© 1994 by Biometrika Trust
Articles |
A fast algorithm for the exact likelihood of stationary and partially nonstationary vector autoregressive-moving average processes
E.T.S. de Ingenieros de Caminos, University of Cantabria 39005 Santander, Spain
Received for publication 1 July 1993.
Revision received 1 February 1994.
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Abstract |
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An expression for the likelihood function of a stationary vector autoregressive-moving average process is developed. The expression is very efficient numerically and applies to any stationary but not necessarily invertible model. In particular, when the multivariate process is autoregressive, the exact likelihood can be evaluated with a small number of operations depending on the order of the autoregressive operator and the process dimension, but not on the size of the observed series. The expression also provides an efficient method for the evaluation of the exact likelihood of a partially nonstationary vector autoregressive-moving average process, for which the determinant of the autoregressive operator has at least one unit root and the remaining roots are outside the unit circle. This method does not require differencing the series, so that complications caused by over-differencing the series, such as noninvertibility and parameter identifiability problems, are avoided. The results for autoregressive models are also applied to testing the stationarity and invertibility of any autoregressive-moving average model with given parameter values.
Key Words: Co-integration • Error correction model • Noninvertible model • Over-differencing • Parameter identifiability • Time series estimation