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Modelling Day Ahead Nord Pool Forward Price Volatility: Realized Volatility versus GARCH Models

Erik Haugoma, Sjur Westgaardb, Per Bjarte Solibakke", Gudbrand Liend

Abstract-- Traditionally, and still within electricity

futures/forward markets, daily data has been utilized as the unit of analyses when modelling and making predictions of volatility. However, over the recent past it is argued that better volatility estimates can be obtained by using standard time series techniques on non-parametric volatility measures constructed from high-frequency intradaily returns. Liquidity in financial electricity markets has increased rapidly over the recent years, which make it possible to apply these relatively new methods for measuring market volatility. In this paper high-frequency data and the concept of realized volatility is utilized to make day ahead predictions of Nord Pool forward price volatility. Such short term volatility predictions are especially important for operators and other participants in the electricity sector. We compare the results obtained from standard time-series techniques with the more traditional GARCH-framework which utilizes daily returns. Additionally, we examine whether different approaches of decomposing the total variation into a continuous - and jump measure improves the model fit or not. The paper provides new insights to how the financial electricity market at Nord Pool works, and how we efficiently can model and make predictions of the price movements in this market.

Index Terms-High-frequency, Realized Volatility, GARCH, Electricity Forward Prices, Nord Pool, Long Memory

I. INTRODUCTION

L IBERALIZATION of electricity sectors around the world has lead to a rapid grow in electricity derivatives trading over the recent years. On the one hand, power producers

wanting to hedge a certain quantity of electricity can do this by selling an amount equal to a portion of their total production on the forward market, while big power consumers on the other hand can buy a specific amount of the power they need at the same market place. By doing this, producers and consumers can buy or sell electricity at a risk free price today with delivery over a specified period in the future. Additionally, this financial market is open to speculators that want to "make a bet" that the price of various derivatives goes one way or the other in the future. Hence, financial electricity markets function in almost the same way as other financial commodity markets.

aE. Haugom is PhD-student at Lillehammer University College. E-mail: erik.haugom@hiLno, Adress: Postboks 952, 2604 Lillehammer, Norway ( corresponding author). bS. Westgaard is Associate Professor at Trondheim Business School. E-mail: sjur.westgaard@hist.no cpo B. Solibakke is Associate Professor at Molde University College. E-mail: per. b.solibakke@hiMolde.no dG. Lien is Professor at Lillehammer University College. E-mail: gudbrand.lien@hiLno

When operating in financial markets in general, asset holders are especially interested in the return volatility over the holding period and not necessarily over some historical period. In order to build adequate model of the series under consideration however, the available historical data are used to obtain estimates of the future risk. Over the last two decades, the well known ARCHIGARCH approach and various generalizations of this, has been found to well describe the non-linear features of financial returns [18], [28]. This framework has also been applied in electricity volatility modelling and forecasting [14], [20], [22], [25], [26], [27] [34], [35], [39]. Most of these studies have utilized electricity spot prices and returns as the unit of analyses, which exhibit several distinctive features that are not found in traditional commodity markets due to the non-storability of electric power. However, for power producers, big power consumers, traders and other players in the electricity sector, the movements of various derivatives are perhaps of more interest than the underlying spot returns, since these are the prime tools for hedging risk.

Since 2005 liquidity in the Nord Pool financial market has been growing rapidly and made it possible to model the movements of derivatives in this market. When having such tick-by-tick, high-frequency transaction data at hand, new methods for modelling and making predictions of volatility become available to researchers and practitioners. That is, daily volatility estimates can be computed non-parametrically by a simple method called realized volatility [2]. By building on continuous time theory, they show that this convenient quantity also can be used for process modelling by relatively simple time-series techniques. Reference [13] and [37] have applied this concept on electricity spot prices. However, since the true intention of the concept is based on continuous time theory, the aspect of applying data that is usually determined once a day for each hourlhalf hour the following day are not quite in line with the underlying theory of realized volatility. Additionally, as electricity spot prices exhibit extreme movements on a day to day and intraday basis, data pre­processing must be carried out in order to remove any deterministic drift in the series.l In contrast to the underlying electricity spot prices, electricity futures/forward prices exhibit far less variation. This is especially the case for the longer contracts as less relevant information of the actual conditions at delivery is available as the horizon increases (Samuelson effect). [38] have utilized energy futures data and the concept of realized volatility and realized correlation for sweet crude

I Chan et al. (2008) do this by demeaning the price changes in the Australian electricity spot prices before calculating the various volatility measures.

978-1-4244-6840-9/10/$26.00 © 2010 IEEE

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oil and natural gas. In this paper the authors find that many of the stylized properties of traditional financial assets also are found in the energy sector. This include: (1) non-Gaussian distribution of unconditional daily returns and daily realized variance and (2) long memory in volatility. They also find evidence of asymmetric volatility in one of the examined contracts (natural gas) even though the absolute value of the skewness is below 1. Additionally, they argue that different results are obtained by utilizing the concept of realized volatility and the GARCH approach regarding the effect of lagged returns (of either sign). Overall, they find the use of realized volatility and realized correlation highly appropriate for energy data, and suggest that this concept should be further examined within and between other energy and financial markets.

In this paper we briefly examine the stylized properties of financial electricity return volatility and suggest various models that can be used for day ahead volatility modelling of two of the most liquid financial contracts traded at Nord Pool. In order to test the performance of the day a head estimates utilizing standard time-series techniques on realized volatility, the results are compared with volatility estimates produced with GARCH-type models utilizing daily returns. We can then compare the traditional approach of modelling volatility in financial markets with standard time series techniques, and examine whether an increase in accuracy is observed when taking the step up to higher frequencies and the concept of realized volatility. Put differently; do the additional costs and complications in data gathering and data pre-processing provide benefits as measured by more accurate volatility forecasts in financial electricity markets?

The various models ability of capturing the true day ahead volatility (as measured by realized volatility at time t+1) is examined by a simple bivariat regression for the whole sample period and two widely used accuracy measures for the last 250 and 100 trading days, respectively.

The rest of this paper is organized as follows: In section II theories on realized volatility and GARCH-type models are provided. Section III describes the market, the data and preliminary analysis. In section IV the results are presented and in section V conclusions, implications and suggestion for further research are provided.

II. GARCH-FRAMEWORK AND REALIZED VOLATILITY

A. GARCH-framework Reference [19] first introduced an autoregressive model to

capture the conditional variance of a given time-series. His hypothesis was that volatility as measured by the square of the mean-adjusted relative change in the dependent variable at time t was related to its values in previous periods, t-i, which can be expressed as in (1):

Et = Ztat Zt-i. i. d. D(O, 1) (1)

at = Wt + L{=l aiEt-i

where Zt is an independently and identically distributed (i. i. d.) with E(zt) = 0 and Var(zt) = 1. at is the conditional

2

variance (that may change over time), wt is a constant and Et is the innovation of the process. For notation simplification, we use W for both wand Wt in all subsequent equations as done in [28].

By employing this model on quarterly United Kingdom inflation data, he found a highly significant relationship, and thus dependency in volatility. In order to reduce the number of estimated parameters in the ARCH(q) model, [10] suggested the Generalized ARCH (GARCH) model:

(2)

where a and � is parameters to be estimated. In this case, the conditional variance at time t depends not

only on the squared error term from previous time periods (as in the ARCH(q) models)) but also on its conditional variance in previous time periods. For both the ARCH- and GARCH­model specification cases, at has to be positive for all time points, t. To ensure this, sufficient conditions are given by Wt > 0 and ai � 0 (for i = 1, ... , q) in equation 1 and additionally {3j � 0 (for j = 1, ... , p) in equation 2. Exogenous variables can additionally be introduced in the conditional variance equation in order to improve the forecasts. The positivity constraints are then not valid anymore, but the conditional variance must still be non-negative [28]. The GARCH-model has been widely examined on different financial data and it is claimed that even in its simplest form (GARCH 1,1) this model provides a good first approximation to the observed temporal dependencies in daily data [2].

Since the introduction of the ARCHIGARCH-model framework, a vast number of generalizations have been suggested in the literature. There are mainly two properties of financial time series that have gained a lot of attention in this matter. This is the aspect of that volatility seems to react differently to big price increases than big price drops also referred to as a leverage effect and that volatility is exhibiting a long term dependence, so called long-memory [31]. The former aspect has among other things lead to the development of the Exponential GARCH-model (EGARCH) originally introduced by [30], and the Asymmetric Power ARCH-model (AP ARCH) of [17]. To capture the long memory in volatility various fractionally integrated GARCH-models (FIGARCH) has been proposed in the literature (e.g. [5], [11], [36]).2

Even though it would be possible to run GARCH-type models at higher frequencies than one business day (see [32] for a comparison of such models with the concept of realized volatility), [16] point out that even when the intradaily seasonal patterns are accounted for, the aggregation properties of the GARCH-model break down at these higher frequencies. Microstructure effects are also observed when time intervals shorter than about 90 minutes are used. Hence, in order to avoid these practical problems, daily returns will be used when constructing volatility forecasts for Nord Pool forward prices.

2 For a nice review of various GARCH-models applied on financial data, see [31].

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B. Realized Volatility In the GARCH models we use daily returns to provide

volatility forecasts one day ahead. However, in the high­frequency data framework, this approach means that we are ignoring all transactions between opening and closing of the given day. In order to utilize the available information provided in ticker-data, [2] proposed an ex-post measure of the volatility based on the cumulative intradaily squared returns, known as realized variance. This nonparametric measure has shown to provide a consistent estimate of the price variability and could thus be used for process modelling. The daily realized volatility at the end of day t is defined in the following way:

(3)

where Tt is log price difference between t and t - 1 (returns at time t).

Hence, realized variation is defined as the sum of the intradaily squared returns, with !:J. equal to the return period (e.g. 30 minutes). When !:J.-t 0 and in the absence of jumps, RVt+1 (!:J.) measures the latent integrated volatility perfectly [28]:

In most practical circumstances however, empirical studies have shown that a simple continuous diffusion model have its limitations when trying to explain some characteristics of asset returns [1], [7]. That is, the total variation may better be described by a continuous component and a discontinuous jump component as illustrated by the following jump-diffusion process for the logarithmic price p(t) = log(s(t)):

dp(t) = Jl(t)dt + a(t)dW(t) + K(t)dq(t),O :s; t :s; T, (5)

where Jl(t)is the drift, aCt) is a strictly positive and caglad (left continuous with right limits) stochastic volatility process, Wet) is a standard Wiener process and q(t) is a counting process assuming the value 1 if a jump occurs at time t and 0 otherwise, with possibly time-varying intensity J(t) and size K(t).

From quadric variation theory it can be shown that realized volatility is a consistent estimator of the integrated volatility in the absence of jumps. In presence of jumps, the realized variance measure may be specified as a continuous integrated variance component and a discontinuous jump component:

RVt+1 (!:J.) -t ftt+l a2(i)di + Lt<ist+l K2 (i) (6)

Reference [8] has shown that by decomposing the total quadric variation into two components through the bi-power measure we obtain a consistent estimator of the integrated variance under the presence of jumps. The bi-power variation measure could be defined like this:

(7)

3

where Jll == ,jZ/Ti == 0.79788. Hence, in the presence of jumps and for !:J.-t 0 the probability of picking up discontinuities disappears in the limit, and we obtain:

r t+l 2 BVt+1(!:J.) -t h a (s)ds (8)

The discontinuous jump-component can now consistently be estimated as the difference of the realized variance and bi­power variance measures:

(9)

However, since the bi-power measure in practice could exceed the realized variation measure, a positivity constrain should be imposed for the jump component:

It+l (!:J.) == max [RVt+l (!:J.) - BVt+l (!:J.), 0] (10)

In this paper, we will apply a maxlogZ-test in order to determine whether a jump is present as advocated by [9]. This test statistic is computed in the following way:

(11)

where we use the Tri-power quadrati city TQt(!:J.) to estimate integrated quarticity IQt as suggested by [3]:

_ -3 M 1 14/31 1

4/31 14/3 TQt(!:J.) = MJl4/3 Li=4 Tt,i Tt,i-l Tt,i-2 ,

with Jl4/3 == 2�r(Z)r(�rl 6 2

(12)

However, the bi-power variation measure utilized in the above have some drawbacks [12], [4]. First, for extremely short sampling intervals (small !:J.), there is a small chance that jumps affect two neighbouring returns, and the impact of jumps on the BV measure is almost non-existent. Moreover, the bi-power measure is sensitive to the presence of "zero" returns in the sample, which is highly probable in markets where liquidity is low, such as the Nord Pool Financial markets (even with sufficiently low sampling frequencies). Alternatives to the bi-power variation measure have been proposed in the literature. The nearest neighbor truncation estimator by [4], the range based variance estimator by [6], and the realized outlyingness weighted quadratic variation procedure, (ROWVar ) recently introduced by [12] and [28], have been included as an nonparametric estimator for the integrated variance (IV(t)). In this paper we apply the latter method and examine whether this way of constructing the continuous component can help improving the day ahead volatility predictions. This method weights down returns that are local outliers relative to neighboring returns. As more "jumpy" a local return window is, as lower weights it receives on the continuous outlyingness estimator. According to [28] the computation of the realized outlyingness weigthed variance involves two steps. First the outlyingness of return Tt,i is calculated:

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d . =

(rt•i)2 t.! at.i (13)

which is the square of the robustly studentized return. Here, at•i is a robust estimate of the instaneous volatility computed from all the returns belonging to the same local window as Tt,i. Due to strong intradaily seasonality in volatility, [12] strongly suggest to compute dt.i on filtered returns instead of raw returns. Thus, we additionally adjust for this periodicity in volatility and examine whether this can help improve volatility predictions. The median absolute deviation (MAD) method is applied and is defined as follows:

(14)

where 1.486 is a correction factor to make sure that the MAD is a consistent scale estimator of the normal distribution. The periodicity factor of Tt,i is then equal to:

(IS)

The second step in this process involves choosing an appropriate weight function. Reference [12] here strongly recommends the use of the Soft Rejection Huber function defined as follows:

WSR (Z) = min{1, k/z} (16)

where k is a tuning parameter to be selected. They also show that the outlyingness measure is asymptotically chi-squared distributed with 1 degree of freedom. Hence, the rejection threshold k can be set to xf({3), the � quantile of the xf distribution function. � = 9S% is suggested as a compromise between robustness and efficiency in this matter. Given the above, the ROWVar measure can be computed as follows:

(17)

where Cw is a correction factor to ensure that ROWVar is consistent for the IVar under the Brownian Semi martingale Model. A jump measure to test for significant jumps has also

max been proposed. It can be shown that._ .jd;;. follow a

!-l . . ... M Gumbel distribution under the null of no jump during day t. Hence, we can reject the null hypothesis if

._ max

.jd;;. > C-1(1 - a)Sn + Cn !-l ..... M (18)

where C-1(1 - a) is the I-a quantile of the standard Gumbel

distribution C = (210gn)O.5 - log(rr)+log (Iogn) and S = , n 2(2Iogn)0.5 n

( I 1

)05' Hence, according to [28] a better estimate of the 20gn . realized jumps is:

(19)

4

where It.a(fl) is a dummy variable assuming the value one if a jump is present on day t. Hence, in this study we will use the various approaches for computing the continuous and jump component in realized volatility and examine whether one approach is able to outperform the other methods when predicting day ahead realized volatility for Nord Pool electricity forward prices.

III. THE NORD POOL FINANCIAL MARKET AND THE DATA

The Nord Pool power exchange has offered trade in financial contracts without physical delivery since 1994. Such financial contracts were designed to meet the needs of producers, retailers and end-users who use the products for risk management. Additionally, the financial market is open for traders who profit on continuous price movements in the various contracts. High-quality ticker data for all the financial contracts traded at the Nord Pool exchange has been centrally recorded since 200S. From these data it's clear that the market seems to prefer short term futures and nearest quarter and year forward contracts. As the number of ticks is crucial in order to construct high-quality, high-frequency data, we will examine nearest quarter and year forward contracts in this paper. The data-range is from Januray 3rd 200S to May 29th for the yearly contracts and June 15t 200S to May 29th 2009 for the quarterly contracts. The quarterly contracts series had a different organization for the first quarter in 200S (tertial contracts), which is why the data-range for these contracts is a bit shorter than for the yearly contracts.

Table I illustrates the number of transactions for each contract over the sample period. As clearly illustrated in the table, the quarterly contract is more liquid than the yearly contract, ranging from IOS.2 to 267.2 unique transactions on average over the business day, while the yearly contract is ranging from 26.2 to 142.9 unique transactions. Both contracts are getting increasingly popular over the sample period as measured by the number of ticks.

Contract typ e and year 2005 Quarterly 2006Qu3rterty 2007 Quarterly 2008Quarterly 2009 Quarterly 2005 Yearly 2006yearly 2007 yearly 2008yearly

2009 yearly

TABLE I THE DATA SET

Numberof unigue time-point ticks Numberof business days 13570 U9 33795 249 48378 248 66805 250 18862 101 6498 248 9710 249

11316 248 2nl1 250 14435 101

Business day freguency 105.2 135.7 195.1 267.2 186.8 26.2 39.0 45.6

110.8

142.9

Number of unique time point transactions. The data is filtered using only one observation for each specified unique time-point. Hence, if more than one seller or buyer occurs at the same time point with the same price, only one observation is calculated.

In order to avoid the unwanted side-effect caused by unusual high returns (in absolute values) at contract roll-over, these are set to the expected value for a stationary series with a zero mean. This is done for both the high-frequency observations and the daily returns observations. Moreover, as previous research has found significant differences between overnight and trading hours returns, and that including these overnight returns introduce more noise than useful information

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[29], we will only use trading hours returns in this study. The daily returns are calculated as the logarithmic difference of the mean daily prices as calculated of the 16 intradaily prices (constructed for the high-frequency analysis). As a starting point we present descriptive statistics and preliminary tests of the variables of interest. These results are presented in Table II.

First, consistent with findings from traditional financial markets we find that none of the examined variables are normally distributed. That is, they are mostly right skewed (except the returns) with high excess kurtosis (fat tails). Second, most of the examined variables exhibit significant autocorrelation for the first lag (except two of the jump volatility measures). This has the implication that when running GARCH models, we need to adjust the mean by running an AR(I) model. For the constructed realized volatility measures this means that current and recent past volatility most likely can be used for estimating tomorrow's volatility. A weekly seasonality can also be observed as ACF­estimates at lag 5 are significant for most variables (which is equal to one business week). Hence, this aspect needs to be accounted for in the models. The Augmented Dickey Fuller test (ADF-test) in most cases reject the null hypothesis of that a unit root exists. However, the KPSS-test (which test the opposite hypothesis - 1(0», provide a somewhat different conclusion. These results are consistent with findings in traditional financial futures markets and provide evidence of long-memory and thus hyperbolic decay of the autocorrelation function [15].

Based on this evidence, the fractionally integrated parameter, d, is calculated with the method of [23]. Nearly all variables (with one exception) have a significant d parameter varying between 0.07 (JVOW) and 0.29 (RV). Surprisingly, even though the return variable for the quarterly contract series are stationary based on the ADF-test and conventionally stationary on the basis of the KPSS-test, the d parameter is significantly different from zero (0.15) and higher than for the squared returns. A comparison of the autocorrelogram of the returns and squared returns for both the quarterly and yearly contract series does not however, give any indication of why the quarterly contract returns should exhibit a hyperbolic decay and the yearly returns should not. In order to examine this aspect further the method proposed by [33] were also applied, but with the same conclusion. Either way, such a significant d parameter in the quarterly contract returns series suggests that the mean should be adjusted with an ARFIMA (l,d,O) model before applying any GARCH models. However, after running a simple ARFIMA model of the returns, no evidence of a significant fractionally integrated parameter is found and the mean is on the basis of this just corrected for first order autocorrelation.

TABLE II DESCRIPTIVE STATISTICS AND PRELIMINARY TESTS ON THE VARIOUS

VARIABLES RV

Quarterly #Cbs 1000 1000 1000

Mean -0_001 0,001 0,023

Min, -0,128 0,000 0,000

Max, 0,085 0,016 0,141

std, Dev, 0,025 0,001 0,014

CV lV CVOW JVow CVOWo JVOWo

1000 492 1000

0,014 0,010 0,016

0,000 0,000 0,000

0,141 0,097 0,141

0,012 0,014 0,013

275 1000 86

0,007 0,022 0,002

0,000 0,000 0,000

0,105 0,141 0,059

0,014 0,014 0,007

5

Skew. Kurt. JB

-0.150· 5.183*** 1.984*"'* 2.233*** 1.904***

2.080*** 42.385*** 7.523*** 12.093*** 4.743***

184*** 79332*** 3013*** 6924*** 1541***

1.906*** 2.563*** 1.952*** 4.853***

9.366*** 7.an"'** 7.758*** 26.883***

4260*** 3679*** 3142*** 34037***

ACF Lag 1

ACF lag 2

ACF lag 5

Q5

Q20

ADF KPss lMl-2

0.225* 0.158* 0.492* 0.376* 0.039

0,006 0,057 0.419' 0,360' 0.D18

0.007 0.128* 0.456* 0.268* 0.210*

50*** 68***

98*** 104***

-20.11*** -8.65***

0.17 1.11***

13.27*** 6.11***

910*** 471*** 47***

1872*** 961*** 121***

-3.24** -5.50*** -7.69***

1.31*** 2.17*** 0.86***

66.02*** 55.06*** 3.83**

0,395* 0.053 0.406* 0.071*

0.362* 0.030 0.391'" 0.074*

0.258* 0.195* 0.402* 0.030

532*** 45*** 713*** 14**

853*** 105*** 1393*** 32**

-6.50*** -7.64*** -6.74*** -16.06***

1.41*** 1.46*** 3.99*** 0.24 82.10*** 3.44** 56.37*** 3.33**

lM 1-10 6.00*** 2.28** 21.17*** 12.42*** 9.74*** 18.60*** 8.61*** 16.60*** 0.94

dGPH 0.15*** 0.12*** 0.29*** 0.26*** 0.07** 0.29*** 0.07** 0.24*** 0.11***

Yearly #Cbs 1100 1100 1100 1100 679 1100 263 1100 109

Mean 0.000 0.000 0.017 0.009 0.008 0.012 0.005 0.015 0.002

Min. -0.110 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000

Max. o.on 0.012 0.136 0.136 0.059 0.136 0.071 0.136 0.049

Std. Dev. 0.018 0.001 0.011 0.010 0.009 0.010 0.010 0.011 0.005

Skew. -0.431*** 7.100*** 2.469*** 3.322*** 1.511*** 3.013*** 2.441*** 2.412*** 4.252***

Kurt. JB ACF Lag 1

ACF Lag 2

3.099***

474***

0.153*

-0,034

87.047*** 13.980*** 25.564*** 2.861*** 22.199*** 6.570*** 13.499*** 21.040***

356530*** 10085*** 32006*** 794*** 24273*** 3073*** 9427*** 23625***

0.230* 0.605* 0.361* 0.088* 0.361* 0.112* 0.544* 0.086*

0.158* 0.496* 0.349* 0.128* 0.320* 0.110* 0.435* 0.037

ACF lagS 0.002 0.168* 0.4n* 0.297* 0.158* 0.241* 0.111* 0.427* -0.007

Q5 29*** 168*** 13n*** 521*** 55*** 453*** 43*** 1107*** 16***

Q20 61*** 474*** 3481*** 1125*** 166*** 1061*** 110*** 2767*** 40***

ADF -6.90*** -4.36*** -3.32** -7.03*** -7.46*** -3.20** -3.24** -4.46*** -11.21***

KPSS 0.26 1.37** 1.76*** 3.85*** 0.86*** 1.48*** 0.78*** 3.08*** 0.56**

lM 1-2 37.82*** 12.48*** 138.05*** 22.57*** 21.31*** 31.25*** 12.95*** 129.59*** 4.31**

lM 1-10 15.98*** 2.76*** 32.57*** 5.48*** 12.45*** 7.87*** 5.03*** 30.32*** 1.74*

dGPH 0.043 0.18*** 0.44*** 0.34*** 0.21*** 0.33*** 0.16*** 0.38*** 0.13***

r = returns, r2 = squared returns, RV = Realized Volatility, CV = Continuous Volatility, JV = Jump Volatility, CVOW = Continuous Volatility based on Outiyingness weighted procedure, JVOW = Jump Volatility based outiyingness weighted procedure, CVOWa = Continuous Volatility based Outiyingness weighted procedure adjusted for intraday seasonality, JVOWa = Jump Volatility based on Outiyingness Weigthed procedure adjusted for intraday seasonality, Q51Q20 = The Box-Pierce statistic with 5 and 20 lags respectively, ADF = Augmented Dickey Fuller test (HO: I( I )), The number of lagged difference terms included in the models is based on Akaike's Information Criterion and the following numbers where used for the quarterly contract series: r = I, r = 7, RV = 25, CV = 9, JV = 9, CVOW = 8, JVOW = 9, CVOWa = 4, JVOWa = 2, and for the yearly contract series: r = 18, r = 20, RV= 19, CV= 7, JV= 8, CVOW= 24, JVOW= 35, CVOWa= II, JVOWa = 6, KPSS = Kwiatrowski, Phillips, Schmidt, and Shin test (HO: I(O)). The same numbers as in ADF are used for bandwith, LM 1-2/1-10 is the LM ARCH test of[19] using 2 and 10 of its own lags respectively (HO: no ARCH effects), dGPH is the long-memory test of [23]. *** = p<O,OI, ** = p<0,05,

*=p<0, I (for the autocorrelation function *=that the parameter is outside the 95%-confidence bands,

Both the Box-Pierce statistics and the LM ARCH test of [19] clearly provide evidence of serial correlation in most of the examined series. Hence, when considering the returns, this suggests that volatility forecasting with the GARCH­framework is suitable. For the various realized volatility measures, these results clearly show that standard autoregressive models (and fractionally integrated autoregressive models) easily can be applied in order to construct day ahead volatility estimates for the Nord Pool forward prices.

Finally, one should note the effect of the various ways of calculating the continuous and jump component of realized volatility. As seen of the jump variables of interest, the number of detected jumps with the different approaches varies much. While the Z-test utilizing the traditional bipower

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variation measure as an estimate of the integrated variance detects 492 and 679 jumps (49.2% and 61.7%) for the quarterly and yearly contract series respectively, the equivalent number when applying the jump test with outlyingness measure is 275 (27.5%) and 263 (23.9%) without controlling for intradaily seasonality and 86 (8.6%) and 109 (9.9%) when taking this periodicity into account.

IV. MODELLING DAY AHEAD VOLATILITY

In order to make one day ahead forecasts of volatility for the quarterly and yearly Nord Pool forward prices we compare the performance of RiskMetrics™ and a Fractionally Integrated GARCH (FIGARCH) model with the performance of standard time series models utilizing lagged values of the constructed realized volatility measures. Additionally, as the descriptive statistics suggests that realized volatility exhibit long memory, models including a fractional integrated parameter, d, will be included to examine whether this can improve volatility predictions. The models that will be used for predicting day ahead volatility is presented in (20) to (23) below:

RiskMetrics™:

(20)

FIGARCH:

(21)

HAR-CV-JV:

(22)

HAR-CV-JV-d:

The first model (21) is the internal market risk management methodology applied by J.P. Morgan. This model is actually a special case of an IGARCH(1,I) model where the ARCH (1 - A) and GARCH (A) coefficients are fixed to 0.06 and 0.94 respectively. The FIGARCH model (22) is the Fractionally Integrated GARCH process proposed by Chung (1999). Here, (J2 is the unconditional variance and L is the backshifioperator. For a discussion on the technical details on this model, see Laurent (2009). In both the RiskMetrics™ model and the FIGARCH model, the mean is adjusted with an AR(1) model. The square root of the obtained conditional variance at time t+ 1 from the two GARCH models is used when comparing the performance to the other models. The HAR-CV-JV Model in (23) is the heterogeneous autoregressive model of continuous and jump variation as

proposed by Chan et al. (2008). Here, CVt•5,t = � L�=I CVt-s

and JVt_5,t = � L�=IJVt-S' That is, the average of the

6

continuous and jump volatility measures over the last 5 trading days. (24) provides regression estimates based on a fractionally differenced series to account for the long memory. Both model (23) and (24) use the different variations of continuous and jump volatility as discussed in section II.

Moreover, previous research has found a clear evidence of higher volatility on Mondays [24], and a clear evidence of trading volume's impact on volatility [21], these variables will be entered to the various models in order to examine whether inclusion of these can help improving day ahead volatility predictions.

In order to compare the ability of the various models to predict day ahead volatility we apply two different approaches. First we apply the method proposed by [2] on the whole sample:

RVt+I = Po + PI VModelX,t+1 + Et+I (24)

Here, RVt+I is actually realized volatility at time t+ 1 constructed of the half-hourly squared returns as specified in section 2, and PI VModelX,

t+1 is the volatility estimate obtained by one of the specified models. Hence, the model with the highest explained variance (R2) is performing best. In addition to this method, we calculate the traditional mean absolute percentage error (MAPE) and the symmetric version of this (sMAPE) for the last year of the sample (250 days), and the last 100 days in order to examine the likely forecasting abilities of the various methods. The MAPE and sMAPE accuracy measures are calculated in the following way:

MAPE =

CIRVt+1-VModeIX,t+1D RVt+l

sMAPE =

CIRVt+1-VModeIX,t+1D CIRVt+1 +VModeIX,t+1D/2

(25)

(26)

The results of the various models ability to predict day ahead volatility are presented in Table III.

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TABLE III RESULTS ON THE VARIOUS MODELS PERFORMANCES

Adj. R' MAPE·250 sMAPE·250 MAPE-1OO sMAPE-1OO Quarterly RiskMetrics"" 26.7% 28.5% 27.9% 27.3% 26.0%

FIGARCH 30.5% 27.2% 27.9% 25.5% 25.5%

HAR-CV-JV 35.5% 29.0% 27.6% 26.3% 25.5%

HAR-CVOW-JVOW 34.8% 28.6% 27.3% 26.1% 25.4% HAR-CVOWa-JVOWa 33.3% 29.6% 28.2% 26.9% 25.8%

HAR-CV-JV-d 35.5% 29.3% 27.7% 27.2% 25.9'"

HAR-CVOW-JVOW-d 34.8% 29.2% 27.6% 27.2% 25.9%

HAR-CVOWa-JVOWa-d 33.4% 30.4% 28.5% 28.3% 26.4%

Yearly RiskMetrics"" 36.6% 30.9% 29.3% 29.5% 26.4% FIGARCH 40.0% 30.0% 29.2% 28.7% 26.9%

HAR-CV-JV 44.6% 30.2% 28.5% 27.5% 26.9'"

HAR-CVOW-JVOW 44.3% 30.6% 28.9% 28.1% 27.2%

HAR-CVOWa-JVOWa 43.8% 31.6% 29.6% 29.4% 28.1%

HAR-CV-JV-d 45.0% 30.1% 28.5% 28.3% 26.4% HAR-CVOW-JVOW-d 44.8% 30.5% 28.8% 28.7% 26.8%

HAR-CVOWa-JVOWa-d 44.2% 31.5% 29.4% 29.9% 27.5%

Quarterly with Ex. RiskMetricsT'" 27.0% 30.3% 28.4% 29.4% 27.1%

FIGARCH 33.8% 26.1% 27.2% 23.5% 24.0%

HAR-CV-JV 42.4% 26.5% 25.4% 23.1% 22.7%

HAR-CVOW-JVOW 41.8% 26.3% 25.3% 22.7% 22.5% HAR-CVOWa-JVOWa 40.4% 26.9% 25.8% 23.1% 22.5% HAR-CV-JV-d 42.6% 27.1% 25.6% 24.2% 23.2%

HAR-CVOW-JVOW-d 42.0% 27.1% 25.7% 24.1% 23.2%

HAR-CVOWa-JVOWa-d 40.8% 27.9'" 26.2% 24.7% 23.3%

Yearly with Ex. RiskMetrics™ NC NC NC NC NC

FIGARCH 41.2% 29.0% 29.4% 27.0% 26.2%

HAR-CV-JV 48.0% 30.7% 28.4% 27.2% 26.1% HAR-CVOW-JVOW 48.5% 30.9% 28.7% 27.5% 26.4%

HAR-CVOWa-JVOWa 47.8% 31.6% 29.0% 28.5% 27.0%

HAR-CV-JV-d 48.0% 30.3% 28.8% 27.6% 26.2%

HAR-CVOW-JVOW-d 48.6% 31.2% 28.5% 27.6% 26.2%

HAR-CVOWa-JVOWa-d 47.9% 31.8% 28.8% 28.5% 26.8%

HAR-CV-JV = heterogeneous autoregressive model of continuous and jump variation where the continuous and jump component are constructed with the traditional bipower variation measure (d = fractionally integrated parameter included). HAR-CVOW-JVOW = heterogeneous autoregressive model of continuous and jump variation where the continuous and jump component are constructed on the basis of the realized outiyingness measure (d = fractionally integrated parameter included). HAR-CVOWa-JVOWa = Heterogeneous Autoregressive model of continuous and jump variation where the continuous and jump component are constructed with the realized outiyingness measure adjusted for intradaily seasonality (d = fractionally integrated parameter included). The adj. R2 is obtained from (24) using the sample period from January 12 2005 to May 29 2009 for the yearly contract data and from June 10 2005 to May 29 2009 for the quarterly contract data. For the MAPE-250 and sMAPE 250 the sample period is from June 2 2008 to May 29 2009, and for MAPE-JOO and sMAPE-JOO data from January 5 2009 to May 29 2009 are used for both contract series. NC = did not converge.

When comparing the models in the upper panel of the table (models without exogenous effects), we see that the various models utilizing current and past realized volatility measures to construct next day volatility predictions in general fits better to the actual realized volatility at time t+ 1 than the applied GARCH-models when looking at the whole sample period. In this case, the HAR-CV-JV model with a fractionally integrated parameter, d, is performing best for both the quarterly and yearly contract series with an explained variance of 35.5% and 45.0% respectively. Hence, the results indicate that the new proposed methods of separating the total variation into its continuous and jump component (realized outlyingness weigthed variance and realized outlyingness weighted variance adjusted for intradaily seasonality), do not improve

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the accuracy in the volatility predictions for these models. The traditional RiskMetrics™ method and the FIGARCH method obtain an explained variance of 26.7% and 30.5% respectively. These results clearly indicate that utilizing simple time series techniques and volatility measures constructed from high-frequency data seems useful.

However, when looking at the constructed accuracy measures for the last 250 and 100 trading days, the results are not that clear as for the whole sample period. That is, the differences in accuracy on the basis of the MAPE and sMAPE measure are much smaller than for the obtained R2,

s. Actually, for the last 100 business days in the sample, the FIGARCH model is performing best on the basis of the MAPE measure for the quarterly contract series (25.5%), while the HAR­CVOWa-JVOWa-d model is berforming worst (28.3%). The FIGARCH model is also performing best on the basis of the MAPE measure over the last 250 days for the yearly contract series, which again indicate that volatility predictions constructed with daily returns measures are able to compete with high-frequency volatility predictions. Surprisingly, this is further supported when looking at the symmetric MAPE measure over the last 100 days in the sample, where the RiskMetrics™ method is performing equally well as the HAR­CV-JV-d model (26.4%), and thus ties the number one ranking on the basis of this measure. Overall, there is no clear evidence of a superior model when predicting day ahead volatility for electricity forward contracts when not controlling for exogenous effects in these results. Even though we obtain a higher R2 for the models utilizing past values of realized volatility, the accuracy measures at the end of the sample period provide no clear indications of improved volatility predictions utilizing high-frequency data. One should note however, that the amount of data in this study is to some extent limited, and this aspect should be examined further in the future as more data will be available for analyses.

In the lower panel of Table III, the results of the models with exogenous effects are presented. The major finding here is that including exogenous effects (Monday dummy and current day trading volume) in general improves the accuracy of the day ahead volatility predictions (with exception of the RiskMetric™ method). The best performing model on the basis of the explained variance is the HAR-CV-JV-d model (42.6%) for the quarterly contract series, and the HAR­CVOW-JVOW-d (48.6%) for the yearly contract series. The two GARCH models obtained a R2 value of 27.0% (RiskMetrics™) and 33.8% (FIGARCH) for the quarterly contracts and 41.2% (FIGARCH - RiskMetrics™ did not converge here) for the yearly contracts. Again however, we see that the FIGARCH model is able to perform almost equally well as the other applied models on the basis of the other accuracy measures. Considering the yearly contract series first, we note that for the both last 250 and 100 days in the sample, the FIGARCH model obtains the lowest MAPE measure (29.0% and 27.0%). On the basis of the symmetric MAPE measure the HAR-CVOW-JVOW-d model obtain the lowest (best) score for the last 250 days for the yearly contract series, while the HAR-CV-JV model, applying the traditional bipower measure when constructing the continuous and jump component, performs better (26.1 %) for the yearly contracts. For the quarterly contracts, FIGARCH is again performing

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best over the last 250 days in sample, when looking at the MAPE measure. However, when examining the sMAPE measure, the HAR-CVOW-JVOW is obtaining the lowest score for both horizons. This model also obtains the lowest MAPE score for the last 100 days.

Overall, the results in this paper provide some interesting and perhaps surprising findings. First, even though models utilizing past values of realized volatility and its various components constructed from high-frequency data are able to fit the actually realized volatility one day ahead better than the two GARCH models examined, there is little evidence that they perform substantially better at the end of the sample period (which is a good starting point for any practitioner wanting to forecast day ahead volatility). Second, no evidence is found on that models accounting for the long-memory in the series (as found in Table II), perform better than the models not accounting for this. Third, the two new approaches for constructing the continuous and jump volatility examined in this paper (realized outlyingness weighted variance and realized outlyingness variance adjusted for intradaily seasonality), do not seem to increase the accuracy of volatility predictions mentionable. Hence, even though reducing the number of jumps may be correct from a theoretical perspective, it does not seem to have any significant impact on the prediction performance in practice for the Nord Pool forward price volatilities.

V. SUMMARY AND CONCLUSION

In this paper we have compared volatility predictions of Nord Pool forward prices for two contracts traded at the central Nord Pool exchange. The comparison has used a total of 8 different models, whereas two have represented the GARCH-framework, three have represented a traditional autoregressive framework and the last three models are autoregressive models accounting for long-memory. We have also examined the usefulness of different approaches to calculate the continuous and jump variation in the constructed realized volatility measure.

The results suggest that when comparing the one day ahead volatility predictions over the whole sample period, the various models utilizing past values of the realized volatility components outperform the traditional GARCH-approach. The explained variance for these models is on average 8.6%­points above the average explained variance of the two GARCH-models for the quarterly contract series and 7.0%­points for the yearly contract series. These results clearly indicate improved accuracy in day ahead volatility predictions when utilizing standard time series techniques on the constructed realized volatility measures. However, risk managers, traders and other players in the electricity sector would probably have more interest in looking at the accuracy of the volatility predictions at the end of the sample period, as these results provide more information on how the various models are likely to perform for volatility forecasting out-of­sample. Even though most of the models outperform the various GARCH models on the basis of the symmetric MAPE measure, which probably is a better forecasting accuracy measure than the traditional MAPE measure, the differences in

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accuracy are very small and in most cases almost negligible. Hence, the GARCH-framework utilizing daily returns is clearly compatible with models applying lagged values of various realized volatility measures based on high-frequency returns (this finding is also concistent with [13]).

Another interesting aspect examined in this paper is whether alternative methods of calculating the continuous and jump variation can help us improving the day ahead volatility estimates for Nord Pool forward price volatility. However, our results provide in general no evidence of improved accuracy for neither of the contracts price volatility.

From the descriptive statistics we found evidence of long memory in almost all examined variables. This suggested that movements in volatility should be described by fractionally integrated models. When comparing the results however, both for the whole sample period, and for the last 250 and 100 days, there is no concluding evidence of more accurate predictions with these models.

Inclusion of exogenous effects seems important when predicting day ahead volatility. That is, for almost all models examined the explained variance within the whole sample increases and additionally, the accuracy for the last 250 and 100 trading days improves.

A number of interesting areas remains unexplored for future research within this area. First, as this paper mainly focuses on modelling day ahead volatility, a natural extension in this matter would be to estimate models and use them for post­sample forecasting with various horizons and post-sample sizes. The analyses carried out in this paper, should also be extended to the multivariate case. Comparing a multivariate GARCH-approach with realized covariance is thus a natural extension. Moreover, inclusion of other exogenous effects, such as news announcements, weather forecasts, and more is interesting future research areas.

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VII. BIOGRAPHIES

Erik Haugom was born in Lillehammer in Norway, on February 7, 1982. He graduated in Economic Marketing at Copenhagen Business School in 2007.

His employment experience includes Taxi Driver in Lillehammer, Child Welfare, and Research Planner at Omnicom Media Group in Oslo. Today he is employed as a Research Fellow at Lillehammer University College.

His special fields of interest includes energy price and volatility modeling and forecasting, risk

analysis, risk management, and practical financial modeling and forecasting in general.

technology management.

Sjur Westgaard was born in Sarpsborg in Norway, on November II, 1967. He is a MSc and Phd in Industrial Economics from Norwegian University of Science and Technology and a MSc in Finance from Norwegian School of Business and Economics.

He has worked as a investment portfolio manager for an insurance company, a project manager for a consultant company and as a credit analyst for a bank. His is now a Associate Professor at Trondheim Business School and an Adjunct Associate Professor at NTNU department of industrial economics and

His research interest includes empirical finance / financial econometrics in general. A the time being he is involved in a project analyzing empirical issues of the Nordic Electricity Market involving two local energy companies.

Per Bjarte Solibakke is Dr.Oecon from NHH, Norway. From 1983 to 1984 economic advisor at Statoil, Stavanger. From 1984 to 1988 researcher at NTNU/Sintef Trondheim. From 1988 to 1993 associate professor at T0H, Trondheim and from 1994 associate professor at Molde University College.

Gudbrand Lien was born in Hamar in Norway, on May 21, 1966. He graduated in Agricultural Economics at Agricultural University of Norway in Aas in 1993, and he got in 2001 a Dr.Oecon in Business Economics from Norwegian School of Economics and Business Administration (NHH) in Bergen.

His employment experience includes Senior Executive Officer at Ministry of Agriculture in Oslo and Senior Researcher at Eastern Norway Research

Institute in Lillehammer. Today he is a Professor in Business Economics at Lillehammer University College in Lillehammer, and holds a part-time position as a Senior Researcher at Norwegian Agricultural Economics Research Institute in Oslo.

His special field of interest includes energy price and volatility modelling and forecasting, risk analysis and risk management in general, agricultural economics and innovation policy studies. Both operations research methods and econometric time series and panel data methods are applied.