This thesis examines scaling properties of petrol and diesel prices in the Czech Republic and a crude oil price over the period from January 2004 to February 2013. Using generalised Hurst exponent and multifractal detrended fluctuation analysis techniques we find out that crude oil market is efficient, do not contain long memory and the returns exhibit monofractal behaviour. On the other hand, petrol and diesel markets in the Czech Republic are not efficient, because their returns contain long-range dependence in autocorrelations and exhibit multifractal behaviour caused mostly by fat-tailed distribution. Thus, fuels can be modelled by complex methods like Markov switching multifractal model. JEL Classification C15, C16, C46 Keywords petrol, diesel, crude oil, long memory, multifrac- tality, GHE, MF-DFA Author's e-mail martin.baletka@ies-prague.org Supervisor's e-mail kristoufek@ies-prague.org Abstrakt Tato práce zkoumá škálování cen benzínu a motorové nafty v České repub- lice a ceny ropy na datech v období od ledna 2004 do února 2013. Použitím metod zobecněného Hurstova exponentu a multifraktální detrendované fluk- tuační analýzy jsme zjistili, že trh s ropou je efektivní, bez přítomnosti dlouhé paměti v autokorelacích a výnosy na trhu s ropou vykazují monofraktální...

O objetivo deste trabalho foi estudar o comportamento da volatilidade do preço da soja negociada em contratos futuros na BM&FBOVESPA (série SFI). O estudo foi realizado por meio da comparação entre duas abordagens: na primeira, foi utilizada a série de retornos absolutos da série em questão para representar a volatilidade da mesma, que se mostrou persistente ao longo do tempo, comprovando o fato de que a série possui o comportamento de memória longa. Por ter apresentado tal comportamento, fez-se necessária a utilização de modelos ARFIMA (\"Autorregressivos Fracionários Integrados de Médias Móveis\") estes, que são capazes de capturar de maneira efetiva tal comportamento. Ainda dentro desta abordagem, os modelos foram estimados de duas maneiras distintas: a primeira, em que todos os parâmetros foram estimados simultaneamente e a segunda, em que primeiramente foi estimado o parâmetro de memória longa, diferenciada a série e, posteriormente, foram ajustados os modelos ARIMA nos dados diferenciados. Por fim, a segunda abordagem utilizada no trabalho é a mais comum em pesquisas acadêmicas: foi realizada a estimação dos modelos GARCH (\"Autorregressivos Generalizados de Heteroscedasticidade Condicional\") diretamente na série de retornos. Neste estudo, concluímos que a primeira abordagem se mostrou mais eficiente, dados os critérios de comparação utilizados. / The purpose of this article was to study the volatility of the soybean price traded in futures contracts on the BM&FBOVESPA (SFI series). The study was conduct by comparison between two approaches: first, was use the series of absolute returns of the respective series, to represent its volatility, which was persistent over time, proving the fact that the series has a long memory behavior. Because of such behavior, it was necessary to use ARFIMA models (\"Autoregressive Fractional Integrated Moving Average\"), which are able to capture effectively such behavior. Still using this approach, the models were estimate in two different ways: first, which all parameters were estimate simultaneously, and the second one, that was first estimated the long memory parameter, differentiated the series and, later, adjusted the ARIMA models in differentiated data. Finally, the second approach used in this work is the most common in academic research: the estimation of GARCH models (\"Generalized Autoregressive Conditional Heretoscskedasticity\") directly in the returns series of the studied series. In this study, we conclude that the first approach was more effective, given the comparison criteria used.

O objetivo deste trabalho é apresentar e comparar diferentes métodos de modelagem da volatilidade (variância condicional) de séries temporais financeiras. O modelo ARFIMA é empregado para capturar o comportamento de memória longa observado na volatilidade de séries financeiras. Por sua vez, o modelo GARCH é utilizado para modelar a volatilidade variando no tempo destas séries. Finalmente, o modelo FIGARCH é utilizado para modelar a dinâmica dos retornos de séries temporais financeiras juntamente com sua volatilidade. Serão apresentados alguns estimadores para os parâmetros dos modelos estudados. Foram realizadas simulações dos três tipos de modelos com o objetivo de comparar o comportamento dos estimadores para diferentes valores dos parâmetros. Por fim, serão apresentadas aplicações em séries reais. / The goal of this project is to present and compare differents methods of modeling volatility (conditional variance) in financial time series. ARFIMA model is applied to capture long memory behavior of volatility in financial time series. GARCH model is used to model the temporal variation in financial volatility. Finally, FIGARCH model is used to model dynamic of financial time series returns as well as its volatility behavior. We present some estimators for the studied models. Estimators behavior of the three types of models for different parameters is assessed through a simulation study. At last, applications to real data are presented.

La mémoire longue, aussi appelée la dépendance à long terme (LRD), est couramment détectée dans l’analyse de séries chronologiques dans de nombreux domaines, par exemple,en finance, en économétrie, en hydrologie, etc. Donc l’étude des séries temporelles à mémoire longue est d’une grande valeur. L’introduction du processus ARFIMA (fractionally autoregressive integrated moving average) établit une relation entre l’intégration fractionnaire et la mémoire longue, et ce modèle a trouvé son pouvoir de prévision à long terme, d’où il est devenu l’un des modèles à mémoire longue plus populaires dans la littérature statistique. Précisément, un processus à longue mémoire ARFIMA (p, d, q) est défini comme suit : Φ(B)(I − B)d (Xt − µ) = Θ(B)εt, t ∈ Z, où Φ(z) = 1 − ϕ1z − · · · − ϕpzp et Θ(z) = 1 + · · · + θ1zθpzq sont des polynômes d’ordre p et q, respectivement, avec des racines en dehors du cercle unité; εt est un bruit blanc Gaussien avec une variance constante σ2ε. Lorsque d ∈ (−1/2,1/2), {Xt} est stationnaire et inversible. Cependant, l’hypothèse a priori de la stationnarité des données réelles n’est pas raisonnable. Par conséquent, de nombreux auteurs ont fait leurs efforts pour proposer des estimateurs applicables au cas non-stationnaire. Ensuite, quelques questions se lèvent : quel estimateurs doit être choisi pour applications, et à quoi on doit faire attention lors de l’utilisation de ces estimateurs. Donc à l’aide de la simulation de Monte Carlo à échantillon fini, nous effectuons une comparaison complète des estimateurs semi-paramétriques, y compris les estimateurs de Fourier et les estimateurs d’ondelettes, dans le cadre des séries non-stationnaires. À la suite de cette étude comparative, nous avons que (i) sans bonnes échelles taillées, les estimateurs d’ondelettes sont fortement biaisés et ils ont généralement une performance inférieure à ceux de Fourier; (ii) tous les estimateurs étudiés sont robustes à la présence d’une tendance linéaire en temps dans le niveau de {Xt} et des effets GARCH dans la variance de {Xt}; (iii) dans une situation où le probabilité de transition est bas, la consistance des estimateurs quand même tient aux changements de régime dans le niveau de {Xt}, mais les changements ont une contamination au résultat d’estimation; encore, l’estimateur d’ondelettes de log-regression fonctionne mal dans ce cas; et (iv) en général, l’estimateur complètement étendu de Whittle avec un polynôme locale (fully-extended local polynomial Whittle Fourier estimator) est préféré pour une utilisation pratique, et cet estimateur nécessite une bande (i.e. un nombre de fréquences utilisés dans l’estimation) plus grande que les autres estimateurs de Fourier considérés dans ces travaux. / Long memory, also called long range dependence (LRD), is commonly detected in the analysis of real-life time series data in many areas; for example, in finance, in econometrics, in hydrology, etc. Therefore the study of long-memory time series is of great value. The introduction of ARFIMA (fractionally autoregressive integrated moving average) process established a relationship between the fractional integration and long memory, and this model has found its power in long-term forecasting, hence it has become one of the most popular long-memory models in the statistical literature. Specifically, an ARFIMA(p,d,q) process X, is defined as follows: cD(B)(I - B)d X, = 8(B)c, , where cD(z)=l-~lz-•••-~pzP and 8(z)=1-B1z- .. •-Bqzq are polynomials of order $p$ and $q$, respectively, with roots outside the unit circle; and c, is Gaussian white noise with a constant variance a2 . When c" X, is stationary and invertible. However, the a priori assumption on stationarity of real-life data is not reasonable. Therefore many statisticians have made their efforts to propose estimators applicable to the non-stationary case. Then questions arise that which estimator should be chosen for applications; and what we should pay attention to when using these estimators. Therefore we make a comprehensive finite sample comparison of semi-parametric Fourier and wavelet estimators under the non-stationary ARFIMA setting. ln light of this comparison study, we have that (i) without proper scale trimming the wavelet estimators are heavily biased and the y generally have an inferior performance to the Fourier ones; (ii) ail the estimators under investigation are robust to the presence of a linear time trend in levels of XI and the GARCH effects in variance of XI; (iii) the consistency of the estimators still holds in the presence of regime switches in levels of XI , however, it tangibly contaminates the estimation results. Moreover, the log-regression wavelet estimator works badly in this situation with small and medium sample sizes; and (iv) fully-extended local polynomial Whittle Fourier (fextLPWF) estimator is preferred for a practical utilization, and the fextLPWF estimator requires a wider bandwidth than the other Fourier estimators.

Os modelos ARFIMA de memória longa mostraram-se nesse trabalho mais versáteis à análise da persistência em séries temporais em comparação aos modelos ARIMA. As funções impulso-resposta dos modelos de integração fracionária indicam que essa classe de modelos capta mais adequadamente as informações contidas nas baixas freqüências das séries e, portanto, estes modelos são mais capacitados para avaliar como os choques econômicos são acomodados no médio e longo prazo. Os estudos simulatórios mostraram que os testes de raiz unitária aplicados a processos com memória longa possuem baixo poder, e que os estimadores por máxima verossimilhança e os baseados no espectro de ondaletas são eficientes para estimar o parâmetro de integração fracionária. Os estudos empíricos encontraram componentes altamente persistentes nas séries brasileiras do produto, desemprego e consumo. A análise de co-integração fracionária refutou os resultados do arcabouço I(1)-I(0) que sugerem a não co-integração entre as séries consumo das famílias e renda disponível. A variabilidade relativa dessas séries foi analisada por meio da análise em multiresolução de ondaletas. Concluiu-se que, nas baixas escalas, a variabilidade entre as séries varia em função da escala temporal envolvida. A doutrina da paridade do poder de compra com dados brasileiros foi revisitada por meio da análise de co-integração fracionária. / The long-memory ARFIMA models proved to be more versatile in this study to the analysis of endurance in time series compare to the ARIMA models. The impulse-response functions of the fractionally integrated models indicate that this class of models more adequately gathers the data enclosed in the low frequencies of the series and thus these models are more befitted to evaluate how economic shocks are settled in the medium and long terms. Simulation studies unveiled that the unit root tests applied to long-memory processes have low power, and that the maximum likelihood estimators as well as those based on wavelet spectrum are efficient in estimating the fractional difference parameter. Empirical studies have found highly persistent components in the Brazilian series of the product, unemployment and consumption. The fractional co-integration analysis rebutted the results of the I(1)-I(0) framework, which suggest the non co-integration between the series of families\' consumption and the disposable income. The relative variability of these series was investigated through a wavelet multiresolution analysis. It was concluded that, in small scales, the variability between the series changes according to the time scale involved. The Purchasing Power Parity doctrine with Brazilian data has been revisited through the fractional co-integration analysis.

Aggregated data appears in many areas such as econimics, sociology, geography, etc. This motivates an importance of studying the (dis)aggregation problem. One of the most important reasons why the contemporaneous aggregation become an object of research is the possibility of obtaining the long memory phenomena in processes. The aggregation provides an explanation of the long-memory effect in time series and a simulation method of such series as well. Accumulation of short-memory non-ergodic random processes can lead to the long memory ergodic process, that can be used for the forecasts of the macro and micro variables. We explore the aggregation scheme of AR(1) processes and nearest-neighbour random fields with infinite variance. We provide results on the existence of limit aggregated processes, and find conditions under which it has long memory properties in certain sense. For the random fields on Z^2, we introduce the notion of (an)isotropic long memory based on the behavior of partial sums. In L_2 case, the known aggregation of independent AR(1) processes leads to the Gaussian limit. While we describe a new model of aggregation based on independent triangular arrays. This scheme gives the limit aggregated process with finite variance which is not necessary Gaussian. We study a discrete time risk insurance model with stationary claims, modeled by the aggregated heavy-tailed process. We establish the asymptotic properties of the ruin probability and the dependence structure... [to full text] / Agreguoti duomenys naudojami daugelyje mokslo sričių tokių kaip ekonomika, sociologija, geografija ir kt. Tai motyvuoja tirti (de)agregavimo uždavinį. Viena iš pagrindinių priežasčių kodėl vienalaikis agregavimas tapo tyrimų objektu yra galimybė gauti ilgos atminties procesus. Agregavimas paaiškina ilgos atminties atsiradima procesuose ir yra vienas iš būdų tokius procesus generuoti. Agreguodami trumpos atminties neergodiškus atsitiktinius procesus, galime gauti ilgos atminties ergodišką procesą, kuris gali būti naudojamas mikro ir makro kintamųjų prognozavimui. Disertacijoje nagrinėjama AR(1) procesų bei artimiausio kaimyno atsitiktinių laukų, turinčių begalinę dispersiją, agregavimo schema, randamos sąlygos, kurioms esant ribinis agreguotas procesas egzistuoja, ir turi ilgąją atmintį tam tikra prasme. Atsitiktinių laukų atveju, įvedamas anizotropinės/izotropinės ilgos atminties apibrėžimas, kuris yra paremtas dalinių sumų elgesiu. Baigtinės dispersijos atveju yra gerai žinoma nepriklausomų AR(1) procesų schema, kuri rezultate duoda Gauso ribinį agreguotą procesą. Disertacijoje aprašoma trikampio masyvo agregavimo modelis, kuris baigtinės dispersijos atveju duoda nebūtinai Gauso ribinį agreguotą procesą. Taip pat disertacijoje nagrinėjama bankroto tikimybės asimptotika, kai žalos yra aprašomos sunkiauodegiu agreguotu procesu, nusakoma priklausomybė tarp žalų, apibūdinama žalų ilga atmintis.

Agreguoti duomenys naudojami daugelyje mokslo sričių tokių kaip ekonomika, sociologija, geografija ir kt. Tai motyvuoja tirti (de)agregavimo uždavinį. Viena iš pagrindinių priežasčių kodėl vienalaikis agregavimas tapo tyrimų objektu yra galimybė gauti ilgos atminties procesus. Agregavimas paaiškina ilgos atminties atsiradima procesuose ir yra vienas iš būdų tokius procesus generuoti. Agreguodami trumpos atminties neergodiškus atsitiktinius procesus, galime gauti ilgos atminties ergodišką procesą, kuris gali būti naudojamas mikro ir makro kintamųjų prognozavimui. Disertacijoje nagrinėjama AR(1) procesų bei artimiausio kaimyno atsitiktinių laukų, turinčių begalinę dispersiją, agregavimo schema, randamos sąlygos, kurioms esant ribinis agreguotas procesas egzistuoja, ir turi ilgąją atmintį tam tikra prasme. Atsitiktinių laukų atveju, įvedamas anizotropinės/izotropinės ilgos atminties apibrėžimas, kuris yra paremtas dalinių sumų elgesiu. Baigtinės dispersijos atveju yra gerai žinoma nepriklausomų AR(1) procesų schema, kuri rezultate duoda Gauso ribinį agreguotą procesą. Disertacijoje aprašoma trikampio masyvo agregavimo modelis, kuris baigtinės dispersijos atveju duoda nebūtinai Gauso ribinį agreguotą procesą. Taip pat disertacijoje nagrinėjama bankroto tikimybės asimptotika, kai žalos yra aprašomos sunkiauodegiu agreguotu procesu, nusakoma priklausomybė tarp žalų, apibūdinama žalų ilga atmintis. / Aggregated data appears in many areas such as econimics, sociology, geography, etc. This motivates an importance of studying the (dis)aggregation problem. One of the most important reasons why the contemporaneous aggregation become an object of research is the possibility of obtaining the long memory phenomena in processes. The aggregation provides an explanation of the long-memory effect in time series and a simulation method of such series as well. Accumulation of short-memory non-ergodic random processes can lead to the long memory ergodic process, that can be used for the forecasts of the macro and micro variables. We explore the aggregation scheme of AR(1) processes and nearest-neighbour random fields with infinite variance. We provide results on the existence of limit aggregated processes, and find conditions under which it has long memory properties in certain sense. For the random fields on Z^2, we introduce the notion of (an)isotropic long memory based on the behavior of partial sums. In L_2 case, the known aggregation of independent AR(1) processes leads to the Gaussian limit. While we describe a new model of aggregation based on independent triangular arrays. This scheme gives the limit aggregated process with finite variance which is not necessary Gaussian. We study a discrete time risk insurance model with stationary claims, modeled by the aggregated heavy-tailed process. We establish the asymptotic properties of the ruin probability and the dependence structure... [to full text]

O objetivo deste trabalho foi estudar o comportamento da volatilidade do preço da soja negociada em contratos futuros na BM&FBOVESPA (série SFI). O estudo foi realizado por meio da comparação entre duas abordagens: na primeira, foi utilizada a série de retornos absolutos da série em questão para representar a volatilidade da mesma, que se mostrou persistente ao longo do tempo, comprovando o fato de que a série possui o comportamento de memória longa. Por ter apresentado tal comportamento, fez-se necessária a utilização de modelos ARFIMA (\"Autorregressivos Fracionários Integrados de Médias Móveis\") estes, que são capazes de capturar de maneira efetiva tal comportamento. Ainda dentro desta abordagem, os modelos foram estimados de duas maneiras distintas: a primeira, em que todos os parâmetros foram estimados simultaneamente e a segunda, em que primeiramente foi estimado o parâmetro de memória longa, diferenciada a série e, posteriormente, foram ajustados os modelos ARIMA nos dados diferenciados. Por fim, a segunda abordagem utilizada no trabalho é a mais comum em pesquisas acadêmicas: foi realizada a estimação dos modelos GARCH (\"Autorregressivos Generalizados de Heteroscedasticidade Condicional\") diretamente na série de retornos. Neste estudo, concluímos que a primeira abordagem se mostrou mais eficiente, dados os critérios de comparação utilizados. / The purpose of this article was to study the volatility of the soybean price traded in futures contracts on the BM&FBOVESPA (SFI series). The study was conduct by comparison between two approaches: first, was use the series of absolute returns of the respective series, to represent its volatility, which was persistent over time, proving the fact that the series has a long memory behavior. Because of such behavior, it was necessary to use ARFIMA models (\"Autoregressive Fractional Integrated Moving Average\"), which are able to capture effectively such behavior. Still using this approach, the models were estimate in two different ways: first, which all parameters were estimate simultaneously, and the second one, that was first estimated the long memory parameter, differentiated the series and, later, adjusted the ARIMA models in differentiated data. Finally, the second approach used in this work is the most common in academic research: the estimation of GARCH models (\"Generalized Autoregressive Conditional Heretoscskedasticity\") directly in the returns series of the studied series. In this study, we conclude that the first approach was more effective, given the comparison criteria used.

Os modelos ARFIMA de memória longa mostraram-se nesse trabalho mais versáteis à análise da persistência em séries temporais em comparação aos modelos ARIMA. As funções impulso-resposta dos modelos de integração fracionária indicam que essa classe de modelos capta mais adequadamente as informações contidas nas baixas freqüências das séries e, portanto, estes modelos são mais capacitados para avaliar como os choques econômicos são acomodados no médio e longo prazo. Os estudos simulatórios mostraram que os testes de raiz unitária aplicados a processos com memória longa possuem baixo poder, e que os estimadores por máxima verossimilhança e os baseados no espectro de ondaletas são eficientes para estimar o parâmetro de integração fracionária. Os estudos empíricos encontraram componentes altamente persistentes nas séries brasileiras do produto, desemprego e consumo. A análise de co-integração fracionária refutou os resultados do arcabouço I(1)-I(0) que sugerem a não co-integração entre as séries consumo das famílias e renda disponível. A variabilidade relativa dessas séries foi analisada por meio da análise em multiresolução de ondaletas. Concluiu-se que, nas baixas escalas, a variabilidade entre as séries varia em função da escala temporal envolvida. A doutrina da paridade do poder de compra com dados brasileiros foi revisitada por meio da análise de co-integração fracionária. / The long-memory ARFIMA models proved to be more versatile in this study to the analysis of endurance in time series compare to the ARIMA models. The impulse-response functions of the fractionally integrated models indicate that this class of models more adequately gathers the data enclosed in the low frequencies of the series and thus these models are more befitted to evaluate how economic shocks are settled in the medium and long terms. Simulation studies unveiled that the unit root tests applied to long-memory processes have low power, and that the maximum likelihood estimators as well as those based on wavelet spectrum are efficient in estimating the fractional difference parameter. Empirical studies have found highly persistent components in the Brazilian series of the product, unemployment and consumption. The fractional co-integration analysis rebutted the results of the I(1)-I(0) framework, which suggest the non co-integration between the series of families\' consumption and the disposable income. The relative variability of these series was investigated through a wavelet multiresolution analysis. It was concluded that, in small scales, the variability between the series changes according to the time scale involved. The Purchasing Power Parity doctrine with Brazilian data has been revisited through the fractional co-integration analysis.

Earthquakes are amongst the most destructive of natural phenomena and have been the subject of significant research effort over many decades, to predict the onset of seismic events. Electromagnetic emissions detected prior to earthquakes provide a potential data source for seismic predictions and research suggests that specific pre-seismic electromagnetic activity can be directly related to specific earthquakes although it is still an open issue as to the precise links between these electromagnetic emissions and subsequent earthquakes. In this research, findings of the long memory or the self-organization of several pre-earthquake MHz electromagnetic time-series provide significant outcomes regarding the earthquake prediction. It is also recognised that enhanced radon gas emission has an equally long history as being associated with seismic activity. In general, several anomalous soil radon emissions have been observed prior to earthquakes and this has been recorded all over the world. The abnormal soil radon exhalation from the interior of the earth has been associated with earthquakes and is considered as an important field of research. The research reported in this thesis compared and contrasted the merits of combining electromagnetic emission data and radon exhalation data as precursors of earthquakes with the aim of enhancing earthquake prediction methodology. The findings from the long-memory analysis of radon disturbances in the soil indicated a very significant issue: the radon disturbances in the soil prior to earthquakes exhibit similar behaviour as the MHz RF disturbances of general failure. So, the radon precursors and the MHz electromagnetic correspond to the same pre-earthquake phase. Geological explanations were proposed in view of the asperity model. Persistent and anti-persistent MHz anomalies were due to the micro-cracking of the heterogeneous medium of the earth's crust which may have led the system's evolution towards the global failure. Fractal methods have been used on historical data, to investigate MHz electromagnetic time-series spectra on emissions preceding major earthquakes over the period 2007 to 2014 and the characteristics of enhanced radon emissions have been studied over the period 2008 to 2015 for seismic events occurring in the Aegean Region. It has been found that both the electromagnetic emissions and the radon exhalation data exhibit similar fractal behaviour and are associated with impending seismic activity. Hence both phenomena are relevant to earthquake predictions and should both be employed in any systematic approach to this problem as the varying geological and geographic conditions under which earthquakes can occur, might preclude one or other data from being measurable. According to the several techniques applied in this thesis, all should be employed in sequential steps, albeit the power-law spectral fractal analysis is the most significant to trace long-memory patterns of 1/f processes as those of the processes of earthquakes.