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

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  • Jean-Marie Dufour

    ()

  • Tarek Jouini

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).

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Bibliographic Info

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

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Date of creation: 01 Feb 2011
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Handle: RePEc:cir:cirwor:2011s-25

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Keywords: echelon form; linear estimation; generalized least squares; GLS; two-step linear estimation; three-step linear estimation; asymptotically efficient; maximum likelihood; ML; stationary process; invertible process; Kronecker indices; simulation;

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  1. Tsay, Ruey S, 1989. "Parsimonious Parameterization of Vector Autoregressive Moving Average Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(3), pages 327-41, July.
  2. Christian Kascha, 2007. "A Comparison of Estimation Methods for Vector Autoregressive Moving-Average Models," Economics Working Papers ECO2007/12, European University Institute.
  3. Lutkepohl, Helmut, 2006. "Forecasting with VARMA Models," Handbook of Economic Forecasting, Elsevier.
  4. Maravall, Agustin, 1993. "Stochastic linear trends : Models and estimators," Journal of Econometrics, Elsevier, vol. 56(1-2), pages 5-37, March.
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  8. Poskitt, D. S., 2003. "On the specification of cointegrated autoregressive moving-average forecasting systems," International Journal of Forecasting, Elsevier, vol. 19(3), pages 503-519.
  9. Lutkepohl, Helmut & Claessen, Holger, 1997. "Analysis of cointegrated VARMA processes," Journal of Econometrics, Elsevier, vol. 80(2), pages 223-239, October.
  10. Jean-Marie Dufour & Tarek Jouini, 2005. "Asymptotic distribution of a simple linear estimator for VARMA models in echelon form," CIRANO Working Papers 2005s-06, CIRANO.
  11. Kapetanios, George, 2003. "A note on an iterative least-squares estimation method for ARMA and VARMA models," Economics Letters, Elsevier, vol. 79(3), pages 305-312, June.
  12. 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.
  13. Gallego, Jose L., 2009. "The exact likelihood function of a vector autoregressive moving average process," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 711-714, March.
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  15. 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.
  16. 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.
  17. 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.
  18. Konstantinos Metaxoglou & Aaron Smith, 2007. "Maximum Likelihood Estimation of VARMA Models Using a State-Space EM Algorithm," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(5), pages 666-685, 09.
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  20. D. Poskitt & H. Lütkepohl, 1995. "Consistent Specification of Cointegrated Autoregressive Moving-Average Systems," SFB 373 Discussion Papers 1995,54, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  21. Izquierdo, Segismundo S. & Hernández, Cesáreo & del Hoyo, Juan, 2006. "Forecasting VARMA processes using VAR models and subspace-based state space models," MPRA Paper 4235, University Library of Munich, Germany.
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  23. repec:wop:humbsf:1995-54 is not listed on IDEAS
  24. D.S. Poskitt, . "Specification of echelon form VARMA models," Statistic und Oekonometrie 9305, Humboldt Universitaet Berlin.
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