Chi‐squared portmanteau tests for structural VARMA models with uncorrelated errors

We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving average models with uncorrelated errors. Under the assumption that errors are uncorrelated but non‐independent, it is known that the Ljung–Box (or Box–Pierce) portmanteau test statistic is asymptotically distributed as a weighted sum of chi‐squared random variables which can be far from the chi‐square distribution usually employed. We therefore propose a new portmanteau statistic that is asymptotically chi‐squared even in the presence of uncorrelated but non‐independent errors. Monte Carlo experiments illustrate the finite sample performance for the proposed portmanteau test.     

On modelling and diagnostic checking of vector periodic autoregressive time series models

Abstract.  Vector periodic autoregressive time series models (PVAR) form an important class of time series for modelling data derived from climatology, hydrology, economics and electrical engineering, among others. In this article, we derive the asymptotic distributions of the least squares estimators of the model parameters in PVAR models, allowing the parameters in a given season to satisfy linear constraints. Residual autocorrelations from classical vector autoregressive and moving-average models have been found useful for checking the adequacy of a particular model. In view of this, we obtain the asymptotic distribution of the residual autocovariance matrices in the class of PVAR models, and the asymptotic distribution of the residual autocorrelation matrices is given as a corollary. Portmanteau test statistics designed for diagnosing the adequacy of PVAR models are introduced and we study their asymptotic distributions. The proposed test statistics are illustrated in a small simulation study, and an application with bivariate quarterly West German data is presented.     

Diagnostic checking of nonlinear multivariate time series with multivariate arch errors

Multivariate time series with multivariate ARCH errors have been found useful in many applications. In order to check the adequacy of these models, we define the sum of squared (standardized) residual autocorrelations and derive their asymptotic distribution. The results are used to derive several new multivariate portmanteau tests. Simulation results show that the asymptotic standard errors are quite satisfactory compared with empirical standard errors and that the tests have reasonable empirical size and power. The distribution of the standardized residual autocorrelations is also derived.     

DIAGNOSTIC CHECKING OF PERIODIC AUTOREGRESSION MODELS WITH APPLICATION

Abstract. An overview of model building with periodic autoregression (PAR) models is given emphasizing the three stages of model development:identification, estimation and diagnostic checking. New results on the distribution of residual autocorrelations and suitable diagnostic checks are derived. The validity of these checks is demonstrated by simulation. The methodology discussed is illustrated with an application. It is pointed out that the PAR approach to model development offers some important advantages over the more general approach using periodic autoregressive moving-average models.     

SOME ASPECTS OF THE PERFORMANCE OF DIAGNOSTIC CHECKS IN BIVARIATE TIME SERIES MODELS

Abstract. This paper is concerned with the use of score, or Lagrangian multiplier and portmanteau tests of fitted model adequacy in vector autoregressive-moving average processes. The relation between these alternative diagnostic checking devices is discussed from an asymptotic theoretic standpoint. Some finite sample properties of the tests are investigated in the context of bivariate models using Monte Carlo methods. Asymptotic theory is used to help determine the simulation design and also proves useful in appraising the experimental outcomes. The results provide evidence on the likely relative performance of the two procedures in practice and suggest that the score test is to be preferred.     

A complete VARMA modelling methodology based on scalar components

Abstract. This article proposes an extension to scalar component methodology for the identification and estimation of VARMA models. The complete methodology determines the exact positions of all free parameters in any VARMA model with a predetermined embedded scalar component structure. This leads to an exactly identified system of equations that is estimated using full information maximum likelihood.     

Testing Non‐Correlation and Non‐Causality between Multivariate ARMA Time Series

Abstract. Haugh [Journal of the American Statistical Association (1976) Vol. 71, pp. 378–85] developed an approach to the problem of testing non‐correlation (at all leads and lags) between two univariate time series. Haugh's tests however have low power against two series which are related over a long distributed lag when individual lag coefficients are relatively small. As a remedy, Koch and Yang [Journal of the American Statistical Association (1986) Vol. 8, pp. 533–44] proposed an alternative method that performs better than Haugh's under such dependencies. A multivariate extension of Haugh's procedure was proposed by El Himdi and Roy [The Canadian Journal of Statistics (1997) Vol. 25, pp. 233–56], but suffers the same weaknesses as the original univariate method. We develop here an asymptotic test generalizing Koch and Yang's method to the multivariate case. Our method includes El Himdi and Roy's as a special case. Based on the same idea, we also suggest a generalization of the El Himdi and Roy procedure for testing causality in the sense of Granger [Econometrica (1969) Vol. 37, pp. 424–38] between two multivariate series. A Monte Carlo study is conducted, which indicates that our approach performs better than El Himdi and Roy's for a wide range of models. Both procedures are applied to the problem of testing the absence of correlation between Canadian and US economic indicators, and to a brief study of causality between money and income in Canada.     

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 Improvement of the Portmanteau Statistic

Abstract. The portmanteau statistic is based on the first m‐residual autocorrelations, and is used for diagnostic checks on the adequacy of fit of a model. In this article, we propose a modified portmanteau statistic with a correction term that allows for the use of small values of m for the chi‐squared test. For this modification, we take a different approach to that suggested by Ljung [Biometrika (1986), Vol. 73, pp. 725–30]. Their empirical behaviour is clarified using asymptotic theory.     

Time-varying autoregressions with model order uncertainty

We explore some aspects of the analysis of latent component structure in non-stationary time series based on time-varying autoregressive (TVAR) models that incorporate uncertainty on model order. Our modelling approach assumes that the AR coefficients evolve in time according to a random walk and that the model order may also change in time following a discrete random walk. In addition, we use a conjugate prior structure on the autoregressive coefficients and a discrete uniform prior on model order. Simulation from the posterior distribution of the model parameters can be obtained via standard forward filtering backward simulation algorithms. Aspects of implementation and inference on decompositions, latent structure and model order are discussed for a synthetic series and for an electroencephalogram (EEG) trace previously analysed using fixed order TVAR models.     

Score statistics for testing serial dependence in count data

In this study, we extend earlier work of Freeland (1998) and Jung and Tremayne (2003) , and develop a general formula for a score statistic to test for dependence in an integer autoregressive process with an arbitrary arrivals distribution. We give two statistics that cater for arrivals processes that may be under‐, equi‐ or overdispersed. The first is based on the Katz family which includes Poisson, binomial and negative binomial distributions as special cases. The second uses the generalized Poisson which includes the Poisson distribution as a special case and can also cater for under‐ and over‐ dispersion. The null distribution of the tests is provided and consistency is discussed. Size and power properties are investigated under different model assumptions by Monte Carlo simulations. The autocorrelation coefficient is also investigated as a benchmark for comparison.