Forecasting hourly electricity prices using ARMAX–GARCH models: An application to MISO hubs

Energy Economics 34 (2012) 307–315

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Forecasting hourly electricity prices using ARMAX–GARCH models: An application to MISO hubs Emily Hickey, David G. Loomis ⁎, Hassan Mohammadi 1 Department of Economics, Illinois State University, Normal, IL 61790-4200, United States

a r t i c l e

i n f o

Article history: Received 3 September 2010 Received in revised form 17 August 2011 Accepted 25 November 2011 Available online 3 December 2011 JEL classification: C53 L94 Q4

a b s t r a c t The recent deregulation of the electricity industry and reliance on competitive wholesale markets has generated significant volatility in wholesale electricity prices. Given the importance of short-term price forecasts in this new environment, this paper estimates and evaluates the forecasting performance of four ARMAX– GARCH models for five MISO pricing hubs (Cinergy, First Energy, Illinois, Michigan, and Minnesota) using hourly data from June 1, 2006 to October 6, 2007. Our empirical results reveal three important patterns: (a) electricity price volatility is regional and the optimum volatility model depends in part on the hub location, the forecast horizon, and regulated versus unregulated status of the market; (b) the APARCH model performs well in hubs in deregulated states; and (c) volatility dynamics in regulated states are better captured by a simple GARCH model and thus are less complex. © 2011 Elsevier B.V. All rights reserved.

Keywords: Electricity Pricing MISO GARCH ARMAX

1. Introduction Up until the last two decades or so, short-term electricity price forecasts (STEPF) were rarely performed due to the regulatory nature of these prices (Bunn and Karakatsani, 2003). The popularity of STEPF grew in the late 1990s due to a rise in market risk and price volatility following deregulation. Since then STEPF has become an important risk management tool for regulators, consumers, and producers of electricity alike. Producers, for example, need accurate forecasts of prices in order to optimize the terms of bilateral contracts and to respond efficiently in the day-ahead and real-time markets. Similarly retailers and large consumers need forecasts in order to create an optimal bidding strategy while regulators can benefit by using forecasts to assess market efficiency. The STEPF literature has primarily focused on modeling cyclical and seasonal patterns in electricity prices; volatility has also been explored, but to a lesser extent, which is surprising given the profound impact it has on industry stakeholders. In fact, electricity prices are often more volatile than the prices of other commodities. 2 This is a direct result of the inability to store electricity and complications

⁎ Corresponding author. Tel.: + 1 309 438 7979. E-mail addresses: [email protected] (E. Hickey), [email protected] (D.G. Loomis), [email protected] (H. Mohammadi). 1 Tel.: + 1 309 438 7777. 2 See Johnson and Barz (1999); Escribano et al. (2002); Hadsell et al. (2004). 0140-9883/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.eneco.2011.11.011

surrounding its generation and transmission among other things. Specifically, the inability to store electricity means that inventories cannot be used to smooth out supply and demand shocks, thus contributing to greater price volatility. The utilization of diverse plant types to cater to demand in a market clearing price environment and line congestion have also contributed. Given the importance of understanding electricity price volatility and the fact that the literature has only begun to explore these issues, a thorough investigation of volatility models in this context is a useful contribution to the subject. This paper investigates the performance of four classes of ARMAX–GARCH volatility models (GARCH, EGARCH, APARCH and CGARCH) 3 and evaluates their out-of-sample forecasting performance for five MISO 4 pricing hubs. Currently, there is no consensus in the literature as to which volatility model is the most appropriate. In a recent study, Bowden and Payne (2007) evaluated the forecasting performance of a number of GARCH models for hourly electricity 3 GARCH stands for generalized autoregressive conditional heteroskedasticity; EGARCH is exponential GARCH; APARCH is asymmetric power ARCH; and CGARCH is Component GARCH. 4 MISO is a non-profit organization whose duties include upholding the terms of bilateral contracts and orchestrating the day-head and real-time power market within their region. MISO was established on December 20, 2001 in response to FERC order 888/2000. The geographical area served by MISO covers the Midwest and parts of Canada and includes most of Illinois, Michigan, Indiana, Ohio, Iowa, Minnesota, North Dakota, South Dakota, and Missouri. MISO is composed of roughly 300 market participants who together serve over 40 million people.

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prices in MISO's service territory. However, their study is limited in three important respects. First, their sample is limited to one month (July 9–August 5, 2007) of hourly observations. Second, their modeling evaluation considers only the out-of-sample performance of the conditional mean, rather than the conditional variance. Third, the authors do not formally test for significant differences in forecasting performance across models. As such, we extend Bowden and Payne's (2007) work in four important directions. First, we extend the hourly sample period from June 1, 2006 to September 29, 2007, thus allowing for additional complexities due to seasonal, cyclical and secular variations. Second, we extend and evaluate the forecasting performance of two additional classes of GARCH models, namely APARCH and CGARCH, to better capture persistence in volatility. Third, we evaluate the forecasting performance of the models by examining the outof-sample forecasting performance of their conditional variance rather than their conditional mean. Fourth, in addition to the standard out-of-sample forecasting evaluation criteria (i.e. RMSE and RMAE), we formally evaluate the forecasting performance of the conditional variance equations across models using the procedure proposed by Diebold and Mariano (1995). To our knowledge, no published study to date has evaluated the forecasting performance of the conditional volatility in MISO's service territory. Our findings suggest that the optimal volatility model depends on both the hub and forecasting horizon considered. APARCH performs well in hubs located in states with deregulated prices while volatility dynamics in regulated states appear to be less complex such that a simple GARCH model suffices. The remainder of this paper is organized as follows: section two discusses the time-series literature, section three discusses the theoretical foundation of the ARCH/GARCH models considered, section four covers the MISO market and data, section five details the construction of the empirical model, and section six discusses the empirical results.

2. Literature review In the regulated environment, the utility owned the generation such that prices merely reflected the underlying costs of generation and prevailing regulatory policy. With deregulation has come greater price risk and uncertainty; the desire to hedge against that risk has motivated the literature in recent years. Initially, Black–Scholes and other models traditionally used to forecast commodity prices were used to forecast electricity prices. However, the underlying assumptions (e.g. geometric Brownian motion) proved to be inappropriate (Knittel and Roberts, 2005). This is not entirely surprising given that, in many ways, electricity prices are more complex than prices for other commodities. In fact a general consensus has formed in the literature that electricity price series are subject to price spikes, an inverse leverage effect, positive skewness, leptokurtosis 5, high frequency, multiple seasonality, and high volatility. In recent years a variety of empirical methods have emerged that at least partially address these complexities: neural networks, wavelets, jump diffusion models, and time series techniques, to name a few. Time series techniques have become the most widely used method for short and medium-term forecasts in practice (Bunn and Karakatsani, 2003). Conejo et al. (2005) state that “[time series techniques have] revealed themselves, through many realistic studies, as the most efficacious tools for day-ahead market-clearing price forecasting (436).” Due to their widespread use and relatively successful application in this field, this study explores time-series techniques in greater depth. The next section briefly details the exploration of both conditional mean and variance specifications within the time series literature.

5 Leptokurtosis is a term used to describe data that has a non-normal distribution with a higher peak and fatter tails than normal.

2.1. The mean equation In modeling the conditional mean, one must properly account for the trend, cyclical and seasonal movements of prices. Thus, the time series literature can be envisioned in terms of how it has addressed each component. The trend component has been extensively considered in the literature. In the post-restructuring environment, Johnson and Barz (1999) were among the first to identify the tendency of electricity prices to mean revert. Over the years a growing number of studies have found additional evidence in support of mean reversion.6 In fact, neither the location nor the frequency of the electricity price data appears to affect the finding of stationarity. While the literature generally agrees that mean reversion exists, the means through which studies have come to that conclusion has varied. While some appear to assume prices mean revert, others have formally tested for its existence. Escribano et al. (2002) performed one of the most through investigations of stationarity in the price forecasting literature to date. Alternatively, other studies have found electricity prices to be non-mean reverting including Conejo et al. (2005), Garcia et al. (2005), and Bowden and Payne (2007). Typically, those studies that have found evidence against mean reversion have used hourly data over smaller time periods than those studies that supported mean reversion7. The discussion of seasonality in the literature has been somewhat non-controversial. In fact, no study in the STEPF literature to date has refuted the presence of seasonality. Seasonal fluctuations in electricity prices by and large reflect the seasonal consumption (i.e. load) of electricity8. Generally speaking, time-series load (and price) data contain three seasonal patterns: intradaily9, weekly10, and monthly11 (Hahn et al., 2009). Common approaches to account for seasonality have included the use of deterministic seasonal dummies and/or sinusoidal equations which are seen as virtually equivalent techniques (Escribano et al. 2011). The cyclical behavior of electricity prices can be captured by employing a broad class of ARMA/ARMAX models. The complexity of the cyclical specifications considered in the literature has varied; while some have employed a simple AR (1) process others have performed a more thorough investigation of the autocorrelation and partial autocorrelation plots of price. Not surprisingly, simple AR (1) processes have performed poorly compared to alternative ARMA/ ARMAX specifications. Knittel and Roberts (2005) and Cuaresma et al. (2004), for example, found that the out-of-sample forecasting performance of their respective ARMA model exceeded that of an AR (1). Both studies used hourly data and a one week out-ofsample forecasting horizon 12 for different locations yet had almost identical ARMA models. Each included the first, twenty-fourth, and twenty-fifth lags of price while Cuaresma et al. (2004) included an additional twenty-third lag 13.

6 See Bhanot (2000); Karesen and Husby (2000); Lucia and Schwartz (2002); Knittel and Roberts (2005); Thomas and Mitchell (2007); Guirguis and Felder (2004); Johnson and Barz (1999); Escribano et al. (2002); Cuaresma et al. (2004); Li and Zhang (2007); Leon and Rubia (2001); Higgs and Worthington (2005). 7 Note that the exception is Garcia et al. (2005) who consider two data sets spanning 1 and 1.25 years. 8 Knittel and Roberts (2005); Escribano et al. (2002). 9 The intradaily seasonal pattern reflects the shift in peak and off-peak consumption throughout the day. 10 The weekly pattern reflects changing weekend consumption patterns. 11 The seasonal l pattern reflects changing consumption trends throughout the year as temperature and lifestyle choices change. 12 A one week forecast horizon was chosen by the authors for primarily two reasons; first, the high frequency of the data suggested only short-term forecasts are feasible and also because most contracts made for electricity are short-term ranging from one day to several months. 13 Note that Cuaresma et al. (2004) consider a range of ARMA models and state that MOST models include the first, twenty-third, twenty-fourth, and twenty-fifth lags of price.

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To address the complexity of electricity prices, many studies have augmented their ARMA models with additional terms 14. A frequent extension is that of a jump diffusion 15 term which can simulate the spiking behavior of prices 16. Alternatively, one may address spikes by removing them from the data set. 17 As Escribano et al. (2002) point out; ignoring the presence of spikes can have negative consequences on model estimation; especially in ARCH/GARCH models. Guirguis and Felder (2004) rightly note that “Although [spikes] are of important value, their nature and the probability of their occurrence seems to be unique and non-repetitive. Therefore, including these observations with such unusual high values may produce bias in parameter estimates and hence may deteriorate the efficiency of our forecasts (162).” Given the importance of accounting for price spikes, techniques for detecting and removing them are discussed in greater depth in section five. Besides jump diffusion terms, exogenous variables, like temperature and demand, have also been used within the literature. Generally speaking, findings have been mixed as to whether these variables improve the out-of-sample forecasting performance. Some studies – Knittel and Roberts (2005) and Conejo et al. (2005) for example – have found that the inclusion of these variables results in negligible if not worsened 18 model performance. Others (Weron, 2006) suggest that including variables such as load is imperative since electricity prices are largely influenced by fluctuations in load 19.

2.2. The variance equation Volatility is generally regarded as one of the most important features of electricity prices. In fact, compared to other commodity and financial markets, electricity prices are seen as more volatile 20. This behavior is largely attributed to the instantaneous production process, highly variable demand, the inability to economically store electricity, market power, and the diversity of plant costs among other things 21. GARCH-variety models are particularly adept at modeling electricity price volatility, which is characterized by clusters of nonconstant variance (Escribano et al., 2002). Several studies in the literature have reported that GARCH improved forecasting performance 22. Guirguis and Felder (2004), for example, found that an AR (1) process with GARCH effects outperforms dynamic regression, transfer function, and exponential smoothing specifications in out-of-sample forecasting 23. They also noted that GARCH performance improved when outliers in the dataset were removed; a point that was highlighted earlier (Guirguis and Felder, 2004).

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While a simple GARCH model accounts for time-varying volatility, alternative specifications can accommodate more complex dynamics such as skewness, leptokurtosis, and asymmetry. Of those studies that have considered both basic GARCH and more advanced ARCH/ GARCH processes, all have reported significant improvements in out-of-sample forecasts with the latter 24. APARCH, PARCH, EGARCH, and Threshold ARCH (TARCH) are just a few of the specifications that have been explored in the literature. Volatility models that allow for asymmetry grew in popularity after Hadsell et al. (2004) found that electricity prices exhibit asymmetric time-varying volatility. Using TARCH, the authors reported a negative and statistically significant coefficient on the asymmetry term, implying that positive shocks increase volatility more than negative shocks 25; a condition known as the inverse leverage effect. More recently Bowden and Payne (2007), Knittel and Roberts (2005), and Thomas and Mitchell (2007) have applied EGARCH to support the presence of an inverse leverage effect. Bowden and Payne (2007) also considered EGARH-M which takes into account the impact of volatility on the mean equation. What all three of the aforementioned studies have in common, besides the application of EGARCH, is the finding that EGARCH is the superior specification for some regions, but not for others 26. PARCH is an alternative specification that allows the power parameter to be estimated, rather than imposed, as is the case with traditional ARCH/GARCH models. It has been argued that given the presence of leptokurtosis in electricity price series, other power terms besides the square may be appropriate. Thomas and Mitchell (2007) applied PARCH to five regional pool markets in Australia and found that for all regions considered, the estimated power parameter was positive and statistically different from one and two. Curiously PARCH was only chosen as superior in three of the five regions considered based on the AIC, SIC, and LM test. Rather than apply PARCH, Higgs and Worthington (2005) 27 used an extension of PARCH, namely Asymmetric Power-ARCH (or APARCH), which is also able to capture the asymmetric response to positive and negative shocks. 3. Theoretical background While the primary focus of our investigation is on modeling the conditional variance, it is worth providing a basic theoretical foundation for the cyclical component of the mean equation. In the ARMAX (p,q) specification the price variable is modeled as, ΦðBÞP t ¼  cþΘ ðBÞεt þ βX t ε t eWN 0; σ 2

ð1Þ

14

See Knittel and Roberts (2005); Escribano et al. (2002); Cuaresma et al. (2004); Conejo et al. (2005); and Guirguis and Felder (2004); and Johnson and Barz (1999). 15 Jump diffusion models link changes in price to the arrival of news where there are two types of news; normal news and abnormal news. Abnormal news causes the discrete price jumps and is modeled via a probabilistic discrete time process such as Poisson or a binomial distribution. See Blanco and Soronow (2001) for a discussion of jump diffusion models. 16 The spiking behavior has been largely attributed to supply side shocks (transmission constraints and unexpected outages for example). 17 See Thomas and Mitchell (2007); Guirguis and Felder (2004); Cuaresma et al. (2004). Note that Cuaresma et al. (2004) only remove spikes higher than the vthquartile (here v = 0.997) of the normal distribution and go on to use the jump diffusion process to model spikes remaining in the series. 18 Knittel and Roberts found that out of sample forecasting performance was among the worst during the crisis period (May 1, 2000 to August 31st 2000) with temperature included. 19 While weather fluctuations can also affect prices, Weron (2006) cites that such factors are captured by including a load variable. 20 Johnson and Barz (1999); Escribano et al. (2002); Hadsell et al. (2004). 21 Bunn and Karakatsani (2003); Hadsell and Shawky (2006). 22 See Escribano et al. (2002); Garcia et al. (2005); Li and Zhang (2007); Guirguis and Felder (2004). 23 Mean Error, Mean Absolute Error, and Root Mean Squared Error were the diagnostic tests used to compare models.

where P is the price level, εt are random innovations, Xt is the vector of exogenous variables, and B is the backshift operator such that: ΦðBÞ ¼ 1−ϕ1 B−…−ϕp B

p

ΘðBÞ ¼ 1 þ θ1 B þ … þ θq B

q

24 Thomas and Mitchell (2007); Higgs and Worthington (2005); Bowden and Payne (2007). 25 This, the authors argue, has to do with the definition of good and bad news. For electricity, good news entails negative shocks or returns that fall below the mean. 26 Thomas and Mitchell (2007) found that EGARCH was preferred in two of the five regions considered; Bowden and Payne (2007) found that EGARCH was preferred in four of the five regions considered; and Knittel and Roberts (2005) found that EGARCH was preferred during one of the two periods considered. 27 Note that Higgs and Worthington (2005) also consider Risk Metrics but ultimately find it performs poorly in all four markets as judged by the Ljung-Box statistic. The authors point out that a potential shortfall of the Risk Metrics model is that it cannot take into account non-normal characteristics of the time series.

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Estimation of the above model requires covariance stationary price variables and homoskedastic error terms. In the event that the conditional distribution differs from the unconditional distribution the assumption of constant variance may no longer apply. While the unconditional variance must remain constant, as required of a covariance stationary series, the conditional variance may be conditionally heteroskedastic. Rather than treat heteroskedasticity as a problem to be corrected, GARCH models aim to explicitly model the timevarying volatility process (Engle, 2001). We begin with the GARCH (1,1) specification of Bollerslev (1986), which is the workhorse of the volatility models, 2

2

2

σ t ¼ ω þ αεt−1 þ βσ t−1

ð2Þ

where α þ βb1; α≥0; β≥0 andω > 0 Where α (the ARCH parameter) quantifies the size of the effect – how much volatility increases irrespective of the direction of the shock – and β (the GARCH effect) captures the degree of volatility persistence. A high α suggests that market participants have an exaggerated reaction to prior price errors such that expected volatility is unstable (Hadsell et al., 2004). While a GARCH specification can account for time-varying volatility, it also assumes that only the magnitude and not the direction of lagged residuals can affect conditional variance. Where positive and negative shocks have different impacts on volatility an asymmetric GARCH specification such as EGARCH may be preferred: p r   X X ε ε 2 log σ t ¼ ω þ α i j t−i j þ γi t−i 2 σ σ t−i t−i i¼1 k¼1

! þ

q X

  2 βj log σ t−1

ð3Þ

j¼1

whereα þ βb1; α≥0; β≥0andω > 0 The log of conditional variance implies that the asymmetric effect is exponential and that forecasts of conditional variance are nonnegative (Thomas and Mitchell, 2007). The asymmetry in the model is captured by γk; specifically γk b 0 implies a leverage effect while γk > 0 suggests an inverse leverage effect. In the presence of an inverse leverage effect positive shocks increase volatility more than negative shocks; the opposite is traditionally true for financial assets where a leverage effect exists. In the STEPF literature there has been a general consensus that electricity prices suffer from an inverse leverage effect 28 such that it is worth applying EGARCH to all five MISO hubs. Knittel and Roberts (2005) provide intuitive support for the inverse leverage effect and cite that, in the context of electricity prices, a positive shock to prices is in essence an unexpected positive demand shock. Due to the convexity of marginal costs, a positive demand shock has a larger impact on price changes than negative shocks. A restrictive feature of GARCH models is the assumption of squared power term which may not always be appropriate. The Power ARCH (PARCH) model of Ding et al. (1993) allows the power parameter (δ) to be estimated, rather than imposed, and is specified as follows: δ

σt ¼ ω þ

q X j¼1

δ

βj σ t−j þ

p X

δ

α i εt−i

ð4Þ

i¼1

whereα þ βb1; α≥0; β≥0 andω > 0 PARCH may be appropriate in the context of electricity prices since the choice of the power parameter is not obvious 29. An extension of 28 Bowden and Payne (2007); Thomas and Mitchell (2007); Knittel and Roberts (2005). 29 Thomas and Mitchell (2007); Higgs and Worthington (2005).

PARCH is the Asymmetric PARCH (APARCH) which allows for asymmetry. APARCH can be specified as: δ

σt ¼ ω þ

q X j¼1

δ

βj σ t−j þ

p X

δ

α i ðjεt−i j−γ i εt−i Þ

ð5Þ

i¼1

where α + β b 1; α ≥ 0; β ≥ 0, and ω > 0, and γi is the asymmetric parameter. GARCH and its extensions (EGARCH and PARCH) rely on the assumption that α and β sum to a number less than one. Generally speaking, this is consistent with the empirical evidence in the literature. However, there have been exceptions. Hadsell et al. (2004), for example, found that in all five markets 30 they considered the sum exceeded one which implies that the model is unstable and that the volatility process is highly persistent. Surprisingly the STEPF literature has not thoroughly considered GARCH models that capture persistence and long-run volatility effects. Component GARCH (CGARCH) of Engle and Lee (1999) is one particular model that accounts for persistent volatility dynamics. The model separates the long-run and short-run volatility effects and has been cited as superior to GARCH in describing volatility dynamics (Christoffersen et al., 2004). According to Wei (2009), the primary difference between GARCH and CGARCH is that shocks decay towards the unconditional variance in GARCH whereas shocks to the transitory component revert to the trend with CGARCH. The CGARCH (1,1) specification can be derived from a GARCH (1,1) process by replacing the unconditional variance term (ω) with a time varying component (qt) that represents long-run volatility: 

     σ 2t −ϖ ¼ α ε2t−1 −ϖ þ β σ 2t−1 −ϖ       σ 2t −qt ¼ α ε2t−1 −qt−1 þ β σ 2t−1 −qt−1   2 2 qt ¼ ω þ ρðqt−1 −ωÞ þ φ εt−1 −σ t−1 whereα þ βb1; α≥0; β≥0; ρ > 0; φ > 0 andω > 0 where ρ measures long-run dynamics while α + β captures the transitory or short run dynamics. Here (εt2− 1 − σt2− 1) drives the timedependent movement of the permanent component, which has a zero expected value, while(σt2− 1 − qt − 1) defines the transitory component of conditional variance (Wei, 2009). In the event the long run volatility component is more persistent than the short run volatility component then 0 b α + β b ρ b 1 will hold. 4. Data and descriptive statistics Transactions in the wholesale electricity market occur primarily through two mediums; bilateral contracts and the pool which contains both the day-ahead and real-time market. Bilateral contracts generally range in duration from one day to several months and are independently negotiated between two parties. One of the primary duties of the local Regional Transmission Organization (RTO) or Independent System Operator (ISO) is ensuring that the physical requirements of contractual agreements can be satisfied without compromising system integrity. In the pool market, the RTO/ISO has the additional responsibility of setting the hourly price; currently the majority of market operators employ a market clearing31 algorithm. If the traded electricity can be dispatched from the buyer to the seller without experiencing line 30 The exception was Cinergy and Entergy during the year 2001 and COB during the year 1998. 31 Under the market clearing price (MCP) system electricity producers submit bids specifying the minimum price at which they are willing to produce a given amount of output from each of their plants. Actors on the demand side bid in the maximum price at which they are willing to purchase a given amount of electricity; equilibrium prices are determined through clearing the market.

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congestion, voltage constraints, or thermal limits then the price simply reflects the market clearing cost of the energy itself. Where congestion or line losses are present the price is inflated to reflect those locational circumstances; what is known as the locational marginal price (LMP). The LMP is made up of three components: the cost of energy as determined through the market clearing price auction, the marginal cost of congestion, and the marginal cost of transmission losses (Midwest ISO, 2005). Within the pool itself are two distinct markets that vary in the timing of purchases: the day ahead and real time market. Like most other RTO/ISOs, MISO 32 oversees both a day-ahead and real time (or spot) market where prices are set as stated above. The day-ahead market functions as a short-term futures market where electricity is traded for specific hours of the following day (Longstaff and Wang, 2004). Alternatively, participants make offers on a real-time basis in the spot market; in MISO real-time prices are dispatched to participants every 5 min (Midwest ISO, 2009). In both the day-ahead and real-time markets prices vary by commercial pricing nodes (CP node); physical locations where electricity is bought and sold off of the grid. An aggregation of CP nodes is defined as a hub (Midwest ISO, 2005). We use hourly real-time prices spanning from June 1st, 2006 to September 27th, 2007 across five MISO hubs (Cinergy, First Energy, Illinois, Michigan, and Minnesota). A few criteria were taken into consideration when choosing the frequency, duration, and timing of the sample period. With respect to the frequency, the literature has generally favored hourly data 33. Given that previous studies 34 have found that a year and a half of hourly observations is well-suited, it seems appropriate to pursue a dataset with that quality. The specific time frame of June 1st, 2006 to September 27th, 2007 was chosen for two reasons; first, as documented in the literature, unusually high volatility is generally observed in the first year of a market's operation which can be explained by learning by doing for participants 35. Given that MISO officially met FERC's reliability and continuity standards for real-time market operation in March 2005, the appropriate data sample should begin no earlier than April 1st, 2006. Second, the data sample was chosen to exclude the recent recession which has had an extremely atypical effect on electricity consumption and prices 36. We estimate the models using data from June 1st, 2006 to September 29th, and reserve the last week of our data set (September 30th, 2007–October 6th, 2007) for out-of-sample forecast evaluations. The forecasting horizon of one week was chosen to be in line with the literature 37. Moreover it is appropriate given the data frequency and the fact that the duration of most electricity contracts is short-term. A forecasting horizon of one day (September 30th) was also considered to aid in evaluating the optimal volatility specification. The descriptive statistics of the electricity price data are reported in Table 1. In addition to electricity price data, we obtained data on forecasted load for the entire MISO footprint to be used as an explanatory variable in the conditional mean. Forecasted load (as opposed to actual load) is appropriate since the market clearing price is the outcome of bids placed by market participants with only the knowledge of that hour's forecasted load (Weron, 2006). The data was obtained

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Table 1 Descriptive statistics. Cinergy

First Energy

Illinois

Michigan

Minnesota

42.632 31.645 456.310 − 55.440 30.285 2.396 14.241 72,574 0.000

42.171 31.170 470.290 − 112.070 31.062 2.489 16.097 95,408 0.000

42.158 30.580 456.190 − 72.620 33.252 2.606 15.685 91,410 0.000

45.335 33.610 474.630 − 49.720 32.905 2.616 16.473 101,517 0.000

49.675 35.000 552.830 − 173.610 47.016 2.709 16.400 101,524 0.000

Without Outliers/Recursive Filtered Data: Mean 41.134 40.722 Median 31.625 31.150 Maximum 142.620 145.530 Minimum − 55.440 − 63.610 Std. dev. 25.661 26.150 Skewness 1.313 1.266 Kurtosis 4.359 4.379 Jarque–Bera 4249 4073 Probability 0.000 0.000

39.835 30.575 144.450 − 63.200 26.121 1.213 4.529 3996 0.000

43.341 33.600 150.370 − 49.720 26.774 1.287 4.268 4001 0.000

45.688 34.870 182.810 − 93.280 34.322 1.152 4.803 4158 0.000

With outliers: Mean Median Maximum Minimum Std. dev. Skewness Kurtosis Jarque–Bera Probability

Note: Descriptive statistics are for the in-sample period (June 1st 2006 to September 29th 2007).

from MISO's Look Ahead Report which is published daily and provides hourly load forecasts for the present day and six days into the future. The report is designed to summarize load forecasts and outage situations and is provided to marketers and generation/transmission owners to assist with planning 38. The forecasted load within the report is an aggregate of load across the MISO footprint; while local load at each hub would have been preferable, it is not provided by MISO. 5. Empirical model and estimation method Before developing the empirical model, we tested for the presence of a unit root in prices using six alternative tests: the Augmented Dickey Fuller, Phillips Peron, KPSS 39, ERS-Po, Ng-Perron, and Dickey Fuller GLS. For all five hubs and tests considered, the null of a unit root was strongly rejected beyond a 1% significance level. Furthermore the autocorrelation of prices decline fairly rapidly which is indicative of a stationary process 40. To properly model the price equation we must address a number of issues including price spikes, seasonality, and the choice of other explanatory variables 41. Thomas and Mitchell (2007) define a spike as a change in price that falls four or more standard deviations from the mean while Clewlow and Strickland (2000) use three standard deviations. In our data set 1.48%–2.11% of observations fell three standard deviations from the mean while only 0.65%–1.05% fell four standard deviations or more from the mean. Given that spikes are inherent in electricity prices and that we only want to limit their severity, the four standard deviation metric was found to be more appropriate. We detect spikes using a recursive filter following

32

See footnote 3 for a description of MISO. Bunn and Karakatsani (2003); Higgs and Worthington (2005); Lucia and Schwartz (2002). 34 Knittel and Roberts (2005); Li and Zhang (2007); and Cuaresma et al. (2004) use two, one, and one and a half years of hourly data respectively. 35 Borenstein et al. (2001); Hadsell and Shawky (2006); Bowden and Payne (2007). 36 For the first time in the history of the United States electricity consumption fell for two consecutive years (Axford, 2009). Given Energy Information Administration (2010) projections that electricity demand will continue its upward trend in 2010 and the fact that this study aims to construct a STEPF under normal market conditions it is appropriate to exclude the most recent recession. The National Bureau of Economic Research (2008) has found that December 2007 marks the beginning of the current recession such that 2008 and 2009 data should be excluded. 37 Knittel and Roberts (2005); and Cuaresma et al. (2004). 33

38 This information was provided via e-mail correspondence with Joyce Senzig at MISO on February 1st, 2010. 39 The null hypothesis in KPSS (Kwiatkowski–Phillips–Schmidt–Shin) test is that the variable is stationary. 40 While some lags are significant at longer lag lengths the series still appears to be stationary. This point is reiterated by Knittel and Roberts (2005) who found that while autocorrelations were statistically significant beyond 1000 lags, the correlogram was consistent with the intuition that prices weren't exploding. 41 Knittel and Roberts (2005) and Cuaresma et al. (2004) found including jumps did not uniformly improve model performance.

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Clewlow and Strickland (2000), Cuaresma et al. (2004), and Trueck et al. (2007) 42. Following Trueck et al. (2007), we replace spike prices with similar-day observations. 43 Finally, we capture seasonality using monthly, daily, and hourly dummy variables. The result is the ARMAX specification (Eq. (8)) for modeling the price level in the Cinergy, First Energy, Illinois, and Michigan hubs: 

 fhg P t ¼ at þ β1 t þ β2 lt þ θ10 εt−10 þ θ25 εt−25 ð8Þ 6 11 23 þϕ24 P t−24 at ¼ c þ ∑i¼1 ai DDi þ ∑j¼1 α j MDj þ ∑k¼1 α k HDk 2

3

4

1−B−B −B −B −B

2

3

4

at ¼ c þ

þ



fhg 24 P t ¼ a t þ β 1 t þ β 2 lt 11 ∑j¼1 α j MDj þ ∑23 k¼1 α k HDk 11

1−B−B −B −B −B −B ∑6i¼1 ai DDi

Cinergy

First energy

Illinois

Michigan

Minnesota

− 98.494 (4.460)* 0.001 (0.000)* 1.851 (0.058)* 0.360 (0.009)* 0.084 (0.010)* 0.078 (0.010)* 0.052 (0.009)* 0.113 (0.018)* –

− 95.911 (4.362)* 0.001 (0.000)* 1.800 (0.058)* 0.372 (0.009)* 0.085 (0.010)* 0.080 (0.010)* 0.041 (0.009)* 0.103 (0.017)* –

− 98.490 (4.496)* 0.001 (0.000)* 1.883 (0.059)* 0.338 (0.009)* 0.099 (0.010)* 0.078 (0.010)* 0.046 (0.009)* 0.117 (0.018)* –

− 107.395 (6.265)* 0.001 (0.0116)** 2.136 (0.0812) 0.373 (0.009)* 0.092 (0.010)* 0.076 (0.010)* 0.060 (0.010)* –

AR (11)

− 98.620 (4.313)* 0.001 (0.000)* 1.847 (0.056)* 0.360 (0.009)* 0.085 (0.010)* 0.090 (0.010)* 0.040 (0.009)* 0.115 (0.017)* –

AR (24)

–

–

–

–

SAR (24)

0.164 (0.009)* − 0.071 (0.020)* –

0.164 (0.009)* − 0.077 (0.020)* –

0.159 (0.009)* − 0.079 (0.019)* –

0.174 (0.009)* − 0.071 (0.020)* –

0.058 (0.009)*

0.050 (0.009)*

0.039 (0.009)*

0.046 (0.009)*

0.619 857.962 [0.000]

0.597 747.762 [0.000]

0.605 915.615 [0.000]

0.600 674.764 [0.000]

Constant Trend Load

10

where Pt{h} is the price level in hub h for hour t; B is the backshift operator; t is the time trend; lt is the aggregate load at hour t; pt − 24 is the seasonal autoregressive term; and εt − 10 and εt − 24 are the moving average components; c is a constant; and DD, MD, and HD represent daily, monthly, and hourly dummy variables. As for the Minnesota hub, the AIC and SIC criterion lead to an alternative specification, 

Table 2 Estimates of the conditional mean equation.

þ θ23 εt−23

ð9Þ

As for conditional variance, we evaluate the performance of four alternative volatility models — GARCH (1,1), EGARCH (1,1), APARCH (1,1), and CGARCH (1,1). 6. Empirical results 6.1. Conditional mean and variance The results of estimating the conditional mean equation are contained in Table 2. For all five hubs the autoregressive and moving average terms sum to a number less than one, ranging from 0.788 to 0.842, consistent with a stationary ARMAX process. Furthermore all ARMA terms are statistically significant at the 1% significance level in all five locations considered. Load and the time trend are also statistically significant beyond the 5% significance level in all hubs. The coefficients on the seasonal dummies (not reported) were also jointly significant. Finally, the ARCH-LM test, based on twenty four lags of the squared residuals, strongly rejects the null hypothesis of no ARCH effects at all five hubs, implying that a variant of the GARCH model may appropriately describe the conditional volatility process. The results of estimating the GARCH (1,1), EGARCH (1,1), APARCH (1,1) and CGARCH (1,1) specifications are reported in Table 3. Panel A of the Table reports the results of estimating the GARCH (1,1) model. As expected, the sum of α and β coefficients falls below one (0.890– 0.938) for all hubs which is indicative of a stable volatility process. The Illinois hub, with an α of 0.556, has the largest ARCH effect among the five locations, which implies that a shock to electricity prices impacts volatility the most at that location. This finding is consistent with Bowden and Payne (2007). While the ARCH and GARCH terms are statistically significant beyond the 1% significance level, the ARCH-LM test reveals the persistence of conditional heteroskedasticity even after 24 lags. Applying EGARCH(1,1) to all five hubs (Panel B) causes a further, albeit marginal decline in the range of the LM statistic. As expected the asymmetry term, γ, is statistically significant beyond the 1% level for all five hubs – ranging from 0.016 to.094 – which is evidence of an inverse leverage effect. Similarly the asymmetry term in the APARCH (1,1) specification (Panel C) is statistically significant at the 1% level for all five hubs. 42 The filter is an iterative procedure that is repeated until no more spikes beyond four standard deviations can be identified. 43 Note that Weron (2006) suggests using the median as opposed to the mean, however, the mean works well for the purposes of this study.

AR (1) AR (2) AR (3) AR (4) AR (10)

MA (10) MA (23) MA (25)

Model diagnostics R2 ARCH-LM

0.045 (0.008)* 0.089 (0.008)* – – 0.0525 (0.009)* –

0.569 669.833 [0.000]

Notes: Standard errors are reported in parenthesis. Significance is denoted by * for 1%, ** for 5%. The ARCH-LM statistic tests the null hypothesis of no conditional heteroskedasticity up to 24 lags. Values in brackets are associated p-values.

The estimated power parameter (δ) is quite low – between 0.544 and 0.843 – which implies that not even modeling the standard deviation is appropriate (i.e. where δ = 1). Thomas and Mitchell (2007) note that when δ is significantly different from 1 it is appropriate to model the conditional standard deviation in a non-linear form. Finally, Panel D reports the results of estimating the CGARCH (1,1) model to capture both short and long run volatility dynamics. The value of ρ is high for all five hubs, ranging from 0.996 to 0.999, suggesting that the permanent component of conditional variance is highly persistent. Additionally the sum of α + β is much lower for the CGARCH specification (0.707–0.793) than for the GARCH specification (0.890–0.938) implying that the short-run volatility component is weaker. Note however, the absence of ARCH effects in the Minnesota and Michigan hubs.

6.2. Forecasting performance While many STEPF studies have evaluated the out-of-sample forecasting performance of the mean equation 44, few if any have considered the performance of the variance equation itself. In fact, even those studies that have sought to compare ARCH models have focused on evaluating the forecasting performance of mean equation across locations or time. Given that we wish to identify the optimal model of conditional variance, it seems appropriate to evaluate the out-ofsample forecasting performance of the variance equation. The mean absolute error (MAE) and mean squared error (MSE) are measures

44 Bowden and Payne (2007); Knittel and Roberts (2005); Cuaresma et al. (2004); Li and Zhang (2007); Guirguis and Felder (2004).

E. Hickey et al. / Energy Economics 34 (2012) 307–315 Table 3 Estimates of conditional volatility models. First Energy

Illinois

Michigan

Minnesota

A. GARCH (1,1) model ARCH (α) 0.514 (0.016) GARCH (β) 0.424 (0.010) ARCH-LM 57.490 [0.000]

Cinergy

0.516 (0.016) 0.396 (0.011) 51.529 [0.001]

0.556 (0.016) 0.408 (0.009) 44.910 [0.004]

0.446 (0.015) 0.444 (0.011) 48.808 [0.001]

0.421 (0.013) 0.473 (0.009) 54.696 [0.000]

B. EGARCH (1,1) model: ARCH (α) 0.614 (0.014) Asymmetry (δ) 0.060 (0.011) GARCH (β) 0.744 (0.008) ARCH-LM 55.299 [0.000]

0.588 (0.013) 0.081 (0.010) 0.717 (0.007) 46.313 [0.003]

0.657 (0.013) 0.016 (0.010) 0.761 (0.007) 45.500 [0.003]

0.522 (0.013) 0.068 (0.011) 0.761 (0.007) 43.260 [0.006]

0.477 (0.011) 0.094 (0.009) 0.752 (0.006) 55.258 [0.000]

C. APARCH (1,1) model: ARCH (α) 0.338 (0.010) Asymmetry (γ) − 0.297 (0.017) GARCH(β) 0.494 (0.009) Power(δ) 0.544 (0.031) ARCH-LM 41.486 [0.010]

0.357 (0.012) − 0.290 (0.0180) 0.488 (0.010) 0.833 (0.037) 36.880 [0.034]

0.388 (0.012) − 0.205 (0.017) 0.500 (0.009) 0.737 (0.034) 39.669 [0.017]

0.308 (0.010) − 0.319 (0.022) 0.533 (0.010) 0.693 (0.036) 35.300 [0.049]

0.282 (0.009) − 0.356 (0.023) 0.593 (0.007) 0.843 (0.038) 48.703 [0.001]

D. CGARCH (1,1) model Permanent (ρ) 0.999 (0.001) Transitory(φ) 0.039 (0.003) ARCH (α) 0.344 (0.011) GARCH(β) 0.413 (0.014) ARCH-LM 45.653 [0.003]

0.999 (0.000) 0.031 (0.001) 0.330 (0.009) 0.395 (0.015) 41.136 [0.011]

0.999 (0.001) 0.069 (0.007) 0.346 (0.010) 0.447 (0.014) 39.211 [0.019]

0.999 (0.001) 0.045 (0.004) 0.316 (0.011) 0.413 (0.016) 29.731 [0.157]

0.996 (0.001) 0.067 (0.005) 0.314 (0.009) 0.393 (0.015) 23.487 [433]

Notes: Standard errors are in parenthesis and p-values in brackets. ARCH-LM tests the null hypothesis of no conditional heteroskedasticity at 24 lags.

of forecast accuracy that can be used to evaluate the performance of the conditional variance specification. Table 4 contains estimates of the MAE and MSE for all four volatility models over the course of a seven days (168 h) and one day (24 h) out-of-sample forecasting horizon. Results at the seven day horizon show that the MAE and MSE typically agree as to which model performed optimally at each hub. At the First Energy, Illinois, and Michigan hubs APARCH is consistently selected, whereas GARCH performs optimally at the Minnesota location. Cinergy is unique in that no volatility model appears to be superior. These results suggest that the optimal volatility model may depend in part on the presence of deregulation. The First Energy, Michigan, and Illinois hubs for example – all of which favored APARCH – are located in states that have deregulated 45. Conversely at the Minnesota and Cinergy hubs, which are located in regulated states, the best volatility model is either simple (e.g. GARCH) or inconsistent suggesting that volatility at these locations is less complex. At the one day forecasting horizon, however, the MAE and MSE no longer produce consistent results; in fact they fail to ever agree on a single volatility model. Rather, the MSE unanimously favors APARCH whereas CGARCH is consistently chosen by the MAE at four out of 45 Note that the First Energy hub is located within Ohio, the Michigan hub in Michigan, the Illinois hub in Illinois, the Cinergy hub in Indiana, and the Minnesota hub in Minnesota.

313

the five hubs. It is worth noting that while EGARCH has the lowest MAE at three of those locations, instability concerns precludes it from being chosen as a superior volatility specification 46. While these results provide insight as to which volatility model could be considered ‘optimal’ at each hub, they do not tell us whether the model performed statistically better than the alternatives. It is possible, for example, that APARCH is not statistically better or different from the alternative specifications which it supposedly outperforms. Accordingly, it is crucial to determine if any reduction in forecasting errors – from employing an ‘optimal’ volatility model – are statistically significant. The Diebold and Mariano test (DM test) is one way to assess whether the difference in forecasting accuracy between two models is significant 47. The DM statistics are reported in Table 4; a significant negative DM statistic implies that the optimal model is in fact statistically outperforming the alternative to which it is being compared. In the case of Cinergy, where there was no clear consensus as to which volatility model was superior, all of the DM statistics are statistically insignificant. This implies that GARCH (1,1), EGARCH (1,1), APARCH (1,1), and CGARCH (1,1) have virtually equivalent forecasting capabilities. Given that Cinergy has the lowest standard deviation, skewness, and kurtosis of all hubs considered (see Table 1) it may be that it stood to benefit the least from models that accommodate those characteristics. In other words, due to the less complex pricing dynamics at Cinergy – which may have resulted from the continued presence of regulation in Indiana where Cinergy is located – a simple volatility model (e.g. GARCH) is sufficient. The presence of regulation may have also affected the volatility dynamics at the Minnesota hub where, like Cinergy, all four volatility models have equivalent forecasting capabilities. For hubs located in deregulated states – First Energy, Illinois, and Michigan – the GARCH model generally performs statistically worse than APARCH over the seven day forecasting horizon. At the Illinois hub, for example, CGARCH and GARCH yield estimates statistically worse than APARCH. At the Michigan Hub, GARCH is nearly statistically worse than APARCH 48 and EGARCH is worse than APARCH beyond a 10% significance level. This would suggest that the volatility process in deregulated states is more complex. These findings also imply that there is a high degree of regionalism in deregulated electricity markets such that volatility dynamics vary. Another noticeable trend across all five locations is that GARCH only produced statistically worse results at longer forecasting horizons. This suggests that for longer horizons the asymmetry and complexity of electricity price volatility is more pronounced such that models that account for more complex dynamics (e.g. EGARCH, APARCH and CGARCH) produce superior forecasts. In other words, over the course of one day, pricing and hub dynamics have not had time to mature. This finding also provides insight as to why results from the 7 day forecasting horizon were more consistent across the MAE and MSE than at one day where the MAE and MSE were inconsistent and appeared to be indifferent. 7. Summary and conclusions Over the last decade or so, electricity price forecasting has emerged as an important risk management tool for regulators, consumers, and producers of electricity alike. This paper evaluates the out-of-sample forecasting performance of GARCH models of 46

See Thomas and Mitchell (2007). The DM statistic can be computed by first subtracting the actual and forecasted variance for model one and two respectively, then subtracting the difference for model 2 from model 1 to get a value d. Regressing d on a constant with HAC errors produces the DM statistic. Note that if one is comparing the MSE then: (a − f1)2 − (a − f2)2 gets you d and for MAE d = |a − f1| − |a − f2|. 48 At the Michigan hub we would have rejected the null of equal forecasting capability at an 87% confidence bound for GARCH. 47

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E. Hickey et al. / Energy Economics 34 (2012) 307–315

Table 4 Out-of-sample forecasting results. Seven-Day-Horizon

Cinergy

First Energy

Illinois

Michigan

Minnesota

One-Day-Horizon

MAE

DM

RMSE

DM

MAE

DM

RMSE

DM

GARCH

1540.84

3476.21

–

278.17

1545.31

3533.80

267.83

APARCH

1520.22

541.32

1524.80

3549.95

GARCH

1000.85

EGARCH

978.48

APARCH

968.08

− 0.28 [0.78] − 2.57 [0.01] − 0.63 [0.53] –

− 1.19 [0.25] − 0.01 [0.99] − 1.63 [0.12] –

− 1.22 [0.23] − 0.68 [0.50] –

CGARCH

− 0.98 [0.33] − 0.93 [0.35] − 1.00 [0.32] − 2.05 [0.04] − 1.62 [0.11] –

− 1.50 [0.16] –

555.18

EGARCH

− 0.57 [0.57] − 1.28 [0.20] –

CGARCH

969.50

264.84

GARCH

1080.57

EGARCH

1152.93

APARCH

1017.28

− 1.56 [0.12] − 2.42 [0.02] − 1.20 [0.23] –

CGARCH

1047.68

GARCH

1088.59

EGARCH

1074.69

APARCH

1068.31

CGARCH

1071.81

GARCH

2680.80

EGARCH

2694.53

APARCH

2697.17

CGARCH

2684.87

− 0.11 [0.91] − 2.29 [0.02] −1.35 [0.18] –

3608.88

1693.73 1693.98 1660.70 1690.00 2013.75 2310.62 1956.44

− 1.55 [0.12] − 1.55 [0.12] − 0.46 [0.65] –

2006.01

− 0.35 [0.73] –

1794.83

− 0.25 [0.81] − 0.385 [0.70] −0.18 [0.86]

7121.74

1798.44 1806.34 1771.37

6924.88

7033.91 7010.70

276.83 267.89 275.54 263.52 271.51

284.17 281.37

− 1.14 [0.27] − 0.18 [0.86] − 0.80 [0.43] –

550.18

563.25 560.74 551.10 549.44 562.77 556.87 556.67

− 2.57 [0.01] − 1.53 [0.13] − 1.84 [0.07] –

283.91

− 1.35 [0.18] –

281.36

− 0.67 [0.51] − 0.33 [0.75] − 1.19 [0.24] − 0.32 [0.75] − 0.74 [0.46] –

380.75

–

700.43

− 1.49 [0.14] − 1.37 [0.17] − 1.41 [0.16]

385.43

0.47 [0.64] 0.37 [0.71] 0.11 [0.91]

696.13

284.84

295.23 283.81 291.06

386.48 389.23

556.10 569.61 581.01 577.69 565.33 594.51

680.71 726.02

− 0.85 [0.41] − 1.31 [0.20] − 0.19 [0.85] – − 0.63 [0.53] − 0.27 [0.79] − 0.13 [0.89] – − 0.81 [0.43] − 1.23 [0.24] − 0.78 [0.44] – − 0.92 [0.37] − 1.39 [0.18] − 0.94 [0.35] – − 1.03 [0.31]

Notes: DM p-values are in brackets. The bolded values for MAE and RMSE are associated with best performing volatility model.

conditional variance for five MISO pricing hubs. We also evaluate the out-of-sample forecasting performance of the conditional variance, as opposed to the conditional mean specification. Our specification of the conditional mean properly accommodates price spikes, seasonality, as well as cyclical and secular movements of electricity prices. As for conditional volatility, we consider four alternative classes of models – GARCH (1,1), EGARCH (1,1), APARCH (1,1) and CGARCH (1,1) – to accommodate various complexities in electricity price volatility. We examined the weekly and daily out-of-sample forecasting performance of the conditional variance models using three test statistics (MAE, RMSE, and DM). Generally speaking, APARCH performed well in most hubs, yet was not always able to produce results that were statistically better than GARCH, EGARCH, or CGARCH over a longer forecasting horizon. We also found that the presence of deregulation influences the optimal volatility model at all five hubs; specifically those hubs located in deregulated states appear to benefit more from employing a more complex volatility specification. Differences in the selected volatility model at each hub are also driven by changes in the forecasting horizon. Over a shorter forecasting horizon, for example, all four volatility models have equal forecasting capabilities irrespective of the hub being considered. This is likely due to the fact that price and hub complexity are less mature at shorter horizons such that a simple volatility model is sufficient. The findings of this study may have implications on the way buyers and sellers anticipate and/or model electricity price volatility

and thus their bids in the real-time market. Furthermore, the evidence provided by this study in support of significant electricity price volatility may provide regulators with greater insight into the efficiency of the market at the locations considered. Volatility could be reduced by increasing participation in real-time pricing (RTP) programs wherein the wholesale price is set equal to the retail price confronted by consumers in a given hour. Without real-time pricing, consumers have no incentive to alter consumption in critical load hours such that facilities that employ higher marginal cost fuels will occasionally set the price. A more costly alternative to reducing volatility entails investing in transmission or increasing generation in congested areas.

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