On the correlations of trend-cycle errors

This note provides explanations for an unexpected result, namely, the estimated parameter of the correlation coefficient of the trend shock and cycle shock in the state-space model is almost always (positive or negative) unity, even when the true variance of the trend shock is zero. It is shown that the set of the true parameter values lies on the restriction that requires the variance-covariance matrix of the errors to be nonsingular, therefore, almost always the likelihood function has its (constrained) global maximum on the boundary where the correlation coefficient implies perfect correlation.