TwoEmpiricalMethodsforForecastingSpotPricesand ... · Electricity demand is aﬀected by cycli-cal components which expand over three distinct time horizons linked to global economic

Two Empirical Methods for Forecasting Spot Prices and

Constructing Price Forward Curves in the Swiss Power

Market

A master thesis submitted to

EIDGENOSSISCHE TECHNISCHE HOCHSCHULE ZURICH

Master of Science in Management, Technology and Economics

GREGOIRE CARO

Jointly supervized by:

Centre for Energy Economics and Policy (CEPE)

Department of Management, Technology and Economics

Swiss Federal Institute of Technology

Dr. Carlos Ordas Criado

Prof. Thomas Rutherford

swissQuant Group

MSc. Marcus Hildmann

Dr. Florian Herzog

Zurich, May 2010

Contents

1 Introduction 1

2 Electricity Market Settings 5

2.1 Physical and Economic Features of Electricity . . . . . . . . . 6

2.2 Financial Approach . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 The Spot Price . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Forwards and Futures . . . . . . . . . . . . . . . . . . . . 12

2.2.3 The Hourly Price Forward Curve . . . . . . . . . . . . . 15

2.3 The Swiss Electricity Market . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Power Production . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 A Key-Location . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.3 Financial Data . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Literature Review 21

3.1 Load Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Spot and Forward Price Models . . . . . . . . . . . . . . . . . . 22

4 Estimation Methodology 24

4.1 Presentation of the Models . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 A Short-Term Spot Price Model . . . . . . . . . . . . . 25

i

4.1.2 A Hourly Price Forward Curve Model for electricity

(HPFC model) . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Regression Techniques . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 The Lad-lasso Regression . . . . . . . . . . . . . . . . . 35

4.2.2 The Least-squares Support Vector Machine Regression 37

5 Empirical Analysis 40

5.1 Spot Price Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Hourly Price Forward Curve Model . . . . . . . . . . . . . . . . 48

6 Conclusion 60

Appendices 61

A Two Key Ideas of Least-Squares Support Vector Machine 62

A.1 The Maximum Margin . . . . . . . . . . . . . . . . . . . . . . . 62

A.2 The Kernel Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.3 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . 64

Bibliography 66

List of Tables

2.1 The stakeholders of the power market . . . . . . . . . . . . . . 10

5.1 Backtest result for one day ahead forecast . . . . . . . . . . . . 45

5.2 Backtest results for four days ahead forecasts . . . . . . . . . . 45

5.3 Backtest results for seven days ahead forecast . . . . . . . . . . 45

5.4 Results of the falsifiability tests . . . . . . . . . . . . . . . . . . 47

5.5 Regression on the spot prices . . . . . . . . . . . . . . . . . . . . 50

5.6 Selected variables of the PFC model . . . . . . . . . . . . . . . 54

5.7 Backtesting results for the HPFC model . . . . . . . . . . . . . 59

iii

List of Figures

2.1 A mixed approach . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Structure of the electricity market . . . . . . . . . . . . . . . . . 9

2.3 The economic equilibrium of the power market . . . . . . . . . 11

2.4 Swiss Spot Price in 2008 . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Two typical spot weeks in summer and winter . . . . . . . . . 13

2.6 Implied German Futures curve . . . . . . . . . . . . . . . . . . . 15

2.7 Swiss power production in 2008 . . . . . . . . . . . . . . . . . . 17

2.8 Swiss average reservoirs level . . . . . . . . . . . . . . . . . . . . 18

2.9 The Swiss power trade balance . . . . . . . . . . . . . . . . . . 19

4.1 The short term spot price model . . . . . . . . . . . . . . . . . 27

4.2 The statistical approach for the HPFC model . . . . . . . . . . 34

5.1 Grid search results for the short term spot price model . . . . 42

5.2 In-sample fits of the short term spot price model . . . . . . . . 43

5.3 A one week ahead forecast . . . . . . . . . . . . . . . . . . . . . 44

5.4 Forecasted and observed weather . . . . . . . . . . . . . . . . . 44

5.5 Training set under a seasonal transition: the first heating days 48

5.6 Training set under a vacation period: Christmas holidays . . . 49

5.7 Evolution of the coefficients . . . . . . . . . . . . . . . . . . . . 51

5.8 HPFC: determination of the hyper-parameters for ls-svm . . . 54

iv

5.9 In-sample fit of the HPFC model . . . . . . . . . . . . . . . . . 55

5.10 Extreme in-sample fits . . . . . . . . . . . . . . . . . . . . . . . . 56

5.11 Simulation output . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.12 Hourly Price Forward Curves from the lad-lasso and ls-svm . 57

A.1 The determination process of the classifier hyperplane. . . . . 63

A.2 The Kernel Trick. . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A.3 The primal and the dual . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 1

Introduction

The best way to predict the future is to create it.

- Peter F. Drucker(1905-2005).

Liberalization of the European electricity market was initiated in the

90’s. Its two main goals were to achieve competitive prices and to favor

the integration of the power markets across borders. Since then, the elec-

tricity companies have faced a completely new challenge. Electricity has

become a commodity traded on a European market, creating exporting facil-

ities and energy-linked financial indexes. The local supplier-customer model

has shifted toward banking business where complex financial instruments are

used to achieve an efficient inter-temporal market equilibrium.

In Switzerland, the first steps toward liberalization occurred in 1998 with

the creation of a national forward index (the SWEP), but it officially started

1

Chapter 1. Introduction 2

in January 2008 with application of the Federal Electricity Supply Act1. Be-

cause of this recent start, the market has not yet reached its maturity. For

instance, while a Swiss spot price (the Swissix) has been recently created

(2006), no financial index for futures (standardized long-term contracts) ex-

ists. The Swiss power market is therefore characterized by the lack of some

important financial series and structural specificities such as the relative im-

portance of the hydro-power plants (56,1% in 2008). Moreover, while the

literature on foreign power markets is abundant, forecasts concerning the

Swiss market are less developed. Several standard models for electricity price

forecasts have not been tested with Swiss data despite the strategic position

of the country as a major trading partner located in the center of Europe.

The aim of this thesis is to provide new insights into the Swiss electricity

prices from a financial perspective. We explore two models, one for predict-

ing electricity spot prices, the other one for estimating the current prices of

the long term contracts for electricity, the so-called Price Forward Curve2 or

PFC. As outlined by Fleten and Lemming [2003], PFCs are important infor-

mation carriers for operational and investment decisions. Practitioners often

need to estimate forward prices, i.e. the price of non-standardized futures, for

more maturity dates than are observed in the market. For that purpose, high

resolution Price Forward Curves are of great interest. We provide such an

estimate for Switzerland and construct hourly forward prices, i.e. an Hourly

Price Forward Curve (HPFC henceforth).

The spot price model estimated here is a standard autoregressive equation

which includes exogenous bottom-up components such as seasonal trends and

1Act published in the Swiss Federal Gazette the 3rd April 2007.2Some authors prefer the term Forward Price Curve.


temperature variables. We explore its ability to predict the Swiss spot prices

for electricity over different time horizons. Two different regression tech-

niques are tested : the least-absolute-deviation least-absolute shrinkage-and-

selection-operator (lad-lasso) and the least-squares support vector machine

(ls-svm) regression approach. These estimators have major advantages com-

pared to the standard linear regression model. Though the lad-lasso remains

a linear model, the estimation procedure is robust to outliers. Moreover, non-

significant coefficients can be ‘shrinked’ toward zero through a procedure that

optimally balances the bias-variance trade-off. Therefore, the lad-lasso re-

gression includes a variable selection mechanism that leaves the researcher

with only the most relevant predictors to interpret. These properties are

particularly interesting when the explained variable displays ‘peaks’ as it is

often the case for daily electricity prices and when a large amount of vari-

able are involved in the estimation process. From another perspective, the

ls-svm approach proposes a highly flexible estimator which performs partic-

ularly well when nonlinear patterns need to be estimated in the presence of

highly correlated explanatory factors. Its main drawback is that it is a ‘black

box’ estimator in the sense that the researcher cannot recover the original

structural parameters and disentangle the importance of each explanatory

component.

Regarding the Price Forward Curve model, an ad-hoc equation widely

employed in the financial industry is investigated with the lad-lasso and the

ls-svm regression techniques. The estimation procedure in this case requires

several steps and adjustments due to the lack of some fundamental series for

the Swiss market. In particular, Swiss futures prices are estimated based on

futures prices observed in the most influential neighboring electricity mar-


kets, Germany and France.

Our main conclusions are that the robust linear (lad-lasso) estimator ex-

hibits better predictive performance for Swiss electricity spot under atypical

weather conditions or when the observed price pattern departs significantly

from past trends. The ls-svm method performs better under ‘regular’ condi-

tions, i.e. when past spot price fluctuations are highly persistent. Regarding

the price curve for the long term contracts, we show that a meaningful Hourly

Price Forward Curve can be build for Switzerland by using data from the

German and French power markets. This does not come as a great surprise

given the strong correlation between the Spot electricity prices in these mar-

kets. Then we also show that the lad-lasso model highlights the particular

importance of hydro-power in Switzerland. However, the support vector ma-

chine estimates catch better seasonal variations of the futures prices.

This thesis is organized as follows. Chapter 2 discusses the theoretical

background on the power market and describes the Swiss market. A brief

review of the relevant literature is proposed in Chapter 3. Chapter 4 presents

the empirical methodology. We start with an overview of the Spot price and

Price Forward Curve models and proceeds with a brief presentation of the

lad lasso and ls-svm regression techniques. The results are given in Chapter 5

and Chapter 6 concludes.

Chapter 2

Electricity Market Settings

The models described in this paper aim at predicting the evolution of

electricity price indexes by integrating basic principles of financial theory

with structural determinants borrowed to the so-called bottom-up models

(see Fig. 2.1). The financial approach generally develops price scenarios

which depend on stochastic factors, market prices, and make use of stochas-

tic differential equation. A fundamental assumption in these models is the

existence of highly liquid markets1. This condition is not yet fulfilled in

most power markets. In addition, most of the electricity financial indexes

are heavily influenced by structural elements such as the weather or business

and seasonal cycles.

Fleten and Lemming [2003] proposed to combine all these determinants

within a single equation in order to ”compensate for the deficiencies that

arise from separate use of either market data or bottom-up models”. This

Bayesian approach is particularly relevant for Switzerland as it allows to in-

1The liquidity of a commodity is its ability to be easily traded without influencing themarket price.

5

Chapter 2. Electricity Market Settings 6

clude into the model all relevant information which may not be fully captured

by the market data available.

Figure 2.1: A mixed approach. Source: Fleten and Lemming [2003].

In this chapter we describe the structural determinants and the relevant

financial indexes for our purpose. We begin by explaining key features of

electricity and proceed with a description of two major financial products:

the spot and futures contracts. The characteristics of the Swiss power market

are explored in the last section.

2.1 Physical and Economic Features of Elec-

tricity

Compared to other commodities, electrical energy has some unique at-

tributes.

A Non-Storable Good. Electricity is economically non-storable, there-

fore most standard commodity models are inadequate to explain the elec-

tricity market. One way to store electricity in the context of hydropower


generation is by pumping water into dams, but only with a significant loss.

Non-storability prevents the market imbalances to be quickly addressed by

quantity adjustments.

A Three-Fold Periodic Pattern. Electricity demand is affected by cycli-

cal components which expand over three distinct time horizons linked to

global economic activity and seasonal variations:

• a yearly cycle linked to seasonal factors. For instance the cooling days

in summer and the heating days in winter have a strong impact on the

market (see Fig. 2.4). There are also some special events taking place

every year, like holidays and vacations.

• a weekly pattern. Demand is low during weekends, high during business

days (see Fig. 2.5). Fridays usually trigger the drop in consumption of

the weekends.

• a daily profile. Demand is very low at night, and experiences a demand

peak at 7pm when everybody is back home from work (see Fig. 2.5).

The peak hours are usually defined over the time interval 8 am : 8 pm

for business days only.

Supply-side constraints. In addition to non-storability and cyclical vari-

ations, the supply of electricity faces two additional major constraints linked

to its source and its distribution network (the grid).

• Electricity has to be supplied at exactly 50Hz, 220V into the grid. A

power grid is like a river that needs to keep the same flow and level.

Therefore, if someone pumps water from it, someone else must at the


same time pour the same amount of water. The interconnectivity of

the different national/regional grids is key for clearing profitably the

market: over-capacity can then be exported abroad, and under-capacity

can be avoided by importing power. Moreover, the grid has a bounded

capacity which cannot be easily increased, and thus congestion might

occur. Many congestion management tools have been put forward by

the EU. The more the national grids are inter-connected, the higher

the flexibility to avoid congestion.

• The generation capacity of a power plant is more or less flexible, de-

pending on the energy source:

– Nuclear, thermal and coal-fired plants cannot be easily stopped or

started and are basically always running. Changing its output is

a lengthy process.

– Power plants based on gas turbines or the hydroelectric dams are

easier to regulate. Gas turbines can be activated on demand at

a high cost, whereas hydroelectric dams can generate power any

time providing the water level in the reservoirs is sufficiently high.

– Solar panels, wind propellers and other ”green plants” are subject

to caution from a grid point of view. Although weather forecasts

exist, their output remains difficult to predict. In some countries

the law might force the regulator of the grid to use ”green energy”

upon any alternative source.

The Market Equilibrium

The market equilibrium results of the interplay between five main agents

(see Fig. 2.2 and Tab. 2.1) and the the time horizon plays an important role.


In the short run, demand for electricity is highly inelastic to prices and it

exhibits strong variations over the day. The supply capacity being extremely

rigid, the short-term prices are essentially demand-driven. This is illustrated

in Fig. 2.3. The supply curve can be plotted as a stepwise and increasing

function where each step reflects the fixed marginal cost associated with a

specific energy source2. This particular demand/supply configuration induces

abrupt changes (peaks or valleys) in the price equilibrium. Although the

supply curve is almost fixed in the short run, it can experience exceptional

shifts due to events such as a temporary maintenance of several nuclear

power plants (left-shift of the supply on Fig. 2.3) or holidays in a neighboring

country (right shift of the supply curve). The Swiss market equilibrium can

be very sensitive to changes in the foreign markets.

Figure 2.2: Structure of the electricity market

Note finally that the price is also influenced by the transporter in charge

of the grid, who takes a small fee for the delivery service, and the electricity

retailers who make profits out of the sales to the final consumers. These

operators may also be public-owned firms which are subject to heavy political

2Green sources are in general more expensive to exploit than nuclear power for instance.


pressure.

Player Actions Examples

Generator• Sells capacity to the retailers• Injects energy into the grid

Alpiq(CH), solarpanel owners

Retailers

• Buy at low prices and sell at a higherprices

• Are often engaged in a war price withother retailers

EDF(FR), Axpo(CH)

Transporter

• State-owned, result of a naturalmonopoly

• Separation to other players due to theunbundling condition of the liberaliza-tion of the market.

• Controls the grid, the auctions, theinter-border transactions...

Swissgrid(CH),National Grid(UK)

Consumers• Are free to chose their retailer• Usually, first criteria is price

Households, firms,state

Regulator

• Supervizes the liberalization of themarket

• Check that the laws (European andNational) are applied

Bundesnetzagentur(DE), CompetitionCommission (CH)

Table 2.1: The stakeholders of the power market


Figure 2.3: The supply and demand curves of the power market. The demandfor electricity is highly inelastic.

2.2 Financial Approach

Electricity markets offer a wide range of standard financial instruments,

whose sophistication depends on the market maturity. The main ones are

spot, futures and forward contracts. More sophisticated derivatives, such as

options, have started to appear in more mature pools, like the Nord Pool.

There are about twenty Electricity exchanges in Europe, which cover sev-

eral countries. The EU Commission has pushed the different exchanges to

cooperate in order to improve the integration of the European electricity

market. The exchanges we are interested in are the European Power Ex-

change (EPEX) in Paris for spot prices, and the European Energy Exchange

Power Derivatives (or EEX Power Derivatives) in Leipzig. The cooperating

countries in these exchanges are France, Germany, Austria and Switzerland.

2.2.1 The Spot Price

The spot market is the one-day ahead market, where the power traded

is delivered the day after. The term ‘spot’ designates the price quoted for


immediate settlement. As explained in Section 2.1, the rigidities of the power

supply are so strong that the three cyclical patterns affecting the demand

side can be easily identified in the spot price curve. We can see on Fig. 2.4

that the prices are on average higher from the end of summer vacation until

the winter vacation. The average lowest price is usually reached in spring,

when there is neither intensive cooling nor long heating periods.

Feb Apr May Jul Aug Oct Dec0

50

100

150

200

Time

Pric

e/M

wh

Figure 2.4: Swiss Spot Price in 2008

The weekly and daily variations for two typical summer/winter weeks are

shown on Fig. 2.5. Note that the level of the curves can fluctuate considerably

within a season. Both series clearly display weekly as well as intra-day cycles.

During summer, the 1pm peak corresponds to air cooling while winter is

rather characterized by the 9am and 7pm peaks due to heating. We also

notice the irregular Fridays pattern which triggers the weekends drop in

electricity demand.

2.2.2 Forwards and Futures

Forward and futures contracts are a type of financial ”derivative” prod-

uct. They consist in an agreement between two parties to deliver a certain


Mon Tue Wed Thu Fri Sat Sun

20

40

60

80

100

120

140

160

180

Day

Pric

e/M

Wh

WinterSummer

Figure 2.5: Swiss electricity spot price for a typical week in summer and inwinter 2008.

amount of a good at a prearranged price (the forward/futures price). Unlike

spot contracts, forwards and futures can be continuously traded until their

delivery date. The difference between a forward and a futures contract is

that the terms and conditions for forwards are not standardized. Forwards

are negotiated to meet the specific business, financial or risk management

needs. Note also that futures are traded in an exchange while forwards are

traded over-the-counter. Before the liberalization of the power market, for-

ward contracts were the only long-term contract available in Switzerland.

Assuming the arbitrage-free condition3 is checked, the classic theory of

rational pricing establishes a relationship between the forward price and the

value of its underlying (storable) asset4:

Ft = S0(1 + rf + storage costs − convenience yield)t (2.1)

3An arbitrage occurs when one can make a profit out of the difference in prices in twomarkets. As an example, if the forward price was not to converge to the spot price atthe delivery date, there would be an arbitrage opportunity. This convergence property isassumed to hold in all models presented in this paper.

4See Harris [2006].


where Ft refers to the market price of a forward delivered in date t, S0 is

the current spot price and rf corresponds to the risk-free rate. This formula

proposes a theoretical link between the price of buying a storable commod-

ity and storing it for further usage and the price of buying a right (forward

contract) of getting the commodity when needed in the future. This equality

simply states that the owner of the commodity must be compensated for all

incurred costs. Note that a commodity bought now can be used any time and

therefore provides a convenience yield. The difference between the storage

costs and the convenience yield is called the net convenience yield. Since

electricity cannot be stored, the net convenience yield becomes the conve-

nience yield which is difficult to estimate if the terms of the contracts are not

standardized [Carmona and Ludkovski, 2004].

Electricity futures contracts are defined according to three main criteria:

the delivery date, the length of the delivery period and a daily component

(8am-8pm vs. 24/7). Delivery starts at the beginning of either a month, a

quarter (January, April, July and October) or a year5. Peak futures are de-

livered only between 8am and 8pm whereas base futures are delivered 7/24.

Note that country-specific standards exist. In Germany, the upper limits for

each delivery range are seven months, seven quarters and six years ahead.

In France, these limits are three months, four quarters and three years ahead.

Fig. 2.6 shows a series of typical Peak and Base futures prices where the

horizontal regions indicate delivery period intervals. We employ a representa-

tion called ”implied Futures curve” which allows to build a none overlapping

and continuous sequence of prices for futures with different delivery periods.

5The only exception is for the first monthly contract which covers a delivery for theon-going month.


It consists in plotting the price of the futures contracts by starting with con-

tracts with the shortest delivery period over the permitted trading horizon

(monthly delivery starting at the beginning of months 1 to 7 for the Ger-

man case), followed by those with longer delivery ranges (quarterly contracts

starting in the next 2nd to 7th quarters and finally yearly contracts delivered

in next 2nd to 6th year).

Aug09 Dec10 May12 Sep13 Feb15 Jun1630

40

50

60

70

80

90

100

Pric

e/M

Wh

Futures BaseFutures Peak

Figure 2.6: Implied German Futures.

2.2.3 The Hourly Price Forward Curve

Futures curves such as the one in Fig. 2.6 provide information about

the long-term expectations of the market players. However, products with

maturities exceeding 6-12 months are rather scarce and interpolating a small

amount of points with similar (but not identical) characteristics is not an

optimal way to price futures. Looking at forward prices would be of little

help. As noted by Fleten and Lemming [2003], forward contracts are usually

exchanged in large chunks which involves problems for finding prices for

specific maturity times or for the more general purpose of constructing a high-

resolution term-structure curve. This is why futures or forward curves gain at

being complemented with high-resolution Price Forward Curves. The Hourly

Price Forward Curves (HPFC) is such a curve and it describes the prices as


of today for the delivery of electricity at each hour in the future. It represents

the term-structure of forward prices in hourly resolution. It is important to

remember that the HPFC is not a forecast of the spot price. One can hardly

say whether today’s futures price will reflect the spot price at the delivery

date. This depends on many uncertain factors such as the global economic

climate, oil prices and agents irrational behaviour. Therefore, assessing the

quality of the generated forward curves is not straightforward.

2.3 The Swiss Electricity Market

Compared to other West-European electricity markets, the Swiss market

has three major differences: the relative importance of hydraulic power, its

particular position in the middle of Europe and its relative youth.

2.3.1 Power Production

Due to the federal system, the Swiss Power Market is highly fragmented.

About 900 companies are active in production and retailing, most of them

working at a cantonal or regional scale. However, only one transporter is in

charge of the grid: Swissgrid. Like 75% of the power utilities, Swissgrid is

state-owned.

With 56 % of electricity produced from hydropower plants, Switzerland

has a particularly high share of hydropower in its total electricity output.

Combined with 5 nuclear plants representing about 35% of the total produc-

tion, Switzerland has almost CO2-free power generation.


Figure 2.7: Swiss power production in 2008. Source: Swiss Federal Office forEnergy - Schweizerische Gesamtenergiestatistik 2008.

2.3.2 A Key-Location

The location of Switzerland in the heart of Europe makes it an important

transit country. The main transit axis is the French-Italian one (see Fig. 2.9).

Switzerland has been for years a net power exporter, but has become in 2007

a net importer. With a total production of 64 TWh and a total consumption

of 63 TWh in 2008, the Swiss trade balance of energy is almost null, but there

are strong seasonal and daily variations. Switzerland cannot rely entirely on

hydropower from winter to late spring because of the snow that starts melt-

ing only in spring (see Fig. 2.8). The reservoirs reach their lowest level in

May and get filled in summer when precipitations are important. In order

to keep a safety margin for spring in case of winters with excessive heating,

the Swiss market is a net importer in winter (-4.5 TWh). By contrast, once

the reservoir starts to fill up again in late spring Switzerland becomes a net

exporter until the end of summer (+5.5 TWh for summer 2008).

Interestingly enough, the large hydropower capacity of Switzerland give

it a comparative advantage with respect to its neighbors. Since the only way

to store electrical power is by pumping the water up to the reservoirs, and

since water plants can deliver electricity at any time, Switzerland has become


Dec07 Apr08 Jul08 Oct08 Jan09 May09 Aug090

0.2

0.4

0.6

0.8

1

Time

Leve

l (%

)

Figure 2.8: Average reservoirs level in Switzerland. The filling is done inSummer, when precipitations are the most important. To avoid shortage inSpring the Swiss market is a net importer in Winter. Source: Swiss FederalOffice of Energy - Schweizerische Gesamtenergiestatistik 2008.

a supplier for the EU-neighbours at peak hours. Indeed, as one can hardly

slow down a nuclear or a thermal plant, the energy price drops significantly

during off-peak hours. Swiss generators take advantage of this situation by

pumping water into their dams during off-peak hours and sell their ‘stored’

electricity during peak hours. With a 8.5 TWh water storage capacity, Swiss

generators have enough capacity to meet demand from home and from abroad

during peak hours. This profit scenario applies especially to France which

relies heavily on nuclear power: as shown in Fig. 2.9, France is the only net

exporter of electricity to Switzerland. This interdependence between France

and Switzerland will be exploited to build the HPFC in section 5.2

2.3.3 Financial Data

The only financial indexes available for the Swiss power market are the

Swiss Electricity Price Index (SWEP) and the Swiss Electricity Index (Swis-

six). The SWEP is a local indicator of one-day-ahead over-the-counter prices.

It was launched in 1998 and it became the first wholesale electricity price in-

dex published on the European continent. The Swissix is the average price


Figure 2.9: Trade balance (yearly-based and winter-based). Source: Centrefor Energy Policy and Economics, Swiss Federal Institute of Technology -Report: Electricity gap; ways to face the challenge.

at the European Energy Exchange (EEX) for next-day deliveries in the Swiss

grid, hourly based, with base and peak series. It was launched in 2006. Rel-

atively to the SWEP it has a wider range and is not affected by local effect6.

The Swissix is the reference spot price for this study.

Although these two indicators give a good idea of the historical evolution

of electricity prices, they capture only around 10%7 of the Swiss market. The

remaining 90% are over-the-counter contracts (not referenced by the SWEP),

largely influenced by public service obligations and therefore with biased, un-

referenced prices. This is a usual problem often encountered in the rest of

the European countries as well.

Finally, the main issue with the Swiss financial data, relatively to France

and Germany, is the absence of futures products. The only financial data

we possess are the spot price which is past-oriented. so that there is no

6Since there is no official price for over-the-counter trading, the SWEP is just thevolume-weighted average at the 380-kV Laufenburg’s grid hub.

7IEA [2007].


long-term expectations indicators. The models developed in the literature to

build Price Forward Curves therefore cannot be directly applied. This issue

is addressed in Section 5.2.

Chapter 3

Literature Review

Since the beginning of the liberalization of the European Energy market,

papers related to predicting energy consumption or prices have flourished

with the purpose of improving risk management. The HPFC is a key tool

to optimize power plants production capacities and better estimate firms’

upcoming income. Energy producers can also use load models to make real-

time scheduling of electricity generation. In this literature review, we focus

on the papers related to price forecasting and the construction of forward

curves.

3.1 Load Models

Given the strong correlation existing between the electrical load and the

electricity spot prices, load models can help in identifying the main drivers

of electricity prices. These models generally assume a deterministic path and

employ hourly data to forecast up to seven days ahead. Based on load data

from Brazil, Soares and Medeiros [2008] compares a purely stochastic-trend

model (SARIMA-type) with an autoregressive model with a flexible deter-

21

Chapter 3. Literature Review 22

ministic trend component (TLSAR-type) and conclude that a deterministic-

based approach performs better for short run forecasts1. They also find no

evidence that nonlinear models are better in terms of predictive performance.

Therefore. capturing explicitly the deterministic trend seems to be important

for short-run load forecasts. These authors do not include weather variables

in their model but they emphasize that they can significantly improve the fit.

Taylor [2008] uses a very short-term model (ten minutes ahead) based

only on past load data. He finds that for forecasts longer than four hours

ahead, models with weather variables are superior to the purely autoregres-

sive models based on last week data. He then emphasizes the importance of

the accuracy of the weather forecasts for the predictive performance.

Finally, Amaral et al. [2008] compared different linear and non-linear

methods, based on Australian load data. They propose a specific treatment

for special days like holidays, and point out that non-linear methods can be

more efficient than linear ones for short-term forecasting (one day ahead)

while basics linear models are better for longer time-spans.

3.2 Spot and Forward Price Models

Most of the energy price models consist in short-term prediction models

of the price on the spot market (the day-ahead market). They usually focus

on the most liberalized markets, where data are abundant and rigidities are

low, such as the North Pool market (which includes Sweden, Norway and

Finland). Weron and Misiorek [2008] use this market to contrast paramet-

1SARIMA stands for Seasonal Integrated Autoregressive Moving Average and TLSTARis a Two-Level Seasonal Autoregressive model.

Chapter 3. Literature Review 23

ric versus semi-parametric models and show that the semi-parametric models

perform better and are more sensitive to exceptional market conditions (peak

demand, weather conditions). They conclude that autoregressive models for

short-term forecasts on an hourly basis are the best in terms of predictive

power.

Not all price models focus on short-term predictions. Some recent mod-

els aim at constructing forward prices, i.e the Price Forward Curve itself.

Based on the Nordic market, Fleten and Lemming [2003] use bid-ask data

of futures products to construct the long-term product and emphasize that

the method performs well in the range of four to ten months ahead. Jump-

diffusion models are also often employed to capture the spiky behavior of

the spot price. However, Chan et al. [2008] argue that traditional financial

approaches (like the jump-diffusion model) are unsuccessful in capturing the

spot price dynamics.

Chapter 4

Estimation Methodology

As outlined in Chapter 2, electricity prices display multiple cyclical be-

haviors which must be taken into account in the modeling process. These

structural determinants (e.g.: daily/monthly/yearly seasonal patterns) may

be heavily correlated among themselves and with other independent vari-

ables (e.g. weather indicators). Moreover, daily transactions generate a small

amount of peak values (extreme spot prices) which may influence excessively

the fits. Section 4.1 presents the spot price and the forward price models

while section 4.2 details the regression methods used to estimate them. The

lad-lasso estimator combines regularization techniques (shrinkage regression)

that allow to control for the bias-variance trade-off with a robust approach

(LAD minimization). The ls-svm regression is particularly appropriate to

estimate non-linear relationships in the presence of strongly correlated pre-

dictors.

24

Chapter 4. Estimation Methodology 25

4.1 Presentation of the Models

In this section we describe two models based on the mixed approach in-

troduced by Fleten and Lemming [2003], see page 6. The short-term spot

price model is directly inspired from the short term vertical load model de-

scribed in Espinoza et al. [2007]. This HPFC model has been developed by

swissQuant Group and Axpo and it corresponds to a model widely used in

the power industry.

4.1.1 A Short-Term Spot Price Model

Methodology

Espinoza et al. [2007] use an ls-svm regression technique to estimate an

autoregressive equation (called ar-lssvm in JAK and Vandewalle [2000]) for

predicting short term loads. Their model mixes a load dynamics component

(autoregressive part) with daily trends and weather indicators to predict fu-

ture loads. Although vertical load series display a similar pattern to the spot

price, the former ones are smoother and less noisy in general.

Let’s consider the following model:

yt = f(xt) + et,

where yt denotes the electricity price at time t (each hour), f(xt) is an

unknown (possibly non-linear) function and xt ∈ Òn is the regressors’ matrix:

xt = {yt−1, ..., yt−j ,Ht,Dt,Wt} ,


with

• Wt: weather forecasts indicators composed of the temperature FTt and

heating and cooling indicators defined respectively as FHt =max(18 −Tt,0) and FCt =max(Tt − 20,0);

• Dt ∈ {0,1}7 a binary-valued vector which captures the effects of each

day of a week;

• Ht ∈ {0,1}24 a binary-valued vector which captures the effects of each

hour in a day.

The parameter j denotes the size of the auto-regressive part. Let ∆ be the

number of days forming the auto-regressive part (j = 24 ×∆). When hourly

spot price curves are estimated, hourly temperature forecasts are needed in

Wt. In general, temperature for the forthcoming days is predicted in terms

of expected mean, maximum and minimum (as it is the case in Switzerland).

Hourly forecasts can be reconstructed in some way with the help of profiles

stemming from temperatures observed on a hourly basis in the past. The

global estimation procedure is illustrated on Fig. 4.1.

Validation of the Model

Performance Indicators Since our spot price model provides spot price

forecasts, standard indicators can be used to assess the quality of the fit out-

of-the-sample (this is called ‘backtesting the model’ in finance). Denoting the

observed data y, the fit y, the number of observations N and the arithmetic

mean of z as z, the following four performance indicators are considered :

• The correlation coefficient:

σy,y

σyσy

= ∑i (yi − y)(yi − ¯y)√∑i(yi − y)2

√∑i (yi − ¯y)2

,


Figure 4.1: The short term spot price model framework.

• The mean error:1

N ∑i ∣yi−yi∣1

N ∑i yi× 100,

• The mean absolute prediction error:

1

N∑i

∣yi − yi∣∣yi∣ ,

• The mean standard deviation error:

var(y − yy) × 100 = 1

N

N

∑i=1

(ǫi − ǫ)2 × 100,

with y > 0 and ǫ the relative error: ǫi = yi−yiyi

.

Falsifiability Tests In addition to the statistical measures of fit, we sub-

mit the spot model to a ‘falsifiability test’ a la Popper. Indeed, the true

data generation process of the spot prices is expected to depend on weather


predictions, among other determinants. The reason for this is that the trans-

actions on the spot market are settled one day before delivery. The traders

use weather forecasts to make up their mind on the quantity to purchase

or sell and at which price. Therefore we would expect the spot model to

perform better when weather predictions for the next day are used as predic-

tor as compared to the use of, say, the true weather observed the next day.

Indeed several weather scenarios can be tested against the weather forecasts

variable in that perspective:

• the observed (true) weather

• a normalized weather (seasonal weather)

• a random weather.

Letting ℘(.) be the prediction accuracy of a model, we expect the follow-

ing preference order to hold

℘(forecasted weather) ≻ ℘(seasonal expectations) ≻ ℘(random walk weather).

We also expect that

℘(forecasted weather) ⪰ ℘(observed weather).

In the latter preference order, the identity relationship arises in case of perfect

weather forecasts. These tests may be useful to discriminate two estimation

methods which perform similarly in terms of out-of-sample predictions.


4.1.2 A Hourly Price Forward Curve Model for elec-

tricity (HPFC model)

A fundamental component of the HPFC is futures price series. No fu-

tures products for electricity exist in Switzerland. However, the German and

French markets propose a small set of them. Figure 2.6 in page 15 shows the

only forward curve that can be observed in the market. Note the September

and the winter peaks which correspond to periods of the year where firms

need to secure power supply. After the first seven months, we notice that the

curve becomes flat and its shape provides no precise guidance on future fluc-

tuations (see [Espinoza et al., 2006]). Note the upward trend of long-term

contracts. The further away from the delivery date, the higher the cost of

hedging.

The construction of the HPFC relies on a combination of characteristics

extracted from the spot price series and observed futures prices. The sea-

sonal, weekly and daily variations are taken from the historical spot prices,

whereas the average values of the HPFC is provided by observed futures prod-

uct. In other words, the guideline is to apply a complex coefficients structure

to the observed futures prices1 to obtain hourly values. This approach relies

on structural links between the short-term (spot) and long-term markets.

Although equation 2.1 does not apply formally to electricity products, we

assume that an approximate link between the spot and the forward prices

exists.

1The construction of the futures curve is described in page 15.


Methodology

Our goal is to estimate P (t, h) the ”hourly forward price” for every day

t and hour h over the time horizon defined by the futures curve. The Peak

and Base futures curve can be denoted by (FBasei , F Peak

i ) with i ∈ [1,NF ]and where NF represents the number of products used to build the curve.

These two curves can be alternatively represented as a single stepwise func-

tion fluctuating between Peak and Base values. The latter representation is

denoted F (t, h). In order for the HPFC to be fully consistent with F (t, h),the following arbitrage-free conditions must hold

E(t,h)[P (t, h)] = FBasei for (t, h) ∈ TFBase

i

(4.1)

E(t,h)[P (t, h)] = F Peaki for (t, h) ∈ TFPeak

i

(4.2)

where TF represents the time horizon of a specific contract represented

in the implied curve. In order to capture the seasonal patterns in the high-

resolution representation of the price forward curve, we introduce the hourly

coefficient s(t, h) and set

P (t, h) = F (t, h) × s(t, h),

where s(.) must comply with conditions (4.1) and (4.2). To determine s(.),we adopt a bottom-up approach which captures the weather factors and

daily and seasonal components likely to influence F (.). The details of the

four steps procedure for building s(.) are outlines below:

1. We first estimate the following regression over two years of data and we

get m, i.e. daily estimates of the spot price based on purely bottom-up


components:

S(t) =m(W (t),D(t),M(t)) + ǫt,with

• S(t): historical daily spot prices;

• W (t): daily weather consisting of the mean, the highest and the

lowest temperature of the day, the precipitation, the wind-speed,

the relative humidity, and the heating and cooling indicators;

• D(t): matrix of dummy variables with 0/1 values for each different

day of the week. A single variable is used for Tuesday, Wednesday

and Thursday and holidays are treated as Sundays.

• M(t): matrix of dummy variables with 0/1 values for each differ-

ent month of the year.

2. The second step consists in an out-of-sample simulation over a time

horizon given by the time range of the implied futures curve, where

each component of W (t) is set to its daily mean value over the last

40 years to capture expected values for the season and where the daily

and monthly indicators (Dfuture(t),Mfuture(t)) are projected over the

pertinent time horizon :

S(t)future =m(Wnorm(t),Dfuture(t),Mfuture(t)).

3. In the third step, we transform the daily predicted values S(t)futureinto hourly profiles. This is done with the coefficients p(t, h) which are

build from daily means of hourly spot prices observed during the last

two years2. Note that for each day, the profile is normalized so that

2Here, we use month-day clusters. As an example, an historical profile for the Mondays


the following condition is met3:

∀t,Eh[p(t, h)] = 1.

Then, an estimate for s(.) can be obtained by setting

s(t, h) = S(t) × p(t, h).

4. In the last step, once the predicted St are hourly-based, they are cal-

ibrated or adjusted to each of the steps of the implied futures curve,

so that they fluctuate around these steps and match an arbitrage free

condition. More precisely, for each futures product (e.g. Month 2 Peak

product4), we normalize the coefficients s(t, h) to ˆsTF(t, h) in order to

fulfill the following arbitrage-free conditions:

E(t,h)∈TF[sTF(t, h)] = 1.

Recall that TF is the time horizon of a specific contract represented

in the implied futures curve, i.e the length of one of the steps in the

implied futures curve. Denoting F (t, h) the price of the futures for thedelivery hour h at day t and P (t, h) the estimated HPFC at day t and

hour h, we have:

P (t, h) = F (t, h) × ˆsTF(t, h).

of September is obtained by averaging for each hour all spot values the eight Mondays ofSeptember for the last 2 years.

3This is done by simply dividing the mean hourly-based daily profiles by their meanover the whole day.

4Starting arbitrarily in September 15, 2009, since the first monthly product concernsthe current month, the Month 2 Peak product would have a delivery period from the 1stto the 31st of October, from Monday to Friday, 8 am to 8 pm.


Some words on the arbitrage-free condition used at stage 4 in the HPFC

procedure may be necessary. This condition guarantees that the mean value

of the HPFC over the delivery period of a certain product is equal to the

product value itself. The global estimation steps of the HPFC are illustrated

on Fig. 4.2.

Validation of the HPFC Model

Since the HPFC is not a prediction of the spot price, evaluating the

quality of the fit is not straightforward. A qualitative assessment can be done

based on criteria such that the plausibility of HPCF shape, i.e. its ability to

capture seasonal patterns usually observed with high-frequency price series,

or stylized facts such that low/high prices during holiday/working periods.

In addition, the first stage of the HPFC estimation procedure provides results

that can be contrasted with facts observed in the market under study. Finally,

the HPFC can also be tested in the short run in the same spirit as the spot

price model. Indeed, for a long term contract close to its delivery date, the

non-arbitrage condition implies convergence toward the spot price and we

expect the contract value to be close to the spot price.


Figure 4.2: The statistical approach for the HPFC model


4.2 Regression Techniques

4.2.1 The Lad-lasso Regression

According to Tibshirani [1996], the ordinary least squares estimator (OLS)

suffers from two main drawbacks:

• its lack of accuracy: OLS estimates achieve low bias at the expense of

a high variance. The prediction accuracy can be improved by tolerat-

ing a small amount of bias, i.e chopping the less significant factors or

”shrinking” some coefficient toward zero.

• Interpretation: as all factors have non-zero coefficients, a smaller subset

of coefficients (with the strongest effect) may suffice to achieve a model

that performs well and provide sensible interpretation.

One of the main ideas to address these two drawbacks is to introduce

a Tikhonov regularization term [Tikhonov, 1963] that controls the bias-

variance trade-off in the regression. This procedure has originated the ridge

regression, a variant of the least squares technique intended to fight the effects

of collinearity among the regressors. More recently Tibshirani [1996] intro-

duced the Lasso (Least absolute shrinkage and selection operator), which

combines in a single procedure the nice features from ridge regression and

subset selection, i.e. a continuous process of ”shrinking coefficients” that re-

sults in setting some coefficients to zero. Later, Wang et al. [2007] suggested

a robust regression method based on a combination of the least absolute de-

viation and the lasso, giving the lad-lasso. The use of the absolute deviation

instead of the squared deviation makes the model more robust to outliers,

and the lasso guarantees a good variable selection and therefore a good bias-

variance trade-off.


Consider the usual linear model equation:

Y = Xβ + ǫ,

where β is the unknown coefficients vector, X represents the matrix of the

explanatory variables, Y is the response or explained variable and ǫ is an

error term. The lad-lasso optimization problem is given under its canonical

form by:

β(s) = argmin∑j∈[1,p] ∣βj ∣≤s

∣Y −Xβ∣,where p is the dimension of the factor matrix X and s is the regularization

coefficient which controls the amount of shrinkage that is applied to the

estimates. This parameter can be determined by cross-validation5. However,

since there are p tuning parameters, the search can be computationally heavy.

To address this issue, Wang et al. propose the BIC-type objective function :

n

∑i=1

∣yi − xiβ∣ + n p

∑j=1

λj ∣βj ∣ − log(5nλj)log(n),

which avoids the lengthy cross-validation process and sets the λj parameters

to

λj = log(n)n∣βj ∣ ,

where n is the number of sample points and λj is the tuning parameters

controlling the shrinkage. It corresponds to a lasso relatively ‘tight’ and

therefore very robust. The code used for the lad-lasso has been developed by

swissQuant Group.

5A numeric criteria measuring the forecasting power of the regression.


4.2.2 The Least-squares Support Vector Machine Re-

gression

A major development in the field of non-linear regression is the statis-

tical learning theory developed by Vapnik. This author gives a framework

based on the concept of empirical risk minimization (how to minimize the

loss in data modeling from an empirical sample), leading to the support vec-

tor machine (svm) theory. Suykens and Vandewalle [1999] propose a slightly

modified version of this theory, the least-squares svm, which introduces the

least squares term in the optimization problem. Support vector machine is

currently considered as being the state-of-the-art technique in classification

problems. Subsequent developments include the least-squares svm (ls-svm)

with a symmetric part [Espinoza et al., 2005] or the fixed-size least squares

svm [Espinoza et al., 2006]. It is important to point out that the ls-svm

estimator behaves like a black box, i.e. the non-linear transformations in-

volved do not allow to recover parameters associated with each predictor.

The philosophy here is therefore very different compared to the lad-lasso.

For the computation, the toolbox LS-SVMLab developed by Suykens et al.

[2002] was used.

Overview of the Objective Function

The key idea of the ls-svm regression is to map the regression space into a

higher dimensional space and find a linear hyperplane with the help of kernel

functions (the so-called kernel trick). We introduce briefly here the objective

function, but the optimization problem as well as the two key ideas of the

ls-svm theory are presented in the Appendix A.


Let’s consider the sample of points (xt, yt) where t ∈ [1,N], M indepen-

dent variables (∀t ∈ [1,N], xt = [xt1, ...xtM ]) entering the unknown function

f , and let

yt = f(xt) + et, t ∈ [1,N].In the absence of prior information about the structure of f(), this func-

tion can parametrized in a primal space based on ls-svm, that is

yt = ωTϕ(xt) + b + etwhere ω ∈ Ò is an unknown coefficient vector and b a bias term. The

feature map ϕ is unknown and transforms the input data into a higher di-

mensional vector. Introducing the least squares cost function, we can define

the objective function of the problem and its constraints:

minω,b,et

1

2ωTω + γ 1

2

N

∑i=1

e2t ,

with yt = ωTφ(xt) + b + et.The parameter γ is a regularization parameter.

As in the lad-lasso, we use a Tikhonov regularization. Introducing Lagrange

multipliers and using Mercer’s theorem (see Appendix A), the ls-svm theory

shows that f() can be expressed in terms of a positive-definite kernel function

K() without having to compute the feature map ϕ. The new objective

function expressed in this dual space is:

yt =N

∑t=1

αiKσ(xt, xi) + b + et,

where σ is an exogenous parameter. The kernel function can be any standard

kernel, like the polynomial or the uniform kernel. In our application, we use


the Gaussian radial basis function kernel given by

K(x, z) = exp(− ∣∣x − z∣∣22σ2

) .

The estimates of the model depend heavily on the two hyper-parameters

(γ,σ).

Determination of the Optimal Parameters (γ,σ)

The two parameters (γ,σ) control the bias-variance trade-off. If the bias

is low the fit on the training set will be very good but the variance of the

prediction might be very high. The better the fit, the higher the sensitivity

to outliers and the risk of overfitting6. Increasing the smoothness of the fits

surely reduce the variance but increases the bias. There is therefore an opti-

mal pair (γ,σ) to find.

One popular way to optimize the predictive power of the model for a given

set of parameters is cross-validation (CV). So once the kernel matrix has

been found, the leave-one-out cross-validation score can be easily obtained

for different pairs (γ,σ) defined over a grid. Another method proposed by

Keerthi et al. [2007] is to perform a gradient search on the same grid, like

a Newton’s algorithm. According to Cawley and Talbot [2007], the cross-

validation criteria prevents overfitting if there are only two hyper-parameters.

For more than two parameters, they recommend a Bayesian regularization

approach. In this paper, we use cross-validation to find the optimal pair.

6Overfitting occurs when a regression captures irrelevant features of a particular sample.

Chapter 5

Empirical Analysis

The main goal of the thesis is to apply the models presented in the pre-

vious section to Switzerland. This chapter describes the data employed in

each model as well as the results obtained with the lad-lasso and the ls-svm

approaches. Note that all data come from Bloomberg.

5.1 Spot Price Model

The equation of the spot price model is given by:

St = f(St−1, ..., St−j ,Ht,Dt,Wt).

The first decision we need to make is about the time horizon and the fre-

quency of the observations. We estimate hourly spot prices based on 30 days

data taken from August 1-31, 2009, i.e. the St vector is of size 24 × 30. We

use the Swiss spot price index traded on EPEX (the Swissix).

Regarding the weather indicator, the only hourly-based temperatures

40

Chapter 5. Empirical Analysis 41

available in Switzerland are those measured at Geneva and Zurich airports.

The mean of these two series is used as the Swiss hourly temperature indi-

cator. We build hourly temperature forecasts by combining the latter mean

with the expected daily mean, maximum and minimum temperatures from

MeteoSuisse1.

The daily dummy matrix Dt defined in Section 4.1.1 is used with the

following slight modification: Swiss national holidays are coded as Sundays.

Finally, the spot model is tested for different lag orders and its predic-

tive performance is evaluated over various forecasting horizons (1, 4 and 7

days ahead). Before presenting the empirical results, we analyze the cross-

validation procedure for the ls-svm regression model.

Cross-Validation Score for the ls-svm Model

The cross-validation score of the ls-svm fits is computed for a given grid

of plausible values for the pair (γ,σ). Fig. 5.1 presents the results for two

different specification of the spot model: the left-hand side plot refers to

an auto-regressive model with 24 lags (a full day), while the right-hand side

model has no lag. We notice that the auto-regressive component is key to get

accurate in-sample predictions. Though the global shape of the CV function

is quite similar, the CV magnitudes are much lower for the auto-regressive

1Denoting Thigh(d) and Tlow(d) the minimum and maximal predicted temperaturesfor day d, z(h) the hourly mean temperature for each hour over the 30 training days andFT (d,h) the hourly forecasts, we have:

FT (d,h) =(Thigh − Tlow)(d)

maxh z(h) −minh z(h)× (z(h) −min

hz(h)) + Tlow(d).

Note that z(h) is been first normalized by dividing each observed hourly temperature bythe mean over the 30 days.


model.

0200

400600

8001000

0200

400600

8001000

26

28

30

32

34

36

38

sigmalambda

0

200

400

600

800

1000

0

200

400

600

800

1000

195

200

205

210

215

220

sigmalambda

Figure 5.1: Grid search for a training set (a) with a one day autoregressivepart and (b) without autoregressive part

Lad-lasso vs Ls-svm In-sample Fits

We notice in Fig. 5.2 that the in-sample fit is excellent for both the lad-

lasso and ls-svm when an auto-regressive part is included in the model. The

visual inspection of the graphs also seems to indicate that increasing the lag

order does not improve spectacularly the fit.

Out-of-sample Predictive Performance

A typical out-of-sample prediction is shown in Fig. 5.3. The prediction

starts on 10/30 and ends on 11/3. November 1st and 2nd correspond to a

Saturday and a Sunday. We notice that the fit performs well in the short-run

(day one and two), but the week-end peaks are not well captured. A more ac-

curate picture of the out-of-sample predictive performance of the spot model

is given in Tables 5.1 to 5.3 for the lad-lasso and ls-svm regressions and for

different lag orders over three forecast horizons. It is important to note that

the performance indicators are mean values. In order to be able to evaluate

the predictive performance of the model more robustly, we performed 1, 3,

and 7 days-ahead predictions every three days starting from Sept 1, 2009,


(a)

(b)

(c)

Figure 5.2: In-sample fit with parameters (15,900) for three window-sizes (inday): (a) ∆ = 0, (b) ∆ = 1, (c) ∆ = 4.

until Dec 15, 2009. This gave us 33 measures for each performance indicator

whose average2 is reported in Tables 5.1 and 5.3.

We first notice on Tables 5.1 to 5.3 that ls-svm performs generally bet-

2We are aware that a confidence interval may have been provided.


Figure 5.3: A typical one week ahead forecast with parameters: (∆, γ, σ) =(1,50,500)ter than lad-lasso whatever the performance indicator. Second, when the

forecast horizon is extended, the quality of the prediction increases if the

length of the auto-regressive part is also increased. Finally, the best per-

formance in terms of correlation and mean standard deviation is given by

the ls-svm 1-day-ahead/2-lags model, the ls-svm 4-days-ahead/2-lags model

has the lowest MAPE while the lowest mean error is given by ths ls-svm

7-days-ahead/4-lags model.

28−Oct 30−Oct 01−Nov 03−Nov 05−Nov 07−Nov 09−Nov

0

2

4

6

8

10

12

14

16

18

Time

Deg

ree

Cel

sius

Weather ForecastObserved weather

Figure 5.4: The weather forecast and the observed weather (Fall 2009). By

construction, the peak values of the forecast model are the one day ahead

forecast.


Lags (∆)Performance

0 Day 2 Days 4 Days 6 Daysladlasso lssvm ladlasso lssvm ladlasso lssvm ladlasso lssvm

Correlation 0.82 0.83 0.83 0.85 0.83 0.84 0.83 0.84Mean Error (%) 21.68 21.57 20.50 18.09 20.27 18.79 21.04 20.39MAPE 5.63 5.49 5.50 4.72 5.46 4.92 5.55 5.35Mean Std Dev. (%) 19.87 19.75 19.62 18.17 19.54 18.58 19.75 19.46

Table 5.1: Forecast one day ahead




Table 5.2: Forecast four days ahead




Table 5.3: Forecast seven days ahead

Falsifiability test

As indicated in Section 4.1.1, we can apply various falsifiability tests to

check whether or not the model behaves as the true model should do. The

most powerful way to verify this hypothesis is by testing :

℘(forecasted weather) ⪰ ℘(observed weather).

We also test the two less stringent conditions described in Section 4.1.1.

Two independent quality indicators are used for these falsifiability tests: the


MAPE and the correlation coefficient. We apply the following rule:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

℘(A) ≻ ℘(B) if the two indicators of A perform better,

℘(A) ∼ ℘(B) if only one indicator of A performs better,

℘(A) ≺ ℘(B) if none performs better.

Fig. 5.4 simply compares the true or the predicted hourly-based tem-

perature series with those observed a posteriori. As expected, the observed

temperature exhibits a more wiggly pattern due to the variations induced by

the wind, precipitation and snowfall. The results of the falsifiability tests are

presented in Table 5.4 for both the auto-regressive ls-svm and the lad-lasso.

The ls-svm model passes successfully the test while the lad-lasso model per-

forms better when the observed temperature is used as predictor. We can

therefore state that the ls-svm estimator is likely to capture the true data

generation process. Regarding the lad-lasso results, we notice that the model

performs better when real temperatures are employed as predictor in lieu of

temperature forecasts. Therefore, this estimation technique seems less ade-

quate in terms of the falsifiability criteria. This seems to indicate that the

true model is non-linear.

Performance under transitions/vacation periods

All the above results were obtained with a dataset that mixes regular

patterns with more noisy ones such as those encountered during transition

phases from fall to winter or official/local holiday periods. As an example,

the regression based on September values may include a no-heating period,

but the forecasts are done for the heating period. Another example is the

forecasts for early January which are based on a training set that includes

Christmas and New Year’s Eve and erratic patterns in between.


AR-lssvm:

• ℘(Forecast one day ahead) ≻ ℘(Observed (real) weather)• ℘(Forecast one day ahead) ≻ ℘(Seasonal expectations)• ℘(Seasonal expecations) ≻ ℘(Random walk weather)

Lad-lasso:

• ℘(Forecast one day ahead) ≺ ℘(Observed weather)• ℘(Forecast one day ahead) ≻ ℘(Seasonal expectations)• ℘(Seasonal expecations) ≻ ℘(Random walk weather)

Table 5.4: Falsifiability test: results

Empirical evidence show that under these particular periods, the ls-svm

approach can perform very bad whereas the lad-lasso is more robust and

displays lower volatility. And this is indeed what we find in our data. In

Fig. 5.5, the presence of the heating days at the beginning of the simulation

set in Exhibit (a) biases the forecasts in Exhibit (b), though the lad-lasso fits

are better than the ls-svm ones. Another example is given in Fig. 5.6, where

the training set is includes Christmas holidays. The past spot price displays

a pretty irregular pattern (note the midnight peak of New Years Eve) due to

the drop in many economic activities. Again the ls-svm fits are misled while

the lad-lasso is closer to the true curve.


(a)

(b)

10/08 10/09 10/10 10/11 10/12 10/13 10/14 10/15 10/16 10/17 10/18 10/190

20

40

60

80

100

120

Pric

e [E

UR

/MW

h]

Ls−svmLad−lassoObserved

Figure 5.5: Training set under a seasonal transition: the first heating days.

5.2 Hourly Price Forward Curve Model

The data used to perform the pertinent regressions for the HPFC cover

the period Jan 2007 to Dec 2009. More precisely, given that the estimates

for the HPFC start the 1st of January 2009, we use the two previous year of

data as training set. Then the 2-years windows changes as the HPFC horizon

is increased. The variables involved in our estimation are the following:

• The Swiss, German and French spot prices, hourly-based, in Euro/MWh

from EPEX.

• The German and French futures contracts, base, peak, monthly, quar-

terly and yearly-based, traded in Euro/MWh from EEX.

• The daily-based Swiss weather indicators which comprise eight compo-

nents: the mean, highest and lowest temperature; the relative humidity

level, precipitation in mm, the wind speed in m/h and the cooling and


12/27 12/28 12/29 12/30 12/31 01/01 01/02 01/03 01/04 01/05 01/06 01/070

20

40

60

80

Pric

e [E

UR

/MW

h]

Ls−svmLad−lassoObserved

Figure 5.6: Training set under a vacation period: Christmas holidays

the heating days defined as follows:

Cooling Days =max(Mean Temp. − 18,0),

Heating Days =max(18 −Mean Temp.,0).The seasonal variations are calculated out of the historical data of the

40 past years.

• The Swiss national average reservoirs level in meters.

• The Swiss, French and German national holidays. A holiday is handled

as a Sunday in the regression (in the day indicator dummy matrix).

HPFC Estimation Steps for Switzerland

The HPFC model presented in section 4.1.2 can be applied in all markets

which possess spot and futures data. Since there is no futures contracts in

Switzerland for electricity, we need a plausible model to estimate their price.

The simplest way to do it to use information from the most influential neigh-


boring electricity markets, i.e. the French and German ones (see Fig. 2.9

page 19 or Section 2.3.2). We use the relationship between the Swiss spot

prices for electricity and the French and German spot prices to build futures

contracts for Switzerland under an arbitrage-free condition. The underlying

assumption of this approach is that the impact of the French and German

spot markets on the Swiss spot market would translate symmetrically the the

Swiss futures contract if they were to exist. As official holiday in Germany

and France also influence the spot price in Switzerland, the specific effect of

official holiday is also taken into account.

The relationship between the Swiss electricity spot prices and the German

and French electricity spot prices, all daily-based, is given by

SCH = α0 +αDESDE +αFRSFR + βCHHCH + βCHHDE + βFRHFR (5.1)

where SX and HX denote the country-specific spot prices and dummy

variables which capture holiday attributes. A lad-lasso regression is per-

formed over the period 2007/12/11 to 2009/12/10 and we get the following

results:

Variables SDE SFR HCH HDE HFR constCoefficients 0.22 0.78 1.90 0 -0.58 3.20Student’s t 45.60 175.70 5.41 ∅ 0.99 22.71

Table 5.5: Results of the regression (R2 = 0.77)

We notice in Table 5.5 that 77% of the total variance is explained by

this regression. The Swiss spot price is much more sensitive to the French


spot price than to the German spot price. This is also confirmed by the very

high t-statistic value of the associated with SFR. Moreover, the coefficient

for the German holiday variable happen to be shrunk to zero and is therefore

dropped out of the regression. The French holidays have a decreasing impact

on the Swiss spot price but note that the coefficient is not significant at the

10% level. The negative impact can be easily explained by the fact that the

French nuclear plants cannot be stopped for for short periods of time, so that

France experiences a temporary over-supply during the French holidays that

can be sold to Switzerland. The negative sign of the Swiss holidays’ coeffi-

cient was not expected and is pretty difficult to interpret given that holidays

in Switzerland are set at the canton’s level.

We further investigate the stability of coefficients of the French and Ger-

man spot price (αDE , αFR) for 2-years and 100-days periods ending from

2009/01/01 to 2009/12/10.

Jan09 Mar09 May09 Jun09 Aug09 Oct09 Nov090

0.2

0.4

0.6

0.8

1

Coe

ffici

ents

Calculation date

DE − 100 days regression

FR − 100 days regression

DE − 2 years regression

FR − 2 years regression

Figure 5.7: Evolution of the coefficients

Fig. 5.7 shows that the 2-years-based coefficients remain pretty stable and


fluctuate close to the values given in Table 5.5. In contrast, the 100-days-

based coefficients exhibit strong variations but the changes remain pretty

smooth. This indicate that the sensitivity of the Swiss spot price to the

French and German spot markets is not constant over time. Interestingly

enough, the steady drop of the 100-days-based coefficient of the French spot

from end 2008 until March 2009 coincides with the most productive period

of the German wind turbines. Then, the increase in both the French and

German 100-days-based coefficients after March 2009 correspond to a period

of pretty empty dams in Switzerland (see Fig. 2.8). Finally the Oct. 2009

change in both the German and French 100-days and 2-years spot coefficients

occurs at a time when France shut down exceptionally several power plants

for maintenance reasons.

The relationship 5.1 is then exploited to build Swiss futures by setting:

FCH(t) = αDEFDE(t) + αFRFFR(t) + α0. (5.2)

In the following, we apply the 4-stages procedure in conjunction with

equation 5.2 to build the Swiss HFPC. Recall that this procedure requires

estimating the bottom-up determinants of the spot prices with a regression

technique (see stage 1 of the procedure presented in section 4.1.2). We use the

lad-lasso and the ls-svm regressions and compare their results. Therefore, two

HPFC are estimated with the two regression techniques. In the following, we

comment some interesting results of the spot regressions without mentioning

stages 2-4 because they are of little interest.


Bottom-up Model for the Spot Prices: the Lad-lasso Variables Se-

lection

Since the lad-lasso estimator of the bottom-up spot model performs a

variable selection procedure, it is interesting to show the list of the spot

predictors with none-zero coefficients retained in the case of Switzerland,

Germany and France. Note that the estimates of the German and French

spot models are of no use for building the Swiss HPFC. Table 5.6 presents

the results. Globally, it may be surprising to see that the mean temperature

is selected in none of the spot models. In case of strong correlation between

the weather regressors, we should expect the drop of some highly correlated

determinants. However, we notice that some country specificities are well-

captured by the lad-lasso estimator. Germany, with 24 GWh produced in

2008, is one of the major producers of wind energy in the world (behind the

United States). The lad-lasso has selected wind speed as a key predictor of

the German spot prices. For Switzerland, the average level of the national

dams has been selected among many other determinants. Since France relies

heavily on nuclear power, it is harder to interpret the selection of only one

weather variable, heating. In the three cases, the week-end days Saturday

and Sunday are selected. It means the lad-lasso correctly singles these days

out.

Bottom-up Model for the Spot Prices: Lad-lasso vs Ls-svm In-

sample Fits

Before turning to the in-sample performance of the lad-lasso and ls-svm

estimators, note that the coefficients (γ,σ) for the ls-svm regression are deter-

mined with a grid search. The cross-validation function is shown in Fig. 5.8.

We notice the wide flat zone with a CV score almost constant.


Switzerland:

• Saturday, Sunday

• September, December

• Heating Days, Cool-ing Days, Reservoirslevel

France:


• June, September

• Heating Days

Germany:


• September

• Heating Days, WindSpeed

Table 5.6: Variable selection: results

100

200

300

400

0500

10001500

2000

0.041

0.042

0.043

0.044

0.045

0.046

0.047

sigma

Grid search results

lambda

Leav

e−on

e−ou

t CV

sco

re

Figure 5.8: Determination of (γ,σ).

The in-sample fits are shown on Fig. 5.9. The regression being daily-

based, we can identify the U-inverted shape weekly pattern, the top values

being the week days and the lowest values being week-ends. These patterns


are later converted to hourly profiles for the HPFC estimates.

Figure 5.9: In-sample fit

We also notice on Fig. 5.9 that the fit is better with the ls-svm regression

than with the lad-lasso one, particularly for the cooling-days and the winter

peaks. The lad-lasso produces smoother estimates. We can also distinguish

the seasonal and weekly variations for both curves, as well as the impact of

the weather predictors. Regarding the latter effect, we clearly see that the

spot prices are lower in winter 2008 (Q1-08) than in winter 2009 (Q1-09)3.

The only difference in the training set between these two seasons lies in the

weather indicators. This underlines the ability of both regression methods to

capture well the impact of the weather. A forecast based only on past trends

(monthly clustering) would result in similar levels for the Q1-08 and Q1-09

fits.

As already mentioned in the methodological section, the choice of the

parameters (γ,σ) is key to optimally balance the bias and variance of the

fits. This trade-off is illustrated in Fig. 5.10. A very small σ would give a

perfect fit and at the cost of bad out-of-sample performance. Inversely a high

σ and a low γ would lead to a curve shaped like the lad-lasso or totally flat

3Winter 2009 was colder than winter 2008.


for extreme values.

Figure 5.10: Extreme behaviour of ls-svm: (a) perfect fit achieved with a σ

of 1(γ = 1000) and (b): horizontal curve with a σ of 50,000 (γ = 10).

03−Mar−2010 19−Sep−2010 07−Apr−2011 24−Oct−2011 11−May−2012 27−Nov−2012 15−Jun−2013

0.7

0.8

0.9

1

1.1

1.2

1.3

Time

Wei

ghtin

g co

effic

ient

Output of the regression on the simulation test

LinearNon−linear

Figure 5.11: Simulation Output from the HPFC Estimation Procedure.

In the second and third stages of the HPFC estimation procedure, sim-

ulation are performed with the bottom-up model for Swiss spot prices and


a normalization is performed. Fig.5.11 shows the resulting patterns for the

lad-lasso and ls-svm regressions once the data have been normalized. The

impact of each variable selected by the lad-lasso appears very clearly in the

related curve, while the pattern for the ls-svm estimator is much more wiggly.

The jumps are linked to the seasonal indicators and the sinusoidal shape is

due to the weather indicators. The application of the arbitrage-free condi-

tions smooths the curve.

(a)

(b)

Figure 5.12: Hourly Price Forward Curve get from (a) the lad-lasso and (b)

ls-svm. The second curve looks simply more realistic. On average, these two

curves are both equals to the monthly, quarterly and yearly Futures.


HPFC curves

Fig. 5.12 shows the HPFC obtained when the lad-lasso (a) and ls-svm (b)

estimators are applied at the first stage of the estimation procedure. Let’s

first focus on the lad-lasso HPFC in (a). The first thing to notice is the sea-

sonal pattern: the prices of the long term contracts are high in winter and low

in summer. Second, we clearly see a September peak over the whole period

estimated (cluster of high peaks before the end of the years 2011, 2012, 2013

and 2014). These peaks are expected given that September corresponds to

the return period from the summer vacation and all sectors of the economy

are reactivated. Note however that this peak may be over-amplified due to

the exceptionally high prices reached in September 2009.

Regarding the ls-svm HPFC on Fig. 5.12(b), we notice first that the

global shape is similar to the one derived with the lad-lasso. The main dif-

ference comes from the fact that the ls-svm HPFC possess more pronounced

monthly variations. The Christmas drop and the cooling-days appear very

clearly. May and August prices, when neither cooling or heating are usually

required, are both very low. The September peak is also present but to a

less explicit compared to the lad-lasso results.

Based on these consideration, our impression is that the methodology

proposed to estimate the Hourly Price Forward Curve for Switzerland pro-

vides a meaningful shape which captures the most prominent stylized facts

that characterize the Swiss electricity prices.

Finally, according to the backtest results reported in Table 5.7, ls-svm

performs better than the lad-lasso. The vast majority of the quality indi-


cators for the ls-svm fits outperform the lad-lasso ones whatever the fore-

casting horizon. Note also departures of the forward prices from the spot

prices increase as the forecasting horizon increases. This is expected as the

convergence of the forward value toward the spot price is expected hold in

the short run.

Forecasting range1D ahead 5D ahead 30D ahead

ladlasso lssvm ladlasso lssvm ladlasso lssvm

Correlation 0.82 0.82 0.76 0.78 0.75 0.76

Mean Error (%) 24.19 23.35 25.90 25.02 25.73 25.10

MAPE 6.12 5.9 6.52 6.33 6.66 6.43

Mean Std Dev. (%) 17.78 17.70 22.26 21.50 25.90 25.40

Table 5.7: Backtesting results

Chapter 6

Conclusion

In this master thesis, two different models have been employed to predict

electricity prices in Switzerland, a spot price model and a model for estimat-

ing hourly prices for long term contracts (the so-called Hourly Price Forward

Curve or HPFC). The spot price model consists in a standard autoregressive

equation with bottom-up variables (weather indicators and seasonal vari-

ables). Two recent regression techniques, the least absolute deviation lasso

(lad-lasso) and the least squares support vector machine estimator (ls-svm),

have been used for the estimation. The former method is more robust than

the standard OLS estimator and includes a variable selection procedure while

the latter one is more appropriate in the presence of nonlinearities but ex-

clude structural interpretations.

Our results for the spot price model indicate that ls-svm regression out-

performs the lad-lasso estimator in terms of out-of-sample performance. This

holds over the three forecasting horizons investigated, i.e. 1, 3 and 7 days

ahead. However, the lad-lasso is more reliable when the predictions are done

over holiday periods or during transitions between seasons. Therefore, these

60

Chapter . Conclusion 61

two estimators are complementary.

Regarding the HPFC curve, we overcome the fundamental problem of the

absence of prices for standardized long term contracts (futures) in Switzer-

land by using German and French futures prices and building Swiss prices

through a statistical relationship based on spot prices. We show that the

HPFC obtained exhibits a meaningful shape. The multi-step procedure used

to build the HPFC for Switzerland also indicates that underlying relation-

ships clearly capture important stylized facts linked to the Swiss, the German

or French electricity markets. For example, the lad-lasso regression outlines

the great impact of the reservoirs levels or the wind speed on the electricity

spot prices in Switzerland and Germany respectively. However, the ls-svm

regression technique appears again as being superior to the lad-lasso for build-

ing the most reliable HPFC curve.

Clearly, further work remains to be done before using our HPFC for pric-

ing long term contracts or hedging in the Swiss electricity market. Here,

we show that in the absence of some fundamental financial instruments in a

recently liberalized electricity market, we can combine arbitrage-free condi-

tions and simple statistical relationships with other interconnected electricity

markets to build fundamental missing financial instruments.

Appendix A

Two Key Ideas of

Least-Squares Support Vector

Machine

A.1 The Maximum Margin

The support vector machine theory is historically a classification theory.

It suggest to separate two classes with an the hyperplane with the maximum

margin. The margin is the distance between the separation hyperplane and

the closer sample points. This definition implies that some points are not

useful. The closest points to the classifier hyperplane are called support

vectors. On Fig.A.1 H1 and H2 are both linear classifiers, but H1 has the

farthest margin to the closest sample points. According to the theory, H2 is

therefore a more reliable classifier. The circled points become the support

vectors. This classifier is easy to find for a 2 dimensional plane, but its

usefulness is limited if it applies only to linearly separable data. When the

sample size and the dimensions increase, the classifier is less obvious. There

62

Chapter A. Two Key Ideas of Least-Squares Support Vector Machine 63

comes the second main idea of the support vector machine theory: the Kernel

Trick.

Figure A.1: The determination process of the classifier hyperplane.

A.2 The Kernel Trick

This trick was introduced first by Aizerman in 1964 and consists in map-

ping the original non-linear observations into a higher dimensional space

where a linear classifier is available. Finding a linear classifier in this dual

space is equivalent to finding a non-linear classifier in the primal space. To

illustrate the procedure, one can consider the mapping of a power function in

a log-scale and in a regular scale. In Fig. A.2 a group of data is surrounded by

a different group. Since a linear classifier would not be able to separate these

two distinct groups, the problem is projected into a higher-dimensionality

space. We go from a two-dimensional problem to a three-dimensional prob-

lem with the non-linear mapping function

Φ ∶⎧⎪⎪⎪⎨⎪⎪⎪⎩

Ò2→ Ò3

(x1, x2)→ (x2

1,√2x1x2, x

2

2)


Figure A.2: The Kernel Trick.

A.3 General Framework

The mapping of the sample points into the higher-dimensional space is

done with symmetric and semi-positive Kernel functions with bandwidths

σ. The sample points are projected on a kernel-based algebra with the help

of Mercer’s theorem. The optimization problem is then solved in this dual

space with a Thikonov regularization (with parameter γ). It avoids to derive

the initial non-linear function ϕ. The return to the primal space is done with

the Nystrom approximation, which is a scalar product with the kernel base

vectors. The framework is illustrated in Fig. A.3.


Figure A.3: The primal-dual approach of the LS-SVM. Source:

Suykens and Vandewalle [1999]

.

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