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.     

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.     

Maximum likelihood estimation for nearly non‐stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near‐unit root autoregressive process Y_t = ρ_nY_t?1 + ?_t with α‐stable noise {?_t} is studied in this paper. Herein ρ_n = 1 ? γ/n, γ ≥ 0 is a constant, Y₀ is a fixed random variable and ε_t is an α‐stable random variable with characteristic function φ(t,θ) for some parameter θ. It is shown that when 0 < α < 1 or α > 1 and E?₁ = 0, the limit distribution of the MLE of ρ_n and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α > 1 and E?₁ ≠ 0, the limit distribution of the MLE of ρ_n and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1.     

Second-order properties of locally stationary processes

Abstract.  In this article, we investigate an optimal property of the maximum likelihood estimator of Gaussian locally stationary processes by the second-order approximation. In the case where the model is correctly specified, it is shown that appropriate modifications of the maximum likelihood estimator for Gaussian locally stationary processes is second-order asymptotically efficient. We also discuss second-order robustness properties.     

Sequential estimation for the parameters of a stationary auto regressive model

This paper considers the problem of sequential point estimation and fixed accuracy confidence set procedures of autoregressive parameters in a ρ-th order stationary autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be risk efficient as the cost of estimation error tends to infinity. Furthermore, the proposed procedure for fixed-width confidence set is shown to be both asymptotically consistent and asymptotically efficient as the width approaches zero.     

Tests against stationary and explosive alternatives in vector autoregressive models

Abstract. The article proposes new tests for the number of unit roots in vector autoregressive models based on the eigenvalues of the companion matrix. Both stationary and explosive alternatives are considered. The limiting distributions of test statistics depend only on the number of unit roots. Size and power are investigated, and it is found that the new test against some stationary alternatives compares favourably with the widely used likelihood ratio test for the cointegrating rank. The powers are prominently higher against explosive than against stationary alternatives. Some empirical examples are provided to show how to use the new tests with real data.     

Bootstrapping Time Series Regressions with Integrated Processes

This paper studies the bootstrap procedures for time series regressions with integrated processes. Both estimation and hypothesis testing are studied. It is shown that the suggested bootstrap approximations to the distribution of the least squares estimator and the regression test statistic are asymptotically valid. A Monte Carlo experiment is conducted to evaluate the finite sample performance of these bootstrap procedures. The simulation results indicate that the bootstrap method provides reasonably good approximation to the distribution of the least squares estimator, and gives proper size and satisfactory power.     

The zero-crossing rate of pth-order autoregressive processes

He and Kedem have studied the relationship between the zero- crossing rate (ZCR) of a second-o rder autoregressive process and its characteristic roots and have found that, when the roots are on the unit circle, the ZCR converges in mean square to θ/π very quickly regardless of the noise level. In this paper, the ZCR of a p th-order autoregressive process ((AR) p ) is investigated. The relationships betwe en the ZCR and the one-step asymptotic correlation function (ACF) and between the one-step ACF and the characteristic roots of the AR( p ) model are discussed, and some links between the convergence rate of the ZCR and the characte ristic roots are considered.     

DISTRIBUTIONS OF LEAST SQUARES ESTIMATORS OF AUTOREGRESSIVE PARAMETERS FOR A PROCESS WITH COMPLEX ROOTS ON THE UNIT CIRCLE

Abstract. Asymptotic distributions of the autoregressive parameters in the AR(2) model are derived, when the characteristic polynomial has a pair of complex roots on the unit circle. Percentage points are tabulated based on simulations from the asymptotic formulae. The usefulness of the asymptotic results in finite sample situations is investigated by a Monte Carlo study, and an illustrative example is given.     

Transformation to approximate independence for locally stationary Gaussian processes

We provide new approximations for the likelihood of a time series under the locally stationary Gaussian process model. The likelihood approximations are valid even in cases when the evolutionary spectrum is not smooth in the rescaled time domain. We describe a broad class of models for the evolutionary spectrum for which the approximations can be computed particularly efficiently. In developing the approximations, we extend to the locally stationary case the idea that the discrete Fourier transform is a decorrelating transformation for stationary time series. The approximations are applied to fit non‐stationary time‐series models to high‐frequency temperature data. For these data, we fit evolutionary spectra that are piecewise constant in time and use a genetic algorithm to search for the best partition of the time interval.     

A note on maximum autoregressive processes of order one

We consider estimating the coefficient of a maximum autoregressive process of order one. Under a parametric assumption for innovations, the exact distribution of this estimate is calculated using a recursion method while, under the assumption that the distribution for the innovations has a regularly varying tail at infinity, we derive its limiting distribution.     

Unit root testing with stationary covariates and a structural break in the trend function

The issue of testing for a unit root allowing for a structural break in the trend function is considered. The focus is on the construction of more powerful tests using the information in relevant multi‐variate data sets. The proposed test adopts the generalized least squares detrending approach and uses correlated stationary covariates to improve power. As it is standard in the literature, the break date is treated as unknown. Asymptotic distributions are derived, and a set of asymptotic and finite sample critical values are tabulated. Asymptotic local power functions show that power gains can be large. Finite sample results show that the test exhibits small‐size distortions and power that can be far beyond what is achievable by univariate tests.     

Recursive adjustment,unit root tests and structural breaks

In this paper, we introduce unit root tests for time series with a potential structural break computed from test regressions in which the deterministic components have been recursively adjusted. We present finite sample critical values as well as Monte Carlo results on the size and power performance of the new procedures, and compare these with other available tests in the literature, such as OLS and quasi‐differenced based tests (see, for instance, Perron, (1997) Perron and Rodriguez, (2003) and Carrion‐i‐Silvestre et al. (2009) ). The small sample behaviour of the tests is evaluated in a known and an unknown break date context allowing for negligible and non‐negligible initial conditions. In the unknown break date case, two break date estimation procedures are considered, one based on the minimum unit root t‐statistic and the other based on the minimum sum of squared residuals obtained from a regression on a set of deterministic variables. The size and power performance of the recursive adjustment based procedure in the unknown break date case is encouraging. A further result of this paper relates to the aditional finite sample evidence on the performance of quasi‐differenced unit root tests, complementing the results in Perron and Rodriguez (2003) .     

Limiting distributions of unconditional maximum likelihood unit root test statistics in seasonal time–series models

Abstract.  The likelihood function of a seasonal model, Y _t  =  ρ Y _{t − d}  +  e _t as implemented in computer algorithms under the assumption of stationary initial conditions is a function of ρ which is zero at the point ρ  = 1. It is a smooth function for ρ in the above seasonal model with a well-defined maximum regardless of the data-generating mechanism. Gonzalez-Farias (PhD Thesis, North Carolina State University, 1992) proposed tests for unit roots based on maximizing the stationary likelihood function in nonseasonal time series. We extend it to seasonal time series. The limiting distribution of seasonal unit root test statistics based on the unconditional maximum likelihood estimators are shown. Models having a single mean, seasonal means, and a single-trend variable across the seasons are considered.     

ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES

Abstract. It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving-average (ARMA) representation. Under the usual assumptions, it is deduced that a process Y described by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH( z ) has no roots on the unit circle. Given an invertible process Y which has an ARMA representation, it is finally shown that the process YT , where YT , =ε _i =0^l S _i Y _t-i , is invertible if and only if the matrix S ( z ) =ε _{i =0}^l S _i z ⁱ is of full rank for all z of modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.     

A study on misspecified nonstationary autoregressive time series with a unit root

The effects of order misspecification in nonstationary autoregressive time series estimations are investigated. The true process is assumed to be stationary if differenced. The ordinary least squares estimator is shown to be weakly convergent and its probability limit is derived. Expressions for the dominating terms of the prediction error and of the prediction mean squared error are derived. Using the expressions and Monte Carlo simulations, we compare prediction errors in the misspecified models based on the observation series and those based on the differenced series.     

Testing non‐parametric hypotheses for stationary processes by estimating minimal distances

In this article, new tests for non‐parametric hypotheses in stationary processes are proposed. Our approach is based on an estimate of the L²‐distance between the spectral density matrix and its best approximation under the null hypothesis. We explain the main idea in the problem of testing for a constant spectral density matrix and in the problem of comparing the spectral densities of several correlated stationary time series. The method is based on direct estimation of integrals of the spectral density matrix and does not require the specification of smoothing parameters. We show that the limit distribution of the proposed test statistic is normal and investigate the finite sample properties of the resulting tests by means of a small simulation study.     

A note on mean squared prediction error under the unit root model with deterministic trend

Assume that observations are generated from the first‐order autoregressive (AR) model with linear time trend and the unknown model coefficients are estimated by least squares. This article develops an asymptotic expression for the mean squared prediction error (MSPE) of the least squares predictor in the presence of a unit root. As a by‐product, we also obtain a connection between the MSPE and the growth rate of the Fisher information. The key technical tool used to derive these results is the negative moment bound for the minimum eigenvalue of the normalized Fisher information matrix.