A Note on the Paper by H.J. Bierens: “Complex Unit Roots and Business Cycles: Are They Real?”

Abstract

This note discusses the concept of aliasing and its use in the paper by H.J. Bierens (2001, Econometric Theory 17, 962–983), in the framework of a second-order autoregression with complex unit roots. The condition on the range of the angular frequency φ is extended to φ ∈ (0,2π) − {π}.Financial support from UPV-EHU research project 9/UPV-00038.321-13503/2001, Basque Government project PI9970 and Ministerio de ciencia y tecnología BEC2003-02028 is gratefully acknowledged.

On the Behaviour of Phillips–Perron Tests in the Presence of Persistent Cycles

“…The polynomial generates a persistent cycle of 2 π / ϕ periods. Indeed, and as noted by Díaz‐Emparanza (), it also generates an aliased cycle of 2 π /(2 π − ϕ ) periods, and as such our analysis in fact covers ϕ ∈ (0,2 π ) − { π }.…”

The long swings in the spot exchange rates and the complex unit roots hypothesis

“…Note that we consider the case suggested by Diaz-Emparanza (2004) where (φ j ) ∈ (0, 2π) − {π} to take into consideration the cycle caused by the root cos (φ j ) + sin (φ j ) and its alias. 7 The generated cycle has periods of 2π/φ j and periods of 2π/(2π − φ j ) for its alias.…”

“…8Diaz-Emparanza (2004) concludes that both cycles have the same cyclical behavior if the data generating process is discrete but it is impossible to distinguish them from their alias.…”

Cyclical and Persistent Carry Trade Returns and Forward Premia

“…; n. So that k corresponds to possible cycle period. 4 To take into account that returns would display multiple peaks in the cycle frequencies, we follow Bierens (2001) and Diaz-Emparanza (2004) by considering testing the null hypothesis that the process under consideration, rx t , is complex unit root process of a form…”

“…(0, I) with Eðj" j;t j 2þ Þ < 0 for some > 0. Note that we consider the case suggested by Diaz-Emparanza (2004), where ð j Þ 2 ð0; 2Þ À fg to take into account the cycle generated by the root cosð j Þ þ sinð j Þ and its alias. 5 The generated cycle has periods of 2= j and corresponding periods of 2=ð2 À j Þ for its alias.…”