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).     

A k-Factor GARMA Long-memory Model

Long-memory models have been used by several authors to model data with persistent autocorrelations. The fractional and fractional autoregressive moving-average (FARMA) models describe long-memory behavior associated with an infinite peak in the spectrum at f = 0. The Gegenbauer and Gegenbauer ARMA (GARMA) processes of Gray, Zhang and Woodward (On generalized fractional processes. J. Time Ser. Anal. 10 (1989), 233–57) can model long-term periodic behavior for any frequency 0 ≤ f ≤ 0.5. In this paper we introduce a k -factor extension of the Gegenbauer and GARMA models that allows for long-memory behavior to be associated with each of k frequencies in [0, 0.5]. We prove stationarity conditions for the k -factor model and discuss issues such as parameter estimation, model iden- tification, realization generation and forecasting. A two-factor GARMA model is then applied to the Mauna Loa atmospheric CO₂ data. It is shown that this model provides a reasonable fit to the CO₂ data and produces excellent forecasts.     

SOME ASYMPTOTIC PROPERTIES OF THE SAMPLE COVARIANCES OF GAUSSIAN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. The paper deals with the asymptotic variances of the sample covariances of autoregressive moving average processes. Using state-space representations and some matrix Lyapunov equation theory, closed-form expressions are derived for the asymptotic variances of the sample covariances and for the Cramer-Rao bounds on the process covariances. The main results obtained from these expressions are as follows: For ARMA ( p, q ) processes with p ≥ q , the sample covariance of order n is asymptotically efficient if and only if 0 ≤ n ≤ p – q .
For ARMA ( p, q ) processes with p < q , none of the sample covariances is asymptotically efficient.     

REGRESSION OF SPECTRAL ESTIMATORS WITH FRACTIONALLY INTEGRATED TIME SERIES

Abstract. Assuming a normal distribution we supplement the proof of periodogram regression suggested by Geweke and Porter-Hudak ( J. Time Ser. Anal. 4 (1983) 221–38) in order to estimate and test the difference parameter of fractionally integrated autoregressive moving-average models. The procedure proposed by Kashyap and Eom ( J. Time Ser. Anal. 9 (1988) 35–41) arises as a special case and is found to be correct if the true parameter value is negative. Regression of the smoothed periodogram yields estimators for the difference parameter with much faster vanishing variance; no asymptotic distribution can be derived, however. In computer experiments we find that the smoothed periodogram regression may be superior to pure periodogram regression when we have to discriminate between autoregression and fractional integration     

Valid Edgeworth Expansions of Some Estimators and Bootstrap Confidence Intervals in First-order Autoregression

We derive the third-order valid Edgeworth expansions for the standardized and the Studentized versions of some estimators in first-order autoregression without Gaussianity. As a special case of a Gaussian process, the validity of the expansion obtained by Ochi (Asymptotic expansions for the distribution of an estimator in the first-order autoregressive process. Journal of Time Ser. Anal. 4 (1983), 57–67) is demonstrated. By applying the second-order Edgeworth expansion to the bootstrap procedure, we construct the confidence intervals for the autoregressive coefficient.     

THE DETECTION OF A SINGLE ADDITIVE OUTLIER OF UNKNOWN POSITION

Abstract. Kudo (On the testing of outlying observations. Sankhya 17 (1956), 67–73) has derived an optimal invariant detector of a single additive outlier of unknown position in the context of an underlying Gaussian process consisting of independent and identically distributed random variables. We show how this author's arguments can be extended to derive an invariant detector of an additive outlier of unknown position for an underlying zero-mean Gaussian stochastic process. This invariant detector depends on the parameters of this process; its properties are analysed further for the particular case of an underlying zero-mean Gaussian AR( p ) process. It provides an upper bound on the performance of any invariant detector based solely on the data and it may be 'bootstrapped' to provide an invariant detector based solely on the data. A plausibility argument is presented in favour of the proposition that the bootstrapped detector is nearly optimal for sufficiently large data length n. The truth of this proposition has been confirmed by simulation results for zero-mean Gaussian AR(1) and AR(2) processes (for certain sets of possible outlier positions). The bootstrapped detector is shown to be closely related to the detector based on the approximate likelihood ratio criteria of Fox (Outliers in time series. J. Roy. Statist. Soc. Ser. B 34 (1972), 350–63) and the leave-one-out diagnostic of Bruce and Martin (Leave- k -out diagnostics in time series. J. Roy. Statist. Soc. Ser B 51 (1989), 363–424). It is also shown how the case of an underlying Gaussian process with arbitrary mean can be reduced to the case of an underlying zero-mean Gaussian process.     

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.     

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.     

Simulating a class of stationary Gaussian processes using the Davies–Harte algorithm, with application to long memory processes

We demonstrate that the fast and exact Davies–Harte algorithm is valid for simulating a certain class of stationary Gaussian processes – those with a negative autocovariance sequence for all non-zero lags. The result applies to well known classes of long memory processes: Gaussian fractionally differenced (FD) processes, fractional Gaussian noise (fGn) and the nonstationary fractional Brownian Motion (fBm).     

Plug-in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long-memory Time Series

We consider the problem of selecting the number of frequencies, m , in a log-periodogram regression estimator of the memory parameter d of a Gaussian long-memory time series. It is known that under certain conditions the optimal m , minimizing the mean squared error of the corresponding estimator of d , is given by m ^(opt)= Cn ^4/5, where n is the sample size and C is a constant. In practice, C would be unknown since it depends on the properties of the spectral density near zero frequency. In this paper, we propose an estimator of C based again on a log-periodogram regression and derive its consistency. We also derive an asymptotically valid confidence interval for d when the number of frequencies used in the regression is deterministic and proportional to n ^4/5. In this case, squared bias cannot be neglected since it is of the same order as the variance. In a Monte Carlo study, we examine the performance of the plug-in estimator of d , in which m is obtained by using the estimator of C in the formula for m ^(opt) above. We also study the performance of a bias-corrected version of the plug-in estimator of d . Comparisons with the choice m = n ^1/2 frequencies, as originally suggested by Geweke and Porter-Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37), are provided.     

Robust estimation of the scale and of the autocovariance function of Gaussian short‐ and long‐range dependent processes

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time‐series modelling. In this article, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Ma and Genton (2000) are established for Gaussian processes, with either short‐ or long‐range dependence. It is shown in the short‐range dependence setting that this robust estimator is asymptotically normal at the rate , where n is the number of observations. An explicit expression of the asymptotic variance is also given and compared with the asymptotic variance of the classical autocovariance estimator. In the long‐range dependence setting, the limiting distribution displays the same behaviour as that of the classical autocovariance estimator, with a Gaussian limit and rate when the Hurst parameter H is less than 3/4 and with a non‐Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when H ∈ (3/4,1). Some Monte Carlo experiments are presented to illustrate our claims and the Nile River data are analysed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.     

Banded and tapered estimates for autocovariance matrices and the linear process bootstrap

We address the problem of estimating the autocovariance matrix of a stationary process. Under short range dependence assumptions, convergence rates are established for a gradually tapered version of the sample autocovariance matrix and for its inverse. The proposed estimator is formed by leaving the main diagonals of the sample autocovariance matrix intact while gradually down‐weighting off‐diagonal entries towards zero. In addition, we show the same convergence rates hold for a positive definite version of the estimator, and we introduce a new approach for selecting the banding parameter. The new matrix estimator is shown to perform well theoretically and in simulation studies. As an application, we introduce a new resampling scheme for stationary processes termed the linear process bootstrap (LPB). The LPB is shown to be asymptotically valid for the sample mean and related statistics. The effectiveness of the proposed methods are demonstrated in a simulation study.     

A Note on Diagnosing Multivariate Conditional Heteroscedasticity Models

In this paper we consider several tests for model misspecification after a multivariate conditional heteroscedasticity model has been fitted. We examine the performance of the recent test due to Ling and Li ( J. Time Ser. Anal. 18 (1997), 447–64), the Box–Pierce test and the residual-based F test using Monte Carlo methods. We find that there are situations in which the Ling–Li test has very weak power. The residual-based diagnostics demonstrate significant under-rejection under the null. In contrast, the Box–Pierce test based on the cross-products of the standardized residuals often provides a useful diagnostic that has reliable empirical size as well as good power against the alternatives considered.     

NON-LINEAR TIME SERIES MODELLING AND DISTRIBUTIONAL FLEXIBILITY

Abstract. Most of the existing work in non-linear time series analysis has concentrated on generating flexible functional models by specifying non-linear specifications for the mean of a particular process, without much, if any, attention given to the distributional properties of the model. However, as Martin ( J. Time Ser. Anal. 13 (1992), 79–94) has shown, greater flexibility in perhaps a more natural way can be achieved by consideration of distributions from the generalized exponential class. This paper represents an extension of the earlier work of Martin by introducing a flexible class of non-linear time series models which can capture a wide range of empirical behaviour such as skewed, fat-tailed and even multimodal distributions. This class of models is referred to as generalized exponential non-linear time series. A maximum likelihood algorithm is given for estimating the parameters of the model and the framework is applied to estimating the distribution of the movements of the exchange rate.     

ESTIMATION OF THE MULTIVARIATE AUTOREGRESSIVE MOVING AVERAGE HAVING PARAMETER RESTRICTIONS AND AN APPLICATION TO ROTATIONAL SAMPLING

Abstract. The vector autoregressive moving average model with nonlinear parametric restrictions is considered. A simple and easy-to-compute Newton-Raphson estimator is proposed that approximates the restricted maximum likelihood estimator which takes full advantage of the information contained in the restrictions. In the case when there are no parametric restrictions, our Newton-Raphson estimator is equivalent to the estimator proposed by Reinsel et al. (Maximum likelihood estimators in the multivariate autoregressive moving-average model from a generalized least squares view point. J. Time Ser. Anal. 13 (1992), 133–45). The Newton-Raphson estimation procedure also extends to the vector ARMAX model. Application of our Newton-Raphson estimation method in rotational sampling problems is discussed. Simulation results are presented for two different restricted models to illustrate the estimation procedure and compare its performance with that of two alternative procedures that ignore the parametric restrictions.     

Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Consistent Estimation for Non-Gaussian Non-Causal Autoregessive Processes

Traditional estimation based on least squares or Gaussian likelihood cannot distinguish between causal and non-causal representation of a stationary autoregressive (AR) process. Breidt et al . (Maximum likelihood estimation for non-causal autoregressive processes. J. Multivariate Anal. 36 (1991), 175–98) proved the existence of a consistent likelihood estimation of possibly non-causal AR processes; however, in this case an existence result is not very useful since the likelihood function generally exhibits multiple maxima. Moreover the method assumes full knowledge of the distribution of the innovation process. This paper shows a constructive proof that a modified L ₁ estimate is consistent if the innovation process has a stable law distribution with index α∈ (1, 2). It is also shown that neither non-Gaussianity nor infinite variance is sufficient to ensure consistency.     

Gaussian Semiparametric Estimation of Non-stationary Time Series

Generalizing the definition of the memory parameter d in terms of the differentiated series, we showed in Velasco (Non-stationary log-periodogram regression, Forthcoming J. Economet. , 1997) that it is possible to estimate consistently the memory of non-stationary processes using methods designed for stationary long-range-dependent time series. In this paper we consider the Gaussian semiparametric estimate analysed by Robinson (Gaussian semiparametric estimation of long range dependence. Ann. Stat . 23 (1995), 1630–61) for stationary processes. Without a priori knowledge about the possible non-stationarity of the observed process, we obtain that this estimate is consistent for d ∈ (−½, 1) and asymptotically normal for d ∈ (−½,¾) under a similar set of assumptions to those in Robinson's paper. Tapering the observations, we can estimate any degree of non-stationarity, even in the presence of deterministic polynomial trends of time. The semiparametric efficiency of this estimate for stationary sequences also extends to the non-stationary framework.     

ASYMPTOTIC EFFICIENCY OF THE SAMPLE COVARIANCES IN A GAUSSIAN STATIONARY PROCESS

Abstract. This paper deals with the asymptotic efficiency of the sample autocovariances of a Gaussian stationary process. The asymptotic variance of the sample autocovariances and the Cramer–Rao bound are expressed as the integrals of the spectral density and its derivative. We say that the sample autocovariances are asymptotically efficient if the asymptotic variance and the Cramer–Rao bound are identical. In terms of the spectral density we give a necessary and sufficient condition that they are asymptotically efficient. This condition is easy to check for various spectra.     

Exactly/Nearly Unbiased Estimation of Autocovariances of a Univariate Time Series With Unknown Mean

This article proposes an exactly/nearly unbiased estimator of the autocovariance function of a univariate time series with unknown mean. The estimator is a linear function of the usual sample autocovariances computed using the observed demeaned data. The idea is to stack the usual sample autocovariances into a vector and show that the expectation of this vector is a linear combination of population autocovariances. A matrix that we label, A , collects the weights in these linear combinations. When the population autocovariances of high lags are zero (small), exactly (nearly) unbiased estimators of the remaining autocovariances can be obtained using the inverse of upper blocks of the A matrix. The A ‐matrix estimators are shown to be asymptotically equivalent to the usual sample autocovariance estimators. The A ‐matrix estimators can be used to construct estimators of the autocorrelation function that have less bias than the usual estimators. Simulations show that the A ‐matrix estimators can substantially reduce bias while not necessarily increasing mean square error. More powerful tests for the null hypothesis of white noise are obtained using the A ‐matrix estimators.