Distributions for residual autocovariances in parsimonious periodic vector autoregressive models with applications

The asymptotic distribution of the residual autocovariance matrices in the class of periodic vector autoregressive time series models with structured parameterization is derived. Diagnostic checking with portmanteau test statistics represents a useful application of the result. Under the assumption that the periodic white noise process of the periodic vector autoregressive time series model is composed of independent random variables, we demonstrate that the finite sample distributions of the Hosking‐Li‐McLeod portmanteau test statistics can be approximated by those of weighted sums of independent chi‐square random variables. The quantiles of the asymptotic distribution can be computed using the Imhof algorithm or other exact methods. Thus, using the (single) chi‐square distribution for these test statistics appears inadequate in general, although it is often recommended in practice for diagnostic methods of that kind. A simulation study provides empirical evidence.     

ON THE SQUARED RESIDUAL AUTOCORRELATIONS IN NON-LINEAR TIME SERIES WITH CONDITIONAL HETEROSKEDASTICITY

Abstract. Time series with a changing conditional variance have been found useful in many applications. Residual autocorrelations from traditional autoregressive moving-average models have been found useful in model diagnostic checking. By analogy, squared residual autocorrelations from fitted conditional heteroskedastic time series models would be useful in checking the adequacy of such models. In this paper, a general class of squared residual autocorrelations is defined and their asymptotic distribution is obtained. The result leads to some useful diagnostic tools for statisticians using conditional heteroskedastic time series models. Some simulation results and an illustrative example are also reported.     

Asymptotics of Quantiles and Rank Scores in Nonlinear Time Series

This paper extends the concept of regression and autoregression quantiles and rank scores to a very general nonlinear time series model. The asymptotic linearizations of these nonlinear quantiles are then used to obtain the limiting distributions of a class of L-estimators of the parameters. In particular, the limiting distributions of the least absolute deviation estimator and trimmed estimators are obtained. These estimators turn out to be asymptotically more efficient than the widely used conditional least squares estimator for heavy-tailed error distributions. The results are applicable to linear and nonlinear regression and autoregressive models including self-exciting threshold autoregressive models with known threshold.     

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.     

THRESHOLD TIME SERIES MODELS AS MULTIMODAL DISTRIBUTION JUMP PROCESSES

Abstract. Recent contributions by Tong and others in modelling time series exhibiting threshold points have generally been based on approximating non-linear processes by piecewise linear time series models. In this paper we provide an alternative framework in which to model time series displaying jump behaviour by using a multimodal conditional distribution to capture the jump process. Each subordinate model of the distribution is determined by an autoregressive process, and jump behaviour occurs when the relative heights of the modes of the distribution change whilst the threshold points are identified by the antimodes of the distribution. This class of models is referred to as multipredictor autoregressive time series (MATS).     

Bayesian Inference on Periodicities and Component Spectral Structure in Time Series

We detail and illustrate time series analysis and spectral inference in autoregressive models with a focus on the underlying latent structure and time series decompositions. A novel class of priors on parameters of latent components leads to a new class of smoothness priors on autoregressive coefficients, provides for formal inference on model order, including very high order models, and leads to the incorporation of uncertainty about model order into summary inferences. The class of prior models also allows for subsets of unit roots, and hence leads to inference on sustained though stochastically time-varying periodicities in time series. Applications to analysis of the frequency composition of time series, in both time and spectral domains, is illustrated in a study of a time series from astronomy. This analysis demonstrates the impact and utility of the new class of priors in addressing model order uncertainty and in allowing for unit root structure. Time-domain decomposition of a time series into estimated latent components provides an important alternative view of the component spectral characteristics of a series. In addition, our data analysis illustrates the utility of the smoothness prior and allowance for unit root structure in inference about spectral densities. In particular, the framework overcomes supposed problems in spectral estimation with autoregressive models using more traditional model-fitting methods.     

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.     

Improved multivariate portmanteau test

A new portmanteau diagnostic test for vector autoregressive moving average (VARMA) models that is based on the determinant of the standardized multivariate residual autocorrelations is derived. The new test statistic may be considered an extension of the univariate portmanteau test statistic suggested by Peňa and Rodríguez (2002) . The asymptotic distribution of the test statistic is derived as well as a chi‐square approximation. However, the Monte–Carlo test is recommended unless the series is very long. Extensive simulation experiments demonstrate the usefulness of this test as well as its improved power performance compared to widely used previous multivariate portmanteau diagnostic check. Two illustrative applications are given.     

Mixed Portmanteau Tests for Time‐Series Models

Abstract. This paper obtains the joint limiting distribution of residuals and squared residuals of a general time‐series model. Based on this, we propose a mixed portmanteau statistic for testing the adequacy of fitted time‐series models. In some cases, it is shown that this statistic can be simply approximated by the sum of well‐known portmanteau statistics. The finite‐sample performance of the new test is compared with those of well‐known tests through simulations.     

On Residual Variance Estimation in Autoregressive Models

In this paper we consider time series models belonging to the autoregressive (AR) family and deal with the estimation of the residual variance. This is important because estimates of the variance are involved in, for example, confidence sets for the parameters of the model, estimation of the spectrum, expressions for the estimated error of prediction and sample quantities used to make inferences about the order of the model. We consider the asymptotic biases for moment and least squares estimators of the residual variance, and compare them with known results when available and with those for maximum likelihood estimators under normality. Simulation results are presented for finite samples     

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.     

Generalized autoregressive moving average models with GARCH errors

One of the important and widely used classes of models for non-Gaussian time series is the generalized autoregressive model average models (GARMA), which specifies an ARMA structure for the conditional mean process of the underlying time series. However, in many applications one often encounters conditional heteroskedasticity. In this article, we propose a new class of models, referred to as GARMA-GARCH models, that jointly specify both the conditional mean and conditional variance processes of a general non-Gaussian time series. Under the general modeling framework, we propose three specific models, as examples, for proportional time series, non-negative time series, and skewed and heavy-tailed financial time series. Maximum likelihood estimator (MLE) and quasi Gaussian MLE are used to estimate the parameters. Simulation studies and three applications are used to demonstrate the properties of the models and the estimation procedures.     

An Extended Portmanteau Test for VARMA Models With Mixing Nonlinear Constraints

Abstract. The portmanteau test is a widely used diagnostic tool for univariate and multivariate time‐series models. Its asymptotic distribution is known for the unconstrained vector autoregressive moving‐average (VARMA) case and for VAR models with constraints on the autoregressive coefficients. In this article, we give conditions under which the test can be applied to constrained VARMA models. Unfortunately, it cannot generally be applied to models with constraints that simultaneously affect the ARMA polynomial coefficients and the covariance matrix of the innovations (mixing constraints). This happens in latent‐variable models such as dynamic factor models (DFM). In addition, when there are constraints on the covariance matrix it seems convenient to check the goodness of fit using the zero‐lag residual covariances. We propose an extended portmanteau test that not only checks the autocorrelations of the residuals but also whether their covariance matrix is consistent with the constraints. We prove that the statistic is asymptotically distributed as a chi‐square for ARMA models under the assumption that the innovations have Gaussian‐like fourth‐order moments. We also show that the test is appropriate for the DFM, Peña–Box model and factor‐structural vector autoregression (FSVAR).     

Testing for parameter constancy in non‐Gaussian time series

This paper investigates testing for parameter constancy in models for non‐Gaussian time series. Models for discrete valued count time series are investigated as well as more general models with autoregressive conditional expectations. Both sup‐tests and CUSUM procedures are suggested depending on the complexity of the model being used. The asymptotic distribution of the CUSUM test is derived for a general class of conditional autoregressive models.     

ESTIMATION OF THE MOVING-AVERAGE REPRESENTATION OF A STATIONARY PROCESS BY AUTOREGRESSIVE MODEL FITTING

Abstract. The Hannan-Rissanen procedure for recursive order determination of an autoregressive moving-average process provides 'non-parametric' estimators of the coefficients b ( u ), say, of the moving-average representation of a stationary process by auto-regressive model fitting, and also that of the cross-covariances, c ( u ), between the process and its linear innovations. An alternative 'autoregressive' estimator of the b ( u ) is obtained by inverting the autoregressive transfer function. Some uses of these estimators are discussed, and their asymptotic distributions are derived by requiring that the order k of the fitted autoregression approaches infinity simultaneously with the length T of the observed time series. The question of bias in estimating the parameters is also examined.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

Conway–Maxwell–Poisson Autoregressive Moving Average Model for Equidispersed,Underdispersed, and Overdispersed Count Data

In this work, we propose a dynamic regression model based on the ConwayŮMaxwell–Poisson (CMP) distribution with time-varying conditional mean depending on covariates and lagged observations. This new class of ConwayŮMaxwell–Poisson autoregressive moving average (CMP-ARMA) models is suitable for the analysis of time series of counts. The CMP distribution is a two-parameter generalization of the Poisson distribution that allows the modeling of underdispersed, equidispersed, and overdispersed data. Our main contribution is to combine this dispersion flexibility with the inclusion of lagged terms to model the conditional mean response, inducing an autocorrelation structure, usually relevant in time series. We present the conditional maximum likelihood estimation, hypothesis testing inference, diagnostic analysis, and forecasting along with their asymptotic properties. In particular, we provide closed-form expressions for the conditional score vector and conditional Fisher information matrix. We conduct a Monte Carlo experiment to evaluate the performance of the estimators in finite sample sizes. Finally, we illustrate the usefulness of the proposed model by exploring two empirical applications.     

TESTING FOR GAUSSIANITY AND LINEARITY OF A STATIONARY TIME SERIES

Abstract. Stable autoregressive (AR) and autoregressive moving average (ARMA) processes belong to the class of stationary linear time series. A linear time series { } is Gaussian if the distribution of the independent innovations {ε( t )} is normal. Assuming that E ε( t ) = 0, some of the third-order cumulants c_xxx= Ex ( t ) x ( t + m ) x ( t + n ) will be non-zero if the ε( t ) are not normal and E ε³( t )≠O. If the relationship between { x ( t )} and {ε( t )} is non-linear, then { x ( t )} is non-Gaussian even if the ε( t ) are normal. This paper presents a simple estimator of the bispectrum, the Fourier transform of { c _xxx( m, n )}. This sample bispectrum is used to construct a statistic to test whether the bispectrum of { x ( t )} is non-zero. A rejection of the null hypothesis implies a rejection of the hypothesis that { x ( t )} is Gaussian. Another test statistic is presented for testing the hypothesis that { x ( t )} is linear. The asymptotic properties of the sample bispectrum are incorporated in these test statistics. The tests are consistent as the sample size N →-∞     

A Space-Time Bilinear Model and its Identification

In this paper we propose a class of space–time bilinear (STBL) models which can be used to model space–time series which exhibit bilinear behavior. The STBL model is shown to be an extension of a space–time autoregressive moving-average model and a special form of the multiple bilinear model. We focus on the identification procedure of the models. Some results about stationarity and the covariance structure of these models are also discussed. An identification procedure based on the squared observations is established for the simplest pure bilinear model and some illustrative examples are provided.     

MULTIPLICATIVE EXPONENTIAL MODELS FOR STATIONARY TIME SERIES

Abstract. A class of models for one dimensional time series is presented. The spectrum of such a model is obtained by raising the spectrum of a known parameterized model to an exponent, allowed to attain arbitrary real values. For a moving average model this for example means that the roots of the moving average operator are allowed to have any real order. This method adds a further flexibility to the model which for example allows us to model long memory time series using only a few parameters. The exponent is parameterized in a special way to make the estimation of the parameter determining the exponent asymptotically independent of the estimation of the other model-parameters. The asymptotic distribution of the estimators is derived. The idea is also used for multiplicative models with an exponent for each seasonal factor. In this case the estimators are only approximately independent for a large season length. Finally an application of the model is given using the Beveridge wheat price index.