The size of the nonstationary component and its effect on tests for unit roots

We consider a nonstationary time series that is composed of a stationary and nonstationary component. Monte Carlo experiments show that common unit root tests have probabilities of committing a type I error that significantly exceed the level of significance. We find that the probabilities vary according to the relative size of the nonstationary component.     

A time series approach to life table construction

This paper deals with the construction of the life table. A discussion of basic facts about the life table is followed by the proposal of a nonstationary, autoregressive model for the life table. The moment structure of the nonstationary, autoregressive model is developed. Some estimation procedures are proposed followed by several examples.     

A note on general quadratic forms of nonstationary stochastic processes

In this paper, general quadratic forms of nonstationary, α-mixing time series are considered. Under mixing and moment assumptions, asymptotically normality of these forms are derived. These results do not assume that the variance of the generalized quadratic form has a limit, thus allowing for general types of nonstationarity. However, without well-defined limits, it is not possible to understand the differences in sampling properties of quadratic forms of nonstationary and stationary processes. To understand these differences, the nonstationary process is placed within the locally stationary framework. Under the assumption that the nonstationary process is locally stationary the asymptotic expectation and variance of the weighted sample covariance of the discrete Fourier transforms (an important class of quadratic forms) is derived and shown to be very different to its stationary counterpart.     

Sample path properties of an explosive double autoregressive model

This article studies sample path properties of an explosive double autoregressive (DAR) model. After suitable renormalization, it is shown that the sample path converges weakly to a geometric Brownian motion. This further strengthens our understanding of sample paths of nonstationary DAR processes. The obtained results can be extended to nonstationary random coefficient autoregressive (RCA) models. Simulation studies are carried out to support our results.     

On the nearly nonstationary seasonal time series

A time series is said to be nearly nonstationary if some of its characteristic roots are close to the unit circle. For a seasonal time series, such a notion of near-nonstationarity is studied in a double-array setting. This approach not only furnishes a natural transition between stationarity and nonstationarity, but also unifies the corresponding asymptotic theories in a seasonal-time-series context. The general theory is expressed in terms of functionals of independent diffusion processes. The asymptotic results have applications to estimation and testing in a nearly nonstationary situation and serve as a useful alternative to the common practice of seasonal adjustment.     

Nonparametric Estimation of Average Growth Curve with General Nonstationary Error Process

The nonparametric estimation of the growth curve has been extensively studied in both stationary and some nonstationary particular situations. In this work, we consider the statistical problem of estimating the average growth curve for a fixed design model with nonstationary error process. The nonstationarity considered here is of a general form, and this article may be considered as an extension of previous results. The optimal bandwidth is shown to depend on the singularity of the autocovariance function of the error process along the diagonal. A Monte Carlo study is conducted in order to assess the influence of the number of subjects and the number of observations per subject on the estimation.     

Nonstationary panel data analysis: an overview of some recent developments

This paper overviews some recent developments in panel data asymptotics, concentrating on the nonstationary panel case and gives a new result for models with individual effects. Underlying recent theory are asymptotics for multi-indexed processes in which both indexes may pass to infinity. We review some of the new limit theory that has been developed, show how it can be applied and give a new interpretation of individual effects in nonstationary panel data. Fundamental to the interpretation of much of the asymptotics is the concept of a panel regression coefficient which measures the long run average relation across a section of the panel. This concept is analogous to the statistical interpretation of the coefficient in a classical regression relation. A variety of nonstationary panel data models are discussed and the paper reviews the asymptotic properties of estimators in these various models. Some recent developments in panel unit root tests and stationary dynamic panel regression models are also reviewed.     

Nonstationary panel data analysis: an overview of some recent developments

This paper overviews some recent developments in panel data asymptotics, concentrating on the nonstationary panel case and gives a new result for models with individual effects. Underlying recent theory are asymptotics for multi-indexed processes in which both indexes may pass to infinity. We review some of the new limit theory that has been developed, show how it can be applied and give a new interpretation of individual effects in nonstationary panel data. Fundamental to the interpretation of much of the asymptotics is the concept of a panel regression coefficient which measures the long run average relation across a section of the panel. This concept is analogous to the statistical interpretation of the coefficient in a classical regression relation. A variety of nonstationary panel data models are discussed and the paper reviews the asymptotic properties of estimators in these various models. Some recent developments in panel unit root tests and stationary dynamic panel regression models are also reviewed.     

Multivariate autoregressive time semes modeling: one scalar autoregressive model at-A-time

Multivariate (or interchangeably multichannel) autoregressive (MCAR) modeling of stationary and nonstationary time series data is achieved doing things one channel at-a-time using only scalar computations on instantaneous data. The one channel at-a-time modeling is achieved as an instantaneous response multichannel autoregressive model with orthogonal innovations variance. Conventional MCAR models are expressible as linear algebraic transformations of the instantaneous response orthogonal innovations models. By modeling multichannel time series one channel at-a-time, the problems of modeling multichannel time series are reduced to problems in the modeling of scalar autoregressive time series. The three longstanding time series modeling problems of achieving a relatively parsimonious MCAR representation, of multichannel stationary time series spectral estimation and of the modeling of nonstationary covariance time series are addressed using this paradigm.     

The role of information in nonstationary regression

The role of standard likelihood-based measures of information and efficiency is unclear when regressions involve nonstationary data. Typically the standardized score is not asymptotically Gaussian and the standardized Hessian has a stochastic, rather than deterministic limit. Here we consider a time series regression involving a deterministic covariate which can be evaporating, slowly evolving or nonstationary. It is shown that conditional information, or equivalently, profile Kullback–Leibler and Fisher information remain informative about both the accuracy, i.e. asymptotic variance, of profile maximum likelihood estimators, and the power of point optimal invariant tests for a unit root. Specifically, these information measures indicate fractional, rather than linear trends that may minimize inferential accuracy. Such is confirmed in a numerical experiment.     

Quasi-likelihood ratio tests for cointegration,cobreaking, and cotrending

We consider a set of variables with two types of nonstationary features, stochastic trends and broken linear trends. We develop tests that can determine whether there is a linear combination of these variables under which the nonstationary features can be canceled out. The first test can determine whether stochastic trends can be eliminated and thus whether cointegration holds, regardless of whether structural breaks in linear trends are eliminated. The second test can determine whether both stochastic trends and breaks in linear trends are simultaneously removed and thus whether cointegration and cobreaking simultaneously hold. The third test can determine whether not only breaks in linear trends but also linear trends themselves are eliminated along with stochastic trends and thus whether both cointegration and cotrending hold.     

Harrison–Stevens Forecasting and the Multiprocess Dynamic Linear Model

This is an expository article. The Harrison–Stevens forecasting algorithm using the multiprocess dynamic linear model is a robust method for forecasting in a nonstationary time series. The purpose of this article is to help statisticians become familiar with the method.     

Double AR model without intercept: An alternative to modeling nonstationarity and heteroscedasticity

This paper presents a double AR model without intercept (DARWIN model) and provides us a new way to study the nonstationary heteroscedastic time series. It is shown that the DARWIN model is always nonstationary and heteroscedastic, and its sample properties depend on the Lyapunov exponent. An easy-to-implement estimator is proposed for the Lyapunov exponent, and it is unbiased, strongly consistent, and asymptotically normal. Based on this estimator, a powerful test is constructed for testing the ordinary oscillation of the model. Moreover, this paper proposes the quasi-maximum likelihood estimator (QMLE) for the DARWIN model, which has an explicit form. The strong consistency and asymptotic normality of the QMLE are established regardless of the sign of the Lyapunov exponent. Simulation studies are conducted to assess the performance of the estimation and testing, and an empirical example is given for illustrating the usefulness of the DARWIN model.     

A bootstrap procedure in linear regression with nonstationary errors

In the context of linear regression with dependent and nonstationary errors, the classical moving-block bootstrap (MBB) fails to capture the nonstationarity of the errors. A new bootstrap procedure called the blocking external bootstrap (BEB) is proposed to overcome the problem. The consistency of the BEB in estimating the variance of the least-squares estimator is studied in the case of α-mixing and nonstationary sequence of errors. It is shown that the BEB only achieves partial correction if the block size is fixed. Complete consistency is achieved by the BEB when the block size is allowed to go to infinity. We also study the first-order consistency of the least squares estimator based on the BEB. A simulation study is carried out to assess the performance of the BEB versus the MBB in estimating the variance of the least-squares estimator. Finally, some open problems are discussed.     

Nonparametric Inference for Time-Varying Coefficient Quantile Regression

The article considers nonparametric inference for quantile regression models with time-varying coefficients. The errors and covariates of the regression are assumed to belong to a general class of locally stationary processes and are allowed to be cross-dependent. Simultaneous confidence tubes (SCTs) and integrated squared difference tests (ISDTs) are proposed for simultaneous nonparametric inference of the latter models with asymptotically correct coverage probabilities and Type I error rates. Our methodologies are shown to possess certain asymptotically optimal properties. Furthermore, we propose an information criterion that performs consistent model selection for nonparametric quantile regression models of nonstationary time series. For implementation, a wild bootstrap procedure is proposed, which is shown to be robust to the dependent and nonstationary data structure. Our method is applied to studying the asymmetric and time-varying dynamic structures of the U.S. unemployment rate since the 1940s. Supplementary materials for this article are available online.     

A Note on the Estimation of Missing Values in Time Series

This paper derives an expression for the likelihood function of the parameters in an autoregressive-moving average model when some values are missing from the observed time series. The estimation of the missing values and their mean squared errors is discussed. Stationary as well as nonstationary models are considered.     

Fractionally integrated ARMA for crude palm oil prices prediction: case of potentially overdifference

Dealing with stationarity remains an unsolved problem. Some of the time series data, especially crude palm oil (CPO) prices persist towards nonstationarity in the long-run data. This dilemma forces the researchers to conduct first-order difference. The basic idea is that to obtain the stationary data that is considered as a good strategy to overcome the nonstationary counterparts. An opportune remark as it is, this proxy may lead to overdifference. The CPO prices trend elements have not been attenuated but nearly annihilated. Therefore, this paper presents the usefulness of autoregressive fractionally integrated moving average (ARFIMA) model as the solution towards the nonstationary persistency of CPO prices in the long-run data. In this study, we employed daily historical Free-on-Board CPO prices in Malaysia. A comparison was made between the ARFIMA over the existing autoregressive-integrated moving average (ARIMA) model. Here, we employed three statistical evaluation criteria in order to measure the performance of the applied models. The general conclusion that can be derived from this paper is that the usefulness of the ARFIMA model outperformed the existing ARIMA model.     

Adaptive estimating function inference for nonstationary determinantal point processes

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this article, we establish asymptotic normality of estimating function estimators in a very general setting of nonstationary point processes. We then adapt this result to the case of nonstationary determinantal point processes, which are an important class of models for repulsive point patterns. In practice, often first‐ and second‐order estimating functions are used. For the latter, it is a common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance, which is usually specified in an ad hoc manner. We suggest instead a data‐driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non‐stationary determinantal point processes and by an application to a real dataset.     

A vector of quarters representation for bivariate time series

In this paper it is shown that several models for a bivariate nonstationary quarterly time series are nested in a vector autoregression with cointegration restrictions for the eight annual series of quarterly observations. Or, the Granger Representation Theorem is extended to incorporate, e.g., seasonal and periodic cointegration.     

A vector of quarters representation for bivariate time series

In this paper it is shown that several models for a bivariate nonstationary quarterly time series are nested in a vector autoregression with cointegration restrictions for the eight annual series of quarterly observations. Or, the Granger Representation Theorem is extended to incorporate, e.g., seasonal and periodic cointegration.