In applications, the true underlying model of an observed time series is typically unknown or has a complicated structure. A common approach is to approximate the true model by autoregressive (AR) equation whose orders are chosen by information criterions such as AIC, BIC and Parsen's CAT and whose parameters are estimated by the least square (LS), the Yule Walker (YW) or other methods. However, as sample size increases, it often implies that the model order has to be refined and the parameters need to be recalculated. In order to avoid such shortcomings, we propose the Regularized AR (RAR) approximation and illustrate its applications in frequency detection and long memory process forecasting. The idea of the RAR approximation is to utilize a “long" AR model whose order significantly exceeds the model order suggested by information criterions, and to estimate AR parameters by Regularized LS (RLS) method, which enables to estimate AR parameters with different level of accuracy and the number of estimated parameters can grow linearly with the sample size. Therefore, the repeated model selection and parameter estimation are avoided as the observed sample increases. We apply the RAR approach to estimate the unknown frequencies in periodic processes by approximating their generalized spectral densities, which significantly reduces the computational burden and improves accuracy of estimates. Our theoretical findings indicate that the RAR estimates of unknown frequency are strongly consistent and normally distributed. In practice, we may encounter spurious frequency estimates due to the high model order. Therefore, we further propose the robust trimming algorithm (RTA) of RAR frequency estimation. Our simulation studies indicate that the RTA can effectively eliminate the spurious roots and outliers, and therefore noticeably increase the accuracy. Another application we discuss in this thesis is modeling and forecasting of long memory processes using the RAR approximation. We demonstration that the RAR is useful in long-range prediction of general ARFIMA(p,d,q) processes with p > 1 and q > 1 via simulation studies.

In applications, the true underlying model of an observed time series is typically unknown or has a complicated structure. A common approach is to approximate the true model by autoregressive (AR) equation whose orders are chosen by information criterions such as AIC, BIC and Parsen's CAT and whose parameters are estimated by the least square (LS), the Yule Walker (YW) or other methods. However, as sample size increases, it often implies that the model order has to be refined and the parameters need to be recalculated. In order to avoid such shortcomings, we propose the Regularized AR (RAR) approximation and illustrate its applications in frequency detection and long memory process forecasting. The idea of the RAR approximation is to utilize a “long" AR model whose order significantly exceeds the model order suggested by information criterions, and to estimate AR parameters by Regularized LS (RLS) method, which enables to estimate AR parameters with different level of accuracy and the number of estimated parameters can grow linearly with the sample size. Therefore, the repeated model selection and parameter estimation are avoided as the observed sample increases. We apply the RAR approach to estimate the unknown frequencies in periodic processes by approximating their generalized spectral densities, which significantly reduces the computational burden and improves accuracy of estimates. Our theoretical findings indicate that the RAR estimates of unknown frequency are strongly consistent and normally distributed. In practice, we may encounter spurious frequency estimates due to the high model order. Therefore, we further propose the robust trimming algorithm (RTA) of RAR frequency estimation. Our simulation studies indicate that the RTA can effectively eliminate the spurious roots and outliers, and therefore noticeably increase the accuracy. Another application we discuss in this thesis is modeling and forecasting of long memory processes using the RAR approximation. We demonstration that the RAR is useful in long-range prediction of general ARFIMA(p,d,q) processes with p > 1 and q > 1 via simulation studies.

In this paper, we study the parameter estimation of the multiple linear time series regression model with long memory stochastic regressors and innovations. Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002) proposed a class of frequency-domain weighted least squares estimates. Their estimates are shown to achieve the Gauss-Markov bound with standard convergence rate. In this study, we proposed a time-domain generalized LSE approach, in which the inverse autocovariance matrix of the innovations is estimated via autoregressive coefficients. Simulation studies are performed to compare the proposed estimates with Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002). The results show the time-domain generalized LSE is comparable to Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002) and attains higher efficiencies when the autoregressive or moving average coefficients of the FARIMA models have larger values. A variance reduction estimator, called TF estimator, based on linear combination of the proposed estimator and Hidalgo and Robinson (2002)'s estimator is further proposed to improve the efficiency. Bootstrap method is applied to estimate the weights of the linear combination. Simulation results show the TF estimator outperforms the frequency-domain as well as the time-domain approaches.

Šiame darbe atliekami elektros energijos kainų analizė ir modeliavimas. Elektros kainų kitimui ir tokioms jų charakteringoms savybėms, kaip sezoniškumas, vidurkio reversija, darbo dienų, savaitgalio ir švenčių efektas, kintamumo klasterizacija, aprašyti taikomi SARIMA-TGARCH ir SARFIMA-TGARCH modeliai. Tyrimui naudojami kasvalandiniai Prancūzijos elektros energijos biržos kainų stebėjimai. Darbą sudaro dvi dalys – bendroji (teorinė) ir tiriamoji dalys. Pirmoje dalyje apžvelgiama literatūra bei aptariami teoriniai modelių aspektai: aprašomi ilgos atminties modeliai. Antroje dalyje pristatomi modelių empiriniai rezultatai: SARIMA-TGARCH ir SARFIMA-TGARCH modelių taikymas ir adekvatumo tikrinimas. / In this paper an econometric modelling and forecasting of electricity spot prices is presented. The aim of this work is to examine SARIMA-TGARCH and SARFIMA-TGARCH models for describing volatility of electricity spot prices and their characteristics such as season, mean reversion, volatility cauterization, and effects of workdays, weekends or holidays. The data of France electricity stock prices are used for analysis. This paper contains two parts – theoretical and empirical. In the first part the short review of literature is presented. Moreover, the theoretical aspects of long memory models are discussed. In the following part the empirical results are presented: application and adequacy examination of SARIMA-TGARCH and SARFIMA-TGARCH models.

The thesis is made up of a number of studies involving long-range dependence (LRD), that is, a slow power-law decay in the temporal correlation of stochastic models. Such a phenomenon has been frequently observed in practice. The models with LRD often yield non-standard probabilistic and statistical results. The thesis includes in particular the following topics: Multivariate limit theorems. We consider a vector made of stationary sequences, some components of which have LRD, while the others do not. We show that the joint scaling limits of the vector exhibit an asymptotic independence property. Non-central limit theorems. We introduce new classes of stationary models with LRD through Volterra-type nonlinear filters of white noise. The scaling limits of the sum lead to a rich class of non-Gaussian stochastic processes defined by multiple stochastic integrals. Limit theorems for quadratic forms. We consider continuous-time quadratic forms involving continuous-time linear processes with LRD. We show that the scaling limit of such quadratic forms depends on both the strength of LRD and the decaying rate of the quadratic coefficient. Behavior of the generalized Rosenblatt process. The generalized Rosenblatt process arises from scaling limits under LRD. We study the behavior of this process as its two critical parameters approach the boundaries of the defining region. Inference using self-normalization and resampling. We introduce a procedure called "self-normalized block sampling" for the inference of the mean of stationary time series. It provides a unified approach to time series with or without LRD, as well as with or without heavy tails. The asymptotic validity of the procedure is established.

Le but principal de cette thèse est de développer de nouveaux modèles non linéaires à longue mémoire pour modéliser des rendements financiers et leur estimation statistique. En plus de la longue mémoire, ces modèles sont capables de mettre en lumière d’autres faits stylisés comme l’asymétrie ou l’effet de levier. Les processus étudiés dans la thèse sont des solutions stationnaires de certaines équations aux différences stochastiques non linéaires impliquant un “bruit” i.i.d. Outre le fait de résoudre ces équations, qui est non trivial en lui-même, nous prouvons que leur solutions sont dépendantes à longue portée. Enfin pour un modèle non linéaire particulier à longue portée (GQARCH) nous prouvon la consistence et la normalité asymptotique de l’estimateur du quasi-maximum de vraisemblance (QMLE). / The thesis introduces new nonlinear models with long memory which can be used for modelling of financial returns and statistical inference. Apart from long memory, these models are capable to exhibit other stylized facts such as asymmetry and leverage. The processes studied in the thesis are defined as stationary solutions of certain nonlinear stochastic difference equations involving a given i.i.d. “noise”. Apart from solvability issues of these equations which are not trivial by itself, it is proved that their solutions exhibit long memory properties. Finally, for a particularly tractable nonlinear parametric model with long memory (GQARCH) we prove consistency and asymptotic normality of quasi-ML estimators.

Orientador: Mauricio Enrique Zevallos Herencia / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Ciencia da Computação / Made available in DSpace on 2018-08-11T01:10:13Z (GMT). No. of bitstreams: 1 Santos_BrunoReisdos_M.pdf: 1935776 bytes, checksum: f3579f38b051dcbc18a4a0f79c2d6ab2 (MD5) Previous issue date: 2008 / Resumo: Nesta dissertação é discutido o uso da metodologia de diagnóstico de Influência Local em modelos de séries temporais. Especificamente, serão estudados os modelos autoregressivos de ordem um, os modelos de regressão com erros autoregressivos de ordem um e modelos de longa-memória. As medidas de influência local consideradas são: Inclinação de Billor e Loynes e Curvatura de Cook. As principais contribuições nesta dissertação são duas. Primeiro, a utilização da metodologia de limiares (benchmarks) nos modelos mencionados para determinar se uma observação é influente. Isto permite ter uma ferramenta estatística para identificar pontos influentes a diferença da simples análise exploratória que é o mais comum na literatura. Como segunda contribuição, serão obtidas as expressões para o cálculo das medidas de Inclinação de Billor e Curvatura de Cook nos modelos ARFIMA. Finalmente, as metodologias descritas são ilustradas através de dados simulados e da análise de dados reais / Abstract: This work is about Time Series Diagnostics using Local Influence. Specifically, firstorder autoregressive models, regression models with first-order autoregressive errors and long-memory models are studied. In order to assess Local Influence two statistics are considered: the Slope of Billor and Loynes and Cook¿s Curvature. The main contributions are two. First, apply a methodology based on benchmarks calculated by simulation on the aforementioned models for determining influential observations. This permits to have a statistical tool to identify influential points instead of the simple exploratory analysis, which is the most common device in the literature. Second, expressions for Billor and Loynes Slope and Cook¿s Curvature in ARFIMA models are derived. Finally, all methodologies are illustrated using simulated data and the analysis of real data / Mestrado / Series Temporais / Mestre em Estatística

Ao se tratar de mercado de capitais, dentre seus principais fatores de análise, encontra-se a discussão a respeito da teoria de eficiência de mercado que é uma teoria que diverge em relação ao comportamento do preço dos ativos, no que diz respeito à sua linearidade ou não. Neste sentido, este trabalho teve como objetivo analisar o comportamento dos principais índices das dez maiores bolsas de valores do mercado, durante o período de junho de 1999 a junho de 2009. Para realização de tal análise foi utilizada a estatística R/S e o cálculo do Expoente de Hurst, por sua vez, validado pelo Teste Estatístico de Wald. A utilização desta metodologia permitiu investigar a presença da memória longa persistente, anti-persistente ou a identificação de um passeio aleatório. Os resultados evidenciaram que, de modo geral, os índices apresentaram presença de memória longa persistente na maior parte do período analisado, devendo-se ressaltar que apenas no período próximo à crise financeira de 2008 foi possível identificar forte presença de um comportamento aleatório. Assim, foi possível aceitar a hipótese de que os mercados são ineficientes na maioria das séries históricas de retornos dos índices. / When it comes to the capital market, among its main factors of analysis, it´s found the debate concerning the market efficiency theory, which is a theory that differs in relation to the behavior of the asset´s price, concerning its linearity or not. In this way, this work aims to analyse the behavior of the main index of the ten major stock market, from june 1999 until june 2009. To the achievement of such analysis it was used the R/S statistics and the Hurst Exponent, which was, validadet by the Wald Test Statistics. The employment of such methodology allowed to investigate the presence of the long persistence memory, anti-persistence or the identification of a random walk. The results showed that, on the whole, the indexes showed the presence of the long persistence memory most of the analysed period, saying the only in the period close to the financial crisis of 2008, it was possible to identify a relevant presence of a random behavior. So, it was possible to accept the hipothesis that the markets are ineffectual in most of the historical series of restoration of indexes.

Cette thèse porte sur l'identification et l'estimation des ruptures structurelles pouvant affecter des données économiques et financières à mémoire longue. Notre étude s'est limitée dans les trois premiers chapitres au cadre univarié où nous avons modélisé la dépendance de long terme et les changements structurels simultanément et séparément au niveau de la moyenne ainsi que la volatilité. Dans un premier temps nous n'avons tenu compte que des sauts instantanés d'état ensuite nous nous sommes intéressés à la possibilité d'avoir des changements graduels et lisses au cours du temps grâce à des modèles nonlinéaires plus complexes. Par ailleurs, des expériences de simulation ont été menées dans le but d'offrir une analyse comparative des méthodes utilisées et d'attester de la robustesse des tests sous certaines conditions telle que la présence de la mémoire longue dans la série. Ce travail s'est achevé sur une extension aux modèles multivariés.Ces modèles permettent de rendre compte des mécanismes de propagation d'une variation d'une série sur l'autre et d'identifier les liens entre les variables ainsi que la nature des ces liens. Les interactions entre les différentes variables financières ont été analysées tant à court terme qu'à long terme. Bien que le concept du changement structurel n'a pas été abordé dans ce dernier chapitre, nous avons pris en compte l'effet d'asymétrie et de mémoire longue dans la modélisation de la volatilité. / This dissertation deals with the detection and the estimation of structural changes in long memory economic and financial time series. Within the rest three chapters we focused on the univariate case to model both the long range dependence and structural changes in the mean and the volatility of the examined series. In the beginning we just take into account abrupt regime switches but after we use more developed nonlinear models in order to capture the smooth time variations of the dynamics. Otherwise we analyse the efficiency of various techniques permitting to select the number of breaks and we assess the robustness of the used tests in a long memory environment via simulations. Last, this thesis was completed by an extension to multivariate models. These models allow us to detect the impact of some series on the others and identify the relationships among them. The interdependencies between the financial variables were studied and analysed both in the short and the long range. While structural changes were not considered in the last chapter, our multivariate model takes into account asymmetry effects and the long memory behaviour in the volatility.

Cette thèse fait appel à la théorie des ondelettes pour estimer le paramètre de mémoire longue dans le cadre stationnaire et non stationnaire lors de la modélisation des séries financières, et pour l'estimation non paramétrique de la copule lors de l'examen de leurs interdépendances. L'avantage de la méthode des ondelettes comparée à l'analyse de Fourier est d'être parfaitement localisée dans le domaine temporel et celui des fréquences.Dans une première étape, nous nous sommes intéressés à la modélisation de la dynamique des séries de variations. À cette fin, nous proposons un modèle économétrique qui permet de tenir compte, en plus de la composante mémoire longue dans la moyenne, une dépendance de long terme dans la variance conditionnelle.D'une part, nous étudions les liens de causalité entre les séries obtenus par décomposition en ondelettes à chaque niveau de résolution. D'autre part, nous nous orientons vers la théorie de cointégration fractionnaire. À cet égard,nous estimons un modèle vectoriel à correction d'erreur dans lequel la vitesse d'ajustement à l'équilibre de long terme est plus lente que la vitesse associée à la cointégration linéaire. L'atout de cette approche est de détecter la présence d'une relation de long terme en plus des fluctuations de court terme et de mettre en œuvre des liens de causalité durables.Dans une deuxième étape, nous proposons une analyse des dépendances multivariées entre les risques financiers et leurs impacts sur les mesures de risques communément rencontrées en gestion des risques. Nous exposons une application de la théorie des copules à l'analyse de la structure des dépendances entre différentes séries financières. Les résultats empiriques obtenus montrent que la structure de dépendance devient accentuée lorsque les séries sont filtrées de l'effet mémoire. Ensuite, nous étudions l'effet du changement de la structure de dépendance dans la frontière d'efficience et dans les mesures du risque sur l'ensemble des portefeuilles optimaux en considérant le modèle moyenne-variance-copules et en élaborant une mesure du risque basée sur l'approche CVaR-copules. Les résultats empiriques prouvent que nous sommes loin des portefeuilles optimaux de Markowitz.Enfin, nous proposons un nouvel estimateur dans le cadre des ondelettes qui constitue une extension de celui de Shimotsu et Phillips (2005, 2010). / This thesis uses wavelet theory to estimate the long memory parameter in the stationary and non stationary framework when modeling financial time series, and non parametric estimation of the copula in the examination of their interdependencies. The advantage of the wavelet method compared to the Fourier analysis is to be fully localized in the time domain and that of the frequency. In a first step, we are interested in modeling the dynamics of series of variations. To this end, we propose an econometric model that takes into account, in addition to the long memory component in the mean, a long term dependence in the conditional variance. On the first hand, we study the causal links between the series obtained by wavelet decomposition at each level of resolution. On the second hand, we are moving towards the theory of fractional cointegration. In this regard, we consider a vector error correction model in which the speed of adjustment to the long run equilibrium is slower than the speed associated with the linear cointegration. The advantage of this approach is to detect the presence of a long term relationship in addition to short term fluctuations and implement long run causal links.In a second step, we deal a multivariate analysis of dependencies between risks and their impacts on financial measures of risk commonly used in risk management. We present an application of copula theory to analyze the structure of dependencies between different financial series. The empirical results show that the dependence structure becomes accentuated when the series are filtered from the memory effect. Next, we investigate the effect of changing the structure of dependence in the efficiency frontier and the risk measures on all optimal portfolios considering the mean-variance-copulas model and developing a risk measure based on the CVaR-copula approach. The empirical results show that we are far from optimal portfolios Markowitz . Finally, we propose a new estimator in the wavelet domain which is an extension of the estimator of the Shimotsu and Phillips (2005, 2010).