(ii) Transportation and storage. (iii) Refining and processing. (iv) Distribution and sales. The entire energy industry revolves around these four… [603884]

1. Introduction 2
(ii) Transportation and storage.
(iii) Refining and processing.
(iv) Distribution and sales.
The entire energy industry revolves around these four functions. Each of these functions
inherently contains complex risks that shape business strategies to mitigate these risks
and their effects. The first two steps in the energy value chain, exploration and production
and transportation and storage, are generally referred to as upstream, and the last two
steps, refining and processing and distribution and sales, as downstream (GARP, 2009).
Energy markets around the world are undergoing rapid deregulation, leading to more
competition, increased volatility in energy prices, and exposing participants to poten-
tially much greater risks. Before then, prices were set by regulators, who were mainly
governments. Energy prices were relatively stable, but consumers had to pay high premia
for costs of inefficiencies such as complex cross-subsidies from areas with surpluses to
areas with shortages or inefficient technology (Clewlow and Strickland, 2000).
As a result of deregulation, a free market with more competitive prices arose that revealed
that energy prices are the most volatile among all commodities. Deregulation impacts
both consumers and producers and has lead to a heightened awareness of the need for risk
management and the use of derivatives for controlling exposure to energy prices (Dooley,
1998; Winston, 1993). However, this is not the only source of the development of risk
management products. Investment banks are also being drawn into the area as they look
for new markets in which to operate. There is also an increasing number of power mar-
keters entering the market. This combination of the two different sides of the market,
along with the sheer size of the market at the sales level, makes energy derivatives one
of the fastest growing of all derivatives markets. For many market participants, energy
derivatives appear to be a new phenomenon.

1. Introduction 3
Due to deregulation, the expansion of the oil market has continually grown and has now
become the world’s biggest commodity market. This market has developed from a primar-
ily physical product activity into a sophisticated financial market. Over the last decade,
crude oil markets have matured greatly, and their range and depth could allow a wide
range of participants, such as crude oil producers, crude oil physical traders, and refining
and oil companies, to hedge oil price risk. Risk in the crude oil commodity market is
likely to occur due to unexpected jumps in global oil demand, a decrease in the capacity
of crude oil production and refinery capacity, petroleum reserve policy, Organization of
Petroleum Exporting Countries (OPEC) share capacity and policy, major regional and
global economic crises, and geopolitical risks.
Crude oil is a fossil fuel made up of a mixture of hydrocarbons that formed from plants
and animals that lived millions of years ago. It exists in liquid form in underground pools
or reservoirs, in tiny spaces within sedimentary rocks, and near the surface in tar (or
oil) sands. Petroleum products are fuels made from crude oil and other hydrocarbons
contained in natural gas. Petroleum products can also be made from coal. After it is
removed from the ground, it is sent to a refinery where different parts of the crude oil are
separated into usable petroleum products. These petroleum products include gasoline,
distillates such as diesel fuel and heating oil, jet fuel, petrochemical feed-stocks, waxes,
lubricating oils, and asphalt.
Oil refining is the process that cracks the crude oil into its constituent products. The
refiner’s profits are tied directly to the crack spread (the difference between the refiner’s
revenue and cost). Refiners are therefore exposed to both sides of the market simulta-
neously and hence they are exposed to greater risks compared to oil producers and fuel
consumers. The “crack spread” is the difference between crude oil prices and whole sale
petroleum product prices (mainly gasoline and distillate fuels). Refiners are caught be-
tween two markets: the raw materials they have to purchase and the finished products
they offer for sale. Factors which contribute to volatility are demand and supply, sea-

1. Introduction 4
sonality, foreign exchange risk, decrease in capacity of crude oil production and refinery
capacity, petroleum reserve policy, OPEC share capacity and policy, major regional and
global economic crisis, political and geopolitical issues, environmental regulations and
stringent product specifications among others (Chang, McAleer, and Tansuchat, 2011).
Figure 1.1 displays the products of the refining process (EIA, 2015).
Figure 1.1: Products of the refining process
Gasoline is mainly used as an engine fuel in vehicles. Most gasoline-fuelled vehicles will
operate on regular gasoline, which is usually the least expensive grade. Diesel fuel and
heating oil are closely related products called distillates. Diesel fuel is used in diesel en-
gines found in most trucks, trains, buses, boats, and farm and construction vehicles and
heating oil is used for heating homes.
Inthissectionwegiveabriefhistoryofenergymarketsandhowtheyhaveevolved; discuss
the risks faced in these markets and how they can be managed. We introduce futures and
crack spread trading; we give a brief insight into hedging against energy market risks using
crack spreads. This section also gives the problem statement, objectives and significance

1. Introduction 5
of the study.
1.1 Futures and Forwards
A futures contract is a contract that calls for payment of a certain asset at a certain price
to be delivered at a certain date in the future. It’s essentially a way to “lock in" a price
now and potentially benefit later if the price rises. Significant changes in demand and sup-
ply, pricing and expansion of global markets have affected most of the world’s commodity
markets over the last 30 years, and especially the energy markets. Dynamic economic
patterns, war, international politics as well as structural changes within the energy sector
have introduced considerable uncertainty with regard to the future direction of market
conditions (Dorsman, Simpson, and Westerman, 2013). This has led to an increase in
market volatility, which has in turn created a need for an effective means to hedge the
risk of adverse price exposures. The main risk management instruments available to the
participants in the energy markets are versatile futures and options contracts listed in
various exchanges around the world.
A futures contract is a commitment to make or accept delivery of a specified quantity and
quality of a commodity during a specific future month at a price agreed upon at the time
the commitment is made. The buyer, who is the long party agrees to take delivery of
the underlying commodity, while the seller, who is the short party agrees to make deliv-
ery of the underlying commodity. Futures are standardized, in terms of the future date,
amount traded, etc. and can be re-traded through time on a futures exchange. Forward
contracts are also agreements to transact on fixed terms at a future date, but these are
direct agreements between two parties, made over the counter.
Although forwards and futures are similar contracts involving an agreement to buy or sell
on a certain date for a certain price, important differences exist. Firstly, futures are ex-
change standardized contracts meaning that they are exchange traded, whereas forward
contracts trade between individual institutions (Over-the-counter). Secondly, the cash

1. Introduction 6
flows of the two contracts occur at different times – futures are daily marked to market
with cash-flows passing between the long and the short position to reflect the daily futures
price change, whereas forwards are settled once at maturity. Despite these differences, if
future interest rates are known with certainty then futures and forwards can be treated
as the same for pricing purposes.
There are two sides to every forward contract. The party who agrees to buy the asset
is said to hold a long forward position, whilst the seller is said to hold a short forward
position. At the maturity of the contract (delivery date) the short party delivers the
asset to the long party in return for the cash amount agreed in the contract (the delivery
price). Most market participants choose to trade their physical supplies through the
existing channels and use futures or options to manage price risk and liquidate their
positions before delivery. Option contracts are the second most important instruments in
derivative markets after futures (Clewlow and Strickland, 2000).
1.2 Hedging using futures
A hedge is a financial position you set up in your investments to offset a big loss (or
gain) that you could incur in your portfolio. One of the most important and practical
applications of Futures is ‘Hedging’. In the event of any adverse movements in the market,
hedging is a simple work around to protect a trader’s trading positions from making a
loss. Futures can be used to make profit.
A futures contract is a commitment to make or take delivery of a specific quantity and
quality of a given commodity at a specific delivery location and time in the future. All
terms of the contract are standardized except for the price, which is discovered via the
supply (offers) and the demand (bids). All contracts are ultimately settled either through
liquidation by an offsetting transaction (a purchase after an initial sale or a sale after an
initial purchase) or by delivery of the actual physical commodity.

1. Introduction 7
Hedging is based on the principle that spot and futures market prices tend to move to-
gether. This movement is not necessarily identical, but it usually is close enough that it
is possible to minimise the risk of a loss in the spot market by taking an opposite position
in the futures market. Taking opposite positions allows losses in one market to be offset
by gains in the other. In this manner, the hedger is able to establish a price level for a
spot transaction that may not actually take place for several months.
A perfect hedge is a strategy that completely eliminates the risk associated with a future
market commitment. To establish a perfect hedge, the trader matches the holding period
to the futures expiration date, and the physical characteristics of the commodity to be
hedged must exactly match the commodity underlying the futures contract. If either of
these features are missing then a perfect hedge is not possible. In such circumstances risk
can still be reduced but not eliminated. Using futures contracts, a trader used short and
long hedges replace price risk with basis risk.
1.2.1 Spread trading
Spread trading involves the simultaneous purchase of a financial instrument and sale of
another instrument with the objective of profiting from the movement of the spread be-
tween the two prices, and not from the movement of the absolute value of the two prices
(Kanamura, Svetlozar T Rachev, and Frank J Fabozzi, 2011).
In the energy markets, beside the temporal spread traders, who try to take advantage of
the differences in prices of the same commodity at two different dates in the future, and
the locational spread traders, who try to hedge transportation/transmission risk exposure
from futures contracts on the same commodity with physical deliveries at two different
locations, most of the spread traders deal with at least two different physical commodities.
In the energy markets spreads are typically used as a way to quantify the cost of produc-

1. Introduction 8
tion of refined products from the complex of raw material used to produce them. The
most frequently quoted spread options are the crack spread options and the spark spread
options. Crack spreads are often called paper refineries while spark spreads are sometimes
called paper plants.
1.2.2 Crack spreads
Oil refining is the process that cracks the crude oil into its constituent products. The
refiner’s profits are tied directly to the crack spread (the difference between the refiner’s
revenue and cost). the “crack spread” therefore, is the difference between crude oil prices
and whole sale petroleum product prices (mainly gasoline and distillate fuels). Refiners
are caught between two markets: the raw materials they have to purchase and the fin-
ished products they offer for sale. It is normal for the processes in these markets to be
independently subject to various levels of supply, demand, transportation costs and other
factors. Refiners are therefore exposed to greater risks compared to oil producers and fuel
consumers, especially when crude oil prices rise while refined product prices stay static
or even drop. A refinery therefore straddles the raw materials it buys and the finished
products it sells.
The crack spread consists of a position comprising the simultaneous purchase and sale
of crude oil and the petroleum products futures contracts. This permits the refiner to
lock in any differentials between refinery input and refinery output prices A crack spread
seller (typically the refiner) would buy crude oil contracts and sell gasoline and heating
oil contracts. The refiner has to observe all these aspects on two sides of the market and
on more than one commodity namely the crude on the costs side and the refined products
on the revenue side. In crack spread hedging, all these are put into consideration simul-
taneously and a profit maximizing portfolio for the refiner is sought in terms of taking
various positions (long and short) in the market. Other than refining and oil companies,
other players in the oil market include crude oil producers and crude or physical traders.

1. Introduction 9
They are all interested in hedging oil price risk.
1.3 Statement of the Problem
Over the past 20-30 years oil has become the biggest commodity market in the world
evolving from a primarily physical activity into a sophisticated financial market with
trading horizons extending to over 10 years forward (Chang, McAleer, and Tansuchat,
2010; Fleming and Ostdiek, 1999). In the process, it has attracted a wide range of par-
ticipants including investment banks, asset managers for mutual funds, pension funds or
endowments, insurance companies, hedge funds as well as the traditional oil majors, the
independents and the physical oil traders.
In parallel, the volatility of the oil price and the hedging needs for industry participants
triggered the development of a financial sphere of derivative contracts (Futures, forwards,
swaps and options) which now dominate the process of worldwide oil price formation.
It is important to understand that oil is physically traded twice: first, as a refinery feed-
stock, and, second, as a finished product. Even if crude and refined products have rather
different characteristics, they are inextricably linked by the technology and economics of
refining. Although product prices may, and do, fluctuate widely relative to each other,
they must together be related to crude oil price because refineries will not continue to op-
eratelongonnegativemarginsandcompetitionwillsetaceilingtohighmarginsinreturn.
This study will be looking at possible ways of coming up with a good instrument to
hedge against exposure to adverse price changes of crude oil and the refined petroleum
products by using the spread differentials while at the same time also incorporating the
effects of time varying volatility as one of the key determinants of the dynamics in the oil
industry.

1. Introduction 10
1.4 Objectives of the Study
In many energy markets, the concept of being able to perfectly replicate contracts by
continuously trading the underlying asset is unrealistic. For example spot electricity
cannot be easily stored, and therefore continuously adjusting the position is not possible.
Many energy derivatives actually depend on the futures prices rather than the spot price
and futures can be used to replicate contract positions allowing the application of the risk
neutral pricing approach. This study will seek to find optimal dynamic hedge ratios for
crack spreads in the oil markets under the GARCH model framework, by incorporating
the effects of volatility spill overs in the crude, gasoline and distillate fuel markets.
1.4.1 Specific objectives
(i) Analysing the trends and patterns in energy markets and investigating the stylised
features of the financial time series.
(ii) Investigate co-integration and multivariate heteroscedasticity in energy markets so
as to capture volatility spill overs or transmissions in the markets.
(iii) Incorporating volatility spill over effects in the futures and spot markets market
using GARCH models.
(iv) Estimating static and dynamic hedge ratios that can account for spillover effects in
energies and compare their hedge effectiveness.
1.5 Significance of study
Oil production and consumption is one of the greatest drivers of most economies the world
over. Oil has virtually no substitute as a transportation fuel and there are no immediate
prospects for that to happen at current prices. The need for a reliable source of energy
is key for any economy as is evidenced by the continuous explorations for oil prospects
by most countries. This seems to be the only path for industrialization. Energy Risk
management is therefore an emerging area and strategies to hedge these risks would also

1. Introduction 11
be worth considering. Due to the heavy reliance of all economies around the world, on
oil products as the main sources of energy, ability to carefully harness this energy, and
handle the risks that come with this by turning them into opportunities will steer growth
in any economy.
This study is also intended to establish the better hedging strategy for international trade
affected by oil price fluctuations, demand and supply, and transportation among other
factors. For an investor wishing to hedge their market risk exposure, crack spread trading
in oil markets would be an interesting alternative.

2. LITERATURE REVIEW
For any economy, investment in energy and management of risks in the energy markets
is of paramount importance. Salman and Atya (2014) in their paper on the role of finan-
cial development and energy consumption on economic growth, state the importance of
the existing causal relationship between energy consumption, financial development and
economic growth, to policy makers as it directly affects production, economic growth and
development. According to Dorsman, Simpson, and Westerman (2013), the oil shocks
in the 1970’s demonstrated how vulnerable the world’s economy was to energy supply
interruptions and price volatility, and the recent increases in energy prices, the steady
rise in global energy demand, instability in energy producing regions and the threat of
terrorist strikes against energy infrastructure have significantly led to a growing concern
over energy security.
ARMA models provide insight into many areas of time series forecasting. An ARMA
model entails order identification, parameter estimation and forecasting (Box and Jenk-
ins, 1976). In ARMA modeling, the time series must be linear and stationary (Chan,
2011), though real life time series data are non-linear and non-stationary in nature. If
the present can be plausibly modelled in terms of only the past values of the independent
inputs, then, forecasting will be possible (Shumway and Stoffer, 2010). They are most
widely used for the prediction of second-order stationary processes (Chan, 2011; Box and
Jenkins, 1976; William and Wei, 1990; Montgomery, Jennings, and Kulahci, 2011; Taylor,
2007).
The ARCH model was introduced by Robert F. Engle (1982) and later Bollerslev (1986)
gave a useful generalization of ARCH, the GARCH model. The family of GARCH volatil-

2. Literature Review 13
ity models has become an important tool kit in empirical asset pricing and financial risk
management. Following the work of Robert F. Engle (1982) and Bollerslev (1986), a volu-
minous econometric literature has developed on volatility estimation and forecasting. Im-
portant papers in empirical finance that apply GARCH models include Bollerslev (1986),
J. Campbell and Hentschel (1992), French, Schwert, and Stambaugh (1987), Glosten, Ja-
gannathan, and Runkle (1993), Pagan and Schwert (1990), and Schwert (1989) among
others.
Since the introduction of the ARCH model by Robert F. Engle (1982) and its generaliza-
tion to GARCH by Bollerslev (1986), this family of models have been extended in various
directions in order to increase the flexibility of the original model. Herwartz and Reimers
(2002) analysed daily log returns of exchange rates. to account for volatility clustering,
they fit a GARCH(1,1)-model with leptokurtic innovations. They found that this model
to accurately described the empirical distribution of foreign exchange returns.
Motivated equally by the success of GARCH models in fitting asset log returns and by the
failure of deterministic volatility models in fitting option prices, researchers have extended
the GARCH model into the domain of option valuation.
With reference to the application of GARCH in the derivatives pricing area, J. C. Duan
(1995) was the first to develop a risk-neutral model within the GARCH framework. He
uses a model called non-linear GARCH in mean (NGARCH-M) developed by Robert F.
Engle and Ng (1993). He characterized the transition between the actual and the risk-
neutral probability distributions if the dynamics of the underlying equity price is given by
a GARCH process, and thus established the foundation for the valuation of options under
GARCH.Inhismodel, therisk-neutraloptionpricedependsuponaleverageriskpremium
parameter, and the variance is negatively correlated with the past equity returns. Such a
negative correlation gives rise to a negative skewness in a risk-neutral distribution. The
leverage risk premium parameter is estimated directly from the empirical data since it is

2. Literature Review 14
part of the conditional expected return of the underlying equity. Hence, Duan’s model
generates option prices that are consistent with the observed volatility skew. Another fea-
ture of the Duan’s model (and all GARCH models) is that it is non-Markovian in nature
and can explain some of the systematic biases associated with the Black-Sholes model.
In particular, the option price depends on the information set generated by the past and
current prices of the underlying equity and its past, current and one-step-ahead values of
the conditional variance process.
J.-C. Duan and Pliska (2004) developed the co-integration option pricing model. Their
approach is based on a discrete-time model where the assets are co-integrated process
with multivariate GARCH (MGARCH) volatility. They showed that the co-integration
variable enters the pricing model only when volatility is time-varying. They performed a
numerical study for European-style spread options, comparing the option prices obtained
with their model to those obtained with a constant volatility model and with those ob-
tained with an MGARCH model without co-integration.
Despite GARCH models being convenient parametric models that capture leptokursis and
volatility clustering, Bollerslev (1987) and Baillie and Bollerslev (1990), and others note
that GARCH models with conditional normality of errors generally fail to capture the
leptokursis evident in asset returns. Benavides (2004) used the univariate GARCH, the
BEKK model (named after Baba, Engle, Kraft and Kroner, Robert F. Engle and Kroner
(1995)), an option implied and a composite forecast model investigate to the accuracy of
volatilityforecastsforcornandwheatfuturesandfoundthattheoptionimpliedmodelwas
superior to the historical models and the composite forecast model was the most accurate.
Cecchetti, Cumby, and Figlewski, 1988; Chang, McAleer, and Tansuchat, 2011; S.-S.
Chen, C.-f. Lee, and Shrestha, 2003; Alizadeh, Kavussanos, and Menachof, 2004; Haigh
and Holt, 2002; De Jong, De Roon, and Veld, 1997 among others discuss the concept of
optimal hedging, optimal hedge ratios and hedging efficiency in futures markets. Haigh

2. Literature Review 15
and Holt (2002) allowed for time-varying volatility spillovers when accounting for the op-
timal hedge ratio and noted that from a risk management perspective, there had been no
attempt to combine the estimation of time-varying hedge ratios while allowing for time-
varying co-variability between related energy prices. They therefore built upon previous
work to examine, for the first time, the effectiveness of using crude oil, heating oil and
unleaded gasoline futures contracts in helping reduce price uncertainty for energy-traders.
In particular, to directly account for the time-varying volatility spillovers between the re-
lated markets they employed a Multivariate GARCH (MGARCH)model that also allowed
for the direct incorporation of the time to maturity effect often found in futures markets.
Alizadeh, Kavussanos, and Menachof (2004) also compared the effectiveness of constant
versus time-varying hedge ratios and noted that differences in hedging effectiveness across
regional markets are attributed to varying regional supply and demand factors in each
market.
Haigh and Holt (2002) note that substantial improvements over all other hedging pro-
cedures may be obtained if informational linkages among energy markets are directly
accounted for in a time varying manner by employing MGARCH model that incorporates
the maturity effect.
Byun and Min (2011) investigated the disparity between physical and risk neutral con-
ditional volatilities. They allowed the risk-neutral one-day ahead conditional volatility
to be different from physical one-day-ahead conditional volatility so as to describe more
accurately, the risk neutral dynamics of asset returns and volatility implied by a cross
section of prices. Their results showed that using one-day-ahead conditional volatility
estimated under the physical measure was not indicative of an advantage of the discrete
GARCH option pricing model over continuous-time stochastic volatility models unless
risk neutral one-day-ahead volatility is adjusted by non-normality and risk premium.
Alternatively, Beck(2001)arguesthatgiventhatcommoditiescanbestored, random-walk

2. Literature Review 16
specifications that allow for conditional heteroscedasticity are compatible with rational
expectationsandriskaversion. Accordingly, shesuggeststhatARCH-in-meantypeframe-
works present a good modelling choice for these commodity prices.
Carmona and Durrleman (2003) surveyed the theoretical and computational problems
associated with pricing and hedging financial instruments for which closed form formulae
cannot be derived such as spread options. They also looked at their role as speculation
devices and risk management tools. In considering risks associated with below-target
returns, Fishburn (1977) considers a mean-risk dominance model where a threshold is
measured by a probability weighted function of deviations below a specified target return
and later De Jong, De Roon, and Veld (1997) uses this to study out of sample hedging
effectiveness for currency futures.

3. TRENDS AND PATTERNS IN ENERGY MARKETS
Most financial data occur sequentially through time forming a time series which is a time-
oriented or chronological sequence of observations on a variable of interest. The first step
in building dynamic econometric models entails a detailed analysis of the characteristics
of the individual time series variables involved so that properties of the individual series
are taken into account when modelling the data generating process (DGP) of a system of
potentially related variables (Lütkepohl and Krätzig, 2004). Econometricians are tasked
with developing reasonably simple models that can predict future values, interpret and
test hypothesis of economic series (Enders, 2008). There are two methods in time se-
ries analysis, namely: frequency-domain which refers to the analysis of mathematical
functions or signals with respect to frequency, and time-domain which is the analysis of
mathematical functions, physical signals or time series of economic or environmental data,
with respect to time. This research is mainly concerned with time-domain.
In this work, we regard an observed price series, p1,p2,…,pT,as a particular realisation
of a stochastic process {pt}. A realization of stochastic process is just a sample path of
Treal numbers which could for example be a stock price process; and if history took a
different course, we would have observed a totally different sample path. A stochastic
process can be described as a model which describes the probability structure of a se-
quence of random data, the simplest being specified as pti.i.d.∼Dfor some distribution D.
The standard time series analysis rests on important concepts such as stationarity, er-
godicity, autocorrelation, white noise, innovation, and on a central family of models, the
ARMA models (Francq and Zakoian, 2011). In addition to these, for the analysis of finan-
cial time series, we introduce the concept of volatility, which is pivotal in finance. Along

3. Trends and Patterns in Energy Markets 18
side volatility we also consider the main stylized facts which include unpredictability of
returns, volatility clustering and hence predictability of squared returns, leptokurticity
of the marginal distributions, asymmetries, etc. concerning financial time series (Francq
and Zakoian, 2011; Tsay, 2010; Zivot and Wang, 2007).
This chapter is structured into three main sections. Section 3.1 introduces time series
concepts and important models together with extensions to model financial time series
data. Section 3.2 discusses financial time series concepts and models and in section 3.3,
an exploratory analysis of the US energy markets is carried out based on the models
discussed in section 3.1.
3.1 Time Series Concepts and Models
A time series is a stochastic process and can be defined as follows:
Definition 3.1.1 (Time series model) .A time series model for the observed data {pt}
is a specification of the joint distributions (or possibly only the means and covariances)
of a sequence of random variables {Pt}of which{pt}is postulated to be a realization
(Brockwell, 2002).
Figure 3.1 shows a time series plot of the daily prices of Cushing OK WTI crude oil
futures for contract 1 for the period 03/01/2006 to 22/05/1015. In this section, we discuss
the basic concepts of time series analysis which are the building blocks for complex or
advanced time series models.
3.1.1 Stationarity
Thestationarityofatimeseriesrelatestoitsstatisticalpropertiesintime. Oneofthemost
basic requirements of any statistical analysis of data is the existence of some statistical
properties of the data which remain stable over time. In a more strict sense, a stationary
time series exhibits similar “statistical behaviour” in time and this is often characterized
byaconstantprobabilitydistributionintime(Montgomery, Jennings, andKulahci, 2011).

3. Trends and Patterns in Energy Markets 19
Figure 3.1: Cushing OK WTI crude oil futures contract 1 daily prices
Stationarity exists in two forms, strict stationarity and covariance or weak stationarity.
Definition 3.1.2 (Strict Stationarity) .A stochastic process {pt}is strictly stationary if
it’s properties are unaffected by a change in the time origin; that is; if the joint probability
distribution associated with tobservations p1,p2,…,pt,at times 1,2,…,tis the same as
that associated with tobservations pk+1,pk+2,…,pk+t,at timesk+ 1,k+ 2,…,k +tfor
k∈Z
The joint distribution of any such set of observations must be unaffected by shifting all
the times of observation forward or backwards by any integer amount k. The only factor
that should affect the relationship between the two sets of observations is the gap between
them.
For a strictly stationary process, since the distribution function is the same for all t, the
mean function µt=µ, a constant provided E[pt]<∞, the variance function σ2
t=σ2
for alltprovidedE[p2
t]<∞, the covariance function γ(i,j) =γ(i+k,j+k)and the
correlation function ρ(i,j) =ρ(i+k,j+k). Also if we let i=t−kandj=t, we will
haveγ(i,j) =γ(t−k,t) =γ(t,t+k) =γ(k)andρ(i,j) =ρ(t−k,t) =ρ(t,t+k) =ρ(k)
sinceγ(k)andρ(k)are even functions and hence symmetric (William and Wei, 1990).

3. Trends and Patterns in Energy Markets 20
The first two moments are also finite and the covariance and correlation between ptand
pt+kdepend only on the time difference k.
Definition 3.1.3 (Covariance Stationarity) .A stochastic process {pt}is covariance sta-
tionary if the unconditional mean and unconditional variance are finite and do not change
over time. This means
E[pt] =µ∀t
V[pt] =σ2<∞ ∀t
E[(pt−µ) (pt−k−µ)] =γk∀tand anyk.(3.1)
From equation (3.1), We notice that a strictly stationary process which has the first two
moments being finite is also covariance stationary. The simplest example of a covariance
stationary process is the white noise. This process is particularly important since it facil-
itates the construction of more complex stationary processes.
The covariance between two random variables depends on their variances as well as the
strength of the linear relationship between them. Covariances are extremely important
as input to, for example, a portfolio analysis, but, in order to understand the relationship
between variables, it is much better to examine their sample correlations.
Definition 3.1.4 (White noise) .The process{at}is called weak white noise if, for some
positive constant σ2:
(i)E[at] = 0∀t.
(ii)V[at] =σ2∀t.
(iii) Cov (at,at+k) = 0∀tand fork/negationslash= 0.
It is sometimes necessary, particularly for financial time series, to replace hypothesis
(iii) by the stronger hypothesis that {at}and{at+k}are independent and identically
distributed. In this case the process {at}is said to be strong white noise. For strong
white noise,{at}cannot be predicted either linearly or non-linearly and it reflects the

3. Trends and Patterns in Energy Markets 21
true innovation in the series. For a general white noise process, the innovation term may
not be predictable linearly, yet it can be probably predictable non-linearly using ARCH
or GARCH models. Figure 3.2 shows a simulated white noise process of 5,000 values with
σ2= 4
Figure 3.2: White noise process
In analysis of time series data, the first step is to convert the non-stationary data into a
stationary one by removing the trend or any other inherent pattern, since most statistical
and econometric models can only be applied to stationary time series. This is achieved
either by de-trending or differencing the data.
The easiest form of non-stationarity to work with is the trend stationary model wherein
the process has stationary behaviour around a trend. We may write this type of model as
xt=µt+yt (3.2)
wherextare the observations, µtdenotes the trend, and ytis a stationary process. Very
often, a strong trend will obscure the behaviour of the stationary process, hence, there is
some advantage to removing the trend as a first step in an exploratory analysis of such