We suggest an explicit data-driven consistent estimator of the optimal smooth trend in a multivariate Hodrick-Prescott filter, when the associated disturbances (i.e., signal and cycle components) follow a moving average, and a vector autoregressive process, respectively. This is done through deriving consistent estimators of the covariance matrices of the signal and the cycle components. We then fit some macroeconomic data to compare the performances of the associated smooth trend and business cycle with the ones obtained using the estimators of the univariate Hodrick-Prescott filter with auto-correlated disturbances.

adaptive estimation, Gaussian process, Hodrick-Prescott filter, noise-to-signal ratio, orthogonal parametrization, business cycle, trend.