Consistent Estimation for Non‐Gaussian Non‐Causal Autoregessive Processes

Traditional estimation based on least squares or Gaussian likelihood cannot distinguish between causal and non-causal representation of a stationary autoregressive (AR) process. Breidt et al. (Maximum likelihood estimation for non-causal autoregressive processes. J. Multivariate Anal. 36 (1991), 175–98) proved the existence of a consistent likelihood estimation of possibly non-causal AR processes; however, in this case an existence result is not very useful since the likelihood function generally exhibits multiple maxima. Moreover the method assumes full knowledge of the distribution of the innovation process. This paper shows a constructive proof that a modified L1 estimate is consistent if the innovation process has a stable law distribution with index α∈ (1, 2). It is also shown that neither non-Gaussianity nor infinite variance is sufficient to ensure consistency.