Long-Memory Errors in Time Series Regressions with a Unit Root

This paper is concerned with estimation and inference in univariate time series regression with a unit root when the error sequence exhibits long-range temporal dependence. We consider generating mechanisms for the unit root process which include models with or without a drift term and we study the limit behavior of least squares statistics in regression models without drift and trend, with drift but no time trend, and with drift and time trend. We derive the limit distribution and rate of convergence of the ordinary least squares (OLS) estimator of the unit root, the intercept and the time trend in the three regression models and for the two different data-generating processes. The limiting distributions for the OLS estimator differ from those obtained under the hypothesis of weakly dependent errors not only in terms of the limiting process involved but also in terms of functional form. Further, we characterize the asymptotic behavior of both the t statistics for testing the unit root hypothesis and the t statistic for the intercept and time trend coefficients. We find that t ratios either diverge to infinity or collapse to zero. The limiting behavior of Phillips's Z _α and Z _t semiparametric corrections is also analyzed and found to be similar to that of standard Dickey– Fuller tests. Our results indicate that misspecification of the temporal dependence features of the error sequence produces major effects on the asymptotic distribution of estimators and t ratios and suggest that alternative approaches might be more suited to testing for a unit root in time series regression.     

Simple Regressions with Linear Time Trends

Simple regressions of two trend stationary time series are considered. As the linear trend dominates the stochastic components the rates of convergence and the limiting distributions of ordinary least squares statistics are exactly the same as in the case of cointegrated regressions with drifts. In particular, asymptotic standard normal t statistics are readily available. Hence, asymptotic inference requires no distinction between simple regressions of trend stationary series and of cointegrated variables with drifts.     

Testing Stochastic Cycles in Macroeconomic Time Series

A particular version of the tests of Robinson (1994) for testing stochastic cycles in macroeconomic time series is proposed in this article. The tests have a standard limit distribution and are easy to implement in raw time series. A Monte Carlo experiment is conducted, studying the size and the power of the tests against different alternatives, and the results are compared with those based on other tests. An empirical application using historical US annual data is also carried out at the end of the article.     

Using Difference‐Based Methods for Inference in Regression with Fractionally Integrated Processes

Abstract. This paper suggests a difference‐based method for inference in the regression model involving fractionally integrated processes. Under suitable regularity conditions, our method can effectively deal with the inference problems associated with the regression model consisting of nonstationary, stationary and intermediate memory regressors, simultaneously. Although the difference‐based method provides a very flexible modelling framework for empirical studies, the implementation of this method is extremely easy, because it completely avoids the difficult problems of choosing a kernel function, a bandwidth parameter, or an autoregressive lag length for the long‐run variance estimation. The asymptotic local power of our method is investigated with a sequence of local data‐generating processes (DGP) in what Davidson and MacKinnon [Canadian Journal of Economics. (1985) Vol. 18, pp. 38–57] call ‘regression direction’. The simulation results indicate that the size control of our method is excellent even when the sample size is only 100, and the pattern of power performance is highly consistent with the theoretical finding from the asymptotic local power analysis conducted in this paper.     

ON BOOTSTRAP PREDICTIVE INFERENCE FOR AUTOREGRESSIVE PROCESSES

Abstract. In this paper we consider bootstrap-based predictive inference for autoregressive processes of order p. We consider both unconditional inference and inference conditional on the last p observed values. We make two contributions. Our first contribution is to point out the best way to apply the bootstrap to unconditional predictive inference when the process is Gaussian. Now, it may be argued that predictive inference for autoregressive processes of order p should be carried out conditional on the last p observed values. When the process is Gaussian, a bootstrap predictive inference conditional on the last p observed values is conveniently computed by 'running' the same autoregressive process backwards in time. This procedure is inappropriate for non-Gaussian autoregressive processes. Our second (and more important) contribution is to present a method (which is not computationally burdensome) for the computation of a bootstrap predictive inference for a non-Gaussian autoregressive process of order p conditional on the last p observed values.     

Bootstrapping unit root tests for integrated processes

Abstract. In this paper, we consider two bootstrap algorithms for testing unit roots under the condition that the observed process is unit root integrated. The first method consists of generating the resampled data after fitting an autoregressive model to the first differences of the observations. The second method consists of applying the stationary bootstrap to the first differences. Both procedures are shown to give methods that approach the correct asymptotic distribution under the null hypothesis of a unit root. We also present a Monte-Carlo study comparing the two methods for some ARIMA models.     

Locally Optimal Tests Against Unit Roots in Seasonal Time Series Processes

This paper builds on the existing literature on tests of the null hypothesis of deterministic seasonality in a univariate time-series process. Under the assumption of independent Gaussian errors, we derive the class of locally weighted mean most powerful invariant tests against unit roots at the zero and/or seasonal frequencies in a seasonally observed process. Representations for the limiting distributions of the proposed test statistics under sequences of local alternatives are derived, and the relationship with tests for corresponding moving average unit roots is explored. We also propose nonparametric modifications of these test statistics designed to have limit distributions which are free of nuisance parameters under weaker conditions on the errors. Our tests are shown to contain existing stationarity tests as special cases and to extend these tests in a number of useful directions.     

Regression Models with Time Series Errors

In models of the form Y_t = r ( X_t ) + Z_t , where r is an unknown function and { X_t } is a covariate process independent of the stationary error { Z_t }, we give conditions under which estimators based on residuals Z ₁, ..., Z _n obtained from linear smoothers are asymptotically equivalent to those based on the actual errors Z ₁, ..., Z_n .     

Asymptotics for stationary very nearly unit root processes

Abstract. This article considers a mean zero stationary first‐order autoregressive (AR) model. It is shown that the least squares estimator and t statistic have Cauchy and standard normal asymptotic distributions, respectively, when the AR parameter ρ_n is very near to one in the sense that 1 ? ρ_n = o(n^?1).     

Change-Point Estimation of Fractionally Integrated Processes

In this paper we analyze the least-squares estimator of the change point for fractionally integrated processes with fractionally differencing parameter −0.5 < d < 0.5. When there is a one-time change, we show that the least-squares estimator is consistent and that the rate of convergence depends on d . When there is no change, we find that the least-squares estimator converges in probability to the set {0, 1} for −0.5 < d ≤ 0 but is likely to suggest a spurious change for 0 < d < 0.5. Simulations are also used to illustrate the asymptotic analysis.     

Comparison of unit root tests for time series with level shifts

Unit root tests are considered for time series which have a level shift at a known point in time. The shift can have a very general nonlinear form, and additional deterministic mean and trend terms are allowed for. Prior to the tests, the deterministic parts and other nuisance parameters of the data generation process are estimated in a first step. Then, the series are adjusted for these terms and unit root tests of the Dickey–Fuller type are applied to the adjusted series. The properties of previously suggested tests of this sort are analysed and modifications are proposed which take into account estimation errors in the nuisance parameters. An important result is that estimation under the null hypothesis is preferable to estimation under local alternatives. This contrasts with results obtained by other authors for time series without level shifts.     

Outlier Detection And Estimation In NonLinear Time Series

Abstract. The problem of identifying the time location and estimating the amplitude of outliers in nonlinear time series is addressed. A model‐based method is proposed for detecting the presence of additive or innovational outliers when the series is generated by a general nonlinear model. We use this method for identifying and estimating outliers in bilinear, self‐exciting threshold autoregressive and exponential autoregressive models. A simulation study is performed to test the proposed procedures and comparing them with the methods based on linear models and linear interpolators. Finally, our results are applied for detecting outliers in the Canadian lynx trappings and in the sunspot numbers data.     

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Robust Estimation For Periodic Autoregressive Time Series

Abstract. A robust estimation procedure for periodic autoregressive (PAR) time series is introduced. The asymptotic properties and the asymptotic relative efficiency are discussed by the estimating equation approach. The performance of the robust estimators for PAR time‐series models with order one is illustrated by a simulation study. The technique is applied to a real data analysis.     

Modelling Count Data Time Series with Markov Processes Based on Binomial Thinning

Abstract. We obtain new models and results for count data time series based on binomial thinning. Count data time series may have non‐stationarity from trends or covariates, so we propose an extension of stationary time series based on binomial thinning such that the univariate marginal distributions are always in the same parametric family, such as negative binomial. We propose a recursive algorithm to calculate the probability mass functions for the innovation random variable associated with binomial thinning. This simplifies numerical calculations and estimation for the classes of time series models that we consider. An application with real data is used to illustrate the models.     

Bayesian Inference for Time Series with Stable Innovations

This paper describes Bayesian inference for a linear time series model with stable innovations. An advantage of the Bayesian approach is that it enables the simultaneous estimation of the parameters characterizing the stable law and the parameters of the linear autoregressive moving-average model. Our approach uses a Metropolis–Hastings algorithm to generate samples from the joint posterior distribution of all the parameters and subsequent inference is based on these samples. We illustrate our approach using data simulated from three linear processes with stable innovations and a real data set     

Estimation of Nonparametric Autoregressive Time Series Models Under Dynamical Constraints

Abstract. A method is introduced to estimate nonparametric autoregressive models under the additional constraint that its regression function has a stable cycle. It is based on a penalty approach that chooses a series expansion approximation taking into account both goodness‐of‐fit and fulfillment of the constraint. Consistency of the proposed estimator is obtained under general hypothesis. Feasibility and effective performance of the introduced method are studied through simulated examples and electro‐encephalographic data collected from a subject suffering from epilepsy.     

A Class of Antipersistent Processes

Abstract. We introduce a class of stationary processes characterized by the behaviour of their infinite moving average parameters. We establish the asymptotic behaviour of the covariance function and the behaviour around zero of the spectral density of these processes, showing their antipersistent character. Then, we discuss the existence of an infinite autoregressive representation for this family of processes, and we present some consequences for fractional autoregressive moving average models.