Long-run dynamic correlation of nonstationary variables when the trends are misspecified

We study long-run comovement of the nonstationary time series variables with a focus on the use of coherency, defined as the long-run dynamic correlation. We pay attention to the effect of specification of trends on the long-run correlations by analyzing the cases that the data are either correctly or incorrectly detrended. Our simulation studies show that when the true process is trend stationary, time-removed long-run correlation estimates perform well, whereas the differenced case fails to generate valid outcomes due to degeneracy of the spectrums at the zero frequency of the series. We also provide empirical applications using unemployment rates of major cities in Korea from 1999 to 2016, and exemplify that false detrending could lead to nocuous outcomes. This work brings attention to correct specification of trends in nonstationary economic data in practice.