—

Ca’ FoscariDorsoduro 324630123 Venezia

UniversitàCa’FoscariVenezia Master’s Degree programme —

Second Cycle (D.M. 270/2004)

in Economics — Models and Methods of Quanti-

tative Economics

Final Thesis

A model-combination approach

to pricing weather derivatives

Supervisor

Ch. Prof. Roberto Casarin

Ch. Prof. Martina Nardon

Graduand

Iuliia Lipanova

Matriculation number 855735

Academic Year

2015/2016

Abstract

This study provides an empirical example of fitting a model that can explain and

predict the behavior of weather time series. Different models of GARCH-type are

fitted on the series of the average temperature. Autoregressive, seasonal and trend

components are included in mean and variance equations. The seasonal components

are constructed as a combination of simple harmonic functions, the frequencies of

which are found with Fast Fourier Transform. Forecast combination approach is

applied to define the best model and thus, this model is used as an underlying to

price weather derivatives. Numerical examples of pricing some insurance contracts

are given, found with Monte Carlo simulations.

Keywords: Weather derivatives, ARMA-GARCH, Fast Fourier Transform, Monte

Carlo, Insurance premium calculation

Acknowledgment

I would like to express my deepest gratitude to my thesis advisor Professor

Roberto Casarin for his support of this Master’s thesis. His profound knowledge

helped me to organize the econometrics part of this thesis, while his personal interest

in the field was a real motivation to me. I would also like to thank my co-advisor

Professor Martina Nardon, for her guidance on finance part of this thesis. She

was always open to communication and has significantly broadened my understand-

ing of the whole field. What is more, I would like to express my gratitude to QEM

Master’s program for providing me with the aid to gain such a unique and life-

changing experience studying in two the most beautiful cities, Paris and Venice.

I place on record, my sincere thank you to my dearest colleagues of this course

Laura Hurtado Moreno, Erdem Yenerdag and Kedar Kulkarni. It was a real

pleasure to work on several course projects with Laura and Erdem. Not only their

contribution was important for the final grade, but more importantly, I significantly

enriched my knowledge in various fields of economics thanks to their contribution.

Finally, I must express my gratitude to my parents for being always supportive

in any decisions I have made in my life so far. I am exceptionally grateful to my

mother Iryna Lipanova for encouraging me to push my boundaries and to my

father, Vyacheslav Lipanov for his constant concern. Special thanks go to my

dearest friends Daniele Genovese and Ksenia Nepomnyashaya. It would have

been much harder for me to adjust to the life abroad without encouragement and

guidance from Daniele and his family.

i

Contents

List of Tables iv

List of Figures v

1 Introduction to weather derivatives 3

1.1 Weather derivatives market . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Premuim calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Asian-style insurance policy . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Parisian-style insurance policy . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Insurance policy based on weather index . . . . . . . . . . . . . . . . 8

2 Models for temperature 10

2.1 Fourier analysis of the temperature . . . . . . . . . . . . . . . . . . . 10

2.2 GARCH model with seasonal components . . . . . . . . . . . . . . . 13

2.3 GARCH in mean model . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Exponential GARCH model . . . . . . . . . . . . . . . . . . . . . . . 14

3 Empirical results 16

3.1 Data and descriptive statistics . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Forecast combination . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 Pricing weather insurance contracts . . . . . . . . . . . . . . . . . . . 32

ii

Appendices 39

Appendix A Discrete Fourier transformation, MATLAB function 40

Appendix B An example of Fourier analysis on simulated series, MAT-

LAB script 41

Appendix C Spectral analysis,MATLAB script 43

Appendix D Diagnostics of the residuals 45

Bibliography 52

iii

List of Tables

3.1 Summary of the Series . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Augmented Dickey-Fuller Test, Results . . . . . . . . . . . . . . . . . 17

3.3 Ljung-Box Test, Results . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Fourier Transformation, Results . . . . . . . . . . . . . . . . . . . . . 22

3.5 The components inclueded in the models . . . . . . . . . . . . . . . . 25

3.6 Jarque-Bera test on residuals, Results . . . . . . . . . . . . . . . . . . 25

3.7 Goodness of forecast, Results . . . . . . . . . . . . . . . . . . . . . . 28

3.8 Weights for the model combination . . . . . . . . . . . . . . . . . . . 31

3.9 Goodness of forecast, Result of Model combination . . . . . . . . . . 31

3.10 Description of the contracts . . . . . . . . . . . . . . . . . . . . . . . 34

3.11 Prices of Asian-style contracts . . . . . . . . . . . . . . . . . . . . . . 35

3.12 Prices of Parisian-style contracts . . . . . . . . . . . . . . . . . . . . . 36

3.13 Prices of Index-based contracts . . . . . . . . . . . . . . . . . . . . . 36

D.1 White test, Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

D.2 Estimation output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

iv

List of Figures

2.1 The magnitude spectrum of the observations versus the frequencies . 12

3.1 The graph of the observations and the histogram of the distribution . 17

3.2 ACF and PACF of the observations . . . . . . . . . . . . . . . . . . . 18

3.3 Fourier analysis of the observations a) Magnitude spectrum; b)Log

transformation of the magnitude spectrum . . . . . . . . . . . . . . . 20

3.4 Graph of the observations and the harmonic functions . . . . . . . . . 21

3.5 Fourier analysis of the squared observations a) Magnitude spectrum;

b)Log transformation of the magnitude spectrum . . . . . . . . . . . 22

3.6 Graph of the squared observations and the harmonic functions . . . . 23

3.7 Q-Q plots: model residuals against Student’s t quantiles . . . . . . . 26

3.8 Actual,fitted,residual of studied models . . . . . . . . . . . . . . . . . 27

3.9 Ex post forecast and credible intervals . . . . . . . . . . . . . . . . . 29

3.10 Forecast combination, Result . . . . . . . . . . . . . . . . . . . . . . . 32

3.11 Simulated trajectories of the combined model over the time . . . . . . 33

D.1 Box-Pierce P-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

v

Introduction

Weather changes have a huge impact on the different branches of business activity.

Nowadays, these changes are not only represented by some periodical patterns (such

as climate oscillations, El Nino pattern, monsoons) or by low probable hazards (as

volcanic activity, earthquakes, wildfires etc.), but, mainly, by the global warming.

The increase of the temperature has direct and indirect effects on such industries

as agriculture, energy, tourism. One of the latest destructive impacts of the climate

change is bleaching of Great Barrier Reef caused both by warm streams of El Nino

and global temperature increase. The result of the bleaching will probably be the

loss of countless marine species, a decrease of scuba-diving tourism and tourism on

the adjacent Australian shore cost. Obviously, in such case, no financial instrument

would neither cover the losses nor fix the problem. However, in other situations,

short-term derivative instruments might cover the possible risks, so that companies

would be able to sustain unpredictable changes. In a long-term the reconsideration

of the technologies is required that is out of the scope of this study. We would like

to construct and analyze insurance contracts that may be used to hedge against

weather-related losses. In the literature, similar empirical case studies are conducted

by Carolyn W. Chang, Jack S. K. Chang and Min-Teh Yu (1996); Calum G. Turvey

(2001).

The insurance contracts on weather are similar to the derivatives. However, due

to specifics of the underlying and its market, these contracts cannot be evaluated

using assumptions of the Black-Scholes model. The first issue that arises is the

need to construct a model for the underlying. The Black-Scholes model assumes

1

that the prices of the underlying follow a geometric Brownian motion process with

Markov’s property that is interpreted into Efficient Market Hypothesis that actual

price of a stock already includes all available market information. Consequently, its

expected value does not depend on historical values thus, it can be explained only

by today’s value. However, Efficient Market Hypothesis cannot be valid for weather

patterns that depend on the seasonality. Another reason why the standard valuation

approaches cannot be used is the lack of evidence towards the assumption of the

risk-neutral valuation principle, as the weather is a non-traded asset and there is no

corresponding spot market. Even though there are some traded weather derivatives,

mainly presented by Chicago Mercantile Exchange, the market is not liquid. All it

calls to consider not a standard derivative instrument on weather, but an insurance

contract that requires premium calculation in order to determine its price.

Models that can possibly explain the behavior of the weather are discussed in the

literature by Sean D. Campbell and Francis X. Diebolt (2005),Timothy J. Richards,

Mark R. Manfredo, Dwight R. Sanders (2004). In this thesis, we use a model sim-

ilar to the one introduced by Campbell (2005) and we consider other specifications

from the family of ARMA-GARCH models. In order to obtain a model with better

forecasting properties, the model combination approach is applied as it is shown by

A.Timmermann (2005). The evaluation of the insurance contract is done by cal-

culating the premiums based on the values of average temperature simulated with

Monte-Carlo method.

The structure of the paper is as follows. Section 2 presents a theoretical analysis

of the weather derivative market. It also defines how the premiums are calculated

and what the possible representations of the loss function are. In Section 3 univariate

structural time series models on temperature are discussed. The empirical results

are given in Section 4.

2

Chapter 1

Introduction to weather

derivatives

In the following section, the market of weather derivatives is described and the place

of insurance contracts on this market. We present different approaches to define the

premium of an insurance contract and we also give representations of possible loss

functions. For a potential claimant, there are several ways to include the expected

loss caused by weather changes. In this thesis, we focus on the insurance policies

that account for some threshold, crossing which may lead to losses. This logic is

similar to barrier options and, indeed, insurance policies constructed in this thesis

have some features of this type of derivatives.

1.1 Weather derivatives market

The market for weather derivatives appeared around 20 years ago that is relatively

short term compared to other markets of financial instruments. Initially weather

as an asset appeared in individually constructed over-the-counter (OTC) contracts

mainly between some companies of the energy sector. The main purpose of such

negotiations was risk-management and so hedging against possible losses. This reason

still can be considered as the main driver of the markets growth.

3

The increase of the demand for the OTC contracts on weather spurred the need to

create standardized derivatives as well as to establish a market that would provide

both some liquidity and regulation. Consequently, Chicago Mercantile Exchange

(CME) introduced weather derivatives in 1999. Nowadays, CME is the biggest

market that offers such types of contracts 1 mainly on CDD and HDD indexes 2.

Derivatives currently traded on CME are Monthly Futures, Seasonal Strip Futures,

Monthly Options, Seasonal Strip Options.

The contracts mentioned above are built on the meteorological data of a quite

limited number of cities. Precisely, there are weather derivatives for New York,

Chicago, Atlanta, Cincinnati, Dallas, Sacramento, Las Vegas, Minneapolis, London

and Amsterdam. In fact, the market for such instruments is rather illiquid and

cannot cover the needs of all the weather dependent sectors.

An alternative solution to cover risks potentially caused by climate changes is

presented by insurance contracts on weather (that we consider as a specific type of

weather derivatives). It has to mention, that the nature of such weather derivatives

is far from standard insurance instruments. For standard insurance contracts, an

issuer has a portfolio of clients insured against the same risk and some historical

data on the claims. To evaluate such contracts, an issuer can use actuarial approach

taking into account statistical distributions of claims and payments. From the point

of view of an individual company the probability that a risky event occurs is rather

low. Yet, in this thesis we consider the contracts uniquely constructed for the needs

of the specific sector and the underlying for which is not an asset, but the weather.

Insurance contracts on weather are historically on demand in agriculture. Weather

changes have a direct and indirect impact on crop yields, need of fertilizers, livestock,

storage life and quality of the production. Events against which farmers usually

consider the purchase of the insurance contracts are temperature changes, extreme

rainfalls, draughts, the number of sunny days etc. Other sectors may be interested

1According to the Weather Product State on April 2016 from CME group web-site:

http://www.cmegroup.com/trading/weather/files/updated-weather-product-slate.pdf2See chapter 2.5 for a detailed explanation about the weather indexes

4

in the derivatives of this type are energy and tourism.

1.2 Premuim calculation

There are various issues that prevent spread usage of these contracts. For instance,

if a farmer requires the creation of unique derivative that covers risks exceptionally

present in his enterprise, an issuer may not have a portfolio of similar cases. Conse-

quently, the premium of such a contract will be quite the same as expected losses.

The farmer may anyway stick to the insurance contract when his own expectation

of the losses is higher than those of the issuer.

Evaluation of the price for such an insurance contract is rather different from

the approach generally used to price non-life insurance contract. As it is mentioned

above, an issuer usually attributes an incoming contract to some portfolio of similar,

but independent contracts. The price of the derivative depends on the equivalence

premium that accounts for the historical number of claims, the time-pattern of a

claim, the amount of the reimbursement, i.e. the loss of an issuer. The issuer also

takes into account competition on the market expressed by services and prices offered

by other insurance companies.

The price of an insurance contract that policyholder has to pay is the premium.

In this thesis, we will apply three different premium calculation principles on the

contracts: the pure premium principle, the standard deviation principle and the

risk-adjusted premium principle.

The pure premium principle is the expectation of losses Y of a policyholder:

Πpure = E[Y ] (1.1)

In order to take into account the variation of the expected loses the standard

deviation principle can be applied:

Πdev = E[Y ] + ϑ1(V ar[Y ])1/2 (1.2)

5

where ϑ1 > 0 is a safety loading that accounts for the volatility risk.

The safety loading can be also applied to adjust the distribution of the expected

loss. It may be useful when the historical data is not available or when there is any

other reason to assume that the real data-generating process has heavier tails. In

such cases the safety loading is directly applied on the probability:

Πrisk.adj =

∫ ∞0

[Pr(Y > y)]1/ϑdy =

∫ ∞0

[1− F (Y )]1/ϑ2dy (1.3)

where ϑ2 > 1. Equation 1.3 can be alternatively shown in discreet form as:

Πrisk.adj =∞∑i=0

p1/ϑ2Yi

Yi (1.4)

In the next sections, we will apply these 3 methods to calculate premiums for

different insurance policies.

1.3 Asian-style insurance policy

In order to create an insurance contract, it is important to make some assump-

tions about a possible loss function. Dealing with temperature as an underlying of

derivatives, losses are assumed to appear when the real grade is rather different from

the expected one. An example of exotic derivatives is barrier options that may be

activated when the price of an underlying crosses some barrier. Asian option is a

particular type of exotic options that compares the average temperature during the

maturity of the contract to some fixed level of the temperature.

In the literature, Asian options are discussed among others by L.C.G. Rogers and

Z. Shi (1995), J. Aase Nielsen and Klaus Sandmann (2003) etc. In this thesis, we

implement a loss function similar to the payoff function of an Asian option.

Let us consider a situation when a policyholder is sensitive to the average tem-

perature. The loss function l(XT ) is assumed to depend on how much the average

temperature XT during the time interval [0,T] exceeds some predefined level of tem-

perature K,

l(XT ) = L ·max(XT −K, 0) (1.5)

6

where L is the loss per 1 ◦C of temperature increase and XT is computed

Analogically, in case a policyholder suffers losses on the decreasing temperature,

the loss function is:

l(XT ) = L ·max(K − XT , 0). (1.6)

A contract built with this approach can be of interest in cheese making, viticulture,

sausage making and other agriculture spheres that have a maturity stage in the

production cycle. In fact, deviations from the average temperature are unlikely to

destroy the production, but it can significantly affect the quality of the output. For

instance, the sort of wine is dependent on the temperature during most of the stages

of wine making process such as harvesting, crushing and pressing, fermentation,

clarification, aging and bottling. Relatively small fluctuations can affect the amount

of sugar in grapes and life of bacteria vital for the wine fermentation. Not only a

producer may not obtain expected quantity, but also the ready product may not

meet its quality standards.

1.4 Parisian-style insurance policy

Another example of a loss function with a barrier is implemented in Parisian

options. The payoff of a Parisian exotic option does not only depend on the fact

whether an underlying crosses some predefined barrier, but it also takes into account

the duration over or under this barrier. In the literature, such derivatives are de-

scribed by Marc Chesney, Monique Jeanblanc-Picque and Marc Yor (1997), Angelos

Dassios and Shanle Wu (2011), etc.

Let us consider a case when a company tries to minimize the costs that could

be potentially caused by extreme temperatures so that either too high or too low

temperature may lead to major loss of the production. For instance, in agriculture,

there may occur a situation such that a proportion of the total crops will be destroyed

each day out of maximum D days that the temperature stays above (or below)

some value H. Time of excursion is the duration of the period of time when the

7

temperature is above the barrier. It can be shown as a function D(Xt, t):

D(Xt, t) =

0, if Xt < H

t− gt, if Xt ≥ H

(1.7)

where gt is the time when temperature Xt reaches barrier H.

gt = sup {τ ≤ t|Xτ = H} (1.8)

Taking into account how far from the barrier the temperature may get, the loss

l(XT , D) can be defined as:

l(Xt, D) =

L ·max(Xt −K, 0), if D(Xt, t) ≤ D,D ≤ t ≤ T

0, otherwise

(1.9)

where L is the loss per 1 ◦C of temperature increase, t ∈ [0, T ].

The loss function described are used to price Parisian-style insurance contracts.

1.5 Insurance policy based on weather index

One of the questions that arises with the need of accounting for weather-related

risks is what exactly should (or even could) be used as an underlying asset. This

problem was solved by the introduction of indexes on the weather. Such indexes can

be built on snowfall, rain, wind speed, etc. Yet the most widely spread are contracts

built on the temperature indexes such as CDD and HDD.

Heating degree days (HDD) is the difference between the temperature measured

at some day and some base temperature (usually 18 ◦C). Cumulative Heating degree

days is the sum of all the deviations from the base level during certain period of time

and it can be represented by the following:

HDD =T∑t=1

max{Xt − 18, 0} (1.10)

Cooling degree days (CDD) is an analog index for the days during which daily

average temperature Xi was below some certain level. And it is defined by the

8

following:

CDD =T∑t=1

max{18−Xt, 0} (1.11)

The loss function can also take into account the deviations from some fixed thresh-

old K instead of using predefined threshold of 18 ◦C. In this thesis, we calculate

expected values of CDD and HDD indexes from simulated series. The loss function

of an indexed-based insurance contract is the same as the payoff function of a stan-

dard option. In order to price such contracts, time to maturity and strike have to be

specified.

Let L be some loss that accrues when the index crosses the threshold, let CDDstrike

and HDDstrike be strikes, than the loss functions based on CDD and HDD indexed

are shown:

lCDD = L ·max(CDD − CDDstrike, 0) (1.12)

lHDD = L ·max(HDDstrike −HDD, 0) (1.13)

In fact, it is hard to imagine practical usage of insurance contracts on weather

indexes, as a potential insurance holder may not easily interpret their value. However,

we create such insurance contracts as alternative instruments to weather derivatives

traded on CME.

9

Chapter 2

Models for temperature

In a univariate structural time series models, the dynamics is decomposed in four

factors: trend, seasonality, cyclicality, and residual. A trend represents a general

tendency in the series. This tendency is present throughout all the time interval of

the observations. Seasonality has a repetitive nature that lasts for some limited time

period. Cyclicity is similar to seasonality, yet it may not exhibit fixed period pattern.

The final components of the structural time series is residual that serves to explain

all the irregularities that other components failed to capture. In case of weather time

series the proper construction of the seasonal component is particularly important.

In the following, the specification of some GARCH type models are studied in

order to apply them to weather time series analysis. We also present the theoretical

notion of Fourier analysis used to detect the seasonal components in the time series.

2.1 Fourier analysis of the temperature

One of the methods used to detect the seasonal components in time series is

Fourier analysis. More specifically, the discrete Fourier transformation allows

a researcher to find the most relevant frequencies that would explain the periodic

behavior of the series. The transformation provides the frequency domain represen-

tation of time series instead of the time domain representation which is commonly

10

used in time series analysis. As a result, it allows to determine at what time certain

frequency occurs. To apply Fourier transformation the power spectrum function has

to be constructed as following:

s(fj) =1

2π

∞∑τ=−∞

γje−ifjτ (2.1)

where γj is the autocovariate of the order j; i=√−1 and the Fourier frequency :

fj =2πj

T, j = 0, 1, ..., T − 1 (2.2)

Applying the theorem of De Moivre one can show that :

e−ifjτ = cos(fjτ)− isin(fjτ) (2.3)

Substituting the result (2.3) into the equation (2.1) and the stationarity condition

of the time series: γj = γ−j one can obtain and the power spectrum:

s(fj) =1

2π

∞∑τ=−∞

γj (cos(fjτ)− isin(fjτ))

=1

2π

(γ0 +

∞∑τ=−∞

γj [cos(fjτ) + cos(−fjτ)− isin(fjτ)− isin(−fjτ)]

) (2.4)

due to the properties of the trigonometric functions cos(θ) = cos(−θ) and −sin(θ) =

sin(−θ) :

s(fj) = 12π

(γ0 + 2

∞∑τ=−∞

γjcos(fjτ)

)(2.5)

The sample power spectrum is used at Fourier transformation, instead of the

theoretical power spectrum one. The sample spectrum I(fj) is found in a similar

way as to find the power spectrum. The difference is that in order to obtain the

sample power spectrum one should use the autocovariances γj of the series:

I(fj) = 12π

(γ0 + 2

T−1∑τ=1

γjcos(fjτ)

)(2.6)

11

After the frequencies are calculated, the models seasonal component can be given as

the combination of the following harmonic functions :

ωt =K∑k=1

(λ1k cos(fkt) + λ2k sin(fkt)) (2.7)

where K is the number of picks in the magnitude spectrum, i.e. the appropriate

number of combinations of the harmonic functions (see Appendix A for the MatLab

function).

As an example, let us simulate a time series that includes an autoregressive part

and the combination of simple harmonic functions:

yt = 0.7yt−1 − 0.36yt−2 + (0.7cos(f1t) + 1.2sin(f1t))+

+(−5.6cos(f2t) + 0.09sin(f2t)) + εt, εt ∼ N(0, 1)(2.8)

for t ∈ [1,1500].The frequencies (f1 and f2) of the harmonic components of the time

series are f1=0.02 and f2=0.075.

Assuming that the true data generating process is unknown, the discrete Fourier

transformation is used in order to determine the frequencies of the harmonic compo-

nent(see Appendix A for the MatLab code). From the graph of magnitude spectrum

( Figure ?? ) one can see that there are present two pics at the frequency bins 5 and

19. The values corresponding to the frequencies bins found are 0.0168 and 0.0754.

These values approximately equal to the frequencies of the harmonic components

specified in (2.8).

Figure 2.1: The magnitude spectrum of the observations versus the frequencies

12

2.2 GARCH model with seasonal components

Due to the nature of the underlying asset i.e. the average temperature, periodic

(or seasonal) components and possibly a nonlinear trend should be considered as

explanatory variables. A similar model was specified by Campbell and Diebold

(2002).

Let Xt for t=1,...,N be the average daily temperature. The mean equation is:

Xt = ωt + µt +I∑i=1

αiXt−i + σtεt; εtiid∼ (0, σ2

ε) (2.9)

The autoregressive part is included in the mean model, while the variables µt and

ωt represent trend and seasonal components, respectively. The trend is given in the

form of M-th order polynomial as following:

µt =M∑m=1

γmtm (2.10)

The seasonal component is assumed to be present due to the nature of the weather.

One of the possible ways to capture it in time series is by constructing external

variables both in the equation for mean and variance. The external variables are

well defined by the simple harmonic functions as it is shown in (2.7).

The model for variance is a periodic GARCH(P,Q) that includes the external

regressor $t that is the component of seasonal nature driving volatility.The spectral

analysis of the squared values of the underlying was performed in order to determine

the frequencies fvar present in the model for the volatility. fvar has the same structure

as the frequency of the mean model specified in (2.2). So the variance model is given

by:

σ2t = ρ+$t +

P∑p=1

βpσ2t−p +

Q∑q=1

ϕq(σt−qεt−q)2 (2.11)

In the study two models of this type are analysed:

Model I: I=14, Kmean = 1, M=1, P=6,Q=0,Kvar = 1;εt ∼ N(0, 1);

Model II: I=14, Kmean = 1, M=1, P=6,Q=2,Kvar = 1;εt ∼ St(ν).

13

2.3 GARCH in mean model

GARCH in mean model (GARCH-M) is one the extensions of classical generalized

autoregressive conditional heteroskedasticity model (Bollerslev 1986).This model un-

derlines a possible correlation between a variance and a mean. It allows not only the

effect of the conditional variance on the mean but also the direct effect of variance.

So that the variance appears two times in the model specification: the direct effect is

represented in terms of an additional covariate in the mean equation, and standard

deviation is present in the error term.

The motivation to use GARCH-M model comes from the assumption about the

dependence between a variance and a mean mentioned above.

GARCH-M (P,Q) has the following representation :

Xt = ωt + µt +I∑i=1

αiXt−i + δσ2t + σtεt (2.12)

σ2t = ρ+

P∑p=1

βpσ2t−p +

Q∑q=1

ϕq(σt−qεt−q)2; εt

iid∼ (0, σ2ε) (2.13)

The specifications of the fitted models of this type are:

Model III: I=14, Kmean = 1, M=1, P=3,Q=0,εt ∼ N(0, 1);

Model IV: I=14, Kmean = 1, M=0, P=1,Q=0,εt ∼ St(ν).

2.4 Exponential GARCH model

Another way to model the correlation between a mean and a variance is by

exponential GARCH model (EGARCH). Precisely, it takes into account a possible

asymmetric relationship between the value of the mean model and the magnitude of

the volatility. It preserves the conditions of the non-negativity of the variance and

zt = σtεt is modeled in the way such that it includes both the magnitude and the

sign effect of the variance:

g(zt) = θzt + λ(|zt|−E(zt)) (2.14)

14

where zt corresponds to the sign effect and the expression (|zt|−E(zt)) for the mag-

nitude.

So the EGARCH(P,Q) model is the following:

Xt = ωt +I∑i=1

αiXt−i + δlog(σ2t ) + zt (2.15)

where log(σ2t ) is logarithm of the GARCH in mean component;

log(σ2t ) = ρ+$t +

P∑p=1

βplog(σ2t−p) +

Q∑q=1

ϕqg(zt−q); εtiid∼ (0, σ2

ε) (2.16)

The specifications of the fitted models of this type are:

Model V: I=6, Kmean = 1, P=2,Q=0,Kvar = 1, εt ∼ N(0, 1);

Model VI: I=6, Kmean = 2, P=1,Q=0,Kvar = 1, εt ∼ St(ν).

15

Chapter 3

Empirical results

In the following, we present the results of fitting the average temperature on the

models described in the previous chapter. The preliminary descriptive data analysis

and spectral analysis are given as well as the goodness of fit is evaluated. In the end,

we make ex-post forecasts for each of the GARCH models and a forecast based on a

model combination approach.

3.1 Data and descriptive statistics

The data used in this thesis is the average daily temperature in London and it

has been extracted from the real-time weather information provider Weather Under-

ground. The decision to use the observations from London is driven by the fact that

there are some weather derivatives traded on Chicago Mercantile Exchange written

on the temperature from this city. So that it may be possible to price derivatives

similar to those currently traded in order to evaluate the adequateness of our pricing

methodology.

The sample consists of the observation from 01/01/2010 till 02/02/2016 i.e.

2224 observed days. The part of the sample from 01/01/2010 till 02/02/2015 is used

for the fitting purpose, while another part from 02/02/2015 till 02/02/2016 is used

for evaluating the goodness of forecast and selecting the best fit model.

16

Figure 3.1: The graph of the observations and the histogram of the distribution

Table 3.1: Summary of the Series

Mean Med Max Min Std.Dev Skew Kurt JB P-Val. Num

Xt 11.233 11.000 29.000 -4.000 5.684 -0.108 2.368 41.320 0.000 2224

Table 3.2: Augmented Dickey-Fuller Test, Results

Variable T-Statistic %1 C.V. %5 C.V. %10 C.V P-Value

Xt -5.078086 -3.433102 -2.862642 -2.567402 0.0000

From the graph of the observations (see Figure 3.1) one can clearly see the

seasonal nature of the time series. The main descriptive statistic is given in Table 3.2

and the histogram of the distribution are shown in Figure 3.1. According to Jarque-

Bera test statistics, we cannot accept the null hypothesis that the data is normally

distributed. From the histogram of the distribution one can conclude multimodality,

precisely, three modes are observed.

17

(a) ACF of the observations (b) PACF of the observations

(c) ACF of the squared observations (d) PACF of the squared observations

Figure 3.2: ACF and PACF of the observations

The stationarity of the given series is checked by applying Augmented Dickey-

Fuller unit root test (the results are shown in Table 3.2). The hypothesis of the unit

root is rejected that means that the series is stationary.

In order to justify the possible use of the AR part in the mean equation of

GARCH model, the presence of serial autocorrelation has to be controlled. The

serial autocorrelation is tested with the Ljung-Box-Pierce Q-Test (see Table 3.3 and

Figure 3.2 ). Graphs of ACF and PACF functions show the pattern typical for the

autoregressive process and the results of the Ljung-Box-Pierce Test prove that the

null hypothesis of independence in a given time series is rejected. Similarly, the ACF

and PACF functions of the squared observations exhibit the behavior common for AR

models. Consequently, all it calls for the use of models that includes autocovariates

both equations for mean and variance.

18

Table 3.3: Ljung-Box Test, Results

Lags Statistic df P-Value

5 3535.101 5 0

10 6946.427 10 0

15 10235.461 15 0

20 13403.745 20 0

25 16452.874 25 0

30 19384.499 30 0

3.2 Spectral analysis

Spectral analysis helps us to determine the frequency of the harmonic functions

with which the seasonality can be possibly explained. It is done by applying the

discrete Fourier transformation on the series. The procedure is formulated in (2.1) -

(2.6).

In order to interpret the result of the Fourier transformation it is important to

remember that the transformation rescales all the data from the time domain into

the domain in radians such that the whole cycle 2π is divided by the total number

of the observations in the sample as it is given in equation (2.6). For the sake of

simplicity the rescaling in terms of years ( the sampling frequency) is also used, so

that the result obtained in radians can be easily interpreted :

fyear =j

fsamp, j = 1, ..., T (3.1)

where fyear is in terms of the sampling frequency, fsamp is the sampling frequency

i.e. the duration of a year in days and T is the total number of the observations in

the sample.

The frequency given in terms of years is interpreted as the effect of the harmonic

function occurs every fyear years. Then the frequency per year (that shows how often

19

the effect of the harmonic function occurs in 1 year) can be also defined:

dcycle =T

fyearfsamp(3.2)

fp.y =dcycle365

(3.3)

where dcycle is the duration of 1 cycle in days; fp.y is the frequency per year.

In order to define the seasonal component in the mean and in the variance, the

spectral analysis is performed on the values of the sample and its squared values

respectively (see Appendix B for the MatLab code).

(a) Magnitude spectrum (b) Log transformation of the magnitude spec-

trum

Figure 3.3: Fourier analysis of the observations a) Magnitude spectrum; b)Log trans-

formation of the magnitude spectrum

Analyzing the magnitude spectrum of the observations (see Figure 3.3) one can

conclude that the magnitude effects are present on the frequency bin 7 and on the

frequency bin 14. The magnitude of the last one is less significant, though.

The transformation of the results obtained in the bins of the frequency into the

frequency in terms of the frequency per year and in radians is given in Table 3.4.

According to the results obtained, the seasonal component can be expressed with two

simple harmonic functions such that the whole cycle of one is of one year and another

is twice in a year. The graphical representation of sine functions built with the both

20

frequencies found and sine function that is the combination of those mentioned above

is given in Figure 3.4. The combination of both frequencies may be able to explain

multimodality present in the distribution of the observations (see Figure 3.1).

Figure 3.4: Graph of the observations and the harmonic functions

Analogically, the frequencies present in the seasonal component for the equation

of the variance is studied. In this case, discrete Fourier transformation is applied to

the squared observations. The magnitude spectrum is given in Figure 3.5.

21

(a) Magnitude spectrum (b) Log transformation of the magnitude

spectrum

Figure 3.5: Fourier analysis of the squared observations a) Magnitude spectrum;

b)Log transformation of the magnitude spectrum

Table 3.4: Fourier Transformation, Results

bin number frequency fp.y f(radians) duration of the cycle dcycle

fr1 7 1.0155 0.0170 370.6667

fr2 14 0.4687 0.0367 171.0769

fr3 13 0.5078 0.0339 185.033

The results are similar to the previously obtained: as before the seasonal

component can be expressed with two simple harmonic functions one with the cycle

of one year and another with half a year cycle. The last weather component though

has slightly smaller frequency fr3 (see Table 3.4). The graphical representation of

sine functions built with the both frequencies found and sine function that is the

combination of those mentioned above is given in Figure 3.6.

22

Figure 3.6: Graph of the squared observations and the harmonic functions

3.3 Model estimation

In order to estimate a coefficient on the explanatory variables the quasi-maximum

likelihood (QML) method is applied.

In the models I, III, V the residuals are assumed to follow a standard Gaussian

distribution. Let us denote the vector of unknown parameters as

θ = (λ11, ...λ1K , λ21, ...λ2K , γ1, ...γ1M , α1, ...αI ,

ρ, λ31, ...λ3Kvar , λ41, ...λ4Kvar , β1, ...βP , ϕ1, ...ϕQ)(3.4)

where K,M,I,Kvar,P,Q are given by model specification.

Conditionally on all the initial valuesX0, ..., Xt−I , σ2t−P , ..., σ

20, εt−Q, ...ε0,the Gaus-

23

sian quasi-likelyhood function is:

L(θ) =T∏t=1

1√2πσ2

t

exp

(− ε2

t

2σ2t

)(3.5)

To make computation easier the log-likelyhood function can be used :

lnL(θ) = −T∑t=1

1

2

(log(2π) + log(σ2

t ) +ε2t

σ2t

)(3.6)

Residuals of the models II,IV,VI are assumed to be of Student’s t distribution. In

this case, we have to use the relevant conditional quasi-likelihood function. Precisely,

it was defined by Bollerslev (1987) in the following form:

Ln(θ) =T∏t=1

Γ[12(υ + 1)]

π12 Γ[1

2υ]

[(υ − 2)σ2

t

]− 12

[1 +

(εtσt)2

(υ − 2)σ2t

]− 12

(υ+1)

(3.7)

where υ is the degrees of freedom parameter.

In order to find the coefficients θ the following maximization problem has to be

solved :

θQMLE = arg maxθ∈Θ

lnLn(θ) (3.8)

3.4 Model selection

For this study, 6 models of GARCH type are fitted on the weather time series.

Table 3.5 gives the detailed information about the external components specified for

each of the models and the assumption on the error terms. The symbol * indicates the

presence of the component in the model, µt corresponds to the trend component in

the mean (given in (2.10)), ωt and $t are seasonal components in mean and variance

respectively (see (2.7)), δ shows whether the variance is present in the specification

of the mean (GARCH-M model).

Having estimated the parameters of the models (see Table D.2 in Appendix D),

the qualitative analysis of the models explanatory and forecasting strength has been

done (see the Diagnostics of the Residuals, Appendix D).

24

Table 3.5: The components inclueded in the models

Model µt ωt $t δ Error distribution

Model I * * * Normal

Model II * * * Student’s t

Model III * * * Normal

Model IV * * Student’s t

Model V * * * Normal

Model VI * * * Student’s t

Jarque-Bera test on normality evidences that the residuals of the models I,III,V

are not normal (see Table 3.6). These results violate the assumptions of GARCH

models.

Table 3.6: Jarque-Bera test on residuals, Results

Model I III V

Jarque-Bera statistics 21.22133 14.21267 15.57638

P-value 0.000025 0.000820 0.000415

In order to avoid the presence of heavy tails of the residuals distributions, the

error terms of the models II,IV,VI are assumed to be of the Student’s t distribution.

However residuals with Student’s t distribution have less heavy tails, it is shown on

Q-Q plots (see Figure 3.7) that some outliers are still present.

Despite the rejection of the null hypothesis about the normality of the residuals,

models show good statistical properties, so it is possible to perform ex-post forecast.

The part of the sample from 02/02/2015 till 02/02/2016 is used for evaluating the

goodness of the forecast and selecting the best fit model.

25

(a) Model II (b) Model IV

(c) Model VI

Figure 3.7: Q-Q plots: model residuals against Student’s t quantiles

26

(a) Model I (b) Model II

(c) Model III (d) Model IV

(e) Model V (f) Model VI

Figure 3.8: Actual,fitted,residual of studied models

Root mean square error (RMSE) is used to compare the forecasts made for the

same time series, but with different fitted models. The general rule to distinguish

the better model in terms of RMSE values is the smaller the RMSE the better the

27

predicting ability.

Theil coefficient is one of the forecast accuracy measures. Its values can possibly

lay between 0 and 1 where the closer to 0 the better the forecast. Theil coefficient

consists of three parts that represent the bias proportion, the variance proportion

and the covariance proportion. From the first two components, one can determine

the systematic error of the forecast and the accuracy of the prediction of the variance,

respectively.

The values of the coefficients mentioned above are listed in Table 3.7. Model

I and Model II are the models with the weakest forecasting accuracy in terms of

RSME and Theil coefficient.However, these models capture the maximum values of

the series (see Figure 3.9). Model III can be considered as the best model as it has

the smallest RMSE and Theil coefficient and shows the smallest systematic error.

Its ex-post forecast explains the seasonal behavior in the series but fails to catch

extreme values.

Table 3.7: Goodness of forecast, Results

Model I II III IV V VI

RMSE 5.906904 5.872250 2.964946 2.999786 2.988514 2.966667

Theil coeficient 0.239421 0.242125 0.118669 0.120621 0.120321 0.119474

Bias proportion 0.097271 0.123517 0.000014 0.001347 0.000889 0.002101

Variance Proportion 0.094629 0.058407 0.044618 0.052574 0.054530 0.042252

28

(a) Model I (b) Model II

(c) Model III (d) Model IV

(e) Model V (f) Model VI

Figure 3.9: Ex post forecast and credible intervals

29

3.5 Forecast combination

The analysis of the estimated models shows that the average value of the observation

is well forecasted. Model I and II are able to catch the maximum picks, while model

IV and V have the widest amplitudes. In the end, it is not possible to state that

there is a model that is definitely better compared to others. All it calls for the

consideration of a forecast combination.

In the literature the advantages and disadvantages of the forecast combination

approach have been widely discussed. One of the arguments against it is that by

combining different models a researcher somewhat diminished the importance of

single models. In this case, there may arise a controversy on whether it is better to

improve a single model or to use a composition of them.

On the contrary, the forecast combination may be a nice method to exploit all

the desired qualities of each single model. In fact, Bates and Granger (1969) showed

that the composite forecast of the models built on the separate information sets may

have higher forecasting accuracy. Moreover, Chan, Stock, Watson (1999) argued

that the improvements are possible even for the models that information sets are not

necessarily exclusive. Hsiao and Wanz (2011) affirmed that by combining different

forecasts, one increases the robustness against possible misspecification and errors.

The reason to combine the models in this thesis is to obtain a model with better

forecasting qualities. The new model is expected to be good in explaining the general

behavior of the series as well as the maximum values and the trend.

One of the ways to perform the forecast combination is by the construction of

a weighted sum of the other models. The most crucial part of this approach is the

assignment of weights. Stock and Watson (2001) use MSE as a defining parameter

to choose the appropriate weights. Precisely, let Xt+h,i be a forecast for h periods

ahead, performed by model i. In our case i ∈ [1,6]. Let υi be a weight assigned to

the model i and Xt+h be the forecast combination. The forecast combination is the

following:

30

Xt+h =6∑i=1

υiXt+h,i, (3.9)

υi =1/MSEi

6∑i=1

1/MSEi

(3.10)

The weights that can be assigned are given in Table 3.8.

Table 3.8: Weights for the model combination

Model Model I Model II Model III Model IV Model V Model VI

MSE 34.89151 34.48332 8.79090 8.99872 8.93122 8.80111

Weight 0.05640 0.05699 0.22356 0.21840 0.22005 0.22330

The new model has average forecast accuracy compared to previously forecasted

models (see Table 3.7 and Table 3.9). Analyzing the graphical representation of the

simulated series (see Figure 3.10), it captures the main behavior of the series and a

slight trend. The trend of this model seems to catch better the behavior of the series

compared to the trends in the Model I and II.

Table 3.9: Goodness of forecast, Result of Model combination

Model RMSE Theil coeficient Bias Proportion Variance proportion

Forcast combination 2.99372 0.12102 0.00990 0.039297

31

Figure 3.10: Forecast combination, Result

3.6 Pricing weather insurance contracts

Let us consider an agriculture company that is exposed to the losses related to

the change of temperature. The expected loss l per 1◦C for temperature increase

and decrease is taken the same as the value of a contract unit of weather derivatives

traded on CME i.e. L = 20 euro. In order to cover some losses, a policyholder can

also specify the number of contracts. In this thesis, we show the prices for a single

contract. Types of the contracts and their time to maturity are listed in Table 3.10.

In order to price contracts we used 30000 trajectories of the series simulated with

Monte Carlo method. The terms of possible insurance policies are the following:

Asian-style insurance policy would insure the losses if the average temperature

during the life of the contract reaches no more than the average temperature

of a specified strike volume during the same period of time. The model built

with model combination approach is used as an underlying model and Model

IV is used as a barrier for both call and put type of contracts.

32

Parisian-style insurance policy can insure the policyholder against extreme tem-

peratures. We use the model built with model combination approach as an

underlying model for all types of contracts. A model for the barrier depends

on whether the company wants to be issued against extreme temperature in-

crease or decrease. In other words, we use different barriers for short and long

contracts. Instead of using some fixed over the time strike volume, we take the

performance of a fitted to the series model as an upper or lower bound of the

contract. Specifically, Model II is assumed to be a proper upper bound and

Model VI is used as a lower bound. In such way, we assure that the barriers

are not constant over the time. A Parisian style insurance contract can cover

occurred losses provided if the underlying crosses the barrier and stays above

or below it for at least five consecutive days.

Insurance policy based on weather index would insure payments with respect

to HDD and CDD indexes on the daily average temperature, such that the

temperature base level is 18 ◦C. The indexes are calculated on the simulated

series of the model combination approach.

Figure 3.11: Simulated trajectories of the combined model over the time

33

Table 3.10: Description of the contracts

Name Type Maturity Start and end dates

March 2015 long 1 month [01.03.2015 - 31.03.2015]

July 2015 long 1 month [01.07.2015 - 31.07.2015]

August 2015 long 1 month [01.08.2015 - 31.08.2015]

September 2015 long 1 month [01.09.2015 - 30.09.2015]

December 2015 short 1 month [01.12.2015 - 31.12.2015]

February 2016 short 1 month [01.02.2016 - 29.02.2016]

March 2016 long 1 month [01.03.2016 - 31.03.2016]

Summer 2015 long 3 months [21.06.2015 - 20.09.2015]

Summer 2016 long 3 months [21.06.2016 - 20.09.2016]

Winter 2016-2017 short 3 months [21.12.2016 - 20.03.2017]

Results of pricing Asian-style contracts are shown in Table 3.11 and the prices

for Parisian-style contracts are listed in Table 3.12. One can see whether a contract

covers the expected losses of a policy holder by analyzing profit of a contract Y :

Y = l(Xt)− Π (3.11)

where l(Xt) is a loss function described in (1.9) and (1.5) (it depends on an insurance

policy applied to the contract) and Π is premium paid by insured party. In order

to calculate final profit or loss of the policyholder, we take the minimum among the

premiums as a price of insurance contracts. Negative profit of a contract such that

Y < 0 shows that the event that potentially could have caused the losses, did not

occur. Using the historical data, the payoffs and profits for the contracts up to May

2016 were calculated.

As one can see the prices of Asian-style contracts are usually more expensive

than those of Parisian-style contracts. It is a logical consequence as there are more

trajectories that can cross an average barrier than those that can cross some upper

or lower bound.

34

Table 3.11: Prices of Asian-style contracts

Πpure Πdev Πrisk.adj Payoff Profit/loss

March 2015 14.171 19.545 23.151 42.471 28.301

July 2015 46.500 55.539 75.971 0 -46.500

August 2015 51.866 61.121 84.738 0 -51.866

September 2015 55.966 65.401 91.436 0 -55.966

December 2015 0.727 2.045 1.187 71.317 70.591

February 2016 0.802 2.222 1.311 31.248 30.446

March 2016 66.177 77.071 108.119 0 -66.177

Summer 2015 49.522 57.903 80.908 0 -49.522

Summer 2016 87.313 96.347 142.651 n/a n/a

Winter 2016-2017 0.027 0.240 0.044 n/a n/a

Index-based contracts have quite different construction of a loss function, so we

consider them separately from other two types of the derivatives. CDD and HDD

indexes are publicly available on CME and some standard derivatives are traded on

the indexes. We, in fact, represent an alternative weather derivative contract of this

type. As an example, we priced contracts on the hottest on the coolest months in

a year that are July and February respectively. The result of pricing is shown in

Table 3.13.

Prices of Index-based contract not only depend on the temperature performance

and the base level of the temperature (18◦C), but also on a strike price. This makes

these insurance contracts more similar to standard derivatives. The value of indexes

may not be intuitively clear to an agriculture company, as neither one-time variation

from some temperature K nor the cumulative number of such variations may cause

any significant loss. However, the indexes are important to determining the need for

heating or cooling, so this type of contracts are more applicable in the energy sector.

35

Table 3.12: Prices of Parisian-style contracts

Πpure Πdev Πrisk.adj Payoff Profit/loss

March 2015 20.167 21.080 32.948 41.043 20.876

July 2015 5.790 6.596 9.458 0 -5.790

August 2015 3.789 4.656 6.190 0 -3.789

September 2015 2.556 3.510 4.175 0 -2.556

December 2015 4.108 5.280 6.710 0 -4.108

February 2016 1.281 2.482 2.094 1.281 10.272

March 2016 1.377 2.516 2.250 0 -1.377

Summer 2015 5.650 6.617 9.231 0 -5.650

Summer 2016 0.253 1.282 0.414 n/a n/a

Winter 2016-2017 0.221 1.359 0.361 n/a n/a

Table 3.13: Prices of Index-based contracts

February 2016 CDD call February 2016 CDD put

Strike Πpure Πdev Πrisk.adj payoff Πpure Πdev Πrisk.adj payoff

300 27.287 37.926 44.582 0 17.210 25.772 28.118 55

350 7.480 13.089 12.221 0 47.403 60.659 77.447 5

370 3.816 7.743 6.234 15 63.739 78.292 104.135 0

390 1.781 4.382 2.909 35 81.703 97.132 133.486 0

410 0.760 2.388 1.242 55 100.683 116.639 164.495 0

July 2015 HDD call July 2015 HDD put

Strike Πpure Πdev Πrisk.adj payoff Πpure Πdev Πrisk.adj payoff

30 22.193 31.369 36.258 0 11.425 16.235 18.666 8

70 12.983 20.129 21.212 32 22.216 29.288 36.296 0

90 6.942 12.128 11.341 52 36.174 45.168 59.101 0

110 3.348 6.859 5.471 72 52.581 63.020 85.906 0

130 1.443 3.661 2.357 92 70.675 82.065 115.469 0

36

Conclusion

Over the last few years, the average temperature has been increasing. If this trend

continues, many economic sectors will become exposed to additional costs and losses.

Such companies as agricultural may even reconsider changes to its production tech-

nologies. In a short term, though, purchasing some insurance contracts on weather

could cover losses.

In this thesis we created insurance contracts on the temperature that include

some hybrid characteristics of both standard derivatives (such as exotic options) and

insurance contracts (in terms of the premium calculation). First of all, we modeled

the underlying (in our case it is average daily temperature) so that it would be

possible to simulate future trajectories of the series. However, instead of geometric

Brownian motion process that is often used to price derivatives, our models are of

ARMA-GARCH type. Consequently, pricing such contracts requires an approach

alternative to Blach-Scholes.

Some underlying models we used are similar to the one presented by Sean D.

Campbell and Francis X. Diebolt (2005), as well as other specifications of ARMA-

GARCH models, are studied to explain and simulate the series. Model II is a GARCH

model with seasonal components in mean and variance with students’t distribution

of errors is the best model to catch the extreme picks. For that reason, it is used in

pricing Parisian-style insurance contracts. The model build with model combination

approach has an average performance both in term of simulated values so that we

use it as a model for the underlying.

Asian-style insurance contracts may be the most interesting for a potential policy

37

holder as exposure to the risk cased by the average temperature increase is increasing

every year. What is more, this contracts will also account for the extreme jumps

in temperature. Parisian-style insurance contract has a more limited application,

however, it is a cheaper alternative to Asian-style contracts.

38

Appendices

39

Appendix A

Discrete Fourier transformation,

MATLAB function

%% f f t c o v c o s i s the d i s c r e t e Four ie r transform (DFT)

% of au tocova r i a t e s o f vec tor X

func t i on [X,X mag , X phase , k ,w]= f f t c o v c o s (x )

T=length (x ) ;

w=0:(1/T)∗2∗ pi :2∗ pi ;

k=length (w) ;

%cons t ruc t c ova r i a t e s

f o r i i =1:T 2 % at T 1 we have only cov (num1 , num2)=0 , so no need

cov matr=cov (x ( 1 :T i i ) , x(1+ i i :T) ) ;

autocov ( i i )=cov matr ( 1 , 2 ) ;

end

f o r l =1:k %number o f f requency bin

X sum=0;

f o r m=1:T 2 %number o f the autocov

X temp=autocov (m)∗ cos (w( l )∗m) ;

X sum=X sum+X temp ;

end

X( l )=(1/2∗ pi )∗ ( var (x)+2∗X sum ) ;

end

X mag=abs (X) ;

X phase=angle (X) ;

40

Appendix B

An example of Fourier analysis on

simulated series, MATLAB script

c l e a r

c l c

t =1:1500;

T=length ( t ) ;

w=l i n spa c e (0 ,2∗ pi ,T) ; %frequency in rad iance

%s imulat i on o f AR proce s s with harmonic component

f1 =0.02;

f 2 =0.075;

a0=5.3;

a1=0.7;

a2=1.2;

a3 = 5 . 6 ;

a4=0.09;

ar=arima ( ’ Constant ’ , 0 . 5 , ’AR’ , { 0 . 7 , 0 . 3 6 } , ’ Variance ’ , . 1 ) ;

x0=s imulate ( ar , 1 5 0 0 ) ’ ;

x=a0∗x0+(a1∗ cos ( f1 ∗ t)+a2∗ s i n ( f1 ∗ t ))+( a3∗ cos ( f2 ∗ t)+a4∗ s i n ( f2 ∗ t ) ) ;

X=f f t c o v c o s (x ) ;

X mag=abs (X) ;

f i g u r e (1 )

p lo t (X mag ) ;

t i t l e ( ’ Magnitude spectrum of the observat ions ’ ) ;

bin1=6; % from the graph o f the magnitude spuctrum

bin2=19;

f r 1=w( bin1 ) ;

f r 2=w( bin2 ) ;

harm1=cos ( f1 ∗ t)+ s in ( f1 ∗ t ) ;

harm2=cos ( f2 ∗ t)+ s in ( f2 ∗ t ) ;

harm3=harm1+harm2 ;

f i g u r e (2)

subplot (4 , 1 , 1 )

p lo t (x , ’ blue ’ )

t i t l e ( ’ Simulated time s e r i e s ’ )

41

hold on

subplot (4 , 1 , 2 )

p lo t (harm1 , ’ red ’ )

t i t l e ( ’ Harmonic component 1 ’ )

hold on

subplot (4 , 1 , 3 )

p lo t (harm2 , ’ red ’ )

t i t l e ( ’ Harmonic component 2 ’ )

hold on

subplot (4 , 1 , 4 )

p lo t (harm3 , ’ red ’ )

t i t l e ( ’The combination o f the harmonic components ’ )

42

Appendix C

Spectral analysis,MATLAB script

%% SPECTRAL ANALYSIS OF THE OBSERVATIONS

c l e a r

c l c

f i l e p a t h 1 = ’D:\ un i v e r s i t y \ !\ Thes is\Data\dataLondon . csv ’ ;

data1=load data ( f i l e p a t h 1 ) ;

T=length ( data1 ) ;

s f =365; %sampling f requency

w=l i n spa c e (0 ,2∗ pi ,T) ; %frequency in rad iance

f=1/ s f : 1/ s f :T/ s f ; %r e s c a l e data wrt to sampling frequency

t=1:T;

f r=w( t ) ;

[X,X mag , X phase , k ,w]= f f t c o v c o s ( data1 ) ;

f i g u r e (1 )

p lo t (X mag)

%t i t l e ( ’ Magnitude spectrum of the observat ions ’ ) ;

f i g u r e (2 )

X mag log=log (X mag ) ;

p l o t ( X mag log )

%t i t l e ( ’ Log t rans fo rmat ion o f the magnitude spectrum ’ ) ;

bin0=1;

bin1=7;% from the p lo t o f the magnitude spectrum

bin2=14;% from the p lo t o f the magnitude spectrum

cyc l e s 1=f ( bin1 )∗ s f ; % # of c y c l e s in the sample

durat ion=T/( f ( bin1 )∗ s f ) ; %durat ion o f 1 cy c l e

f r y e a r=durat ion /365 ; %frequency in year terms

f r 0=w( bin0 )∗ s f ;

f r 1=w( bin1 )∗ s f ; % frequency o f the harmonic s i n+cos func t i on

f r 2=w( bin2 )∗ s f ;

t e s t 1=cos ( f r 1 ∗ f )+ s in ( f r 1 ∗ f );% f o r s ea sona l component o f the from s in+cos

t e s t 2=cos ( f r 2 ∗ f )+ s in ( f r 2 ∗ f ) ; % f o r s ea sona l component o f the from s in+cos

t e s t 3=t e s t 1+t e s t 2 ;

f i g u r e (3 )

subplot ( 4 , 1 , 1 ) ;

p l o t ( data1 , ’ b ’ ) ;

t i t l e ( ’ Graph o f the observat ions ’ ) ;

ax i s ( [ 0 , 2204 , 5 , 3 0 ] ) ;

43

hold on

subplot (4 , 1 , 2 )

p lo t ( te s t1 , ’ r ’ ) ;

t i t l e ( ’ Harmonic component 1 ’ )

ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;

hold on

subplot (4 , 1 , 3 )

p lo t ( te s t2 , ’ r ’ ) ;

t i t l e ( ’ Harmonic component 2 ’ )

ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;

hold on

subplot (4 , 1 , 4 )

p lo t ( te s t3 , ’ r ’ ) ;

t i t l e ( ’ Combination o f the harmonic components ’ )

ax i s ( [ 0 , 2204 , 3 , 3 ] ) ;

% SPECTRAL ANALYSIS OF THE SQUARED OBSERVATIONS

x1 sq=data1 . ˆ 2 ;

[X2 , X mag2 , X phase2 , k2 ,w2]= f f t c o v c o s ( x1 sq ) ;

f i g u r e (4 )

p lo t (X mag2)

%t i t l e ( ’ Magnitude spectrum of the squared observat ions ’ ) ;

f i g u r e (5 )

X mag log2=log (X mag2 ) ;

p l o t ( X mag log2 )

bin11=7;% from the p lo t o f the magnitude spectrum

bin21=13;% from the p lo t o f the magnitude spectrum

cyc l e s 11=f ( bin11 )∗ s f ; % # of c y c l e s in the sample

durat ion11=T/( f ( bin11 )∗ s f ) ; %durat ion o f 1 cy c l e

f r y e a r 1 1=durat ion11 /365 ; %frequency in year terms

f r 11=w( bin11 )∗ s f ; % frequency o f the harmonic s i n / cos func t i on

f r 21=w( bin21 )∗ s f ;

t e s t 11=cos ( f r 11 ∗ f )+ s in ( f r 11 ∗ f );% f o r s ea sona l component o f the from s in+cos

t e s t 21=cos ( f r 21 ∗ f )+ s in ( f r 21 ∗ f ) ;

t e s t 31=t e s t 1+t e s t 2 ;

f i g u r e (6 )

subplot ( 4 , 1 , 1 ) ;

p l o t ( x1 sq , ’ b ’ ) ;

t i t l e ( ’ Graph o f the squared observat ions ’ ) ;

ax i s ( [ 0 , 2204 , 0 , 6 0 0 ] ) ;

hold on

subplot (4 , 1 , 2 )

p lo t ( test11 , ’ r ’ ) ;

t i t l e ( ’ Harmonic component 1 ’ )

ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;

hold on

subplot (4 , 1 , 3 )

p lo t ( test21 , ’ r ’ ) ;

t i t l e ( ’ Harmonic component 2 ’ )

ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;

hold on

subplot (4 , 1 , 4 )

p lo t ( test31 , ’ r ’ ) ;

t i t l e ( ’ Combination o f the harmonic components ’ )

ax i s ( [ 0 , 2204 , 2 . 5 , 3 ] ) ;

44

Appendix D

Diagnostics of the residuals

Additionally to Jack-Berra test on the residuals given in section 3.4, in the fol-

lowing more profound analysis is shown. Particularly the hypothesis tested are the

significance of the explanatory variables with Student’s t-test; the absence of the

serial autocorrelation with Box-Pierce test and the homoscedasticity of the residuals

with White test.

In Figure D.1 one can see that all of the models show the absence of the serial

autocorrelation in residuals: as p-values of Box-Pierce test are always greater than

0.05 we accept the null hypothesis of the residuals independence.

Table D.1: White test, Results

Model I Model II Model III

Stat P-val Stat P-val Stat P-val

F-stat 1.030 0.389 1.120 0.273 1.181 0.073

obs R2 156.281 0.389 49.175 0.274 176.936 0.081

Model IV Model V Model VI

Stat P-val Stat P-val Stat P-val

F-stat 1.212 0.055 1.025 0.153 1.048 0.101

obs R2 161.149 0.062 153.251 0.272 173.610 0.336

45

The results of White test on heteroskedasticity suggest to accept the null hypoth-

esis of absence of the heteroskedasticity (see Table D.1).

The output of the estimation is given in Table D.2. Regarding Student’s t-test, we

tried to include only statistically significant covariates, yet it was not always possible.

For example in autoregressive part, we included from 6 to 14 lags in order to avoid

serial autocorrelation. It appeared quite often that the coefficients corresponding to

the autocorrelation of lag higher than 10 were significant even though the covariates of

smaller lags were not significant. It can be explained as an another sign of seasonality

in the series. The insignificance of the coefficients was also accepted when one of the

parts of the seasonal component given in (2.7) is significant while another is not.

As long as discarding one part of seasonal component (2.7) will contradict Fourier

theorem, we left it as it is. In other cases, insignificant single coefficients show joint

significance.

(a) Model I (b) Model II (c) Model III

(d) Model IV (e) Model V (f) Model VI

Figure D.1: Box-Pierce P-values

46

Table D.2: Estimation output

Model I

Variable Coefficient Std. Error z-Statistic Prob.

α1 0.816331 0.024008 34.00203 0

α2 -0.074036 0.029262 -2.530106 0.0114

α3 0.037956 0.029853 1.271417 0.2036

α4 -0.00538 0.031392 -0.171394 0.8639

α5 0.029624 0.031944 0.927356 0.3537

α6 0.072276 0.031051 2.327686 0.0199

α7 0.010346 0.030288 0.341599 0.7327

α8 0.00205 0.030532 0.067147 0.9465

α9 0.019006 0.030491 0.62332 0.5331

α10 -0.03781 0.03068 -1.232419 0.2178

α11 0.054 0.030664 1.761022 0.0782

α12 -0.012978 0.030046 -0.431934 0.6658

α13 0.016889 0.029027 0.581844 0.5607

α14 0.047825 0.023087 2.071479 0.0383

λ11 -0.266298 0.090805 -2.93264 0.0034

λ12 0.300571 0.070714 4.250531 0

γ 0.000211 8.77E-05 2.402942 0.0163

Variance Equation

ρ 1.598102 0.074747 21.38005 0

β1 1.277811 0.004503 283.792 0

β2 -1.018661 0.00388 -262.5523 0

β3 -0.096077 0.002536 -37.89183 0

β4 0.531661 0.005297 100.3698 0

β5 -0.333428 0.003557 -93.74113 0

β6 0.228513 0.003153 72.47252 0

λ21 0.302219 0.077317 3.908848 0.0001

λ22 -0.090097 0.074289 -1.21279 0.2252

(a) Estimation output: Model I

47

Model II

Variable Coefficient Std. Error z-Statistic Prob.

α1 0.810341 0.023969 33.8079 0

α2 -0.077142 0.031124 -2.478536 0.0132

α3 0.052411 0.032014 1.637124 0.1016

α4 -0.018263 0.030886 -0.591302 0.5543

α5 0.033361 0.029311 1.138192 0.255

α6 0.067604 0.029434 2.296791 0.0216

α7 0.024325 0.031277 0.777718 0.4367

α8 -0.009702 0.031898 -0.304157 0.761

α9 0.024038 0.030216 0.795532 0.4263

α10 -0.032706 0.029915 -1.093301 0.2743

α11 0.059116 0.029609 1.996522 0.0459

α12 -0.017076 0.029275 -0.583312 0.5597

α13 0.011817 0.029379 0.402216 0.6875

α14 0.049594 0.023396 2.119797 0.034

λ11 -0.241109 0.092948 -2.594006 0.0095

λ12 0.274782 6.78E-02 4.051529 0.0001

γ 0.000193 8.84E-05 2.187391 0.0287

Variance Equation

ρ 2.203899 1.931892 1.140798 0.254

φ1 0.016164 0.018543 0.871671 0.3834

φ2 0.044892 0.019562 2.294804 0.0217

β1 0.62895 0.225795 2.785498 0.0053

β2 -0.778208 0.253479 -3.070109 0.0021

β3 -0.195731 0.317636 -0.616211 0.5378

β4 0.694454 0.328699 2.112737 0.0346

β5 -0.555604 0.25648 -2.166268 0.0303

β6 0.583307 0.183175 3.184432 0.0015

λ21 0.451797 0.435522 1.03737 0.2996

λ22 -0.222142 0.179655 -1.23649 0.2163

(b) Estimation output: Model II

48

Model III

Variable Coefficient Std. Error z-Statistic Prob.

δ 0.552785 0.073842 7.486094 0

α1 0.782545 0.023722 32.98835 0

α2 -0.08635 0.028944 -2.983384 0.0029

α3 0.032704 0.029873 1.094755 0.2736

α4 -0.018279 0.030173 -0.605826 0.5446

α5 0.021081 0.030556 0.689897 0.4903

α6 0.063528 0.029513 2.15252 0.0314

α7 0.007537 0.028681 0.262796 0.7927

α8 -0.019515 0.029192 -0.6685 0.5038

α9 0.019895 0.030165 0.659556 0.5095

α10 -0.043961 0.03019 -1.456135 0.1454

α11 0.037424 0.029874 1.252755 0.2103

α12 -0.023564 0.0289 -0.81535 0.4149

α13 0.009581 0.028342 0.338047 0.7353

α14 0.019924 0.023123 0.861647 0.3889

λ11 -1.405722 1.71E-01 -8.2063 0

λ12 -0.055451 8.11E-02 -0.683346 0.4944

γ 0.000131 8.99E-05 1.45397 0.146

Variance Equation

ρ 2.1241 1.015668 2.091332 0.0365

β1 0.73098 0.135747 5.384859 0

β2 -1.007305 0.005079 -198.3274 0

β3 0.711041 0.137096 5.186445 0

(c) Estimation output: Model III

49

Model IV

Variable Coefficient Std. Error z-Statistic Prob.

δ 0.546156 0.073497 7.430961 0

α1 0.783229 0.023717 33.02329 0

α2 -0.086278 0.0289 -2.985441 0.0028

α3 0.036149 0.029933 1.207646 0.2272

α4 -0.013872 0.030291 -0.457941 0.647

α5 0.016728 0.029106 0.574731 0.5655

α6 0.062351 0.024859 2.508159 0.0121

α7 0.01101 0.027809 0.3959 0.6922

α8 -0.015875 0.029196 -0.543758 0.5866

α9 0.015813 0.03018 0.52395 0.6003

α10 -0.044 0.02993 -1.470117 0.1415

α11 0.040415 0.028495 1.418324 0.1561

α12 -0.022481 0.028647 -0.784765 0.4326

α13 0.010763 0.028382 0.379235 0.7045

α14 0.020959 0.023047 0.909419 0.3631

λ11 -1.317104 1.71E-01 -7.687573 0

λ12 -0.045395 8.17E-02 -0.55581 0.5783

Variance Equation

ρ 0.871846 0.971893 0.897059 0.3697

β1 0.769045 0.25746 2.987045 0.0028

(d) Estimation output: Model IV

50

Model

VM

odel

VI

Var

iable

Coeffi

cien

tStd

.E

rror

z-Sta

tist

icP

rob.

Var

iable

Coeffi

cien

tStd

.E

rror

z-Sta

tist

icP

rob.

δ1.

6954

340.

1784

849.

4990

60

δ1.

7437

560.

1885

169.

2499

030

α1

0.78

3913

0.02

3595

33.2

2385

0α

10.

7802

430.

0237

4732

.856

830

α2

-0.0

8122

0.02

9005

-2.8

0023

80.

0051

α2

-0.0

8182

50.

0290

9-2

.812

859

0.00

49

α3

0.01

9394

0.02

933

0.66

1247

0.50

85α

30.

0241

270.

0296

010.

8150

690.

415

α4

-0.0

0796

20.

0301

3-0

.264

246

0.79

16α

4-0

.010

077

0.03

0505

-0.3

3032

60.

7412

α5

0.01

892

0.03

0525

0.61

982

0.53

54α

50.

0176

450.

0309

320.

5704

560.

5684

α6

0.06

8179

0.02

3502

2.90

1021

0.00

37α

60.

0631

150.

0237

532.

6571

750.

0079

λ11

-1.7

0702

80.

1986

28-8

.594

111

0λ

11

-1.8

3041

10.

2121

88-8

.626

366

0

λ12

0.01

2443

0.10

8199

0.11

5003

0.90

84λ

12

-0.0

1661

50.

1132

9-0

.146

659

0.88

34

λ21

-0.1

2028

30.

0646

1-1

.861

691

0.06

26

λ22

-0.0

623

0.06

4384

-0.9

6762

80.

3332

Vari

ance

Equati

on

Vari

ance

Equati

on

ρ0.

0524

170.

0018

528

.330

560

ρ0.

3734

060.

5782

060.

6458

010.

5184

β1

1.94

6724

0.00

6242

311.

8866

0β

10.

7170

480.

4378

451.

6376

760.

1015

β2

-0.9

8673

40.

0061

17-1

61.3

164

0λ

21

0.05

8346

0.09

3208

0.62

5979

0.53

13

λ21

0.00

7408

0.00

1839

4.02

8486

0.00

01λ

22

-0.0

1737

52.

57E

-02

-0.6

7535

90.

4994

λ22

-0.0

0228

90.

0018

74-1

.221

087

0.22

21

(e)

Est

imat

ion

outp

ut:

Model

V,

Model

VI

51

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