No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Abstract
A necessary and sufficient condition for time-reversibility of stationary linear processes is given which, contrary to previous results, does not require existence of moments of order higher than two.
... Moreover, at a conceptual level, it is pertinent to have a complete characterization of time reversible time series. The situation for a univariate stationary linear time series is now completely solved (e.g., Cheng (1999)). However, it is a curious fact that time reversibility for multivariate non-Gaussian time series is, as far as we know, hardly studied in the literature. ...
... , X −t k ). In the univariate case, Weiss (1975) shows that the only time-reversible non-Gaussian ARMA processes are sub-classes of pure moving average processes, and Findley (1986), Hallin, Lefevre and Puri (1988), Breidt and Davis (1992) and Cheng (1990Cheng ( , 1999 studied time-reversibility and related problems in the context of general linear processes. ...
... The following theorem on time-reversibility of univariate linear processes is due to Cheng (1999). ...
We study the time-reversibility of multivariate linear processes, introducing a necessary and sufficient condition related to linear transforms of the multivariate linear process. Conditions analogous to Cheng's for univariate non-Gaussian linear processes are also explored, these are in terms of the noise distribution and the model parameters. The exploration results in an easily verifiable set of necessary and sufficient conditions for a multivariate non-Gaussian linear process driven by a univariate noise, leaving the case of multivariate noise as a challenging open problem.
... A stationary process {X k , k ∈ Z} is said to be time-reversible, in the sense of Cheng [21], if for each n ∈ N and integers k 1 < · · · < k n the vectors (X k 1 , . . . , X kn ) and (X −k 1 , . . . ...
... This condition is fulfilled for sequences of independent and identically distributed random variables. Cheng [21] established necessary and sufficient conditions for stationary Gaussian linear processes to be time-reversible, see also [18, p. 546] for related references. As a consequence of time-reversibility: large sample properties established for the (forward) partial sum process hold in the same way for the "backward" version { n k=1 X −k , n ≥ 1}. ...
... Within a general dependence situation, beyond the scope of [21] and even beyond mixing conditions, the NED condition seems to be indispensable to assure an approximately kind of time-reversibility, which in turn guarantees strong "backward" approximations. In conclusion we believe that our result is the only contribution towards Ling's new direction so far. ...
This thesis concerns dependence issues arising from nonparametric change-point analysis based on weighted approximations. We establish new approximation results under strong mixing conditions. Based on coupling methods, approximations for weighted tied-down partial sum processes by standardized Brownian bridge processes are derived. Moreover, we present some new “backward” strong invariance principles for linear processes with strongly mixing errors. As a consequence, we are able to establish Darling-Erdős type limit theorems for weighted tied-down partial sum processes within a financial time series framework.
... Moreover, at a conceptual level, it is pertinent to have a complete characterization of time reversible time series. The situation for a univariate stationary linear time series is now completely solved (e.g., Cheng (1999)). However, it is a curious fact that time reversibility for multivariate non-Gaussian time series is, as far as we know, hardly studied in the literature. ...
... , X −t k ). In the univariate case, Weiss (1975) shows that the only time-reversible non-Gaussian ARMA processes are sub-classes of pure moving average processes, and Findley (1986), Hallin, Lefevre and Puri (1988), Breidt and Davis (1992) and Cheng (1990Cheng ( , 1999 studied time-reversibility and related problems in the context of general linear processes. ...
... The following theorem on time-reversibility of univariate linear processes is due to Cheng (1999). ...
We derive some readily verifiable necessary and sufficient conditions for a multivariate non-Gaussian linear process to be
time-reversible, under two sets of conditions on the contemporaneous dependence structure of the innovations. One set of conditions
concerns the case of independent-component innovations, in which case a multivariate non-Gaussian linear process is time-reversible
if and only if the coefficients consist of essentially asymmetric columns with column-specific origins of symmetry or symmetric
pairs of columns with pair-specific origins of symmetry. On the other hand, for dependent-component innovations plus other
regularity conditions, a multivariate non-Gaussian linear process is time-reversible if and only if the coefficients are essentially
symmetric about some origin.
... The time-reversible non-Gaussian ARMA processes are sub-classes of pure moving-average processes with symmetric or skewsymmetric constraints on the coefficients. Hallin et al. (1988) extended this result to the case of general linear processes under some conditions (see also Breidt and Davis, 1991; Cheng, 1999). It is clear then that testing for time reversibility is not equivalent to testing for Gaussianity. ...
... There exist stationary nonlinear time processes that are time reversible (see, for example, McKenzie, 1985; Lewis et al., 1989; Rao et al., 1992). 1 Similarly, time irreversible processes are non-Gaussian but the contrary is not necessarily true. A non- Gaussian linear process can be time reversible when it satisfies certain conditions (see Theorem 2 of Cheng, 1999). Giannakis and Tsatsanis (1994) pointed out that any non-Gaussian i.i.d. ...
This paper first introduces the trispectrum-based time reversibility test to complement its bispectrum counterpart introduced earlier in extant literature. Using these frequency domain tests, we then examine whether the returns series of major stock market indices in 48 countries are time reversible. The results consistently show that time irreversibility is the rule rather than the exception for stock market indices. Further investigation on the sources of time irreversibility by applying the bispectrum-based Gaussianity and linearity tests on the original returns series reveals that the rejections in most cases are due to nonlinearity. Implications of the findings are discussed in the paper.
... This has been studied in turbulence by considering the time reversibility of subgrid scales models [6], coherent structures in wall turbulence [7], by following Lagrangian tracers and their dispersion backward and forward [8], by considering the power along Lagrangian trajectories [9][10][11][12][13][14], or by considering their links with singularities, especially using the Eulerian acceleration [15,16]. Time reversibility for nonlinear time series has also been considered in other fields, such as statistical physics [17,18], economy [19][20][21][22] or geosciences [23]. ...
In fully developed turbulence, there is a flux of energy from large to small scales in the inertial range until the dissipation at small scales. It is associated with irreversibility, i.e., a breaking of the time reversal symmetry. Such turbulent flows are characterized by scaling properties, and we consider here how irreversibility depends on the scale. Indicators of time-reversal symmetry for time series are tested involving triple correlations in a non-symmetric way. These indicators are built so that they are zero for a time-reversal symmetric time series, and a departure from zero is an indicator of irreversibility. We study these indicators applied to two fully developed turbulence time series, from flume tank and wind tunnel databases. It is found that irreversibility occurs in the inertial range and has scaling properties with slopes close to one. A maximum value is found around the injection scale. This confirms that the irreversibility is associated with the turbulent cascade in the inertial range and shows that the irreversibility is maximal at the injection scale, the largest scale of the turbulent cascade.
... In the past few decades, many different irreversible measures have been proposed [4][5][6][7][8][9][10][11][12][13][14][15]. However, on the one hand, the reversibility test of time series can merely analyze the irreversibility of time series qualitatively rather than quantitatively. ...
As a practical tool, visibility graph provides a different perspective to characterize time series. In this paper, we present a new visibility algorithm called directed vector visibility graph and combine it with the Kullback–Leibler divergence to measure the irreversibility of multivariable time series. T directed vector visibility algorithm converts the time series into a directed network. Subsequently, the ingoing and outgoing degree distributions of the directed network can be got to calculate the Kullback–Leibler divergence, which will be applied to assess the level of irreversibility of the time series. This is a simple and effective method without any special symbolic process. The numerical results from various types of systems are used to validate that this method can accurately distinguish reversible time series from those irreversible ones. Finally, we employ this method to estimate the irreversibility of financial time series and the results show that our method is efficient to analyze the financial time series irreversibility.
... All univariate, Gaussian stationary or stationary-increment stochastic processes are time-reversible. More generally, in the univariate context, confirmation of time irreversibility is relevant in both theory and modeling because it can be viewed, for example, as evidence of either non-Gaussianity or nonlinearity (see Weiss (1975), Cox (1981), Section 3, Cheng (1999), and De Gooijer (2017), p. 315; see also Jacod and Protter (1988), Cox (1991) and Rosenblatt (2000), chapter 1). In particular, time reversibility is well known to be a topic of central importance in Physics (e.g., Kuśmierz et al. (2016)). ...
In this paper, we construct operator fractional L\'{e}vy motion (ofLm), a broad class of non-Gaussian stochastic processes that are covariance operator self-similar, have wide-sense stationary increments and display infinitely divisible marginal distributions. The ofLm class generalizes the univariate fractional L\'evy motion as well as the multivariate operator fractional Brownian motion (ofBm). The ofLm class can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We characterize time reversibility for ofLm through parametric conditions related to its L\'evy measure, starting from a framework for the uniqueness of finite second moment, multivariate stochastic integral representations. In particular, we show that, under non-Gaussianity, the parametric conditions for time reversibility are generally more restrictive than those for the Gaussian case (ofBm).
... There are many statistical models for discrete time stochastic processes that are time reversible. In particular, time series generated by independent identical distributed random variables, stationary process of independent random variables and stationary Gaussian processes have been proved to be time reversible [20][21][22][23]. ...
In this paper, we suggest a new measure for testing reversibility of time series which combines two different tools: the visibility algorithm and the inversion number. First, the visibility algorithm maps the time series to the network according to a geometric criterion. After that, the degree of irreversibility of the time series can be estimated by the relative asynchronous index (RAI), based on the inverse number, between out and degree sequences of the network (out and represent the outgoing sequence of forward time series and reverse time series, respectively). This method does not need to rely on additional parameters, so it can avoid the error caused by parameter estimation. In addition, we also study the multiscale RAI and find that the optimal scale selection for detection time irreversibility is 1–4. Different types of time series are used to confirm the validity of this metric. Finally, we apply the method to financial time series and find that the financial crisis can be detected by RAI.
... Now, (18) means that the sequence of coefficients in (2) is symmetric to the index n. Hence, using the necessary and sufficient condition of reversibility of [5] (what states, that a linear time series with a.e. positive spectrum is reversible if and only if either the series c in (2) is symmetric to some index, or it is skew-symmetric and the r.v. ...
Let {X(t),t 2 Z} be a stationary time series with a.e. positive spectrum. Two con- sequences of that the bispectrum of {X(t),t 2 Z} is real-valued but nonzero: 1) if {X(t),t 2 Z} is also linear, then it is reversible; 2) {X(t),t 2 Z} can not be causal linear. A corollary of the first statement: if {X(t),t 2 Z} is linear, and the skewness of X(0) is nonzero, then third order reversibility implies reversibility. In this paper the notion of bispectrum is of a broader scope since we do not assume the absolute summability of the third order cumulants.
... After that Chen and Kuan (2001), Fong (2003) and Noel (2003) are the authors who try to show the time series which they considered are time irreversible. Cheng (1999) provided a basic theorem which gave a necessary and sufficient condition for time reversibility of stationary linear processes and did not require existence of moments of order higher than two. ...
In this paper stationary irreducible aperiodic finite state Markov chains are considered. We investigate time reversibility of these chains and a statistical tool for characterizing their time reversibility is pro-posed. It is shown that this test has asymptotically the Chi-squared distribution under null hypothesis. Our simulations also confirm the proposed test. Two empirical examples are given, one of them on gaso-line price markups, involves observed states, and the other on price level series for different countries.
... The synthesis procedure above can be further tuned to the special case of time-reversible series (e.g., [18, 19, 20]). In the Gaussian case, time-reversibility is equivalent to ...
A fast and exact procedure for the numerical synthesis of stationary multivariate Gaussian time series with a priori prescribed and well controlled auto- and cross-covariance functions is proposed. It is based on extending the circulant embedding technique to the multivariate case and can be viewed as a modification and variation around the Chan and Wood algorithm proposed earlier to solve the same problem. The procedure is shown to yield time series possessing exactly the desired covariance structure, when sufficient conditions are satisfied. Such conditions are discussed theoretically and examined on several examples of multivariate time series models. Issues related to prescribing a priori the spectral structure rather than the covariance one are also discussed. Matlab routines implementing this procedure are publicly available at http://www.hermir.org.
... Furthermore, we can show, using Corollary 1 in Cheng (1999), that the noise {ǫ t } is strong if and only if the process {X t } is Gaussian. ...
The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Theorem 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.
... In the case of a Markov process with finite state (discrete time or continuous time), Kolmogorov's criterion for time reversibility is well-known. In the References [9,10,11,12,13], they examined time reversibility in the context of a univariate stationary linear time series (Gaussian or non-Gaussian) and of multivariate linear processes. ...
For the continuous-time and the discrete-time three-state hidden Markov
model, the flux of the likelihood function up to 3-dimension of the observed
process is shown explicitly. As an application, the sufficient and necessary
condition of the reversibility of the observed process is shown.
... Under standard assumptions (see Examples 1 and 2 below), the ARMA processes, and also processes with long memory, satisfy (6)-(7). It is shown in [27] and [34] that the "two-sided" linear representation ...
Tools and approaches are provided for nonlinear time series modelling in econometrics. A wide range of topics is covered, including probabilistic properties, statistical inference and computational methods. The focus is on the applications but the ideas of the mathematical arguments are also provided. Techniques and concepts are illustrated by various examples, Monte Carlo experiments and a real application.
... This is not equivalent to (a) if {y t } is not time-reversible. Except for Gaussian linear processes, very few time series have been shown to be timereversible; see [8]. Thus, (1.1) cannot be used for (b), generally. ...
This paper first establishes a strong law of large numbers and a strong invariance principle for forward and backward sums of near-epoch dependent sequences. Using these limiting theorems, we develop a general asymptotic theory on the Wald test for change points in a general class of time series models under the no change-point hypothesis. As an application, we verify our assumptions for the long-memory fractional ARIMA model.
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.
In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).
By locating the running maxima and minima of a time series, and measuring the current deviation from them, it is possible to generate processes that are relevant for the analysis of the business cycle and for characterizing bull and bear phases in financial markets. First, the measurement of the time distance from the running peak originates a first order Markov chain, whose characteristics can be used for testing time reversibility of economic dynamics and specific types of asymmetries in financial markets. Secondly, the gap processes can be combined to provide a nonparametric measure of the growth cycle. The paper derives the time series properties of the gap process and other related processes that arise from the same measurement context, and proposes new nonparametric tests of time reversibility and new measures of the output gap.
Self‐normalization has been celebrated as an alternative approach for inference of time series because of its ability to avoid direct estimation of the nuisance asymptotic variance. However, when being applied to quantities other than the mean, the conventional self‐normalizer typically exhibits certain degrees of asymmetry, an undesirable feature especially for time‐reversible processes. This paper considers a new self‐normalizer for time series, which (i) provides a time‐symmetric generalization to the conventional self‐normalizer, (ii) is able to automatically reduce to the conventional self‐normalizer in the mean case where the latter is already time‐symmetric to yield a unified inference procedure, and (iii) possibly leads to narrower confidence intervals when compared with the conventional self‐normalizer. For the proposed time‐symmetric self‐normalizer, we establish the asymptotic theory for its induced inference procedure and examine its finite sample performance through numerical experiments.
The distributional properties of forecasts in an integer-valued time series model have not been discovered yet mainly because of the complexity arising from the binomial thinning operator. We propose two bootstrap methods to obtain nonparametric prediction intervals for an integer-valued autoregressive model : one accommodates the variation of estimating parameters and the other does not. Contrary to the results of the continuous ARMA model, we show that the latter is better than the former in forecasting the future values of the integer-valued autoregressive model.
The paper solves the open problem of identification of two-sided moving average representations with i.i.d. summands, for stationary processes in non-Gaussian domains of attraction of α-stable laws. This shows the possibility to identify nonparametrically both the sequence of two-sided moving average coefficients and the distribution of the underlying heavy-tailed i.i.d. process.
For the class of autoregressive-moving average (ARMA) processes, we examine the relationship between the dual and the inverse processes. It is demonstrated that the inverse process generated by a causal and invertible ARMA(p, q) process is a causal and invertible ARMA(q, p) model. Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. More precisely, it is derived that the linear innovation process of the inverse process is an all-pass model. Some examples and applications to time reversibility are given to illustrate the obtained results.
In this paper, we present three nonparametric trispectrum tests that can establish whether the spectral decomposition of kurtosis of high frequency financial asset price time series is consistent with the assumptions of Gaussianity, linearity and time reversiblility. The detection of nonlinear and time irreversible probabilistic structure has important implications for the choice and implementation of a range of models of the evolution of asset prices, including Black-Sholes-Merton (BSM) option pricing model, ARCH/GARCH and stochastic volatility models. We apply the tests to a selection of high frequency Australian (ASX) stocks.
For the class of autoregressive-moving average (ARMA) processes, the relationship between the dual and the inverse processes is examined. It is shown that the inverse process generated by a causal and invertible ARMA(p,q) process is a causal and invertible ARMA(q,p). Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. Some examples and applications to time reversibility are given to illustrate these theoretical results.
Contributions made by Biometrika authors over the past century are described by arranging them into fairly coherent groups, with brief introductions whenever deemed necessary. The paper is concluded with a look at challenges in the new century.
This article considers a simple procedure for assessing whether a weakly dependent univariate stochastic process is time-reversible. Our approach is based on a simple index of the deviation from zero of the median of the one-dimensional marginal law of differenced data. An attractive feature of the method is that it requires no moment assumptions. Instead of relying on Gaussian asymptotic approximations, we consider using subsampling and resampling methods to construct confidence intervals for the time-reversibility parameter, and show that such inference procedures are asymptotically valid under a mild mixing condition. The small-sample properties of the proposed procedures are examined by means of Monte Carlo experiments and an application to real-world data is also presented. Copyright 2008 The Author. Journal compilation 2008 Blackwell Publishing Ltd
A recent result by D. F. Findley, ibid. 73, 520-521 (1986; Zbl 0595.62095), on the uniqueness of moving average representations for non- Gaussian time series is shown to establish a conjecture by G. Weiss, J. Appl. Probab. 12, 831-836 (1975; Zbl 0322.60037), on the time- reversibility of general linear processes.