This article examines asymptotically point optimal tests for parameter instability in realistic circumstances when little information about the unstable parameter process and error distribution is available. We first show that, under a correctly specified error distribution, if the unstable parameter processes converge weakly to a Wiener process, then any asymptotic optimal tests for structural breaks and time-varying parameters are asymptotically equivalent. Our finding is then extended to a semi-parametric set-up in which the error distribution is treated as an unknown infinite-dimensional nuisance parameter. We find that semi-parametric tests can be adaptive without further restrictive conditions on the error distribution.

]]>Multivariate processes with long-range dependent properties are found in a large number of applications including finance, geophysics and neuroscience. For real-data applications, the correlation between time series is crucial. Usual estimations of correlation can be highly biased owing to phase shifts caused by the differences in the properties of autocorrelation in the processes. To address this issue, we introduce a semiparametric estimation of multivariate long-range dependent processes. The parameters of interest in the model are the vector of the long-range dependence parameters and the long-run covariance matrix, also called functional connectivity in neuroscience. This matrix characterizes coupling between time series. The proposed multivariate wavelet-based Whittle estimation is shown to be consistent for the estimation of both the long-range dependence and the covariance matrix and to encompass both stationary and nonstationary processes. A simulation study and a real-data example are presented to illustrate the finite-sample behaviour.

]]>This article discusses filtering, prediction and simulation in univariate and multivariate noncausal processes. A closed-form functional estimator of the predictive density for noncausal and mixed processes is introduced that provides prediction intervals up to a finite horizon H. A state-space representation of a noncausal and mixed multivariate vector autoregressive process is derived in two ways-by the partial fraction decomposition or from the real Jordan canonical form. A recursive BHHH algorithm for the maximization of the approximate log-likelihood function is proposed, which calculates the filtered values of the unobserved causal and noncausal components of the process. The new methods are illustrated by a simulation study involving a univariate noncausal process with infinite variance.

]]>Regularity conditions are given for the consistency of the Poisson quasi-maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer-valued autoregressive (INAR) and integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.

]]>Many studies record replicated time series epochs from different groups with the goal of using frequency domain properties to discriminate between the groups. In many applications, there exists variation in cyclical patterns from time series in the same group. Although a number of frequency domain methods for the discriminant analysis of time series have been explored, there is a dearth of models and methods that account for within-group spectral variability. This article proposes a model for groups of time series in which transfer functions are modelled as stochastic variables that can account for both between-group and within-group differences in spectra that are identified from individual replicates. An ensuing discriminant analysis of stochastic cepstra under this model is developed to obtain parsimonious measures of relative power that optimally separate groups in the presence of within-group spectral variability. The approach possesses favourable properties in classifying new observations and can be consistently estimated through a simple discriminant analysis of a finite number of estimated cepstral coefficients. Benefits in accounting for within-group spectral variability are empirically illustrated in a simulation study and through an analysis of gait variability.

]]>The aim of this article is to introduce new resampling scheme for nonstationary time series, called generalized resampling scheme (GRS). The proposed procedure is a generalization of well known in the literature subsampling procedure and is simply related to existing block bootstrap techniques. To document the usefulness of GRS, we consider the example of model with almost periodic phenomena in mean and variance function, where the consistency of the proposed procedure was examined. Finally, we prove the consistency of GRS for the spectral density matrix for nonstationary, multivariate almost periodically correlated time series. We consider both zero mean and non-zero mean case. The consistency holds under general assumptions concerning moment and *α*-mixing conditions for multivariate almost periodically correlated time series. Proving the consistency in this case poses a difficulty since the estimator of the spectral density matrix can be interpreted as a sum of random matrixes whose dependence grow with the sample size.

The rescaled fourth-order cumulant of the unobserved innovations of linear time series is an important parameter in statistical inference. This article deals with the problem of estimating this parameter. An existing nonparametric estimator is first discussed, and its asymptotic properties are derived. It is shown how the autocorrelation structure of the underlying process affects the behaviour of the estimator. Based on our findings and on an important invariance property of the parameter of interest with respect to linear filtering, a pre-whitening-based nonparametric estimator of the same parameter is proposed. The estimator is obtained using the filtered time series only; that is, an inversion of the pre-whitening procedure is not required. The asymptotic properties of the new estimator are investigated, and its superiority is established for large classes of stochastic processes. It is shown that for the particular estimation problem considered, pre-whitening can reduce the variance and the bias of the estimator. The finite sample performance of both estimators is investigated by means of simulations. The new estimator allows for a simple modification of the multiplicative frequency domain bootstrap, which extends its considerable range of validity. Furthermore, the problem of testing hypotheses about the rescaled fourth-order cumulant of the unobserved innovations is also considered. In this context, a simple test for Gaussianity is proposed. Some real-life data applications are presented.

]]>In blind source separation, one assumes that the observed p time series are linear combinations of p latent uncorrelated weakly stationary time series. To estimate the unmixing matrix, which transforms the observed time series back to uncorrelated latent time series, second-order blind identification (SOBI) uses joint diagonalization of the covariance matrix and autocovariance matrices with several lags. In this article, we find the limiting distribution of the well-known symmetric SOBI estimator under general conditions and compare its asymptotical efficiencies to those of the recently introduced deflation-based SOBI estimator. The theory is illustrated by some finite-sample simulation studies.

]]>An order selection test is proposed to check the equality of two independent stationary time series in their correlation structures. The asymptotic distribution of the order selection test statistic under the null hypothesis is obtained. Furthermore, it is shown that the proposed test is consistent not only under any fixed alternative hypothesis but also under a sequence of local alternative hypotheses. A simulation study is conducted to examine the finite sample performance of the test in comparison to some existing methods. The proposed test is also applied to an analysis of a biomedical data set.

]]>An *r* states random environment integer-valued autoregressive process of order 1, RrINAR(1), is introduced. Also, a random environment process is separately defined as a selection mechanism of differently parameterized geometric distributions, thus ensuring the non-stationary nature of the RrNGINAR(1) model based on the negative binomial thinning. The distributional and correlation properties of this model are discussed, and the *k*-step-ahead conditional expectation and variance are derived. Yule–Walker estimators of model parameters are presented and their strong consistency is proved. The RrNGINAR(1) model motivation is justified on simulated samples and by its application to specific real-life counting data.

Stationary processes are a natural choice as statistical models for time series data, owing to their good estimating properties. In practice, however, alternative models are often proposed that sacrifice stationarity in favour of the greater modelling flexibility required by many real-life applications. We present a family of time-homogeneous Markov processes with nonparametric stationary densities, which retain the desirable statistical properties for inference, while achieving substantial modelling flexibility, matching those achievable with certain non-stationary models. A latent extension of the model enables exact inference through a trans-dimensional Markov chain Monte Carlo method. Numerical illustrations are presented.

]]>Identification and estimation of outliers in time series is proposed by using empirical likelihood methods. Theory and applications are developed for stationary autoregressive models with outliers distinguished in the usual additive and innovation types. Some other useful outlier types are considered as well. A simulation experiment is used for studying the behaviour of the empirical likelihood-based method in finite samples and indicates that the proposed methods are preferable when dealing with the non-Gaussian data. Our simulations suggest that the usual sequential procedure for multiple outlier detection is suitable also for the methods based on empirical likelihood.

]]>The asymptotic local power properties of various fixed *T* panel unit root tests with serially correlated errors and incidental trends are studied. Asymptotic (over *N*) local power functions are analytically derived, and through them, the effects of general forms of serial correlation are examined. We find that a test based on an instrumental variables (IV) estimator dominates the tests based on the within-groups (WG) estimator. These functions also show that in the presence of incidental trends, an instrumental variables test based on the first differences of the model has non-trivial local power in an *N*^{−1/2} neighbourhood of unity. Furthermore, for a test based on the within-groups estimator, although it is found that it has trivial power in the presence of incidental trends, this ceases to be the case if there is serial correlation as well.

This work develops maximum likelihood-based unit root tests in the noncausal autoregressive (NCAR) model with a non-Gaussian error term formulated by Lanne and Saikkonen (2011, *Journal of Time Series Econometrics* 3, Issue 3, Article 2). Finite-sample properties of the tests are examined via Monte Carlo simulations. The results show that the size properties of the tests are satisfactory and that clear power gains against stationary NCAR alternatives can be achieved in comparison with available alternative tests. In an empirical application to a Finnish interest rate series, evidence in favour of an NCAR model with leptokurtic errors is found.

The asymptotic behaviour of nonparametric estimators of the stationary density and of the spectral density function of a stationary process have been studied in some detail in the last 50–60years. Nevertheless, less is known about the behaviour of these estimators when the target function happens to vanish at the point of interest. In the article at hand, we fill this gap and show that asymptotic normality still holds true but with super-efficient and different rates of convergence for the density and for the spectral density estimators that are affected also by the dependence structure of the process.

]]>We propose a new type of periodogram for identifying hidden frequencies and providing a better understanding of the frequency behaviour. The quantile periodogram by Li () provides richer information on the frequency of signal than a single estimation of the mean frequency does. However, it is difficult to find a specific quantile that identifies hidden frequencies. In this study, we consider a weighted linear combination of quantile periodograms, termed 'composite quantile periodogram'. It is completely data adaptive and does not require prior knowledge of the signal. Simulation results and real-data example demonstrate significant improvement in the quality of the periodogram.

]]>The paper addresses the unit root testing when the range of the time series is limited and considering the presence of multiple structural breaks. The structural breaks can affect the level and/or the boundaries of the time series. The paper proposes five unit root test statistics, whose limiting distribution is shown to depend on the number and position of the structural breaks. The performance of the statistics is investigated by means of Monte Carlo simulations.

]]>For autoregressive count data time series, a goodness-of-fit test based on the empirical joint probability generating function is considered. The underlying process is contained in a general class of Markovian models satisfying a drift condition. Asymptotic theory for the test statistic is provided, including a functional central limit theorem for the non-parametric estimation of the stationary distribution and a parametric bootstrap method. Connections between the new approach and existing tests for count data time series based on moment estimators appear in limiting scenarios. Finally, the test is applied to a real data set.

]]>This article studies the effect of market microstructure noise on volatility estimation in the frequency domain. We propose a bias-corrected periodogram-based estimator of integrated volatility. We show that the new estimator is consistent and the central limit theorem is established under a general assumption of the noise. We also provide a feasible procedure for computing the bias-corrected estimator in practice. As a byproduct, we extract a consistent frequency-domain estimator of the long-run variance of market microstructure noise from high-frequency data.

]]>In this article, we propose a nonparametric procedure for validating the assumption of stationarity in multivariate locally stationary time series models. We develop a bootstrap-assisted test based on a Kolmogorov–Smirnov-type statistic, which tracks the deviation of the time-varying spectral density from its best stationary approximation. In contrast to all other nonparametric approaches, which have been proposed in the literature so far, the test statistic does not depend on any regularization parameters like smoothing bandwidths or a window length, which is usually required in a segmentation of the data. We additionally show how our new procedure can be used to identify the components where non-stationarities occur and indicate possible extensions of this innovative approach. We conclude with an extensive simulation study, which shows finite-sample properties of the new method and contains a comparison with existing approaches.

]]>A two-step approach for conditional value at risk estimation is considered. First, a generalized quasi-maximum likelihood estimator is employed to estimate the volatility parameter, then the empirical quantile of the residuals serves to estimate the theoretical quantile of the innovations. When the instrumental density *h* of the generalized quasi-maximum likelihood estimator is not the Gaussian density, both the estimations of the volatility and of the quantile are generally asymptotically biased. However, the two errors counterbalance and lead to a consistent estimator of the value at risk. We obtain the asymptotic behavior of this estimator and show how to choose optimally *h*.

Long-memory effects can be found in many data sets from finance to hydrology. Therefore, models that can reflect these properties have become more popular in recent years. Mandelbrot–Van Ness fractional Lévy processes allow for such stationary long-memory effects in their increments and have been used in different settings ranging from fractionally integrated continuous-time ARMA–GARCH-type setups to general stochastic differential equations. However, their conditional distributions have not yet been considered in detail. In this article, we provide a closed formula for their conditional characteristic functions and suggest several applications to continuous-time ARMA–GARCH-type models with long memory.

]]>This article explores the problem of estimating stationary autoregressive models from observed data using the Bayesian least absolute shrinkage and selection operator (LASSO). By characterizing the model in terms of partial autocorrelations, rather than coefficients, it becomes straightforward to guarantee that the estimated models are stationary. The form of the negative log-likelihood is exploited to derive simple expressions for the conditional likelihood functions, leading to efficient algorithms for computing the posterior mode by coordinate-wise descent and exploring the posterior distribution by Gibbs sampling. Both empirical Bayes and Bayesian methods are proposed for the estimation of the LASSO hyper-parameter from the data. Simulations demonstrate that the Bayesian LASSO performs well in terms of prediction when compared with a standard autoregressive order selection method.

]]>The article reviews methods of inference for single and multiple change-points in time series, when data are of retrospective (off-line) type. The inferential methods reviewed for a single change-point in time series include likelihood, Bayes, Bayes-type and some relevant non-parametric methods. Inference for multiple change-points requires methods that can handle large data sets and can be implemented efficiently for estimating the number of change-points as well as their locations. Our review in this important area focuses on some of the recent advances in this direction. Greater emphasis is placed on multivariate data while reviewing inferential methods for a single change-point in time series. Throughout the article, more attention is paid to estimation of unknown change-point(s) in time series, and this is especially true in the case of multiple change-points. Some specific data sets for which change-point modelling has been carried out in the literature are provided as illustrative examples under both single and multiple change-point scenarios.

]]>We consider a heteroscedastic nonparametric regression model with an autoregressive error process of finite known order *p*. The heteroscedasticity is incorporated using a scaling function defined at uniformly spaced design points on an interval [0,1]. We provide an innovative nonparametric estimator of the variance function and establish its consistency and asymptotic normality. We also propose a semiparametric estimator for the vector of autoregressive error process coefficients that is consistent and asymptotically normal for a sample size *T*. Explicit asymptotic variance covariance matrix is obtained as well. Finally, the finite sample performance of the proposed method is tested in simulations.

In this article we establish a simulation procedure to generate values for a real discrete time multivariate stationary process, based on a factor of spectral density matrix. We prove the convergence of the simulator, at each time epoch, to the actual process, and provide the corresponding rate of convergence. We merely assume that the spectral density matrix is continuous and of bounded variation. By using the positive root factor, we provide an extended version for the Sun and Chaika () simulator, for real univariate stationary processes.

]]>We examine the effects of mixed sampling frequencies and temporal aggregation on the size of commonly used tests for cointegration, and we find that these effects may be severe. Matching sampling schemes of all series generally reduces size distortion, and the nominal size is obtained asymptotically only when all series are skip sampled in the same way – for example, end-of-period sampling. We propose and analyse mixed-frequency versions of the cointegration tests in order to control the size when some high-frequency data are available. Otherwise, when no high-frequency data are available, we discuss controlling size using bootstrapped critical values. We test stock prices and dividends for cointegration as an empirical demonstration.

]]>In stationary time series modelling, the autocovariance function (ACV) through its associated autocorrelation function provides an appealing description of the dependence structure but presupposes finite second moments. Here, we provide an alternative, the Gini ACV, which captures some key features of the usual ACV while requiring only first moments. For fitting autoregressive, moving-average and autoregressive–moving-average models under just first-order assumptions, we derive equations based on the Gini ACV instead of the usual ACV. As another application, we treat a nonlinear autoregressive (Pareto) model allowing heavy tails and obtain via the Gini ACV an explicit correlational analysis in terms of model parameters, whereas the usual ACV even when defined is not available in explicit form. Finally, we formulate a sample Gini ACV that is straightforward to evaluate.

]]>In this article, we propose a first-order integer-valued autoregressive [INAR(1)] process for dealing with count time series with deflation or inflation of zeros. The proposed process has zero-modified geometric marginals and contains the geometric INAR(1) process as a particular case. The proposed model is also capable of capturing underdispersion and overdispersion, which sometimes are caused by deflation or inflation of zeros. We explore several statistical and mathematical properties of the process, discuss point estimation of the parameters and find the asymptotic distribution of the proposed estimators. We also propose a test based on our model for checking if the count time series considered is deflated or inflated of zeros. Two empirical illustrations are presented in order to show the potential for practice of our zero-modified geometric INAR(1) process. This article contains a Supporting Information.

]]>Suppose that
is an i.i.d. symmetric *α*-stable noise, 1 < *α* < 2, and consider the moving average process
given by
. Conditions are obtained for the convergence rate of the moving average series, as well as that of the inverted (autoregressive) representation
. These conditions are expressed in terms of the associated function
and its reciprocal belonging to certain mixed-norm spaces of functions on the open unit disc. Properties of these spaces are explored. Criteria are also derived for the rate of mixing in a certain sense.

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.