Spectral Maximum Likelihood Estimation Of A Signal‐To‐Noise Ratio Lying In The Vicinity Of Zero

. A time series model representing a decomposition into permanent plus transient components contains a deterministic component when the signal‐to‐noise ratio is equal to zero; otherwise, the permanent component is said to be stochastic. This distinction has important consequences in the analysis of economic phenomena. On the other hand, the absence of a stochastic permanent component in residuals from a time series regression may indicate cointegration. This paper considers the frequency domain estimation of the signal‐to‐noise ratio in a representative of the unobserved components model class. The sampling properties of the estimator from the resulting approximate spectral likelihood differ from those observed in the time domain and they vary substantially depending on whether the overall slope must be estimated or not. Further, it is shown that spectral estimates are T‐consistent—instead of T2‐consistent in the time domain. These results may explain some of the differences in estimators from frequency domain approximations to the likelihood and exact maximum likelihood estimators, and may be of use when testing for deterministic trends.