The Power of Weather: Some Empirical Evidence on Predicting Day-ahead Power Prices through Day-ahead Weather Forecasts
ABSTRACT In the literature the effects of weather on electricity sales are well-documented. However, studies that have investigated the impact of weather on electricity prices are still scarce (e.g. Knittel and Roberts, 2005), partly because the wholesale power markets have only recently been deregulated. We introduce the weather factor into well-known forecasting models to study its impact. We find that weather has explanatory power for the day-ahead power spot price. Using weather forecasts improves the forecast accuracy, and in particular new models with power transformations of weather forecast variables are significantly better in term of out-of-sample statistics than popular mean reverting models. For different power markets, such as Norway, Eastern Denmark and the Netherlands, we build specific models. The dissimilarity among these models indicates that weather forecasts influence not only the demand of electricity but also the supply side according to different electricity producing methods.
- [show abstract] [hide abstract]
ABSTRACT: This paper considers forecasting techniques to predict the 24 market-clearing prices of a day-ahead electric energy market. The techniques considered include time series analysis, neural networks and wavelets. Within the time series procedures, the techniques considered comprise ARIMA, dynamic regression and transfer function. Extensive analysis is conducted using data from the PJM Interconnection. Relevant conclusions are drawn on the effectiveness and flexibility of any one of the considered techniques. Furthermore, they are exhaustively compared among themselves.International Journal of Forecasting. 01/2005;
- [show abstract] [hide abstract]
ABSTRACT: A time series model is proposed that describes the day-of-the-week seasonality in returns as well as in volatility of the daily S&P 500 index. The model is a periodic autoregression with periodically integrated GARCH [PAR-PIGARCH]. With this statistically adequate model, positive (negative) autocorrelation is found in the returns on Monday (Tuesday). Day-of-the-week variation in the persistence of volatility is also found. Copyright 2000 by Taylor and Francis GroupApplied Financial Economics. 02/2000; 10(5):483-88.
Article: Comparing Predictive Accuracy[show abstract] [hide abstract]
ABSTRACT: This paper provides an introduction to alternative models of uncertain commodity prices. A model of commodity price movements is the engine around which any valuation methodology for commodity production projects is built, whether discounted cash flow (DCF) models or the recently developed modern asset pricing (MAP) methods. The accuracy of the valuation is in part dependent on the quality of the engine employed. This paper provides an overview of several basic commodity price models and explains the essential differences among them. We also show how futures prices can be used to discriminate among the models and to estimate better key parameters of the model chosen.National Bureau of Economic Research, Inc, NBER Technical Working Papers. 01/1994;
Tinbergen Institute Discussion Paper
The Power of Weather
The Power of Weather
1 Financial Engineering Associates;
2 Erasmus Universiteit Rotterdam, and Tinbergen Institute.
The Tinbergen Institute is the institute for
economic research of the Erasmus Universiteit
Rotterdam, Universiteit van Amsterdam, and Vrije
Tinbergen Institute Amsterdam
1018 WB Amsterdam
Tinbergen Institute Rotterdam
Burg. Oudlaan 50
3062 PA Rotterdam
Most TI discussion papers can be downloaded at
Tinbergen Institute Amsterdam
+31(0)20 551 3500
+31(0)20 551 3555
Tinbergen Institute Rotterdam
+31(0)10 408 8900
+31(0)10 408 9031
The power of weather
Some empirical evidence on predicting day-ahead power prices
through weather forecasts∗
Christian Huurman Francesco Ravazzolo†
April 24, 2007
In the literature the effects of weather on electricity sales are well-documented.
However, studies that have investigated the impact of weather on electricity
prices are still scarce (e.g. Knittel and Roberts, 2005), partly because the
wholesale power markets have only recently been deregulated. We introduce
the weather factor into well-known forecasting models to study its impact.
We find that weather has explanatory power for the day-ahead power spot
price. Using weather forecasts improves the forecast accuracy, and in par-
ticular new models with power transformations of weather forecast variables
are significantly better in term of out-of-sample statistics than popular mean
reverting models. For different power markets, such as Norway, Eastern Den-
mark and the Netherlands, we build specific models. The dissimilarity among
these models indicates that weather forecasts influence not only the demand of
electricity but also the supply side according to different electricity producing
Key words: Electricity prices, forecasting, GARCH models, weather fore-
JEL Classification Code: C53, G15, Q40.
∗The authors would like to thank Philip Hans Franses, Michiel de Pooter, Dick van Dijk, and
Marno Verbeek for extremely helpful comments that improved the clarity of the paper. We also
thank seminar participants at the Tinbergen Institute Amsterdam and Econometric Institute Rot-
terdam. We thank Nord Pool ASA and APX for providing data.
†Corresponding author: Tinbergen Institute, Erasmus University Rotterdam, P.O. Box 1738,
NL-3000 DR Rotterdam, The Netherlands. Tel.: +31-10-4088924, fax: +31-10-4089031, e-mail:
The heatwave in Europe during August 2003 (the warmest summer in Europe since
1500), resulted in extremely high prices in several power markets, as France, Ger-
many and the Netherlands, see for example Figure 1. The fact that not the 15000
casualties due to the heatwave but the technical problems of electricity supply ex-
perienced by´Electricit´ e de France (EDF), the main power supplier in France, were
on the top of the agenda of the French Cabinet meeting held on August 11, 2003,
illustrates the tremendous importance of the functioning of the power system to our
A decade ago, the weather effects would only affect electricity sales, which is
documented in various studies (see for example the special issue of Journal of Econo-
metrics 1979). Back then, the electricity industry was vertically integrated, prices
were regulated and reflected the short-term marginal (production) costs. Hence, the
demand curve that is a function of temporal effects such as seasonality, weather and
business activity did not affect the price at that time. But this all changed when
many governments started reforming their electricity industry as of the 1990s. Var-
ious market places were created with complementary investment horizons to trade
electricity on spot or forward (hour-ahead, day-ahead, month-ahead), and power
prices were based on the economic law of demand and supply. Market participants
are now exposed to the volatile market conditions that stem from the non-storability
of electricity; and the absence of inventories makes that supply and demand of power
must be balanced on every precise moment in time. Hence, the change to a market-
based electricity price system implies that temporal and regional effects such as
seasonality, time-varying volatility and extreme price shocks explain the observed
price behavior rather well.
Many studies have documented these stylized facts from examining the prices
observed at day-ahead markets1, which are by far the most liquid power wholesale
markets, see Escribano, Pena, and Villaplana (2002), Lucia and Schwartz (2002)
and Koopman, Ooms, and Carnero (2007). Bunn and Karakatsani (2003), provide a
thorough review of the stochastic price models presented in these studies and classify
these into three groups, being random walk models, basic mean-reversion models,
and extended mean-reversion models that incorporate time-varying parameters (to
control for seasonality and volatility patterns). They conclude that the idiosyncratic
price structure has not been accurately described. Furthermore, 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.
However, only few studies have recognized the need for modelling weather directly
and addressed some interesting issues. Knittel and Roberts (2005) test stochastic
price models on hourly hour-ahead power prices obtained from the California market
and find that the forecasting performance is superior for price models which account
for seasonal patterns and temperature effects.
In this study we attempt to shed more light on the issue of forecasting perfor-
1On these markets, hourly prices are quoted for delivery of electricity on certain hours on the
mance of stochastic day-ahead price models. We examine six stochastic price models
to forecast day-ahead prices of the two most active power exchanges in the world: the
Nordic Power Exchange and the Amsterdam Power Exchange, see Geman (2005).
Three of these forecasting models extend Knittel and Roberts (2005) by including
new weather variables as price factors. Firstly, considering that operators make de-
cisions today on tomorrow’s electricity, the real weather of tomorrow is unknown at
that moment, and the only available information of weather comes from the weather
forecasts. Therefore, we use weather forecasts as predictors, which are more appro-
priate than real weather. The empirical study agrees with this intuition. Secondly,
for specific weather variables, we consider temperature, total precipitation and wind
speed, which may capture significant and interpretable supply and demand effects.
Thirdly, since we find that the influence of the weather forecasts on the electricity
prices is non-linear, we use non-linear transformations of the weather forecasts in
our new models. Finally, we implement specific models for different power markets
due to their heterogeneity in weather conditions and production plants.
We find that an extended ARMA model, which includes power transformations
of next-day weather forecasts, yields the best forecasting results for predicting one
day-ahead power prices. This model has some predictability power to anticipate
prices jumps. Intuitively, adverse climate conditions often lead to sharp increases
in demand resulting in supply shortages in electricity. We investigate carefully the
relation between prices and weather. We find that the weather forecasts influence
the electricity prices via the demand as well as the supply side, and when produc-
tion is less related to weather, which is the case for Amsterdam Power Exchange,
the weather forecasts play only a minor role. We also show that a GARCH speci-
fication extended with weather forecast variables provides accurate forecasts. This
result contradicts with earlier findings that ‘standard’ GARCH models would predict
electricity prices poorly2.
The remainder of the paper is structured as follows. Section 2 introduces the
day-ahead power markets. Section 3 presents the data. Section 4 describes the
forecasting models. Section 5 discusses the empirical results. Section 6 concludes.
2Day-ahead power markets
On 1 January 1991, the Norwegian government imposed a deregulation process on
its electricity industry that resulted in the establishment of the first national power
market for short-term delivery of power (real-time and day-ahead3) in the world,
the Nordic Power Exchange (NPX). Two years later, in 1993, the range of products
was extended with forward and futures contracts that had longer maturity horizons.
Another few years later, Sweden joined the NPX (1996), soon followed by Finland
(1998), West-Denmark (1999) and East-Denmark (2000). From 2003 all customers
of Scandinavian electricity markets may trade freely in the market.The NPX,
2See for example Knittel and Roberts (2005).
3We remember from section 1 that day-ahead means that prices are quoted at day t for delivery
of electricity on certain hours on the day t + 1.
now also named Nord Pool ASA, is considered as the most liquid wholesale market
worldwide. Nord Pool ASA constitutes of a day-ahead market (Elspot), a financial
market (Elbas), and a clearing service. In the remainder, we mainly focus on the
Elspot market. For more details on Nord Pool ASA we refer to NordPool (2004).
Another country that liberalized its power industry at an early stage onwards, is
the Netherlands. In 1999, here the second electronic power exchange was founded,
being the Amsterdam Power Exchange (APX). The APX is also composed by a
day-ahead market and a financial market. For more details on APX we refer to
In Figure 2 some descriptive statistics of these two markets are listed. The
Nord Pool market is largely dependent on electricity that is generated by renewable
sources. In particular, hydro-plants, which use water stored in reservoirs or lakes,
are dominant in Norway and partly Sweden; wind-plants, which use wind to pro-
duce electricity, are dominant in Denmark. In the APX market oil, coal, gas or a
combination of these fuels is used to generate electricity.
Electricity prices are affected by regional and temporal influences due to the
transportation and transmission limits of electricity. This statement is particular
important in the Nord Pool market. For instance, when a power plant falls out in
the eastern part of Sweden this only affects the power supply in the surrounding
region. Hence, this will not affect 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 allowing to split the market
in several bidding and prices 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 the Nord Pool, being the Oslo area
and Eastern Denmark area. It is interesting to note that these areas are the most
densely populated areas in Scandinavia.
The data set used in this study consists of day-ahead prices in EUR/MWh for Oslo,
Eastern Denmark and the Netherlands from the period December 24, 2003 to March
14, 2006. Oslo and Eastern Denmark are two bidding areas of Nord Pool market;
Dutch electricity prices are obtained from APX market4. Nord Pool provides bidding
area prices both in the local currency and in EUR. We choose EUR to compare
directly to APX prices. Daily prices are computed as the arithmetic mean of the
available 24 hourly prices series on the physical market of each country.
Figure 4 plots the time series, the log transformations and the histograms of the
4Electricity prices may be available for a longer sample, but weather forecasts are available to
us only for this sample.
daily day-ahead electricity prices; Table 1 gives some important descriptive statistics.
As in Wilkinson and Winsen (2002) and Lucia and Schwartz (2002) we start from
a statistical analysis of the data we have5. A first casual look reveals a quite erratic
behavior of the prices. The series follow a small positive increasing trend with
several spikes. Interesting, prices in Oslo have more negative spikes than positive
high spikes. This may indicate that the supply was often higher than the demand and
may support Geman (2005) conclusion that prices in hydropower markets are less
subjects to jumps and more similar to other commodity prices than prices of thermal-
based electricity. Eastern Denmark and the Netherlands prices are sensitively higher,
in particular the Netherlands ones. Higher spikes are more frequent. The maximum
prices in the sample are 235.71 EUR/MWh and 250.69 EUR/MWh for Eastern
Denmark and the Netherlands respectively, which are more than 7 and 5 times higher
than the average prices, 32.85 EUR/MWh and 44.85 EUR/MWh respectively. The
histograms provide similar evidence. Eastern Denmark and the Netherlands prices
are highly non-normally distributed; their volatility is very high such as the kurtosis;
their skewness is positive. Oslo has a more regular distribution, but a Jarque-Bera
test rejects the null hypothesis of normality for each of the three series. The series are
characterized by a weekly pattern: Table 1 reports prices are lower on weekend than
working days. Yearly patterns, well documented in other studies, are more difficult
to notice since the series are not very long, but differences among seasons in Figure
3 may be drawn. Electricity prices are very persistent and possible close to non-
stationary. We do not investigate the hypothesis of non-stationary for reasons which
we discuss in Section 4.1. Table 1 shows that the sample autocorrelations are high
up to 14-day lags. The last stylized fact we notice in Figure 4 is volatility clustering.
Dramatic spikes tend to occur in clusters, mainly as result of consecutively exceeding
the system capacity.
In our application we use log prices and not the level. The log transformation
reduces the spike behavior of the prices and makes moments of the distribution of
electricity prices more similar to standard distributions, in particular for Eastern
Denmark and the Netherlands log prices.
3.2 Weather forecasts
We continue the data analysis by focusing on weather forecasts. Forecasts on the
daily average temperature in degrees Celsius, total precipitation in mm, and wind
power in m/s are applied. Data are obtained from the EHAMFORE index, which is
provided by Meteorlogix (www.meteorlogix.com)6. We assume that market opera-
tors use the weather forecasts provided by Meteorlogix in their decisions. We think
that this assumption is quite realistic considering the market share of Meteorlogix in
providing real-time information services in the agriculture, energy, and commodity
trading markets, and Bloomberg in providing data to operators. Weather forecasts
refer to a square area around the measurement station, which implies that series
5We briefly discuss some stylized facts; we refer for a more detailed analysis, for example, to
Lucia and Schwartz (2002) and Pilipovic (1997).
6Data from the EHAMFORE index are available in Bloomberg.
that cover all the lands of the markets we consider do not exist. The combination
of different stations might be applied, but we exclude it because it might be difficult
to collect data from minor cities, the weather forecast errors might arise introduc-
ing further noise in the forecasting process, and the country/areas that we study
are quite small and quite homogenous in term of weather. Therefore, we only use
weather forecasts for Oslo, Copenhagen and Amsterdam. The weather around Oslo
may well approximate the weather in the area on the south of Oslo along the sea cost
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 Zealand, the main island in
Eastern Denmark. Finally, Amsterdam is located in the middle of the Netherlands.
Figures 5-7 plot the three variables for each country. Temperatures have highly
seasonal patterns, with lower values for Oslo and higher for the Netherlands. Pre-
cipitations are higher in Oslo and the Netherlands than in Eastern Denmark. The
wind is particular strong in Eastern Denmark and the Netherlands. The wind fore-
casts on all the three countries have a quite stable pattern in the initial months
of 2004, because the meteorologic institute applies a different forecasting model on
those months. We decide to keep these forecasts to extend, as much as we can, the
sample period, meanwhile it is what operators got as information for the weather.
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; few days of very low temperature
in Oslo in February 2005 correspond to high prices; strong wind in Eastern Den-
mark at the end of 2004 and beginning of 2006 is associated to low prices. However,
even if the real weather was the weather forecasts, a graphical analysis would not
be satisfactory because the relation between weather variables and electricity prices
is possibly highly nonlinear as we will discuss in Section 5.1. Therefore, we try to
find specific models to interpret the weather influences.
Knittel and Roberts (2005) shows that traditional time series approaches as ARMA
models provide more accurate results in forecasting electricity prices than their con-
tinuous counterparts. Starting from these findings we built several models that may
cope with the stylized facts of electricity prices.
4.1Model 1: ARMA
The first model is a traditional time series approach to model electricity prices, the
autoregressive moving average (ARMA) model (Hamilton (1994)). The ARMA(p,q)
model implies that the current value of the investigated process (say, the log price)
Pt is expressed linearly in terms of its past p values (autoregressive part) and in
terms of the q previous values of the process ?t(moving average part):
where φ(L) and θ(L) are the autoregressive and moving average polynomials in the
lag operator L respectively, defined as:
φ(L) = 1 − φ1L − φ2L2− ... − φpLp
θ(L) = 1 − θ1L − θ2L2− ... − θpLp
and where ?tis an independent and identically distributed (iid) noise process with
zero mean and finite variance σ. The motivation of an ARMA process follows from
the correlogram. Table 1 shows high correlation between the current price and the
previous days’ prices.
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 differentiating. The resulting model is known as the au-
toregressive integrated moving-average model (ARIMA). We do not work with first
difference prices for several reasons. Firstly, the Dickey Fuller test on the series
rejects the null hypothesis of non-stationary. Secondly, we think more appropriate
working on the levels since the final object of the study is modelling and predicting
the pattern of electricity prices and the first difference transformation might drop
out same important characteristics, such as the trend of the series. Thirdly, evidence
in literature are almost unique in favor of the level of prices. For example, Lucia
and Schwartz (2002) find that models based on levels and log levels provide more
accurate results than models based on first differences and log first differences in
forecasting Nord Pool electricity prices. And Weron and Misiorek (2005) find that
ARMA models do better in term of out-of-sample statistics than ARIMA models in
forecasting California electricity prices.
4.2Model 2: ARMAX
The second model is an extension of model 1. ARMA 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. The ARMAX(p,q) can be written as:
φ(L)(Pt− Xt) = θ(L)?t
Following Lucia and Schwartz (2002) we use three explanatory variables: a dummy
with values 0 on working days and 1 on holidays, a seasonal dummy given by the com-
bination of the two variables sin(2πt/365.25) and cos(2πt/365.25). These dummy
variables may be interpreted as proxy for load profiles (higher demand on working
days), and proxy for weather effect (higher demand on cold and warm seasons).
In the empirical application, the ARMAX model will be our benchmark.
i=1ψixi,t, where xt= (x1,x2,...xk)?is the (k×1) vector of explanatory
variables at time t, and where ψ = (ψ1,ψ2,...,ψk)?is a (k ×1) vector of coefficients.
4.3Model 3: ARMAXW
Averse weather conditions may change the demand for electricity, and may also affect
the production. Low amount of precipitation and low wind may cause reduction on
the supply of energy, in particular in electricity markets which depend on renewable
producer plants, such as Norway and Denmark. Furthermore, producer plants may
study future weather conditions to estimate demand and plan their supply optimally.
The third model is an extension of model (4) and is built following the previous
reasoning. Forecasts on the average daily temperature in degrees Celsius, precipi-
tation in mm and wind speed in m/s are applied as further explanatory variables.
The model is:
φ(L)(Pt− Xt− Wt) = θ(L)?t
where Wt =?l
vector of coefficients. This model includes deterministic components that account
for genuine regularities in the behavior of electricity prices and stochastic components
that comes from weather shocks.
Knittel and Roberts (2005) apply a similar model for forecasting California elec-
tricity prices, where the set of weather variables is composed by the level, the square
and the cubic of realized temperature. We think that the weather of tomorrow is
more important of the weather of today to forecast the price of tomorrow. There-
fore, we use weather forecasts and not realized values. Moreover, we add wind speed
since it may play a role both in the feeling of the people - people feel colder with
stronger wind - and in the supply of wind power plants. We also use precipitation
since the variable may be appropriate to approximate the supply of hydroelectric
plants. As in Knittel and Roberts (2005) we allow nonlinearity in the relation be-
tween prices and weather variables by including the level, the square and the cubic
of the temperature forecasts, and the level and the square of the precipitation and
Some objections may be argued for hydroelectric power generator. The water
reservoir is often more important to plan production than the amount of precipita-
tion, see e.g. Koopman, Ooms, and Carnero (2007) and Deng (2004). But water
reservoir is not known in advance and may not be forecasted. Furthermore, we
think that hydroelectric plants incorporate forecasted future precipitations in their
strategic decisions of the amount of water to store.
j=1ϕjwj,t, where wt = (w1,t,w2,t,...,wl,t)?is the (l × 1) vector of
weather forecast variables at time t, and where ϕ = (ϕ1,ϕ2,...,ϕl)?is a (l × 1)
4.4Model 4: ARMAX-GARCH
ARMA models assume homoscedasticity, i.e. constant variance and covariance func-
tion, but the preliminary data analysis has revealed that electricity prices exhibit
volatility clustering. The fourth model extends model 2 by assuming a time varying
conditional variance of the noise term. The heteroskedasticity is modelled by a gen-
eralized autoregressive conditional heteroskedastic GARCH(p,q) model (Bollerslev
7Precipitation and wind forecasts are always positive, therefore we do not consider useful to
include the cubic transformation.
(1986)). Relaxing the assumption of homoscedasticity may change the parameter
estimates of model 2, and consequently the out-of-sample forecast of the investigated
The model is:
φ(L)(Pt− Xt) = θ(L)?t
with ht= α0+
where ?tis an independent and identically distributed (iid) noise process with zero
mean and conditional time varying variance ht, and the coefficients have to satisfy
αi≥ 0 for 1 ≤ i ≤ q, βj≥ 0 for 1 ≤ j ≤ p, and α0> 0 to ensure that the conditional
variance is strictly positive.
4.5Model 5: ARMAXW-GARCH
Following the same reasoning for model (6)-(7), model 3 can be extended by assuming
a noise process with a time varying conditional variance.
Model 5 is:
φ(L)(Pt− Xt− Wt) = θ(L)?t
with ht= α0+
4.6Model 6: ARMAXW-GARCHW
Koopman, Ooms, and Carnero (2007) find that seasonal factors and other fixed
effects in the variance equation are also important to estimate electricity prices. The
fifth model extends model 4 by reformulating model 3 and 4 to incorporate Koopman,
Ooms, and Carnero (2007) results. The conditional variance of the noise term in
model 3 is assumed to be time-varying and modelled with a GARCH expression
where some explanatory variables are added to the ARMA form of equation (7).
The model looks as:
φ(L)(Pt− Xt− Wt) = θ(L)?t
where zt = [x
coefficients. Despite the fact that Koopman, Ooms, and Carnero (2007) assume au-
toregressive fractionally integrated moving average noises which we do not consider,
an important difference with model 6 is the set of (weather) explanatory variables.
Koopman, Ooms, and Carnero (2007) include water reservoir and consumption,
which we think less adequate to forecast future prices.
with ht= α0+
t]?, and where ? = (?1,?2,...,?k+l)?is a ((k + l) × 1) vector of
We apply the models described in Section 4 to our data set, try to figure out which
is the best on forecasting. Before the out-of-sample forecast exercise, we estimate
the set of models using the complete sample to have an ex-post predictability idea.
We describe some assumptions. In estimation we apply nonlinear ordinary least
square (NLS) estimator (Davidson and MacKinnon (1993)) for ARMA type models
and approximate maximum likelihood (QML) estimator (Davidson and MacKinnon
(1993) and Greene (1993)) for GARCH family models.
We restrict our ARMA type models to be ARMA(7,0), where only lags 1 and
7 are considered. We do the same for the level equation of the GARCH models.
Autocorrelation analysis and in-sample criteria would suggest more complex ARMA
forms. However, the risk of over-parametrization and previous studies, for example
Lucia and Schwartz (2002) show that an ARMA(7,0) provide optimal forecasts on
daily day-ahead electricity prices, convince us to restrict the models to the afore-
mentioned specification. Following the same reasoning we choose a GARCH(1,1)
specification for the variance equation of models 4, 5, and 6.
The inclusion of the weather variables follows from statistical evidence. We allow
different transformations of the weather forecast variables on the three markets to
incorporate the fact that the weather may affect only the supply of electricity, which
is different in the three markets. Precisely, since the influence of the weather variables
appears to be non-linear as shown later, the generic initially unrestricted model in
all the three exercises includes the level, the square and the cubic of the temperature
forecasts, and the level and the square of the precipitation and wind forecasts. We
use selection criteria, as the the adjust R-square and Akaike information criteria,
and parameter statistical significance to specify the model.
5.1In-sample analysis: Oslo Case
The in-sample analysis is based on the overall sample, from December 24, 2003 to
March 14, 2006. We start with the ARMAX model which is considered to be a very
accurate forecasting model. The ARMAX model in Lucia and Schwartz (2002) is
Pt= Xt+ φ1(Pt−1− Xt−1) + φ7(Pt−7− Xt−7) + ?t
Xt= c + d1Dhol,t+ d2sin(2πt/365.25) + d3cos(2πt/365.25),
where Ptis the log of the price at day t, and where Dhol,tis a dummy variable with
value 0 if day t is a working day or 1 if day t is not a working day. Taking a close look
at the errors of the ARMAX model, as shown in Figure 8, the errors have non-linear
relations with the daily average temperature and the total precipitation, however, a
linear-like relation with the wind speed. This suggests us to introduce the weather
variables as the the level, the square and the cubic of the temperature forecasts, and
the level and the square of the precipitation and wind forecasts. By the selecting
procedure mentioned above, the reduced specific ARMAXW model for Oslo data is
Pt= Xt+ Wt+ φ1(Pt−1− Xt−1− Wt−1) + φ7(Pt−7− Xt−7− Wt−7) + ?t
Xt= c + d1Dhol,t+ d2sin(2πt/365.25) + d3cos(2πt/365.25),
Wt= a1Tempt+ a2Temp3
t+ b1Prect+ b2Prec2
where Tempt, Prect and Windt are the forecasts on daily average temperature,
total precipitation and wind speed, respectively, on day t. The estimation procedure
indicates that the square of the temperature and the square of the wind can be
excluded. The square of the temperature does not take into account the difference
between very low and very high temperature, which is a serious limitation. The
wind seems to have a direct linear relation with prices. These empirical findings
agree with the above graphical analysis.
Table 2 gives the results of the estimation of model (13) with the chosen Wt
over the complete sample. We discuss the estimated coefficients for the temperature
forecasts, a1and a2. This hopefully explains the nonlinearity in the relation of prices
and weathers, and support our views to introduce them. The temperature forecasts
affect the day-ahead electricity price via the following function:
f(Tempt) = a1Tempt+ a2Temp3
Taking the first order derivative, we get
= a1+ 3a2Temp2
By substituting in the previous equation a1and a2with their empirical estimates,
= 0, we find the roots as ±15. From our data set, the minimum
observed temperature is −15. So the only switch point is Temp∗= 15. When the
temperature is lower than the switch point, it is negatively influenced, i.e. the
lower forecasted temperature, the higher electricity price. On the other hand, when
the temperature forecast is above the switch point, it is positively influenced, i.e.
the higher the forecasted temperature, the higher the electricity price. Intuitively,
it reflects the fact that when temperature forecast is relatively higher or lower,
the consumption of the electricity will arise. Meanwhile the difficulty of producing
electricity is also increased when it is extremely hot or cold. This suggest that the
weather forecasts can influence both the demand and supply side of the power. For
further discussion on which side the weather forecast really affects, see next section.
Comparing to the ARMAX, the improvement of introducing the weather forecast
variables is not impressive for the in-sample analysis as shown in Table 2. The
inclusion of weather variables seems also appropriate in the GARCH specification.
The parameters of the GARCHW equation are less persistent than the GARCH
counterpart and model 5 has the lowest Akaike information criteria.
The estimate of autoregressive terms and of the constant term may suggest the
presence of unit root. As we explained, we do not take this hypothesis into account.
But we notice that we could have spurious regression. Therefore we relax the analysis
of the goodness of fit and proceed with the out-of-sample analysis with the chosen
5.2Out-of-sample analysis: Oslo Case
The object of the out-of-sample analysis is to forecast the electricity price from
January 1, 2005 to March 14, 2006. We repeat the selection procedure in Section
5.1 over the initial in-sample period, from December 24, 2003 to December 31, 2004.
The reduced specific model remains the same as in (13). In forecasting, the model
is re-estimated to make any new forecast, but it is not re-specified. An expanding
window is used, which means that, to forecast the price of one day, all the previous
data are applied.
Two criteria (typically used in the electricity forecasting literature, see e.g.
Conejo, Contreras, Espinola, and Plazas (2005), Knittel and Roberts (2005), Shahideh-
pur, Yamin, and Li (2002), Weron (2006)) are computed to compare the models. The
first one is the Root Mean Square Prediction Error (RMSPE), defined as
where PT+sis the log price at time T + s, whereˆPT+sis the forecasted log price at
time T + s, where n = 438 is the number of days being forecasted. The alternative
criterion is the Mean Absolute Percentage Prediction Error (MAPE), see for example
Misiorek, Trueck, and Weron (2006). It is defined as
|pT+s− ? pT+s|
We apply all five models in section 3 to forecast the daily price for Oslo data, and
calculate the RMSPE and MAPE statistics. For description, we also report results
for the Random Walk (RW) model. The results are given in Table 3.
All the models provide quite superior statistics than the RW model, implying
predictability in the electricity prices. It is also clear that the model 4, ARMAXW,
is the best under both the criteria. For example, for the RMSPE statistics, com-
paring to model 3, ARMAX, which is the best among the non-weather models, the
improvement is 3.8%.
We test whether the difference between two forecasting methods is significant in
order to show precisely how large is the improvement of the new weather forecast
model. We choose the Diebold-Mariano test (Diebold and Mariano (1995)) with loss
function the mean square prediction error (MSPE). The null hypothesis is
H0: The square of the forecast errors are equal.
Statistics are in Table 4. The p-value of this test is p = 0.0024. We conclude
that the ARMAXW model is significantly improving in the sense of out-of-sample
Figure 9 shows the 60-day average RMSPE for the ARMAX model and the
ARMAXW model. From the graph, we find that, when the error of model 3 is in a
relatively lower level, the errors of two models are similar; but when there is a higher
error from model 3 due to possible jumps, our weather forecast model often predicts
better. Price jumps are mainly due to problems of inelasticity of the demand, and of
non-storability of electricity with consequent shortage in the supply. These problems
often arise when the weather conditions are adverse. Empirical results confirm the
theoretical intuition that the weather forecasts help in predicting high prices or
jumps, possible related to extreme adverse climate situations.
Adding weather forecast variables in a GARCH model is also very beneficial.
Forecasts from the model 6, ARMAXW-GARCHW, gives accurate forecasts and
quite similar to model 3, even if marginally lower than it. On contrary, model 4,
ARMAX-GARCH, gives very poor forecasts, and extending the mean equation with
weather variable, as model 5, ARMAXW-GARCH, is not enough. To sum up, a
’classical’ GARCH specification is not adequate to predict electricity prices, but
adding weather variables as shock indicators improve enormously the performance.
Although the weather forecast models show improvement on predicting the day-
ahead prices, it is still mysterious whether this kind of influence is via the demand
of the electricity or the supply. One way to verify this is to introduce the volume
variable into a forecasting model. In principle, the volume indicates the demand of
the electricity. Then, if the weather only influence the consumption of the power,
introducing the volume at time T+s to forecast the electricity price at time T+s into
the ARMAX model will lead to similar results as the ARMAXW model. We stress
that the volume at time T +s is not known in advance, but previous literature form
Engle, Granger, Ramanathan, and Andersen (1979) finds that it may be forecasted
accurately, then we assume to know it. The model with volume (ARMAXV) is given
Pt= Xt+ Vt+ φ1(Pt−1− Xt−1− Vt−1) + φ7(Pt−7− Xt−7− Vt−7) + ?t
Xt= c + d1Dhol,t+ d2sin(2πt/365.25) + d3cos(2πt/365.25)
where Vtis the volume at time t. The calculated RMSPE is 0.0509, the improve-
ment with respect to the ARMAX is 0.04%. We also apply the Diebold-Mariano
test, the p-value is 0.8161. The comparison shows that introducing the volume on
forecasting the day-ahead price is not comparable with the weather forecasts, even
if the future (unknown in practice) volume is applied. In Oslo electricity market,
the weather influence is not only via the demand of the electricity, but even more
via the production of the electricity. This reflects to the producing method in Oslo,
5.3Further Application: Eastern Denmark Case
With the estimation in the in-sample period, we find that the specified model for the
Eastern Denmark data only depends on the temperature and wind speed as follows
Pt= Xt+ Wt+ φ1(Pt−1− Xt−1− Wt−1) + φ7(Pt−7− Xt−7− Wt−7) + ?t