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
because of the interest in obtaining adequate and reliable
predictors, Bhansali (1974, Journal of the Royal Statistical
Society, Series B 36, 61–73) examined the properties
of a nonparametric predictor based on the canonical factorization
of the spectral density function given in Whittle (1963,
Prediction and Regulation by Linear Least Squares)
and known as FLES.However, the preceding work does not cover the so-called strongly
dependent data. Because of the interest in this type of processes,
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.
Second order expansions of estimators in nonparametric moment conditions models with weakly dependent data
