The power of weather

Computational Statistics and Data Analysis 56 (2012) 3793–3807

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Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda

The power of weather Christian Huurman a , Francesco Ravazzolo b , Chen Zhou c,∗ a

Financial Engineering Associates, 75 King William Street, London EC4N 7BE, United Kingdom

b

Norges Bank, Research Department, Bankplassen 2, 0107 Oslo, Norway

c

De Nederlandsche Bank, Postbus 98, 1000 AB Amsterdam, The Netherlands

article

info

Article history: Received 12 May 2010 Received in revised form 21 June 2010 Accepted 22 June 2010 Available online 30 June 2010 Keywords: Electricity prices GARCH models Weather forecasts Point and density forecasts

abstract Weather information demonstrates predictive power in forecasting electricity prices in day-ahead markets in real time. In particular, next-day weather forecasts improve the forecast accuracy of Scandinavian day-ahead electricity prices in terms of point and density forecasts. This suggests that weather forecasts can price the weather premium on electricity prices. By augmenting with weather forecasts, GARCH-type time-varying volatility models statistically outperform specifications which ignore this information in density forecasting. © 2010 Elsevier B.V. All rights reserved.

1. Introduction This paper focuses on the predictive power of weather forecasts for electricity prices. The aim is to assess the performance in terms of both point and density forecasting of stochastic models with weather forecasts relative to the performance of models that ignore this information. We base our evaluation throughout different types of electricity prices (peak hours and daily average) and different markets. Two decades ago, the electricity industry was strongly regulated and electricity prices reflected short-term production costs. Hence, back then, the electricity price did not reflect temporal effects such as seasonality, weather and business activity. But this all changed when many governments worldwide started reforming their electricity industry as of the mid-1990s. Currently, the economic law of supply and demand determines the price in marketplaces where electricity can be traded in spot or forward contract (i.e., hour-ahead, day-ahead, month-ahead, or longer contracts). Trading electricity on liquid power wholesale markets, for example, the day-ahead markets where hourly prices are quoted for delivery of electricity at certain hours on the next day, makes the price series predictable due to the aforementioned temporal effects. Escribano et al. (2002), Lucia and Schwartz (2002) and Koopman et al. (2007) document that these markets are by far the most liquid power wholesale markets. Many studies have documented these stylized facts on in-sample analysis in electricity price models. Bunn and Karakatsani (2003) provide a thorough review of the stochastic price models presented in these studies and classify these in three groups: random walk models, basic mean-reversion models and extended mean-reversion models that incorporate timevarying parameters (to control for seasonality and volatility patterns). They conclude that the idiosyncratic price structure has not been accurately described. Note that the results reported in these studies are often obtained from in-sample tests; hence they do not resolve the issue of the out-of-sample predictive value of power models.

∗

Corresponding author. Tel.: +31 205243189; fax: +31 205242506. E-mail addresses: [email protected] (C. Huurman), [email protected] (F. Ravazzolo), [email protected], [email protected], [email protected] (C. Zhou). 0167-9473/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2010.06.021

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While a few studies have recognized the need for modelling weather directly, these mainly addressed the weather effect on electricity sales (see, for example, the special issue of Journal of Econometrics 1979). Moral-Carcedo and VicnsOtero (2005) study temperature effects on the variability of daily electricity demand in Spain and document a nonlinear relationship between variations in temperature and the demand response. Similar findings are described in Cancelo et al. (2008). More attention to the relationship between prices and weather is, however, given in very recent studies. Knittel and Roberts (2005) compare price models that incorporate seasonal and temperature variables with models that do not include these variables on hour-ahead electricity prices obtained from the California market, and provide preliminary evidence that the former models significantly outperform in terms of forecast accuracy. Kosater (2006) conducts a similar exercise on the European Energy Exchange (the German market), but he extends the set of weather variables to wind velocity, to proxy for windmill producers, and uses nonlinear models in the class of Markov regime-switching models. He finds that the significance of the weather–price relationship for forecasting is confined to certain hours. Huisman (2007) focuses on how temperature influences the probability of a spike and he shows that the difference between the actual and expected temperature significantly influences this probability. As agents submit their bids and offers for delivery of electricity in all hours of the next day, the weather conditions on the delivery day cannot be exactly known at the time when the electricity prices are quoted. Therefore, we introduce weather forecasts, which are available in real time when prices are traded, in stochastic price models to forecast day-ahead prices in two bidding areas of the Nordic Power Exchange, Oslo and Eastern Denmark. We include a set of weather variables (temperature, precipitation and wind speed) to approximate the fact that both electricity demand and supply are subject to weather conditions and we implement specific models for different bidding areas due to the heterogeneity in weather conditions and production plants. To the best of our knowledge, a similar dataset is not shared by any of the other papers focusing on electricity prices. We find that an ARIMA model extended with power transformations of next-day weather forecasts yields the best point forecasting results in terms of root mean square prediction error for predicting day-ahead prices. This model outperforms, in terms of forecast accuracy, ARIMA models extended with actual weather on the delivered day or extended with only determinist trends. The improvements are, however, marginal. Nevertheless, we show that, although subject to forecast errors, weather forecasts seem to have explanatory power to anticipate ex ante price jumps. Furthermore, we show that the relation between prices and weather forecasts is nonlinear and that a model that incorporates weather forecasts must be carefully specified depending on the data of interest. Evaluations of point forecast accuracy are only relevant for highly restricted loss functions. More generally, complete probability distributions over all potential outcomes provide information which is helpful for making economic decisions, for example for spot market participants to hedge risk or for financial institutions to price derivatives on spot prices. Therefore, as a further step we apply our models to density forecasting. We apply a measure of absolute forecast accuracy, relative to the ‘‘true’’ but unobserved density. Like Diebold et al. (1998), we utilize the probability integral transform scores, PITS, of the realization of the variable with respect to the forecast densities and use the Berkowitz (2001) likelihood ratio test for zero mean, unit variance and independence of the PITS to infer the goodness of fit. We also use a measure of relative predictive accuracy, and following Kitamura (2002), Mitchell and Hall (2005), Amisano and Giacomini (2007), Kascha and Ravazzolo (2010) and Caporin and Pres (2010), apply the Kullback–Leibler divergence or Kullback–Leibler Information Criterion (KLIC) in our evaluation. Our results show that weather forecasts substantially improve the density forecast accuracy. Time-volatility GARCH models augmented with weather forecasts produce well-calibrated density forecasts which are statistically superior in terms of the KLIC to models which ignore this information. Therefore, weather forecasts provide relevant information on the distribution of day-ahead electricity prices. The remainder of the paper is structured as follows. Section 2 introduces the day-ahead power markets and presents the data. Section 3 describes the forecasting models. Section 4 discusses the empirical results for point forecasts. Section 5 is devoted to density forecasting. Section 6 concludes. 2. Data On January 1, 1991, the Norwegian government started a deregulation process for its electricity industry which resulted in the establishment of the first national power market for short-term delivery of power (real-time and day-ahead) in the world, the Nordic Power Exchange (NPX). Two years later, in 1993, the range of products was extended to include financial derivative contracts that have longer maturity horizons. A few years later, Sweden joined the NPX (1996), soon followed by Finland (1998), Western Denmark (1999) and Eastern Denmark (2000). Since 2003, all customers of Scandinavian electricity markets may trade freely in this market. The NPX, now also named Nord Pool ASA, is considered to be the most liquid wholesale market worldwide. Nord Pool ASA consists of a physical market, a financial market, and a trading emission allowances market. We only focus on the physical market, i.e., the day-ahead spot market (Elspot). For more details on Nord Pool ASA we refer to NordPool (2004) and www.nordpool.com. The Nord Pool market is largely dependent on electricity generated by renewable sources. In particular, hydro plants, which use water stored in reservoirs or lakes, are dominant in Norway and in parts of Sweden; wind plants, which use wind to produce electricity, are dominant in Denmark. We again refer to NordPool (2004) for more details. Electricity prices are affected by regional and temporal influences due to the transportation and transmission limits of electricity. This statement is crucial in the Nord Pool market. For instance, when a power plant stops in the eastern part

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(c) ∆ log prices. Fig. 1. Electricity prices. Note: The graphs in this figure present in panel (a) prices, in panel (b) log prices, and in panel (c) first differences log prices for Oslo (in the left panel) and for Eastern Denmark (in the right panel).

of Sweden this failure only affects the power supply in the surrounding region and does not affect the power supply in the western part of Sweden and the rest of the market. Similarly, rainfall in the southern part of Norway will potentially affect the regional demand and/or supply curve, but not the bidding curves in other regions. Nord Pool faces the problem by splitting the market into several bidding and price areas. Therefore, we take into account the Nord Pool bidding area prices separately, rather than examining the Elspot system price (which is a weighted average of the bidding prices in all Nord Pool bidding areas). We examine two out of the eleven bidding areas in Nord Pool, i.e., the Oslo area and Eastern Denmark area. It is interesting to note that these areas are the most densely populated areas in Scandinavia. 2.1. Electricity prices The dataset used in this study consists of day-ahead prices in EUR/MWh for Oslo and Eastern Denmark from the period December 24, 2003 to March 14, 2006. Nord Pool provides bidding area prices both in the local currency and in EUR. We choose EUR to compare directly the two area prices. Daily prices are computed as the arithmetic mean of the available 24 hourly price series on the physical market of each country. In our analysis we also use peak prices, which are defined as the average price of hourly prices from 8 am to 8 pm. As in Wilkinson and Winsen (2002) and Lucia and Schwartz (2002), we start from a brief statistical analysis of the data we have; see, for example, Lucia and Schwartz (2002) and Pilipovic (1997) for a more detailed analysis. Fig. 1 plots the

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Table 1 Descriptive statistics. Oslo

Mean Standard deviation Maximum Minimum Skewness Kurtosis Working days Non-working days Mondays

ρ1 ρ7 ρ14 DF

Eastern Denmark

Level

Log

30.43 4.965 52.45 17.16 1.319 6.278 30.90 29.35 31.04 0.938 0.824 0.678 −0.535

3.403 0.154 3.960 2.843 0.513 4.768 3.420 3.366 3.424 0.928 0.78 0.677 −1.678

Diff 0.001 0.055 0.461 −0.252 0.963 12.86 0.011 −0.023 0.070 −0.126 0.346 0.351 −7.134**

Level

Log

Diff

32.85 12.74 29.69 235.7 6.295 79.13 34.81 28.40 36.68 0.651 0.493 0.405 −3.311*

3.446 0.280 5.463 2.127 1.293 8.181 3.5004 3.3232 3.525 0.778 0.671 0.555 −2.691*

0.001 0.185 1.737 −0.902 1.310 17.21 0.0318 −0.069 0.224 −0.191 0.298 0.273 −15.18**

Note: The table reports descriptive statistics on the level and logarithm of electricity prices and of logarithm first difference in Oslo and Eastern Denmark. Lines Mondays, working days and non-working days give the sample average prices on Mondays, working days and non-working days (weekends and holidays) respectively. Lines ρ1 , ρ7 and ρ14 give the 1st, 7th and 14th sample autocorrelation. The last line refers to the augmented Dickey–Fuller (DF) test; one or two asterisks denote significance relative to the asymptotic null hypothesis respectively at 5% and 1% levels.

time series, the log transformations and the first differences of log transformation of the daily day-ahead electricity prices; Table 1 reports some descriptive statistics. A first casual look discloses erratic behavior of the prices. The series follow a slightly positive increasing trend with several spikes. Prices in Oslo have more negative spikes than positive spikes. In Oslo and southern Norway in general, hydro power plants are considered producers of ‘‘cheap storable’’ electricity and for them it is convenient to export electricity to other markets where power is predominantly produced from fossil-fuelled (e.g. gas, oil) plants. Then the Oslo area imports at higher electricity prices to satisfy its demand. However, electricity in thermal markets is also often offered at discount prices to avoid costs for ramping up and down later (Bunn and Karakatsani, 2003), for example during night hours, and the problem of bottlenecks (cable constraints) can occur. This implies that Norwegian producers offer their ‘‘cheap’’ electricity to the Oslo market, decreasing the prices. The wind power drivers of the Danish market make prices substantially different. The price level is higher, their distribution is non-normal, their volatility is as high as their kurtosis and their skewness is positive. Oslo prices have a more regular distribution, but a Jarque–Bera test rejects the null hypothesis of normality for each of the six series. The series are characterized by weekly patterns. From Table 1 we can observe that prices are lower during the weekend than on working days. Moreover, as in Misiorek et al. (2006), we find evidence of higher prices on Monday. Yearly patterns, well documented in other studies, are more difficult to identify probably due to our shorter sample. The logarithmic transformation reduces the spike behavior of the prices and moments of the distribution of logarithmic electricity prices are more similar to standard distributions, in particular for Eastern Denmark. We decide therefore to model log prices and present the analysis as a function of them. Electricity prices are very persistent and possibly close to non-stationary. Table 1 shows that the sample autocorrelations are high for up to 14-day lags. The Dickey–Fuller test on the series does not reject the null hypothesis of non-stationarity at any level of significance for Oslo prices and 1% level of significance for Eastern Denmark prices. This result prompts us to model the first difference prices, even if the Dickey–Fuller test does not account for nonlinear trends which may exist, but are not known to us. Modelling the first difference is also robust to potential structural breaks caused by the emergence of CO2 certificates in 2005. For example, electricity market participants have started to account for CO2 costs when they calculate the marginal costs of coal-fired plants. Because quantifying the increase is difficult, the first difference seems to be a proper solution. From Fig. 1, we can observe another stylized fact: volatility clustering. Dramatic spikes tend to occur in clusters, mainly as a result of consecutively exceeding the system capacity. 2.2. Weather forecasts Our decision to focus on specific bidding areas of Nord Pool is motivated by the nature of weather variables. Weather observations and relative forecasts refer to a square area around the measurement station. Hence, it is impossible to have single observations that cover the entire country of the markets under consideration. Nevertheless, the areas that we study are small and homogenous in terms of weather. Therefore, we use weather observations and forecasts for Oslo and Copenhagen. The weather in the city of Oslo, the most populated zone of this bidding area, should well approximate the weather in the area itself and to the south of Oslo along the seacoast where most of the electricity for South East Norway is produced. The weather in the area of Copenhagen may be a proxy for the weather of the city of Copenhagen, again the most populated zone of this bidding area, and of Zealand, the main island in Eastern Denmark. Daily average temperatures in degrees Celsius, total daily precipitation in mm and daily average wind power in m/s are applied. Weather observations on day t and weather forecasts of the same day made at time t − 1 are obtained from the EHAMFORE index, which is provided by Meteorlogix (www.meteorlogix.com). Data from the EHAMFORE index are available in Bloomberg and we integrate them with values of several different stations around Oslo from the Meteorological

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Institute (www.met.no) to fill in missing points. We assume that market operators use the weather forecasts provided by Meteorlogix in their decisions. This assumption is realistic considering the market share of Meteorlogix in providing realtime information services in the agricultural, energy, and commodity trading markets, and Bloomberg in providing data to operators. Figs. 2 and 3 plot actuals and forecasts of temperature, precipitation and wind in Oslo, and temperature and wind in Copenhagen. Temperatures are highly persistent, have highly seasonal patterns, and forecasts are quite precise. The correlation between temperature observations and their forecasts made the day before is higher than 0.9 for both areas. The wind speed is particularly high in Eastern Denmark, and forecasts are less accurate. The correlation between observations and forecasts is around 0.5 for both areas. We also notice that wind forecasts have a quite stable pattern in the initial months of 2004, because the Meteorological Institute applies a different scaling forecasting model in those months. We decided to keep these forecasts to extend the sample period as far as we could, because we believe that market operators received this information in real time and used it to take their decisions. Actual precipitation and forecasted precipitation in the Oslo area are quite different. First, forecasting precipitation accurately is rather difficult; and second, observations are often zero, but models always forecast positive (possibly small) numbers. Some graphical relations between the forecasted weather variables and electricity prices may be identified. For example, high precipitation in Oslo at the end of May 2004 or in October 2004 corresponds to low prices; a few days of very low temperatures in Oslo in February 2005 correspond to high prices; strong wind in Eastern Denmark at the end of 2004 and beginning of 2006 is associated with low prices. Using real weather does not change the conclusion: a graphical analysis is not satisfactory because the relation between weather variables and electricity prices is possibly nonlinear, as we will discuss in Section 4.1.

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Several studies, see e.g. Koopman et al. (2007) and Deng (2004), argue that the water reservoir is the key variable to plan production, even more than the amount of precipitation. Even if we agree with this view, we emphasize that in this paper we work with local prices and information about local water reservoirs is in most cases not publicly available. This is particularly true when the number of electricity producers is high, as for example in the Oslo area. And we do not yet have a model to construct regional water reservoirs. Therefore, we opt to apply total precipitation as a proxy for water reservoirs. 3. Forecasting models Knittel and Roberts (2005) show that traditional time series approaches such as ARMA and ARIMA models provide more accurate results in forecasting electricity prices than their continuous counterparts. Starting from these findings, we construct several models that can cope with the stylized facts of electricity prices. 3.1. ARIMA ARIMA1 . The first model is a traditional time series approach to model electricity prices, the autoregressive moving average (ARMA) model (see, for example, Hamilton (1994)). The ARMA(p, q) model implies that the current value of the investigated process (say, the logarithmic transformation of the price Pt ) yt is expressed linearly in terms of its past p values (autoregressive part) and in terms of the q previous values of the process ϵt (moving average part). The ARMA modelling approaches assume that the time series under study is (weakly) stationary. If it is not, a transformation of the series to stationarity is necessary, such as first-order differentiating. The resulting model is known as the autoregressive integrated moving average (ARIMA) model. We define the ARIMA1 model as

φ(L)pt = θ0 + θ (L)ϵt ,

(1)

where pt = (yt − yt −1 ), θ0 is a constant term, φ(L) and θ (L) are the autoregressive and moving average polynomials in the lag operator L respectively, defined as

φ(L) = 1 − φ1 L − φ2 L2 − · · · − φp Lp ,

(2)

θ (L) = 1 − θ1 L − θ2 L − · · · − θq L , 2

q

(3)

and ϵt is an independent and identically distributed (i.i.d.) noise process with zero mean and finite variance σ . 2

ARIMA2 . ARIMA models apply information related to the past of the process and do not use information contained in other pertinent time series. However, as the data analysis shows, electricity prices are generally governed by various fundamental factors, such as seasonality and load profiles. Following the analysis in the previous section, we extend the ARIMA1 model as

φ(L)pt = θ0 + Xt + θ (L)ϵt , (4) k ′ ′ where Xt = i=1 ψi xi,t , xt = (x1,t , x2,t , . . . , xk,t ) is the (k × 1) vector of dummies at time t and ψ = (ψ1 , ψ2 , . . . , ψk ) is a (k × 1) vector of coefficients. We consider two variables: a dummy with values 0 on working days and 1 on holidays, and a dummy with values 1 on Monday and 0 elsewhere. In the empirical application, we define it as ARIMA2 model and take it as our benchmark. ARIMAX. Adverse weather conditions may change the demand for electricity, and may also affect production. Low levels of precipitation and/or wind speed may reduce the supply of energy, in particular in electricity markets which depend on renewable resources plants, such as Norway and Denmark. Furthermore, producer plants may study future weather conditions to estimate demand and plan their supply optimally. Forecasts of the average daily temperature in degrees Celsius, precipitation in mm and wind speed in m/s are applied as further explanatory variables in the ARIMA2 model. The ARIMAX model is

φ(L)pt = θ0 + Xt + Wt + θ (L)ϵt , (5) l ′ where Wt = j=1 ϕj wj,t , wt = (w1,t , w2,t , . . . , wl,t ) is the (l × 1) vector of weather forecast variables for time t, and ′ ϕ = (ϕ1 , ϕ2 , . . . , ϕl ) is a (l × 1) vector of coefficients. We recall that weather forecasts for day t are available at time t − 1. This model includes deterministic components that account for genuine regularities in the behavior of electricity prices and stochastic components that come from weather shocks. Knittel and Roberts (2005) apply a similar model for forecasting California electricity prices, where the set of weather variables consists of the level, as well as the square and the cube of the realized temperature. As in Knittel and Roberts (2005), we allow nonlinearity in the relation between prices and weather variables by including the level, the square and the cube of the temperature forecasts. We also add a variable to measure the wind speed since wind may play a role both in the demand for electricity in such markets where electricity is used for heating – people associate stronger wind with colder temperature – and in the supply of wind power plants. And, finally, we introduce precipitation as a further explanatory variable to approximate the supply in hydro-dominated plants. Again, we allow nonlinearity by using the level and the

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square of the precipitation and wind forecasts. Precipitation and wind forecasts are always positive and therefore we do not consider it useful to include their cubic transformations. ARIMAX-A. Our weather forecasts for day t differ from the actual weather on day t, in particular for total precipitation, as we discuss in Section 2. We investigate whether these differences have an impact on electricity price forecasts by replacing in the previous model weather forecasts made at time t − 1 for the weather on day t with actuals on day t. In other words, we assume that when trading the electricity one day ahead, it is possible to produce a completely accurate forecast on the weather of the next day. This is in fact not achievable in practice; but we would like to investigate whether increasing the forecast accuracy for weather may lead to a better forecast for electricity prices. We define this model ARIMAX-A. 3.2. GARCH models ARIMA–GARCH. ARIMA models assume homoscedasticity, i.e., constant variance and covariance functions, but the preliminary data analysis has disclosed that the electricity prices exhibit volatility clustering. We extend the previous models by assuming a time-varying conditional variance of the noise term. The heteroskedasticity is modelled by a generalized autoregressive conditional heteroskedastic GARCH(r , s) model (Bollerslev, 1986). Relaxing the assumption of homoscedasticity may change the parameter estimates of the ARIMA models, and consequently the out-of-sample forecast of the investigated process. Furthermore, GARCH models produce volatility forecasts which may improve the accuracy of density forecasts. The model is

φ(L)pt = θ0 + Xt + θ (L)ϵt , 1/2

ϵ t = ν t ht

and ht = α0 +

(6) s 

αi ϵt2−i +

i =1

r 

βj ht −j ,

(7)

j =1

where νt is an independent and identically distributed (i.i.d.) noise process with zero mean and variance 1. We use the sufficient conditions for the coefficients in (7), that is αi ≥ 0 for 1 ≤ i ≤ s, βj ≥ 0 for 1 ≤ j ≤ r, and α0 > 0, to ensure that the conditional variance is strictly positive. See Nelson and Cao (1992) for a discussion on sufficient and necessary conditions on GARCH coefficients. ARIMAX–GARCH. The ARIMAX model can also be extended by assuming a noise process with a time-varying conditional variance; that is,

φ(L)pt = θ0 + Xt + Wt + θ (L)ϵt , s r   1/2 and ht = α0 + αi ϵt2−i + ϵ t = ν t ht βj ht −j . i =1

(8) (9)

j =1

ARIMAX–GARCHX. When considering a GARCH model on the time-varying variance without incorporating other information as in the ARIMAX–GARCH model, the forecasted variance is derived from past forecasting errors, which is ex post information. Koopman et al. (2007) find that seasonal factors and other fixed effects in the variance equation are also important in estimating electricity prices. We follow these suggestions and consider also the predicting power of weather information on the variances. If weather forecasts are interpreted to improve the forecast accuracy for the level of electricity prices, such a more accurate forecast should be associated with a lower level of forecasting uncertainty. Thus, weather forecasts will have a simultaneous impact on the variance forecasting when considering time-varying variance models and volatility will not only be function of past square price innovations as in (9). In this spirit, we consider the conditional variance of the noise term in the ARIMAX model to be time varying and modelled by an extended GARCH expression containing dummy variables and weather forecasts. The model is specified as

φ(L)pt = θ0 + Xt + Wt + θ (L)ϵt , 1/2

ϵ t = ν t ht

(10)

s

and

ht = α0 +

 i =1

k+l

r

αi ϵt2−i +

 j =1

βj ht −j +



ϱf zf ,t ,

(11)

f =1

where zt = [x′t , wt′ ]′ , and where ϱ = (ϱ1 , ϱ2 , . . . , ϱk+l )′ is a ((k + l) × 1) vector of coefficients. Since Koopman et al. (2007) assume autoregressive fractionally integrated moving average noises, we do consider integration of order one to simplify the estimation. Koopman et al. (2007) include water reservoirs, which are excluded in this study for the aforementioned reasons. 4. Empirical results We apply the models described in Section 3 to our dataset, and first assess which model performs best in terms of point forecast accuracy. We start by estimating the set of models using the complete sample to have an ex post predictability idea.

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Table 2 In-sample estimation. Oslo

Eastern Denmark

ARIMA1

ARIMA2

ARIMAX

ARIMA1

ARIMA2

ARIMAX

0.001 [0.001] −0.214 [0.033] −0.183 [0.0321] −0.164 [0.031] −0.207 [0.031] −0.098 [0.032] –

0.009 [0.003] −0.254 [0.033] −0.193 [0.032] −0.193 [0.030] −0.229 [0.032] −0.116 [0.032] –

0.046 [0.008] −0.449 [0.033] −0.421 [0.034] −0.344 [0.033] −0.267 [0.033] −0.131 [0.032] –

–

a1

–

a2

–

–

a3

–

–

b1

–

–

b2

–

–

γ

–

–

0.039 [0.033] −0.032 [0.004] 0.049 [0.006] −0.001 [0.0003] 1.17E−04 [3.97E−05] −2.84E−06 [1.47E−06] −0.032 [0.009] 0.013 [0.005] −0.002 [0.001]

–

dm

0.047 [0.034] −0.029 [0.004] 0.053 [0.006] –

0.001 [0.002] −0.430 [0.035] −0.461 [0.038] −0.409 [0.040] −0.329 [0.041] −0.273 [0.040] −0.148 [0.038] 0.149 [0.035] –

0.009 [0.003] −0.415 [0.033] −0.396 [0.034] −0.325 [0.033] −0.251 [0.032] −0.127 [0.013] –

dh

0.001 [0.001] −0.254 [0.034] −0.237 [0.035] −0.208 [0.036] −0.240 [0.036] −0.191 [0.036] −0.076 [0.034] 0.240 [0.035] –

Adj. R-squared AIC Arch LM stat.

0.214 −3.172 9.541

0.360 −3.376 10.55

µ φ1 φ2 φ3 φ4 φ5 φ6 φ7

0.393

−3.422 11.01

–

[0.013] 0.132 [0.017] –

–

–

–

–

–

–

0.039 [0.032] −0.133 [0.013] 0.111 [0.018] −0.010 [0.003] 0.001 [4.00E−04] −4.12E−05 [1.74E−05] –

–

–

–

–

–

−0.008 [0.002]

0.287 −0.880 5.165

0.417 −1.082 3.964

–

−0.125

0.437

−1.110 4.077

Note: The table reports the coefficient estimates (and their standard errors between square brackets), and selection criteria tests of the models on Oslo 7 first difference log prices for models in Section 3. The parameter µ is defined as µ = θ0 / i=1 φi .

We apply a nonlinear ordinary least square (NLS) estimator (Davidson and MacKinnon, 1993) for ARIMA-type models and the quasi-maximum likelihood (QML) estimator (Davidson and MacKinnon, 1993; Greene, 1993) for GARCH family models. We restrict our ARIMA-type models to ARIMA(7,0). Autocorrelation analysis and in-sample criteria would suggest more complex ARIMA forms. In particular, lags 14, 21, 28 and 56 still have high correlations, suggesting longer weekly and monthly price behaviors. However, the risk of overparameterization and the evidence presented in previous studies, for example Lucia and Schwartz (2002), show that an ARIMA(7,0) specification provides optimal forecasts on daily day-ahead electricity prices. The MA parameters are often not significant. Following the same reasoning, we choose a GARCH(1,1) specification for the time-varying variance models. The inclusion of the weather variables follows from statistical evidence. We allow different transformations of the weather forecast variables on the two markets as the weather may affect only the supply of electricity, which is different in the two markets. We minimize the Akaike information criterion (AIC) to specify the model. 4.1. In-sample analysis The in-sample analysis is based on the overall sample, from December 24, 2003 to March 14, 2006. Table 2 reports the estimation results. We focus in the discussion on Oslo prices. The coefficient estimate of the variable dh , which is a dummy variable with value 1 if day t is not a working day or 0 if day t is a working day, is statistically significant and negative. The coefficient estimate of the variable dm , which is a dummy variable with value 1 if day t is Monday and 0 elsewhere, is statistically significant and positive. These results confirm the evidence in Table 1 and provide us with some important information on how agents quote prices. Prices are set lower during weekends (or non-working days) when electricity utilization is lower, but when the market is back to standard working conditions, agents seem initially to overreact and then adjust their positions. Even if the fit is quite high, the residuals of the ARIMA2 model are not normally distributed. Moreover, Fig. 4 shows that the errors of the ARIMA2 model have a nonlinear relation with the daily average temperature and the total precipitation,

C. Huurman et al. / Computational Statistics and Data Analysis 56 (2012) 3793–3807

.2

.2

.1

.1 ARIMA_2

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a

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0 10 Celsius degrees

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16

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-.1

-.2 0

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m/s Fig. 4. Scatter plot: Oslo. Note: The graphs in this figure present in panel (a) the scatter plot of the errors of the ARIMA2 model against the forecasts of the daily average temperature (in the left panel), and total precipitation (in the right panel); in panel (b) the scatter plot of the errors of the ARIMA2 model against the forecasts of the wind speed in Oslo.

but have a linear-like relation with the wind speed. This suggests that the weather forecast variables may explain part of the errors. Thus we consider models conditional on the weather forecasts. In the ARIMAX model for temperature forecasts we use the level, the square and the cube; for precipitation and wind forecasts we use the level and the square. The selection criterion results in the following specific set of weather variables: Wt = a1 Tempt + a2 Temp2t + a3 Temp3t + b1 Prect + b2 Prec2t + γ Windt , where Tempt , Prect and Windt are the forecasts of daily average temperature, total precipitation and wind speed, respectively, on day t. The square of the wind is then excluded in the selection procedure. Fig. 4 indicates a direct linear relation between prices and wind speed and the empirical findings are consistent with the graphical analysis. The interpretation of estimated coefficients for the weather variables is not often straightforward due to nonlinear relations. For example, the temperature forecasts affect the day-ahead electricity price via the following function: f (Tempt ) = a1 Tempt + a2 Temp2t + a3 Temp3t . Taking the first-order derivative, we get df (Tempt ) dTempt

= a1 + 2a2 Tempt + 3a3 Temp2t . df (Temp )

By substituting in the previous equation a1 , a2 and a3 with their empirical estimates, and solving dTemp t = 0, we find that t the roots are 5.3 and 22.2. When the temperature is lower than 5.3 or higher than 22.2, the derivative is negative; when the temperature is in the interval [5.3, 22.2], the derivative is positive. From Fig. 2, the switching point 22.2 lies around the upper bound of the temperature forecasts. In fact, only in 1.6% cases in our sample is the temperature forecast higher than 22.2°. Considering the estimation uncertainty, it is likely that such a switching point does not have any economic impact. Thus, we have one unique switching point: 5.3. From Fig. 2, we observe that 5.3° lies in the middle of the range of temperature forecasts. When the forecast is below 5.3°, it has a negative impact on the electricity price. When above 5.3°, its impact becomes positive. Intuitively, this reflects that when the temperature forecast is relative higher or lower than the switching point 5.3°, the consumption of electricity will rise as well as the difficulty in producing electricity.

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Table 3 Out-of-sample forecasting results: RMSPE. Oslo

ARIMA2 ARIMA1 ARIMAX ARIMAX-A ARIMA–GARCH ARIMAX–GARCH ARIMAX–GARCHX

Eastern Denmark

24 h

Peak

24 h

Peak

0.048 1.123 0.973 0.984 1.029 1.003 1.001

0.058 1.136 0.972 0.984 1.011 0.984 0.979

0.169 1.106 0.969 0.979 1.042 1.032 1.036

0.235 1.095 0.972 0.984 1.034 1.025 1.003

Note: The table reports forecasting statistics of the alternative models in the Oslo and Eastern Denmark electricity markets using daily average prices (24 h) or peak hours (8 am–8 pm), (Peak) prices. The first line in the table reports the value of RMSPE for the ARIMA2 model while all the other lines report statistics relative to the ARIMA2 model. Bold numbers indicate the best performing model in term of RMSPE. We compare the forecast accuracy of the given models against that of the ARIMA2 model as in Clark and West (2007). The null hypothesis is that the two forecasts have the same mean square error. There is no significant difference at 5% level.

Following similar reasoning, we find that precipitation influences prices negatively such that higher precipitation leads to lower prices. This is consistent with the hydropower nature of Oslo electricity markets. The coefficient of the wind speed is negative and we do not have strong economic explanations for it. We note that the Akaike criterion is maximized when weather variables are introduced, supporting this strategy. 4.2. Out of sample analysis We forecast day-ahead Oslo log electricity prices from January 1, 2005 to March 14, 2006. We repeat the selection procedure in Section 4.1 over the initial in-sample period, from December 24, 2003 to December 31, 2004. The reduced specific form in this shorter sample for all models remains the same as in full sample analysis. This indicates that relations do not seem to be sample dependent. In forecasting, when new values are available we re-estimate models to produce a new forecast, but we do not respecify it. An expanding window is used, which means that to forecast the price of day t, all the previous data are included. We compare point forecasts from different models using a quadratic loss function, the Root Mean Square Prediction Error (RMSPE), defined as

  t 1  RMSPE =  (yt +1 − yˆ t +1,i )2 , n t =t

where yt +1 is the log price at time s + 1, yˆ t +1,i is the forecasted log price at time s given by model i, and n = (t − t ) = 438 is the number of days being forecasted. The results are given in Table 3. Weather forecasts improve point forecasts. The ARIMAX model provides the most accurate forecasts. The improvement with respect to the ARIMA2 model is, however, not much: around 3% in term of RMSPE. We compare whether the difference is significant by applying the Clark and West (2007) for nested models, but we do not reject the null hypothesis of equal mean square errors. Fig. 5 provides further evidence to interpret the results. Fig. 5 shows the 60-day average RMSPEs for the ARIMA2 model and the ARIMAX model. From the graph, we find that, when the error of the former model is at a relatively lower level, the errors of the two models are similar, but when there is a higher error due to possible jumps, the latter model often predicts better. When the weather conditions are adverse, producing and transferring electricity is more difficult than in normal times. Considering that the electricity cannot be stored in advance and in general there are cable constraints, shortages in the domestic supply can happen. These shortages together with the inelasticity of the demand result in price jumps. Adding GARCH specification to approximate evidence of volatility clustering does not provide more accurate results, even if the Arch LM tests in Table 2 support it. In particular, the ARIMAX–GARCHX model proposed by Koopman et al. (2007) does not outperform the benchmark model. Our ARIMAX–GARCHX model is, however, simpler than the estimated model of Koopman et al. (2007) and the set of variables is different. Finally, augmenting ARIMA models with actual weather does not help. The ARIMAX-A model gives better results than the ARIMA2 model, but the ARIMAX model outperforms it. This confirms that market agents do not have information on the following day’s actual weather when they take decisions, so the information they might use to settle electricity prices is forecasts of the following day’s weather. In other words, the forecasting power of the weather variables on the electricity prices is contained in the weather forecasts rather than in the actual weather. Increasing forecast accuracy on weather does not improve electricity forecasting unless the information is to some extent publicly available. We test the robustness to electricity peak-hour prices. Peak-hour prices are defined as the average of the hourly prices from 8 am to 8 pm. In fact, prices are more likely to be volatile in that specific time-frame. The models for peak-hour prices specified from general to specific using the Akaike selection criterion are the same as for daily prices. We could anticipate this, considering the role of peak hours in determining daily prices. The forecasting results are reported in Table 3. The

C. Huurman et al. / Computational Statistics and Data Analysis 56 (2012) 3793–3807

a

.07

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2005Q3 ARIM A_2

2005Q4

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ARIM AX

.05

2005Q2

2005Q3 ARIM A_2

2005Q4

2006Q1

ARIM AX

Fig. 5. 60-day moving average RMSPE. Note: The graphs in this figure present in panel (a) the 60-day moving average RMSPEs given the ARIMA2 and ARIMAX models in forecasting Oslo log electricity prices; in panel (b) the 60-day moving average RMSPEs given the ARIMA2 and ARIMAX models in forecasting Eastern Denmark log electricity prices.

ARIMAX model is the best model, which seems to confirm our conjecture that weather forecasts help to anticipate sharp movements in prices. We implement our analysis in a different Nord Pool bidding area, Eastern Denmark. We re-estimate and re-specify all the models. We find that the variable precipitation is not significant, which can be explained by the fact that Denmark does not rely on hydropower. The significant negative coefficient of the wind variable is, on the contrary, associated with the supply from wind plants. Our models do not offer a framework to distinguish between demand and supply effects, but we interpret differences in the selection and significance of weather forecasts in Oslo and Eastern Denmark as empirical evidence that both demand and supply drive electricity prices and therefore models for specific bidding areas should consider carefully features of the markets of interest. We develop a forecasting exercise similar to the Oslo study. Again, the ARIMAX model provides the best forecasts. The improvement is higher than 3%, but not statistically significant. Fig. 5 shows, however, that also for the Eastern Denmark sample weather forecasts help to partially anticipate price jumps. Furthermore, from Table 3 we can observe that using actual weather reduces forecast accuracy and results for the ARIMAX model are again robust to peak-hour prices. 5. Density forecasting Complete probability distributions over all potential outcomes provide information which is helpful for making economic decisions. We focus on density forecasting as a general tool to infer the uncertainty of future prices. First we present how we compute and evaluate density forecasts. Then we discuss empirical evidence. 5.1. Computing and evaluating density forecasts We compute one-step ahead (conditional) density forecasts using a normal approximation. Assuming that the past errors and coefficients are known, the conditional expectation corresponds to the point forecast of each individual model. The forecast variance is computed differently for ARIMA models and GARCH models. In the former class of models we approximate the forecast error variance with the in-sample estimate of the error variance σ 2 in Section 3.1. In the latter class we use the conditional time-varying variance ht in Section 3.2. The predictive density given by any of the models in the suite is then ft +1,i (yt +1 ) ∼ N (ˆyt +1,i , σˆ t2+1,i ),

(12)

where yˆ t +1,i is the point forecast for model i, σˆ is the variance forecast for model i made at time t for t + 1, and i = 1, . . . , 7 as in Section 3. The essential problem in evaluating density forecasts is that the true density is never observed—not even after the random variable is drawn. Additional difficulties arise, if one wants to compare multiple models that are misspecified and possibly nested; see Timmermann (2006) for a discussion. We evaluate the ensemble predictive densities using a test of absolute forecast accuracy. Like Diebold et al. (1998), we utilize the probability integral transform scores, PITS, of the realization of the variable with respect to the forecast densities. A forecast density is preferred if the density is correctly calibrated, regardless of the forecasters loss function. The PITS are 2 t +1,i



yt +1

z t +1 =

f (u)du.

−∞

The PITS should be both uniformly distributed, and independently and identically distributed if the forecast densities are correctly calibrated. Hence, calibration evaluation requires the application of tests for goodness of fit. We apply the

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Berkowitz (2001) likelihood ratio test for zero mean, unit variance and independence of the PITS. The test statistic is distributed χ 2 (3), chi-square with three degrees of freedom, with the null of no calibration failure. Turning to our analysis of relative predictive accuracy, we consider a Kullback–Leibler Information Criterion (KLIC)-based test, utilizing the expected difference in the logarithmic scores of the candidate forecast densities; see for example Kitamura (2002), Mitchell and Hall (2005), Amisano and Giacomini (2007), Kascha and Ravazzolo (2010), Caporin and Pres (2010). Geweke and Amisano (2010) and Mitchell and Wallis (2009) discuss the value of information-based methods for evaluating forecast densities that are well calibrated on the basis of PITS tests. The KLIC chooses the model which on average gives higher probability to events that have actually occurred. Specifically, the KLIC distance between the true density ft +1 of a random variable yt +1 and some candidate density ft +1,i obtained from model i is defined as



ft +1 (yt +1 ) ln

KLICt +1 =

ft +1 (yt +1 ) ft +1,i (yt +1 )

dyt +1 ,

= E [ln ft +1 (yt +1 ) − ln ft +1,i (yt +1 )].

(13)

Under some regularity conditions, a consistent estimate can obtained from the average of the sample information, yt +1 , . . . , yt +1 , on ft +1 and ft +1,i : KLIC =

t 1

n t =t

[ln ft +1 (yt +1 ) − ln ft +1,i (yt +1 )].

(14)

Even though we do not know the true density, we can still compare multiple densities, ft +1,i . For the comparison of two competing models, it is sufficient to consider only the latter term in the above sum,

−

t 1

n t =t

ln ft +1,i (yt +1 ),

(15)

for all i and to choose the model for which the expression in (15) is minimal, or as we report in our tables, the opposite of the expression in (15) is maximal. For application of the log predictive score approach on forecasting the density of electricity demand in Japan, see Ohtsuka et al. (2010). An alternative method of relative predictive accuracy, the continuous predictive ranked probability score, is proposed by Panagiotelis and Smith (2008). Differences in the KLIC can be statistically tested. We apply a test of equal accuracy of two density forecasts for nested models similar to Mitchell and Hall (2005) and Amisano and Giacomini (2007). Suppose there are two one-step-ahead density forecasts, ft +1,1 (yt +1 ) and ft +1,2 (yt +1 ), and consider the loss differential dt +1 = ln ft +1,1 (yt +1 ) − ln ft +1,2 (yt +1 ). We apply the following WALD test:

 GW = n n−1

t  t =t

′ ht d t +1

 t +1 n−1 Σ

t 

 ht d t +1

,

(16)

t =t

t +1 is the HAC estimator for the variance of (ht dt +1 ). Under the null of equal predictability, where ht = (1, dt )′ , and Σ GW ∼ χ22 . 5.2. Results The results of the out-of-sample density forecasting are summarized in Tables 4 and 5. The tables report the out-ofsample forecasting performance of the individual models for daily log prices and peak-hour log prices of the two countries. The results are somewhat different from those in the previous section. First we focus on the PITS that are absolute measures for goodness of calibration. Table 4 show that all the models are well calibrated for Oslo, because the null of good calibration is not rejected. The evidence is different in the Eastern Denmark example: the null hypothesis is rejected for the ARIMA models. Only for the ARIMAX model is the null not rejected at the 5% level of confidence for daily average prices. Adding a time-varying volatility, as in the GARCH models, increases the calibration properties substantially and the null is never rejected at the 10% level in either the ARIMAX–GARCH or the ARIMAX–GARCHX models. We use the logarithmic scores to investigate the relative performance and to find which is the most accurate model in terms of density forecasting. We repeat that higher logarithmic scores in Table 5 correspond to better performance. The weather forecasts increase the accuracy in density forecasting. The ARIMAX, ARIMAX–GARCH and ARIMAX–GARCHX models give statistically higher scores than the ARIMA2 model in all four exercises, and in three cases the best results are given when the weather forecasts are used both in the mean equation and in the variance equations, the ARIMAX–GARCHX model. When scores of the ARIMAX–GARCHX model and of the ARIMAX–GARCH model are compared, the difference is statistically significant at the 5% level of confidence for peak-hour Oslo prices and at the 1% level of confidence for day average and peak-hour Eastern Denmark prices.

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Table 4 Out-of-sample forecasting results: PITS. Oslo

ARIMA2 ARIMA1 ARIMAX ARIMAX-A ARIMA–GARCH ARIMAX–GARCH ARIMAX–GARCHX

Eastern Denmark

24 h

Peak

24 h

Peak

0.137 0.395 0.119 0.291 0.989 0.962 0.555

0.440 0.741 0.247 0.594 0.379 0.508 0.212

0.031 0.012 0.051 0.021 0.382 0.904 0.911

0.000 0.000 0.010 0.000 0.081 0.286 0.191

Note: The table reports the likelihood ratio p-value of the Berkowitz (2001) test of zero mean, unit variance and independence of the inverse normal cumulative distribution function transformed PITS, with a maintained assumption of normality for transformed PITS for Oslo and Eastern Denmark electricity density forecasts using daily average prices (24 h) or peak hours (8 am–8 pm), (Peak) prices. Table 5 Out-of-sample forecasting results: Logarithmic score. Oslo

ARIMA2 ARIMA1 ARIMAX ARIMAX-A ARIMA–GARCH ARIMAX–GARCH ARIMAX–GARCHX

Eastern Denmark

24 h

Peak

24 h

Peak

1.503 1.351 1.563∗∗ 1.525∗∗ 1.637∗∗ 1.663∗∗ 1.662∗∗

0.834 0.765 0.974∗∗ 0.833∗∗ 1.186∗∗ 1.120∗∗ 1. 285∗∗

−1.127 −1.108∗ −1.033∗∗ −1.092∗∗ −1.280 −0.488∗∗

−1.922 −1.929 −1.846∗∗ −1.947 −1.726∗∗ −1.803∗∗ −0.707∗∗

0. 211∗∗

Note: The table reports logarithmic score statistics for Oslo and Eastern Denmark electricity density forecasts using daily average prices (24 h) or peak hours (8 am–8 pm), (Peak) prices. Bold numbers indicate the best performing model in term of logarithmic score, whereas asterisks indicate the results of the forecast density comparison test of the given models against those of the ARIMA2 model. One asterisk denotes significance relative to the asymptotic null hypothesis at 5%, two asterisks denotes significance relative to the asymptotic null hypothesis at 1%.

Fig. 6 shows the forecasts of the standard deviations given by the various models. Forecasts given by the ARIMA class of models (in panel (a)) are too flat over time, particularly in periods of high volatility; see for example March and April 2005 in both countries. Estimating GARCH residuals and producing forecasts consequently introduce time-varying patterns which are very useful in density forecasting. The contribution of weather variables in the GARCH volatility is more evident in the Eastern Denmark examples. The plot for Eastern Denmark prices in panel (b) shows that the ARIMAX–GARCHX model gives smaller forecasts of the variances in high-volatility periods than other GARCH models, which results in densities more precisely concentrated around the outturns. Therefore, GARCH residuals augmented with weather forecasts do not necessarily help to anticipate jumps, as we discuss in the previous section, but can provide accurate indications of uncertainty in day-ahead electricity prices. This finding seems to confirm the intuition that weather forecasts can price the weather premium and its effect on uncertainty in electricity prices. 6. Conclusion In this paper, we study whether weather variables have predictive power for electricity prices in real time. We collect forecasts of several weather variables for two bidding areas of Nord Pool, Oslo and Eastern Denmark, and we use them to predict the respective prices and to produce density forecasts. Our empirical results suggest that weather forecasts play a central role in forecasting day-ahead prices. Weather forecasts give relevant information to partially anticipate spikes in prices, providing more accurate point forecasts compared to several alternative approaches. The gains are even larger in density forecasting: weather forecasts model accurately the uncertainty in the distribution of day-ahead electricity prices. How weather forecasts are introduced in econometric models depends on the loss function: when focusing on the quadratic loss function from point forecasting, weather variables should be used only in the mean of the process; when focusing on more general loss functions from density forecasting, weather variables should also be included in modeling higher moments. We also find that the relation between electricity prices and weather forecasts is highly nonlinear and depends on the price drivers associated both with the demand and with the supply side behind each bidding area. Results are robust to peak-hour price applications. There are several topics for further research. First, the set of weather forecasts might include other weather-related variables, such as water reservoir levels. Observations of these variables are not available to us, but results support the construction of specific models to compute approximations and forecasts of them. Second, considering the strong nonlinear relation of prices and weather forecasts, weather forecasts might be included in other nonlinear models, such as Markov regime switching models, jump models or time-varying models; see Karakatsani and Bunn (2008). These specifications may cope better with jumps in forecasting exercises and keep a more realistic value of uncertainty in their predictive densities compared to the method employed in this study.

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a .052

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0.0

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Fig. 6. Standard deviation forecasts. Note: The graphs in this figure present in panel (a) the standard deviation forecasts for the ARIMA1 , ARIMA2 and ARIMAX models in density forecasting daily Oslo log electricity prices (in the left panel) and Eastern Denmark log electricity prices (in the right panel); in panel (b) the standard deviation forecasts for the ARIMA–GARCH, ARIMAX–GARCH and ARIMAX–GARCHX models in density forecasting daily Oslo log electricity prices (in the left panel) and Eastern Denmark log electricity prices (in the right panel).

Acknowledgements We are very grateful to participants at the 3rd International Conference on Computational Econometrics (Cyprus 2009) for their helpful comments, and we are in particular indebted to Qaisar Farooq Akram, Jonas Andersson, Rob Hyndman, two anonymous referees and the associate editor for extremely useful comments. We thank Nord Pool ASA for providing data. The views expressed in this paper are our own and do not necessarily reflect the views of De Nederlandsche Bank and/or Norges Bank. References Amisano, G., Giacomini, R., 2007. Comparing density forecasts via weighted likelihood ratio tests. Journal of Business & Economic Statistics 25, 177–190. Berkowitz, J., 2001. Testing density forecasts, with applications to risk management. Journal of Business & Economic Statistics 19, 465–474. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327. Bunn, D., Karakatsani, N., 2003. Forecasting electricity prices. Discussion Paper. London Business School. Cancelo, J., Espasa, A., Grafe, R., 2008. Forecasting the electricity load from one day to one week ahead for the Spanish system operator. International Journal of Forecasting 24 (4), 588–602. Caporin, M., Pres, J., 2010. Modelling and forecasting wind speed intensity for weather risk management. Marco Fanno Discussion Paper 0106. University of Padova. Clark, T., West, K., 2007. Approximately normal tests for equal predictive accuracy in nested models. Journal of Econometrics 138 (1), 291–311. Davidson, R., MacKinnon, J., 1993. Estimation and Inference in Econometrics. Oxford University Press. Deng, D., 2004. Modelling and investigating the relationship between electricity prices and water reservoir content at Nord Pool power exchange. Discussion Paper. Gothenburg University. Diebold, F., Gunther, T., Tay, A., 1998. Evaluating density forecasts with applications to finance and management. International Economic Review 39, 863–883. Escribano, A., Pena, J., Villaplana, P., 2002. Modeling electricity prices: international evidence. Discussion Paper. Universidad Carlos III de Madrid. Geweke, J., Amisano, G., 2010. Comparing and evaluating Bayesian predictive distributions of asset returns. International Journal of Forecasting 26 (2), 216–230. Greene, W., 1993. Econometric Analysis, 4th ed. Prentice Hall. Hamilton, J., 1994. Time Series Analysis. Princeton University Press. Huisman, R., 2007. The influence of temperature on spike proabability in day-ahead power markets. Energy Economics 30 (5), 2697–2704. Karakatsani, N., Bunn, D., 2008. Forecasting electricity prices: the impact of fundamentals and time-varying coefficients. International Journal of Forecasting 24 (4), 744–763. Kascha, K., Ravazzolo, F., 2010. Combining inflation density forecasts: some cross-country evidence. Journal of Forecasting 29, 231–250. Kitamura, Y., 2002. Econometric comparisons of conditional models. Discussion Paper. University of Pennsylvania. Knittel, C., Roberts, M., 2005. An empirical examination of restructured electricity prices. Energy Economics 27, 791–817. Koopman, S., Ooms, M., Carnero, M., 2007. Periodic seasonal reg-ARFIMA-GARCH models for daily electricity spot prices. Journal of American Statistical Association 102, 16–27. Kosater, P., 2006. On the impact of weather on German hourly power prices. Discussion Paper. University of Cologne.

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