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The Band Pass Filter

The "ideal" band pass filter can be used to isolate the component of a time series that lies within a particular band of frequencies. However, applying this filter requires a dataset of infinite 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.

Keywords: Band pass filter; projection; time series; frequency domain; unit roots; bootstrap; business cycle; Phillips curve

Suggested citation: Christiano, Lawrence, and Terry Fitzgerald, 1999. “The Band Pass Filter,” Federal Reserve Bank of Cleveland, Working Paper, no. 99-06.

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