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Some Exact Formulae for Autoregressive Moving Average Processes

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  • Zinde-Walsh, Victoria

Abstract

This paper demonstrates that for a finite stationary autoregressive moving average process the inverse of the covariance matrix differs from the matrix of the covariances of the inverse process by a matrix of low rank. The formula for the exact inverse of the covariance matrix of the scalar or multivariate process is provided. We obtain approximations based on this formula and evaluate some of the approximate results in the existing literature. Applications to computational algorithms and to the distributions of two-step estimators are discussed. In addition the paper contains the formula for the determinant of the covariance matrix which is useful in exact maximum likelihood estimation; it also lists the expressions for the autocovariances of scalar autoregressive moving average processes.

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  • Zinde-Walsh, Victoria, 1988. "Some Exact Formulae for Autoregressive Moving Average Processes," Econometric Theory, Cambridge University Press, vol. 4(3), pages 384-402, December.
    • Handle: RePEc:cup:etheor:v:4:y:1988:i:03:p:384-402_01
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    Cited by:

    1. vdr Leeuw, J.L., 1997. "Maximum Likelihood Estimation of Exact ARMA Models," Other publications TiSEM a1cdd9b8-93d9-460c-a0c9-1, Tilburg University, School of Economics and Management.
    2. David F. Findley & Demetra P. Lytras & Agustin Maravall, 2016. "Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 7(1), pages 11-52, March.
    3. Choudhury, Askar H. & Power, Simon, 1995. "A new approximate GLS estimator for the linear regression model with ARMA(p, q) disturbances," Economics Letters, Elsevier, vol. 48(2), pages 119-127, May.
    4. Paulina Granados Z., 2004. "Income Function of Chilean Households: Life Cicle and Persistence of Shocks," Working Papers Central Bank of Chile 257, Central Bank of Chile.
    5. McLeod, A. Ian & Yu, Hao & Krougly, Zinovi L., 2007. "Algorithms for Linear Time Series Analysis: With R Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 23(i05).
    6. Galbraith, JohnW. & Zinde-Walsh, Victoria, 1999. "On the distributions of Augmented Dickey-Fuller statistics in processes with moving average components," Journal of Econometrics, Elsevier, vol. 93(1), pages 25-47, November.
    7. McLeod, A.I. & Zhang, Y., 2008. "Faster ARMA maximum likelihood estimation," Computational Statistics & Data Analysis, Elsevier, vol. 52(4), pages 2166-2176, January.
    8. Wilfredo Palma & Mauricio Zevallos, 2004. "Analysis of the correlation structure of square time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(4), pages 529-550, July.
    9. Paulina Granados Z., 2004. "Chilean Household Income Function: Life Cycle and Persistence of Shocks," Journal Economía Chilena (The Chilean Economy), Central Bank of Chile, vol. 7(1), pages 51-89, April.
    10. Peter C.B. Phillips, 1990. "Operational Algebra and Regression t-Tests," Cowles Foundation Discussion Papers 948, Cowles Foundation for Research in Economics, Yale University.

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