Prediction and Signal Extraction of Strong Dependent Processess in the Frequency Domain

We frequently observe that one of the aims of time series analysts is to predict future values of the data. For weakly dependent data, when the model is known up to a finite set of parameters, its statistical properties are well documented and exhaustively examined. However, if the model was misspecified, the predictors would no longer be correct. Motivated by this observation and due to the interest in obtaining adequate and reliable predictors, Bhansali (1974) examined the properties of a nonparametric predictor based on the canonical factorization of the spectral density function given in Whittle (1963) and known as FLES. However, the above work does not cover the so-called strongly dependent data. Due to the interest in this type of process, one of our objectives in this paper is to examine the properties of the FLES for these processes. In addition, we illustrate how the FLES can be adapted to recover the signal of a strongly dependent process, showing its consistency. The proposed method is semiparametric, in the sense that, in contrast to other methods, we do not need to assume any particular model for the noise except that it is weakly dependent.