ii Abstract This thesis focuses on one of the attractive topics of current financial literature, the application of wavelet-based methods in volatility modeling. It introduces a new, wavelet-based estimator (wavelet Whittle estimator) of a FIEGARCH model, ARCH- family model capturing long-memory and asymmetry in volatility, and studies its properties. Based on an extensive Monte Carlo experiment, both the behavior of the new estimator in various situations and its relative performance with respect to two more traditional estimators (maximum likelihood estimator and Fourier-based Whittle estimator) are assessed, along with practical aspects of its application. Possible solutions are proposed for most of the issues detected, including suggestion of a new specification of the estimator. This uses maximal overlap discrete wavelet transform instead of the traditionally used discrete wavelet transform, which should improve the estimator performance in all its applications, not only in the case of FIEGARCH model estimation. The thesis concludes that, after optimization of the estimation setup, the wavelet-based estimator may become an attractive robust alternative to the traditional methods.

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.

Este trabalho tem como objetivo a avaliação de três estimadores do parâmetro de integração fracionária d e de um teste para memória longa compartilhada. Os estimadores a serem avaliados são: o estimador de Geweke e Porter-Hudak, o estimador usando o periodograma suavizado e o estimador semiparamétrico truncado de Whittle. A avaliação dos estimadores será no contexto de processos ARFIMA+ARMA, e em relação a variações nos termos autoregressivos e de médias móveis, tanto do termo de memória curta quanto do termo de memória longa. Além disso, serão introduzidos o conceito de modelos com memória longa compartilhada e um método de identificação através da análise de correlação canônica para séries temporais multivariadas proposto, por Ray e Tsay (1997). Por fim, serão apresentadas três aplicações sobre dados reais dos tópicos estudados: uma para a velocidade do vento em São Paulo e Piracicaba e outras duas para séries das bolsas de valores de Hong Kong, Nova Zelândia, Singapura, Brasil e Reino Unido / The goal of this project is the evaluation of three long memory parameter estimators and a common long range dependence test. The estimators evaluated are: the Geweke and Porter-Hudak, the smoothed periodogram and the semiparametric truncated Whittle estimators. The evaluation is in the context of processes ARFIMA+ARMA, and related to variations in the autoregressive and moving average coefficients, both in the short and long memory terms. Furthermore, we describe common long range dependence processes and an identification approach (Ray and Tsay, 1997) for them, using the canonical correlation analysis. Finally, three applications to real data are presented: the first one to the wind\'s speed in the Brazilian cities of São Paulo and Piracicaba, and the other ones to financial time series of the stock markets of Hong Kong, New Zealand, Singapore, Brazil and the United Kingdom.

Este trabalho tem como objetivo a avaliação de três estimadores do parâmetro de integração fracionária d e de um teste para memória longa compartilhada. Os estimadores a serem avaliados são: o estimador de Geweke e Porter-Hudak, o estimador usando o periodograma suavizado e o estimador semiparamétrico truncado de Whittle. A avaliação dos estimadores será no contexto de processos ARFIMA+ARMA, e em relação a variações nos termos autoregressivos e de médias móveis, tanto do termo de memória curta quanto do termo de memória longa. Além disso, serão introduzidos o conceito de modelos com memória longa compartilhada e um método de identificação através da análise de correlação canônica para séries temporais multivariadas proposto, por Ray e Tsay (1997). Por fim, serão apresentadas três aplicações sobre dados reais dos tópicos estudados: uma para a velocidade do vento em São Paulo e Piracicaba e outras duas para séries das bolsas de valores de Hong Kong, Nova Zelândia, Singapura, Brasil e Reino Unido / The goal of this project is the evaluation of three long memory parameter estimators and a common long range dependence test. The estimators evaluated are: the Geweke and Porter-Hudak, the smoothed periodogram and the semiparametric truncated Whittle estimators. The evaluation is in the context of processes ARFIMA+ARMA, and related to variations in the autoregressive and moving average coefficients, both in the short and long memory terms. Furthermore, we describe common long range dependence processes and an identification approach (Ray and Tsay, 1997) for them, using the canonical correlation analysis. Finally, three applications to real data are presented: the first one to the wind\'s speed in the Brazilian cities of São Paulo and Piracicaba, and the other ones to financial time series of the stock markets of Hong Kong, New Zealand, Singapore, Brazil and the United Kingdom.

[pt] Este trabalho tem como objetivo o estudo das séries de energia natural afluente (ENAs) por meio de modelos de memória longa, no intuito de gerar cenários hidrológicos sintéticos. Séries temporais com memória longa são definidas como séries que apresentam persistente dependência entre observações afastadas por um longo período de tempo. Inicialmente procedeu-se uma análise exploratória através da qual foi possível encontrar características de série temporais com longa dependência. Os modelos empregados nesta dissertação foram os SARFIMA (p,d.q)x(P,D.Q)s em que os parâmetros dˆ e Dˆ assumem valores fracionários, para que seja possível a incorporação de efeitos de longa dependência e/ou cíclicos. Também foi utilizada a técnica de computação intensiva bootstrap em diversas etapas, dentre elas a construção de um teste não paramétrico para significância dos parâmetros fracionários, assim como bootstrap nos resíduos do modelo para a geração de séries hidrológicas sintéticas. Para averiguar a adequabilidade dos cenários gerados, foram realizados testes estatísticos de igualdade de médias, igualdade de variâncias, testes de aderência e análise de sequências. Por meio destes, pode-se concluir que os modelos empregados nesta dissertação conseguiram reproduzir de maneira satisfatória o histórico disponível de ENAs. / [en] The aim of this thesis is to study the series of natural energy surging (NES) through long memory models, whose interest is to fit models capable of generating synthetic hydrological series. Time Series with long memory are defined as a series which have persistent dependence between observations separated by a long period of time. Firstly, we proceed to the exploration analysis where we found particulars of long memory time series. The models employed is this work were SARFIMA (p, d, q)x(P, D,Q)s where parameters d and D assume fractional values so as to incorporate long memory and/or cycles effects. It was also used a intensive computational technique called bootstrap in various stages, among them the construction of a non-parametric test for the significant of fractional parameters and the bootstrap in the residual models for generating synthetic hydrological series. In order verify the accuracy of the scenarios generated, statistical tests were performed for equal means, equal variance, adherence test and sequence analysis. Through these, we can conclude that the models used in this thesis could satisfactorily reproduce the history of natural energy surging available.

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The main goal of this research is to estimate the model parameters and to detect multiple change points in the long memory parameter of Gaussian ARFIMA(p, d, q) processes. Our approach is Bayesian and inference is done on wavelet domain. Long memory processes have been widely used in many scientific fields such as economics, finance and computer science. Wavelets have a strong connection with these processes. The ability of wavelets to simultaneously localize a process in time and scale domain results in representing many dense variance-covariance matrices of the process in a sparse form. A wavelet-based Bayesian estimation procedure for the parameters of Gaussian ARFIMA(p, d, q) process is proposed. This entails calculating the exact variance-covariance matrix of given ARFIMA(p, d, q) process and transforming them into wavelet domains using two dimensional discrete wavelet transform (DWT2). Metropolis algorithm is used for sampling the model parameters from the posterior distributions. Simulations with different values of the parameters and of the sample size are performed. A real data application to the U.S. GNP data is also reported. Detection and estimation of multiple change points in the long memory parameter is also investigated. The reversible jump MCMC is used for posterior inference. Performances are evaluated on simulated data and on the Nile River dataset.

Persistence of stock returns is an extensively studied and discussed theme in the analysis of financial markets. Antipersistence is usually attributed to volatilities. However, not only volatilities but also stock returns can exhibit antipersistence. Antipersistent noise has a somewhat rougher appearance than Gaussian noise. Heuristically spoken, price movements are more likely followed by movements in the opposite direction than in the same direction. The pertaining integrated process exhibits a smaller range – prices seem to stay in the vicinity of the initial value. We apply a widely used test based upon the modified R/S-Method by Lo [1991] to daily returns of 21 German stocks from 1960 to 2008. Combining this test with the concept of moving windows by Carbone et al. [2004], we are able to determine periods of antipersistence for some of the series under examination. Our results suggest that antipersistence can be found for stocks and periods where extraordinary corporate actions such as mergers & acquisitions or financial distress are present. These effects should be properly accounted for when choosing and designing models for inference.

No description available.

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.