Trigonometric Trend Regressions of Unknown Frequencies with Stationary or Integrated Noise

We propose a new procedure to select the unknown frequencies of a trigonometric function, a problem Örst investigated by Anderson (1971) under the assumption of serially uncorrelated noise. We extend the analysis to general linear processes without the prior knowledge of a stationary or integrated model allowing the frequencies to be unknown. We provide a consistent model selection procedure. We Örst show that if we estimate a model with fewer frequencies than those in the correct model, the estimates converge to a subset of the frequencies in the correct model. This opens the way to a consistent model selection strategy based on a speciÖc to general procedure that tests whether additional frequencies are needed. This is achieved using tests based on the feasible ìsuper e¢ cientî(under unit root noise) Generalized Least Squares estimator of Perron, Shintani and Yabu (2017) who assumed the frequencies to be known. We show that the limiting distributions of our test statistics are the same for both cases about the noise function. Simulation results conÖrm that our frequency selection procedure works well with sample sizes typically available in practice. We illustrate the usefulness of our method via applications to unemployment rates and global temperature series.