The Band Pass Filter

The `ideal' band pass ̄lter can be used to isolate the component of a time series that lies within a particular band of frequencies. However, applying this ̄lter requires a dataset of in ̄nite length. In practice, some sort of approximation is needed. Using projections, we derive approximations that are optimal when the time series representations underlying the raw data have a unit root, or are stationary about a trend. We identify one approximation which, though it is only optimal for one particular time series representation, nevertheless works well for standard macroeconomic time series. To illustrate the use of this approximation, we use it to characterize the change in the nature of the Phillips curve and the money-in°ation relation before and after the 1960s. We ̄nd that there is surprisingly little change in the Phillips curve and substantial change in money growth-in°ation relation.