Asymptotic behavior of cross spectral density estimator at the zero frequency in the presence of degeneracy

We analyze the asymptotic properties of nonparametric cross spectral density estimators when the time series process has a degenerate spectrum at the origin in the frequency domain. Degeneracy at the zero frequency caused by over-differencing the series invalidates inferences based on cross spectral methods. We derive the asymptotic mean squared errors of kernel-based cross spectral density estimators at the zero frequency. It is found that degeneracy entails different asymptotics including convergence rates for nonparametric cross spectral density estimators from existing results in time series econometrics literature.