statsmodels.tsa.filters.bk_filter.bkfilter(X, low=6, high=32, K=12)[source]

Baxter-King bandpass filter


X : array-like

A 1 or 2d ndarray. If 2d, variables are assumed to be in columns.

low : float

Minimum period for oscillations, ie., Baxter and King suggest that the Burns-Mitchell U.S. business cycle has 6 for quarterly data and 1.5 for annual data.

high : float

Maximum period for oscillations BK suggest that the U.S. business cycle has 32 for quarterly data and 8 for annual data.

K : int

Lead-lag length of the filter. Baxter and King propose a truncation length of 12 for quarterly data and 3 for annual data.


Y : array

Cyclical component of X


Returns a centered weighted moving average of the original series. Where the weights a[j] are computed

a[j] = b[j] + theta, for j = 0, +/-1, +/-2, ... +/- K
b[0] = (omega_2 - omega_1)/pi
b[j] = 1/(pi*j)(sin(omega_2*j)-sin(omega_1*j), for j = +/-1, +/-2,...

and theta is a normalizing constant

theta = -sum(b)/(2K+1)


Baxter, M. and R. G. King. “Measuring Business Cycles: Approximate
Band-Pass Filters for Economic Time Series.” Review of Economics and Statistics, 1999, 81(4), 575-593.


>>> import statsmodels.api as sm
>>> import pandas as pd
>>> dta = sm.datasets.macrodata.load_pandas().data
>>> index = pd.DatetimeIndex(start='1959Q1', end='2009Q4', freq='Q')
>>> dta.set_index(index, inplace=True)
>>> cycles = sm.tsa.filters.bkfilter(dta[['realinv']], 6, 24, 12)
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> cycles.plot(ax=ax, style=['r--', 'b-'])

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