### Historical Development of X-11

This section briefly describes the historical development of the standard X-11 seasonal adjustment method and the later development of the X-11-ARIMA method. Most of the following discussion is based on a comprehensive article by Bell and Hillmer (1984), which describes the history of X-11 and the justification of using seasonal adjustment methods, such as X-11, given the current availability of time series software. For further discussions about statistical problems associated with the X-11 method, see Ghysels (1990).

Seasonal adjustment methods began to be developed in the 1920s and 1930s, before there were suitable analytic models available and before electronic computing devices were in existence. The lack of any suitable model led to methods that worked the same for any series — that is, methods that were not model-based and that could be applied to any series. Experience with economic series had shown that a given mathematical form could adequately represent a time series only for a fixed length; as more data were added, the model became inadequate. This suggested an approach that used moving averages. For further analysis of the properties of X-11 moving averages, see Cleveland and Tiao (1976).

The basic method was to break up an economic time series into long-term trend, long-term cyclical movements, seasonal movements, and irregular fluctuations.

Early investigators found that it was not possible to uniquely decompose the trend and cycle components. Thus, these two were grouped together; the resulting component is usually referred to as the trend cycle component.

It was also found that estimating seasonal components in the presence of trend produced biased estimates of the seasonal components, but, at the same time, estimating trend in the presence of seasonality was difficult. This eventually lead to the iterative approach used in the X-11 method.

Two other problems were encountered by early investigators. First, some economic series appear to have changing or evolving seasonality. Secondly, moving averages were very sensitive to extreme values. The estimation method used in the X-11 method allows for evolving seasonal components. For the second problem, the X-11 method uses repeated adjustment of extreme values.

All of these problems encountered in the early investigation of seasonal adjustment methods suggested the use of moving averages in estimating components. Even with the use of moving averages instead of a model-based method, massive amounts of hand calculations were required. Only a small number of series could be adjusted, and little experimentation could be done to evaluate variations on the method.

With the advent of electronic computing in the 1950s, work on seasonal adjustment methods proceeded rapidly. These methods still used the framework previously described; variants of these basic methods could now be easily tested against a large number of series.

Much of the work was done by Julian Shiskin and others at the U.S. Bureau of the Census beginning in 1954 and culminating after a number of variants into the X-11 Variant of the Census Method II Seasonal Adjustment Program, which PROC X11 implements.

References for this work during this period include Shiskin and Eisenpress (1957), Shiskin (1958), and Marris (1961). The authoritative documentation for the X-11 Variant is in Shiskin, Young, and Musgrave (1967). This document is not equivalent to a program specification; however, the FORTRAN code that implements the X-11 Variant is in the public domain. A less detailed description of the X-11 Variant is given in U.S. Bureau of the Census (1969).

#### Development of the X-11-ARIMA Method

The X-11 method uses symmetric moving averages in estimating the various components. At the end of the series, however, these symmetric weights cannot be applied. Either asymmetric weights have to be used, or some method of extending the series must be found.

While various methods of extending a series have been proposed, the most important method to date has been the X-11-ARIMA method developed at Statistics Canada. This method uses Box-Jenkins ARIMA models to extend the series.

The Time Series Research and Analysis Division of Statistics Canada investigated 174 Canadian economic series and found five ARIMA models out of twelve that fit the majority of series well and reduced revisions for the most recent months. References that give details of various aspects of the X-11-ARIMA methodology include Dagum (1980, 1982a, 1982c, 1983, 1988), Laniel (1985), Lothian and Morry (1978a), and Huot et al. (1986).

#### Differences between X11ARIMA/88 and PROC X11

The original implementation of the X-11-ARIMA method was by Statistics Canada in 1980 (Dagum, 1980), with later changes and enhancements made in 1988 (Dagum, 1988). The calculations performed by PROC X11 differ from those in X11ARIMA/88, which will result in differences in the final component estimates provided by these implementations.

There are three areas where Statistics Canada made changes to the original X-11 seasonal adjustment method in developing X11ARIMA/80 (Monsell, 1984). These are (a) selection of extreme values, (b) replacement of extreme values, and (c) generation of seasonal and trend cycle weights.

These changes have not been implemented in the current version of PROC X11. Thus the procedure produces results identical to those from previous versions of PROC X11 in the absence of an ARIMA statement.

Additional differences can result from the ARIMA estimation. X11ARIMA/88 uses conditional least squares (CLS), while CLS, unconditional least squares (ULS) and maximum likelihood (ML) are all available in PROC X11 by using the METHOD= option in the ARIMA statement. Generally, parameters estimates will differ for the different methods.