An Extended Review of the X11ARIMA Seasonal Adjustment Package

Stuart Scott

Bureau of Labor Statistics

441 G Street, N.W., Room 2126

Washington, D.C. 20212

(202) 523-1874

(202) 786-9397 (fax)

Abstract.

The revised version of X11ARIMA, Statistics Canada's seasonal

adjustment package, is reviewed. The 1988 package, available

for PC's as well as mainframes, is easy to use and more

flexible than the 1980 version. New features reflecting

research by Estela Dagum and her colleagues are highlighted.

Comparisons are made with some other available X-11 programs.

Keywords: X-11, ARIMA models, Extrapolation
An Extended Review of the X11ARIMA/88 Seasonal Adjustment Package

by Stuart Scott

International Journal of Forecasting 8 (1992), 627-633

The views expressed are those of the author and do not represent

official positions of BLS.

1. Introduction

This paper reviews X11ARIMA/88, the revised version of probably

the most widely used, currently maintained seasonal adjustment

package. It is small and simple enough for anyone interested in

applying seasonal adjustment, handling monthly and quarterly data.

Estela Dagum, the developer, and her associates at Statistics Canada

have advanced the practice of seasonal adjustment through a stream of

research studies, with many of the results reflected in the package.

The new version, which will be called X88 for short, is available

both for mainframes and PC's. Alternative packages include the

original version, released in 1980, the U. S. Bureau of the Census

package X-11.2, and SAS's PROC X11. This review will discuss various

features of the package, emphasizing differences from the original

version, denoted X80, and PC aspects, followed by a brief description

of the other packages mentioned above, which will be denoted X2 and

XSAS, respectively. Some comparisons based on runs with U.S. Bureau

of Labor Statistics time series will be presented.

All these programs are based on Julius Shiskin's X-11 method

developed at Census. In modern statistical terms, it is a

nonparametric, robust procedure for decomposing a time series into

trend-cycle, seasonal, and irregular components. Another program in

this spirit is STL (Cleveland et al, 1990), but initial comparisons

with X-11 were not that favorable. Model-based methods are not widely

used by statistical agencies, because (1) they require care and labor

in identifying models, (2) they are somewhat subjective, and (3) many

series behave too erratically to be described well by simple

univariate time series models.

2. Program features

ARIMA extrapolation

This feature is the major innovation introduced by Dagum:

extending the series with forecasts to lessen the use of X-11's

asymmetric filters. Dagum (1982) gives some measures of potential

(theoretical) reduction in revisions and refers to empirical studies

by herself and others.

The principal changes from X80 are a new set of ARIMA models for

the automatic forecasting option, a relaxing of criteria for fitting

and forecasting, and changes in the program options. The default is

the automatic option, while in X80 the default is no extrapolation.

Two new models have been added to the three available in X80, based on

an empirical study by Chiu et al (1985). Huot et al (1986) describe
sets in the parameter space for each model in which improvement from

extrapolation can be expected.

The criteria for accepting a model under the automatic option are

relaxed by adopting a p-value of .05, rather than .10, as the critical

value of the Ljung-Box statistic and permitting mean absolute

percentage forecasting error as high as 15%, rather than 12%. This

means that the chances are greater that some model will be accepted

and, additionally, that the first model, the airline model, will be

selected.

The new version has an option for forecast horizon, allowing

forecasts up to three years ahead. Theoretical results by Pierce

(1980) suggest "full forecasting," i.e., extending the series with

enough forecasts that the symmetric filters will be applied at the

most current time point, approximated by 36-42 months of forecasts

with the 3×5 seasonal filter and 60-66 months with the 3×9. Huot et

al (1986) tend to recommend one or two years of forecasts. Bobbitt

and Otto (1990) and Buszuwski (1987) suggest full forecasting. The

default is 1 year, the only value in X80.

Another option permits zero estimation iterations for ARIMA

parameters, so that the user can fully specify a model and the

parameter values to be used. This is especially helpful, since the

program's method of estimation is conditional least squares, faster to

compute, but less accurate than maximum or exact likelihood methods.

(See empirical comparisons below).

These days structural models with Kalman filter estimation are

popular, but Findley (1990) gives empirical evidence of ARIMA models

performing better than structural models. At least for now, X88's use

of ARIMA models seems appropriate.

The one serious bug I found has been corrected; it affects only

PC versions released prior to February 1992. When "no extrapolation"

is selected, extrapolation option 1, the forecast seasonal factors

repeated the last year's. They should be computed as last year's,

plus half the difference between a month's factors for the last two

years.

Composition or aggregation

X-11 is not an additive process, i.e., the seasonally adjusted

sum is not the sum of seasonally adjusted components. While X-11 is

composed of linear filters, the replacement of extremes in the program

destroys the additivity. The composite run feature carries out

"indirect adjustment" of an aggregate series by adding up the

seasonally adjusted components. It also computes a direct adjustment

of the aggregate, and provides diagnostic statistics for comparing the

two. Where practicable, government agencies often opt for indirect

adjustment for consistency. For instance, the 12 major U.S. labor

force series are adjusted directly, and then all series derived from

them are adjusted indirectly.
A shortcoming of the program is the limitation to a single set of

weights for computing aggregates. Variable weights have several

applications. In computing an indirect adjustment of quarterly

average household spending from age group components, one needs to

weight these components by their share of the population. Since the

population estimates shift somewhat every quarter, separate weighting

of components is appropriate for each time period. Weighting is

always required in aggregation for indexes, and for non-Laspeyres

formulations, there will be variable weighting. Even for Laspeyres

indexes, the weights change periodically. This can be due to a

weights revision or to a new definition of the components of the

aggregate. BLS has a modified version which permits this.

Other basic options

(1). Seasonal filter

X88 implements automatic selection of the seasonal filter based

on the global I/S ratio and an empirical study by Lothian (1984).

Somewhat more conservative than Lothian in departing from the former

default 3×5 filter, the program has three intervals for accepting a

3×3, 3×5, or 3×9 filter. If the value falls in a "gray area" on

either side of the central interval accepting 3×5, data for one year

are dropped and the ratio re-computed. While automatic selection is

the default, the user can specify a seasonal filter, including stable

estimation of seasonal factors.

(2) Easter option

The program can estimate and adjust for simple Easter effects,

where the effect is principally a function of Easter's falling in

March or April. Often, this simple model is not adequate. Different

holiday procedures are described in Monsell (1989), available in X11.2; McIntire (1990); and, Dagum, Huot, and Morry (1988).

Diagnostics

Dagum and her colleagues deserve praise for stressing the need to

check whether a seasonal adjustment is adequate. Based on 11 "quality

control" measures, labeled M1-M11, and an summary weighted average Q,

the program states whether or not (1) seasonality is "identifiable"

and (2) the seasonal adjustment is "accepted" or "rejected." These

measures can be roughly separated into measures directly related to

the seasonal component (M6-M11), measures related to the seasonally

adjusted series (M1-M3, M5), and a test of randomness of the irregular

component (M4). Analysts usually want to see the ANOVA-style F test

for seasonality used by Shiskin in X-11, which is included in the

output and reflected in the most heavily weighted measure M7. Some of

the other statistics are related to Shiskin's concept of months for

cyclical dominance (MCD). Not strict statistical tests, except for

M4, the passing scores for these statistics were derived from a largescale empirical study.
Ease of use; input/output

The new version has two executable modules, one for processing

the input data and control information (program options) and one for

carrying out one or more seasonal adjustments. The first step uses

NAMELIST-style control input. The information is placed in SAS-style

statements, with command name, options, and final semi-colon. These

statements can either be entered interactively or, my preference, by

specifying a file containing these statements. The user names the

input data set and specifies its format or one of several

"traditional" year×month table formats. The PUNCH option, the

counterpart to "special output" in X80, permits saving output, such as

seasonal factors. For a bad statement, such as one not issued upper

case, the program returns "ERRONEOUS;" if there is a mistake or

mismatch between control information and input data, the program

prints an error message. With preprocessing done, the user types

"XA88" for actual program execution, and is prompted for a file for

placement of the program printout.

All these features make the program easy to use for either a

casual or experienced user. As mentioned above, the syntax is

straightforward, and input/output is fairly flexible. In typical

multiple runs, it is an advantage to place all data in a single data

set. On the other hand, this requires all data, including any prior

adjustment factors, to have the same format.

Documentation

The thorough documentation is divided into two major parts,

technical description of the program and a users manual. The main

users manual has 37 pages of instruction and another 37 pages with a

full seasonal adjustment printout. A separate 8-page guide goes over

installation and execution of the PC version.

I would urge some improvements in the documentation to help the

beginner. To improve the presentation, I would suggest grouping

options under headings to reflect explicitly the program syntax,

similar to SAS manuals. For example, all the appropriate options

could be grouped under a heading "SA Command Options." Furthermore, a

couple of examples of an entire set of command statements are needed.

Only the PC guide has any full examples, and its examples are fairly

limited.

The technical description repeats the old manual, after

conveniently collecting the new features in the first seven pages.

The material repeated, highly useful, includes brief rationales for

seasonal adjustment and aspects of the methodology and a 32-page

layout of the iterative steps of the program.

In addition to the two executable modules and a help function,

the diskette for the PC contains an example, including the input data,

command statements, and a printout. The source code (FORTRAN) does

not come with the PC version, but may be available from Statistics

Canada.
3. Other Packages

Census X-11.2

X-11.2 (X2 for short) lacks (1) the extrapolation option of X88

and (2) composite adjustment. Otherwise, it has most of X88's

features, including the diagnostic statistics M1-M11 and Q. Like X80

and XSAS, the default seasonal filter is 3×3, then 3×5. However, the

program prints an alternate filter recommendation based on the same

criterion from Lothian (1984) that X88's variable filter option is

based on.

The principal X2 feature not found in X88 is the sliding spans

analysis (Findley et al, 1990). More direct and intuitive than M1-

M11, sliding spans statistics seem to me a highly useful diagnostic.

They check for stability of factors, seasonally adjusted values, and

change estimates, usually across three or four spans, formed by adding

a new year and dropping an old one. I find the sliding spans more

informative for comparing direct and indirect adjustment than the

smoothness measures of X80 and X88, which are somewhat difficult to

interpret. X2 has a couple of spectrum plots. This is a tool I've

also found very useful, but I haven't tried X2's routine. A holiday

adjustment method, somewhat more elaborate than X88's, permits special

treatment when Easter, Labor Day, or Thanksgiving affects a series.

Input/output involves NAMELIST input, and is neatly organized.

X2 is more flexible than X88 in expecting separate files and format

specifications for each input series, including series of prior

adjustments factors. This flexibility must be weighed against the

proliferation of input and output files in multiple runs.

Census continues to develop "X12ARIMA" (Findley et al, 1988).

This package potentially offers many desirable features, including a

superior estimation package, composite adjustment, and a flexible

routine for treating outliers and interventions. Test versions,

however, have been too large to adopt easily to PC's.

X2 takes about 530K of memory on my machine, more than X88.

Unlike X88, source code is supplied, an advantage if you have a

FORTRAN compiler and you would like to modify something for your

application .

SAS PROC X11

SAS PROC X11 (or XSAS for short), as one might expect, is an

adequate, but less complete version of X-11 than the others. It lacks

ARIMA extrapolation or composite adjustment. It has the MCD, the

basic F test for stable seasonality, and several summary measures

tables, but not M1-M11 or sliding spans. It is easy to use, and has

many of the program options, including prior adjustment and trading

day adjustment; it outputs specified tables as SAS data sets.

XSAS apparently is an older Census program, modified only for

interface with the SAS language and data sets. Specialists will want

one of the other programs, but, as a "free" program within SAS ETS,

XSAS is quite usable, and, for basic X-11, appears to give results

very similar to the others, especially X2.
4. Empirical Comparisons

Basic X-11

Choosing the "no extrapolation" option in X11ARIMA reduces the

methodology to "basic X-11." Different versions introduce

modifications from the original program, such as X88's variable

seasonal filter selection and improved accuracy of asymmetric

Henderson filter weights. Both X80 and X88 appear to use single

precision, while X2 has double precision.

As a small test for differences among these programs, the 12

major U.S. labor force series are run with X88, X80, X2, and XSAS with

default options (except "no extrapolation" for X88). The time series,

identified below the exhibits, are available from both the author and

this journal. Exhibit 1 shows pairwise summary difference statistics.

These are reported separately for 1981-90, the data input span, and

for 1991, the year of projected seasonal factors. The two left-hand

columns are mean median absolute differences, based first on computing

median absolute differences across time spans for each series, and

then averaging these medians across series. For the right-hand

columns, the maximum absolute difference is computed for each series

and then the median value computed across the twelve maxima. This

choice of statistics depicts typical differences better than taking

means of mean absolute differences and means of maximum absolute

differences, both of which are influenced more by a few large values.

The pairs are listed by increasing size of differences.

It is perhaps surprising that differences are this large for the

same basic methodology: over half the time X88's within-span factors

differ from the other programs by more than .05 and its projected

factors by more than .20. All pairs show differences greater than

0.10 for most series. Many of these differences are nontrivial.

It is in the more volatile teenage series that we find most of

the large differences, with the smallest within-span maximum being .45

for YMU. Exhibit 2 gives maximum absolute differences for each pair

of methods for the 6 teenage series for the within-span period.

X88 chooses the 3×9 filter for three of these series, YMN, YWN,

and YWU, reflected in larger pairwise differences with this method,

especially for YWN and YWU. A detailed examination of YWU shows that,

as expected, the 3×9 gives a smoother, more moderate seasonal pattern

for X88. Across the ten years, July factors vary from 119.0 to 123.6

for X88, a range of 4.6 compared to 6.7 or 6.8 for the other methods.

X2, while using the 3×5, recommends trying the stable filter option,

based on a large I/S ratio.

For YWA, Exhibit 2 shows the largest differences across 1981-90

come from X80. The 7/81 factor for X80 is 200.6 while the other

methods yield values between 206.4 and 207.9. (Seeing the size of

these multiplicative factors for YWA suggests that the differences are

not as severe as they might appear at first). The differences for

July might be attributed to replacement values for extreme SI ratios.

X80 computes substantially smaller replacement values in Table D9 for

7/81 and 7/83.

Summarizing, all four versions of X-11 give very similar seasonal

patterns, but nontrivial differences occur. When one method is
different from the others, sometimes it appears to be better, as when

X88 applies a 3×9 seasonal filter for YWU or when X80 apparently

computes better replacement SI ratios for YWA. X2 and XSAS tend to

give the most similar results. Incidentally, for X88, PC and

mainframe seasonal factors agree to two decimal places.

ARIMA Modeling

As a check on the extrapolation portion of X88, a comparison was

made with other modeling programs. The models and transformations

selected by BLS for the official seasonal adjustment of the twelve

labor force series are estimated by SAS PROC ARIMA with both

conditional least squares (CLS) and maximum likelihood (ML) options,

and by a modeling and signal extraction program written by Peter

Burman (1980). Perhaps surprisingly, the airline model is used for

none of the series. All include a single seasonal moving average

parameter. Eight involve 2-4 moving average parameters, three have AR

parameters, and one is the problematic (212),(011)12 model.

X88 uses conditional least squares estimation, confirmed by close

agreement with PROC ARIMA's CLS option on all but one series.

Burman's exact likelihood results are close to SAS's ML option , so

further comparisons will be between the latter and X88. The most

significant differences in parameter estimates are for the seasonal

moving average parameter, with the ML estimates always higher, and for

half the series substantially higher. Turning to the 12 forecast

values produced by each method, six series have fairly large

differences during the latter half of the year, the longer-range

forecasts. Four of these have large differences in seasonal MA

parameters. For two of these, the shape of the forecasts appears

appreciably different.

Summarizing these results, CLS and ML parameter estimates may

differ quite a bit for seasonal models. As well as modeling the

series outside X88, one may benefit from specifying the parameters

estimated externally and applying X88's "no iterations" option.

Still, in most cases in this small empirical comparison, CLS and ML

forecast patterns are similar.

Conclusion

X11ARIMA/88 stands as the most complete, general purpose seasonal

adjustment program. The revision adds flexibility and more userfriendly input/output. With correction for one significant bug

(discussed in 2a. above), the PC version works easily and about as

well as the mainframe version without excessive space requirements.

Minor changes to the documentation would clarify the syntax for input.
Appendix. Program Information and System Requirements

The X11ARIMA/88 Seasonal Adjustment Method, Estela Bee Dagum, 7-E R.

H. Coats Building, Statistics Canada, Ottawa, CANADA K1A 0T6, 613/951-

9876; PC version in the U.S.: Levenbach Associates, 103 Washington

Street, Suite 348, Morristown, NJ 07960, 201/285-9248. List price:

Mainframe, $900; PC, $450 academic, $650 government, $895 commercial.

PC requirements are stated as the following or better: 8086

microprocessor, DOS 3.1, 8087 math co-processor, and 640K memory;

either one hard disk and one floppy disk drive or two floppy drives.

The program executes easily and quickly on my GRID 386, needing under

450K of memory.

Acknowledgments.

The author is grateful to George Methee for providing the labor

force data and for help in examining and running the programs, and to

James Buszuwski and Robert McIntire for helpful comments.

References

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Revisions of Seasonally Adjusted Data Using the X-11 Adjustment

Procedure," Proceedings of the Business and Economic Statistics

Section of the American Statistical Association, 449-453.

Burman, J. P., 1980, "Seasonal Adjustment by Signal Extraction,"

Journal of the Royal Statistical Society A, 143, 321-337.

Buszuwski, James A., 1987, "Alternative ARIMA Forecasting Horizons

When Seasonally Adjusting Producer Price Index Data with X-11-ARIMA

in Concurrent Mode," Proceedings of the Business and Economic

Statistics Section of the American statistical Association, 488-

493.

Chiu, K., J. Higginson, and G. Huot, 1985, "Performance of ARIMA

Models in Time Series," Survey Methodology, 11, 51-64.

Cleveland, Robert B., William S. Cleveland, Jean E. McRae, and Irma

Terpenning, 1990, "STL: A Seasonal-Trend Decomposition Procedure

Based on Loess," Journal of Official Statistics, 6, 3-73.

Dagum, E. B., 1980, The X-11-ARIMA Seasonal Adjustment Method,

Statistics Canada, Catalogue 12-564E.

Dagum, E. B., 1982, "The Effects of Asymmetric Filters on Seasonal

Factor Revisions," Journal of the American Statistical Association,

77, 732-738.

Dagum, Estela Bee, Guy Huot, and Marietta Morry, 1988, "Seasonal

Adjustment in the eighties: Some problems and solutions," Canadian

Journal of Statistics, 16, 109-126.

Findley, David F., 1990, "Making Difficult Model Comparisons

Graphically," Proceedings of the Survey Research Methods Section of

the American Statistical Association, 622-627.

Findley, David F., Brian Monsell, Mark Otto, William Bell, and Marian

Pugh, 1988, "Toward X-12-ARIMA," Proceedings of the Bureau of the

Census Annual Research Conference, 4, 591-622.

Findley, David F., Brian C. Monsell, Holly B. Shulman, and Marian

Pugh, 1990, "Sliding Spans Diagnostics for Seasonal and Related
Adjustments," Journal of the American Statistical Association, 85,

345-355.

Huot, G., K. Chiu, J. Higginson, and N. Gait, 1986, "Analysis of

Revisions in the Seasonal Adjustment of Data Using X11ARIMA ModelBased Filters," International Journal of Forecasting, 2, 217-229.

Lothian, J., 1984, "The Identification and Treatment of Moving

Seasonality in X11ARIMA," Proceedings of the Business and Economic

Statistics Section of the American Statistical Association, 166-

171.

McIntire, Robert J., 1990, "A Procedure to Control for Moving-Holiday

Effects in Seasonally Adjusting Employment and Hours Series,"

Proceedings of the Business and Economic Statistics Section of the

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Pierce, David A., 1980, "Data Revisions with Moving Average Seasonal

Adjustment Procedures," Journal of Econometrics, 14, 95-114.

SAS/ETS User's Guide, Version 6, 1988, SAS Institute Inc., Cary, NC
Exhibit 1. Pairwise Absolute Differences in Seasonal Factors Between

X-11 Programs Across 12 Labor Force Series

Program Mean Median

Pairs Median Absolute Difference Maximum Absolute Difference

1981-90 1991 1981-90 1991

X2-XSAS .02 .03 .15 .15

X2-X80 .03 .04 .20 .14

X80-XSAS .04 .04 .25 .19

X88-XSAS .08 .21 .25 .34

X2-X88 .07 .22 .30 .38

X80-X88 .07 .21 .38 .40

Exhibit 2. Maximum Absolute Differences, 1981-90, for Teenage Series

Program Pairs YMU YWU YMA YWA YMN YWN

X2-XSAS .26 .42 .53 .60 .19 .13

X2-X80 .38 .11 .88 6.34 .55 .13

X80-XSAS .14 .43 .88 5.86 .36 .14

X88-XSAS .19 1.84 .92 1.51 .57 .58

X2-X88 .45 1.82 .92 1.03 .56 .62

X80-X88 .23 1.85 .45 7.37 .57 .61

Guide to series:

YMU - Unemployment, teenage men

YWU - Unemployment, teenage women

AMU - Unemployment, adult men

AWU - Unemployment, adult women

YMA - Agricultural employment, teenage men

YWA - Agricultural employment, teenage women

AMA - Agricultural employment, adult men

AWA - Agricultural employment, adult women

YMN - Nonagricultural employment, teenage men

YWN - Nonagricultural employment, teenage women

AMN - Nonagricultural employment, adult men

AWN - Nonagricultural employment, adult women