On X-11 Seasonal Adjustment and estimation of its variance 

 

Introduction 

 

 

 

 

 

 

 

Michail Sverchkov, Bureau of Labor Statistics & BAE Systems IT; 

Danny Pfeffermann, Hebrew University & University of Southampton; 

Stuart Scott, Bureau of Labor Statistics 

2 Massachusetts Ave, NE, Suite 1950, Washington, DC 20212, Sverchkov.Michael@bls.gov 

Most official seasonal adjustments use the X-11 method and its extensions, available for instance in the 

Census Bureau’s X-12-ARIMA software (Findley et al, 1998). An important problem with the use of this 

method is how to estimate the variances of the estimators of the seasonal effects and the other components 

that the method produces. Wolter and Monsour (1981) propose an approach to variance estimation that 

uses the linear approximation to X-11. The methods of Pfeffermann (1994) and Bell & Kramer (1999) 

build on and extend this approach. The three methods use different definitions of the variance. 

In this paper we propose new definitions of the seasonal and trend components under which the X-11 

estimators of the trend and the seasonal components are almost unbiased in the central part of the series. 

Next, we define the variance and Mean Square Error (MSE) of the X-11 estimators with respect to the 

newly defined trend and seasonal components and we show that under these definitions the variance 

estimators of Pfeffermann (1994) are unbiased. We investigate the behavior of the X-11 estimators of the 

newly defined trend and seasonal components at the two ends of the observed series where they are biased 

and suggest a bias correction procedure. 

The results are illustrated by a small simulation study based on the “Education and Health Services 

employment” (EDHS) series, obtained as part of the Current Employment Statistics program. Finally we 

estimate the bias corrected MSE of the X-11 estimators for EDHS. 

Bias, Variance and MSE of X-11 estimators and their estimation 

We begin with the usual notion that an economic time series can be decomposed into a trend or trend-cycle 

component T , a seasonal component St , and an irregular term, It ; Y = T + S + I . Here we consider for t t t t t 

simplicity the additive decomposition but the results can be generalized to the multiplicative 

decomposition, Y =×× T I S , using similar considerations as in Pfeffermann et al. (1995). Typically, the t t t t 

data are obtained from a sample survey, such that the observed value, yt , can be expressed as the 

population value, Yt, plus a sampling error, 

yt = Yt + ε t = Tt + St + et , et = It + ε t , E[et | (Sn ,Tn ;n = 1,..., N)] = 0 ; t = 0,..., N . (1) 

The X-11 program applies a sequence of moving averages or linear filters to the observed data. Thus the X11 estimators of the trend and the seasonal components can be approximated as, 

t −1 t−1

ˆ S ˆ T

St = ∑ wkt yt −k , Tt = ∑ wkt yt−k , (2) 

k =−N +t k=−N +t 

where the filters S and Tw are defined by the X-11 program options for the given time interval kt wkt

t =1,..., N . Moreover, at the central part of the series the filters are time-invariant, w

S 

= w

S , for kt k 

a ≤ t ≤ N − a , w

T 

= w

T , for a ≤ t ≤ N − a , where a , a are also defined by the X-11 program S S kt k T T S T 

T T S S options. Note also that wkt = wk = 0 if k ∉[−aT , aT ] and wkt = wk = 0 if k ∉[−aS , aS ], and 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

wkt

T 

= wkt

S 

= 0 if t − k ∉[1,..., N]. To simplify summation indexes, we denote for a given series Z , 

C C C C ∑ w Z = ∑ w Z and ∑ w Z = ∑ w Z , C = S or T . kt t−k kt t−k k t−k kt t−k 

k C k C : ≠0 k:wk k w ≠0

kt 

Remark 1. X-11 and its extensions, like X-12 ARIMA include also “non-linear” operations such as the 

identification and estimation of ARIMA models and the identification and gradual replacement of extreme 

observations. We assume that the time series under consideration is already corrected for outliers. The 

effects of the identification and non-linear estimation of ARIMA models are generally minor, see, e.g., 

Pfeffermann et al. (1995) and Pfeffermann et al. (2000). 

Assuming that St

 and Tt are well defined (although never observed) for −∞ < t < ∞ , define, 

x11 S x11 T x11 x11 St = ∑ wk (Tt −k + St −k ) , Tt = ∑ wk (Tt −k + St −k ) , such that St

 and Tt are the outputs of 

k k 

applying the symmetric filters to the signal of the infinite series at each time point t = 1,..., N. Denote 

x11 S = {St ,−∞ < t < ∞} and T = {Tt ,−∞ < t < ∞}. Note from (1) that St = E[S

ˆ 

t | S, T] when 

x11

a ≤ t ≤ N − a and T = E[T

ˆ

| S, T] when a ≤ t ≤ N − a , which implies the following obvious S S t t T T 

result. 

x11 x11 x11 Result 1. Let e = y −T − S . X-11 decomposes therefore the observed series into the ‘X-11- t t t t 

x11 x11 x11 trend’ Tt

, the ‘X-11-seasonal component’ St, and the ‘X-11 error’, et; 

x11 x11 x11

y = T + S + e , (3) t t t t 

and at the center part of the series, max(aS , aT ) ≤ t ≤ N − max(aS , aT ) , the X-11 estimators of the trend 

and the seasonal components are almost unbiased with respect to the decomposition (3). 

Remark 2. The decomposition defined by (3) into a seasonal component, a trend component and an error 

term is clearly not unique; see, for example, the discussion in Hilmer and Tiao (1982). Bell and Kramer 

(1999) use a similar decomposition: they define the “target” of the seasonal adjustment as the adjusted 

series that would be obtained if there was no sampling error and there are sufficient data before and after 

the time points of interest for the application of the symmetric filter (Bell and Kramer 1999, page 15). 

Thus, the Bell and Cramer seasonal and trend components are defined as, 

Bell,Kramer S Bell,Kramer T St = ∑ wk (Tt−k + St−k + It−k ) and Tt = ∑ wk (Tt−k + St−k + It−k ) . The difference 

k k 

between (3) and the Bell and Kramer decomposition is therefore that the latter decomposition considers the 

irregular term as a part of the signal. As a result, the MSE of the X-11 estimators of the components 

defined by the decomposition (3) ( see below), is generally higher than the MSE of the X-11 estimators of 

the components defined by the Bell and Kramer decomposition. 

The bias, variance and MSE of the X-11 estimators with respect to decomposition (3), conditional on the 

true components S, T are obtained as follows: 

S S Bias[S

ˆ 

t | S,T] = ∑(wkt − wk )(Tt −k + St −k ) (4) 

k 

S S 2

Var[S

ˆ 

t | S, T] = E{[∑ wkt yt −k − E(∑ wkt yt −k | S,T)] | S,T} =

k k 

S 2 S 2 E{[∑ wkt ( yt −k − St −k −Tt −k )] | S,T} = E[(∑ wkte ) | S,T] , (5) t −k

k k 

x11 2 ˆ 2 ˆ

[

ˆ ] = E[(Sˆ MSE S − S ) | S, T] =Var S [| [ S, T ] + Bias S | S, T ], (6) t t t t t 

and similarly for the trend. 

By Eq. 5, the variance of the X-11 estimator of the seasonal component is a linear combination of the 

covariances, Cov(et , ek ) , t, k = 1,..., N . Following Pfeffermann (1994), let 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ˆ ˆ S T S T Rt = yt − St −Tt = ∑ akt yt −k , a0t = 1− w0t − w0t , akt = −wkt − wkt , k ≠ 0 , 

k 

define the linear filter approximation of the X-11 residual term. Then, 

Var(Rt | T,S) = E{[∑ akt ( yt −k − E( yt −k | T,S)]2 | T,S} =Var(∑ aktet −k | T,S) 

k k 

Cov R ( , R | T,S ) = Cov ( a e , a e | T,S ) = a a Cov e , e | T,S )

t m ∑ kt t −k ∑ km m−k ∑∑ kt lm ( t −k m−l . (7) 

k k k l 

It follows from (7) that v = Cov ( , e e | , ) ST , tm , =1,..., N , and u = CovR (, , R | S T) , t m, =1,..., N ,

tm t m tm t m 

are related by the system of linear equations, 

U = DV , (8) 

where the matrix D is defined by the weights a tk kt , , = 1,..., N through (7). Since the X-11 residuals, Rt

are observed for t = 1,..., N , (and assuming et

 is independent of the true trend and the seasonal 

components), Cov R ( ,t Rk | , S T) can be estimated from the observed series at least at the central part of the 

* * * * series, t = t ,..., N − t for some t * > 0 . However, the number of equations in (8) for t = t ,..., N − t is 

smaller than the number of unknown covariates v = Cov ( , e e | , S T) , and therefore (8) can not be solved tm t m 

directly and the solution is very unstable. A possible way to overcome this problem is by assuming that the 

covariances vtk are negligible (and hence set to zero) for | tk − | > C for some constant C, which allows 

then to solve the reduced set of equations obtained from (8). See Pfeffermann (1994), Pfeffermann and 

Scott (1997) and Chen et al. (2003), for different approaches to the estimation of U and V . 

Remark 3. Pfeffermann (1994) developed his variance estimators under the Postulate: 

∑ akt (St −k + Tt −k ) ≅ 0 at the center of the series. Although this assumption seems to hold approximately 

k 

x11 x11 in practice, it is essentially impossible to test it. Note that this Postulate implies that (T,S) = (T ,S ) 

at the center of the series, which is not generally true, see the results of the simulation study below. On the 

other hand, as shown above, Pfeffermann’s (1994) method produces consistent estimators for the variance 

defined by (5). 

Estimation of the MSE of the X-11 estimators is complicated. The error term, et , can be usually assumed 

to be independent of the true trend and the seasonal components, and therefore the variance in (5) does not 

depend on the signal. On the other hand, by (4), the bias of the estimator is a function of S,T and its value 

depends on the particular realization of the signal and therefore estimating of the bias requires strict model 

assumptions that could be hard to validate. Instead of estimating the MSE given the trend and the seasonal 

components, we propose therefore to estimate instead the expected MSE, 

x11 2 ˆ 2 ˆ E{MSE[S S S,T { [ E Bias S,T ]} ˆ ]} = E{E[( ˆ − S ) | ]} = E Var S | S,T ]} + { [S | . (9) t t t t t 

Note that E{MSE[S

ˆ 

t

]} can be considered as the best predictor of MSE[S

ˆ 

t

] under a square loss function. 

Assuming that the error term, et is independent of the true trend and the seasonal components, the first 

term in (9) does not depend on the signal and therefore it can be estimated by use of Pfeffermann (1994) 

method. The second term can be estimated by the following parametric bootstrap procedure, which is 

illustrated in the simulation study: 

(a) Fit a parametric model and estimate the parameters of the separate models identified for the trend, the 

seasonal component, the irregular term and the sampling errors. See Steps 1–3 of the simulation study 

where we use the models identified by SEATS, accounting for the sampling error information. 

(b) Generate B series, y

b 

,b = 1,..., B , each of sufficient length for applying the symmetric filters to the 

central N months of the generated series, by independently generating the four component series, and store 
 

 

 

 

 

 

 

 

 

 

the trend and the seasonal components. For each generated series compute the bias, 

b S S b b Bt = ∑(wkt − wk )(Tt−k + St−k ) , see Steps 4-6 of the simulation study. 

k 

2 ˆ b )2

(c) Estimate E Bias { [St | S,T]} by averaging (B over b = 1,..., B ., see Eq. 10 . t 

Remark 4. Bell and Kramer (1999) estimate the unconditional MSE of the X-11 estimator with respect to 

Bell,Kramer Bell,Kramer Bell, Kramer 2

S ,T , that is, they estimate E[(S

ˆ − S ) ] , instead of estimating t t 

Bell,Kramer 2 E[(S

ˆ − S ) | S,T], which is similar to the use of (9). t t 

Simulation study 

We illustrate the implications of the use of the new definitions of the trend and the seasonal component in 

(3), and their estimation, and compare it to the estimation of the corresponding ‘true’ components by use of 

simulations. The simulations use the models fitted to the series “Education and Health Services 

employment” (EDHS), with observations from January, 1996 through December, 2005, (N=120). 

We also estimate the MSEs of the X-11 trend and seasonally adjusted estimators of the true series. 

Our interest in this series is in the month-to-month change in employment. As explained in Scott, et al. 

(2004), we consider the log ratios of the EDHS series, corrected for outliers as the original series (see 

Remark 1). 

As the main objectives of the study we consider the estimation of the trend and the seasonally adjusted 

(SA) series. Correspondingly to decomposition (3), the definition of “X-11 SA series” is, 

x11 x11 ˆ −

ˆ A At = Yt − St . X-11 estimate of the SA series (SAE) is defined as At = yt St = ∑ wkt yt −k where 

k 

A S A S w = 1− w , w = −w , k ≠ 0 . 0t 0t kt kt 

The study consists of the following steps: 

Step 1. Fit an ARIMA model to the observed series using X-12-ARIMA.. 

Step 2. Re-estimate the parameters of the model by use of REGCOMNT program (Bell, 2003), accounting 

for the presence of the sampling error component. (We model this component using the autocovariance 

estimates of the sampling errors as computed by the Bureau of Labor Statistics.) 

Step 3. Identify the trend, seasonal, and irregular component models by use of signal extraction, available 

in the experimental software X-12-SEATS developed by the Census Bureau. The models and parameter 

estimates used in the simulation study are as follows: 

Trend- Tt ; ARIMA(1,1,2) with parameters -.90, .06, -.94 and disturbance variance 0.5; 

Seasonal component- St ; ARIMA(11,0,11) model with AR-coefficients equal to 1, MA-coefficients equal 

to, .70, .42, .17, -.04, -.20, -.30, -.37, -.39, -.38, -.34, -.28, and disturbance variance 4.5; 

Irregular component- I

t ; white noise with disturbance variance 18.0; 

Sampling error- ε t ; MA(1) with MA-coefficient -.15 and disturbance variance 58.68. 

Step 4. Generate independently 3,000 series from the component models developed in Step 3 and add them 

up to form new original series yt

b . Each generated series has length N+96, N = 120 , so that the application 

of X-11 to the full series gives values approximately equal to the “final” X-11 values for the central N 

points. Store the series yt

b and their components, b = 1,...,3,000 . 
 

 

 

 

 

 

 

 

 

 

Estimation of MSE of X-11 Seasonally Adjusted and Trend Estimators for EDSH series 

 

 

 

Step 5. Fix the form of the X-11-ARIMA model for the original series. For each series generated in Step 4 

estimate the parameters of the ARIMA models and compute the filter weights wkt

A and wktT

, reflecting 

extrapolation with forecasts based on the model identified for the series. 

Step 6. For each series generated in Step 4 and the weights obtained in Step 5 define, 

ˆ b A b b T b T A ˆ

t = ∑ wkt yt−k , t = ∑ wkt yt −k , 

k k 

~b b b A b b b b ~ b T b b At = E(A

ˆ 

t | T

b ,S ) = ∑ wkt (Tt−k + St−k ) , Tt = E(T

ˆ 

t | T

b

,S ) = ∑ wkt (Tt −k + St −k ) , 

k k 

x11,b A b b x11,b T b b At = ∑ wk (Tt−k + St−k ) , Tt = ∑ wk (Tt −k + St −k ), 

k k 

, b b A A b b Tb b b T T b b [

ˆ [ B ˆ

t

Ab 

= Bias A t | S ,Tb ] = ∑(wkt − wk )( Tt −k + St −k ) , Bt 

, 

= Bias T t | S ,Tb ] = ∑(wkt − wk )( Tt −k + St −k ) , 

k k 

t = 48,..., N + 48 ; b = 1,...,3,000 . 

Step 7. Compute the Empirical Root MSE of the X-11 estimators with respect to the ‘true components’ 

b x11,b x11,b Ab 

,T and with respect to the ‘X-11 components’ A ,T , and the Empirical Standard Deviation 

(SD) of the X-11 estimators 

3,000 3,000 b 1 b b b 1 b b RMSET,S (A

ˆ 

t ) = ∑ (A

ˆ 

t − At )

2 

, RMSET,S (T

ˆ 

t ) = ∑ (T

ˆ 

t −Tt )

2 ;

3,000 b=1 3,000 b=1 

3,000 3,000 b 1 b x11,b 2 b 1 b x11,b 2 RMSEx11(A

ˆ 

t ) = ∑ (A

ˆ 

t − At ) , RMSEx11(T

ˆ 

t ) = ∑ (T

ˆ 

t −Tt ) ;

3,000 b=1 3,000 b=1 

3,000 3,000 ˆ b 1 ˆ b % b 2 b 1 b ~b )

2

SD A () = (A − A ) , SD(T

ˆ

) = (T

ˆ − T ,

t ∑ t t t ∑ t t 3,000 b=1 3,000 b=1 

Step 8. For each series generated in Step 4 estimate the variances of the X-11 estimators by the method 

developed in Pfeffermann (1994); see Pfeffermann (1994) and Pfeffermann and Scott (1997) for details. 

b b Denote the estimates by Vˆ

(A

ˆ 

t

) and Vˆ

(T

ˆ 

t ) , and define, 

∧ 3,000 ∧ 3,000 

ˆ b 1 

ˆ ˆ b ˆ b 1 ˆ ˆ b

SD A V A () = ( ) , SD T () = V T ( ) .

t ∑ t t ∑ t

3,000 b=1 3,000 b=1 

For the original EDSH series estimate the variances of the X-11 estimators by the method developed in 

A b, , T b Pfeffermann (1994). Denote the estimates by Vˆ

(A

ˆ 

t

) and Vˆ(T

ˆ 

t ) . Using the estimates Bt

 and Bt

obtained in Step 6 of the simulation study compute the estimates of the squared bias when estimating the 

‘X-11 components’, 

3,000 3,000 

ˆ{

2 ˆ 1 A b, 2 ˆ

2 ˆ 1 T b, 2 E Bias [At | S,T ]} = ∑ (Bt ) , E Bias { [Tt | S,T ]} = ∑ (Bt ) , (10) 

3,000 b=1 3,000 b=1 

and define the estimate of (9) for the original series as, 

∧

ˆ ˆ ˆ ˆ 2 ˆ ∧

ˆ ˆ ˆ ˆ 2 ˆ RMSE A V A () = () + E Bias { [A | S,T ]}, RMSE T () ( { = V T ) + E Bias [ | T S,T]}. (11) t t t t t t 

The results of the study are summarized in Figures 1–4. 
 

Figure 1. Empirical SD and Root MSE of X-11 SA Estimators when estimating the true SA series, 

b b b x11,b b x11,b A = Y − S and when estimating the X-11 SA series A = Y − S . 

∧

b b b b SD(A

ˆ 

t ) is drawn in red, SD(A

ˆ 

t ) in blue, RMSET,S (A

ˆ 

t ) in black and RMSEx11(A

ˆ 

t ) in green. 

Figure 2. Empirical SD and Root MSE of X-11 Trend Estimators when estimating the true Trend, T

b

 and 

T

x11,b when estimating the X-11 Trend . 

∧

b b b b SD(T

ˆ 

t

) is drawn in red, SD(T

ˆ 

t ) in blue, RMSET,S (T

ˆ 

t ) in black and RMSEx11(T

ˆ 

t ) in green. 
 

 

 

 

x11,b b x11,b

Figure 3. Empirical Root MSE of X-11 SAE when estimating the X-11 SA series A = Y − S , 

and estimates of the SD and Root MSE of X-11 SAE when estimating the X-11 SA series for the original 

EDHS series (see Eq. 11). 

∧ ∧ b RMSEx11(A

ˆ 

t ) is drawn in green, RMSE(A

ˆ 

t ) in red and SD(A

ˆ 

t ) in blue. 

Figure 4. Empirical Root MSE of the X-11 trend estimator when estimating the X-11 trend component 

T

x11,b 

, and estimates of the Root MSE and Standard Deviations of the X-11 trend estimator when 

estimating the X-11 Trend component for the original EDHS series (see Eq. 11). 

∧ ∧ b RMSEx11(T

ˆ 

t ) is drawn in green, RMSE(T

ˆ 

t ) in red and SD(T

ˆ 

t ) in blue. 

Conclusions. 

1) In this study the Empirical RMSEs of the X-11 estimators when estimating the true components are 

higher than the Empirical RMSEs when estimating the X-11 components defined by (3), which illustrates 

that the X-11 decomposition (3) is different from the ‘true’ decomposition used in the simulations. 
 

 

 

 

 

 

2) For the 5 years in the center of the series the X-11 estimators are almost unbiased when estimating the 

newly defined X-11 components but at the beginning and at the end of the series there are nonnegligible 

biases. 

3) Pfeffermann (1994) estimate of the variance approximates closely the empirical variance of the X-11 

SAE when estimating the X-11 SA component, and overestimates slightly the empirical variance of the X11 Trend estimator when estimating the X-11 trend. 

4) Figures 3 and 4 illustrate that the proposed RMSE estimator (Eq. 10) corrects the bias incurred by the 

use of Pfeffermann (1994) variance estimators at the ends of the series when estimating the newly defined 

X-11 components. More extensive simulations are needed in order to test that the bias corrections are of the 

right magnitude. 

References 

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2003, Statistical Research Division, U.S. Census Bureau. 

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Procedure: Spectrum Approach and Comparison,” Statistics Canada Report BSMD-2003-001E 

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