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.     

Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series

Abstract. We provide a direct proof for consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible autoregressive moving‐average (ARMA) time series models, which were initially established by Hannan [Journal of Applied Probability (1973) vol. 10, pp. 130–145] via the asymptotic properties of a Whittle's estimator. This also paves the way to establish similar results for spatial processes presented in the follow‐up article by Yao and Brockwell [Bernoulli (2006) in press].     

ASYMPTOTIC DISTRIBUTIONS OF LIKELIHOOD RATIOS FOR OVERPARAMETRIZED ARMA PROCESSES

Abstract. This paper is devoted to an extension of a classical problem of statistics to the asymptotic distribution of likelihood ratios. Two main types of likelihood ratios are considered for Gaussian ARMA processes. It is assumed in both cases that the asymptotic Fisher information matrix of estimation is singular in the higher order models. It is proved that the asymptotic distributions of the log likelihood ratios are invariant with respect to the parameters generating the process. A simulation shows that the sample distribution of the log likelihood ratio approaches the asymptotic one. Finally, the likelihood ratio test is proposed for model order reduction.     

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.     

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.     

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.     

A mixed portmanteau test for ARMA‐GARCH models by the quasi‐maximum exponential likelihood estimation approach

This paper investigates the joint limiting distribution of the residual autocorrelation functions and the absolute residual autocorrelation functions of ARMA‐GARCH models. This leads a mixed portmanteau test for diagnostic checking of the ARMA‐GARCH model fitted by using the quasi‐maximum exponential likelihood estimation approach in Zhu and Ling (2011) . Simulation studies are carried out to examine our asymptotic theory, and assess the performance of this mixed test and other two portmanteau tests in Li and Li (2008) . A real example is given.     

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.     

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.     

Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models

This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving-average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one-step-ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an efficient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included.     

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.     

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.     

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.     

ORDER IDENTIFICATION STATISTICS IN STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS:VECTOR AUTOCORRELATIONS AND THE BOOTSTRAP

Abstract. In this paper we consider the vector autocorrelation approach for identifying ARMA ( p, q ) models and use a bootstrap procedure in order to evaluate the distribution of the corresponding sample statistics by means of a resampling scheme for the residuals when p and q are unknown. The asymptotic validity of the bootstrap procedure applied to the vector autocorrelation estimates is established. Some simulations and examples demonstrating the appropriateness of the proposed bootstrap procedure in comparison with large-sample Gaussian approximations are included.     

Robust Estimation in Vector Autoregressive Moving-Average Models

Bustos and Yohai proposed a class of robust estimates for autoregressive moving-average (ARMA) models based on residual autocovariances (RA estimates). In this paper an affine equivariant generalization of the RA estimates for vector ARMA processes is given. These estimates are asymptotically normal and, when the innovations have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the covariance matrix corresponding to the maximum likelihood estimate. A Monte Carlo study confirms that the RA estimates are efficient under normal errors and robust when the sample contains outliers. A robust multivariate goodness-of-fit test based on the RA estimates is also obtained.     

An approximate likelihood function for panel data with a mixed ARMA(p, q) remainder disturbance model

Abstract. An approximate likelihood function for panel data with an autoregressive moving‐average (ARMA)(p, q) model remainder disturbance is presented and Whittle's approximate maximum likelihood estimator (MLE) is used to yield an asymptotic estimator. Although an asymptotic approach, the power test is quite successful for estimating and testing. In this approach, we do not need to calculate the transformation matrix in exact form. Through the Riemann sum approach, we can construct a simple approximate concentrated likelihood function. In addition, the model is also extended to the restricted maximum likelihood (REML) function, in which the package of Gilmour, Thompson and Cullis [Biometrics (1995) Vol. 51, pp. 1440–1450] is applied without difficulty. In the case study, we implement the model on the characteristic line for the investment analysis of Taiwanese computer motherboard makers.     

FREQUENCY-DOMAIN ESTIMATION OF BILINEAR TIME SERIES MODELS

Abstract. Two frequency-domain methods of estimation of the parameters of linear time series models–one based on maximum likelihood, called the 'Whittle criterion', and the other based on least squares, called the 'Taniguchi criterion'–are discussed in this paper. A heuristic justification for their use in models such as bilinear models is given. The estimation theory and associated asymptotic theory of these methods are numerically illustrated for the bilinear model BL( p ,0, p , 1). For that purpose, an approach based on the calculus of Kronecker product matrices is used to obtain the derivatives of the spectral density function of the state-space form of the model.     

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.     

A METHOD FOR DIAGNOSTIC CHECKING OF TIME SERIES MODELS

Abstract. Suppose a tentative ARMA ( p, q )-model has been fitted to a stationary time series. A diagnostic check for this model is suggested, using the estimated cross correlation function (CCF) between the observed series and the residuals. The CCF may also indicate how the model can be improved. The method is applied to the Wolfer sunspot series. For AR ( p )-processes the asymptotic covariance matrix of the estimated cross correlations is obtained.