Asymptotic Distributions for Some Quasi-Efficient Estimators in Echelon VARMA Models
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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- Hannan, E. J., 1981. "Estimating the dimension of a linear system," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 459-473, December.
- Maravall, Agustin, 1993. "Stochastic linear trends : Models and estimators," Journal of Econometrics, Elsevier, vol. 56(1-2), pages 5-37, March.
- Lutkepohl, Helmut, 2006.
"Forecasting with VARMA Models,"
Handbook of Economic Forecasting,
- 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.
- D.S. Poskitt, .
"Specification of echelon form VARMA models,"
Statistic und Oekonometrie
9305, Humboldt Universitaet Berlin.
- Jean-Marie Dufour & Tarek Jouini, 2005.
"Asymptotic distribution of a simple linear estimator for VARMA models in echelon form,"
CIRANO Working Papers
- DUFOUR, Jean-Marie & TAREK, Jouini, 2005. "Asymptotic Distribution of a Simple Linear Estimator for VARMA Models in Echelon Form," Cahiers de recherche 2005-09, Universite de Montreal, Departement de sciences economiques.
- 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.
- 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.
- Athanasopoulos, George & Vahid, Farshid, 2008.
"VARMA versus VAR for Macroeconomic Forecasting,"
Journal of Business & Economic Statistics,
American Statistical Association, vol. 26, pages 237-252, April.
- 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.
- Holger Bartel & Helmut Lutkepohl, 1998.
"Estimating the Kronecker indices of cointegrated echelon-form VARMA models,"
Royal Economic Society, vol. 1(Conferenc), pages C76-C99.
- 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.
- 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.
- George Athanasopoulos & Farshid Vahid, 2006.
"A Complete VARMA Modelling Methodology Based on Scalar Components,"
Monash Econometrics and Business Statistics Working Papers
2/06, Monash University, Department of Econometrics and Business Statistics.
- George Athanasopoulos & Farshid Vahid, 2008. "A complete VARMA modelling methodology based on scalar components," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(3), pages 533-554, 05.
- 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.
- Christian Kascha, 2007. "A Comparison of Estimation Methods for Vector Autoregressive Moving-Average Models," Economics Working Papers ECO2007/12, European University Institute.
- 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.
- 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.
- 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.
- Christian Kascha & Karel Mertens, 2006.
"Business Cycle Analysis and VARMA models,"
Economics Working Papers
ECO2006/37, European University Institute.
- 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.
- 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.
- Lutkepohl, Helmut & Claessen, Holger, 1997. "Analysis of cointegrated VARMA processes," Journal of Econometrics, Elsevier, vol. 80(2), pages 223-239, October.
- Deistler, M. & Hannan, E. J., 1981. "Some properties of the parameterization of ARMA systems with unknown order," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 474-484, December.
- Solo, Victor, 1984. "The exact likelihood for a multivariate ARMA model," Journal of Multivariate Analysis, Elsevier, vol. 15(2), pages 164-173, October.
- 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.
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