oints.
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.