Order selection and inference with long memory dependent data

In empirical studies selection of the order of a model is routinely invoked. A common example is the order selection of an autoregressive model via Akaike's AIC, Schwarz's BIC or Hannan and Quinn's HIC. The criteria are based on the conditional sum of squares, CSS. However the computation of the CSS might be difficult for some models such as Bloomfield's exponential model and/or when we allow for long memory dependence. The main aim of the paper is thus to propose an alternative way to compute the criterion by using the decomposition of the variance of the innovation errors in terms of its frequency components. We show its validity to obtain the correct order the model. In addition, as a by-product, we describe a simple (two-step) estimator of the parameters of the model.