
4.1 Extension
Although the main focus of this paper is predicting the day ahead price of the Dutch electricity base load future, it is interesting to extend this to a week ahead. As the model described in the five steps above can only forecast one step ahead due to the inclusion of first lagged regressors, a different approach has to be adopted. One solution for this is the use of a univariate time series model ARIMA(p,1,q) with GARCH(1,1) errors. Forecasting five time periods ahead equals the week ahead prediction. With this method extra information that other time series may contain is not included. To still being able to incorporate that, another method is converting daily data points into weekly data points such that the closing value of the week (essentially the closing value on Friday) is used. With these weekly data points a model37 is
estimated using the three variable selection techniques discussed.
4.2 Expectations
For the day ahead prediction four models are constructed, the AR(1) model is the simple baseline model. It is expected to perform reasonably because of the conservative predictions based on the previous value in combination with the
stationarity of the series. The other three “models38” (from here named by the variable selection technique); AIC-s, BIC-s and adLASSO-s39 use more variables and hence more information. An improved forecast compared to that of the baseline model is therefore expected. Of the two models involving the information criteria the BIC-s will be more parsimonious predicting less extreme values than the AIC-s. The more conservative forecasts are expected to yield a lower MASE but will also exclude zero from its prediction interval fewer times, making it less conclusive. The adLASSO-s is harder to place, in theory the adaptive LASSO can select bigger models than the AIC and at other
37
Without ARCH/GARCH errors as too few data points were available for R to calculate it 38 Although every step a new model is constructed they will be referred to as a certain model 39
24
times smaller models than BIC. The consistency of variable selection by the adaptive LASSO that is done in one step compared to backward elimination tends us to expect superior performance compared to the three other models.
5 Results
In this chapter the results are shown for the day ahead and the week ahead forecasts, the last part covers the cointegration relationships found in our dataset.
5.1 Day ahead
In Table 3 the results of the four models is shown, the optimal rolling window length40 is chosen based on the MASE. For all holds that the use of a rolling window improved the forecasting performance over the use of all available data points. The prediction interval is in all the cases two standard errors wide (point forecast ± one estimated standard error).
Model AR(1) AIC-s BIC-s adLASSO-s
Rolling window 200 450 450 250
MDA 0.550 0.563 0.547 0.590
Interval 0.713 0.633 0.697 0.663
MASE 0.753 0.770 0.752 0.730
Inter. ex. zero 0 0.083 0.03 0.053
Contained real 0 0.043 0.02 0.033
Table 3. Results shown for optimal (based on MASE) rolling window length Looking at the MASE we see the AIC-s forecasting the worst, AR(1) and BIC-s close together and the adLASSO-s performing the best although the values are not far apart. Comparing the model without exogenous variables (AR(1)) and with (the other three) doesn’t show a clear improvement in MASE. This is in line with the comparison of including and excluding explanatory variables in predicting the hourly price in the Spanish and California electricity markets (Contreras, Espinola, Nogales, & Conejo, 2003).
The MDA is highest for the adLASSO-s is performing slightly better than the rest with 59%. “Interval” tells us which part of the real values lay inside the prediction interval.
40
25
The last two rows gives us the fraction of the time zero was not included in the prediction interval and the fraction in which the real value was present in such an interval respectively. These last two can be of particular interest as it can be used as a kind of measure for the prediction interval. The ratio between the two values should be close to the value given in the row “Interval”. Because the prediction interval of the baseline model always included zero it strengthen the expectation that AR(1) has more conservative predictions which are closer to zero. The ratios for the last three41
columns are 0.52, 0.67 and 0.62 respectively. Although the fraction of prediction intervals not including zero is low, the ratios of the BIC-s and adLASSO-s are reasonably close the “Interval” values. Thus supporting the validity of the intervals even for more extreme estimates.
According to the QLR test there is evidence42 of a structural break in the forecast period, this gives reason to look at the performance of the models when combining different window lengths. These optimal combinations are shown below in Table 4.
Model AR(1) AIC-s BIC-s adLASSO-s
Rolling window 200 200 – 450 (6) 200 – 450 (6) 250 – 450 (5)
MDA 0.550 0.546 0.563 0.573
Interval 0.713 0.630 0.643 0.690
MASE 0.753 0.756 0.750 0.726
Inter. ex. zero 0 0.107 0.037 0.01
Contained real 0 0.067 0.020 0.01
Table 4. Results shown for optimal (based on MASE) combination of different rolling window lengths
No combination of the AR(1) model with different window lengths performed better than the one already used. For the other models it did cause a marginal improvement of the MASE. To look if the point forecasts of our final adLASSO-s model are
significantly better than those of the baseline model we use the Diebold-Mariano test (Diebold & Mariano, 1995). This test shows that the improvement of our forecasting model over the simple AR(1) model is significant at a 10% level (p = .0947).
Based on the results in the “Interval”, “Inter. ex. zero” and “Contained real” we see a change the adLASSO-s model. Although the ratio of the last two rows is 1, only 1% of
41
As 0/0 from the first column can’t be done obviously
42 The test is conducted using one static model produced by using the AIC/backward elimination technique over all available data
26
the prediction intervals didn’t contain zero, compared to 5,3% before combining the different estimation windows. In Figure 6 the day ahead forecasts for the Dutch electricity futures price is plotted using the last 100 forecasts of the adLASSO-s model with the optimal window length combination.
Figure 6. The day ahead forecasts with one sigma upper and lower bound (grey shaded area) together with the real values for the last 100 points.
5.2 Week ahead
The performance of the week ahead forecasts is given in Table 5. Not surprisingly the MASE values are higher compared to the day ahead forecasts. An interesting point in this table that, although it has the lowest MASE, the AR(1) model prediction the direction of change worse than a fair coin toss. Judging the model on their forecasting ability depends even more heavily on what aspect one finds more important.
27
Model AR(1) AIC-s BIC-s adLASSO-s
Rolling window 20043 80 80 80
MDA 0.485 0.621 0.576 0.5
Interval 0.697 0.621 0.682 0.697
MASE 0.854 1.000 0.878 0.872
Inter. ex. zero 0 0.061 0 0
Contained real 0 0.061 0 0
Table 5. Results shown for the week ahead forecast for the four different models
5.3 Cointegration
Besides the forecasts, we have looked at the possible cointegration relationships for comparison of our results with other papers focussed solely on cointegration. For this analysis the whole dataset is used. Although the Johansen test confirms the presence of a few cointegration vectors, it doesn’t give very interpretable results. The Engle and Granger approach is more useful for this, it has been used to find cointegration among two, three and most four time series shown in the Table 6 below.
CI vectors 1 2 3 4 ADF stat. PP stat. KPSS stat.
1. EBL N.EBL Gas/Coal - 3.81** 3.84** 0.38
2. EBL EUA EUA.Gas/Coal Coal.Coal/Gas 5.1*** 4.44** 1.36*** Table 6. Significant cointegration relations found us ing all available data points. “EBL” is the Dutch base load futures price, similar holds for “N.EBL”, “Gas/Coal” is the ratio between the two futures contract prices, “EUA.Gas/Coal” is the futures price of EUA multiplied by the ratio of the two fossil fuel futures prices. * 10%, ** 5%,*** 1% significance
We find cointegration relationships between the Dutch year ahead electricity futures price, that of the Nordics and the Gas/Coal ratio. The relationships we found for the Dutch market are not as intuitive as the ones found in other papers (de Jong & Schneider, 2009; Asche, Osmundsen, & Sandsmark, 2006; Zachmann, 2008). Our findings may not be very intuitive but a theory could be that the Dutch energy price is following the Nordic energy price closely as long as the gas/coal ratio is reasonably constant. The second relationship consists of four different variables and an exact
43 The AR(1) model uses all the data until a certain Friday to forecast the price for the next Friday. The other models only use the Friday prices to forecast the next Friday price
28
explanation is hard to give. But the connection between gas, coal and EUA is obviously the carbon dioxide which is emitted and the cost that has to be paid for it.
6 Conclusion
Although forecasting energy markets of different countries have received a lot of attention by researchers, the majority of the literature is univariate (with and without explanatory variables) and not focussed on the Dutch market. This paper uses multiple explanatory variables and combines this with the adaptive LASSO method and a combination of different rolling window lengths. The numerous connection to neighbouring markets makes the Dutch electricity price influenced by neighbouring electricity prices plausible. Using this multivariate approach, stationarity is tested and the property of cointegration is exploited.
The prediction process is meant to be automated, so for each step the most useful variables and cointegration relationships are used. A variable selection method is used after which a model is set up with GARCH errors. Based on the MASE, the adaptive LASSO method proves to be slightly superior to backward elimination in combination with AIC or BIC and a “simple” AR(1) model. Even though the improvement to the AR(1) model is small based on the MASE, it is significant according to the DM test. With these results we show that the explanatory variables contain useful information for forecasting, but the variable selection method is very important in this process. As the QLR test suggests the presence of a break, a reason for this could be the flow based market coupling which is active since 2015. But regardless of the source, the use of a combination of different window lengths improves the forecast. This is stated by theory as it gives more recent observations more weight. The GARCH component provides a prediction interval which can be of particular interest when zero is not included. The prediction intervals used are two sigma’s wide and, comparable to the normal distribution, around 68% percent of the real values fall into this interval. The situations where zero is not included are sparse but even then the interval is correctly specified in around the same fraction of the cases. This makes the sigma prediction seem well specified.
The week ahead point forecasts are, as expected, less accurate. Predicting a longer period into the future gives more uncertainty. The AR(1) model performs the best in terms of MASE, this can be expected as it uses all the end of day prices compared to
29
only the Friday end of day prices for the other selection methods. In contrast to the MASE, the MDA was the worst for the AR(1) model. The AIC-s model, having the highest MASE, does have a MDA higher than 0.6.
Furthermore cointegration relationships are found in our data, but in a more complex structure than found in other papers. A theory for their existence could be thought of but a simple intuitive one is hard to give.
In an efficient market available information is incorporated in the price rather quickly. Focusing on end of day prices means that there is a whole day between the last data point and the forecasted point. In an active energy market like that of Holland, the prices can be expected to adjust to new information within a day. But the forecasting model introduced in this paper still shows to have predictive abilities based on one day ago information.
Where we propose a forecasting model including a few exogenous regressors, a further improvement in predictive power can possibly be made by including more variables. Capturing political decisions44 or geographic numbers45 about fossil fuels could be valuable. This can be illustrated by the idea that changes in fossil fuel prices will be incorporated in the electricity price within a day, but political decisions may have a delayed effect hence have more predictive potential. Being able to include this kind of data in the model will most likely lead to improved forecasts.
44 For example the allowing of more drilling for gas or the construction of more coal plants 45
30
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Appendix A
List of variables with description and any special additional information EBL.Cal Dutch market electricity base load
futures year contract in €/MWh
Base load G.EBL.Cal German market electricity base load
futures year contract in €/MWh
Base load N.EBL.Cal Nordic market electricity base load
futures year contract in €/MWh
Base load Gas.Cal Gas futures year contract in €/m3 Base load Coal.Cal Coal futures year contract in $/tonne API4 Oil.Dec Oil futures December month contract in
$/barrel
Brent oil EUA.Dec European Union allowances futures
December month contract in €/tonne of CO2 emitted
FX.Dec Future exchange rate December month contract in $/€
From CME
Appendix B
Table with results of three different stationarity tests for the first differenced series. Significance level: 0.10*, 0.05** and 0.01***
Stationarity tests ADF PP KPSS
dEBL.Cal -18.30*** -24.30*** 0.22 dGas.Cal -19.42*** -26.75*** 0.12 dCoal.Cal -6.95*** -26.71*** 1.11*** dOil.Dec -20.01*** -30.81*** 0.19 dEUA.Dec -16.01*** -30.21*** 0.16 dG.EBL.Cal -9.02*** -25.71*** 0.50** dN.EBL.Cal -17.95*** -25.37*** 0.68** dFX.Dec -16.76*** -28.39*** 0.15
