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A fast and robust statistical test based on likelihood ratio with Bartlett correction to identify Granger causality between gene sets

FUJITA, Andre; KOJIMA, Kaname; PATRIOTA, Alexandre G.; SATO, Joao R.; SEVERINO, Patricia; MIYANO, Satoru
Fonte: OXFORD UNIV PRESS Publicador: OXFORD UNIV PRESS
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.32%
We propose a likelihood ratio test ( LRT) with Bartlett correction in order to identify Granger causality between sets of time series gene expression data. The performance of the proposed test is compared to a previously published bootstrapbased approach. LRT is shown to be significantly faster and statistically powerful even within non- Normal distributions. An R package named gGranger containing an implementation for both Granger causality identification tests is also provided.; RIKEN, Japan; RIKEN, Japan; Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); FAPESP, Brazil

Modelo 'export-led growth' : evidências empíricas em uma perspectiva não linear; Model "Export-led growth" : empirical evidence in a non-linear perspective

Faleiros, João Paulo Martin
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 25/02/2008 Português
Relevância na Pesquisa
26.25%
Esse trabalho faz uma avaliação não linear sobre "Export-Led Growth" (ELG), por meio do modelo MR-STVAR. O tratamento não linear aqui desenvolvido assumiu que a trajetória da taxa de crescimento do produto, ao longo do tempo, pode alternar entre quatro diferentes tipos de regimes. Cada um destes se caracteriza como uma combinação entre altas e baixas taxas de crescimento, tanto do produto, como das exportações. Realizando o teste de causalidade de Granger, nessa estrutura não linear, é possível verificar se a taxa trimestral de crescimento do valor das exportações aumenta a capacidade preditiva do crescimento do PIB. Portanto, esse enfoque possibilita expandir a análise, até então realizada, de que as contribuições das taxas de crescimento das exportações, às taxas de crescimento do produto, são lineares ao longo do tempo. E essa última perspectiva, implicitamente assume uma dinâmica temporal uniforme, bastante restritiva em termos da complexidade que ronda o padrão de desenvolvimento econômico de uma nação. O modelo MR-STVAR foi aplicado para um conjunto de 7 países, Estados Unidos, Canadá, Japão, Hong Kong, Coréia do Sul, Brasil, Chile e México, além de Hong Kong.; The aim of this thesis is to evaluate Export-Led Growth hypothesis through MR-STVAR. If its model is assumed...

Recurrent Activity in Higher Order, Modality Non-Specific Brain Regions: A Granger Causality Analysis of Autobiographic Memory Retrieval

Lou, Hans C.; Joensson, Morten; Biermann-Ruben, Katja; Schnitzler, Alfons; Østergaard, Leif; Kjaer, Troels W.; Gross, Joachim
Fonte: Public Library of Science Publicador: Public Library of Science
Tipo: Artigo de Revista Científica
Publicado em 27/07/2011 Português
Relevância na Pesquisa
26.42%
It has been proposed that the workings of the brain are mainly intrinsically generated recurrent neuronal activity, with sensory inputs as modifiers of such activity in both sensory and higher order modality non-specific regions. This is supported by the demonstration of recurrent neuronal activity in the visual system as a response to visual stimulation. In contrast recurrent activity has never been demonstrated before in higher order modality non-specific regions. Using magneto-encephalography and Granger causality analysis, we tested in a paralimbic network the hypothesis that stimulation may enhance causal recurrent interaction between higher-order, modality non-specific regions. The network includes anterior cingulate/medial prefrontal and posterior cingulate/medial parietal cortices together with pulvinar thalami, a network known to be effective in autobiographic memory retrieval and self-awareness. Autobiographic memory retrieval of previous personal judgments of visually presented words was used as stimuli. It is demonstrated that the prestimulus condition is characterized by causal, recurrent oscillations which are maximal in the lower gamma range. When retrieving previous judgments of visually presented adjectives, this activity is dramatically increased during the stimulus task as ascertained by Granger causality analysis. Our results confirm the hypothesis that stimulation may enhance causal interaction between higher order...

Granger Causality Analysis of Steady-State Electroencephalographic Signals during Propofol-Induced Anaesthesia

Barrett, Adam B.; Murphy, Michael; Bruno, Marie-Aurélie; Noirhomme, Quentin; Boly, Mélanie; Laureys, Steven; Seth, Anil K.
Fonte: Public Library of Science Publicador: Public Library of Science
Tipo: Artigo de Revista Científica
Publicado em 05/01/2012 Português
Relevância na Pesquisa
26.41%
Changes in conscious level have been associated with changes in dynamical integration and segregation among distributed brain regions. Recent theoretical developments emphasize changes in directed functional (i.e., causal) connectivity as reflected in quantities such as ‘integrated information’ and ‘causal density’. Here we develop and illustrate a rigorous methodology for assessing causal connectivity from electroencephalographic (EEG) signals using Granger causality (GC). Our method addresses the challenges of non-stationarity and bias by dividing data into short segments and applying permutation analysis. We apply the method to EEG data obtained from subjects undergoing propofol-induced anaesthesia, with signals source-localized to the anterior and posterior cingulate cortices. We found significant increases in bidirectional GC in most subjects during loss-of-consciousness, especially in the beta and gamma frequency ranges. Corroborating a previous analysis we also found increases in synchrony in these ranges; importantly, the Granger causality analysis showed higher inter-subject consistency than the synchrony analysis. Finally, we validate our method using simulated data generated from a model for which GC values can be analytically derived. In summary...

Confounding Effects of Phase Delays on Causality Estimation

Vakorin, Vasily A.; Mišić, Bratislav; Krakovska, Olga; Bezgin, Gleb; McIntosh, Anthony R.
Fonte: Public Library of Science Publicador: Public Library of Science
Tipo: Artigo de Revista Científica
Publicado em 21/01/2013 Português
Relevância na Pesquisa
26.48%
Linear and non-linear techniques for inferring causal relations between the brain signals representing the underlying neuronal systems have become a powerful tool to extract the connectivity patterns in the brain. Typically these tools employ the idea of Granger causality, which is ultimately based on the temporal precedence between the signals. At the same time, phase synchronization between coupled neural ensembles is considered a mechanism implemented in the brain to integrate relevant neuronal ensembles to perform a cognitive or perceptual task. Phase synchronization can be studied by analyzing the effects of phase-locking between the brain signals. However, we should expect that there is no one-to-one mapping between the observed phase lag and the time precedence as specified by physically interacting systems. Specifically, phase lag observed between two signals may interfere with inferring causal relations. This could be of critical importance for the coupled non-linear oscillating systems, with possible time delays in coupling, when classical linear cross-spectrum strategies for solving phase ambiguity are not efficient. To demonstrate this, we used a prototypical model of coupled non-linear systems, and compared three typical pipelines of inferring Granger causality...

Short run and long run causality in time series: Inference

DUFOUR, Jean-Marie; PELLETIER, Denis; RENAULT, Éric
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 223846 bytes; application/pdf
Português
Relevância na Pesquisa
46.26%
We propose methods for testing hypotheses of non-causality at various horizons, as defined in Dufour and Renault (1998, Econometrica). We study in detail the case of VAR models and we propose linear methods based on running vector autoregressions at different horizons. While the hypotheses considered are nonlinear, the proposed methods only require linear regression techniques as well as standard Gaussian asymptotic distributional theory. Bootstrap procedures are also considered. For the case of integrated processes, we propose extended regression methods that avoid nonstandard asymptotics. The methods are applied to a VAR model of the U.S. economy.; Nous proposons des méthodes pour tester des hypothèses de non-causalité à différents horizons, tel que défini dans Dufour et Renault (1998, Econometrica). Nous étudions le cas des modèles VAR en détail et nous proposons des méthodes linéaires basées sur l’estimation d’autorégressions vectorielles à différents horizons. Même si les hypothèses considérées sont non linéaires, les méthodes proposées ne requièrent que des techniques de régression linéaire de même que la théorie distributionnelle asymptotique gaussienne habituelle. Dans le cas des processus intégrés...

Contractual Savings or Stock Market Development—Which Leads?

Catalan, Mario; Impavido, Gregorio; Musalem, Alberto R.
Fonte: World Bank, Washington, DC Publicador: World Bank, Washington, DC
Português
Relevância na Pesquisa
26.3%
The authors study the relationship between the development of contractual savings (assets of pension funds, and life insurance companies) and non-life insurance, and, the development of stock markets (market capitalization and value traded). Their contribution lies in providing time-series evidence on a hypothesis that is very popular - but had not been substantiated - among supporters of fully funded pension systems in which funds invest large shares of their portfolios in tradable securities (equities, bonds). The literature is not clear on its assumption regarding causality between contractual savings, and capital market development. A one-way or two-way relationship is assumed, usually inter-changeably; the authors address the questions of which leads empirically. They present the evidence, including descriptive statistics, and the results of Granger causality tests, for OECD countries, and such countries as Chile, Malaysia, Singapore, South Africa, and Thailand. They do not present a theoretical framework, but do explain how the growth of the contractual savings sector, is thought to promote financial development. The authors find evidence in the data that causality between institutions, and markets either does not exist, or, if it exists...

Nonlinear Causality Testing with Stepwise Multivariate Filtering

BEKIROS, Stelios D.
Fonte: Instituto Universitário Europeu Publicador: Instituto Universitário Europeu
Tipo: Trabalho em Andamento Formato: application/pdf; digital
Português
Relevância na Pesquisa
46.38%
This study explores the direction and nature of causal linkages among six currencies denoted relative to United States dollar (USD), namely Euro (EUR), Great Britain Pound (GBP), Japanese Yen (JPY), Swiss Frank (CHF), Australian Dollar (AUD) and Canadian Dollar (CAD). These are the most liquid and widely traded currency pairs in the world and make up about 90% of total Forex trading worldwide. The data covers the period 3/20/1987-11/14/2007, including the Asian crisis, the dot-com bubble and the period just before the outbreak of the US subprime crisis. The objective of the paper is to test for the existence of both linear and nonlinear causal relationships among these currency markets. The modified Baek-Brock test for nonlinear non-causality is applied on the currency return time series as well as the linear Granger test. Further to the classical pairwise analysis causality testing is conducted in a multivariate formulation, to correct for the effects of the other variables. A new stepwise multivariate filtering approach is implemented. To check if any of the observed causality is strictly nonlinear, the nonlinear causal relationships of VAR/VECM filtered residuals are also examined. Finally, the hypothesis of nonlinear non-causality is investigated after controlling for conditional heteroskedasticity in the data using GARCH-BEKK...

A nonparametric copula based test for conditional independence with applications to granger causality

Bouezmarni, Taoufik; Rombouts, Jeroen V. K.; Taamouti, Abderrahim
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: Trabalho em Andamento Formato: application/pdf
Publicado em /06/2009 Português
Relevância na Pesquisa
56.44%
This paper proposes a new nonparametric test for conditional independence, which is based on the comparison of Bernstein copula densities using the Hellinger distance. The test is easy to implement because it does not involve a weighting function in the test statistic, and it can be applied in general settings since there is no restriction on the dimension of the data. In fact, to apply the test, only a bandwidth is needed for the nonparametric copula. We prove that the test statistic is asymptotically pivotal under the null hypothesis, establish local power properties, and motivate the validity of the bootstrap technique that we use in finite sample settings. A simulation study illustrates the good size and power properties of the test. We illustrate the empirical relevance of our test by focusing on Granger causality using financial time series data to test for nonlinear leverage versus volatility feedback effects and to test for causality between stock returns and trading volume. In a third application, we investigate Granger causality between macroeconomic variables

Nonparametric tests for conditional independence using conditional distributions

Bouezmarni, Taoufik; Taamouti, Abderrahim
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: info:eu-repo/semantics/draft; info:eu-repo/semantics/workingPaper Formato: text/plain; application/pdf
Publicado em 06/01/2012 Português
Relevância na Pesquisa
66.46%
The concept of causality is naturally defined in terms of conditional distribution, however almost all the empirical works focus on causality in mean. This paper aim to propose a nonparametric statistic to test the conditional independence and Granger non-causality between two variables conditionally on another one. The test statistic is based on the comparison of conditional distribution functions using an L2 metric. We use Nadaraya-Watson method to estimate the conditional distribution functions. We establish the asymptotic size and power properties of the test statistic and we motivate the validity of the local bootstrap. Further, we ran a simulation experiment to investigate the finite sample properties of the test and we illustrate its practical relevance by examining the Granger non-causality between S&P 500 Index returns and VIX volatility index. Contrary to the conventional t-test, which is based on a linear mean-regression model, we find that VIX index predicts excess returns both at short and long horizons.; Financial support from the Natural Sciences and Engineering Research Council of Canada and from the Spanish Ministry of Education through grants SEJ 2007-63098 are also acknowledged

Specification and casualty of distribution models

Troster, Víctor Emilio
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: Tese de Doutorado
Português
Relevância na Pesquisa
36.57%
Many important economic and finance hypotheses are investigated through testing the specification of restrictions on the conditional distribution of a time series, such as conditional goodness-of- t (Box and Pierce (1970)), conditional quantiles (Koenker and Machado (1999)), and distributional Granger non-causality (Taamouti, Bouezmarni, and El Ghouch, 2014). This PhD Thesis contributes to the study of specification and causality tests that provide a more flexible and detailed approach to evaluate economic relationships, which are useful in many relevant empirical applications. In the first chapter, we propose a practical and consistent specification test of conditional distribution models for dependent data in a very general setting. Our approach covers conditional distribution models possibly indexed by function-valued parameters, which allows for a wide range of important empirical applications, such as the linear quantile auto-regressive, the CAViaR, and the distributional regression models. Our test statistic is based on a comparison between the estimated parametric and the empirical distribution functions. The new specification test (i) is valid for general linear and nonlinear dynamic models under parameter estimation error...

Statistical inference of nonlinear Granger causality: a semiparametric time series regression analysis.

Lee, Sooyoung
Fonte: Universidade de Adelaide Publicador: Universidade de Adelaide
Tipo: Tese de Doutorado
Publicado em //2013 Português
Relevância na Pesquisa
36.76%
Since the seminal work of Granger (1969), Granger causality has become a useful concept and tool in the study of the dynamic linkages between economic variables and to explore whether or not an economic variable helps forecast another one. Researchers have suggested a variety of methods to test the existence of Grangercausality in the literature. In particular, linear Granger causality testing has been remarkably developed; (see, for example, Toda & Philips (1993), Sims, Stock & Watson (1990), Geweke (1982), Hosoya (1991) and Hidalgo (2000)). However, in practice, the real economic relationship between different variables may often be nonlinear. Hiemstra & Jones (1994) and Nishiyama, Hitomi, Kawasaki & Jeong (2011) recently proposed different methods to test the existence of any non-linear Granger causality between a pair of economic variables under a α-mixing framework of data generating process. Their methods are general with nonparametric features, which however suffer from curse of dimensionality when high lag orders need to be taken into consideration in applications. In this thesis, the main objective is to develop a class of semiparametric time series regression models that are of partially linear structures, with statistical theory established under a more general framework of near epoch dependent (NED) data generating processes...

TMS-EEG combined with granger causality: an innovative information flow approach over the full brain connectivity

Fernandes, Tiago José Cardoso Pires Timóteo
Fonte: Universidade de Lisboa Publicador: Universidade de Lisboa
Tipo: Dissertação de Mestrado
Publicado em //2015 Português
Relevância na Pesquisa
36.45%
Tese de mestrado integrado em Engenharia Biomédica e Biofísica, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2015; Atualmente, no mundo das neurociências, a conectividade cerebral é um tema em destaque. Este conceito encontra-se dividido em conetividade estrutural (relações anatómicas entre estruturas cerebrais), conectividade funcional (dependências estatísticas entre estruturas cerebrais) e conectividade efetiva (relações de causalidade entre estruturas cerebrais). Esta tese debaterá fundamentalmente sobre o último destes conceitos, tentando oferecer uma interpretação para o fluxo de informações entre as áreas do cérebro. Muitas técnicas podem ser utilizadas na sua análise, entre os quais a Causalidade de Granger (GC) ou a estimulação magnética transcraniana em combinação com eletroencefalografia (TMS-EEG). Por um lado, a GC permite uma interpretação das ligações diretas dentro e fora das mesmas áreas cerebrais, sendo uma abordagem explicativa sobre os dados, onde não é necessária nenhuma hipótese sobre o comportamento das relações causais. No entanto, os resultados de GC são muito sensíveis, uma vez que dependem de sinais não-estacionários e não colineares, aspetos bastante presentes em sinais de eletroencefalografia (EEG). Desta forma...

Robust Non-linear Wiener-Granger Causality For Large High-dimensional Data

Jafari-Mamaghani, Mehrdad
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/10/2015 Português
Relevância na Pesquisa
26.39%
Wiener-Granger causality is a widely used framework of causal analysis for temporally resolved events. We introduce a new measure of Wiener-Granger causality based on kernelization of partial canonical correlation analysis with specific advantages in the context of large high-dimensional data. The introduced measure is able to detect non-linear and non-monotonous signals, is designed to be immune to noise, and offers tunability in terms of computational complexity in its estimations. Furthermore, we show that, under specified conditions, the introduced measure can be regarded as an estimate of conditional mutual information (transfer entropy). The functionality of this measure is assessed using comparative simulations where it outperforms other existing methods. The paper is concluded with an application to climatological data.

Measures of Causality in Complex Datasets with application to financial data

Zaremba, Anna; Aste, Tomaso
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.33%
This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert--Schmidt norm of the cross-covariance operator) and transfer entropy, examining each method and comparing their theoretical properties, with special attention given to the ability to capture nonlinear causality. We also present the theoretical benefits of applying non-symmetrical measures rather than symmetrical measures of dependence. We apply the measures to a range of simulated and real data. The simulated data sets were generated with linear and several types of nonlinear dependence, using bivariate, as well as multivariate settings. An application to real-world financial data highlights the practical difficulties, as well as the potential of the methods. We use two real data sets: (1) U.S. inflation and one-month Libor; (2) S$\&$P data and exchange rates for the following currencies: AUDJPY, CADJPY, NZDJPY, AUDCHF, CADCHF, NZDCHF. Overall, we reach the conclusion that no single method can be recognised as the best in all circumstances, and each of the methods has its domain of best applicability. We also highlight areas for improvement and future research.; Comment: 40 pages; 13 figures

Neural Networks with Non-Uniform Embedding and Explicit Validation Phase to Assess Granger Causality

Montalto, Alessandro; Stramaglia, Sebastiano; Faes, Luca; Tessitore, Giovanni; Prevete, Roberto; Marinazzo, Daniele
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.37%
A challenging problem when studying a dynamical system is to find the interdependencies among its individual components. Several algorithms have been proposed to detect directed dynamical influences between time series. Two of the most used approaches are a model-free one (transfer entropy) and a model-based one (Granger causality). Several pitfalls are related to the presence or absence of assumptions in modeling the relevant features of the data. We tried to overcome those pitfalls using a neural network approach in which a model is built without any a priori assumptions. In this sense this method can be seen as a bridge between model-free and model-based approaches. The experiments performed will show that the method presented in this work can detect the correct dynamical information flows occurring in a system of time series. Additionally we adopt a non-uniform embedding framework according to which only the past states that actually help the prediction are entered into the model, improving the prediction and avoiding the risk of overfitting. This method also leads to a further improvement with respect to traditional Granger causality approaches when redundant variables (i.e. variables sharing the same information about the future of the system) are involved. Neural networks are also able to recognize dynamics in data sets completely different from the ones used during the training phase.

Graphical models for marked point processes based on local independence

Didelez, Vanessa
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 31/10/2007 Português
Relevância na Pesquisa
35.88%
A new class of graphical models capturing the dependence structure of events that occur in time is proposed. The graphs represent so-called local independences, meaning that the intensities of certain types of events are independent of some (but not necessarily all) events in the past. This dynamic concept of independence is asymmetric, similar to Granger non-causality, so that the corresponding local independence graphs differ considerably from classical graphical models. Hence a new notion of graph separation, called delta-separation, is introduced and implications for the underlying model as well as for likelihood inference are explored. Benefits regarding facilitation of reasoning about and understanding of dynamic dependencies as well as computational simplifications are discussed.; Comment: To appear in the Journal of the Royal Statistical Society Series B

A New Approach to Testing PPP using VECM: Evidence from the Yen

Brailsford, Timothy John; Penm, Jammie H C; Terrell, Richard
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
36.16%
Conventional methods to test for long-term PPP based on the theory of cointegration are typically undertaken in the framework of vector error correction models (VECM). The standard approach in the use of VECMs is to employ a model of full-order, which assumes nonzero entries in all the coefficient matrices. But, the use of full-order VECM models may lead to incorrect inferences if zero entries are required in the coefficient matrices. Specifically, if we wish to test for indirect causality, instantaneous causality, or Granger non-causality, and employ "overparameterised" full-order VECM models that ignore entries assigned a priori to be zero, then the power of statistical inference is weakened and the resultant specifications can produce different conclusions concerning the cointegrating relationships among the variables. In this paper, an approach is presented that incorporates zero entries in the VECM analysis. This approach is a more straightforward and effective means of testing for causality and cointegrating relations. The paper extends prior work on PPP through an investigation of causality between the U.S. Dollar and the Japanese Yen. The results demonstrate the inconsistencies that can arise in the area and show that bi-directional feedback exists between prices...

Do demand and profitability stimulate capital accumulation? An analysis for Brazil

Morrone, Henrique
Fonte: CEPAL - Comissão Econômica para a América Latina e o Caribe Publicador: CEPAL - Comissão Econômica para a América Latina e o Caribe
Tipo: Texto
Português
Relevância na Pesquisa
36.3%
This article tests whether the profit share of gdp and capacity utilization affect capital accumulation in Brazil in the period 1950-2008 (in the sense of Granger causality). The methodology developed by Toda and Yamamoto (1995) is used to verify the Granger non-causality hypothesis. The results show that capacity utilization “Granger-causes” capital accumulation in the Brazilian economy and, also that the profit share of gdp does not “Granger-cause” the national investment-capital ratio. This corroborates the Kaleckian proposal based on the fundamental role of the accelerator, and suggests that the Brazilian economy can grow with either a concentration or a de-concentration of income, provided a suitable institutional arrangement is in place.

Measures of causality in complex datasets with application to financial data

Zaremba, Anna; Aste, Tomaso
Fonte: Multidisciplinary Digital Publishing Institute (MDPI) Publicador: Multidisciplinary Digital Publishing Institute (MDPI)
Tipo: Article; PeerReviewed Formato: application/pdf
Publicado em //2014 Português
Relevância na Pesquisa
26.33%
This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert-Schmidt norm of the cross-covariance operator) and transfer entropy, examining each method and comparing their theoretical properties, with special attention given to the ability to capture nonlinear causality. We also present the theoretical benefits of applying non-symmetrical measures rather than symmetrical measures of dependence. We apply the measures to a range of simulated and real data. The simulated data sets were generated with linear and several types of nonlinear dependence, using bivariate, as well as multivariate settings. An application to real-world financial data highlights the practical difficulties, as well as the potential of the methods. We use two real data sets: (1) U.S. inflation and one-month Libor; (2) S&P data and exchange rates for the following currencies: AUDJPY, CADJPY, NZDJPY, AUDCHF, CADCHF, NZDCHF. Overall, we reach the conclusion that no single method can be recognised as the best in all circumstances, and each of the methods has its domain of best applicability. We also highlight areas for improvement and future research.