Baxter-King Band-Pass Filter
Also known as: Baxter-King Filter, Band-Pass Filter (Baxter-King), BK Band-Pass Filter, Bant Geçiren Süzgeç
The Baxter-King (BK) band-pass filter, introduced by Marianne Baxter and Robert King in 1999, is a linear symmetric moving-average filter designed to isolate cyclical fluctuations in macroeconomic time series that fall within a specified range of periodicities. It removes both very low-frequency trends and very high-frequency noise, retaining only the business-cycle component—typically oscillations with a period of six to thirty-two quarters for quarterly data.
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When to use it
Use the BK filter when you need to extract the business-cycle component from a macroeconomic time series such as GDP, consumption, or investment for purposes of stylized-fact documentation, co-movement analysis, or model evaluation. The method assumes the series is integrated of order at most one (I(0) or I(1)) and that business-cycle fluctuations are adequately characterized by a fixed frequency band. It is less appropriate for short samples (fewer than roughly 50 observations) due to endpoint data loss of 2K observations. When endpoint preservation matters, consider the Hodrick-Prescott filter or the Christiano-Fitzgerald filter; for spectral flexibility, consider the Fourier transform approach.
Strengths & limitations
- Isolates a precisely defined frequency band, giving transparent control over which periodicities are retained.
- Symmetric weight structure introduces no phase shift, so cyclical turning points in the filtered series align with those in the original series.
- Automatically eliminates deterministic trends and unit roots through the zero-sum weight constraint.
- Well-grounded in spectral theory and directly linked to the ideal band-pass filter, making its properties analytically tractable.
- Loses K observations at each end of the sample, which can be severe in short macroeconomic datasets.
- The results are sensitive to the choice of the truncation parameter K; the default of K = 12 for quarterly data is conventional but not always optimal.
- Assumes stationarity within the retained frequency band; nonlinear trends or structural breaks can distort the extracted cycle.
- The fixed symmetric window means it cannot update in real time as new data arrive without losing a new endpoint observation.
Frequently asked
How do I choose the truncation parameter K?
Baxter and King (1999) recommend K = 12 for quarterly data and K = 3 for annual data as defaults that balance frequency isolation against sample loss. Increasing K improves the approximation to the ideal filter but sacrifices more endpoint observations. In practice, researchers often check robustness by comparing results across K = 8, 12, and 16 for quarterly series.
How does the BK filter differ from the HP filter?
The HP filter is a two-sided high-pass filter that minimizes a penalized sum of squared deviations and retains all frequencies above a threshold implicitly determined by the smoothing parameter lambda. The BK filter explicitly targets a frequency band with defined upper and lower cut-offs, providing sharper frequency isolation and no phase shift, but at the cost of data loss at both ends of the sample that the HP filter avoids.
Can the BK filter be applied to monthly data?
Yes. For monthly data the standard business-cycle band is typically set to p_l = 18 and p_u = 96 months (corresponding to 1.5–8 years) and the truncation parameter is conventionally set to K = 36 months. The same formulas apply; only the band limits and K change to reflect the higher sampling frequency.
Sources
- Baxter, M., & King, R. G. (1999). Measuring business cycles: Approximate band-pass filters for economic time series. Review of Economics and Statistics, 81(4), 575–593. DOI: 10.1162/003465399558454 ↗
How to cite this page
ScholarGate. (2026, June 2). Baxter-King Band-Pass Filter. ScholarGate. https://scholargate.app/en/econometrics/bk-filter
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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