---

Empirical Evaluation of X-11 and Model-based Seasonal Adjustment 

Methods 

Richard Tiller, Daniel Chow, and Stuart Scott 

Bureau of Labor Statistics. 

Massachusetts Avenue, N.E, Washington, D.C. 20212, tiller_r@bls.gov 

Introduction

For over three decades X11 has been the standard approach used to seasonally adjust time series at the Bureau of the Labor 

Statistics (BLS). In recent years the model based approach has gained ground as an alternative approach. This study 

compares X-11 and model-based seasonal adjustments for 82 series produced by the BLS. We use “X-11” to refer to the 

seasonal adjustment method developed by Shiskin et. al ( 1967) as implemented in the enhanced version X-12 ARIMA 

(Findley et. al., 1998). The model based approach we use is known as SEATS (Signal Extraction of ARIMA Time Series),

originally implemented by the Bank of Spain (Gomez and Marvall, 1997). This paper was developed from a larger study

conducted at BLS (Scott, Tiller, and Chow, 2007) to evaluate SEATS as a potential supplement to X11.

This study has greatly benefited by the availability of a beta version of the X12-SEATS program under development by the 

Bureau of the Census in collaboration with Agustin Maravall (Findley, Monsell, et al, 2005). This program provides a

unified model-based pre-adjustment framework within which either approach may be invoked and provides a common set of

diagnostic and evaluative tools for comparing results from the two methods. 

Design of Study

Data

Our series are selected from three BLS programs: Current Employment Statistics (CES), the Consumer Price Index (CPI),

and the Producer Price Index (PPI). The aim has been to select a diverse set with respect to classification (industry or item

structure), level of aggregation, and difficulty in seasonal adjustment. All are national series. We use 12 years of recent data 

where practical. For some price series, fewer years are available or important interventions are not available for earlier years, 

so data spans as short as 8 years are used. 

The X12-SEATS software (Findley, Monsell, et al, 2005) provides a controlled environment for comparing the 

decompositions from X-11 and SEATS. Both methods start from identical “RegARIMA” fits of the original series, which 

includes identical ARIMA models and pre-adjustment options such as outliers and calendar effects. The difference between 

the two methods lies in how the filters or moving averages are produced to perform the decomposition of the series into 

trend, seasonal and irregular components. SEATS filters are derived directly from the ARIMA model and hence are tailored 

to the specific properties of the series as reflected in the estimated model. In contrast, X11 selects from a set of 

predetermined filters that have been shown to work well for a wide variety of series. 

There is a well known fundamental conceptual issue that must be considered in comparisons of estimated unobserved

components from alternative seasonal adjustment methods. Because the components are in general unidentified, all methods 

must impose specific assumptions, either explicitly or implicitly, in order to decompose a series. When two approaches are

based on different but plausible assumptions about the behavior of the unobserved components, no objective criteria exist for

judging which empirical decomposition is best (Bell and Hillmer, 1984). Depoutot and Planas (1998) provide conditions

under which X-11 and SEATS filters are similar and thus comparisons of the decompositions are meaningful. Filter 

comparability is data specific in the sense that it depends on the properties of the time series under analysis. We draw on this 

work in our evaluation of the methods applied to BLS data.

In the first stage of our work, X12/SEATS is run with automatic selection of X11 filters, ARIMA models, outliers, and 

adjustment mode (multiplicative or additive). Denoted as AUTO, this stage allows the software to adjust automatically each
Residual Seasonality 

Significant residual seasonal peaks in the spectra of the 

differenced seasonally adjusted series, the irregular, and

the ARIMA residuals 

 rsdpks 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I ..... 

,-----------~ 

'---------------' 

i 

series, in order for us to gain insight into how well X11 and SEATS perform with as little user input as possible. There are

some exceptions. Known calendar effects are included in the case of several CES employment series. For difficult price

series, such as energy-related series, interventions and outliers used in the official adjustments are our starting point.

Experience has shown that automatic detection is inadequate for these effects; for instance, an intervention variable

commonly used for price series is not part of automatic detection. Also, CES calendar effects differ from calendar effects 

options provided by the software. 

In the second evaluation stage, called the analyst adjustment stage and denoted ANALY we have selected a subset of series 

from each program for more in-depth analysis. Comparisons are made across X11 and SEATS results. When X11 quality 

control (QC) statistics or ARIMA model diagnostics show problems or defects we attempt to address them. Other times, in

order to find a feasible decomposition, SEATS discards the model initially identified, in which case we review the model

choice. In other cases, we try to improve on the choice of outliers from AUTO. 

Diagnostic Testing

When performing seasonal adjustment for any method it is important to test for three basic conditions: (1) the observed series

is seasonal, (2) the seasonal effects can be estimated reliably, and (3) no residual seasonality is left in the adjusted series. 

A variety of diagnostic tools are available some of which are method specific and others of a generic nature that can be used 

across methods. For X-11 we have the F-test from the original X11 and the more extensive M and Q tests from X-11 

ARIMA. X12 provides a set of generic tests that include sliding span diagnostics, frequency spectrum estimates, and

revision history statistics that are suitable for comparing methods. 

The individual M statistics and Q2 are scaled to lie between 0 and 3 with smaller values indicating a better adjustment. We 

adopt the following guidelines for interpreting these statistics: 

less than 0.8 diagnostics favorable 

between 0.8 and 1.2 gray area 

greater than 1.2 diagnostic unfavorable 

SEATS provides a theoretical framework for evaluating various diagnostics (Maravall, 2003). However, much of this work is 

still in the experimental stage and not ready for implementation (Findley et al, 2004 and Evans, Holan, and McElroy, 2006). 

Other SEATS specific diagnostics assess the quality of the decomposition: for each component in terms of the estimation 

error, the forecast errors, variances and rate of damping of the revisions as new data are added. For this study the most 

important SEATS diagnostics are the ones related to the overall adequacy of the ARIMA model fit. While model formulation 

and model adequacy are important for X11 adjustment, they are central to SEATS. 

Table 1 below presents the key diagnostics we use to compare across methods and assess suitability of adjustment by the 

individual methods. 

Table 1: Summary of Major Diagnostics

Type Description Abbreviation 

Cross-method 

peaks at seasonal frequencies in the spectrum of the 

differenced observed series Ori Pks 

Presence of Seasonality stable F statistic from a one-way analysis of variance 

carried out on the SI ratios Stable F 

Stability of Seasonal Factors Stable F applied to the final seasonal component Overall F 

Reliability of Seasonal 

adjustment 

Distribution of absolute percent revisions of the monthly 

seasonal adjustments, concurrent to near final
ARIMA Model Diagnostics The version of the AUTOMODEL option based on

TRAMO provides a test for the presence of seasonality 

(seasonal differencing) and the stability of seasonality 

depends on the magnitude of the seasonal MA

parameter 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-- -- -- -- -- -- -- -- -- -- -- -- ·-- -- -··- !---------< 

------------------------------------------------------------------~-------< 

·················· ,------,-------,-------< 

Table 1: Summary of Major Diagnostics

Type 

Median 

Description Abbreviation 

Rmed 

75th percentile 

maximum of revisions 

R75p

Rmax 

Sliding spans –distribution of maximum per cent 

difference (MPD) of seasonally adjusted monthly

change across 2-4 span values

Median Cmed 

60th percentile C60

maximum of revisions Cmax 

Smoothness Root mean square of the first difference of seasonally RMSD adjusted series or trend series

Stable F, described above Stable F 

X11 Diagnostics 

M7 measures amount of stable seasonality present M7 relative to amount of moving seasonality 

Q2 weighted average of M1 and M3-M11 from the Q2

original set of 11 statistics 

Ljung-Box statistic for autocorrelations in model 

residuals for first 12 and 24 lags provides an overall

goodness of fit test

LB 

average absolute in sample percent forecasting error for AAPE the last 3 years

SEATS Diagnostics The most important diagnostics specific to SEATS are

the ARIMA model diagnostics listed above. 

Results

Table 2 presents summary information on the adequacy of the ARIMA models automatically selected for each of the series in 

this study. Recall that both X11 and SEATS begin with identical REGARIMA models. Thirty-four percent of the series (28

of 82) fail at least one of the 4 principal diagnostic tests shown below. In the analyst (ANALY) stage acceptable models are

found in most of the cases studied. For 4 of these series SEATS automatically rejects the initial model as unsuitable for 

seasonal adjustment and actually finds an alternative model with an adequate fit. 

Table 2

Goodness of Fit Tests of Automatically Selected ARIMA Models 

LB12 LB24 AAPE RSDPKS One or more 

# Models Failing Test 13 14 3 8 28

Key:

LB12[24] Ljung-Box test over first 12 [24] lags

AAPE Average absolute percent forecast error test 

RSDPKS test for seasonality in the residuals
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The overall quality of the initial seasonal adjustments by X11 is shown in Table 3. A total of 22 series (27%) are marginal or

failing based on one or more of the 4 quality control tests shown below. There is a large gray area for these statistics, and

some adjustments of these series are found to be adequate after further analysis. Also, in the analyst stage of this study we 

were able to make a number of improvements. 

Table 3

X11 Quality Control Statistics

Stable F 

( < 10 ) 

M7 

( ≥ 0.8 )

Q2 RSDPKS One or 

more 

# Series Marginal or 

Failing 20 13 3 1 22 

Key:

Stable F- test for stable seasonality 

M7 - amount of stable seasonality relative to moving seasonality 

Q2 - weighted average of 10 quality control statistics, exclude cases where 

M7 ≥ 0.8

RSDPKS- test for seasonality in the seasonally adjusted series

The characteristics of the ARIMA models used by SEATS are shown in Table 4. In all but 6 cases these models are identical 

to the ones used by X11 for outlier detection and forecast extension. Note that 7 series are non-seasonal (no seasonal

differencing or seasonal parameters). SEATS does not seasonally adjust these series but does estimate a trend and irregular

component. X11 automatically seasonally adjusts all series, whether seasonal or not, and it is left to the analyst to determine 

the appropriateness of the adjustment based on an extensive set of diagnostic tests. 

Interestingly, 80 percent of the series have airline models (011)(011), which take the following form, 

∇∇

12 yt

 = (1+ θ1B)(1+ θ12B

12)at

where ∇ 

k

is the difference operator, ∇ 

k

yt = yt - yt -k , B is the backshift operator, B

k

yt = yt–yt-k and atis the white noise 

disturbance. 

With only an ordinary θ1 and seasonal θ12 moving average coefficient, this class of models is simple and relatively easy to

interpret. The MA coefficients are related to the stability of the trend and seasonal components, respectively. Specifically,

the closer these coefficients are to -1.0 the more stable are their related components. Of particular interest for this study is

that for many airline models there is a close correspondence between the model-based seasonal adjustment and the X11 

adjustment. Planas and Depoutot (1998) demonstrated that a large set of X11 trend/seasonal filter combinations can be

closely approximated with the Wiener-Kolmogorov filter of SEATS derived from the airline model with appropriately

selected regular and seasonal moving average parameters. There are also major differences between the two approaches 

particularly when seasonality is deterministic or rapidly changing which corresponds to extreme values of the seasonal MA 

parameter, θ12 near -1.0 and at the other extreme close to zero or positive. We will examine this correspondence in more 

detail shortly. 

Table 4 also shows the distribution of the MA coefficients. The bin values represent the interval greater than the adjoining 

lower bin and less than or equal to the current bin number. Focusing on the seasonal coefficient, we see that 22 percent of the 

models have values between -0.75 and -0.99, which implies stable stochastic to fixed seasonal patterns. In fact, 10 series 

have seasonal coefficients between -0.99 and -1.0, which implies a deterministic pattern where the seasonal factor for any 

given month is fixed across all years. Thirty-six percent have rapidly changing seasonal patterns ( -0.50 to 0.00). The trends 

are much less stable than the seasonal patterns since no more than 6 series have MA coefficients smaller than -0.50. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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I I I I I I 

I LI I 

: I ! 1 

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Table 4

Characteristics of SEATS ARIMA Models

(6 models changed from initial identification used by X11)

Model Form

Distribution of MA Coefficients 

θ1 θ12

Type # % Bin Freq % Bin Freq % 

(011)(011) 66 80.5 -0.99 1 1.25 -0.99 10 13.33 

(111)(011) 4 4.9 -0.74 1 1.25 -0.87 11 14.67 

(110)(011) 1 1.2 -0.50 4 5.00 -0.74 14 18.67 

(012)(011) 1 1.2 -0.25 9 11.25 -0.62 13 17.33 

(021)(011) 1 1.2 0.00 16 20.00 -0.50 14 18.67 

(211)(011) 1 1.2 0.25 27 33.75 -0.37 5 6.67 

(311)(011) 1 1.2 0.50 9 11.25 -0.25 4 5.33 

Non-seasonal 7 8.5 0.74 5 6.25 -0.12 3 4.00 

Total 82 100.0 1.00 8 % 0.00 1 % 

How similar the SEATS and X11 seasonal adjustments are in practice depends not only on how comparable their filters are 

but also on how each method automatically selects a filter. SEATS filter selection is automatically determined by the values

of the estimated MA parameters. Since these parameters vary continuously, SEATS provides an infinite range of filters from

which to choose. 

X12 ARIMA optionally implements automatic seasonal filter selection based on the so-called moving seasonality ratio 

(MSR) procedure in ARIMA/88 (Dagum, 1988). This procedure selects from a 3×3 (5 year length), 3×5 (7 year length) or

3×9 (11 year length) seasonal moving average. The choice of which moving averages to use depends on the global MSR 

ratio which is the average absolute year-to-year percentage change in the irregular ( I ) to that in the seasonal component 

( S ). 

MSR = I S = ∑njI

j ∑nj

S j

j j

where,

 ⎟ 

⎞

⎟

⎠

- ⎜ ⎜ ⎟ I ⎛

nj ⎛

nj

I

j = ∑Ii, j

I

i−1, j

nj −1, S j = ∑Si, j

Si−1, j

⎞

nj −1, j = 1K12 nj =

⎜ ⎜ ⎟ ⎝ i=2 ⎝ i=2 ⎠

 number of years with data for month j.

The larger is MSR the longer the seasonal filter tends to be and the smoother is the evolution of the estimated seasonal 

factors. This makes intuitive sense because a stable seasonal will have a small over-the-year change and thus tend to have a

large MSR. Conversely, a small MSR implies the seasonal is changing rapidly requiring a shorter filter. The actual selection

process is implemented by dividing the potential MSR values into broad zones as shown in Table 5. Two buffer zones, B and

D, have been added to prevent frequent switches in filter length for MSR values near the boundaries. If a value falls into 

either of these zones, up to 5 years of the latest data may be dropped, the MSR recomputed and if it continues to be in either 

B or D the 3x5 filter will be selected. 

Table 5 Criteria for selection of the X11 seasonal moving average based on MSR

A 

3×3 

B C 

3×5 

D E 

3×9 

0 2.5 3.5 5.5 6.5 

MSR 
Seasonal Filter Comparisons

MA12 parameter 

intervals 

Frequency of 

values

% of

seasonal 

series 

X11 filter 

closest to 

SEATS 

Frequency of 

MSR 

selection of 

X11 filters 

% of all 

series (82)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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-

• ..... .. - ----,--,- . , 

4-----~ ~ .. -• ... ~_._.'----

______-_ 

•• •I ff 

.... ,-•~•- •-•-·--·-

To see how this process relates to SEATS selection of seasonal filters we show in Figure 1 a scatter plot of the estimated

seasonal MA coefficients with the empirical MSR values for our series. As expected, large values of the MSR tend to be 

associated with small values of θ12. The correlation coefficient is -0.71. This negative relationship, however, is weakest for

θ12 values near -1.0 where the associated MSR values range from about 3.3 to 6.6.

Figure 1: Scatter Plot of Seasonal MA coefficients and MSR values 

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Seasonal MA coefficient

0 1 2 3 4 5 6 7

MSR

Matching the selection criteria in Table 5 with the range of MSR values in Figure 1, we see that the X12 automatic selection 

procedure is strongly oriented to short filters. In Table 6 we make a direct comparison of the X11 and SEATS seasonal 

filters selected for each of our series. Specifically we compare the frequency of the length of the X11 seasonal filter closest 

to the SEATS filter with the frequency of the actual seasonal filter lengths selected by the MSR procedure. 

Of the 75 series identified by SEATS as seasonal 28 percent have highly stable seasonality implying the use of the long 3×15

seasonal filter. An additional 18.7 percent have fairly stable seasonality implying the use of a 3×9 seasonal filter. 

X11 relies on the 3×5 filter for 65 series in contrast to SEATS use of this filter for 27 series. The two methods do closely

agree on the use of the 3×3 filter. The longest seasonal filter selected automatically by the MSR procedure is the 3×9 but for

only two series. One of these series appears to be non-seasonal from the SEATS perspective since it has an ARIMA model 

with no seasonal part. Nevertheless, it generated a very high MSR value. The 3×15 filter cannot be selected by X11 since it 

requires 20 years of observations which exceeds the length of our longest series. The stable seasonal option, a simple

average of the detrended values for each month separately, is available but is excluded from the MSR selection procedure.

X11 will default to this filter only for short series. 

Table 6

-0.88 to -1.00 21 28.0 3×15 0 0 

-0.75 to -0.87 14 18.7 3×9 2 2.44 

-0.51 to -0.74 27 36.0 3×5 65 79.27 

0.00 to -0.50 13 17.3 3×3 15 18.29 

Seasonal 75 

Non-seasonal 7 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Total 82 100.00 82 100.00

We investigate the appropriateness of SEATS selection of longer seasonal filters by looking at the goodness of fit of those

airline models with seasonal coefficients of -0.9 or lower and more generally by examining their spectral test for residual

seasonality in the corresponding seasonally adjusted series. Only 4 out of the 17 airline models with small seasonal MA

coefficients showed clear evidence of lack of fit. There is also the possibility that the use of a long filter could lead to under 

adjustment resulting in residual seasonality in the adjusted series. Table 7 reports no residual seasonality in almost all of the

X11 and SEATS adjustments (this is also true for the irregular series). A minor exception is the three SEATS adjusted PPI

series. These are special series having ARIMA models with no seasonal part. In this case SEATS performs no seasonal 

adjustment while X11 does but with poor quality control statistics. 

Table 7: Number of Seasonally Adjusted Series with 

Residual Seasonality in the Spectrum of their First 

Differences 

CES CPI PPI 

X11 0 1 0 

SEATS 0 1 3 

The CPI for eggs illustrates how the two methods differ in over-the-year variability in seasonal factors when θ12 is near -1.0.

The SEATS adjustment is deterministic while X-11 selects a 3×5 moving average for the seasonal filter (MSR = 4.9). The 

seasonal sub-plots are shown in Figures 2 and 3. Even though the X11 factors are changing rapidly its average seasonal 

pattern is very similar to SEATS. Looking at the spectrum of the first differenced original and adjusted series in Figure 4, 

SEATS compares favorably with X11. Both methods successfully remove the seasonal peaks in the original series but X11

removes more variation around the seasonal frequencies as a consequence of using the shorter 3x5 filter. The seasonally 

adjusted series are compared in Figure 5. Seasonality is not particularly strong in the original series and the two adjusted

series appear fairly close even with very different seasonal stability properties. 

For a few series in the analysts stage of our study we experimented with changing the automatic MSR choice of a 3x5

seasonal filter to a 3x9 when θ12 was close to -1.0. Basing the X11 seasonal filter choice on θ12 tended to result in

improvements over the MSR procedure as shown in Table 8 below. 

Table 8 : X11 Quality Control Statistics for 3×5 and 3×9 Seasonal Filters

#2 Diesel Fuel Mixed Fertilizers Publication & Printed 

Matter Switching Equipment 

Filter length 3×5 3×9 3×5 3×9 3×5 3×9 3×5 3×9 

stable F 7.60 8.5 22.7 35 57.3 94.3 6.9 10.1 

M7 0.83 0.78 0.69 0.44 0.31 0.22 0.96 0.7 

Q2 0.78 0.55 0.47 0.31 0.25 0.13 0.66 0.59 

Table 9 presents smoothness comparisons for the seasonal factors and the trend. Our focus here is on seasonality; we discuss

the trend shortly. Our measure of smoothness is the stable F-statistic applied to the seasonal factors (Table 1). This table

shows that overall SEATS seasonal factors are more stable for about two-thirds of our series. This follows, of course, from

X11’s use of shorter seasonal filters. Other studies have found similar results. Depoutot & Planas found that out of 7,372

Airline series, the closest approximating X11 filter was either the 3×15 or 3×9 for 56 percent of the series and only 26% for

the 3×5. 

Table 9: Smoothness Comparisons for 75 Seasonal Series

Number % of total 

X11 seasonal factors smoother 

than SEATS 22 29.3 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 9: Smoothness Comparisons for 75 Seasonal Series

Number % of total 

X11 trend smoother than SEATS 75 100.0 

Having illustrated how the MSR procedure of X11 is prone to selecting shorter filters than SEATS with series where the

seasonal MA coefficients were close to -1.0 we now look at the opposite extreme where seasonality is rapidly changing. 

Even here, SEATS sometimes shows an advantage with its ability to select from a much wider range of filters than X11. 

Scenic and Sightseeing Transportation series presents an extreme case of model failure resulting from the introduction of

NAICS in 2001 and the reconstruction of the original SIC based series to reflect NAICS for previous years. In 2001 there is 

a clearly visible break in seasonality (see Figure 6). The AUTOMDL spec of X12/SEATS initially fits a (3,0,1) (0,1,1)

model with parameter values that yield an invalid decomposition. SEATS first tries to approximate this model with an airline

model and estimate its parameters from the data. This fails, so SEATS goes to default values of 0.99 for the regular MA

coefficient and 0 for the seasonal MA. The first value implies an unstable trend and the second rapidly changing seasonality.

The seasonal factors derived from this model are much less stable than the X11 factors (Figures 7 and 8), but this is an 

advantage. It helps SEATS adjust to the abrupt change in seasonal pattern, which results in a better adjustment than that

provided by X11 (Figure 6). Of course, the proper way to seasonal adjust this series is to split it into two parts and adjust

each part separately, but this exercise illustrates the flexibility of SEATS filters to adapt to unstable patterns. 

Trend

We now consider comparisons of the trends for both methods. We use as our smoothness measure the root mean square of

the first difference of the trend series. Based on this measure, Table 8 reports that X11 produces smoother trends than

SEATS for all series. Neither approach, however, produces very smooth trends. This is well known from studies of the

spectral properties of the frequency response functions for the respective trend filters. Both methods produce trend filters that

have poor cut off properties in the high frequency range. For long run analysis further smoothing of the trend may be 

necessary for both approaches (see Dagum, 1996 and Kaiser and Maravall, 2001).

Reliability 

We present sliding span statistics for the initial X11 adjustments in Table 10 as an additional evaluative tool. Our focus is on

the 60th percentile of the distribution of the month-to-month change in the seasonally adjusted estimates across overlapping 

spans. The rule of thumb is that three percent is too high for C60. Based on this critical value we could not identify a single 

inadequate adjustment. This seems due to the fact that most of our series do not have very large seasonal swings over the 

year. Because of difficulties in calibrating the critical value for the C60 statistic, we have not found this test useful for 

distinguishing between adequate and inadequate seasonal adjustments. It is, however, useful for comparing methods. 

Table 10 

60 th Percentile (C60) of Change in X11 

Adjusted Series Across Overlapping

Spans

interval 

Frequency

of C60

0 to 0.81 67 

0.82 to1.08 5 

1.09 to 1.62 1 

1.63 to 2.20 2 

2.21 to 3.00 0 

Total 75 

In Table 11 we compare the sliding spans and revision statistics for X11 and SEATS. In order to have a fair comparison, we

restrict our analysis to those series with seasonal coefficients no smaller than -0.695 since the corresponding model filters are 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

····-··- ··---·····--·· ~---~~---------, 

···-···---·····--·-·--· ~ -----~-----~----~· 

very close to the X11 filters in terms of their effective filter length (Findley et al, 2003). For the sliding spans percentiles 

X11 has smaller values than SEATS for less than one third of the series. For the revision statistics, the 75th percentile for

X11 is smaller in slightly over half of the series but for the median and the maximum percent revision SEATS produces

smaller values in more than half the series. The actual numerical percentage point differences between X11 and SEATS 

(Table 12) are not very large, however.

Table 11 

Stability Comparisons, X11 Smaller than SEATS 

(33 series with airline ARIMA models with parameter values allowing sliding span

comparisons) 

cmed c60p cmax rmed r75p rmax 

Number 10 8 8 13 17 16 

% 30.3 24.2 24.2 39.4 51.5 48.5 

Table 12 

Distribution of Sliding Span Differences (X11 minus SEATS)

cmed c60p cmax rmed r75p rmax 

median 0.01 0.01 0.09 0.00 -0.01 0.00 

mean 0.02 0.03 0.18 0.07 0.03 0.03 

max 0.42 0.50 2.85 1.90 1.06 1.22 

min -0.08 -0.09 -1.16 -0.18 -0.07 -0.64 

Robustness features 

The dependence of SEATS on the overall ARIMA model is its defining characteristic. This raises two potential problems. 

First, what happens if an adequately fitting model can not be found for a series? Inadequate models may have an adverse 

effect on X11 but the potential damage is much greater for SEATS. Second, the best-fitting ARIMA model is not necessarily

decomposable into trend, seasonal and irregular components. Certain restrictions have to be imposed and not all models will 

satisfy these restrictions. What happens if the ARIMA model identified is not suitable for deriving a seasonal

decomposition?

These two problems, poorly fitting models and models inappropriate for seasonal adjustment, would seem to limit the 

applicability of SEATS. In practice, these two problems are often related: a bad fitting model is often non-decomposable.

When this happens, SEATS seeks to find a decomposable model that adequately fits the data. If this search fails, SEATS will 

default to an airline model (with judicious selection of parameter values) which can produce filters that rival the X11 filters in 

terms of good empirical properties.

Most of the time X12/SEATS’ automatic selection of model and outliers works acceptably well, but by no means always. 

Close to 20% of the seasonal series exhibit lack of model fit. Out of the 5 series where an adequate but non-decomposable

model was selected, 4 had failing or marginal diagnostics. SEATS found better fitting decomposable models for 3 of the

series. Surprisingly, in a few examples the default filter chosen by SEATS, when no adequate model fit could be found,

actually outperformed what AUTOMODEL selected. From these examples as well as others we did not find consistent 

evidence, in the presence of model failure or the lack of a decomposable initial model selection, that SEATS performed 

worse than X11 in terms of quality of the seasonal adjustment. An issue that remains, however, is gauging confidence in the

results when model fit criteria are not met. 

Simple moving averages like those used by X11 are not robust against outliers. Prior to the introduction of intervention

models and automated outlier detection, the original developers of X11 provided a partial solution to this problem which is 

still routinely implemented. The software computes a moving standard deviation with which it checks a preliminary estimate 

of the irregular or noise term for outliers. With default settings, X11 invariably identifies a number of points as extreme and 

down weights them to avoid their affecting trend and seasonal estimates. While the technique used is old-fashioned, it still 

provides X11 with a considerable level of robustness against the occurrence of one-time outliers in real time. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summary 

The major strength of SEATS is that it selects from an infinite number of filters based on the estimated characteristics of the

individual series. In contrast, X11 has a much smaller set of filters to choose from. The X11 automated filter selection 

procedure is further limited to only three seasonal filters (and three Henderson filters). While X11 is capable of handling a 

wide range of patterns, SEATS wider flexibility and its continuous filter selection process does provide important additional

capabilities but it does not come without costs. Each method has its own strengths and weaknesses. 

Identification of deterministic seasonality by SEATS in some of the series is a major difference with X11. Except for series 

shorter than 5 years, the X11 selection rule always assumes seasonality is stochastic. In terms of our generic tests this does

not result in unsatisfactory decompositions. If the user is not convinced that a deterministic result from SEATS is

appropriate, perhaps due to a suspect model, X11 may be the better choice. The use of shorter filters does protect against

under adjustment. 

The danger with shorter filters is that the seasonal component will absorb additional irregular variation and remove more 

variability from the original series than necessary for an adequate adjustment. For many of the series where SEATS

estimates fixed seasonality, X11’s automatic choice of the 3×5 filter appears to over-adjust the series as evidenced by both

QC and spectral diagnostics. In the X11 ANALY adjustments, use of the 3×9 filter allows for moving seasonality, but gives 

a more stable seasonal and improved diagnostics. 

Most often SEATS views seasonality as stochastic but still tends to produce more stable seasonal factors than X11. SEATS 

identifies 28 series as highly stable but still stochastic as compared to only two series identified by X11 with filters longer

than the 3×5. In these cases X11 filters tend to be in close agreement with SEATS filters but the X11 automated procedure

selects shorter filters. Using SEATS model parameters to suggest a longer seasonal filter for X11 may be a helpful solution. 

In particular, for some difficult-to-adjust price series, this has potential for reducing the amount of intervention treatment 

required. For many series, using the 3×9 filter reduces variability in the X11 seasonal component to an acceptable level, as

evidenced by diagnostics and graphs. 

Full flexibility of the SEATS filter selection may not always be desirable. This may be the case when observed data is

frequently revised or for short series where the ARIMA parameter estimates may be unreliable. Even though the airline 

model is robust a careful check of the adequacy of the model is important particularly when the estimated coefficients take 

extreme values. Special problems can arise with non airline models. For example, a model with one or more ordinary AR

parameters may lead to undesirable decompositions particularly where complex roots are assigned to the seasonal 

component. This can lead to unstable seasonal factors which in some cases may be avoided by allocating the roots to the

transitory component. 

Taking into account the strengths and limitation of both X11 and SEATS, our study concludes that relying solely on one 

method is undesirable. The combined use of both methods provides the user with more tools than either separately to tackle

the difficult problems encountered in seasonal adjustment. This point has been long recognized, if not often practiced, as

evidenced by a quote from early developers of the model based approach (Box, Hillmer, and Tiao, 1978),

“The empirical method [X11] and the model-based method ... are sometimes thought of as rivals. 

But they are only rivals in the same sense that the two sexes are rivals. In both cases, isolation is 

necessarily sterile, while interaction can be fruitful.” 

With the availability of X11 and SEATS in a truly integrated package we are now capable of putting this into practice at

BLS.

References 

Bell, William R. and Hillmer, Steven C. (1984), "Issues Involved with the Seasonal Adjustment of Economic Time Series"

with comments and reply, Journal of Business & Economic Statistics 2, 291-349 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Box, G.E.P., HILLMER, S.C. and TIAO, G.C. (1978). "Analysis and Modeling of Seasonal Time Series," in Seasonal

Analvsis of - Economic Time Series, ed. A. Zellner. Washington. D.C.: U.S. Dept. of Commerce. Bureau of the Census.

309-334.

Dagum, E. B. (1988). X-11-ARIMA/88 seasonal adjustment method - foundations and users manual. Statistics Canada. 

Dagum, E. B. (1996)”A New Method to Reduce Unwanted Ripples and Revisions in Trend-Cycle Estimates from X11

ARIMA”, Survey Methodology, 22, n.1, 77-83.. 

Depoutot, R. and C. Planas (1998), “Comparing Seasonal Adjustment and Trend Extraction Filters with Application to a 

Model-based Selection of X11 Linear Filters,” Tech. Paper TP361, Joint Research Centre, Ispra, Italy

Evans, T. D., S. H. Holan, and T. S. McElroy (2006), "Evaluating Measures for Assessing Spectral Peaks," 2006 Proceedings

of the American Statistical Association, Business and Economics Section [CD-ROM], Alexandria, VA, forthcoming. 

Findley, David F., Brian C. Monsell, William R. Bell, Mark C. Otto, and Bor-Chung Chen (1998), “New Capabilities and 

Methods of the X-12-ARIMA Seasonal Adjustment Program,” with discussion, Journal of Business and Economic Statistics 

16, 127-177

Findley, David F., Brian C. Monsell, et al (2005), X-12-SEATS Reference Manual, Version 0.3, Bureau of the Census, 

Washington, DC. 

Gomez, V. and A. Maravall (1996), Programs TRAMO (Time series Regression with ARIMA noise, Missing values, and

Outliers) and SEATS (Signal Extraction in ARIMA Time Series): Instructions for the User, Working Paper 9628, Servicio

de Estudios, Banco de Espana

Kaiser, R. Maravall A., 2001. Measuring Business Cycles in Economic Time Series, Lecture Notes in Statistics, 154, 

Springer-Verlag, New York. 

Maravall, A. (2003), “A Class of Diagnostics in the ARIMA-Model-Based Decomposition of a Time Series,” memorandum, 

Bank of Spain 

Scott, Stuart, Richard Tiller, and Daniel Chow (2007), " Empirical Evaluation of X-11 and Model-based Seasonal 

Adjustment Methods," Joint Statistical Meetings 2007, Business & Economic Statistics Section 

Shiskin, J., Young, A. H., and Musgrave, J. C.(1967), "The X-11 Variant of the Census Method II Seasonal Adjustment

Program," Bureau of the Census Technical Paper 15, U. S. Department of Commerce, Washington, DC
 

 

 

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Figure 2: Eggs , X11 Seasonal subplots)

108

106

104

102

100

98

96

94

92

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Figure 3: Eggs , SEATS Seasonal subplots)

108

106

104

102

100

98

96

94

92

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
 

 

 

 

 

 

# 

I 

I 

Figure 4: Eggs, Spectrum of Original and Seasonally Adjusted Series

Decibels

-15

-20

-25

-30

-35

-40

-45

Original X11 SEATS

0.000 0.083 0.167 0.250 0.333 0.417 0.500

Cycle frequency

Figure 5: Eggs, Original and Seasonally Adjusted Series

170

160

150

140

130

120

110

100

SEATS X11 

AO Dec96

Original

Jan-93 Jan-95 Jan-97 Jan-99 Jan-01
 

 

I 

,, • 

I 

I 

II I I 

I I I I 

•• 

,• 

I I 

I 

I I I 

I 

I I I 

I 

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Figure 6: Scenic Transportation, Original Seasonally Adjusted Series

38,000

34,000

30,000

26,000

22,000

18,000

Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03

SEATS X11 Original
 

 

 

J 

1.4

Figure 7: Scenic Transportation , X11 Seasonal subplots

1

0.6

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Figure 8: Scenic Transportation , SEATS Seasonal subplots)

1.4

1

0.6

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec