Dual estimation of the poles and zeros of an ARMA(p,q) process

Identification of the p poles and q zeros of an autoregressive mov- ing average process ARMA(p,q) is considered. The method described departs from approaches frequently reported in the litterature in two main respects. First, it substitutes the sample covariance lags by the sequence of estimated reflection coef- ficients. Second, it provides a direct estimation procedure for the MA component, which is not contingent upon prior identification of the AR structure. In distinc- tion to current practice, both tasks are directly addressed, avoiding that errors in one contaminate the other. The algorithm explores the linear dependencies between corresponding coefficients of successively higher order linear filters fitted to the time series: linear predictors are used for the estimation of the MA component and linear innovation filters for the identification of the AR part. The overdimensioned system of linear equations derived from these dependencies provides statistical stability to the procedure. The paper establishes these dependencies and derives from them a recursive algorithm for ARMA identification. The recursiveness is on the number of (sample) reflection coefficients used. As it turns out, the MA procedure is asymp- totic in nature, the rate of convergence being established in terms of the second power of the zeros of the process. Simulation examples show the behavior of the algorithm, illustrating how peaks and valleys of the power spectrum are resolved. The quality of the estimates is established in terms of the bias and mean square error, whose leading terms are shown to be of order T-', where T is the data lenght. 1 The work of the first author was partially supported by INIC (Portugal). The work of the second author was