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Surprisingly little is known regarding the asymptotic properties of estimators for cyclical long memory models such as the Gegenbauer autoregressive moving average (GARMA) model. In this paper, we review the GARMA process and study the properties of several estimation methods with an emphasis on inference related to  η , the parameter governing the length of long memory cycles. We present extensive simulation evidence to show that both Whittle and constrained sum of squares (CSS) estimators yield satisfactory results in terms of mean bias and RMSE for all parameters, although there are serious inferential concerns. Most notably, under the null of an infinitely long cycle ( η  = 1), proposed distributional results for the CSS estimator yield very over-sized tests. For the existing semiparametric estimator with more complete distribution theory, there are unresolved practical issues including complicated bandwidth choices that we address. As an alternative approach for applied researchers wishing to employ a parametric estimator, we propose and validate a parametric bootstrap method using a likelihood ratio test statistic for the hypothesis  η  = 1. We illustrate estimation and inference issues with an application to the US unemployment rate, where evidence of stationary long memory cycles ultimately emerges.

Inference for estimators of generalized long memory processes