The Henderson filters are a widely used set of trend filters, or smoothers. They were initially derived by Robert Henderson (1916) for use in actuarial work. They are also used within the X-11 family of seasonal adjustment packages, e.g. X-12-ARIMA. The Henderson filters can be used separately, for example, the Australian Bureau of Statistics uses them to calculate trend estimates from the seasonally adjusted estimates.
The requirement used by Henderson to derive the filters was that they must follow a local cubic polynomial without distortion. The Henderson filters are derived by minimizing the sum of squares of the third difference of the moving average series. Henderson's criteria ensures that when these filters are applied to third degree polynomials, the resulting smoothed output will fit exactly on these parabolas. The Henderson filters are suitable for smoothing economic time series as they allow the cycles typical of the trend to pass through unchanged. They also have the property that they will eliminate almost all the irregular variations that are of very short frequencies of six months or less.
The filter weights applied in the middle of a time series are symmetric, while the end filter weights are asymmetric. This is due to the standard end point problem where at the start and end of the series there is not enough data on either side of the data points to generate symmetric filter weights. In practice, the impact of this can be reduced by generating seperate forecasts and then applying the symmetric filter weights, e.g. by using ARIMA methods.
There are a number of ways to derive the Henderson symmetric filter weights and the Henderson asymmetric filter weights. Laniel (1985) compared five different methods to generate Henderson asymmetric filter weights and each give different values for the asymmetric filter weights. The methods considered were, a Best Linear Unbiased Estimate (BLUE) method, minimizing the mean square revision between the final estimate and a preliminary estimate, the original X-11 criterion by Shiskin, Young and Musgrave (1967), Kenny and Durbin (1982) presented a minimisation of the sum of squares approach and also an optimisation method, while Cholette (1983) used a modified version of Kenny and Durbin's method. Doherty (2001) provided a relationship between two of the more common methods, the original Musgrave approach and that of Kenny and Durbin (1982). Laniel (1985) also independently rediscovered criteria used by Musgrave which accurately generated the Henderson asymmetric filter weights.
An excellent paper that describes families of filters, of which the Henderson filter is one subset, is by Gray and Thomson (1996).
Using Henderson filters
- Trend estimates should only be calculated on estimates where seasonality has been removed, e.g. the seasonally adjusted estimates. If the original estimates do not have any identifiable seasonality, then you can apply trend filters directly to the original estimates.
- The framework used in Gray and Thomson (1996) is very useful for programming up the Henderson filters. We have implemented this into R and it can be used to generate your own Henderson filters. Contact us if you'd like to see the free source code.
- An example of the standard 13 term Henderson weights are given here.
A noise to trend ratio (denoted as I/C) is used when calculating the asymmetric filter weights.
For this example, the symmetric weight for the 13 term Henderson is a vector:
(-0.01935, -0.02786, 0, 0.06549, 0.14736, 0.21434, 0.24006, 0.21434, 0.14736, 0.06549, 0, -0.02786, -0.01935)
These weights would be applied in the middle of the time series by multiplying each observation by the weight. For example, a filtered time point in the middle of the time series y_N-6 would be found by: (-0.01935 * y_N-12) + ... + (0.24006 * y_N-6) + ... + (-0.01935 * y_N). This means that six time points are required on either side of the time point to estimate the filtered value. This illustrates that in this case the single filtered time point at time N-6 is a combination of 13 time points.
At the ends of the time series there are not enough observations on both sides of the filtered time point to use all of this information. This means that the following asymmetric weights for the 13 term Henderson would be used:
The filtered value at the current end of the series for time point N would be found by: (0.42113 * y_N) + (0.35315 * y_N-1) + ... + (-0.09186 * y_N-6). The calculation of filtered values at the beginning of the time series is done in an identical way, e.g. imagine that the time series is reversed. The other asymmetric weights are applied in a similar way.
Forecasting the y_t series at the beginning and current end of the series could be used to give enough time points so that the symmetric filters can be used. In this example, we could forecast and backcast for 6 time periods. The forecasted estimates could then be discarded once the filtered estimates are derived.
- The length of the Henderson filter use should depend on the variability of your data. Typically: monthly series would use a 13 term Henderson filter and quarterly series use a 5 or 7 term Henderson filter. In both cases the asymmetric weights can be tailored for each individual series using an I/C ratio calculated from the original estimates.
- The Henderson filters are biased at the end of the series. This is because there is an inbuilt linear assumption in the calculation of the asymmetric filters. Also, be careful when interpreting trend (and seasonally adjusted) estimates at the current end of the series.
- You can calculate your own Henderson filter weights with variable noise to trend ratio (I/C), and also other filter weights within the same framework on the myfilter page, or you can get a few prepared versions here: Henderson 5, Henderson 7, Henderson 9 and Henderson 13.
Where can I find out more information on Henderson filters?
The Henderson filters are used extensively within the widely used seasonal adjustment packages: X-11, X-11-ARIMA, X-11-ARIMA88 and X-12-ARIMA. The documentation for X-12-ARIMA is very comprehensive. You can download X-12-ARIMA for free at the United States Bureau of the Census website.
Some references which describe the mathematics behind the Henderson filters are listed here.