Second-order moments of frequency asymmetric cycles

Second-order moments, as even functions in time, are conventionally regarded as containing no information about the time irreversible nature of a sequence and therefore about its frequency asymmetry. However, this paper shows that the frequency asymmetry produces a clearly distinct behaviour in second-order moments that can be observed in both the time domain and the frequency domain. In addition, a frequency domain method of estimation of the differing lengths of the recessionary and expansionary stages of a cycle is proposed and its finite sample performance evaluated. Finally, the asymmetric patterns in the waves of the US unemployment rate and in the sunspot index are analysed.