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Keywords:

  • ARMA process;
  • bootstrap;
  • infinite variance;
  • M-estimation;
  • non-causality;
  • non-invertibility;
  • stable distribution;
  • time series

Abstract.  General autoregressive moving average (ARMA) models extend the traditional ARMA models by removing the assumptions of causality and invertibility. The assumptions are not required under a non-Gaussian setting for the identifiability of the model parameters in contrast to the Gaussian setting. We study M-estimation for general ARMA processes with infinite variance, where the distribution of innovations is in the domain of attraction of a non-Gaussian stable law. Following the approach taken by Davis et al. (1992) and Davis (1996), we derive a functional limit theorem for random processes based on the objective function, and establish asymptotic properties of the M-estimator. We also consider bootstrapping the M-estimator and extend the results of Davis & Wu (1997) to the present setting so that statistical inferences are readily implemented. Simulation studies are conducted to evaluate the finite sample performance of the M-estimation and bootstrap procedures. An empirical example of financial time series is also provided.