Forecasting spot price volatility using the short-term forward curve

Energy Economics 34 (2012) 1826–1833

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Energy Economics journal homepage: www.elsevier.com/locate/eneco

Forecasting spot price volatility using the short-term forward curve Erik Haugom a, 1, Carl J. Ullrich b,⁎ a b

Lillehammer University College, NO-2624 Lillehammer, Norway, and the Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Virginia Tech, Pamplin College of Business, Department of Finance, Insurance, and Business Law, Blacksburg, VA, 24061, United States

a r t i c l e

i n f o

Article history: Received 3 June 2011 Received in revised form 23 July 2012 Accepted 24 July 2012 Available online 4 August 2012 JEL classification: G17 C52 C14 Q47 L94

a b s t r a c t We use high frequency real time spot prices and day-ahead forward prices from the Pennsylvania–New Jersey– Maryland wholesale electricity market to calculate, describe, and forecast spot price volatility. We introduce the concept of forward realized volatility calculated from day-ahead forward prices. Forward realized volatility improves forecasts of spot price volatility – in the sense of higher R2s and significantly lower forecast errors – when compared with forecasts based solely upon historical volatility. The largest forecast improvements obtained when the change in forward realized volatility is large in magnitude. Splitting total volatility into its continuous and jump components is crucial for forecasting volatility at weekly and monthly horizons. © 2012 Elsevier B.V. All rights reserved.

Keywords: Volatility forecasting Realized volatility Implied volatility Forward prices Electricity markets

1. Introduction An understanding of volatility is fundamental to risk management and derivative valuation. Because electricity is not storable, electricity prices are known to be the most volatile price series extant. Accurate forecasts of electricity volatility are crucial for traders, portfolio managers, policy makers, and other market participants. Power plants are call options written on the spark spread – the spread between the cost of fuel required to produce electricity and the value of the electricity produced. An understanding of volatility is necessary for accurate valuation of power plants, both existing and proposed. More generally, the ability to forecast volatility is crucial for anyone who has a position in options written on the spot price of electricity. Chan et al. (2008) emphasize the importance of forecasting electricity volatility when they write (p.731) “… power-market participants stand to benefit greatly from any improvement that can be achieved in volatility forecasting accuracy.”2

⁎ Corresponding author. E-mail addresses: [email protected] (E. Haugom), [email protected] (C.J. Ullrich). 1 This work was completed while Haugom was a Visiting Scholar at Virginia Tech. 2 Other work on electricity volatility includes Solibakke (2002), Goto and Karolyi (2004), Higgs and Worthington (2005, 2008), and Haugom et al. (2011). Hadsell et al. (2004) make clear the importance of modeling electricity volatility when they write (p.24) “Understanding the volatility dynamics of electricity markets is important in evaluating the deregulation experience, in forecasting future spot prices, and in pricing electricity futures and other energy derivatives.” 0140-9883/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.eneco.2012.07.017

Realized volatility calculated from high frequency data is easy to implement, model-free, and lends itself to relatively simple forecasting schemes which capture the long-memory feature of volatility. Another natural candidate for forecasting future volatility is the volatility implied by option prices. Option implied volatility incorporates market participants' expectations about the future and therefore contains information beyond that found in historical realized volatility. There is a large literature comparing volatility forecasts based upon realized versus implied volatility. Examples include Busch et al. (2011), Christensen and Prabhala (1998), Corrado and Miller (2005), Giot and Laurent (2007), and Martens and Zein (2004). In general, predictions obtained from implied volatility tend to be more accurate than those obtained from historical realized volatility. For most electricity markets there are no traded options so the traditional option implied volatility is not available. However, the unique structure of electricity markets provides us with an alternative measure of volatility which incorporates market participants expectations about the future. Many wholesale electricity markets consist of an hourly (or half-hourly) day-ahead, short term forward market, and an hourly (or half-hourly) spot market that operates in real time.3 For the Pennsylvania–New Jersey–Maryland (PJM) market, 3 There is an inconsistency in the literature regarding the term ‘spot’ as applied to electricity prices. Many authors refer to day-ahead prices as spot prices. In this paper, spot will refer to the real time hourly prices, not to the day-ahead forward prices. Dayahead prices make up a short-term forward curve.

E. Haugom, C.J. Ullrich / Energy Economics 34 (2012) 1826–1833

once each afternoon a new set of 24 forward prices becomes available, one for each hour of the following day. We use these day-ahead forward prices to calculate realized volatility and refer to it as forward realized volatility. To our knowledge, this is the first paper to use volatility calculated from the forward curve to forecast spot price volatility. Including forward realized volatility improves forecasts of future spot price volatility when compared to forecasts based solely upon historical realized volatility. The econometric specification used herein is based upon the Heterogeneous AutoRegressive model of Realized Volatility (HAR-RV) developed by Corsi (2009). Andersen et al. (2007) extend the work of Corsi (2009) by separating the continuous and jump portions of total realized volatility, which they refer to as the HAR-RV-CJ model. For electricity markets, Chan et al. (2008) apply HAR-type model specifications to realized volatility based upon high frequency Australian electricity price data, while Haugom et al. (2011) use HAR specifications to model Nord Pool forward data. This paper extends the work of these authors by including forward realized volatility calculated from day-ahead electricity forward prices as a right-hand-side variable, designated HARRV-F and HAR-RV-CJ-F in what follows. We make several contributions. First, we show that including forward realized volatility as a predictor improves forecasts of future spot price volatility relative to forecasts based solely upon historical spot price volatilities. We compare forecast errors (both root mean square errors and mean absolute errors) using the specification of Diebold and Mariano (1995) and find that, for a one day forecast horizon, errors from models which include forward realized volatility are significantly less than errors from the corresponding models without forward realized volatility. The forecast improvement due to the use of forward realized volatility declines as we move to one week and one month forecast horizons. Second, our results show that, at one week and particularly one month forecast horizons, splitting total realized volatility into (i) that portion which is due to continuous price movements, and (ii) that portion which is due to jumps is crucial for forecasting accuracy. As in other markets, electricity volatility is persistent. One can think of overall volatility as consisting of long term patterns and idiosyncratic deviations from the pattern. Including forward realized volatility as an independent variable at the one day horizon is helpful because it impounds market participants' expectation about deviations away from the pattern. The new information contained in forward realized volatility is less helpful for forecasting at weekly and monthly horizons. At longer forecast horizons, idiosyncratic deviations effectively are averaged away. Unlike realized volatility in financial markets, both the continuous and the jump portions of realized electricity volatility are persistent. In financial markets jumps are due to unexpected information and such information tends to be random. In electricity markets, while jumps can be caused by “unexpected information” such as the loss of a baseload power plant, jumps can also be the result of extremely hot weather and persistent supply and demand conditions. Importantly, though both the continuous and jump portions of electricity volatility are persistent, they are not highly correlated. As a result, separating out the continuous and jump components of total volatility improves forecast performance. Finally, we demonstrate that forward realized volatility is most helpful for forecasting future volatility when market participants expect changes in system patterns. Because volatility is persistent, any change in volatility signaled by forward realized volatility can persist into the future. Thus forward realized volatility can be helpful for forecasting at longer horizons when market participants expect a change to the long term system pattern. This paper is structured as follows. Section 2 reviews the calculation of realized volatility based on high frequency data. Section 3 introduces the concept of forward realized volatility. Section 4 describes the data. Section 5 details the forecasting models and Section 6 presents the

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results. In Section 7 we address the issue of the conditions under which forward realized volatility is most helpful in forecasting future spot volatility. Section 8 concludes. 2. Realized volatility Let Pt be the price of an asset at time-t. Let the natural logarithm of the price be denoted by yt ≡log(Pt). Consider the following continuous-time (finite activity) jump-diffusion process for yt, dyt ¼ μ t dt þ σ t dZ t þ κ t dK t ;

ð1Þ

where μt is the drift, σt is the local price volatility, Zt is a Wiener process, κt is the jump size, and Kt is a Poisson counting process with (possibly) time-varying intensity λt. Suppose that the price is observed at discrete times j = 1, …, M within each day t = 1, …, T and let rt,j ≡ yt,j − yt,j − 1 be the Δ≡ M1 period return. 4 Realized variance can then be calculated as M X

RV t ¼

2

r t;j :

ð2Þ

j¼2

In practice microstructure noise limits the observation frequency at which the data are informative. The relative coarseness (hourly) of electricity prices, combined with the fact that hourly electricity prices are averages of five-minute prices, means that microstructure noise is not an issue in our data. 2.1. Continuous versus jump volatility As the observation frequency increases, RVt as defined in Eq. (2) converges in probability to the sum of (i) integrated variance (IVt), i.e., the variance due to continuous price changes, and (ii) variance due to discontinuous jumps. The ability to detect jumps in discrete data relies upon being able to accurately estimate integrated variance. The most common estimate of IVt used in the literature is bipower variation as proposed by Barndorff-Nielsen and Shephard (2004) −2

BV t ¼ μ 1



 X M    M−1    r r ; M−ð2 þ iÞ j¼2þðiþ1Þ t;j t;j−ðiþ1Þ

ð3Þ

qffiffi where μ 1 ≡ π2 and i is the lag length (or stagger) in the multiplication of returns. In order to determine the occurrence of a jump, Huang and Tauchen (2005) suggest the use of a ratio statistic Zt, defined as 1 ½RV t −BV t =RV t Z t ¼ pffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  −4   ffi ; 2 Δ μ 1 þ 2μ −2 1 −5 max 1; TQ t =BV t

ð4Þ

where TQt is the standardized realized tri-power quarticity ! ! M2 1  M−2ð1 þ iÞ μ 34=3 4 M  3  4  4 X       r tþjΔ;Δ  r tþðj−ð1þiÞÞΔ;Δ 3 r tþðj−2ð1þiÞÞΔ;Δ 3 ;

TQ t ¼

ð5Þ

j¼1þ2ð1þiÞ

4 Electricity data is available for M = 24 h each day. We do not include the return for hour 1. Spot markets for electricity operate around the clock, thus the return for hour 1 is not an overnight return as in the case of stock prices. However, the forward ‘return’ for hour 1 is an overnight return. The forward price for hour 24 is determined today, while the forward price for the following hour 1 is determined tomorrow. We drop this return in spot and forward markets. Our summation begins with j = 2 and we have a total of (M − 1) = 23 return observations for each day.

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μ4/3 ≡ 2 2/3Γ(7/6)Γ(1/2) −1 ≈ 0.8308609, and i is as before the lag length in returns. The ratio statistic Zt defined in Eq. (4) is asymptotically normal under the null of no jumps. Significant jumps are those for which Zt exceeds a critical value. The jump portion of variance is then given by J t;α ¼ IZ t >Φα ðRV t −BV t Þ;

ð6Þ

where IZt >Φα is the indicator function which takes a value of one if Zt >Φα and zero otherwise, and Φα is the value of the inverse cumulative standard normal distribution evaluated at α. Consistent with previous work, we set α=0.99. The continuous (Ct) portion of RVt is then given by C t ¼ RV t −J t :

ð7Þ

2.1.1. The case of electricity prices The jump detection statistic in Eq. (4) is based upon the difference between total realized variance and bipower variation. In financial markets microstructure noise induces negative autocorrelation in high frequency returns and causes bipower variation calculated using a lag length of i = 0 to be upward biased. The jump test statistic in Eq. (4) is therefore downward biased and the test underestimates the occurrence of jumps. Huang and Tauchen (2005) show that increasing the lag between returns in the calculation of bipower variation can help to mitigate the negative autocorrelation induced by microstructure noise in high frequency financial returns. Price spikes in electricity markets – a large positive return followed shortly thereafter by a large negative return – also induce negative serial correlation in returns. The reason is different than in financial markets, but the result is the same. The jump test statistic underestimates the occurrence of jumps. Ullrich (2012-this issue) shows that increasing the lag length i greatly improves jump detection in electricity prices. We use three different estimates of IVt and the associated jump detection statistics. In order to ensure comparability with the realized volatility literature, and to serve as a reference case, our first proxy for IVt is BVt with i = 0, referred to as BV0 in the following. Our second estimate of IVt is BVt with the lag length increased to i = 4 (BV4). Our third and final proxy for IVt uses the threshold bipower variation (TBV) measure developed by Corsi et al. (2010). Threshold bipower variation is specifically designed to deal with multiple, possibly consecutive jumps and thus should work well in the case of electricity prices. TBV t ¼

−2 μ1

  X M    M−1    r t;j r t;j−1 Ir2 bθ I r2 bθ ; j j−1 M−2 j¼2þðiþ1Þ t;j t;j−1

3.1. Statistical versus fundamental models Efforts to model electricity prices and volatility can be split into two broad classes. The first class relies upon purely statistical models with no reference to underlying market fundamentals. Examples include Deng (2000, 2005), Gjolberg and Brattested (2011), Knittel and Roberts (2005), and Zareipour et al. (2007). The second class of electricity models eschews the purely statistical and instead explicitly includes factors thought to be fundamental drivers of electricity prices and/or volatility. Examples include Mount et al. (2006) and Karakatsani and Bunn (2008, 2010). Fundamental factors include, but are not limited to, system demand, reserve margin, fuel prices, and operating characteristics of the generating system. Karakatsani and Bunn (2010) persuasively argue that, because of the unique properties of electricity, fundamental factors are necessary for modeling electricity volatility. In particular they write (p.3) that models including fundamental factors are important because “… they represent subtle aspects of pricing within which agents may have different expectations about market fundamentals.” Day-ahead forward prices, and by extension forward realized volatility, are based upon market participants' bids and therefore incorporate expectations about fundamentals. Hence, forward realized volatility incorporates expectations about future electricity spot price volatility in much the same way that option implied volatility incorporates expectations about future equity price volatility. Forward realized volatility thus provides the best of both worlds – it allows the researcher to remain in the realm of statistical models while still including market participants' expectations about fundamental factors. Further, forward realized volatility does not require assumptions regarding the specific functional form through which fundamentals affect volatility. 4. Data

ð8Þ

2 where the threshold function θj ≡cθ V^ t , and V^ t is an auxiliary estimate of the ‘local’ spot variance.5

3. Forward realized volatility The concept of implied volatility is well-established in finance. Option prices can be inverted to determine the volatility of the underlying asset implied by the option price. Often the Black–Scholes option price formula is used to map prices into volatilities. Recently nonparametric methods which utilize the entire cross section of available strikes have been developed. See, for example, Britten-Jones and Neuberger (2000) and Jiang and Tian (2005). The VIX volatility index based upon one month S&P 500 options is one prominent example of implied volatility. For most wholesale electricity markets there are no traded options so the traditional option implied volatility is not available. However, the We use the iterative procedure for determining V^ t suggested in Appendix B of Corsi et al. (2010). We experimented with other specifications with varying levels of success. We chose to stay with the the original specification. This issue is the subject of ongoing research. We utilize the corrected jump test statistic C-Tz defined in Eq. (3.5) in Corsi et al. (2010). 5

unique structure of electricity markets provides an alternative measure of future volatility. The PJM market (and many other wholesale electricity markets) consists of an hourly day-ahead forward market, and an hourly real time spot market. Each afternoon at four o'clock a new short term forward curve transpires, with 24 hourly forward prices – one for each hour of the following day. The shortest of these forward prices has a maturity of 9 h and the longest has a maturity of 32 h. We calculate realized volatility using these 24 hourly day-ahead forward prices and refer to it as forward realized volatility.6

We use both day-ahead forward prices and real time spot prices from the eastern hub of the PJM wholesale electricity market. Our sample period is June 2002 through July 2010. We calculate returns as the log difference of prices. Days with missing returns are dropped from the sample. 7 Our final sample has 2793 days or 67,032 hourly prices (64,239 hourly returns). Electricity prices have intraday patterns that vary by day of week and time of year. Before applying the realized volatility apparatus we need to account for seasonality in the raw data. Returns are demeaned by month of year, day of week, and hour of day. Because return distributions have non-zero skew and large excess kurtosis the ‘demeaning’ is done using hourly median returns. μ^t;j ¼ r mn;dy;hr ;

ð9Þ

6 Forward realized volatility is not calculated from prices of any one particular forward contract. It is based upon hourly contracts with different maturities. Because all 24 prices are determined simultaneously these short term forward prices are not a time series in a strict sense. 7 Electricity prices can be negative. Negative prices render the notion of log returns nonsensical, hence days with negative prices are dropped from the sample.

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where r mn;dy;hr is the median return for day t in month mn, on day of the week dy, and in hour j = hr. In Eqs. (2), (3), and (5) rt,j is then replaced by 

r t;j ¼ r t;j −μ^t;j :

ð10Þ

Summary statistics for prices (in $/MWh) and returns can be found in Table 1 and demonstrate all the usual properties associated with electricity prices and returns – large maximum prices (due to price spikes), high standard deviations(relative to the mean), positive skewness, and extreme excess kurtosis. Tables 2 and 3 present summary statistics for the natural logarithms of spot price realized pffiffiffiffiffiffiffiffi volatility and forward realized volatility respectively, i.e., ln RV t . 8 Spot price realized volatility is much larger and more variable than forward realized volatility. The likely reason is because spot prices include sudden, unexpected changes due to, e.g., transmission constraints, outages, etc. Spot price volatility and forward realized volatility are highly persistent, as demonstrated by the autocorrelations and the Ljung–Box statistics. 5. Forecasting models Our basic regression framework is the Heterogeneous AutoRegressive model of Realized Volatility (HAR-RV) of Corsi (2009). Begin by defining the h-day mean log realized volatility from day t through day (t+h−1) as

qffiffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n pffiffiffiffiffiffiffiffi RV tþh−1 : ð11Þ ln RV t;tþh−1 ≡ ln RV t þ ln RV tþ1 þ … þ ln h

Define similarly the portions ofh-day realized volatility due to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi continuous price movements ln C t;tþh−1 and discontinuous  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  jumps in prices ln J t;tþh−1 þ 1 . We let h = 1, 7, 30 (for h = 1, RVt,t + 1 − 1 ≡ RVt) and refer to these as daily, weekly, and monthly realized volatility. 9 The HAR-RV-F model specification includes forward realized volatility as a forecasting variable. pffiffiffiffiffiffiffiffiffiffiffiffiffi 8 β0 þ βD ln RV t−2 > > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > RV t−8;t−2 þβW ln > > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > þβM ln RV t−31;t−2 > > > > > > > þt;ðtþh−1Þ; > > > > > > > > > <

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffiffiffiffiffiffiffiffiffi ln RV t;tþh−1 ¼ β0 þ βD ln RV t−2 > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > RV t−8;t−2 þβW ln > > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > þβM ln RV t−31;t−2 > > > > > pffiffiffiffiffiffiffiffiffiffiffi > f > þβD ln FRV t > > > > > > > > þt;ðtþh−1Þ; > > > :

HAR−RV ð12Þ

HAR−RV−F

for h = 1, 7, 30. FRVt is the forward realized volatility for day t. 10 8 We chose to take the natural logarithm of volatility because, as is the case for financial assets, the distribution of the log of realized volatility is much closer to being normally distributed than is realized volatility itself. See, for example, Andersen et al. (2001). The OLS forecasting regressions in the following section are ill-suited for variables for which the distribution is far from normal. We have repeated the regressions pffiffiffiffiffiffiffiffi using realized volatility RV t and the results are very similar. 9 Because electricity markets operate around the clock, the data include weekends and holidays. We use 7-day weeks and 30-day months. 10 Notice that the forecast object (the left-hand-side variable) is the forward-looking spot price volatility beginning with day t, but the most recent spot price volatilities which appear on the right-hand-side are for day t − 2. Forward realized volatility for day t (FRVt) is observed at 4 pm on day t − 1. Because the spot market operates around the clock, spot volatility for day t − 1 is not known until midnight, i.e., the beginning of day t. At 4 pm on day t − 1 the most recent observation of daily spot price volatility available to market participants is RVt−2.

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Table 1 Price and return summary statistics. The table presents summary statistics for hourly electricity prices and log returns. Price data have units of dollars per megawatt hour ($/MWh). The raw data are hourly electricity prices from the PJM market. The return data have been demeaned by month of year, day of week, and hour of day. The sample period extends from June 2002 to July 2010. Spot

nobs Mean Stdev Skew Kurt Min Max

Forward

Price

Return

Price

Return

67,032 $54.80 $38.46 3.308 32.900 $0.04 $1032

64,239 −0.004 0.319 −0.056 16.536 −6.024 4.981

67,032 $54.76 $30.30 2.124 13.002 $0.52 $444.0

64,239 0.002 0.096 0.729 13.198 −1.722 1.638

Because forward realized volatility is based upon day-ahead forward prices, it should improve forecast accuracy at the daily horizon. Electricity volatility is persistent, so changes in the historical pattern may well continue into the future, thus forward realized volatility has the potential to improve longer term forecast accuracy as well. Andersen et al. (2007) extend the HAR-RV model by separating the continuous and jump portions of realized volatility, which they refer to as the HAR-RV-CJ model. Our encompassing regressions include the continuous and jump portions of forward realized volatility as forecasting variables, the HAR-RV-CJ-F model. 8 pffiffiffiffiffiffiffiffiffiffi β þ βD ln C xxx > t−2 > > 0 >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > þβW ln C xxx > t−8;t−2 > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > xxx > C t−31;t−2 > > þβM ln > > > p ffiffiffiffiffiffiffiffiffi   > > > þβJ xxx D ln J xxx > t−2 þ 1 > >

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q > > > > þβJ xxx W ln J xxx > t−8;t−2 þ 1 > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > > > > J xxx þβJxxx M ln > t−31;t−2 þ 1 > > > > > > þt;ðtþh−1Þ ; > > > > > > > HAR−RV−CJ > > <

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffiffiffiffiffiffi  xxx ð13Þ ln RV t;tþh−1 ¼ β0 þ βD ln C t−2 > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > xxx > þβW ln C t−8;t−2 > > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > þβM ln C xxx > t−31;t−2 > > > p ffiffiffiffiffiffiffiffiffi   > > > þβJ xxx D ln J xxx > t−2 þ 1 > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > > > þβJ xxx W ln J xxx > t−8;t−2 þ 1 > > >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > > > > J xxx þβJxxx M ln > t−31;t−2 þ 1 > >

pffiffiffiffiffiffiffiffiffiffiffi > > > > FC xxx þβfD ln > t > > >

 > pffiffiffiffiffiffiffiffiffiffi > f > > þβJ xxx D ln FJ xxx þ 1 þ t;ðtþh−1Þ ; > t > > > : HAR−RV−CJ−F where xxx = BV0, BV4, or TBV refers to the estimate of integrated variance used to separate the continuous and jump portions of total volatility. The continuous and jump portions of FRVt are given by FCt and FJt, respectively. 11 Standard error calculations use heteroskedasticity and autocorrelation consistent Newey–West corrections with seven lags for the daily regressions, 14 lags for the weekly regressions, and 60 lags for the monthly regressions. 11 The continuous and jump portions of forward realized volatility are measured in exactly the same way as the continuous and jump portions of spot volatility, i.e., using BV0, BV4, and TBV.

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Table 2 Spot realized volatility summary statistics. The table presents summary statistics for the natural logarithm of spot price realized volatility and its continuous and jump components pffiffiffiffiffi  pffiffiffiffiffiffiffiffi pffiffiffiffi  ln RV t ; ln C t and ln J t þ 1 . The headings refer to the estimate of integrated variance used to in the determination of jumps. BV0 is bipower variation with lag i = 0, BV4 is bipower variation with lag i = 4, and TBV is threshold bipower variation. See Eqs. (3) and (8) in the text. The raw data are hourly electricity prices from the PJM market. The sample period extends from June 2002 to July 2010. AC is the autocorrelation at # lags. LB# is the Ljung–Box portmanteau test for significant autocorrelations with # lags.

nobs Mean Stdev Skew Kurt Min Max AC1 AC7 AC30 LB30

pffiffiffiffiffiffiffiffi ln RV t

BV0 pffiffiffiffiffi ln C t

pffiffiffiffi  ln J t þ 1

BV4 pffiffiffiffiffi ln C t

pffiffiffiffi  ln J t þ 1

TBV pffiffiffiffiffi ln C t

ln

2793 0.261 0.401 −0.112 3.799 −1.265 2.076 0.355*** 0.282*** 0.214*** 4927***

2793 0.250 0.406 −0.107 3.803 −1.265 2.076 0.351*** 0.278*** 0.216*** 4492***

2793 0.021 0.120 5.905 37.419 0.000 1.076 0.008 −0.000 −0.005 30.75

2793 0.182 0.404 −0.493 3.362 −1.265 1.568 0.450*** 0.364*** 0.295*** 8875***

2793 0.138 0.322 2.388 8.333 0.000 2.713 0.072*** 0.067*** 0.020 113***

2793 0.180 0.427 −0.689 4.031 −1.971 1.377 0.485*** 0.365*** 0.294*** 9682***

2793 0.082 0.287 3.718 16.843 0.000 2.187 0.054*** 0.048** −0.001 66.32***

pffiffiffiffi  Jt þ 1

Table 3 Forward realized volatility summary statistics. The table presents summary statistics for the natural logarithm of forward price realized volatility and its continuous and jump compffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi  ponents ln FV t ; ln FC t and ln FJ t þ 1 The headings refer to the estimate of integrated variance used to in the determination of jumps. BV0 is bipower variation with lag i = 0, BV4 is bipower variation with lag i = 4, and TBV is threshold bipower variation. See Eqs. (3) and (8) in the text. The raw data are hourly electricity prices from the PJM market. The sample period extends from June 2002 to July 2010. AC is the autocorrelation at # lags. LB# is the Ljung–Box portmanteau test for significant autocorrelations with # lags. pffiffiffiffiffiffiffiffi FV t

BV0 pffiffiffiffiffiffiffiffi ln FC t

ln

2793 −0.924 0.371 0.210 3.095 −1.921 1.067 0.710*** 0.532*** 0.282*** 15,908***

2793 −0.948 0.373 0.217 3.142 −2.007 1.067 0.666*** 0.488*** 0.251*** 13,427***

2793 0.019 0.075 4.162 20.315 0.000 0.584 0.105*** 0.304 −0.008 130.7***

ln nobs Mean Stdev Skew Kurt Min Max AC1 AC7 AC30 LB30

pffiffiffiffiffiffi  FJ t þ 1

6. Results The results for the daily (h=1), weekly (h=7), and monthly (h=30) regressions are presented in Tables 4, 5, and 6, respectively. The first two columns of coefficients are for the HAR-RV and HAR-RV-F models. The next six columns present results for the HAR-RV-CJ and HAR-RV-CJ-F models based upon BV0, BV4, and TBV. 6.1. Daily regressions Table 4 presents coefficient estimates, adjusted R 2s, root mean square errors (RMSEs), and mean absolute errors (MAEs) for the daily (h = 1) regressions. The main result to emerge from Table 4 is that including information from the forward market (in the form of either FRV, or FC and FJ) significantly improves the accuracy of the forecast of the next day's spot price volatility, measured either by RMSE or MAE. The differences between the forecast errors with and without FRV are significant at the 1% level in every case. The coefficient (βD) on the most recent observation of spot price realized volatility (or its continuous part in the BV0, BV4, and TBV models) is small and insignificant. However, the coefficient (βDf) on FRV (or its continuous part for the BV0, BV4, and TBV models) is significant at the 1% level for every specification. The coefficients (βW and βM) on the most recent observations of weekly and monthly realized volatility (or it's continuous part for the BV0, BV4, and TBV models) are positive and highly significant. As in other markets, electricity volatility is highly persistent. Market participants thus incorporate into forward prices two types of information — (i) knowledge of longer term system patterns, i.e.,

BV4 pffiffiffiffiffiffiffiffi ln FC t

ln

2793 −1.021 0.376 0.176 2.882 −2.088 0.433 0.636*** 0.444*** 0.246*** 11,488***

2793 0.074 0.138 1.808 6.500 0.000 1.262 0.235*** 0.130*** 0.052** 777***

pffiffiffiffiffiffi  FJ t þ 1

TBV pffiffiffiffiffiffiffiffi ln FC t

ln

2793 −1.023 0.390 0.018 3.039 −2.509 0.433 0.564*** 0.306*** 0.152*** 5908***

2793 0.050 0.136 2.860 11.905 0.000 1.353 0.264*** 0.074*** 0.013 606***

pffiffiffiffiffiffi  FJ t þ 1

information about historical spot price volatility, and (ii) expectations about deviations from the long term pattern, i.e., new information. Weekly and monthly historical spot price volatility carry information about (i) system patterns. At the one day horizon, forward realized volatility improves forecast performance because it includes (ii) market participants' expectations for the following day. While including FRV improves our forecasts of future spot price volatility, splitting out total realized volatility into continuous and jump portions has very little effect. Each of the four forecast models that include FRV perform about the same. The forecast errors from f the ‘F’ models are nearly identical. The coefficient (βJD ) on the jump portion of daily realized volatility is significantly different from zero at the 10% level in the BV4 specification and at the 1% level in the TBV specification, but overall these specifications perform no better than the simple RV-F specification. These results are consistent with the results of Chan et al. (2008) who find that splitting out total volatility into its continuous and jump components improves forecasts only marginally at the one day horizon.

6.2. Weekly and monthly regressions Tables 5 and 6 present results for weekly (h = 7) and monthly (h = 30) regressions, respectively. 12 While the forecast improvements that result from including forward realized volatility decline as the forecast horizon increases, splitting total volatility into its continuous and jump components results in large forecast improvements 12

Chan et al. (2008) do not attempt to forecast weekly and monthly volatility.

E. Haugom, C.J. Ullrich / Energy Economics 34 (2012) 1826–1833

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Table 4 pffiffiffiffiffiffiffiffi One Day Forecast (h = 1). The table presents OLS regression estimates for the logarithm of one day realized volatility ln RV t . See Eqs. (12) and (13) in the text. For the coefficients, significance levels are indicated by *** (1%), ** (5%), and * (10%) and based on the Newey–West heteroskedasticity and autocorrelation consistent covariance matrix with seven lags. Significant differences in forecast errors between models with and without forward realized volatility are determined using the method of Diebold and Mariano (1995) and also are indicated by *** (1%), ** (5%), and * (10%). The raw data are hourly electricity prices from the PJM market. The data have been demeaned by month of year, day of week, and hour of day. The sample period extends from June 2002 to July 2010.

β0 βD βW βM βJD βJW βJM f βD f βJD AdjR2 RMSE MAE

RV

RV-F

BV0

BV0-F

BV4

BV4-F

TBV

TBV-F

0.038*** 0.034 0.194*** 0.628***

0.193*** 0.029 0.154*** 0.583***

0.037** 0.033 0.192*** 0.626*** 0.049 0.093 0.403

0.199*** 0.028 0.153*** 0.582*** 0.050 0.093 0.277 0.144*** 0.011 0.224 0.353*** 0.268***

0.109*** 0.018 0.215*** 0.539*** 0.027 −0.017 0.082

0.256*** 0.012 0.180*** 0.499*** 0.024 −0.041 0.108 0.137*** 0.095* 0.227 0.353*** 0.267***

0.102*** 0.036 0.192*** 0.519*** −0.004 0.105 0.200

0.243*** 0.031 0.156** 0.490*** −0.006 0.084 0.237 0.134*** 0.179*** 0.227 0.353*** 0.266***

0.142*** 0.209 0.357 0.271

0.224 0.353*** 0.268***

0.209 0.357 0.271

0.213 0.356 0.270

0.214 0.356 0.270

Table 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi One week forecast (h = 7). The table presents OLS regression estimates for the natural logarithm of one week realized volatility ln RV t;tþ7 . See Eqs. (12) and (13) in the text. For the coefficients, significance levels are indicated by *** (1%), ** (5%), and * (10%) and based on the Newey–West heteroskedasticity and autocorrelation consistent covariance matrix with 14 lags. Significant differences in forecast errors between models with and without forward realized volatility are determined using the method of Diebold and Mariano (1995) and also are indicated by *** (1%), ** (5%), and * (10%). The raw data are hourly electricity prices from the PJM market. The data have been demeaned by month of year, day of week, and hour of day. The sample period extends from June 2002 to July 2010.

β0 βD βW βM βJD βJW βJM f βD f βJD AdjR2 RMSE MAE

RV

RV-F

BV0

BV0-F

BV4

BV4-F

TBV

TBV-F

0.043*** 0.025*** 0.171*** 0.642***

0.112*** 0.023** 0.153*** 0.623***

0.047*** 0.025*** 0.169*** 0.639*** 0.044* −0.014 0.322

0.118*** 0.023** 0.151*** 0.620*** 0.045* −0.042 0.266 0.063*** −0.004 0.521 0.173* 0.137

0.110*** 0.027** 0.185*** 0.545*** 0.007 −0.056 0.154

0.174*** 0.024** 0.170*** 0.529*** 0.006 −0.065 0.165 0.059*** 0.025 0.533 0.171 0.135

0.103*** 0.032*** 0.170*** 0.532*** 0.013 0.044 0.274*

0.164*** 0.030*** 0.157*** 0.522*** 0.012 0.036 0.291* 0.056*** 0.029 0.531 0.172 0.136

0.062*** 0.513 0.175 0.138

0.520 0.174 0.138

0.514 0.175 0.138

at these longer horizons. The single best specification for weekly and monthly volatility (as measured by R 2, RMSE, and MAE) is BV4-F. 13 The results again are consistent with the interpretation that electricity volatility can be decomposed into long term patterns and idiosyncratic shocks. At the monthly forecast horizon the effect of idiosyncratic shocks is significantly reduced. While both the continuous and jump portions of spot price volatility are persistent (see Table 2), they are not highly correlated. The sample correlation coefficient between the continuous and jump portions of overall volatility is 0.007. Hence separately measuring the individual components of overall volatility is very helpful in forecasting weekly and monthly volatility. These results are important because they suggest that we should be able to forecast jump volatility. Ongoing research is aimed separately modeling the continuous and jump portions of overall volatility with the ultimate goal of forecasting price spikes in electricity markets. 7. When does forward realized volatility matter? The primary result of this paper is to show that forward realized volatility improves forecasts of future spot price volatility. In this section we seek to answer the question, “Under what conditions does forward realized volatility matter the most for forecasting future volatility?” Intuitively, we expect that forward realized volatility should contain more new information when market participants expect that system 13 Tables 5 and 6 highlight the inability of the traditional bipower variation measure (BV0) to adequately separate jumps from continuous price movements in the case of electricity prices. The results for the BV0 specification are not a significant improvement over the results for the HAR-RV case.

0.527 0.172 0.136

0.525 0.173 0.137

conditions tomorrow will be different from today. If no significant changes are anticipated, then there should be little or no new information in forward realized volatility and as a result it should not contribute much to the forecast of tomorrow's volatility when compared to forecasts based solely on historical spot price volatility. However, when market participants anticipate that there will be changes tomorrow, e.g., due to impending weather changes or scheduled generator maintenance outages, then those expectations will be impounded into forward prices, and by extension into forward realized volatility. Under these circumstances, the inclusion of forward realized volatility as a forecasting variable should significantly improve forecasts of tomorrow's volatility. Perhaps more importantly, in this case the inclusion of forward realized volatility may also improve forecast performance at longer horizons. Because volatility is persistent, changes in forward realized volatility may signal changes in system patterns which will persist into the future. 7.1. Changes in forward realized volatility We begin by calculating the absolute value of the change in forward realized volatility from yesterday to today. 14

pffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   FRV t − ln FRV t−1  Δ ln FRV t ¼  ln

ð14Þ

14 Ideally, we would like to know the difference between the volatility implied today from forward prices for delivery tomorrow and today's spot price volatility. However, as discussed above, today's spot price volatility is not available at the time FRVt is known.

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Table 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi One month forecast (h = 30). The table presents OLS regression estimates for the natural logarithm of one month realized volatility ln RV t;tþ30 . See Eqs. (12) and (13) in the text. For the coefficients, significance levels are indicated by *** (1%), ** (5%), and * (10%) and based on the Newey–West heteroskedasticity and autocorrelation consistent covariance matrix with 60 lags. Significant differences in forecast errors between models with and without forward realized volatility are determined using the method of Diebold and Mariano (1995) and also are indicated by *** (1%), ** (5%), and * (10%). The raw data are hourly electricity prices from the PJM market. The data have been demeaned by month of year, day of week, and hour of day. The sample period extends from June 2002 to July 2010.

β0 βD βW βM βJD βJW βJM f βD f βJD AdjR2 RMSE MAE

RV

RV-F

BV0

BV0-F

BV4

BV4-F

TBV

TBV-F

0.063*** 0.017*** 0.179*** 0.571***

0.113*** 0.015*** 0.165*** 0.556***

0.072*** 0.016*** 0.175*** 0.571*** 0.021* 0.225* −0.272

0.124*** 0.015*** 0.163*** 0.556*** 0.022** 0.203* −0.315 0.046** 0.025 0.611 0.133 0.104

0.126*** 0.020*** 0.190*** 0.482*** −0.002 −0.034 0.110

0.176*** 0.018*** 0.180*** 0.471*** −0.003 −0.039 0.119 0.044** −0.005 0.629 0.130 0.101

0.123*** 0.024*** 0.177*** 0.466*** 0.004 0.066 0.163

0.161*** 0.023*** 0.169*** 0.460*** 0.004 0.061 0.174 0.035* 0.013 0.628 0.130 0.102

0.676*** 0.601 0.134 0.104

0.607 0.134 0.104

0.605 0.134 0.104

0.624 0.131 0.102

0.625 0.130 0.103

Table 7 Volatility forecasts in the top quartile of changes in FRV.

The table presents OLS regression estimates for the logarithm of one day, one week, and one month realized volatility for pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi  days on which the change in forward realized volatility Δln FRV t ¼ ln FRV t −ln FRV t−1  is in the top quartile, i.e., when there are large changes. For the coefficients, significance levels are indicated by *** (1%), ** (5%), and * (10%) and based on the Newey–West heteroskedasticity and autocorrelation consistent covariance matrix with seven, 14, and 60 lags. Significant differences in forecast errors between models with and without forward realized volatility are determined using the method of Diebold and Mariano (1995) and also are indicated by *** (1%), ** (5%), and * (10%). The raw data are hourly electricity prices from the PJM market. The data have been demeaned by month of year, day of week, and hour of day. The sample period extends from June 2002 to July 2010. We use BV4 to separate out the continuous and jump portions of total volatility. See Eqs. (12) and (13) in the text. One day (h = 1)

β0 βD βW βM βJD βJW βJM f βD f βJD AdjR2 RMSE MAE

One week (h = 7)

One month (h = 30)

RV-CJ

RV-CJ-F

RV-CJ

RV-CJ-F

RV-CJ

RV-CJ-F

0.107*** 0.022 0.165 0.562*** −0.002 −0.018 0.174

0.340*** 0.021 0.096 0.521*** −0.006 −0.018 0.154 0.204*** 0.076 0.202 0.369*** 0.279***

0.130*** 0.029 0.133* 0.556*** 0.034* −0.201*** 0.260

0.234*** 0.027 0.106 0.540*** 0.033 −0.196*** 0.252 0.089*** −0.010 0.470 0.179** 0.140

0.133*** 0.027*** 0.092** 0.553*** 0.016 −0.095 0.102

0.195*** 0.024** 0.080* 0.545*** 0.015 −0.085 0.098 0.051*** −0.062 0.596 0.129 0.098

0.163 0.378 0.286

0.450 0.182 0.142

pffiffiffiffiffiffiffiffiffiffiffi We split the sample into quartiles based upon Δ ln FRV t . We estimate the full HAR-RV-CJ and HAR-RV-CJ-F models given in Eq. (13) within each quartile. Table 7 presents results for the top quartile, i.e., pffiffiffiffiffiffiffiffiffiffiffi when Δ ln FRV t is greatest in magnitude. 15 We use BV4 to separate out the continuous and jump portions of total volatility. Table 7 shows that volatility is harder to forecast when large changes are expected. For example, the R 2 (RMSE) [MAE] for the daily (h = 1) HAR-RV-CJ-F regressions is 20.2% (0.369) [0.279] compared with 22.7% (0.353) [0.267] (see Table 4) for the full sample. However, Table 7 also shows that, when large changes are expected, including forward realized volatility results in a larger improvement in forecast performance. From Table 4, including forward realized volatility in the full sample regressions increases the R 2 by 1.4 percentage points, from 21.3% to 22.7%. From Table 7, for the quartile regession the R 2 increases by 3.9 percentage points, from 16.3% to 20.2%. Forecast errors show a similar pattern. Importantly, these results also hold for the weekly and monthly horizons.

0.585 0.130 0.100

volatility. We extend the literature by including the volatility derived from day-ahead forward prices as a predictor of future spot volatility. Forward realized volatility is based upon market participants' bids and thus incorporates expectations about fundamental factors that might affect future spot price volatility. Including forward realized volatility significantly improves forecast accuracy at the daily horizon; the improvement declines as the forecast horizon increases. Splitting realized volatility into its continuous and jump components improves forecast accuracy for longer forecast horizons. The forecast improvement that results from using forward realized volatility is most pronounced when large changes in volatility are expected. In this case, including forward realized volatility significantly improves forecast accuracy even at the one month horizon. These results suggest that it may be possible to use HAR-type models to predict jump volatility, at least at longer forecast horizons. Ongoing research is aimed at separately predicting the continuous and jump portions of overall volatility and perhaps forecasting spikes in electricity prices.

8. Conclusions This paper uses high frequency electricity price data from PJM to calculate, describe, and forecast spot price electricity realized 15

Differences in forecast performance with and without FRV are small and insignificant in the lower three quartiles. These results are available upon request.

Acknowledgments This paper has been improved by discussions with Raman Kumar, Jostein Lillestøl, Gudbrand Lien, and the participants at FIBE 2011 in Bergen, Norway.

E. Haugom, C.J. Ullrich / Energy Economics 34 (2012) 1826–1833

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