# Long-Run Forecasting of Energy Prices: Analysing natural gas, steam coal and crude oil prices for DELTA Energy

## Long-Run Forecasting of Energy Prices: Analysing natural gas, steam coal and crude oil prices for DELTA Energy

**Long-Run Forecasting of Energy Prices: Analysing natural gas, steam coal and crude oil prices for DELTA Energy As presented at the Energy Forum ”Modelling & Measuring Energy Risk” conference, London, UK, 24th November 2008 Master Thesis in Applied Econometrics By Koen den Blanken 279550kb@student.eur.nl Erasmus University Rotterdam Commissioned by: Portfolio Analytics DELTA Energy B.V. Supervisor DELTA A. Bosschaart MSc. Coreader DELTA F. Schlichter MSc. Supervisor EUR Dr. L. Hoogerheide Coreader EUR Dr. C. Heij 16th December 2008**

3 Executive Summary DELTA Energy relies for its investment decisions on long-run forecasts and scenarios.

Fuel costs are an extremely important component of total costs of most energy generating units and reliable forecasts for fuel prices are therefore needed. Besides a thorough analysis of the fuel commodity and emission allowance prices, research needs to be conducted in the long-run availability, supply and sustainability of the various resources. Despite of the rapid development of renewable energy sources, it is expected that in the next 30 years fossil fuels remain the main portion of power generating capacity. The research question dealt by this report is therefore: to what extent is the long-run behaviour of various commodities used for energy production predictable and what are the implications for DELTA Energy’s asset investment strategy? The focus in the research is on modelling the interdependencies of the fossil fuel price series and assess investment implications.

The main goal is to develop a framework for DELTA Energy to simulate realistic long-run scenarios for natural gas, crude oil and steam coal prices. Besides an application of these price scenarios is given by comparing a coal and gas fired power plant investment.

To answer the research question quarterly price data for natural gas, crude oil and steam coal applicable to the Northwest European region over the period 1970:Q1-2008:Q3 are used. Predictability of the price level is tested by the Augmented Dickey-Fuller, Variance Ratio and KPSS random walk tests. The results are not conclusive though and rule out nor long-run mean-reversion nor the random walk hypothesis. From depletable resource theory it can be expected that long-run prices revert in the long-run to the unobservable marginal cost of extraction, which follows a trend and has continuous fluctuations in both the level and slope of that trend.

Deviations from that trend can be severe due to shortrun supply and demand shocks.

The prices series are modelled by a univariate stochastic long-run mean reverting model, a vector error correction model (VECM) and a multivariate unobserved components model (MUCM). The VECM exploits the significant cointegration relationships between the commodities while the unobserved component models use unobserved underlying (common) factors as explanatory variables. By comparing various methodologies it is possible to judge the models applicability, reducing model uncertainty and likewise improve forecasting performance.

The results indicate, especially from 1978:Q1 onward, strong cointegration between natural gas and crude oil prices and cointegration with steam coal prices as well.

Deviations from the price equilibrium however have been significant and long lasting. The univariate stochastic long-run model is not capturing the common trend effect and therefore results

4 in unsatisfactory out-of-sample performance as a result of uncertainty in the trend parameter. This is evidence in favour of a multivariate approach in energy price modelling. The VECM performs better in capturing the common trend in the long-run energy prices. A negative trend in the long-run relationship between crude oil and natural gas versus steam coal is identified, implying that in the long-run crude oil and natural gas are expected to become relatively more expensive relative to steam coal. The multivariate unobserved components model confirms these long-run relationship between the fuels without assuming cointegration of the price series a priori.

It seems to succeed in capturing the slow mean-reversion to long-run equilibrium better than the VECM.

Long-run forecasts for the main energy prices are generated by multiple commercial and government institutions. This makes it possible to compare forecasts from these institutions with the model forecasts. The fuels prices and thereby the interdependencies of the fuels prices from the models are compared to primarily IEA forecasts from the World Energy Outlook from 1994 onward. These results reveal that forecast performance of the models relative to IEA predictions is mixed. It seems that IEA does not predict much changes in the prices ratios of the fuels in most periods. The models do succeed in describing most of the fuel price interdependencies accurately.

This is evidence in favour of an approach in which the fuel prices are modelled coherently as is done in this research using multivariate models.

High economic growth and quick depletion of crude oil reserves are the most likely explanations for crude oil prices increasing more quickly than natural gas and steam coal prices. Supply of natural gas, uranium and steam coal is currently less problematic. A significant increase in energy demand from emerging markets could in the long-run result in an increasing depletion rate of steam coal reserves and likewise increase coal prices. Greenhouse gas reduction targets and emission trading on the other hand increase total coal usage costs and can potentially decrease total steam coal demand in favour of natural gas demand.

The limited availability of natural gas reserves is an issue which could lead to political instability and security problems resulting in supply disruptions and additional price volatility for this commodity. From a sustainable point of view natural gas has an absolute advantage over steam coal, which currently has higher greenhouse gas emissions per MWh power produced. Technological developments could potentially overcome these difficulties but will most likely increase fuel switching opportunities and further strengthen common trends in prices as well. Uranium poses an interesting alternative to fossil fuels as it is not associated with greenhouse gas emissions.

**The costs of uranium represent only a small part of total operation costs of a nuclear power plant. Political concerns on safety and waste disposal on the other hand make investment in nuclear power highly uncertain. Based on the price forecasts generated by the various models and the qualitative aspects four scenarios are constructed. By identifying these scenarios it is possible to describe the likely future development of the fuel prices in a stylized and practical fashion. The**

5 scenarios are constructed along two dimensions, energy price levels and trends in relative prices. High versus low crude oil prices representing general energy price levels and a stable versus a continuation of the negative trend in natural gas prices relative to steam coal prices. Fuel price uncertainty combined with technological change and high capital cost of investment pose an interesting challenge to energy companies willing to invest in new generating capacity. By constructing a portfolio of assets it is possible to diversify and hedge risks associated to this investment decision.

A physical investment in a power plant is irreversible and can only be adjusted at the margin. The investment decision should be based therefore not only on expectations, but on the associated risks as well. Fuel price risk is a major component of total risk and should therefore be considered if the investment decision is made. An important result is that fuel price variance or technological variance with respect to some technology, reduces investment in that technology. The higher fuel price variance of natural gas is therefore an important disadvantage. If total costs are considered though uncertainty will push towards flexible, less-capital intensive technologies, which is in favour of natural gas fired power plants.

If DELTA Energy could reduce fuel price uncertainty by hedging or using long-term contracts then this could potentially increase power plant value. Another important option is considering to delay the investment decision until uncertainties are reduced. The described framework in this research allows DELTA Energy to make careful investment decisions. Given the four earlier constructed scenarios and by making assumptions for the technological development of natural gas and coal fired power plants, it can from a costs perspective be concluded that a coal fired power plant will probably have over the long-run the lowest power generation costs.

Only if natural gas prices are substantially lower relative to steam coal prices or if emission costs are extremely high, natural gas fired power plants could potentially have lower generation costs. Furthermore, the additional volatility in natural gas prices could potentially outweigh the impact of emission price uncertainty. It is therefore expected that the historic merit order with coal power plants having lower marginal costs than natural gas fired power plants remains intact. The decision for DELTA Energy to invest in a specific type of power plant depends, besides costs, highly on the power price development.

The growth of renewables and energy savings can have a significant influence on long-run power price levels and volatility. It is expected that renewables will increase demand for peak-load capacity which could be generated by flexible natural gas fired power plants. A phase out of nuclear power generating capacity on the other hand could increase demand for base-load capacity as well. A further in detail analysis of the power market and the development of generating capacity is therefore needed to facilitate a strategic investment decision.

6 Preface This master thesis is the result of my efforts at DELTA Energy B.V. in Middelburg, which started in April 2008 and is the completion of my Master of Science degree in Econometrics & Management Science at the Erasmus University Rotterdam. The idea for this research came after the observation that long-run energy price assumptions in many energy studies and scenarios were highly influencing results and likewise research conclusions. To improve long-term decisions making therefore in my opinion a more in depth analysis was needed.

I would like to thank the following persons for being my supervisor: André Bosschaart at DELTA Energy and Lennart Hoogerheide at Erasmus University Rotterdam.

**Also thanks to Felix Schlichter for comments regarding my thesis and discussions about energy related topics in general. I want to thank Koen Minderhoud for giving me the opportunity to present my results at the Energy Forum conference about Measuring & Modelling Energy Risk in London in November 2008. Finally, I would also like to thank Ruben van den Berg and Pier Stapersma for feedback on this report and Robert S. Pindyck for generously providing his dataset.**

Middelburg, December 15th, 2008. Koen den Blanken

CONTENTS 7 Contents Executive Summary 3 Preface 6 Contents 7 1 Introduction 9 1.1 Context of the Research ___ 9

1.2 Research Objective and Problem Statement ___ 10

1.3 Research Questions ___ 10

1.4 Structure and Contents of the Report ___ 12

2 Data 13 3 Trend line for a Depletable Resource 17 4 Econometric Methodology 19 4.1 Random Walk Tests ___ 19

4.1.1 Augmented Dickey-Fuller test ___ 19

4.1.2 Variance Ratio Test ___ 20

4.1.3 KPSS Stationarity Test ___ 20

4.2 Stochastic Long-Run Mean-Reverting Model ___ 21

4.3 Vector Error Correction Model ___ 22

4.3.1 Johansen Cointegration Test ___ 23

4.3.2 Granger Causality ___ 24

4.4 Multivariate Unobserved-Components Model ___ 24

5 Results Forecast Models 26 5.1 Random Walk Tests ___ 26

5.2 Stochastic Long-Run Mean-Reverting Model ___ 28

5.3 Vector Error Correction Model ___ 30

5.4 Multivariate Unobserved-Components Model ___ 37

5.5 Summary ___ 41

6 Comparing Long-Run Forecasts 42 6.1 Stochastic Long-Run Mean-Reverting Model ___ 44

6.2 Vector Error Correction Model ___ 46

6.3 Multivariate Unobserved Components Model ___ 49

6.4 Conclusion and Summary .

CONTENTS 8 7 Qualitative Aspects 54 7.1 Crude Oil ___ 54

7.2 Natural Gas ___ 56

7.3 Steam Coal ___ 57

7.4 Uranium ___ 58

7.5 Emission allowances ___ 59

7.6 Summary ___ 61

8 Scenario Analysis 63 8.1 Expert Scenarios ___ 63

8.2 DELTA Energy’s Scenarios ___ 65

9 Investment under uncertainty 68 9.1 Real Options Theory ___ 68

9.2 DELTA Energy’s perspective ___ 70

9.3 Conclusion and Summary ___ 75

10 Research Conclusion 76 10.1 Problem Statement Revisited ___ 76

10.2 Research Conclusion ___ 76

10.3 Further Research ___ 78

References 80 A State Space format and the Kalman filter 84 A.1 Stochastic long-run mean-reverting model ___ 84

A.2 Multivariate unobserved components model ___ 85

B Abbreviations and Definitions 86

1 INTRODUCTION 9 1 Introduction In this chapter an introduction to the research is presented. First the context of the research is described, followed by the research objective and problem statement. Then several specific research questions to guide the research and their approach are formulated. Finally, a short description of the structure and content of the report is supplied. 1.1 Context of the Research Faced with ageing power utilities and increasing demand for electricity energy companies investigate opportunities to invest in new power plants. The choice on the type of power plant is very difficult because of uncertainty surrounding energy prices, fuel supply, regulation and investment costs.

Reducing these uncertainties would help energy companies to reduce investment risk and accelerates the development of new power generating capacity. In recent years commodity prices increased dramatically as a result of high growth expectation for the world economy and depletion of resources. Crude oil prices peaked in July 2008 at 147 USD, but came down quickly thereafter as a result of the downward adjustment of economic growth expectations. The high volatility in recent energy fuel prices clearly reveals the uncertainty in fuel prices and therefore indicates the importance for energy companies to develop a framework for long-run forecasting of energy prices.

Meanwhile, energy markets in Europe are changing rapidly as a result of market liberalization and the search for alternative sources of energy. By an EU directive designed in 1996 the liberalization of the European energy market began. In January 2005 the European Union introduced the Emissions Trading Scheme (EU ETS) which for the first time put a price on CO2 emissions for the power sector. Currently, the EU is considering the emission reduction targets for 2020. Furthermore, technological developments are changing the market with, for example, the rapid growth of the liquified natural gas (LNG) market, increased power plant efficiency and decreasing capital costs of renewables.

All these developments are further complicating the investment decision. DELTA Energy owns several large power generating facilities and targets to expand power generation to 2000 MW in 2015 by investing in nuclear, biomass, wind and solar power. For the long-run DELTA wants to expand power generation capacity even further and is continuously investigating the various investment opportunities. DELTA is already investing in a natural gas power with the construction of the Sloe plant which is expected to be completed in 2009.

1 INTRODUCTION 10 1.2 Research Objective and Problem Statement DELTA is planning to extend power generating capacity in the near future. To do so a detailed analysis of the various fuel options is needed. Besides a thorough analysis of the fuel commodity and emission allowance prices, research is conducted in the long-run availability, supply and sustainability of the various resources. The long-run in this research is defined as the period until 2040. This is based on the assumption that the life span of a power plant is approximately 30 years. The fuels considered in this research are: Natural Gas, Steam Coal, Crude Oil and Uranium.

The research is limited to these fuels because fossil fuels will, despite of the rapid development of renewable energy sources, remain the main portion of power generating capacity in the long-run as defined above. In IEA (2008a), it is expected that by 2050 only 46% of global power in the BLUE Map (most optimistic) scenario comes from renewables. Uranium is included in this research as well because it is an important alternative for coal power and DELTA Energy currently has a significant part of nuclear power in its production mix. The objective of the research presented here is to provide DELTA Energy with insight into future developments of the main fuels for power generation and to outline implications for DELTA Energy’s assets and power plant investment planning decisions.

To what extent is the long-run behaviour of the various commodities used for energy production predictable and what are the implications for DELTA Energy’s asset investment strategy? Fuel availability and prices may have significant impact for DELTA Energy due to changes in the merit order of the power plants and altering operations. These effects can be direct via higher fuel costs, for example through the purchase of coal for power plants, effecting short-run marginal costs. Or emerging indirect via higher fuel costs effecting long-run marginal cost and changing power plant investments. Overall, DELTA Energy’s future asset allocation varies with the long-run outlook for the different fuels.

1.3 Research Questions Five research questions guide the research and subsequently build a framework to handle the problem statement.

1. What are the long-run fuel price behaviours and their interdependencies? 2. What are the long-run effects of availability, supply and sustainability of the fuel types? 3. Which scenarios are realistic for the long-run fuel price developments? 4. What is the effect of fuel price uncertainty on power plant investment decisions?

1 INTRODUCTION 11 5. What are the implications for DELTA Energy’s assets and investment decisions? The first research question is answered by analysing the historic price series of the various commodities. This analysis is carried out from a time series modelling perspective.

First the individual commodities are analysed in univariate models. By modelling the price series development through time it is possible to better understand the historic developments and to identify the key drivers. Special attention is given to the stochastic long-run mean-reverting process as suggested by Pindyck (1999). To establish this, first long-run mean-reversion opposite to the random walk has to be tested. Subsequently, an analysis of the interdependencies between the series is conducted by analysing the fuel switching opportunities, cointegration tests and by modelling the price series using vector error correction models.

The second research question addresses both historic and future availability and supply limitations. Furthermore, a summarizing analysis of the sustainability of the various fuel types is presented. All these factors add insights in the risks to the use of a specific fuel and helps identifying potential structural changes in the energy market supplementing the statistical analysis. The second research question is answered using literature review and relevant expert reports. The third research question dealt with by this report is the construction of several scenarios for the future. Using the statistical models constructed in the first section and the qualitative remarks several forecasts are generated.

These forecasts are compared with other expert views and judged on realism. By doing so four plausible scenarios, which are easier to interpret, are identified. By creating these scenarios it is easier to describe the impact of long-run developments in stylized examples.

The fourth research question is focused on the investment decision in a new to build power plant of given fuel type. Using the existing literature on investment under uncertainty the impact of the long-run fuel price behaviour on investment is assessed. This research question is investigated by considering several articles from portfolio and the real option theory. The fifth research question aims at answering the problem statement. The focal point lies in the impact for DELTA Energy’s assets and future investments plans. In this part specific characteristics of the Dutch Energy market are included.

1 INTRODUCTION 12 1.4 Structure and Contents of the Report First the process of data collection and transformation is described in Section 2. Subsequently, in section 3 the economic theory behind the long-run price development of depletable resources is presented. In this way the expected behaviour and price drivers of long-run energy prices are identified. In the econometric methodology section 4, thereafter, statistics are formulated to test these hypotheses and models are proposed to describe the long-run price paths. Results and forecasts from this analysis are given in section 5. Thereby the first research question is answered in the results summary.

In section 6 out-of-sample forecasts from the models are compares with historic expert views. In this way the forecasting performance of the models can be determined. To improve current forecasts with qualitative aspects a discussion of possible future developments in the energy markets is included in section 7, answering the second research question in the summary.

Consequently, scenarios from experts are discussed and to answer the third research question several scenarios for DELTA Energy are formulated in section 8. The fourth and fifth research question are answered in section 9, where the theory and consequences for DELTA Energy of investment under uncertainty are discussed. The conclusive answer to the research question is formulated in section 10 as well as a discussion of the research and suggestions for further research. In the appendix a detailed description of the state space models and the Kalman filter is presented as well as an overview of the abbreviations and definitions used in this research.

2 DATA 13 2 Data Data from various sources are collected to answer the research questions as formulated in the introduction section. In this section a brief description of the data and its origination is given. All series are prices applicable to the Northwest European energy market. An overview of the data and sources is given in Table 1. Data on fuel prices were collected on a quarterly basis starting in the first quarter of 1970 till the third quarter of 2008. Quarterly data have the benefit over yearly data that it increases the number of observations significantly. Although quarterly data potentially induces seasonality, regressions using dummy variables for specific quarters indicate that this effect is present but negligible, see Table 2 and 3 for results on natural gas and crude oil.

An explanation for this seasonality effect is that in the winter more natural gas is used for heating as a result of the denser population of the Northern hemisphere. In the summer the demand for crude oil would be higher due to holiday traffic. Increasing the observation frequency to monthly data is not possible simply because of limited availability of high frequency data over the entire sample period. The seasonality was removed and the abbreviation SA was included to those series.

The prices series of the most traded fossil fuel series by DELTA Energy for natural gas, steam coal and crude oil are used in the statistical analysis of this research. Quarterly average prices are calculated by taking the average price over the mid-day-prices over the months with delivery in the concerning quarter. The 1 month forward price is considered to be a good approximation of the spot price for most commodities. In practise this implies that to obtain for example the Q1 price the average prices of the December, January and February contracts are used. Crude oil price data for West Taxes Intermediate (WTI) and North Sea Brent are available over the entire sample period.

The natural gas and steam coal series from the traded data are only available starting from 2000:Q1. To obtain the natural gas price series the Dutch gas hub prices, Title Transfer Facility (TTF), are used. This price is over the entire sample period strongly related to the UK National Balancing Point (NBP) and Belgium Zeebrugge (ZB) natural gas prices. For the period 2000:Q1- 2000:Q4, where data for TTF were not yet available, ZB prices were used instead. For steam coal prices the API]2 price is used which denotes coal imported from Columbia including freight and insurance charges for delivery in Amsterdam, Rotterdam or Antwerp.

Alternatively, API]4 for coal imports from South-Africa could have been used. The relationship between these two series is extremely tight and choice of either one of these two coal series is therefore not relevant for the long-run analysis. For crude oil North-Sea Brent is selected as proxy for European oil prices. To cover the period 1978:Q1-1999:Q4 quarterly data from the OECD/IEA publications Energy End-Use Prices for the sector Electricity and Import Costs and Indices by Importing Country for the Netherlands and Belgium were used. This data is available up to current quarter and has high correlation with the market traded series.

Data over the period 1970:Q1-1977:Q4 were constructed

2 DATA 14 Period Series Source Natural Gas (EUR / MWh) 1970:Q1-1977:Q4 U.S. natural gas PPI and ECN points DOE/EIA, ECN 1978:Q1-1999:Q4 Dutch import prices OECD/IEA 2000:Q1-2000:Q4 ZB market data Bloomberg 2001:Q1-2008:Q3 TTF market data Bloomberg Steam Coal (EUR / tonnes) 1970:Q1-1977:Q4 US Coal PPI and ECN points DOE/EIA, ECN 1978:Q1-1999:Q4 Dutch import prices OECD/IEA 2000:Q1-2008:Q3 API]2 market data Bloomberg Crude Oil (EUR / brl) 1970:Q1-2008:Q3 Crude North-Sea Brent Bloomberg Uranium (EUR / kgU), annual data 1970-2008 NUEXCO EV (Spot) OECD Red Book CO2 Emission (EUR / tonnes), daily data 2005-2008 EU ETS Bloomberg Producer Price Index 1970:Q1-2008:Q3 Netherlands PPI All Items OECD Table 1: Data series and sources.

by single point observations from various sources and linked by assuming high correlation with U.S. commodity prices indexes. The 1970:Q1-1977:Q4 price data are not used in the ultimate model estimation and therefore only have illustrative purposes. All prices are excluding taxes and transportation costs to the end-user.

Furthermore, for illustration annual prices of uranium were collected from the OECD Red Book, as well as prices of emission allowances which are included in section 7. Sufficient historic data of electricity prices are not available due to the non-existence of a free market for electricity in Europe until the late nineties. This is also the reason why this research focusses only on the interdependencies between the various fossil fuels without considering the long-run relationship to electricity prices directly.

After having constructed the price series for natural gas, steam coal and crude oil from the different sources, the series are deflated to 2008:Q1 chained euro’s using the OECD Netherlands PPI All Items.

Average annual producer price inflation in the Netherlands over the period 1970-2008 denoted 3.0%, Figure 1. As a next step the price series, which will be indicated by Pt, are transformed by the natural logarithm. This transformation is done to remove skewness in the returns and to prevent models from predicting negative values. The transformed series will be denoted by the lower case version pt. The real price series before transformation to log prices are visible in Figure 3. The standard deviation of the returns ∆pt over the sample period 1978:Q1-2008:Q3 are σ = 5.9%, 2.0% and 4.9% per quarter for natural gas, steam coal and crude oil respectively.

2 DATA 15 Dependent Variable: PGASLOG Method: Least Squares Sample (adjusted): 1978Q2 2007Q4 Included observations: 119 after adjustments Coefficient Std. Error t-Statistic Prob. DUMMY*PGASLOG(-1) -0.0222 0.0113 -1.966 0.052 PGASLOG(-1) 1.0113 0.0080 126.173 0.000 R-squared 0.8335 Mean dependent var 2.3692 Adjusted R-squared 0.8321 S.D. dependent var 0.3588 S.E. of regression 0.1470 Akaike info criterion -0.9797 Sum squared resid 2.5293 Schwarz criterion -0.9330 Log likelihood 60.2919 Hannan-Quinn criter. -0.9607 Durbin-Watson stat 1.7240 Table 2: Seasonality regression - Natural Gas. The dummy is 1 in Q2,Q3 (summer) and 0 in Q4,Q1 (winter).

Dependent Variable: POILLOG Method: Least Squares Sample (adjusted): 1978Q2 2007Q4 Included observations: 119 after adjustments Coefficient Std. Error t-Statistic Prob. DUMMY*POILLOG(-1) 0.0203 0.0076 -2.684 0.008 POILLOG(-1) 0.9910 0.0053 185.316 0.000 R-squared 0.9093 Mean dependent var 3.1744 Adjusted R-squared 0.9085 S.D. dependent var 0.4369 S.E. of regression 0.1322 Akaike info criterion -1.1930 Sum squared resid 2.0434 Schwarz criterion -1.1463 Log likelihood 72.9839 Hannan-Quinn criter. -1.1740 Durbin-Watson stat 1.6416 Table 3: Seasonality regression - Crude Oil. The dummy is 1 in Q2,Q3 (summer) and 0 in Q4,Q1 (winter).

2 DATA 16 Figure 1: Producer Price Index All Items - Annual percentage change. Figure 2: U.S. dollars per euro (1970:Q1-2008:Q3). Figure 3: Real 2008:Q1 chained quarterly prices in euro’s of natural gas, steam coal, crude oil and uranium (1970:Q1-2008:Q3).

3 TREND LINE FOR A DEPLETABLE RESOURCE 17 3 Trend line for a Depletable Resource Price fluctuations for depletable resources are the result of shocks to supply and demand. For a standard competitive produced good this would imply that the price in equilibrium is equal to the marginal costs of production.

Commodities though are not produced but mined. This leaves the producer with the option to store the commodity in the ground rather than extracting it. The price path should therefore exceed marginal costs and follow a specific dynamic behaviour special to depletable resources, Hotelling (1931). In this section a brief introduction to the economic theory behind price fluctuations for depletable resources is given.

Price changes can emerge from fluctuations in demand, extraction costs and reserves. These shocks affect both the log price level and its slope over time. To see this consider the basic model of a depletable resource produced in a competitive market with constant marginal costs of extraction c, Hotelling (1931). In his model the price trajectory, the change of the price P through time t is dP/dt = r(P − c), where r is the interest rate or the desired return on investment. Hence the price level is given by Pt = P0 0ert + c (1) where P0 0 = P0 −c is the net price. The gap between the price Pt and the marginal costs, c, is also known as the scarcity rent.

This difference can be seen as the compensation to the producer for selling the nonreproducible resource which leaves him indifferent between producing now or producing in the future. At lower prices the producer could decide that it is more profitable to store the resource in the mine or reservoir instead of selling it to the market. If the market price is above the desired compensation than naturally production will be increased. In principle this scarcity rent is increasing over time by the rate of interest.

If the demand function is isoelastic, of the form Qt = AP−η t then the trajectory of the production rate is given by Qt = A(c + P0 0ert )−η , (2) where η denotes the elasticity of demand, so the sensitivity of demand to price changes. Using the fact that cumulative production over the life of the resource must equal initial reserves, R0, the initial net price can be obtained by solving R0 = Z ∞ Qtdt = Z ∞ A(c + P0 0ert )−η dt . (3) This function can be evaluated numerically for any given value of η. If unitary elasticity

3 TREND LINE FOR A DEPLETABLE RESOURCE 18 of demand (η = 1) is assumed the function reduces analytically to R0 = A rc log c + P0 P0 (4) or P0 0 = c ercR0/A − 1 .

(5) The price level for any time t is now given by Pt = c + cert ercR0/A − 1 , (6) with slope dPt dt = rcert ercR0/A − 1 (7) and log price trajectory d log Pt dt = rc (ercR0/A − 1)ce−rt + c . (8) These last three equations show that an increase in either the level of extraction costs c, an upward shift in the demand curve A or depletion of the resources, lower R, leads to a rise in the price level Pt, but also to increases in the slope and the log price trajectory. Although the scarcity rent would in general induce an upward sloping trend it is possible that changes in the underlying factors distort this pattern significantly.

For most depletable resources, one would expect demand, extraction costs and reserves to fluctuate continuously and unpredictable over time. In this way it is for example possible that an unexpected discovery of large resource fields lead to a significant drop in the scarcity rent. Another example would be the recent unexpected high economic growth in emerging markets which has potentially let to much higher scarcity rents for various depletable resources.

Notice that the net price level P0 0 is itself unobservable. This implies that if the price level reverts to its long-term trend, which represents the long-run total marginal costs, that this trend is unobservable. If we want to model the price process under the belief that it reverts towards the long-run marginal cost, the marginal costs and its development through time needs to be estimated from the data. The next chapter introduces several econometric methodologies to test the properties of the historic price series and model the data for forecasting purposes. These tests and models are based on suggestions from depletable resource theory that fuel prices should follow a long-run marginal costs based trend and have continuous random fluctuations in both the level and slope of that trend.

4 ECONOMETRIC METHODOLOGY 19 4 Econometric Methodology In this section the various methodologies used in this report will be discussed. First in subsection 4.1 several tests to distinguish long-run mean reversion from the random walk are introduced. Next in subsection 4.2 the stochastic long-run mean-reverting process is handled. Definition of the vector error correction model and cointegration testing to determine the interdependencies between the variables are set-up in subsection 4.3. Subsequently, as alternative in subsection 4.4 the multivariate unobserved components model is introduced.

4.1 Random Walk Tests The purpose of random walk tests is to determine whether or not changes in time series are predictable.

If the random walk hypothesis (RWH) is valid then the concept is that the price serie pt wanders in an unpredictable manner. It is, therefore, for modelling purposes extremely important to know whether series are random walk or are predictable and consequently can be modelled using explanatory variables. The simple random walk hypothesis without deterministic trend is defined as E[pt] = E[pt+τ ] and Cov(∆pt, ∆pt+τ (9) Which implies that the expectation of future prices is equal to the current price and current prices changes are uncorrelated with future prices changes for all time periods t and lags τ.

This is equal to stating that the series is covariance non-stationary or has a unit root. Hence, the best forecast for future prices under the RWH without trend would be the price in the current period.

Alternatively, the series is covariance stationary, has no unit root and is (long-run) meanreverting. To distinguish the random walk from mean-reversion the ADF, Variance ratio and KPSS test are presented in the following subsections. Different tests are used to avoid limitations in each of the individual tests and establish a robust conclusion. 4.1.1 Augmented Dickey-Fuller test The Augmented Dickey-Fuller test looks whether lagged levels supply any information helpful in explaining changes over and above that contained in lagged changes, Hamilton (1994b). The testing procedure is based on the model ∆pt = α + βt + γpt + δ1∆pt−1 + δp∆pt−m + εt , (10) where m is the lag order of the autoregressive process.

The unit root test is performed by testing the null hypothesis γ = 0 versus the alternative hypothesis γ > 0. Critical values can be obtained by simulation from non-standard distributions and are supplied

4 ECONOMETRIC METHODOLOGY 20 by MacKinnon (1991). 4.1.2 Variance Ratio Test The Variance Ratio test is based on work by Cochrane (1988) and Campbell and Mankiw (1987). If the series follows a random walk then the variance of n-lags differences should grow linearly with n. The n-th order variance ratio is defined as V R(n) = 1 n Var[pt+n − pt] Var[pt+1 − pt] . (11) The ratio of the variance of the 1-lag difference and n-lags difference should, if the series is a random walk, be close to one. 4.1.3 KPSS Stationarity Test A different approach to test the random walk hypothesis was suggested by Kwiatkowski et al.

(1992), hereafter KPSS. The difference in their approach is that they take the mean-reversion case as the null hypothesis and the random walk as the alternative. They take the following unobserved components time-series model as starting point, where they decompose the series in a deterministic trend, a random walk and a stationary error: pt = δt + rt + εt . (12) Here rt is a random walk: rt = rt−1 + ηt , (13) where ηt is a stationary error as well. Under mild regularity assumptions, they suggest that testing the null hypothesis of covariance stationary of the time series pt is equal to whether or not σ2 η is equal to zero by employing the test statistic: LM = T−2 PT t=1 S2 t ˆ ση 2 , (14) In which, the partial sum process St = Pt i=1 ηi, where ηi is the least squares residual obtained from the regression of the de-trended time series pt.

Furthermore, ˆ ση 2 in the denominator of equation (14) is a variance estimator that is intended to relieve the asymptotic distribution of the LM statistic from nuisance parameters under the null hypothesis. As suggested by KPSS, this variance can estimated by the long-run variance estimator of the residuals ηt.

- Reversion to an unobservable long-run marginal cost, which follows a trend;
- Continuous random fluctuations in both the level and slope of that trend. Both observations are based on depletable resource theory as discussed in section 3. Pindyck’s model for annual U.S. data (1870-1996) renders quite reasonable results for crude oil, but experiences some model identification difficulties for coal and natural gas. Later Bernard et al. (2004), using the same data set, find using Monte Carlo simulations instabilities in the mean-reversion parameters for natural gas and coal, but not for crude oil. Furthermore, an extension to Pindyck’s model has been developed by Radchenko (2005) using Bayesian estimation methods and model combination, again for U.S. data. In this research the stochastic long-run mean-reverting model is imputed with European data and results function as reference to the alternative models of section 4.3 and 4.4. The stochastic long-run mean-reverting process as used by Pindyck with two state equations is defined as: pt = α1 + β1pt−1 + φ1t + φ2tt + εt (16) φ1t = α2 + γ1φ1t−1 + η1t (17) φ2t = α3 + γ2φ2t−1 + η2t , (18) where Pindyck sets α2 = α3 = 0, for identification of the steam coal model γ2 = 1 and for identification of the natural gas model he removes the entire first state equation. This model can be estimated re-writing it in state space format and applying the standard Kalman filter algorithm, Hamilton (1994a). General linear state space format is

- 4 ECONOMETRIC METHODOLOGY 22 described in detail in appendix A. In the same appendix also the state space format of the stochastic long-run mean-reverting model is given. Forecasts from the model are constructed from the quarters 1998:Q3, 2003:Q3 and 2008:Q3 to reveal the effect of changes in the parameters over time. To reflect forecast uncertainty a 90% confidence bound is used. The parameter uncertainty is simulated using 5000 simulation from the multivariate normal distribution over the parameter space. Forecast uncertainty is simulated by random drawing from the estimation residuals. 4.3 Vector Error Correction Model If the energy commodity prices have a common stochastic trend then it is maybe possible to improve the forecast for these series by using a vector error correction model (VECM). Besides by modelling the series in a multivariate model, common features can be exploited and if cointegration exists the number of parameters to be estimated remains limited. A common stochastic trend would give a more sophisticated way to describe the independencies between the various fuels. To establish this first an investigation into the existence of common stochastic trends or cointegration of the series has to carried out by testing for a unit root, Granger (1981) and Engle and Granger (1987), as presented in subsection 4.3.1. Furthermore, it is possible to include additional lags of the variables in the VECM to explain additional variation in the returns of the series. By the Granger Causality test, as described in subsection 4.3.2, the usefulness of this approach can be tested. Cointegration is a familiar technique in energy price series modelling. This can be explained from economic theory by the explanation that fuels are competitive substitutes and complements in electricity generation as well as in industrial applications. Siliverstovs et al. (2005), Villar and Jouth (2006), Hartley et al. (2007) and Benchivenga et al. (2008) are just a glimpse of recent articles which find cointegration between crude oil and natural gas prices over various time periods for U.S. and European markets. Bachmeier and Griffin (2006) investigate, using U.S. data since 1990, cointegration of coal prices with crude oil and natural gas prices, but conclude that these markets are weakly integrated and therefore adjust only slowly towards long-run equilibrium. To define the VECM, first consider the vector autoregressive (VAR) model: Yt = µ + Φ1Yt−1 + Φ2Yt−2 + ΦpYt−p + et , (19) where in this application Yt =
- pt,gas pt,coal pt,oil
- 0 and the Φ’s are 3 × 3 coefficient matrices. In error correction format, that leads to the VECM ∆1Yt = µ + Γ1∆1Yt−1 + Γp−1∆1Yt−p+1 + ΠYt−1 + et, (20)

4 ECONOMETRIC METHODOLOGY 23 where ∆1 denote first differences and where Γi = (Φ1 + Φ2 + Φi) − Im , for i = 1, (21) Π = Φ1 + Φ2 + Φp − Im , (22) where Π is the 3×3 matrix containing the information on possible cointegration relations between the elements of Yt and I is the identity matrix. Note that due to the logarithmic specification of the price series pt, the first differences ∆1Yt represent quarterly returns. Expanding parameter estimates are calculated by estimating the VECM with maximum likelihood in Matlab version R2007a using a customized version of the LeSage Toolbox1 .

Forecasts from the VECM are constructed from the quarters 1998:Q3, 2003:Q3 and 2008:Q3 to reveal the effect of changes in the parameters over time.

To reflect parameter and forecast uncertainty a 90% confidence bound is used. The parameter uncertainty is simulated using 5000 simulation from the multivariate normal distribution over the parameter space. Forecast uncertainty is simulated by random drawing from the estimation residuals. 4.3.1 Johansen Cointegration Test By first regressing equation (20) by row wise OLS, the residuals rt,1, rt,2 and rt,3 and the 3 × 3 residual product matrices Sij = (1/n) n X t=1 rt,irt,j , for i, j = 1, 2, 3 (23) are created. The Johansen (1988) maximum likelihood cointegration test focusses on the rank of Π.

If it is possible to construct the rows of Π by a linear combination of the other rows the rank of the matrix is reduced. If the rank of Π = 0, all series are nonstationary and have no common stochastic trend, but there are as many stochastic trends as variables. If on the other hand the matrix has full rank then all series are stationary and have no common stochastic trend as well. If a common stochastic trend or common trends are present the rank is between these two cases. The rank r is tested by the Trace likelihood ratio test, under the null hypothesis of the existence of at most r unit roots and is given by Trace = −n m X i=r+1 log(1 − λ̂i) , (24) where λi represent the i-th eigenvalue of the matrix S.

A value of the test above the critical value implies the rejection of the null hypothesis.

1 The Econometrics Toolbox by James P. LeSage is freely available at www.spatial-econometrics.com.

4 ECONOMETRIC METHODOLOGY 24 4.3.2 Granger Causality Granger causality implies that a variable contains leading information on other variables. This does not mean that the variable actually drives the underlying data generating process and actually causes the other variable to change, but only in the statistical sense. It contains information in addition to the information in the series own history, which is potentially useful for forecasting purposes when specifying the correct lag structure, Granger (1969).

Granger causality can be tested for normal-distributed, stationary series by regressions of the form y1t = α + β11y1t−1 + β21y2t−1 + βp1y1t−p + βp2y2t−p, for lags p = 1, 2 , T, (25) where y2 is the variable that possibly Granger causes y1. Due to the near-random walk properties of the natural gas, crude oil and steam coal series in our sample the data are transformed to first differences to construct stationarity. Stationarity is required to perform the standard Granger causality test. Subsequently, the Granger causality test is performed on all six possible combinations of the three return series.

4.4 Multivariate Unobserved-Components Model As alternative to the standard cointegration analysis a multivariate unobserved components model (MUCM) can be proposed. This model can be seen as a multivariate extension of the stochastic long-run mean reverting model as defined in section 4.2. An important drawback of standard cointegration analysis as discussed in section 4.3 is that all variables constituting the stationary equilibrium relation should be included in the analysis. Additionally, Engel (2000) shows that standard cointegration tests are biased toward accepting the cointegration hypothesis in the presence of omitted integrated variables.

Furthermore as Morley (2007) suggests the speed of adjustment in terms of restoring their long-run equilibrium relationship is the same for all variables. That means speed of adjustment measured as the half-life response of a variable to a shock. So although for example the natural gas price adjusts in terms of magnitude more than crude oil the speed of adjustment of both is equal in the VECM. This limits the ability of the VECM to discriminate between alternative price behaviours.

In Section 3 it is argued that commodity prices are determined by a combination of shocks in demand, changing extraction costs and depletion of the resource. The energy prices are in the long-run determined by changes in the underlying long-run marginal cost factors. It can be expected that changes in the marginal costs for crude oil will correspond with

4 ECONOMETRIC METHODOLOGY 25 similar hikes in marginal costs for natural gas and steam coal. For example, cost increases in extraction costs as a result of rising building and labor costs, resource increases as a result of intensified search by energy companies or in case of demand shocks by increased growth of the global economy.

In the unobserved components framework these omitted integrated variables can be filtered from the long-run relations between the price series. For the set up of the multivariate unobserved components model I follow the methodology in Morley (2007). He expands the work of Stock and Watson (1988) who suggest decomposing time series into common stochastic trends (random walks) and idiosyncratic stationary components. To apply this model to the fuel price series I add a level shift α and the possibility to have a deterministic drift in the common stochastic trend relationship δ. The multivariate unobserved components model with p3 as base series is than defined as p1,t = β1p1,t−1 + α1 + γ1τt + δ1t + ε1,t (26) p2,t = β2p2,t−1 + α2 + γ2τt + δ2t + ε2,t (27) p3,t = β3p3,t−1 + τt + ε3,t , (28) where τt is the common stochastic trend, βipi the idiosyncratic stationary component and εi ∼ N(0, σ2 εi ).

The common trend follows an unobservable random walk and is allowed to have deterministic drift µ to capture positive growth in long-run marginal costs: τt = µ + τt−1 + ηt , (29) where ηt ∼ N(0, σ2 η). To allow for additional dynamics of the model, as explained in Morley (2007), I facilitate the innovations in the MUCM to be correlated: ρηεi = corr(ηt, εi,t) for i = 1, 2, 3 (30) ρεiεj = corr(εi,t, εj,t) for i, j = 1, 2, 3 and i 6= j . (31) For estimation, the model is casted into state space format as defined in appendix A. Subsequently the Kalman filter and maximum likelihood estimation is applied to obtain expanding parameter and state variable estimates.

The Matlab optimisation routine for the Kalman filter is available from the author upon request. Forecasts from the MUCM are constructed from the quarters 1999:Q4, 2004:Q4 and 2008:Q3 to reveal the effect of changes in the parameters over time. To reflect parameter and forecast uncertainty a 90% confidence bound is used in which the parameter uncertainty is simulated using a multivariate normal distribution over the parameter space.