— Ca’ Foscari Dorsoduro 3246 30123 Venezia
Università
Ca’Foscari
Venezia
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
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.
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
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
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
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
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 Ni˜no 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 Ni˜no 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
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.
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.
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.pdf
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:
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 = Z ∞ 0 [P r(Y > y)]1/ϑdy = Z ∞ 0 [1 − F (Y )]1/ϑ2dy (1.3)
where ϑ2 > 1. Equation 1.3 can be alternatively shown in discreet form as:
Πrisk.adj = ∞ X i=0 p1/ϑ2 Yi 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,
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-Picqu´e 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
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
X
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
following: CDD = T X 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 threshthresh-old 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.
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 represenrepresen-tation which is commonly
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 ∞ X 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(f
jτ ) − 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 ∞ X τ =−∞ γj(cos(fjτ ) − isin(fjτ )) = 1 2π γ0+ ∞ X 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) = 2π1 γ0+ 2 ∞ P 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) = 2π1 γ0+ 2 T −1 P τ =1 γjcos(fjτ ) (2.6)
After the frequencies are calculated, the models seasonal component can be given as the combination of the following harmonic functions :
ωt = K
X
k=1
(λ1kcos(fkt) + λ2ksin(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).
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 X i=1 αiXt−i+ σtεt; εt iid ∼ (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
X
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 fvarpresent in the model for the volatility. fvarhas the same structure
as the frequency of the mean model specified in (2.2). So the variance model is given by: σt2 = ρ + $t+ P X p=1 βpσ2t−p+ Q X 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);
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 X i=1 αiXt−i+ δσt2+ σtεt (2.12) σt2 = ρ + P X p=1 βpσt−p2 + Q X q=1 ϕq(σt−qεt−q)2; εt iid ∼ (0, σε2) (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:
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 X i=1 αiXt−i+ δlog(σt2) + zt (2.15) where log(σ2
t) is logarithm of the GARCH in mean component;
log(σt2) = ρ + $t+ P X p=1 βplog(σt−p2 ) + Q X q=1 ϕqg(zt−q); εt iid ∼ (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);
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.
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.
(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.
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
the effect of the harmonic function occurs in 1 year) can be also defined: dcycle = T fyearfsamp (3.2) fp.y = dcycle 365 (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
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.
(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
f r1 7 1.0155 0.0170 370.6667
f r2 14 0.4687 0.0367 171.0769
f r3 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 f r3 (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.
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.
Gaus-sian quasi-likelyhood function is: L(θ) = T Y t=1 1 p 2π˜σ2 t exp − ε 2 t 2˜σ2 t (3.5)
To make computation easier the log-likelyhood function can be used :
lnL(θ) = − T X t=1 1 2 log(2π) + log(˜σ2t) + ε 2 t ˜ σ2 t (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 Y t=1 Γ[12(υ + 1)] π12Γ[1 2υ] (υ − 2)σ2 t −12 1 + (εtσt) 2 (υ − 2)σ2 t −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 : ˆ θQM LE = arg max θ∈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)), ωtand $tare 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).
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.
(a) Model II (b) Model IV
(c) Model VI
(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
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
(a) Model I (b) Model II
(c) Model III (d) Model IV
(e) Model V (f) Model VI
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 bXt+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 bXt+h be the forecast combination. The forecast combination is the
b Xt+h = 6 X i=1 υiXbt+h,i, (3.9) υi = 1/M SEi 6 P i=1 1/M SEi (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
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.
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 dein-crease. 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.
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.
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.
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
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 ARMA-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.
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.
Appendix A
Discrete Fourier transformation,
MATLAB function
%% f f t c o v c o s i s t h e d i s c r e t e F o u r i e r t r a n s f o r m (DFT) % o f a u t o c o v a r i a t e s o f v e c t o r X f u n c t i o n [ X, X mag , X p h a s e , k , w]= f f t c o v c o s ( x ) T=l e n g t h ( x ) ; w= 0 : ( 1 /T) ∗ 2 ∗ p i : 2 ∗ p i ; k=l e n g t h (w ) ; %c o n s t r u c t c o v a r i a t e s f o r i i =1:T 2 % a t T 1 we h a v e o n l y c o v ( num1 , num2) = 0 , s o no n e e d c o v m a t r=c o v ( x ( 1 : T i i ) , x(1+ i i : T ) ) ; a u t o c o v ( i i )= c o v m a t r ( 1 , 2 ) ; end f o r l =1: k %number o f f r e q u e n c y b i n X sum =0; f o r m=1:T 2 %number o f t h e a u t o c o v X temp=a u t o c o v (m) ∗ c o s (w( l ) ∗m) ; X sum=X sum+X temp ;end
X( l ) = ( 1 / 2 ∗ p i ) ∗ ( v a r ( x )+2∗X sum ) ; end
X mag=a b s (X ) ; X p h a s e=a n g l e (X ) ;
Appendix B
An example of Fourier analysis on
simulated series, MATLAB script
c l e a r c l c t = 1 : 1 5 0 0 ; T=l e n g t h ( t ) ; w=l i n s p a c e ( 0 , 2 ∗ p i , T ) ; %f r e q u e n c y i n r a d i a n c e %s i m u l a t i o n o f AR p r o c e s s w i t h h a r m o n i c component f 1 = 0 . 0 2 ; f 2 = 0 . 0 7 5 ; a0 = 5 . 3 ; a1 = 0 . 7 ; a2 = 1 . 2 ; a3 = 5 . 6 ; a4 = 0 . 0 9 ; a r=a r i m a ( ’ C o n s t a n t ’ , 0 . 5 , ’ AR’ , { 0 . 7 , 0 . 3 6 } , ’ V a r i a n c e ’ , . 1 ) ; x0=s i m u l a t e ( a r , 1 5 0 0 ) ’ ; x=a0 ∗ x0+( a1 ∗ c o s ( f 1 ∗ t )+ a2 ∗ s i n ( f 1 ∗ t ) ) + ( a3 ∗ c o s ( f 2 ∗ t )+ a4 ∗ s i n ( f 2 ∗ t ) ) ; X= f f t c o v c o s ( x ) ; X mag=a b s (X ) ; f i g u r e ( 1 ) p l o t ( X mag ) ; t i t l e ( ’ Magnitude s p e c t r u m o f t h e o b s e r v a t i o n s ’ ) ; b i n 1 =6; % f r o m t h e g r a p h o f t h e m a g n i t u d e s p u c t r u m b i n 2 =19; f r 1=w( b i n 1 ) ; f r 2=w( b i n 2 ) ; harm1=c o s ( f 1 ∗ t )+ s i n ( f 1 ∗ t ) ; harm2=c o s ( f 2 ∗ t )+ s i n ( f 2 ∗ t ) ; harm3=harm1+harm2 ; f i g u r e ( 2 ) s u b p l o t ( 4 , 1 , 1 ) p l o t ( x , ’ b l u e ’ ) t i t l e ( ’ S i m u l a t e d t i m e s e r i e s ’ )
h o l d on s u b p l o t ( 4 , 1 , 2 ) p l o t ( harm1 , ’ r e d ’ ) t i t l e ( ’ Harmonic component 1 ’ ) h o l d on s u b p l o t ( 4 , 1 , 3 ) p l o t ( harm2 , ’ r e d ’ ) t i t l e ( ’ Harmonic component 2 ’ ) h o l d on s u b p l o t ( 4 , 1 , 4 ) p l o t ( harm3 , ’ r e d ’ ) t i t l e ( ’ The c o m b i n a t i o n o f t h e h a r m o n i c components ’ )
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: \ u n i v e r s i t y \ ! \ T h e s i s \ Data \ dataLondon . c s v ’ ; d a t a 1=l o a d d a t a ( f i l e p a t h 1 ) ; T=l e n g t h ( d a t a 1 ) ; s f = 3 6 5 ; %s a m p l i n g f r e q u e n c y w=l i n s p a c e ( 0 , 2 ∗ p i , T ) ; %f r e q u e n c y i n r a d i a n c e f =1/ s f : 1 / s f : T/ s f ; %r e s c a l e d a t a w r t t o s a m p l i n g f r e q u e n c y t =1:T ; f r =w( t ) ; [ X, X mag , X p h a s e , k , w]= f f t c o v c o s ( d a t a 1 ) ; f i g u r e ( 1 ) p l o t ( X mag ) % t i t l e ( ’ Magnitude s p e c t r u m o f t h e o b s e r v a t i o n s ’ ) ; f i g u r e ( 2 ) X m a g l o g=l o g ( X mag ) ; p l o t ( X m a g l o g ) % t i t l e ( ’ Log t r a n s f o r m a t i o n o f t h e m a g n i t u d e s p e c t r u m ’ ) ; b i n 0 =1; b i n 1 =7;% f r o m t h e p l o t o f t h e m a g n i t u d e s p e c t r u m b i n 2 =14;% f r o m t h e p l o t o f t h e m a g n i t u d e s p e c t r u m c y c l e s 1=f ( b i n 1 ) ∗ s f ; % # o f c y c l e s i n t h e s a m p l e d u r a t i o n=T/ ( f ( b i n 1 ) ∗ s f ) ; %d u r a t i o n o f 1 c y c l e f r y e a r =d u r a t i o n / 3 6 5 ; %f r e q u e n c y i n y e a r t e r m s f r 0=w( b i n 0 ) ∗ s f ; f r 1=w( b i n 1 ) ∗ s f ; % f r e q u e n c y o f t h e h a r m o n i c s i n+c o s f u n c t i o n f r 2=w( b i n 2 ) ∗ s f ; t e s t 1=c o s ( f r 1 ∗ f )+ s i n ( f r 1 ∗ f ) ; % f o r s e a s o n a l component o f t h e f r o m s i n+c o s t e s t 2=c o s ( f r 2 ∗ f )+ s i n ( f r 2 ∗ f ) ; % f o r s e a s o n a l component o f t h e f r o m s i n+c o s t e s t 3=t e s t 1+t e s t 2 ; f i g u r e ( 3 ) s u b p l o t ( 4 , 1 , 1 ) ; p l o t ( d a t a 1 , ’ b ’ ) ; t i t l e ( ’ Graph o f t h e o b s e r v a t i o n s ’ ) ; a x i s ( [ 0 , 2 2 0 4 , 5 , 3 0 ] ) ;
h o l d on s u b p l o t ( 4 , 1 , 2 ) p l o t ( t e s t 1 , ’ r ’ ) ; t i t l e ( ’ Harmonic component 1 ’ ) a x i s ( [ 0 , 2 2 0 4 , 1 . 5 , 1 . 5 ] ) ; h o l d on s u b p l o t ( 4 , 1 , 3 ) p l o t ( t e s t 2 , ’ r ’ ) ; t i t l e ( ’ Harmonic component 2 ’ ) a x i s ( [ 0 , 2 2 0 4 , 1 . 5 , 1 . 5 ] ) ; h o l d on s u b p l o t ( 4 , 1 , 4 ) p l o t ( t e s t 3 , ’ r ’ ) ; t i t l e ( ’ C o m b i n a t i o n o f t h e h a r m o n i c components ’ ) a x i s ( [ 0 , 2 2 0 4 , 3 , 3 ] ) ;
% SPECTRAL ANALYSIS OF THE SQUARED OBSERVATIONS x 1 s q=d a t a 1 . ˆ 2 ; [ X2 , X mag2 , X p h a s e 2 , k2 , w2]= f f t c o v c o s ( x 1 s q ) ; f i g u r e ( 4 ) p l o t ( X mag2 ) % t i t l e ( ’ Magnitude s p e c t r u m o f t h e s q u a r e d o b s e r v a t i o n s ’ ) ; f i g u r e ( 5 ) X m a g l o g 2=l o g ( X mag2 ) ; p l o t ( X m a g l o g 2 ) b i n 1 1 =7;% f r o m t h e p l o t o f t h e m a g n i t u d e s p e c t r u m b i n 2 1 =13;% f r o m t h e p l o t o f t h e m a g n i t u d e s p e c t r u m c y c l e s 1 1=f ( b i n 1 1 ) ∗ s f ; % # o f c y c l e s i n t h e s a m p l e d u r a t i o n 1 1=T/ ( f ( b i n 1 1 ) ∗ s f ) ; %d u r a t i o n o f 1 c y c l e f r y e a r 1 1=d u r a t i o n 1 1 / 3 6 5 ; %f r e q u e n c y i n y e a r t e r m s f r 1 1=w( b i n 1 1 ) ∗ s f ; % f r e q u e n c y o f t h e h a r m o n i c s i n / c o s f u n c t i o n f r 2 1=w( b i n 2 1 ) ∗ s f ; t e s t 1 1=c o s ( f r 1 1 ∗ f )+ s i n ( f r 1 1 ∗ f ) ; % f o r s e a s o n a l component o f t h e f r o m s i n+c o s t e s t 2 1=c o s ( f r 2 1 ∗ f )+ s i n ( f r 2 1 ∗ f ) ; t e s t 3 1=t e s t 1+t e s t 2 ; f i g u r e ( 6 ) s u b p l o t ( 4 , 1 , 1 ) ; p l o t ( x 1 s q , ’ b ’ ) ; t i t l e ( ’ Graph o f t h e s q u a r e d o b s e r v a t i o n s ’ ) ; a x i s ( [ 0 , 2 2 0 4 , 0 , 6 0 0 ] ) ; h o l d on s u b p l o t ( 4 , 1 , 2 ) p l o t ( t e s t 1 1 , ’ r ’ ) ; t i t l e ( ’ Harmonic component 1 ’ ) a x i s ( [ 0 , 2 2 0 4 , 1 . 5 , 1 . 5 ] ) ; h o l d on s u b p l o t ( 4 , 1 , 3 ) p l o t ( t e s t 2 1 , ’ r ’ ) ; t i t l e ( ’ Harmonic component 2 ’ ) a x i s ( [ 0 , 2 2 0 4 , 1 . 5 , 1 . 5 ] ) ; h o l d on s u b p l o t ( 4 , 1 , 4 ) p l o t ( t e s t 3 1 , ’ r ’ ) ; t i t l e ( ’ C o m b i n a t i o n o f t h e h a r m o n i c components ’ ) a x i s ( [ 0 , 2 2 0 4 , 2 . 5 , 3 ] ) ;
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
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
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 0.07428947 -1.21279 0.2252
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 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
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
Mo del V Mo del VI V ariable Co efficien t Std. Error z-Statistic Prob. V ariable Co efficien t Std. Error z-Statistic Prob. δ 1.695434 0.178484 9.49906 0 δ 1.743756 0.188516 9.249903 0 α1 0.783913 0.023595 33.22385 0 α1 0.780243 0.023747 32.85683 0 α2 -0.08122 0.029005 -2.800238 0.0051 α2 -0.081825 0.02909 -2.812859 0.0049 α3 0.019394 0.02933 0.661247 0.5085 α3 0.024127 0.029601 0.815069 0.415 α4 -0.007962 0.03013 -0.264246 0.7916 α4 -0.010077 0.030505 -0.330326 0.7412 α5 0.01892 0.030525 0.61982 0.5354 α5 0.017645 0.030932 0.570456 0.5684 α6 0.068179 0.023502 2.901021 0.0037 α6 0.063115 0.023753 2.657175 0.0079 λ11 -1.707028 0.198628 -8.594111 0 λ11 -1.830411 0.212188 -8.626366 0 λ12 0.012443 0.108199 0.115003 0.9084 λ12 -0.016615 0.11329 -0.146659 0.8834 λ21 -0.120283 0.06461 -1.861691 0.0626 λ22 -0.0623 0.064384 -0.967628 0.3332 V ariance Equation V ariance Equation ρ 0.052417 0.00185 28.33056 0 ρ 0.373406 0.578206 0.645801 0.5184 β1 1.946724 0.006242 311.8866 0 β1 0.717048 0.437845 1.637676 0.1015 β2 -0.986734 0.006117 -161.3164 0 λ21 0.058346 0.093208 0.625979 0.5313 λ21 0.007408 0.001839 4.028486 0.0001 λ22 -0.017375 2.57E-02 -0.675359 0.4994 λ22 -0.002289 0.001874 -1.221087 0.2221 (e) Estimation ou tput: Mo del V, Mo del VI
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