A PARAMETER‐DRIVEN LOGIT REGRESSION MODEL FOR BINARY TIME SERIES

We consider a parameter‐driven regression model for binary time series, where serial dependence is introduced by an autocorrelated latent process incorporated into the logit link function. Unlike in the case of parameter‐driven Poisson log‐linear or negative binomial logit regression model studied in the literature for time series of counts, generalized linear model (GLM) estimation of the regression coefficient vector, which suppresses the latent process and maximizes the corresponding pseudo‐likelihood, cannot produce a consistent estimator. As a remedial measure, in this article, we propose a modified GLM estimation procedure and show that the resulting estimator is consistent and asymptotically normal. Moreover, we develop two procedures for estimating the asymptotic covariance matrix of the estimator and establish their consistency property. Simulation studies are conducted to evaluate the finite‐sample performance of the proposed procedures. An empirical example is also presented.     

Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Semiparametric Bayesian Inference of Long-Memory Stochastic Volatility Models

Abstract.  In this paper, a semiparametric, Bayesian estimator of the long-memory stochastic volatility model's fractional order of integration is presented. This new estimator relies on a highly efficient, Markov chain Monte Carlo (MCMC) sampler of the model's posterior distribution. The MCMC algorithm is set forth in the time-scale domain of the stochastic volatility model's wavelet representation. The key to and centerpiece of this new algorithm is the quick and efficient multi-state sampler of the latent volatility's wavelet coefficients. A multi-state sampler of the latent wavelet coefficients is only possible because of the near-independent multivariate distribution of the long-memory process's wavelet coefficients. Using simulated and empirical stock return data, we find that our algorithm produces uncorrelated draws of the posterior distribution and point estimates that rival existing long-memory stochastic volatility estimators.     

Autoregressive coefficient estimation in nonparametric analysis

The article considers the Yule‐Walker estimator of the autoregressive coefficient based on the observed time series that contains an unknown trend function and an autoregressive error term. The trend function is estimated by means of B‐splines and then subtracted from the observations. The Yule‐Walker estimator is obtained from the residual sequence. Asymptotic properties of this estimator are derived. The performance of the estimator is illustrated by simulation studies and real data analysis.     

Robustness of Zero Crossing Estimator

Zero crossing (ZC) statistic is the number of zero crossings observed in a time series. The expected value of the ZC specifies the first‐order autocorrelation of the processes. Hence, we can estimate the autocorrelation by using the ZC estimator. The asymptotic consistency and normality of the ZC estimator for scalar Gaussian processes are already discussed in 1980. In this article, first, we derive the joint asymptotic distribution of the ZC estimator for ellipsoidal processes. Next, we show the variance of the ZC estimator does not attain the Cramer–Rao lower bound (CRLB). However, it is shown that the ZC estimator has robustness when the process is contaminated by an outlier. In contrast with this, we observe that the quasi‐maximum likelihood estimator (QMLE) attains the CRLB. However, we can see that QMLE is sensitive for the outlier.     

Common Breaks in Means for Cross‐Correlated Fixed‐T Panel Data

This note considers a panel data model in which the variable of interest has undergone a common structural break in the mean. The object of interest is the unknown breakpoint. The challenge is to device an estimator that is consistent when the data are cross‐correlated and the number of time periods T is fixed and cannot be increased without bound. The proposed solution involves taking an already existing estimator initially proposed for cross‐section uncorrelated panels and applying it to defactored data. Consistency is established as the number of cross‐section units N grows large, and is verified in small samples using Monte Carlo simulation.     

Independent block identification in multivariate time series

In this‐30 work we propose a model selection criterion to estimate the points of independence of a random vector, producing a decomposition of the vector distribution function into independent blocks. The method, based on a general estimator of the distribution function, can be applied for discrete or continuous random vectors, and for i.i.d. data or dependent time series. We prove the consistency of the approach under general conditions on the estimator of the distribution function and we show that the consistency holds for i.i.d. data and discrete time series with mixing conditions. We also propose an efficient algorithm to approximate the estimator and show the performance of the method on simulated data. We apply the method in a real dataset to estimate the distribution of the flow over several locations on a river, observed at different time points.     

Generalized Resampling Scheme With Application to Spectral Density Matrix in Almost Periodically Correlated Class of Time Series

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.     

Inference for the Fourth‐Order Innovation Cumulant in Linear Time Series

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.     

Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

Density estimation with locally identically distributed data and with locally stationary data

Abstract. We consider multivariate density estimation when the assumptions of identically distributed data or stationary data are relaxed to the assumptions of locally identically distributed data or locally stationary data. We assume that the distribution of the data is changing continuously as function of time. To estimate densities non‐parametrically with these local regularity conditions, we need time localization in addition to the usual space localization. We define a time‐localized kernel estimator that estimates the density non‐parametrically at any given point of time. The consistency of the time‐localized kernel estimator is proved and the rates of convergence of the estimator are derived under conditions on the β‐and α‐mixing coefficients. Both the time‐series setting and spatial setting are covered.     

Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.     

An Extended Portmanteau Test for VARMA Models With Mixing Nonlinear Constraints

Abstract. The portmanteau test is a widely used diagnostic tool for univariate and multivariate time‐series models. Its asymptotic distribution is known for the unconstrained vector autoregressive moving‐average (VARMA) case and for VAR models with constraints on the autoregressive coefficients. In this article, we give conditions under which the test can be applied to constrained VARMA models. Unfortunately, it cannot generally be applied to models with constraints that simultaneously affect the ARMA polynomial coefficients and the covariance matrix of the innovations (mixing constraints). This happens in latent‐variable models such as dynamic factor models (DFM). In addition, when there are constraints on the covariance matrix it seems convenient to check the goodness of fit using the zero‐lag residual covariances. We propose an extended portmanteau test that not only checks the autocorrelations of the residuals but also whether their covariance matrix is consistent with the constraints. We prove that the statistic is asymptotically distributed as a chi‐square for ARMA models under the assumption that the innovations have Gaussian‐like fourth‐order moments. We also show that the test is appropriate for the DFM, Peña–Box model and factor‐structural vector autoregression (FSVAR).     

Bootstrapping confidence intervals for the change‐point of time series

Abstract. We study an at‐most‐one‐change time‐series model with an abrupt change in the mean and dependent errors that fulfil certain mixing conditions. We obtain confidence intervals for the unknown change‐point via bootstrapping methods. Precisely, we use a block bootstrap of the estimated centred error sequence. Then, we reconstruct a sequence with a change in the mean using the same estimators as before. The difference between the change‐point estimator of the resampled sequence and the one of the original sequence can be used as an approximation of the difference between the real change‐point and its estimator. This enables us to construct confidence intervals using the empirical distribution of the resampled time series. A simulation study shows that the resampled confidence intervals are usually closer to their target levels and at the same time smaller than the asymptotic intervals.     

LAG WINDOW ESTIMATION OF THE DEGREE OF DIFFERENCING IN FRACTIONALLY INTEGRATED TIME SERIES MODELS

Abstract. In this paper we consider the estimation of the degree of differencing d in the fractionally integrated autoregressive moving-average time series model ARFIMA ( p, d, q ). Using lag window spectral density estimators we develop a regression type estimator of d which is easy to calculate and does not require prior knowledge of p and q. Some large sample properties of the estimator are studied and the performance of the estimator for small samples is investigated using the simulation method for a range of commonly used lag windows. Some practical recommendations on the choice of lag windows and the choice of the window parameters are provided.     

A New Recursive Estimation Method for Single Input Single Output Models

This article is devoted to a new recursive estimation method for dynamic time series models, more precisely for single input single output models. In that method, the recurrence for updating the Hessian is avoided, but the recurrence for updating the estimator makes use of the Fisher information matrix. The asymptotic properties, consistency and asymptotic normality, of the new estimator are obtained under weak assumptions. Monte Carlo experiments and examples indicate that the estimates converge well, comparatively with alternative methods.     

Poisson QMLE of Count Time Series Models

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.     

Polynomial Trend Regression With Long‐memory Errors

Abstract. For a time series generated by polynomial trend with stationary long‐memory errors, the ordinary least squares estimator (OLSE) of the trend coefficients is asymptotically normal, provided the error process is linear. The asymptotic distribution may no longer be normal, if the error is in the form of a long‐memory linear process passing through certain nonlinear transformations. However, one hardly has sufficient information about the transformation to determine which type of limiting distribution the OLSE converges to and to apply the correct distribution so as to construct valid confidence intervals for the coefficients based on the OLSE. The present paper proposes a modified least squares estimator to bypass this drawback. It is shown that the asymptotic normality can be assured for the modified estimator with mild trade‐off of efficiency even when the error is nonlinear and the original limit for the OLSE is non‐normal. The estimator performs fairly well when applied to various simulated series and two temperature data sets concerning global warming.     

A test for second‐order stationarity of a time series based on the discrete Fourier transform

We consider a zero mean discrete time series, and define its discrete Fourier transform (DFT) at the canonical frequencies. It can be shown that the DFT is asymptotically uncorrelated at the canonical frequencies if and only if the time series is second‐order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic has approximately a chi‐square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a generalized non‐central chi‐square, where the non‐centrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power.     

Exactly/Nearly Unbiased Estimation of Autocovariances of a Univariate Time Series With Unknown Mean

This article proposes an exactly/nearly unbiased estimator of the autocovariance function of a univariate time series with unknown mean. The estimator is a linear function of the usual sample autocovariances computed using the observed demeaned data. The idea is to stack the usual sample autocovariances into a vector and show that the expectation of this vector is a linear combination of population autocovariances. A matrix that we label, A , collects the weights in these linear combinations. When the population autocovariances of high lags are zero (small), exactly (nearly) unbiased estimators of the remaining autocovariances can be obtained using the inverse of upper blocks of the A matrix. The A ‐matrix estimators are shown to be asymptotically equivalent to the usual sample autocovariance estimators. The A ‐matrix estimators can be used to construct estimators of the autocorrelation function that have less bias than the usual estimators. Simulations show that the A ‐matrix estimators can substantially reduce bias while not necessarily increasing mean square error. More powerful tests for the null hypothesis of white noise are obtained using the A ‐matrix estimators.