Trend estimation, signal-noise ratios and the frequency of observations

The implied signal extraction filters in unobserved components models depend on key signal-noise ratios. This paper examines how these ratios change with the observation interval. The analysis is based on continuous time models and is carried out for both stocks and flows. As a by-product, a connection is established between continuous time flow models and the canonical decomposition. The implications of using the Hodrick-Prescott filter to extract cycles at annual and monthly frequencies are discussed. Many of the arguments used in the literature to set the smoothing constant are shown to be flawed. The analysis suggests that a model-based approach is the best way to proceed. A model formulated in continuous time, or in discrete time at a fine time interval, automatically adapts to any observation interval if it is set up in state space form. Concerns about the change in the shape of the filter and the way in which the signal-noise ratio adapts are then no longer an issue