TEMPORAL AGGREGATION IN THE ARIMA PROCESS

Abstract. The effect of temporal aggregation on ARIMA models is investigated. The paper discusses the change of model form resulting from aggregation. For the IMA model it is noted that reduction of model order may occur, due to aggregation, which takes an arbitrarily high order IMA ( d , q) process to an IMA ( d , 0) process for the aggregates. For the AR process, we derive the exact order for the aggregate model and show that aggregation of an AR (p) series does not necessarily produce an ARMA (p, q) aggregate series as has been suggested in the literature. In particular, the AR order of AR (p) and ARIMA (p, d, q) can be reduced upon aggregation. The paper gives the general conditions under which this reduction of order may occur.     

A METHOD FOR GENERATING INDEPENDENT REALIZATIONS OF A MULTIVARIATE NORMAL STATIONARY AND INVERTIBLE ARMA(p, q) PROCESS

Abstract. A method for generating finite independent realizations of a normal multivariate stationary ARMA( p, q ) process is proposed. It is based on an AR (1) representation of an ARMA( p, q ) process allowing for an exact generation of the initial values of the simulation algorithm. Input facilities are supplied in order to assure stationarity and invertibility of the considered process.     

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.     

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 closed form of the covariance matrix and its inverse of the causal ARMA process

Abstract.  Derivation of the theoretical autocovariance function of a causal autoregressive moving-average process of order ( p ,  q ), ARMA( p ,  q ), when q  ≥ 1 is considered. A recursive relationship is established between the covariance matrices of an ARMA( p ,  q ) process and its associated ARMA( p ,  q −1) process. The obtained recursion is shown to produce the inverse of the covariance matrix and its determinant. Moreover, the introduced method can be easily implemented in any programming environment.     

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.     

STATIONARY DISCRETE AUTOREGRESSIVE-MOVING AVERAGE TIME SERIES GENERATED BY MIXTURES

Abstract. Two simple stationary processes of discrete random variables with arbitrarily chosen first-order marginal distributions, DARMA ( p, N + 1) and NDARMA ( p, N ), are given. The correlation structure of these processes mimics that of the usual linear ARMA ( p, q ) processes. The relationship of these processes to mover-stayer models, and to models for discrete time series given separately by Lindqvist and Pegram is discussed. Ad hoc nonparametric estimators for the parameters in the DARMA ( p, N + 1) and NDARMA ( p, N ) are given. A simulation study shows them to be as good as maximum likelihood estimators for the first-order autoregressive case, and to be much simpler to compute than the maximum likelihood estimators.     

Autocovariance Structure of Markov Regime Switching Models and Model Selection

We show that the covariance function of a second-order stationary vector Markov regime switching time series has a vector ARMA( p , q ) representation, where upper bounds for p and q are elementary functions of the number of regimes. These bounds apply to vector Markov regime switching processes with both mean–variance and autoregressive switching. This result yields an easily computed method for setting a lower bound on the number of underlying Markov regimes from an estimated autocovariance function.     

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).     

LAGRANGE MULTIPLIER TESTS FOR FRACTIONAL DIFFERENCE

Abstract. This paper develops Lagrange multiplier tests of ARMA( p, q ) models against fractional ARIMA( p, d, q ) alternatives. The performance of the tests is investigated for moderate-sized samples. It is concluded that fractional difference will be difficult to detect when the orders ( p, q ) are over-specified in an autoregressive moving-average (ARMA) analysis. The importance of distinguishing between the mean known and mean estimated cases in fractional difference models is illustrated in the context of these tests.     

ON EMBEDDING A DISCRETE-PARAMETER ARMA MODEL IN A CONTINUOUS-PARAMETER ARMA MODEL

Abstract. It is shown that a real-valued discrete-parameter Gaussian ARMA ( p. q ) model with q < p can be embedded in a real-valued continuous-parameter Gaussian ARMA( p', q' ) model with q' < p' . The problem of embedding a real-valued discrete-parameter Gaussian AR( p ) into a real-valued continuous-parameter Gaussian AR( p ) is also discussed.     

TIME-REVERSIBILITY, IDENTIFIABILITY AND INDEPENDENCE OF INNOVATIONS FOR STATIONARY TIME SERIES

Abstract. Weiss ( J. Appl. Prob. 12 (1975) 831–36) has shown that for causal autoregressive moving-average (ARMA) models with independent and identically distributed (i.i.d.) noise, time-reversibility is essentially unique to Gaussian processes. This result extends to quite general linear processes and the extension can be used to deduce that a non-Gaussian fractionally integrated ARMA process has at most one representation as a moving average of i.i.d. random variables with finite variance. In the proof of this uniqueness result, we use a time-reversibility argument to show that the innovations sequence (one-step prediction residuals) of an ARMA process driven by i.i.d. non-Gaussian noise is typically not independent, a result of interest in deconvolution problems. Further, we consider the case of an ARMA process to which independent noise is added. Using a time-reversibility argument we show that the innovations of the ARMA process with added independent noise are independent if and only if both the driving noise of the process and the added noise are Gaussian.     

A State space approach to bootstrapping conditional forecasts in arma models

A bootstrap approach to evaluating conditional forecast errors in ARMA models is presented. The key to this method is the derivation of a reverse-time state space model for generating conditional data sets that capture the salient stochastic properties of the observed data series. We demonstrate the utility of the method using several simulation experiments for the MA( q ) and ARMA( p, q ) models. Using the state space form, we are able to investigate conditional forecast errors in these models quite easily whereas the existing literature has only addressed conditional forecast error assessment in the pure AR( p ) form. Our experiments use short data sets and non-Gaussian, as well as Gaussian, disturbances. The bootstrap is found to provide useful information on error distributions in all cases and serves as a broadly applicable alternative to the asymptotic Gaussian theory.     

ON THE UNBIASEDNESS PROPERTY OF AIC FOR EXACT OR APPROXIMATING LINEAR STOCHASTIC TIME SERIES MODELS

Abstract. A rigorous analysis is given of the asymptotic bias of the log maximum likelihood as an estimate of the expected log likelihood of the maximum likelihood model, when a linear model, such as an invertible, gaussian ARMA ( p, q ) model, with or without parameter constraints, is fit to stationary, possibly non-gaussian observations. It is assumed that these data arise from a model whose spectral density function either (i) coincides with that of a member of the class of models being fit, or, that failing, (ii) can be well-approximated by invertible ARMA ( p, q ) model spectral density functions in the class, whose ARMA coefficients are parameterized separately from the innovations variance. Our analysis shows that, for the purpose of comparing maximum likelihood models from different model classes, Akaike's AIC is asymptotically unbiased, in case (i), under gaussian or separate parametrization assumptions, but is not necessarily unbiased otherwise. In case (ii), its asymptotic bias is shown to be of the order of a number less than unity raised to the power max { p, q } and so is negligible if max { p, q } is not too small. These results extend and complete the somewhat heuristic analysis given by Ogata (1980) for exact or approximating autoregressive models.     

Influence of structure of iron nanoparticles in aggregates on their magnetic properties

Zero-valent iron nanoparticles rapidly aggregate. One of the reasons is magnetic forces among the nanoparticles. Magnetic field around particles is caused by composition of the particles. Their core is formed from zero-valent iron, and shell is a layer of magnetite. The magnetic forces contribute to attractive forces among the nanoparticles and that leads to increasing of aggregation of the nanoparticles. This effect is undesirable for decreasing of remediation properties of iron particles and limited transport possibilities. The aggregation of iron nanoparticles was established for consequent processes: Brownian motion, sedimentation, velocity gradient of fluid around particles and electrostatic forces. In our previous work, an introduction of influence of magnetic forces among particles on the aggregation was presented. These forces have significant impact on the rate of aggregation. In this article, a numerical computation of magnetic forces between an aggregate and a nanoparticle and between two aggregates is shown. It is done for random position of nanoparticles in an aggregate and random or arranged directions of magnetic polarizations and for structured aggregates with arranged vectors of polarizations. Statistical computation by Monte Carlo is done, and range of dominant area of magnetic forces around particles is assessed.     

A GOODNESS-OF-FIT TEST FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS BASED ON THE STANDARDIZED SAMPLE SPECTRAL DISTRIBUTION OF THE RESIDUALS

Abstract. A stochastic process derived from the standardized sample spectral density of the residuals of a causal and invertible ARMA( p, q ) model is introduced to construct a goodness-of-fit procedure. The test statistics considered have a proper limiting distribution which is free of unknown parameters and which, unlike some well-known goodness-of-fit statistics based on the residuals, does not depend on the sample size.     

A UNIFIED APPROACH TO THE STUDY OF SUMS, PRODUCTS, TIME-AGGREGATION AND OTHER FUNCTIONS OF ARMA PROCESSES

Abstract. Conditions under which sums, products and time-aggregation of ARMA processes follow ARMA models are derived from a single theorem. This characterizes these processes in terms of difference equations satisfied by their autocovariance function. From this we obtain necessary and sufficient conditions for a function of a Gaussian ARMA process and the product of two possibly dependent Gaussian ARMA processes to be ARMA. We show that the sum and product of two ARMA processes related by a Box and Jenkins transfer function model belong to the ARMA family.     

A NOTE ON THE EMBEDDING OF DISCRETE-TIME ARMA PROCESSES

Abstract. Let {X_n, n= 0, 1, 2,…} be a discrete-time ARMA(p, q) process with q < p whose autoregressive polynomial has r (not necessarily distinct) negative real roots. According to a recent result of He and Wang (On embedding a discrete-parameter ARMA model in a continuous-parameter ARMA model. J. Time Ser. Anal. 10 (1989), 315–23) there exists a continuous-time ARMA (p', q') process {Y(t), t≥0} with q' < p'=p+r such that {Y(n), n= 0, 1, 2,…} has the same autocorrelation function as {X_n}. In this paper we show that this result is false by considering the case when {X_n} is a discrete-time AR(2) process whose autoregressive polynomial has distinct complex conjugate roots. We identify the proper subset of such processes which are embeddable in a continuous-time ARMA(2, 1) process. We show that every discrete-time AR(2) process with distinct complex conjugate roots can be embedded in either a continuous-tie ARMA(2, 1) process or a continuous-time ARMA(4, 2) process, or in some cases both. We derive an expression for the spectral density of the process obtained by sampling a general continuous-time ARMA(p, q) process (with distinct autoregressive roots) at arbitrary equally spaced time points. The expression clearly shows that the sampled process is a discrete-time ARMA (p', q') process with q' < p.     

Contemporaneous aggregation of GARCH processes

Abstract. In this article, the effect of contemporaneous aggregation of heterogeneous generalized autoregressive conditionally heteroskedastic (GARCH) processes, as the cross‐sectional size diverges to infinity is studied. We analyse both cases of cross‐sectionally dependent and independent individual processes. The limit aggregate does not belong to the class of GARCH processes. Dynamic conditional heteroskedasticity is only preserved when the individual processes are sufficiently cross‐correlated, although long memory for the limit aggregate volatility is not attainable. We also explore more general forms of cross‐sectional dependence and various types of aggregation schemes.     

ESTIMATION OF MULTIVARIATE TIME SERIES

Abstract. The algorithm proposed here is a multivariate generalization of a procedure discussed by Pearlman (1980) for calculating the exact likelihood of a univariate ARMA model. Ansley and Kohn (1983) have shown how the Kalman filter can be used to calculate the exact likelihood function when not all the observations are known. In Shea (1983) it is shown that this algorithm is much quicker than that of Ansley and Kohn (1983) for all ARMA models except an ARMA (2, 1) and a couple of low-order AR processes and therefore when we have no missing observations this algorithm should be used instead. The Fortran subroutine G13DCF in the NAG (1987) Library fits a vector ARMA model using an adaptation of this algorithm. Experience in the use of this routine suggests that having reasonably good initial estimates of the ARMA parameter matrices, and in particular the residual error covariance matrix, can not only substantially reduce the computing time but more important improve the convergence properties of the minimization procedure. We therefore propose a method of calculating initial estimates of the ARMA parameters which involves using a generalization of the concept of inverse cross covariances from the univariate to the multivariate case. Finally theory is put into practice with the fitting of a bivariate model to a couple of real-life time series.