The Henderson filters are a widely used set of trend filters. They were initially derived by Robert Henderson (1916) for use in actuarial work. They are also used within the X-11, X-11-ARIMA, X-11-ARIMA88 and X-12-ARIMA seasonal adjustment packages. The Australian Bureau of Statistics uses the Henderson moving averages to calculate trend 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 the Henderson filters are applied to third degree polynomials, the resulting smoothed output will fit exactly on these parabolas. The Henderson filters are suitable for 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 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.

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.

1. Trend estimates should only be calculated on seasonally adjusted estimates. If the original estimates do not have any identifiable seasonality, then you can apply trend filters (ie. Henderson filters) directly to the original estimates.



2. The framework used in Gray and Thomson (1996) is very useful for programming up the Henderson filters. I have implemented this into SPlus. Email me if you'd like to see the source code.



3. An example of the standard 13 term Henderson weights are given below. These use an I/C ratio (noise to trend ratio) of 3.5. The I/C ratio is used when calculating the asymmetric filter weights.
   
   

   	N-12	N-11	N-10	N-9	N-8	N-7	N-6	N-5	N-4	N-3	N-2	N-1	N
   sym	-0.01935	-0.02786	0.00000	0.06549	0.14736	0.21434	0.24006	0.21434	0.14736	0.06549	0.00000	-0.02786	-0.01935
   asym6	0.00000	-0.01643	-0.02577	0.00127	0.06594	0.14698	0.21314	0.23803	0.21149	0.14368	0.06099	-0.00532	-0.03401
   asym5	0.00000	0.00000	-0.01099	-0.02204	0.00330	0.06626	0.14559	0.21004	0.23324	0.20498	0.13547	0.05108	-0.01694
   asym4	0.00000	0.00000	0.00000	-0.00813	-0.02019	0.00413	0.06608	0.14441	0.20784	0.23002	0.20076	0.13024	0.04483
   asym3	0.00000	0.00000	0.00000	0.00000	-0.01603	-0.02487	0.00267	0.06784	0.14939	0.21605	0.24144	0.21540	0.14810
   asym2	0.00000	0.00000	0.00000	0.00000	0.00000	-0.04271	-0.03863	0.00182	0.07990	0.17436	0.25392	0.29223	0.27910
   asym1	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	-0.09186	-0.05811	0.01202	0.11977	0.24390	0.35315	0.42113

   
   i.e., the symmetric weight for the 13 term Henderson is:
   
   (-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 weight by the observation. For example, a filtered time point in the middle of the time series at time y_t would be found by: (-0.01935 * y_t-6) + ... + (0.24006 * y_t) + ... + (-0.01935 * y_t+6). This means that six time points are required on either side of the time to estimate the filtered value.
   
   At the ends of the time series there are not enough observations on both sides of the filtered time point. This means that the following asymmetric weights for the 13 term Henderson would be used:
   
   (-0.09186,-0.05811,0.01202,0.11977,0.24390,0.35315,0.42113)
   
   The filtered value at the current end of the series would be found by: (0.42113 * y_t) + (0.35315 * y_t-1) + ... + (-0.09186 * y_t-6). The calculation of filtered values at the beginning of the time series is done in an identical way, ie. imagine that the time series is reversed. The other asymmetric weights are applied in a similar way.
   
   Forecasting of the y_t series could be used to give enough time points so that the symmetric filters can be used.



4. The length of the Henderson filter depends on the variability of your data. Typically: monthly series would use a 13 term Henderson filter an 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.



5. The Henderson filters are biased at the end of the series. There is an inbuilt linear assumption in the calculation of the asymmetric filters. Be careful when interpreting trend (and seasonally adjusted!) estimates at the current end of the series.



6. Henderson filter weights with variable noise to trend ratio (I/C) are given here: 5 term Henderson, 7 term Henderson, 9 term Henderson, 13 term Henderson, 23 term Henderson.

1. 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. You can download X12ARIMA for free at the United States Bureau of the Census website.



2. Australian Bureau of Statistics (2003). Information Paper: A Guide to Interpreting Time Series - Monitoring Trends. Australian Bureau of Statistics, cat. 1349.0, Canberra, Australia. Note: this contains a reproduction of Henderson (1916) paper in Appendix 2.



3. Australian Bureau of Statistics (1993). A Guide to Interpreting Time Series Monitoring ``Trend'' - an Overview. Australian Bureau of Statistics, cat. 1348.0, Canberra, Australia. (email: timeseries@abs.gov.au to request a copy).



4. Australian Bureau of Statistics (1987). A Guide to Smoothing Time Series - Estimates of ``Trend''. Australian Bureau of Statistics, cat. 1316.0, Canberra, Australia. This has a list of the Henderson filters in the back of the article. (email: timeseries@abs.gov.au to request a copy).



5. Dagum, E. B. and Luati, A. (2002), Smoothing Seasonally Adjusted Time Series, American Statistical Association, Proceedings of the Joint Statistical Meetings, p665- 670.



6. Doherty, M. (2001) The Surrogate Henderson Filters in X11, Australia & New Zealand Journal of Statistics, Vol 43, No. 4, 385-392.



7. Gray, A. and Thomson, P. (1996). Design of moving-average trend filters using fidelity, smoothness and minimum revisions criteria, Bureau of the Census, RR96/01.



8. Gray, A. and Thomson, P. (1996). Design of moving-average trend filters using fidelity and smoothness criteria in Vol 2: Time Series Analysis in Memory of E.J. Hannan. ed. P. Robinson and M. Rosenblatt. Springer Lecture Notes in Statistics 115, 205-219.



9. Gray, A. and Thomson, P. (1996). On a family of moving-average trend filters for the ends of series. Proceedings of the American Statistical Association, Section on Survey Research Methods, 1996.



10. Henderson, R. (1916). Note on Graduation by Adjusted Average. Transactions of the American Society of Actuaries, 17, 43-48. Note: a reproduction of this paper is available in Appendix 2 of Australian Bureau of Statistics (2003).



11. Kenny, P.B., and Durbin, J. (1982). Local Trend Estimation and Seasonal Adjustment of Economic and Social Time Series. Journal of the Royal Statistical Society, Series A, 145, 1-41.



12. Ladiray, D. and Quenneville, B. (2001). Seasonal Adjustment with the X-11 method, New York: Springer Verlan, Lecture notes in statistics, 158. This is an excellent book for an overview of the X-11 method.



13. Laniel, N. (1985). Design criteria for 13 term Henderson end-weights. Technical Report Working paper TSRA-86-011, Statistics Canada, Ottawa K1A 0T6.



14. McLaren, C.H. (1999). "Designing Rotation Patterns and Filters for Trend Estimation in Repeated Surveys" Unpublished PhD Thesis, School of Mathematics and Applied Statistics, University of Wollongong, Australia.



15. McLaren, C.H. and Steel, D.G. (2001). "Rotation patterns and trend estimation for repeated surveys using rotation group estimates", Statistica Neerlandica, Vol. 55, no. 2, pages 221-238. (abstract) (paper)



16. Shiskin, J., Young, A.H., and Musgrave, J.C. (1967). The X-11 Variant of the Census Method II Seasonal Adjustment Program. Technical Paper 15, Bureau of the Census, U.S. Department of Commerce, Washington, D.C.



17. Introductory Course on Time Series Analysis (Australian Bureau of Statistics publication)