Roch Roy

Diagnostic Checking in ARMA Models With Uncorrelated Errors

We consider tests for lack of fit in ARMA models with nonindependent innovations. In this framework, the standard Box–Pierce and Ljung–Box portmanteau tests can perform poorly. Specifically, the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration. In this article we derive the asymptotic covariance matrix of a vector of autocorrelations for residuals of ARMA models under weak assumptions on the noise. The asymptotic distribution of the portmanteau statistics follows. A consistent estimator of , and a modification of the portmanteau tests are proposed. This allows us to construct valid asymptotic significance limits for the residual autocorrelations, and (asymptotically) valid goodness-of-fit tests, when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference. A set of Monte Carlo experiments, and an application to the Standard & Poor 500 returns, illustrate the practical relevance of our theoretical results.

Testing Causality between Two Vectors in Multivariate Autoregressive Moving Average Models

Abstract In the analysis of economic time series, a question often raised is whether a vector of variables causes another one in the sense of Granger. Most of the literature on this topic is concerned with bivariate relationships or uses finite-order autoregressive specifications. The purpose of this article is to develop a causality analysis in the sense of Granger for general vector autoregressive moving average (ARMA) models. We give a definition of Granger noncausality between vectors, which is a natural and simple extension of the notion of Granger noncausality between two variables. In our context, this definition is shown to be equivalent to a more complex definition proposed by Tjostheim. For the class of linear invertible processes, we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis. This result is then specialized to the class of stationary invertible ARMA process...

On the application of integer-valued time series models for the analysis of disease incidence

Statistical time series models are practical tools in public health surveillance. Their capacity to forecast future disease incidence values exemplifies their usefulness. Using these forecasts, one can develop strategies to trigger alerts to public health officials when irregular disease incidence values have occurred. Clearly, the better the forecasting performance of the model class used in the time series analysis, the more realistic are the alerts triggered. The time series analysis of disease incidence values has often entailed the Box and Jenkins model class. However, this class was designed to model real-valued variables whereas disease incidences are integer-valued variables. A new class of time series models, called integer-valued autoregressive models, has been developed and studied over the past decade. The objective of this paper is to introduce this new class of models to the application of time series analysis of infectious disease incidence, and to demonstrate its advantages over the class of real-valued Box and Jenkins models. The paper also presents a bootstrap method developed for the calculation of forecast interval limits.

Simplified conditions for noncausality between vectors in multivariate ARMA models

This article derives necessary and sufficient conditions for noncausality between two vectors of variables in stationary invertible ARMA processes. Earlier conditions proposed by Boudjellaba, Dufour, and Roy (1992a) are shown to hold under weaker regularity assumptions and then generalized to cover the important case where the two vectors do not necessarily embody all the variables considered in the analysis. The conditions so obtained can be considerably simpler and easier to implement than earlier ones. Testing of the conditions derived is also discussed and the results are applied to a model of Canadian money, income, and interest rates.

Stochastic modeling of empirical time series of childhood infectious diseases data before and after mass vaccination

The goal of this paper is to analyze the stochastic dynamics of childhood infectious disease time series. We present an univariate time series analysis of pertussis, mumps, measles and rubella based on Box-Jenkins or AutoRegressive Integrated Moving Average (ARIMA) modeling. The method, which enables the dependency structure embedded in time series data to be modeled, has potential research applications in studies of infectious disease dynamics. Canadian chronological series of pertussis, mumps, measles and rubella, before and after mass vaccination, are analyzed to characterize the statistical structure of these diseases. Despite the fact that these infectious diseases are biologically different, it is found that they are all represented by simple models with the same basic statistical structure. Aside from seasonal effects, the number of new cases is given by the incidence in the previous period and by periodically recurrent random factors. It is also shown that mass vaccination does not change this stochastic dependency. We conclude that the Box-Jenkins methodology does identify the collective pattern of the dynamics, but not the specifics of the diseases at the biological individual level.

Examination of biomechanical principles in a patient handling task

Handling patients in bed using a pique (a waterproof padded sheet placed under the patient) with, in particular, the activity of pulling and turning the patient, is associated with a high incidence of risks for the spine. Six female subjects, not experienced with the task, were evaluated for spinal loadings at the L5/S1 joint, for selected muscular activities in the trunk and shoulders and for work-energy factors. Films, force platforms and EMG recordings supplied the data; dynamic segmental analyses were performed to calculate reaction forces at L5/S1, and a planar single-muscle equivalent was used to estimate internal loads. Three treatments were administered which allowed comparisons to be made for two hand grip positions on the pique (close to the patient vs. a 15 cm distance) and two movement patterns (continuous vs. interrupted with a pause). It was hypothesized that moving the hand grips away from the patient would favour a straighter-back position and a reduction of spinal loadings; it was also hypothesized that non-interrupted motions involving changes of direction of efforts would be more strenuous for the spine. Analyses of variance with repeated measures were conducted and the locations of significant differences were made with Scheffe method of multiple comparisons. Conflicting results were obtained for the hand grip positions but the results suggest that the partition of a task into several operations (with pauses) is indicated. Recommendations are made to examine more thoroughly trunk postures or back curvatures in relation to spinal loadings.

On consistent testing for serial correlation of unknown form in vector time series models

Multivariate autoregressive models with exogenous variables (VARX) are often used in econometric applications. Many properties of the basic statistics for this class of models rely on the assumption of independent errors. Using results of Hong (Econometrica 64 (1996) 837), we propose a new test statistic for checking the hypothesis of non-correlation or independence in the Gaussian case. The test statistic is obtained by comparing the spectral density of the errors under the null hypothesis of independence with a kernel-based spectral density estimator. The asymptotic distribution of the statistic is derived under the null hypothesis. This test generalizes the portmanteau test of Hosking (J. Amer. Statist. Assoc. 75 (1980) 602). The consistency of the test is established for a general class of static regression models with autocorrelated errors. Its asymptotic slope is derived and the asymptotic relative efficiency within the class of possible kernels is also investigated. Finally, the level and power of the resulting tests are also studied by simulation.

On the identification of ARMA echelon-form models*

An identification procedure for multivariate autoregressive moving average (ARMA) echelonform models is proposed. It is based on the study of the linear dependence between rows of the Hankel matrix of serial correlations. To that end, we define a statistical test for checking the linear dependence between vectors of serial correlations. It is shown that the test statistic TN considered is distributed asymptotically as a finite linear combination of independent chi-square random variables with one degree of freedom under the null hypothesis, whereas under the alternative hypothesis, TN/N converges in probability to a positive constant. These results allow us, in particular, to compute the asymptotic probability of making a specification error with the proposed procedure. Links to other methods based on the application of canonical analysis are discussed. A simulation experiment was done in order to study the performance of the procedure. It is seen that the graphical representation of TN, as a function of N, can be very useful in identifying the dynamic structure of ARMA models. Furthermore, for the model considered, the proposed identification procedure performs very well for series of 100 observations or more and reasonably well with short series of 50 observations.

On long term air quality trends and intervention analysis

Abstract In this project, we apply the model of intervention analysis by Box and Tiao (1975) to quantify the impact of the regulation of sulphur content of heating and industrial oil (by law 9 of the Montreal Urban Community adopted in 1970) on the evolution of sulphur dioxide concentrations over the past several years. To this end, we have analysed the series of monthly average concentrations of sulphur dioxide observed at 11 monitoring sites in the Montreal area for the period from January 1968 to December 1977. We describe a time series model which (with appropriate parameters) is valid for each site and we conclude with a comparative analysis of the effects of the by-law at each site. The model allows us to conclude that the regulation has had a significant impact on the SO 2 concentrations at 10 of the 11 locations considered. In particular for the winter period, the reduction attributable to the regulation varies from about 1 to 8 pphm and leads to a geographical grouping of the stations.

Tests for Non-Correlation of Two Cointegrated Arma Time Series

In multivariate time series modelling, we are often led to investigate the existence of a relationship between two time series. Here, we generalize the procedure proposed by Haugh (1976) and extended by El Himdi and Roy (1997) for multivariate stationary ARMA time series to the case of cointegrated (or partially nonstationary) ARMA series. The main contribution consists in showing that, in the case of two uncorrelated cointegrated time series, an arbitrary vector of residual cross-correlation matrices asymptotically follows the same distribution as the corresponding vector of cross-correlation matrices between the two innovation series. The estimation method from which the residuals are obtained can be the conditional maximum likelihood method as discussed in Yap and Reinsel (1995) or some other which has the same convergence rate. From this result, it follows that the considered test statistics, which are based on residual cross-correlation matrices, asymptotically follow χ-super-2 distributions. The finite sample properties, under the null hypothesis, of the test statistics are studied by simulation. Copyright 2003 Blackwell Publishing Ltd.