Inference for likelihood-based estimators of generalized long-memory processes

Despite a recent proliferation of research using cyclical long memory, surprisingly little is known regarding the asymptotic properties of likelihood-based methods. Estimators have been studied in both the time and frequency domains for the Gegenbauer autoregressive moving average process (GARMA). However, a full set of asymptotic results for all parameters has only been proposed by Chung (1996a,b), who present somewhat tenuous results without an initial consistency proof. In this paper, we review the GARMA process and the properties of frequency and time domain likelihood-based estimators using Monte Carlo analysis. The results demonstrate the strong efficacy of both estimators and generally sup- port the proposed theory of Chung for the parameter governing the cycle length. Important caveats await. The results show that asymptotic confidence bands can be unreliable in very small samples under weak long memory, and the distribution theory under the null of an infinitely long cycle appears to be unusable. Possible solutions are proposed, including the use of narrower confidence bands and the application of theory under the alternative of finite cycles.