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The integrated copula spectrum
... Tests based on necessary and sufficient conditions, thus, are highly desirable. This is what the copula spectrum developed by Goto et al. (2022), which offers a model-free characterization of pairwise timereversibility, is allowing for. In this paper, we illustrate the performance of the copulaspectrum-based tests proposed by Goto et al. (2022) in comparison with the recent modelbased approach by Giancaterini et al. (2022), who are using it to assess the time-reversibility of climate-related time-series data. ...
... This is what the copula spectrum developed by Goto et al. (2022), which offers a model-free characterization of pairwise timereversibility, is allowing for. In this paper, we illustrate the performance of the copulaspectrum-based tests proposed by Goto et al. (2022) in comparison with the recent modelbased approach by Giancaterini et al. (2022), who are using it to assess the time-reversibility of climate-related time-series data. ...
... The paper is organized as follows: Section 2 reviews the approaches developed by Goto et al. (2022) and by Giancaterini et al. (2022), respectively. Section 3 presents our numerical results, and Section 4 compares the conclusions of the two approaches on the climate-related application considered in Giancaterini et al. (2022). ...
Time-reversibility is a crucial feature of many time series models, while time-irreversibility is the rule rather than the exception in real-life data. Testing the null hypothesis of time-reversibilty, therefore, should be an important step preliminary to the identification and estimation of most traditional time-series models. Existing procedures, however, mostly consist of testing necessary but not sufficient conditions, leading to under-rejection, or sufficient but non-necessary ones, which leads to over-rejection. Moreover, they generally are model-besed. In contrast, the copula spectrum studied by Goto et al. ( 2022, : 3563--3591) allows for a model-free necessary and sufficient time-reversibility condition. A test based on this copula-spectrum-based characterization has been proposed by authors. This paper illustrates the performance of this test, with an illustration in the analysis of climatic data.
We provide a commented bibliography that describes the contributions made by Marc Hallin in the papers he authored or coauthored between 1972 and 2023. Citations using the “name (year)” format correspond to the list of references on page 14, whereas numbers in square brackets refer to Marc’s full list of publications to date, which is provided at the end of this text.
Marc Hallin was born in Ghent, Belgium, on 23 April 1949. He holds a Licence en Sciences mathématiques (1971), a Licence en Sciences actuarielles (1972), and a Doctorat en Sciences (1976) from the Université libre de Bruxelles . He then rose through the professorial ranks at the same institution, being successively Premier Assistant (1977–1978), Chargé de Cours associé (1978–1984), Chargé de Cours (1984–1988), Professeur ordinaire (1988–2009), and Professeur ordinaire émérite upon retirement in 2009. Throughout his career, he supervised 25 PhD students and held invited positions at many institutions of high standing in Austria, Belgium, England, France, Hong Kong, Italy, Portugal, Spain, Switzerland, and the USA (most notably Princeton). A renown expert in time series analysis, econometrics, and non‐parametric inference, Marc is the author or coauthor of over 250 research papers, for which he received numerous awards, including the Medal of the Faculty of Mathematics and Physics of Charles University in Prague (2006), a Humboldt Forschungspreis from the Alexander von Humboldt Foundation (2012), the Pierre‐Simon de Laplace Award of the Société française de Statistique (2022), and the Gottfried E. Noether Distinguished Scholar Award of the American Statistical Association (2022). He gave several distinguished lecture series, including the 2017 Hermann Otto Hirschfeld Lecture Series at the Humboldt Universität zu Berlin , and the 2018 Mahalanobis Memorial Lecture at the Indian Statistical Institute. Over the years, he co‐edited a dozen books and proceedings, and served on the editorial boards of several journals, including the Journal of Time Series Analysis (1994–2009), the Journal of Econometrics (2013–2019), the Journal of Business and Economic Statistics (2018–), and the Theory and Methods Section of the Journal of the American Statistical Association (2005–). He is a Fellow of the Institute of Mathematical Statistics (1990) and the American Statistical Association (1997), as well as a member of the Classe des Sciences of the Royal Academy of Belgium (1999). Marc has been a member of the International Statistical Institute since 1985 and was (co‐) Editor‐in‐Chief of the International Statistical Review from 2010 to 2015.
This study examines the problem of robust spectral analysis of long-memory processes. We investigate the possibility of using Laplace and quantile periodograms for a non-Gaussian distribution structure. The Laplace periodogram, derived by the least absolute deviations in the harmonic regression procedure, demonstrates its superiority in handling heavy-tailed noise and nonlinear distortion. In this study, we discuss an asymptotic distribution of the Laplace periodogram for long-memory processes. We also derive an asymptotic distribution of the quantile periodogram. Through numerical experiments, we demonstrate the robustness of the Laplace periodogram and the usefulness of the quantile periodogram in detecting the hidden frequency for the spectral analysis of the long-memory process under non-Gaussian distribution. Moreover, as an application of robust periodograms under the long-memory process, we discuss the long-memory parameter estimation based on a log periodogram regression approach.
Quantile-based approaches to the spectral analysis of time series have recently attracted a lot of attention. Despite a growing literature that contains various estimation proposals, no systematic methods for computing the new estimators are available to date. This paper contains two main contributions. First, an extensible framework for quantile-based spectral analysis of time series is developed and documented using object-oriented models. A comprehensive, open source, reference implementation of this framework, the R package quantspec, was recently contributed to CRAN by the author of this paper. The second contribution of the present paper is to provide a detailed tutorial, with worked examples, to this R package. A reader who is already familiar with quantile-based spectral analysis and whose primary interest is not the design of the quantspec package, but how to use it, can read the tutorial and worked examples (Sections 3 and 4) independently.
Quantile- and copula-related spectral concepts recently have been considered by various authors. Those spectra, in their
most general form, provide a full characterization of the copulas
associated with the pairs (Xt;Xt-k) in a process (Xt)t2Z, and account
for important dynamic features, such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes, that their traditional counterpart cannot capture.
Despite various proposals for estimation strategies, no asymptotic distributional results are available so far for the proposed estimators, which constitutes an important obstacle for their practical application.
In this paper, we provide a detailed asymptotic analysis of a class of smoothed rank-based cross-periodograms associated with
the copula spectral density kernels introduced in Dette et al. (2011).
We show that, for a very general class of (possibly non-linear) processes, properly scaled and centered smoothed versions of those crossperiodograms, indexed by couples of quantile levels, converge weakly,
as stochastic processes, to Gaussian processes. A first application of those results is the construction of asymptotic confi dence intervals for copula spectral density kernels. The same convergence results also provide asymptotic distributions (under serially dependent observations) for a new class of rank-based spectral methods involving the Fourier transforms of rank-based serial statistics such as the Spearman, Blomqvist or Gini autocovariance coefficients.
The goal of this paper is two-fold: (1) We review classical and recent measures of serial extremal dependence in a strictly stationary time series as well as their estimation. (2) We discuss recent concepts of heavy-tailed time series, including regular variation and max-stable processes.Serial extremal dependence is typically characterized by clusters of exceedances of high thresholds in the series. We start by discussing the notion of extremal index of a univariate sequence, i.e. the reciprocal of the expected cluster size, which has attracted major attention in the extremal value literature. Then we continue by introducing the extremogram which is an asymptotic autocorrelation function for sequences of extremal events in a time series. In this context, we discuss regular variation of a time series. This notion has been useful for describing serial extremal dependence and heavy tails in a strictly stationary sequence. We briefly discuss the tail process coined by Basrak and Segers to describe the dependence structure of regularly varying sequences in a probabilistic way. Max-stable processes with Fréchet marginals are an important class of regularly varying sequences. Recently, this class has attracted attention for modeling and statistical purposes. We apply the extremogram to max-stable processes. Finally, we discuss estimation of the extremogram both in the time and frequency domains.
We propose a new diagnostic tool for time series called the quantilogram. The tool can be used formally and we provide the inference tools to do this under general conditions, and it can also be used as a simple graphical device. We apply our method to measure directional predictability and to test the hypothesis that a given time series has no directional predictability. The test is based on comparing the correlogram of quantile hits to a pointwise confidence interval or on comparing the cumulated squared autocorrelations with the corresponding critical value. We provide the distribution theory needed to conduct inference, propose some model free upper bound critical values, and apply our methods to S&P500 stock index return data. The empirical results suggest some directional predictability in returns. The evidence is strongest in mid range quantiles like 5–10% and for daily data. The evidence for predictability at the median is of comparable strength to the evidence around the mean, and is strongest at the daily frequency.
In this paper we present an alternative method for the spectral analysis of a
strictly stationary time series . We define a "new" spectrum
as the Fourier transform of the differences between copulas of the pairs
and the independence copula. This object is called {\it copula
spectral density kernel} and allows to separate marginal and serial aspects of
a time series. We show that it is intrinsically related to the concept of
quantile regression. Like in quantile regression, which provides more
information about the conditional distribution than the classical
location-scale model, the copula spectral density kernel is more informative
than the spectral density obtained from the autocovariances. In particular the
approach provides a complete description of the distributions of all pairs
. Moreover, it inherits the robustness properties of classical
quantile regression, because it does not require any distributional assumptions
such as the existence of finite moments. In order to estimate the copula
spectral density kernel we introduce rank-based Laplace periodograms which are
calculated as bilinear forms of weighted -projections of the ranks of the
observed time series onto a harmonic regression model. We establish the
asymptotic distribution of those periodograms, and the consistency of
adequately smoothed versions. The finite-sample properties of the new
methodology, and its potential for applications are briefly investigated by
simulations and a short empirical example.
Weak convergence of the empirical copula process is shown to hold under the
assumption that the first-order partial derivatives of the copula exist and are
continuous on certain subsets of the unit hypercube. The assumption is
non-restrictive in the sense that it is needed anyway to ensure that the
candidate limiting process exists and has continuous trajectories. In addition,
resampling methods based on the multiplier central limit theorem, which require
consistent estimation of the first-order derivatives, continue to be valid.
Under certain growth conditions on the second-order partial derivatives that
allow for explosive behavior near the boundaries, the almost sure rate in
Stute's representation of the empirical copula process can be recovered. The
conditions are verified, for instance, in the case of the Gaussian copula with
full-rank correlation matrix, many Archimedean copulas, and many extreme-value
copulas.
We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.
We study the asymptotic behavior of the integrated periodogram for -stable linear processes. For we prove a functional limit theorem for the integrated periodogram. The limit is an -stable analogue to the Brownian bridge. We apply our results to investigate some specific goodness-of-fit tests for heavy-tailed linear processes.
Let X( t ) ( t = 0, � 1,…) be a zero mean, r vector-valued, strictly stationary time series satisfying a particular assumption about the near-independence of widely separated values. Given the values X( t ) ( t = 0, 1,…, T − 1), we construct the statistics: I(T)/XX(γ)(-∞λlt;∞),the matrix of second-order periodograms, F T /XX(λ),the matrix of sample spectral measures, f T /XX(λ), the matrix of sample spectral densities and c( T )/( u ) ( u = 0,� 1,…), the matrix of sample covariances. In the paper expressions are derived for the first- and second-order moments and the asymptotic distributions of I T /XX(λ), F( T )/XX(λ), f( T )/XX(λ) and c( T )/XX( u ). Our purpose is to determine the form of these moments and to indicate the appearance of the Wishart distribution as an exact limiting distribution for f(T)/XX(λ). It has previously been suggested as an approximation.
Weak convergence of the empirical copula process has been established by Deheuvels in the case of independent marginal distributions. Van der Vaart and Wellner utilize the functional delta method to show convergence in ℓ∞([a,b]2) for some 0<a<b<1, under restrictions on the distribution functions. We extend their results by proving the weak convergence of this process in
ℓ∞([0,1]2) under minimal conditions on the copula function, which coincides with the result obtained by Gaenssler and Stute. It is argued that the condition on the copula function is necessary. The proof uses the functional delta method and, as a consequence, the convergence of the bootstrap counterpart of the empirical copula process follows immediately. In addition, weak convergence of the smoothed empirical copula process is established.
We propose a method for the construction of simultaneous confidence bands for a smoothed version of the spectral density of
a Gaussian process based on nonparametric kernel estimators obtained by smoothing the periodogram. A studentized statistic
is used to determine the width of the band at each frequency and a frequency-domain bootstrap approach is employed to estimate
the distribution of the supremum of this statistic over all frequencies. We prove by means of strong approximations that the
bootstrap estimates consistently the distribution of the supremum deviation of interest and, consequently, that the proposed
confidence bands achieve asymptotically the desired simultaneous coverage probability. The behaviour of our method in finite-sample
situations is investigated by simulations and a real-life data example demonstrates its applicability in time series analysis.
We examine Italian inflation rates and the Phillips curve with a very long-run perspective, one that covers the entire existence of the Italian lira from political unification (1861) to Italy's entry in the European Monetary Union (end of 1998). We first study the volatility, persistence and stationarity of the Italian inflation rate over the long run and across various exchange-rate regimes that have shaped Italian monetary history. Next, we estimate alternative Phillips equations and investigate whether nonlinearities, asymmetries and structural changes characterize the inflation-output trade-off in the long run. We capture the effects of structural changes and asymmetries on the estimated parameters of the inflation-output trade-off, relying partly on sub-sample estimates and partly on time-varying parameters estimated via the Kalman filter. Finally, we investigate causal relationships between inflation rates and output and extend the analysis to include the US and the UK for comparison purposes. The inference is that Italy has experienced a conventional inflation-output trade-off only during times of low inflation and stable aggregate supply.
We consider quantile autoregression (QAR) models in which the au- toregressive coefficients can be expressed as monotone functions of a single, scalar random variable. The models can capture systematic influences of conditioning variables on the location, scale and shape of the conditional distribution of the response, and therefore constitute a significant extension of classical constant co- efficient linear time series models in which the effect of conditioning is confined to a location shift. The models may be interpreted as a special case of the general random coefficient autoregression model with strongly dependent coefficients. Sta- tistical properties of the proposed model and associated estimators are studied. The limiting distributions of the autoregression quantile process are derived. Quantile autoregression inference methods are also investigated. Empirical applications of the model to the U.S. unemployment rate and U.S. gasoline prices highlight the potential of the model.
This paper examines, in the time and frequency domains, the quantile dependence and directional predictability of investor attention to cryptocurrency returns. We find that there is significant tail dependence between investor attention and cryptocurrency returns. When market pays very high or very low attention to cryptocurrencies, there is an increased likelihood to have very large positive gains and suffer from very large negative results. Our results indicate that the quantile dependence between investor attention and cryptocurrency returns has a higher statistical significance in the long-term than medium-term and short-term. This implies that quantile dependence between investor attention and cryptocurrency returns is mainly dominated by low-frequency components. These findings have important implications for cryptocurrency investors.
Nonlinear dynamic volatility has been observed in many financial time series. The recently proposed quantile periodogram offers an alternative way to examine this phenomena in the frequency domain. The quantile periodogram is constructed from trigonometric quantile regression of time series data at different frequencies and quantile levels, enabling the quantile‐frequency analysis (QFA) of nonlinear serial dependence. This paper introduces some spectral measures based on the quantile periodogram for diagnostic checks of financial time series models and for model‐based discriminant analysis. A simulation‐based parametric bootstrapping technique is employed to compute the p‐values of the spectral measures. The usefulness of the proposed method is demonstrated by a simulation study and a motivating application using the daily log returns of the S&P 500 index together with GARCH‐type models. The results show that the QFA method is able to provide additional insights into the goodness of fit of these financial time series models that may have been missed by conventional tests. The results also show that the QFA method offers a more informative way of discriminant analysis for detecting regime changes in financial time series.
Spectral analysis of multivariate time series has been an active field of methodological and applied statistics for the past 50 years. Since the success of the fast Fourier transform algorithm, the analysis of serial auto- and cross-correlation in the frequency domain has helped us to understand the dynamics in many serially correlated data without necessarily needing to develop complex parametric models. In this work, we give a nonexhaustive review of the mostly recent nonparametric methods of spectral analysis of multivariate time series, with an emphasis on model-based approaches. We try to give insights into a variety of complimentary approaches for standard and less standard situations (such as nonstationary, replicated, or high-dimensional time series), discuss estimation aspects (such as smoothing over frequency), and include some examples stemming from life science applications (such as brain data).
Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 7 is March 7, 2020. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
We propose three methods for estimating the joint tail probabilities based on a d-variate copula with dimension d≥2. For the first two methods, we use two different tail expansions of the copula which are valid under mild regularity conditions. We estimate the coefficients of these expansions using the maximum likelihood approach with appropriate data beyond a threshold in the tail. For the third method, we propose a family of tail-weighted measures of multivariate dependence and use these measures to estimate the coefficients of the second tail expansion using regression. This expansion is then used to estimate the joint tail probabilities when the empirical probabilities cannot be used because of lack of data in the tail. The three proposed methods can also be used to estimate tail dependence coefficients of a multivariate copula. Simulation studies are used to indicate when the methods give more accurate estimates of the tail probabilities and tail dependence coefficients. We apply the proposed methods to analyze tail properties of a data set of financial returns.
In this paper, we introduce quantile coherency to measure general dependence structures emerging in the joint distribution in the frequency domain and argue that this type of dependence is natural for economic time series but remains invisible when only the traditional analysis is employed. We define estimators which capture the general dependence structure, provide a detailed analysis of their asymptotic properties and discuss how to conduct inference for a general class of possibly nonlinear processes. In an empirical illustration we examine the dependence of bivariate stock market returns and shed new light on measurement of tail risk in financial markets. We also provide a modelling exercise to illustrate how applied researchers can benefit from using quantile coherency when assessing time series models.
Time series with mixed spectra are characterized by hidden periodic components buried in random noise. Despite strong interest in the statistical and signal processing communities, no book offers a comprehensive and up-to-date treatment of the subject. Filling this void, Time Series with Mixed Spectra focuses on the methods and theory for the statistical analysis of time series with mixed spectra. It presents detailed theoretical and empirical analyses of important methods and algorithms. Using both simulated and real-world data to illustrate the analyses, the book discusses periodogram analysis, autoregression, maximum likelihood, and covariance analysis. It considers real-and complex-valued time series, with and without the Gaussian assumption. The author also includes the most recent results on the Laplace and quantile periodograms as extensions of the traditional periodogram. Complete in breadth and depth, this book explains how to perform the spectral analysis of time series data to detect and estimate the hidden periodicities represented by the sinusoidal functions. The book not only extends results from the existing literature but also contains original material, including the asymptotic theory for closely spaced frequencies and the proof of asymptotic normality of the nonlinear least-absolute-deviations frequency estimator.
Finding parametric models that accurately describe the dependence structure of observed data is a central task in the analysis of time series. Classical frequency domain methods provide a popular set of tools for fitting and diagnostics of time series models, but their applicability is seriously impacted by the limitations of covariances as a measure of dependence. Motivated by recent developments of frequency domain methods that are based on copulas instead of covariances, we propose a novel graphical tool that allows to access the quality of time series models for describing dependencies that go beyond linearity. We provide a thorough theoretical justification of our approach and show in simulations that it can successfully distinguish between subtle differences of time series dynamics, including non-linear dynamics which result from GARCH and EGARCH models.
This book provides an overview of the current stat-of-the-art of nonlinear time series analysis, richly illustrated with examples, pseudocode algorithms and real-world applications. Avoiding a "theorem-proof" format, it shows concrete applications on a variety of empirical time series. The book can be used in graduate courses in nonlinear time series and at the same time also includes interesting material for more advanced readers.
The book covers time-domain and frequency-domain methods for the analysis of both univariate and multivariate (vector) time series. It makes a clear distinction between parametric models on the one hand, and semi- and nonparametric models/methods on the other. This offers the reader the option of concentrating exclusively on one of these nonlinear time series methods.
To make the book as user friendly as possible, major supporting concepts and specialized tables are appended at the end of every chapter. In addition, each chapter concludes with a set of key terms and concepts, as well as a summary of the main findings. Lastly, the book offers numerous theoretical and empirical exercises, with answers provided by the author in an extensive solutions manual.
DOI 10.1007/978-3-319-43252-6
Classical spectral methods are subject to two fundamental limitations: they can account only for covariance-related serial dependences, and they require second-order stationarity. Much attention has been devoted lately to quantile-based spectral methods that go beyond covariance-based serial dependence features. At the same time, covariance-based methods relaxing stationarity into much weaker local stationarity conditions have been developed for a variety of time series models. Here, we combine those two approaches by proposing quantile-based spectral methods for locally stationary processes. We therefore introduce a time varying version of the copula spectra that have been recently proposed in the literature, along with a suitable local lag window estimator. We propose a new definition of local strict stationarity that allows us to handle completely general non-linear processes without any moment assumptions, thus accommodating our quantile-based concepts and methods. We establish a central limit theorem for the new estimators and illustrate the power of the proposed methodology by means of a simulation study. Moreover, in two empirical studies (namely of the Standard & Poor's 500 series and a temperature data set recorded in Hohenpeissenberg), we demonstrate that the new approach detects important variations in serial dependence structures both across time and across quantiles. Such variations remain completely undetected and are actually undetectable, via classical covariance-based spectral methods.
This paper proposes the cross-quantilogram to measure the quantile dependence between two time series. We apply it to test the hypothesis that one time series has no directional predictability to another time series. We establish the asymptotic distribution of the cross-quantilogram and the corresponding test statistic. The limiting distributions depend on nuisance parameters. To construct consistent confidence intervals we employ a stationary bootstrap procedure; we establish consistency of this bootstrap. Also, we consider a self-normalized approach, which yields an asymptotically pivotal statistic under the null hypothesis of no predictability. We provide simulation studies and two empirical applications. First, we use the cross-quantilogram to detect predictability from stock variance to excess stock return. Compared to existing tools used in the literature of stock return predictability, our method provides a more complete relationship between a predictor and stock return. Second, we investigate the systemic risk of individual financial institutions, such as JP Morgan Chase, Morgan Stanley and AIG.
A general approach to constructing confidence intervals by subsampling was presented in Politis and Romano (1994). The crux of the method is recomputing a statistic over subsamples of the data, and these recomputed values are used to build up an estimated sampling distribution. The method works under extremely weak conditions, it applies to independent, identically distributed (i.i.d.) observations as well as to dependent data situations, such as time series (possibly nonstationary), random fields, and marked point processes. In this article, we present some theorems showing: a new construction for confidence intervals that removes a previous condition, a general theorem showing the validity of subsampling for data-dependent choices of the block size, and a general theorem for the construction of hypothesis tests (not necessarily derived from a confidence interval construction). The arguments apply to both the i.i.d. setting and the dependent data case.
We propose a unified framework for testing a variety of assumptions commonly made about the structure of copulas, including symmetry, radial symmetry, joint symmetry, associativity and Archimedeanity, and max-stability. Our test is nonparametric and based on the asymptotic distribution of the empirical copula process. We perform simulation experiments to evaluate our test and conclude that our method is reliable and powerful for assessing common assumptions on the structure of copulas, particularly when the sample size is moderately large. We illustrate our testing approach on two datasets.
A threshold extreme value distribution for modeling standardized financial returns is investigated. The main theme is tail asymmetry, which means that the left and right tails of the standardized return distribution are not identical. The peak-over-threshold idea in extreme value theory is adopted to construct the threshold extreme value distribution with two generalized Pareto tails for modeling tail asymmetry. The estimation of unknown parameters is performed within the Bayesian paradigm. Bayesian tail asymmetry tests are set up and Chib's marginal likelihood approach is found to be most reliable. In the empirical analysis of nine securities, strong evidence of tail asymmetry is observed in equities, whereas modest evidence is documented in currencies and Gold futures. Oil futures is very volatile but shows weak evidence of tail asymmetry. Equity indices show a thinner than normal right tail in volatile periods, contradicting the usual fat-tail assumption in financial return modeling. One striking result is that all securities exhibit an increasing propagation of tail asymmetry during financial crises, suggesting that the level of tail asymmetry can be an indicator of the occurrence of extreme financial events. In terms of risk calculation, the threshold extreme value distribution is superior to its symmetric version and Student's t distribution in forecasting multiple-period value at risk, especially when the right tail of the return distribution, i.e. in the short position, is of interest. The proposed method performs particularly well in 10-day-1% and 10-day-99% value at risk forecasting, which are Basel requirements for capital adequacy calculation.
Three tail asymmetry measures for bivariate copulas are introduced using two different approaches—univariate skewness of a projection and distance between a copula and its survival/reflected copula. We compare the asymmetry measures based on certain desirable properties. Bounds for each measure are obtained and also copulas which attain these extreme values are identified. Two data examples show the amount of asymmetry that might be expected in practice.
Economic and financial time series frequently exhibit time irreversible dynamics. For instance, there is considerable evidence of asymmetric fluctuations in many macroeconomic and financial variables, and certain game theoretic models of price determination predict asymmetric cycles in price series. In this paper we make two primary contributions to the econometric literature on time reversibility. First, we propose a new test of time reversibility, applicable to stationary Markov chains. Compared to existing tests, our test has the advantage of being consistent against arbitrary violations of reversibility. Second, we explain how a circulation density function may be used to characterize the nature of time irreversibility when it is present. We propose a copula-based estimator of the circulation density, and verify that it is well behaved asymptotically under suitable regularity conditions. We illustrate the use of our time reversibility test and circulation density estimator by applying them to five years of Canadian gasoline price markup data.
Two periodogram-like functions, called quantile periodograms, are introduced for spectral analysis of time series. The quantile periodograms are constructed from trigonometric quantile regression and motivated by different interpretations of the ordinary periodogram. Analytical and numerical results demonstrate the capability of the quantile periodograms for detecting hidden periodicity in the quantiles and for providing an additional view of time-series data. A connection between the quantile periodograms and the so-called level-crossing spectrum is established through an asymptotic analysis.
Let be a sample time series drawn from the real stationary Gaussian process , with unknown spectral distribution function (s.d.f.) and spectral density function . The problem of estimating of s.d.f. is discussed and the estimate of s.d.f. is considered, whereIn § the asymptotic properties of expressions likeare investigated.
The main section of this paper is § 5. Letand let be a Gaussian stochastic process with We denote by the probability measure induced in by , and by P the probability measure induced in by . The following is proved in §5: Letthen, where the signdenotes weak convergence of the measures.
In § 8 some estimates are given for probabilities of large deviations from .
In § 9 it is shown that all results of §§ are valid for continuous time.
Several concepts of bivariate symmetry are discussed, and relationships between and among these various concepts are explored. One of the concepts, radial symmetry, is shown to be nonparametric in the sense that it is a property of the copula of the random variables. Examples of distributions possessing the various types of symmetry are also presented.
Let X be a linear process having a finite fourth moment. Assume is a class of square-integrable functions. We consider the empirical spectral distribution function Jn,X based on X and indexed by . If is totally bounded then Jn,X satisfies a uniform strong law of large numbers. If, in addition, a metric entropy condition holds, then Jn,X obeys the uniform central limit theorem.
We derive functional limit theorems for the integrated periodogram of linear processes whose innovations may have finite or infinite variance, and which may exhibit long memory. The results are applied to obtain corresponding Kolmogorov-Smirnov and Cramér-von Mises goodness-of-fit tests.
This paper is concerned with the estimation of the spectral measure of a stationary process. Empirical spectral processes indexed by classes of functions are considered and an equicontinuity condition and a weak convergence result for the resulting spectral process are proved. Furthermore, some applications to time series analysis are given.
In this paper we propose a class of new tests for time reversibility. It is shown that this test has an asymptotic normal distribution under the null hypothesis and non-trivial power under local alternatives. A novel feature of this test is that it does not have any moment restriction, in contrast with other time reversibility and linearity tests. Our simulations also confirm that the proposed test is very robust when data do not possess proper moments. An empirical study of stock market indices is also included to illustrate the usefulness of the new test.
A nonparametric bootstrap procedure is proposed for stochastic processes which follow a general autoregressive structure. The procedure generates bootstrap replicates by locally resampling the original set of observations reproducing automatically its dependence properties. It avoids an initial nonparametric estimation of process characteristics in order to generate the pseudo-time series and the bootstrap replicates mimic several of the properties of the original process. Applications of the procedure in nonlinear time-series analysis are considered and theoretically justified; some simulated and real data examples are discussed.
In this paper I introduce quantile spectral densities that summarize the
cyclical behavior of time series across their whole distribution by analyzing
periodicities in quantile crossings. This approach can capture systematic
changes in the impact of cycles on the distribution of a time series and allows
robust spectral estimation and inference in situations where the dependence
structure is not accurately captured by the auto-covariance function. I study
the statistical properties of quantile spectral estimators in a large class of
nonlinear time series models and discuss inference both at fixed and across all
frequencies. Monte Carlo experiments illustrate the advantages of quantile
spectral analysis over classical methods when standard assumptions are
violated.
We consider a strictly stationary sequence of random vectors whose finite-dimensional distributions are jointly regularly varying with some positive index. This class of processes includes, among others, ARMA processes with regularly varying noise, GARCH processes with normally or Student-distributed noise and stochastic volatility models with regularly varying multiplicative noise. We define an analog of the autocorrelation function, the extremogram, which depends only on the extreme values in the sequence. We also propose a natural estimator for the extremogram and study its asymptotic properties under -mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram. Comment: Published in at http://dx.doi.org/10.3150/09-BEJ213 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
The spectral distribution function of a stationary stochastic process standardized by dividing by the variance of the process is a linear function of the autocorrelations. The integral of the sample standardized spectral density (periodogram) is a similar linear function of the autocorrelations. As the sample size increases, the difference of these two functions multiplied by the square root of the sample size converges weakly to a Gaussian stochastic process with a continuous time parameter. A monotonic transformation of this parameter yields a Brownian bridge plus an independent random term. The distributions of functionals of this process are the limiting distributions of goodness of fit criteria that are used for testing hypotheses about the process autocorrelations. An application is to tests of independence (flat spectrum). The characteristic function of the Cramer-von Mises statistic is obtained; inequalities for the Kolmogorov-Smirnov criterion are given. Confidence regions for unspecified process distributions are found.
The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and [phi]-mixing sequences and linear transformations of martingale differences.
The problem of business-cycle symmetry is addressed within the context of time reversibility. To this effect, the authors introduce a time domain test of time reversibility, the TR test. In an application, they show that time irreversibility is the rule rather than the exception for two well-known representative macroeconomic data sets. This shows that many components of the business cycle have asymmetric fluctuations. The characterization of asymmetry provided by the TR test shows that many series exhibit steepness asymmetry. A few series appear to be either deep or sharp. Copyright 1996 by Ohio State University Press.
Two tests for serial dependence are proposed using a generalized spectral theory in combination with the empirical distribution function. The tests are generalizations of the Cramér-von Mises and Kolmogorov-Smirnov tests based on the standardized spectral distribution function. They do not involve the choice of a lag order, and they are consistent against all types of pairwise serial dependence, including those with zero autocorrelation. They also require no moment condition and are distribution free under serial independence. A simulation study compares the finite sample performances of the new tests and some closely related tests. The asymptotic distribution theory works well in finite samples. The generalized Cramér-von Mises test has good power against a variety of dependent alternatives and dominates the generalized Kolmogorov-Smirnov test. A local power analysis explains some important stylized facts on the power of the tests based on the empirical distribution function.
A new type of periodogram, called the Laplace periodogram, is derived by replacing least squares with least absolute deviations in the harmonic regression procedure that produces the ordinary periodogram of a time series. An asymptotic analysis reveals a connection between the Laplace periodogram and the zero-crossing spectrum. This relationship provides a theoretical justification for use of the Laplace periodogram as a nonparametric tool for analyzing the serial dependence of time series data. Superiority of the Laplace periodogram in handling heavy-tailed noise and nonlinear distortion is demonstrated by simulations. A real-data example shows its great effectiveness in analyzing heart rate variability in the presence of ectopic events and artifacts.