A GENERAL METHOD FOR ESTIMATING THE

VARIANCES OF X11-ARIMA ESTIMATORS

Luisa Burck, Israel Central Bureau of Statistics

Michail Sverchkov, Hebrew University

Abstract

The X11-ARIMA method, or X12 method based on the latter procedure, with their various variants are the most

commonly procedures used for estimating the seasonally adjusted data and the trend-cycle. Both of these procedures

fail to provide estimates for the variances of the estimators that they produce. In this paper we propose a simple

general method, based on linear approximation, for estimating the variances of the X11-ARIMA estimators. This

method is based on Pfeffermann (1994) but is extended for any multi-stage run. The variances account for the

sampling error of the survey estimators and for the variability of the trend, seasonal and irregular components

defining the decomposition model. We demonstrate the application of this suggested method to Israeli time series.

Keywords: X11-ARIMA, seasonal adjustment, decomposition model, linear approximation.

Background

Statistical Agencies throughout the world publish each month seasonally adjusted figures for a large number of

series. In recent years, some statistical agencies started to publish trend-cycle estimates as a complement to

seasonally adjusted data to help reveal better the movements in the series and the occurrence of turning points. The

X11-ARIMA method, or X12 method based on the latter procedure, with their various variants are the most

commonly procedures used for estimating the seasonally adjusted data and the trend-cycle. However, both of these

procedures fail to provide estimates for the variances of the estimators that they produce. Thus, while national

statistical institutes publish estimates for the variances of the unadjusted estimators, no estimates are usually

published for the variances of the seasonally adjusted estimators and the trend-cycle estimators.

Various approaches for obtaining variances for X11 or X12 estimators have been proposed. Wolter and Monsour

(1981) suggested two approaches, one that accounts only for the effect of the sampling error and one that reflects

also the uncertainty due to stochastic time series variation. Burridge and Wallis (1985) investigated the use of the

steady-state Kalman filter for calculation of model based seasonal adjustment variances for approximating the X11

filters. Pfeffermann (1994) developed an approach that recognizes the contributions of sampling error and irregular

variation to the X11 seasonal adjustment variances. Bell and Kramer (1999) developed an approach to obtain

variances for X11 adjustments accounting for sampling error and the errors from forecasting extension.

Since May 1996, the Central Bureau of Statistics of Israel uses a specific sequential application of X11-ARIMA

method in order to produce seasonally adjusted data and trend estimators: the modified Henderson procedure (see

Dagum (1996)). This specific application can be described by the following two steps. In the first step, we apply

X11-ARIMA with default options to the original data and estimate the combined linear effect of the trading day and

moving Jewish festivals from the irregular estimators. Using these prior adjustment factors, we apply X11-ARIMA

in order to obtain the best estimators of the seasonally adjusted data. The second step is carried out through another

X11-ARIMA run, with different variants, on the modified adjusted series from step one and produces the final trend

estimators. Thus, unlike one-step application of X11-ARIMA, the seasonal component is obtained from step one and

the trend-cycle component is obtained from step two. The problem of estimating the variances of the estimators

become more difficult in the specific sequential application described above.
One-stage Estimation of the Variances of X11 Estimators by Pfeffermann (1994)

Let { denote the observed series of the survey estimates. First, we assume the additive

decomposition model

yt: t = 1,...,T } 

yt = Tt + St + et(1)

where Tt

is the trend-cycle level, Stis the seasonal effect and etis the compound error term (usually et = I

t + et

where I is the irregular component and etis the survey error). t

It is assumed that et: t = 1,...,T are stationary and E(et

) = 0 where the expectation is with respect to all random

sources.

Following Young (1968) and Wallis (1982), the seasonal component estimator S

ˆ ˆ

t

and the trend-cycle estimator Tt

,

computed by application of X11 linear filters, can be represented by the linear approximation in the form

T −t

Tˆ

t ≈ ∑ wkt yt +k = wt

′y

k=− (t −1)

(2)

T−t

~ ~ S

ˆ

t ≈ ∑wkt yt+ k = wt

′y

k =−(t −1)

(3)

W

~

for t=1,...,T. The matrices and W can be obtained from respective filters of X11 (programs which calculate

these matrices were developed in Statistics Canada and later in BLS).

Under the assumption that the estimators of the seasonal component and the trend level are unbiased we can define

the variances that we want to estimate as:

ˆ

T −t

var(Tt) ≈ var( ∑ wktet +k) , t=1,...,T

k =−(t−1)

(4)

T −t

~ var(S

ˆ

t

) ≈ var( ∑wktet+ k) , t=1,...,T

k= −(t −1)

(5)

T−t

var(Tˆ

t − Tˆt+ s

) ≈ var( ∑(wkt − wk −s,t+ s)et+k) , t=1,...,T-s

k =−(t−1)

(6)

T −t

~ var(S

ˆ

t − S

ˆ

t +s

) ≈ var( ∑(w

~

kt − wk−s,t+ s

)et+ k) , t=1,...,T-s

k=− (t −1)

~ W and W are 

(7)

Since the matrices known and non-random then the problem of estimating the variances above

reduces to estimation of all covariances cov(e , e ). t k

Let R = y − T − S t t t t

ˆ ˆ ˆ define the linear filter approximation to the X11 residuals. Then Rtˆ can be expressed as

T−t T −t T −t

Rˆ

t ≈ ∑akt yt +k = ∑akt(Tt +k + St +k

) + ∑ aktet+ k

k =−(t−1) k=− (t −1) k=− (t −1)

(8)

~ a = 1− w − w if t=k and a = −w − w~ if t ≠ k . where kt kt kt kt kt kt

Pfeffermann (1994) uses the following two properties in order to estimate the variances of the linear combinations of

the error terms et

from the variances of Rˆ

t

.

1. In the center of the series, postulate:

T−t

∑a (Tt + S ) ≈ 0 .kt + k t +k

k =−(t−1)
2. The X11 residual series Rˆ

t

,t = 25,...,T − 24 , is stationary.

Since Rˆ are produced by X11, autocovariances of the series Rˆ can easily be estimated, for example, as t t

T −25

coˆv(Rˆ

t

,Rˆ

t− k

) =

1

∑(Rˆ

t − R)(Rˆt −k − R)

T − 48 t =25+k

and therefore the autocovariances of the series etcan be obtained by solving the resulting equations for the

unknown do not

depend on t. In practice, these autocovariances damp to zero so that it can be assumed that, for k>C (cutoff value),

cov(et, es) . Note that since {e is assumed to be stationary then t,t = 1,...,T } cov(et,et+k)

cov(et,et+k) = 0 . This reduces the number of equations required for the estimation of the variance and the

autocovariances of the error terms to C+1 (see Pfeffermann (1994)).

Now, the variances defined by (4)-(7) can be estimated by:

 

T−t T−t

var(Tˆ

t

) ≈ ∑ ∑wktwlt coˆv(et +k,et+l) , t=1,...,T

k =−(t −1) l= −(t −1)

(9)

 

T−t T −t

~ ~ var(S

ˆ

t

) ≈ ∑ ∑ wktwlt coˆv(et+ k,et +l) , t=1,...,T

k =−(t−1) l =−(t−1)

(10)

T−t T−t

var(Tˆ

t −Tˆt+ s

) ≈ ∑ ∑(wkt − wk −s,t+ s)(wlt − wl− s,t +s)coˆv(et +k,et +l) , 

k =−(t −1) l =−(t −1)

t=1,...T-s (11)

T−t T −t

~ ~ ~ var(S

ˆ

t − S

ˆ

t +s

) ≈ ∑ ∑(wkt − wk −s,t+ s)(w

~

lt − wl− s,t +s

)coˆv(et +k,e

k =−(t−1) l =−(t−1)

t=1,...T-s.

t +l

) ,

(12)

Now consider the multiplicative decomposition . It is known that the multiplicative model of X11

gives similar results to the log-additive decomposition, i.e. applying the additive model to the new series

~

yt = Tt Stet

ˆ ~ ~

)

~yt = log( yt

. The method considered above yields and ˆ

var(T ) and var(S ) . Define T = exp(T )

t t

S = exp(S t t)

~

. Assume that have approximately normal distribution. Utilizing the relationship between

~ ~ T and S t t

ˆ ˆ

the variance of the normal and the lognormal distributions we can write

ˆ

2 ⎡  ̆ var(Tˆ

) = T exp(2 var(T

~ˆ

)) − exp(var(T

~ˆ

)) t t t t

⎢⎣  ̇ ̊ (13)

~ ~ ˆ 2 ⎡  ̆ var(S

ˆ

) = S exp(2 var(S

ˆ

)) − exp(var(S

ˆ

)) t t t t

⎢⎣  ̇ ̊ (14)

(See Pfeffermann (1995) for more details).

Multi-stage Estimation of the Variances of X11-ARIMA Estimators

As described above, the Central Bureau of Statistics of Israel uses a specific sequential application of X11-ARIMA

method in order to produce seasonally adjusted data and trend estimators. Thus, the estimates S

ˆ and Tˆ are the t t

results of sequential application with different options on each step. Assuming linear approximation is valid on each

W W

~

step, the variances of these estimators can also be estimated by (2) and (3) but with some other matrices and .

These matrices can be obtained analytically if one knows the respective weight matrices on each step but, even in
 

 

this case, calculations of these matrices can be not trivial. Here, we propose a simple general procedure in order to

W

~

obtain and W for any given series { yt}.

1. For any given series { t

y , t=1,...,T} and some vector c = (c1,...,cm) define T new series {y

m

t

, t=1,...,T,

m=1,...,T} by y

m m

t = yt

if m ≠ t and yt = yt − cmif m = t

m

. Note that for the series { yt} all the

observations except the m observation coincide with the observations of the original series { yt

}.

2. m Apply all sequence of X11-ARIMA steps independently to all series { yt} as defined above in order to obtain

the estimates ˆ),(Tˆ

1

(T , S

ˆ

, S

ˆ

1 T T ),...,(Tˆ

, S

ˆ

)

t t t t t t

are obtained from the

sequential application of X11-ARIMA to the original series 

. Note that the estimates (Tˆ, S

ˆ

t t )

from the same application to series

{

1

yt} ,etc.

{ y

1 1

t

}, (Tˆ, S

ˆ

t t )

3. Define the required weights by:

ˆ ˆt +k

c Tt − Tt w = , k=-(t-1),...,T-t kt

ct+k

(15)

ˆ ˆt+k

~c St t − S

w = , k=-(t-1),...,T-t . kt

ct +k

(16)

Under the assumption that (2) and (3) hold for the input series {yt

}, one can easily verify that for any cm,

m=1,...,T the weights w

c

and ~c

kt wkt obtained by this procedure coincide with the required weights.

With real data some or all steps of the estimation usually include some non-linear options such as identification and

gradual replacement of outliers, identification and estimation of Arima models for augmenting the series by one year

(or two years) of extrapolated values, pre-adjusting for trading day variation, etc. As a result, in general, equations

(2) and (3) need not be satisfied with the same weights for any input series. On the other hand even under all the

above non-linear options one can expect that for small enough oscillation of input series (2) and (3) are fulfilled with

W

~

almost the same matrices and W .

Now we can write the properties essential for the estimation of the variances from Pfeffermann (1994) as follows:

T −t

c

Tˆ

t ≈ ∑ wkt yt +k

k=−(t−1)

(17)

T−t

~c

S

ˆ

t ≈ ∑wkt yt +k

k =−(t −1)

(18)

T −t T−t

c c Rˆ

t = yt − Tˆt − S

ˆ

t = ∑akt yt+ k ≈ ∑aktet +k

k= −(t −1) k =−(t−1)

(19)

where . The residual series {Rˆ } is stationary in the t

center of the series.

a

c

= 1− w

c

− w

~c

kt kt kt if t = k and a

c

= −w − w

c ~

kt kt

c

if t ≠ k kt

In the one-stage additive case, the respective weights are known and thus it is very convenient to reduce the problem

of variance estimation of multiplicative X11-ARIMA estimators to estimating their log-additive analogue. In our

case, since we do not assume that the weights for log-additive analogue of the series to be known, a more simple and

direct method can be considered. Following the arguments in the previous section, the multiplicative multi-stage

X11-ARIMA estimates Tˆ and S

ˆ may be by t t approximated 
T −t

c

log(Tˆ

t

) ≈ ∑wkt log( yt +k)

k=− (t −1)

(20)

T−t

~c

log(S

ˆ

t

) ≈ ∑wkt log( yt+ k).

k =−(t −1)

(21)

Note that the above procedure for estimating the weights is applied to the logarithms of the original series and the

estimates.

Testing the Estimated Matrices of Weights

Let

T−t

dt(T ) = Tˆ

t − ∑wkt

c

yt+ k

k =−(t −1)

and

T−t

~c

dt(S) = S

ˆ

t − ∑wkt yt +k

k =−(t−1)

define the error terms associated with the approximations (17) and (18). Consider

1/ 2 ⎛ 1

T

2 ˆ 

s (T ) = ⎜ ∑d t

(T)÷

⎝ T t =1  ̄ 

and

1/ 2 ⎛ 1

T

2 ˆ 

s (S) = ⎜ ∑dt

(S)÷

⎝ T t=1  ̄ 

as measures of exactness of these approximations respectively. These measures can easily be used in order to verify

the properties (17) and (18). For example, if these measures are comparably small with respect to the variation in the

original series given t, then the weights can be regarded as satisfactory.

Let us now consider (19). Since et

is not accessible, the exactness of the approximation can not be checked directly.

On the other hand, for any sufficiently smooth function gt

(candidate for trend) it is assumed that

T−t

c ∑akt gt +k ≈ 0 .

k =−(t−1)

Fit a regression model for the input series yt, on the data suitable (not very complicated) assuming that the

residuals are independent. Let e

~

t

denote to the residuals of the regression. Note that e ~ ≠ e but there exist a t t

sufficiently smooth function g

~

t

so that e ~ = e + g

~

t t t

. Let

T−t T−t

c c ~ dt

(e) = ∑akt yt +k − ∑aktet +k

k =−(t−1) k =−(t −1)

define the error term associated with approximation (19). Consider

1/ 2

⎛ 1

T

2 ˆ 

s (e) = ⎜ ∑dt

(e)÷

⎝ T t =1  ̄ 

as a measure of exactness of the approximation (19).
 

The above analysis can help to define the vector c and thus the required weights. Note that for any cm

if the

statistics suggested above have high values then linear approximation cannot be used for the analysis.

Real Data Examples

This section contains two real data examples that illustrate the application of the method. For each data set the

following steps were carried out:

1. Consider sequentially five vectors of constants c=(c,...,c) where c=1.1, 1.01, 1.001, 1.0001, 1.00001.

2. For any given series { yt

} and each vector c define auxiliary series { if myt}, m=1,...,t such that y

m

t = yt

m m ≠ t and ym = ym

/ c.

3. Apply all the sequence of X11-ARIMA steps independently to all series myt{ } in order to obtain the multistage X11-ARIMA estimators Tt

, 

m m T , S and S for t=1,...,T and m=1,...,T. t t t

ˆ ˆ ˆ ˆ

4.

~

For each vector c calculate matrices W and W as

c t +k

w = (log(Tˆ) − log(Tˆ))/ log(c )

kt t t t+k

 and

~c t +k

w = (log(S

ˆ

) − log(S

ˆ

)) /log(c ) .

kt t t t+ k

W

~

 Five pairs of matrices and W corresponding to five vectors c are calculated.

5. Calculate the statistics d (T ), d (S) and d (e)

t t t

as defined above. Use the residuals of 3-d order polynomial

regression of log( yt

) against t , t

2

and t

3

in order to computed (e)

t

. If all of the following conditions:

(One) in the center of

the series the weights are time invariant:

 

c c ~c ~ wi ,k ≈ wi+ j ,k + j

, wi ,k ≈ wi+ j ,k + jfor 25<i,k and i+j, k+j<T-25;

(Two) the s -

statistics are comparably small with respect to variation of { yt

\ t}:

max(s (T ),s (S),s (e)) < std(e

~

t

\ t)

~

are satisfied for a vector c then the corresponding weight matrices W and W are used and the final variances are

calculated by the Pfeffermann (1994) method. If the above conditions are satisfied for more than one vector c, then

the weights corresponding to the vector with lowest values of the statistics are used.

6. If one of the conditions of the previous step is not satisfied then the steps 1 - 5 are repeated with linear options

of X11-ARIMA. Let

T−t

d (lin) (T ) = Tˆ − w(lin)

c

y ,

t t ∑ kt t +k

k =−(t−1)

T −t

c

d (lin)

t

(S) = S

ˆ

t − ∑w

~(lin)

kt yt+ k

k= −(t −1)

define the error terms where Tˆ and S

ˆ are trend and seasonal component estimates obtained in step 3. Consider the t t

statistics:

⎛ 1

T

2 ˆ

1/ 2

s (lin)(T ) = ⎜ ∑d (lin)

t

(T )÷

⎝ T t=1  ̄ 

and
1/ 2 ⎛ 1

T

2 ˆ 

s (lin)(S) = ⎜ ∑d (lin)

t

(S)÷ .

⎝ T t =1  ̄ 

If max(s (lin)(T),s (lin)(S)) < std(e

~

t

\ t)then the weights W (lin) and W (lin)

~

are used and the final

variances are calculated by the Pfeffermann (1994) method. If not, then we conclude that for a given series the

method described above cannot be used.

Total Industrial Production Index. The data was processed through X11-ARIMA program for the time span from

January 1990 to November 1999 (119 observations) using (i) multiplicative decomposition model; (ii) the best

options for seasonal adjustment (including adjustments for trading day and moving festivals) and (iii) the modified

Henderson trend estimation.

In step 4, for c=0.0001 the following values of the statistics were obtained: s (T) = 0.0249, s (S) = 0.0306 ,

s (e) = 0.0005. These values are small enough with respect to std(e

~

t

\ t) ≈ 0.05 so that the second condition

in 5 is satisfied. This condition is not satisfied for c ≠ 0.0001. On the other hand, the first condition concerning

the weights being time invariant is not satisfied for the weights when c=0.0001. In step 6, the following values were

obtained: s (lin)(T) = 0.0264 , s (lin)(S) = 0.0351and they are almost equal for all vectors c. The values of

the weights almost coincide in the center of the series with the respective weights obtained by the program

developed in BLS. Clearly, the condition concerning the weights being time invariant is satisfied and therefore we

~

can apply the weights W (lin) and W (lin). We should mention that we have compared the X11-ARIMA

~

estimators with the estimators based on W and W , and the estimators based on W (lin)

~

and W (lin). These

comparisons are not demonstrated here for reasons of space. Figure 1 displays the plot of the trend-cycle estimators

bounded by 2 times plus or minus the estimates of the standard deviation of the trend-cycle estimators for the series

Total Industrial Production Index. Figure 2 shows the corresponding plot for the seasonal component.

Total Tourist Arrivals by Air. The data was processed through X11-ARAMA program for the time span from

January 1980 to December 1994 (168 observations) using (i) multiplicative decomposition model; (ii) the best

options for seasonal adjustment (including adjustments for trading day and moving festivals) and (iii) the modified

Henderson trend estimation.

In step 5, none of the conditions were satisfied for any vector c. In step 6, the following values were obtained:

s (lin)(T) = 0.0146 , s (lin)(S) = 0.0074 and these values are small enough with respect to

std(e

~

t

\ t) ≈ 0.015. Clearly, the condition concerning the weights being time invariant is satisfied and therefore

~

we can apply the weights W (lin) and W (lin). Figure 3 displays the plot of the trend-cycle estimators bounded

with 2 times plus or minus the estimates of the standard deviation of the trend-cycle estimators for the series Total

Tourist Arrivals by Air for the first 132 points. Figure 4 shows the corresponding plot for the seasonal component.

Concluding Remarks

This paper presented a simple general method, based on Pfeffermann (1994) but extended for any multi-stage run ,

for estimating the variances of the X11-ARIMA estimators. The variances estimated account for the sampling error

and the irregular variation.

Under the assumption that the linear approximation holds, we have presented an alternative procedure for estimating

~

the weight matrices W and W . One notable feature of this procedure is that it is very simple and can be applied to

any length of series processed through X11-ARIMA program (or X12). On the other hand, the use of very small cconstants may cause, for example, calculation errors, therefore some checking is needed.

As mentioned before, the X11-ARIMA algorithm contains several non-linear options. Even under some non-linear

options, one can expect that the linear approximation hold with very similar weight matrices. On the other hand,

some non-linear options, for example, the replacement of extreme values, can lead to terrible behavior of the

weights but the final estimator can be very close to the respective one obtained through linear operations. We have
suggested several statistics in order to determine when the linear weights can be used for the estimation and

demonstrated this for real data examples.
Figure 1. Total Industrial Production Index 

Trend-cycle Estimator

130

120

110

100

90

80

70

60

1 7 13 19 25 31 37 43

Trend

49 55 61 67 73

t

Trend+2SD

79 85 91 97

Trend-2SD

103 109 115

1

1.1

Figure 2. Total Industrial Production Index 

Seasonal Component Estimator

1.05

0.95

0.9

0.85

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115

t

Seasonal C. Seasonal C.+2SD Seasonal C.-2SD
Figure 3. Total Tourist Arrivals by Air 

Trend-cycle Estimator

160000

140000

120000

100000

80000

60000

1 9 17 25 33 41

Trend

49 57 65 73

t

Trend+2SD

81 89 97

Trend-2SD

105 113 121 129

Figure 4. Total Tourist Arrivals by Air 

 Seasonal Component Estimator

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

1 9 17 25 33 41 49

Seasonal C.

57 65 73 81

t

Seasonal C.+2SD

89 97 105

Seasonal C.-2SD

113 121 129
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