RESIDUAL AUTOCOVARIANCES AND UNIT ROOT TESTS BASED ON INSTRUMENTAL VARIABLE ESTIMATORS FROM TIME SERIES REGRESSION MODELS

Abstract. Hall (Testing for a unit root in the presence of moving average errors. Biometrika 76 (1989), 49–56; Joint hypothesis tests for a random walk based on instrumental variable estimators. J. Time Ser. Anal. 13 (1992), 29–45), Pantula and Hall (Testing for unit roots in autoregressive moving average models:an instrumental variable approach. J. Econometrics 48 (1991), 325–53) and Lee and Schmidt (Unit root tests based on instrumental variable estimation. Int. Econ. Rev. 39 (1994), 449–62) proposed instrumental variable (IV) based tests for a unit root in an ARMA(p+ 1, q) time series. To perform the tests it is essentially necessary to know (p, q) but in many cases this information is unknown. In practice a natural solution to this problem is to estimate (p, q) from the data using a strategy based on the residual autocovariances from the IV regression. In this paper we examine the properties of these residual autocovariances under various assumptions about the true nature of the time series. This analysis allows us to propose a model selection procedure which has desirable asymptotic and finite sample properties whether the time series is stationary or possesses a unit root. A sideproduct of our analysis is that we extend Box and Pierce's (Distribution of residual autocorrelations in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 65 (1970), 1509–26) analysis of the least squares residual autocorrelations to the residual autocovariances from IV regressions.     

AN EXACT TEST FOR A STOCHASTIC COEFFICIENT IN A TIME SERIES REGRESSION MODEL

An exact small-sample test is developed for testing the hypothesis that a regression coefficient is constant against the alternative that it is generated by a random walk process. The test is mean- and scale-invariant and approximates the most powerful invariant test against any specific alternative. It thus outperforms tests previously given in the literature. Computationally efficient algorithms are given to compute the test statistic and its distribution using a modified version of the Kalman filter.     

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.     

RESULTS ON ESTIMATION AND TESTING FOR A UNIT ROOT IN THE NONSTATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODEL

Abstract. We review the limiting distribution theory for Gaussian estimation of the univariate autoregressive moving-average (ARMA) model in the presence of a unit root in the autoregressive (AR) operator, and present the asymptotic distribution of the associated likelihood ratio (LR) test statistic for testing for a unit root in the ARMA model. The finite sample properties of the LR statistic as well as other unit root test procedures for the ARMA model are examined through a limited simulation study. We conclude that, for practical empirical work that relies on standard computations, the LR test procedure generally performs better than other standard procedures in the presence of a substantial moving-average component in the ARMA model.     

Seasonal Moving-average Unit Root Tests in the Presence of a Linear Trend

To distinguish stochastic from deterministic seasonality, test procedures are developed for a unit root in the integrated seasonal moving-average (SMA) model when an underlying deterministic trend is present. Locally best invariant unbiase (LBIU) and point optimal invariant tests are considered. Their asymptotic distributions are developed and are found to differ from those for the no-linear-trend case. The limiting distribution of the LBIU statistic is expressed as a functional of Brownian motions. The procedures are extended to more general seasonal autoregressive moving-average (ARMA) models, and to the inclusion of exogenous regressors. Finite-sample distributions are also derived for the SMA(1) model. Simulations suggest that these distributions provide accurate approximations for more general ARMA models. A numerical example is included to illustrate the tests.     

Book Review

Abraham Boyarsky and Pawel Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in one Dimension, Probability and Its Applications     

NONLINEAR TRANSFORMATIONS OF INTEGRATED TIME SERIES

Abstract. In this paper we consider the effects of nonlinear transformations on integrated processes and unit root tests performed on such series. A test that is invariant to monotone data transformations is proposed. It is shown that series are generally not cointegrated with nonlinear transformations of themselves, but the same transformation applied to a pair of cointegrated series can result in cointegration between the transformed series.     

DETERMINING THE DEGREE OF DIFFERENCING FOR TIME SERIES VIA THE LOG SPECTRUM

Abstract. While many time series require differencing before a model may be fitted it has been shown that 'overdifferencing' may result in a fitted model with poor long term forecasting properties. This may present real problems when the degree of differencing which is appropriate is fractional. We show that the log spectrum is a natural quantity to consider when attempting to determine the degree of differencing required and outline the distribution theory required. The ideas are shown to extend to the seasonal case and can be used to assess whether seasonal differencing is appropriate.     

ESTIMATION AND INTERPOLATION OF MISSING VALUES OF A STATIONARY TIME SERIES

Abstract. A complete solution of the important problem of estimating (interpolating) the missing values of a stationary time series is obtained by decomposing it into a prediction plus regression problem. This makes it possible to estimate the missing values by finding the multistep-ahead predictors and using the existing computer packages for time series analysis. Such a solution is vital for the E step of the EM algorithm, and it is shown how this algorithm can be used to develop a simultaneous procedure for estimating the parameters and missing values of a time series.     

SEARCHING FOR ADDITIVE OUTLIERS IN NONSTATIONARY TIME SERIES*

Abstract. Recently, Vogelsang (1999) proposed a method to detect outliers which explicitly imposes the null hypothesis of a unit root. It works in an iterative fashion to select multiple outlier in a given series. We show, via simulations, that, under the null hypothesis of no outliers, it has the right size in finite samples to detect a single outlier but, when applied in an iterative fashion to select multiple outliers, it exhibits severe size distortions towards finding an excessive number of outliers. We show that his iterative method is incorrect and derive the appropriate limiting distribution of the test at each step of the search. Whether corrected or not, we also show that the outliers need to be very large for the method to have any decent power. We propose an alternative method based on first‐differenced data that has considerably more power. We also show that our method to identify outliers leads to unit root tests with more accurate finite sample size and robustness to departures from a unit root. The issues are illustrated using two US/Finland real‐exchange rate series.     

AN APPLICATION OF THE SCHUR-COHN ALGORITHM TO TIME SERIES ANALYSIS

Abstract. Standard least squares analysis of autoregressive moving-average (ARMA) processes with errors-in-variables entails the construction of a new set of parameters which are functions of the original ARMA parameters, and requires that derivatives of these new parameters of order three or less with respect to the ARMA parameters exist and be bounded. The boundedness of these derivatives in turn depends critically on the nonsingularity of a matrix B which is a function of the ARMA parameters via the new parameters in the model. A particular version of the classical Schur–Cohn algorithm enables us to establish this nonsingularity.     

ESTIMATION OF COEFFICIENTS OF TIME SERIES REGRESSION WITH A NONSTATIONARY ERROR PROCESS

Abstract. We treat a problem of estimating unknown coefficients of a time series regression when the variance of the error changes with time, i.e. when a process which the error term obeys is nonstationary. First, we show the weak consistency of the ordinary least squares estimator for the coefficients of a polynomial regression under some assumptions on the covariance structure of the error process. Next, we propose a nonparametric method for estimating the variance of the error process and a weighted least squares estimator of the regression coefficients, which is constructed by using the estimator of the variance. We investigate statistical properties of our proposed estimator in the following way. We consider the prediction of a future value of a linear trend by using our proposed estimator and evaluate its prediction error. By simulation studies, we compare the prediction error of the predictor constructed by using our proposed estimator with the prediction errors obtained for other estimators including the ordinary least squares estimator when the variance of the error process increases with time and the sample sizes are small. As a result, our proposed estimator seems to be reasonable.     

SOME SIMPLE TESTS OF THE MOVING-AVERAGE UNIT ROOT HYPOTHESIS

Abstract. This paper deals with three test statistics for a moving-average (MA) unit root. The spectral test is based on the estimate of the spectral density at frequency zero. The variance difference statistic compares the sample variance of the integrated series with the estimated variance imposing the MA unit root constraint. Furthermore, Tanaka's score type test statistic is modified to improve the power in higher order models. The asymptotic power of the tests is considered and Monte Carlo experiments are performed to investigate the small sample properties of the tests. Finally, the tests are applied to a number of economic time series to determine the degree of integration.     

TESTING FOR A UNIT ROOT IN AN AR(1) TIME SERIES USING IRREGULARLY OBSERVED DATA

Abstract. For an AR(1) model having a unit root with nonconsecutively observed or missing data we consider the ordinary least squares estimator, the one-step Newton-Raphson estimator and an ordinary least squares type estimator which is a simple approximation of the Newton-Raphson estimator. It is shown that the limiting distributions of these estimators of the unit root are the same as those of the regression estimators as tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Assoc. 74 (1979), 427–31) for the complete data situation. Simulation results show that our proposed unit root tests perform very well for small samples.     

ASYMPTOTIC RELATIVE EFFICIENCY OF SOME TESTS OF FIT IN TIME SERIES MODELS

Abstract. Two frequency domain tests of fit for autoregressive moving average time series models are considered. The tests are slight generalizations of those introduced by Cameron (1978) and Milhøj (1981). It is shown that according to asymptotic relative efficiency the test by Milhøj outperforms the test by Cameron. However, if asymptotic relative efficiency is used as a standard of comparison, both of these tests are extremely poor as compared to the well-known time domain test of Box and Pierce (1970), for the asymptotic relative efficiency of the frequency domain tests as compared to the Box-Pierce test is zero.     

TESTING EQUALITY OF MEANS WHEN THE OBSERVATIONS ARE FROM FUNCTIONAL TIME SERIES

There are numerous examples of functional data in areas ranging from earth science to finance where the problem of interest is to compare several functional populations. In many instances, the observations are obtained consecutively in time, and thus, the classical assumption of independence within each population may not be valid. In this article, we derive a new, asymptotically justified method to test the hypothesis that the mean curves of multiple functional populations are the same. The test statistic is constructed from the coefficient vectors obtained by projecting the functional observations into a finite dimensional space. Asymptotics are established when the observations are considered to be from stationary functional time series. Although the limit results hold for projections into arbitrary finite dimensional spaces, we show that higher power is achieved by projecting onto the principle components of empirical covariance operators that diverge under the alternative. Our method is further illustrated by a simulation study as well as an application to electricity demand data.     

SEQUENTIAL METHODS FOR DETECTING CHANGES IN THE VARIANCE OF ECONOMIC TIME SERIES

In this paper EWMA charts and CUSUM charts are introduced for detecting changes in the variance of a GARCH process. The moments of the EWMA statistics are calculated. They permit a better understanding of the underlying control procedure. In an extensive simulation study all control schemes are compared with each other. “Optimal” smoothing parameters and “optimal” reference values are tabulated. It is shown how these charts can be applied to monitor stock market returns.     

BANDWIDTH SELECTION IN KERNEL SMOOTHING OF TIME SERIES

Abstract. The kernel smoothing method has been considered as a useful tool for identification and prediction in time series models. In practice this method is to be tuned by a smoothing parameter. For selection of the smoothing parameter, Härdle and Vieu (Kernel regression smoothing of time series. J. Time Ser. Anal. 13(1992), 209–32) considered a cross-validation rule and proved its asymptotic optimality. In this paper we strengthen their result for a wider use of the kernel smoothing of time series.     

SOME ASPECTS OF THE PERFORMANCE OF DIAGNOSTIC CHECKS IN BIVARIATE TIME SERIES MODELS

Abstract. This paper is concerned with the use of score, or Lagrangian multiplier and portmanteau tests of fitted model adequacy in vector autoregressive-moving average processes. The relation between these alternative diagnostic checking devices is discussed from an asymptotic theoretic standpoint. Some finite sample properties of the tests are investigated in the context of bivariate models using Monte Carlo methods. Asymptotic theory is used to help determine the simulation design and also proves useful in appraising the experimental outcomes. The results provide evidence on the likely relative performance of the two procedures in practice and suggest that the score test is to be preferred.     

(MIS)SPECIFICATION OF LONG MEMORY IN SEASONAL TIME SERIES

Abstract. We present a complete generalization of fractional differencing with seasonal processes. The contribution of each seasonal frequency to the variance of a process may be modelled by separate difference parameters. The regression of the log-periodogram allows estimation of the difference parameters. We approximate the bias and the variance of these estimators with large samples. Numerical examples illustrate the risk of fractional misspecification.