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.     

Testing for a Unit Root in Autoregressive Moving-average Models with Missing Data

Testing for a single autoregressive unit root in an autoregressive moving-average (ARMA) model is considered in the case when data contain missing values. The proposed test statistics are based on an ordinary least squares type estimator of the unit root parameter which is a simple approximation of the one-step Newton–Raphson estimator. The limiting distributions of the test statistics are the same as those of the regression statistics in AR(1) models tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc . 74 (1979), 427–31) for the complete data situation. The tests accommodate models with a fitted intercept and a fitted time trend.     

Unit Root Tests Based on Unconditional Maximum Likelihood Estimation for the Autoregressive Moving Average

Unconditional maximum likelihood estimation is considered for an autoregressive moving average that may contain an autoregressive unit root. The limiting distribution of the normalized maximum likelihood estimator of the unit root is shown to be the same as that of the estimator for the first-order autoregressive process. A likelihood ratio test based on unconditional maximum likelihood estimation is proposed. In a Monte Carlo study for the autoregressive moving-average model of order (1, 1), the new test is shown to have better size and power than those of several other tests.     

Some Results on Specification Search and Pre-testing in an ARMA(1,1) Process

In this study we consider simple autoregressive moving-average (ARMA) models of order at most 1. Pre-testing, on the moving-average coefficient θ, is used to choose between an ARMA(1,1) and an AR(1) in a Monte Carlo design. We find that the pre-test estimator is not always dominated by the others, and that the bias and the mean square error of the estimate of the autoregressive coefficient φ very often depend on the sign of the autoregressive and moving-average parameters of the ARMA(1,1) model in the data-generating process. Further, we note that the degrees of size and power distortion of the t test on φ, after pre-testing for θ, are generally associated with model misspecification.     

A Note on Bootstrapping M-Estimators in ARMA Models

Kreiss and Franke (Bootstrapping stationary autoregressive moving-average models. J. Time Ser. Anal. 13 (1992), 297–317) proposed bootstrapping a linear approximation to the M-estimator in autoregressive moving-average (ARMA) models. In this paper, it is argued that it may be better to apply the bootstrap principle directly to the M-estimator itself. A number of simulation results are presented to compare the two procedures for estimating the sampling distribution of an M-estimator. The theoretical asymptotic validity of the standard bootstrap applied to the M-estimator is established.     

DIFFERENTIAL GEOMETRY OF ARMA MODELS

Abstract. A general approach for the development of a statistical inference on autoregressive moving-average (ARMA) models is presented based on geometric arguments. ARMA models are characterized as members of the curved exponential family. Geometric properties of ARMA models are computed and used to suggest parameter transformations that satisfy predetermined properties. In particular, the effect on the asymptotic bias of the maximum likelihood estimator of model parameters is illustrated. Hypothesis testing of parameters is discussed through the application of a modified form of the likelihood ratio test statistic.     

ASYMPTOTIC INFERENCE FOR NON-INVERTIBLE MOVING-AVERAGE TIME SERIES

Abstract. This paper is concerned with statistical inference of nonstationary and non-invertible autoregressive moving-average (ARMA) processes. It makes use of the fact that a derived process of an ARMA( p, q ) model follows an AR( q ) model with an autoregressive (AR) operator equivalent to the moving-average (MA) part of the original ARMA model. Asymptotic distributions of least squares estimates of MA parameters based on a constructed derived process are obtained as corresponding analogs of a nonstationary AR process. Extensions to the nearly non-invertible models are considered and the limiting distributions are obtained as functionals of stochastic integrals of Brownian motions and Ornstein-Uhlenbeck processes. For application, a two-stage procedure is proposed for testing unit roots in the MA polynomial. Examples are given to illustrate the application.     

ON THE LAGRANGE MULTIPLIER TEST FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. This paper examines the score or Lagrange multiplier statistic for testing the adequacy of a fitted autoregressive moving-average model and gives a simple closed-form expression for this test statistic. Some singularities arising as the order of the alternative model is increased are examined.     

A Class of Non-Embeddable ARMA Processes

We show that a stationary ARMA( p , q ) process { X _n = 0, 1, 2, ...} whose moving-average polynomial has a root on the unit circle cannot be embedded in any continuous-time autoregressive moving-average (ARMA) process { Y }( t ), t ≥ 0}, i.e. we show that it is impossible to find a continuous-time ARMA process { Y }( t )} whose autocovariance function at integer lags coincides with that of { X _n }. This provides an answer to the previously unresolved question raised in the papers of Chan and Tong ( J. Time Ser. Anal. 8 (1987), 277–81), He and Wang ( J. Time Ser. Anal. 10 (1989), 315–23) and Brockwell ( J. Time Ser. Anal. 16 (1995), 451–60).     

RANDOM AGGREGATION OF UNIVARIATE AND MULTIVARIATE LINEAR PROCESSES

Abstract. The paper is devoted to random aggregation of multivariate autoregressive moving-average (ARMA) processes. We derive second-order characteristics of random aggregate models. We show that random aggregation preserves the ARMA structure. Moreover, we specify a functional relation between the initial model poles and aggregate ones. We then examine the case of univariate ARMA processes. Theorem 4 shows that, if the initial process is ARMA( p, q ), the random aggregate process is an ARMA( p*, q* ) model with p* at most equal to p ; * depends, among other things, on the sampling distribution L . This theorem generalizes the well-known results on the topic of time interval aggregation without overlapping.     

PERIODIC CORRELATION IN STRATOSPHERIC OZONE DATA

Abstract. A 50-year time series of monthly stratospheric ozone readings from Arosa, Switzerland, is analyzed. The time series exhibits the properties of a periodically correlated (PC) random sequence with annual periodicities. Spectral properties of PC random sequences are reviewed and a test to detect periodic correlation is presented. An autoregressive moving-average (ARMA) model with periodically varying coefficients (PARMA) is fitted to the data in two stages. First, a periodic autoregressive model is fitted to the data. This fit yields residuals that are stationary but non-white. Next, a stationary ARMA model is fitted to the residuals and the two models are combined to produce a larger model for the data. The combined model is shown to be a PARMA model and yields residuals that have the correlation properties of white noise.     

A NOTE ON THE GENERATION OF INDEPENDENT REALIZATIONS OF A VECTOR AUTOREGRESSIVE MOVING-AVERAGE PROCESS

Abstract. Barone has described a method for generating independent realizations of a vector autoregressive moving-average (ARMA) process which involves recasting the ARMA model in state space form. We discuss a direct method of computing the initial state covariance matrix T ₀ which, unless the number of time series is large, is usually faster than using the doubling algorithm of Anderson and Moore. Our numerical comparisons are particularly valuable because T ₀ must also be computed when calculating the likelihood function. A number of other computational refinements are described. In particular, we advocate the use of Choleski factorizations rather than spectral decompositions. For a pure moving-average process computational savings can be achieved by working directly with the ARMA model rather than with its state space representation.     

A NOTE ON SQUARE ROOT FILTERING FOR VECTOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. A simplified version of the square root Kalman filter is obtained for a vector autoregressive moving-average (VARMA) model. The algorithm is computationally more efficient that the standard square root algorithm and its output can be used to compute the likelihood of a VARMA model accurately.     

ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE MODELS VIA HIGH-ORDER AUTOREGRESSIVE APPROXIMATIONS

Abstract. In this paper the problem of estimating autoregressive moving-average (ARMA) models is dealt with by first estimating a high-order autoregressive (AR) approximation and then using the AR estimate to form the ARMA estimate. We show how to obtain an efficient ARMA estimate by allowing the order of the AR estimate to tend to infinity as the number of observations tends to infinity. This approach is closely related to the work of Durbin. By transforming the approach into the frequency domain, we can view it as an L ²-norm model approximation of the relative error of the spectral factors. It can also be seen as replacing the periodogram estimate in the Whittle approach by a high-order AR spectral density estimate. Since L ²-norm approximation is a difficult task, we replace it by a modification of a recent model approximation technique called balanced model reduction. By an example, we show that this technique gives almost efficient ARMA estimates without the use of numerical optimization routines.     

ON THE ASYMPTOTIC DISTRIBUTION OF THE GENERALIZED PARTIAL AUTOCORRELATION FUNCTION IN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. It has been conjectured and illustrated that the estimate of the generalized partial autocorrelation function (GPAC), which has been used for the identification of autoregressive moving-average (ARMA) models, has a thick-tailed asymptotic distribution. The purpose of this paper is to investigate the asymptotic behaviour of the GPAC in detail. It will be shown that the GPAC can be represented as a ratio of two functions, known as the θ function and the Λ function, each of which itself has a useful pattern for ARMA model identification. We shall show the consistencies of the extended Yule-Walker estimates of the three functions and present their asymptotic distributions.     

Boundary Limit Theory for Functional Local to Unity Regression

This article studies functional local unit root models (FLURs) in which the autoregressive coefficient may vary with time in the vicinity of unity. We extend conventional local to unity (LUR) models by allowing the localizing coefficient to be a function which characterizes departures from unity that may occur within the sample in both stationary and explosive directions. Such models enhance the flexibility of the LUR framework by including break point, trending, and multidirectional departures from unit autoregressive coefficients. We study the behavior of this model as the localizing function diverges, thereby determining the impact on the time series and on inference from the time series as the limits of the domain of definition of the autoregressive coefficient are approached. This boundary limit theory enables us to characterize the asymptotic form of power functions for associated unit root tests against functional alternatives. Both sequential and simultaneous limits (as the sample size and localizing coefficient diverge) are developed. We find that asymptotics for the process, the autoregressive estimate, and its t‐statistic have boundary limit behavior that differs from standard limit theory in both explosive and stationary cases. Some novel features of the boundary limit theory are the presence of a segmented limit process for the time series in the stationary direction and a degenerate process in the explosive direction. These features have material implications for autoregressive estimation and inference which are examined in the article.     

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.     

ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE ORDER GIVEN AN INFINITE NUMBER OF MODELS AND APPROXIMATION OF SPECTRAL DENSITIES

Abstract. A modification of the minimum Akaike information criterion (AIC) procedure (and of related procedures like the Bayesian information criterion (BIC)) for order estimation in autoregressive moving-average (ARMA) models is introduced. This procedure has the advantage that consistency for the order estimators obtained via this procedure can be established without restricting attention to only a finite number of models. The behaviour of these newly introduced order estimators is also analysed for the case when the data-generating process is not an ARMA process (transfer function/spectral density approximation). Furthermore, the behaviour of the order estimators obtained via minimization of BIC (or of related criteria) is investigated for a non-ARMA data-generating process.     

BOOTSTRAPPING STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we develop an asymptotic theory for application of the bootstrap to stationary stochastic processes of autoregressive moving-average (ARMA) type, with known order ( p, q ). We give a proof of the asymptotic validity of the bootstrap proposal applied to M estimators for the unknown parameter vector of the process. For this purpose we derive an asymptotic expansion for M estimators in ARMA models and construct an estimate for the unknown distribution function of the residuals which in principle are not observable. A small simulation study is also included.