Signal Extraction for Non‐Stationary Multivariate Time Series with Illustrations for Trend Inflation

This article advances the theory and methodology of signal extraction by developing the optimal treatment of difference stationary multivariate time‐series models. Using a flexible time‐series structure that includes co‐integrated processes, we derive and prove formulas for minimum mean square error estimation of signal vectors in multiple series, from both a finite sample and a bi‐infinite sample. As an illustration, we present econometric measures of the trend in total inflation that make optimal use of the signal content in core inflation.     

Negative Binomial Quasi‐Likelihood Inference for General Integer‐Valued Time Series Models

Two negative binomial quasi‐maximum likelihood estimates (NB‐QMLEs) for a general class of count time series models are proposed. The first one is the profile NB‐QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two‐stage NB‐QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two‐stage NB‐QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB‐QMLEs are compared with the Poisson QMLE asymptotically and in finite samples for various well‐known particular classes of count time series models such as the Poisson and negative binomial integer‐valued GARCH model and the INAR(1) model. Application to a real dataset is given.     

Time‐Transformed Unit Root Tests for Models with Non‐Stationary Volatility

Abstract. Conventional unit root tests are known to be unreliable in the presence of permanent volatility shifts. In this paper, we propose a new approach to unit root testing which is valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. The new tests are based on a time transformation of the series of interest which automatically corrects their form for the presence of non‐stationary volatility without the need to specify any parametric model for the volatility process. Despite their generality, the new tests perform well even in small samples. We also propose a class of tests for the null hypothesis of stationary volatility in (near‐) integrated time‐series processes.     

Bartlett Correction of Empirical Likelihood for Non‐Gaussian Short‐Memory Time Series

Bartlett correction, which improves the coverage accuracies of confidence regions, is one of the desirable features of empirical likelihood. For empirical likelihood with dependent data, previous studies on the Bartlett correction are mainly concerned with Gaussian processes. By establishing the validity of Edgeworth expansion for the signed root empirical log‐likelihood ratio statistics, we show that the Bartlett correction is applicable to empirical likelihood for short‐memory time series with possibly non‐Gaussian innovations. The Bartlett correction is established under the assumptions that the variance of the innovation is known and the mean of the underlying process is zero for a single parameter model. In particular, the order of the coverage errors of Bartlett‐corrected confidence regions can be reduced from O(n^?1) to O(n^?2).     

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

In several arenas of application, it is becoming increasingly common to consider time series of curves or functions. Many inferential procedures employed in the analysis of such data involve the long‐run covariance function or operator, which is analogous to the long‐run covariance matrix familiar to finite‐dimensional time‐series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite‐dimensional setting, in particular Newey and West ( 1994 ), we propose a bandwidth selection method that aims to minimize the estimator's asymptotic mean‐squared normed error (AMSNE) in L²[0,1]². As the AMSNE depends on unknown population quantities including the long‐run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.     

Zero‐Modified Geometric INAR(1) Process for Modelling Count Time Series with Deflation or Inflation of Zeros

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.