This paper investigates the maximum likelihood estimator (MLE) for unstable autoregressive moving-average (ARMA) time series with the noise sequence satisfying a general autoregressive heteroscedastic (GARCH) process. Under some mild conditions, it is shown that the MLE satisfying the likelihood equation exists and is consistent. The limiting distribution of the MLE is derived in a unified manner for all types of characteristic roots on or outside the unit circle and is expressed as a functional of stochastic integrals in terms of Brownian motions. For various types of unit roots, the limiting distribution of the MLE does not depend on the parameters in the moving-average component and hence, when the GARCH innovations reduce to usual white noises with a constant conditional variance, they are the same as those for the least squares estimators (LSE) for unstable autoregressive models given by Chan and Wei (1988). In the presence of the GARCH innovations, the limiting distribution will involve a sequence of independent bivariate Brownian motions with correlated components. These results are different from those already known in the literature and, in this case, the MLE of unit roots will be much more efficient than the ordinary least squares estimation.
"Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroscedastic errors." Ann. Statist. 26 (1) 84 - 125, February 1998. https://doi.org/10.1214/aos/1030563979