Practical methods for modelling weak VARMA processes: identification, estimation and specification with a macroeconomic application

North Carolina State University; Department of Economics, McGill University, H3A 2T7, Montréal, Québec, Canada; Department of Economics, North Carolina State University, 27695-8110, Raleigh, NC, USA

ABSTRACT program on Mathematics of Information Technology and Complex Systems (MITACS)], the Canada Council for the Arts (Killam Fellowship), the CIREQ, the CIRANO, and the Fonds FCAR (Government of Québec). William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse des organisations (CIRANO), and Centre interuniversitaire de recherche en économie quantitative (CIREQ). Mailing address: ABSTRACT In this paper, we develop practical methods for modelling weak VARMA processes. In a first part, we propose new identified VARMA representations, the diagonal MA equation form and the final MA equation form, where the MA operator is diagonal and scalar respectively. Both of these representations have the important feature that they constitute relatively simple modifications of a VAR model (in contrast with the echelon representation). In a second part, we study the problem of estimating VARMA models by relatively simple methods which only require linear regressions. We consider a generalization of the regression-based estimation method proposed by Hannan and Rissanen (1982). The asymptotic properties of the estimator are derived under weak hypotheses on the innovations (uncorrelated and strong mixing) so as to broaden the class of models to which it can be applied. In a third part, we present a modified information criterion which gives consistent estimates of the orders under the proposed representations. To demonstrate the importance of using VARMA models to study multivariate time series we compare the impulse-response functions and the out-of-sample forecasts generated by VARMA and VAR models.

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    ABSTRACT: We study two linear estimators for stationary invertible VARMA models in echelon form – to achieve identification (model parameter unicity) – with known Kronecker indices. Such linear estimators are much simpler to compute than Gaussian maximum-likelihood estimators often proposed for such models, which require highly nonlinear optimization. The first estimator is an improved two-step estimator which can be interpreted as a generalized-least-squares extension of the two-step least-squares estimator studied in Dufour and Jouini (2005). The setup considered is also more general and allows for the presence of drift parameters. The second estimator is a new relatively simple three-step linear estimator which is asymptotically equivalent to ML, hence asymptotically efficient, when the innovations of the process are Gaussian. The latter is based on using modified approximate residuals which better take into account the truncation error associated with the approximate long autoregression used in the first step of the method. We show that both estimators are consistent and asymptotically normal under the assumption that the innovations are a strong white noise, possibly non-Gaussian. Explicit formulae for the asymptotic covariance matrices are provided. The proposed estimators are computationally simpler than earlier "efficient" estimators, and the distributional theory we supply does not rely on a Gaussian assumption, in contrast with Gaussian maximum likelihood or the estimators considered by Hannan and Kavalieris (1984b) and Reinsel, Basu and Yap (1992). We present simulation evidence which indicates that the proposed three-step estimator typically performs better in finite samples than the alternative multi-step linear estimators suggested by Hannan and Kavalieris (1984b), Reinsel et al. (1992), and Poskitt and Salau (1995).
    Computational Statistics & Data Analysis 03/2011; · 1.15 Impact Factor
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    ABSTRACT: We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. The structural forms are mainly used in econometrics to introduce instantaneous relationships between economic variables. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the G. M. Ljung and G. E. P. Box [Biometrika 65, 297–303 (1978; Zbl 0386.62079)] (or G. E. P. Box and D. A. Pierce, J. Am. Stat. Assoc. 65, 1509–1526 (1970; Zbl 0224.62041)) portmanteau statistics in this framework. It is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte Carlo experiments illustrate the finite sample performance of the modified portmanteau test.
    Journal of Statistical Planning and Inference 08/2011; 8(8). · 0.60 Impact Factor

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