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Asymptotic Distributions for Some Quasi-Efficient Estimators in Echelon VARMA Models

  • Jean-Marie Dufour
  • Tarek Jouini

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).

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2011s-25.

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Length: 71 pages
Date of creation: 01 Feb 2011
Date of revision:
Handle: RePEc:cir:cirwor:2011s-25
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  14. Bartel, Holger & Lütkepohl, Helmut, 1997. "Estimating the Kronecker indices of cointegrated echelon form VARMA models," SFB 373 Discussion Papers 1997,2, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  15. L. Kavalieris & E. J. Hannan & M. Salau, 2003. "Generalized Least Squares Estimation Of Arma Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(2), pages 165-172, 03.
  16. DUFOUR, Jean-Marie & JOUINI, Tarek, 2005. "Asymptotic Distribution of a Simple Linear Estimator for VARMA Models in Echelon Form," Cahiers de recherche 10-2005, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  17. George Athanasopoulos & Farshid Vahid, 2006. "VARMA versus VAR for Macroeconomic Forecasting," Monash Econometrics and Business Statistics Working Papers 4/06, Monash University, Department of Econometrics and Business Statistics.
  18. Kohn, R., 1981. "A note on an alternative derivation of the likelihood of an autoregressive moving average process," Economics Letters, Elsevier, vol. 7(3), pages 233-236.
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  22. Mauricio, Jose Alberto, 2006. "Exact maximum likelihood estimation of partially nonstationary vector ARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 50(12), pages 3644-3662, August.
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