We present a filtering technique to estimate trends of multivariate time series. This methos is based on a vector signal-plus-noise representation of Penalized Least Squares that requires only the first two sample moments, and introduces an index of smoothness.

This index allows setting in advance a desired amount of smoothness to achieve. It is also a function of the correlation between the noises of the series and the sample size.

Our proposal arises from a statistical solution to a multivariate GLS problem. Such a solution leads to an index of smoothness that is applicable in the general multivariate case, but we pay special attention to the bivariate situation.

Here we show the closed-form expressions for calculating trend estimates with their corresponding variance-covariance matrices, and present the proposed algorithm for smoothing bivariate time series.

We discuss the results on simulated data and a real application.