Baxter-King Filter

The Baxter-King band pass filter is a method of smoothing a time series, that is a modification of the Hodrick-Prescott filter that provides wider opportunities for removing cycle component from a time series.

The filter procedure consists of singling out the repeated component of a time series by setting the width for oscillations of periodic component. The Baxter-King filter is a band pass filter that removes the cycle component S from the time series Y based on weighted moving average with specified weights. To calculate weights, the user should set cutoff frequencies that describe permissible non-seasonal oscillations of the smoothed series.

Let wu, wl are the upper and the lower limits of the cutoff frequency. Then the Bj weights for the specified lag/lead K are calculated by the following formulas:

Seasonal (cycle) component of the source series is calculated by the formula:

where:

Bj. Weight value that corresponds to value of the source series Y, positioned at the distance j from the current element.
K. Sets lead or lag value, with which the moving average is calculated.

The result of smoothing is the source series with removed seasonal (cycle) component.

A generalized Baxter-King filter is applied to non-stationary time series. Non-stationarity is accounted for in the matrix of weights that depend on the observation number in generalized model. In this case the following formula is applied to calculate the seasonal component:

where:

Bj,t. Weight value that corresponds to value of the source series Y positioned at the distance j from the element Yt.

K. Sets lead or lag value, with which the moving average is calculated.

There are recommended values for lead, lag, upper and lower limits of bandwidth depending on calendar frequency of the source series:

Frequency	Lead/lag	Lower limit	Upper limit
Annual	3	2	8
Semi-annual	6	3	16
Quarterly	12	6	32
Monthly	36	8	96
Weekly	156	78	416
5 Days	783	391.5	2088
7 Days	1095	547.5	2920