Consistent estimation for non-Gaussian non-causal autoregressive processes
The estimation of possibly non-causal order-\(p\) autoregressive (AR) models NEWLINE\[NEWLINEx_{t}-\sum_{j=1}^{p}\phi_{j}x_{t-j}=u_{t}NEWLINE\]NEWLINE is considered, where the \(u_{t}\) are independent and identically distributed r.v.s with common distribution function \(F.\) It is assumed that \(\phi(z)=1-\sum_{j=1}^{p}\phi_{j}z^{j}\) has no roots on the unit circle. Traditional estimation based on least squares or Gaussian likelihood cannot distinguish between causal and non-causal representations of a stationary autoregressive process. Because of second-order equivalence it is not possible to identify a non-causal Gaussian process. Furthermore, one cannot use the Gaussian likelihood (or the related least squares procedure) to estimate the parameters of a non-causal process consistently. \textit{F.J. Breidt, R.A. Davis, K.-S. Lii} and \textit{M. Rosenblatt} [J. Multivariate Anal. 36, No.2, 175-198 (1991; Zbl 0711.62072)] showed that there is a consistent likelihood estimation of possibly non-causal AR processes. However, this is only an existence result which is not very useful since the likelihood function in this problem generally exhibits multiple maxima.NEWLINENEWLINENEWLINEThis paper shows by constructive proofs that a modified \(L_{1}\) estimate is consistent if the innovation process has a stable law distribution with exponent \(1<\alpha <2.\) On the basis of simulation results it is conjectured that the result also holds for \(\alpha\leq 1.\) It is also shown that neither non-Gaussianity nor infinite variance is sufficient to ensure consistency.
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