Pre-adjustment in X12-ARIMA 

Luisa Burck and Yuri Gubman 

Israel Central Bureau of Statistics 

66, Kanfey Nesharim, Jerusalem, Israel. E-mail: louiza@cbs.gov.il 

Abstract 

The X12-ARIMA method, allows modeling either the irregular series or the original series for pre-adjustment. These models 

are used for outlier detection and correction, for missing observation treatment, for testing and correcting for calendar effects 

and for forecasting. Being an almost uncorrelated series, the irregular series has the appealing simplicity of being a candidate 

for ordinary least squares regression estimation of the above components simultaneously. On the other hand, regression 

models for time series with non-stationary ARIMA residuals enable to model the effects directly rather than as a residual 

component accounting for the correlation structure of the observed series. The estimation of the moving Jewish festivals and 

the trading day effects has always posed a difficult problem in Israel. In the past, a special method was developed at the 

Statistical Analysis Sector of the CBS for the simultaneous estimation of these effects while using X11-ARIMA procedure 

for seasonal adjustment. Currently, the CBS is in the process of implementing the X12-ARIMA method as the standard 

method of seasonal adjustment. In this study, we show how these effects can be estimated either modeling the observed series 

or the irregular series in X12-ARIMA. Selection of number of regressors in the regARIMA model and comparison of several 

models are based on various statistics such as AICC and out-of-sample forecast performance, and the results are further 

checked by diagnostic checking statistics including sliding spans analysis. Furthermore, the model is extended to include the 

estimation of trend breaks as well as outlier detection processes. An empirical study has been carried out for the illustration of 

the application of this method to three main Israeli indicator series. The empirical results support this approach and use of 

holiday and trading-day regressors indeed produce a better seasonal decomposition. 

Keywords: Seasonal Adjustment, Calendar Effects, RegARIMA Models. 

1. Introduction 

Festival date movements are typical of festivals whose dates are fixed according to the lunar year, but vary according to the 

Georgian calendar (e.g., Jewish festivals, Easter, the Chinese New Year). Jewish festivals usually move between two 

consecutive solar months; the date of the Passover festival moves between March and April, and the dates of the Jewish New 

Year, the Day of Atonement (Yom Kippur) and the Feast of Tabernacles (Succoth) move between September and October. 

The Feast of Weeks (Shavuoth) and the Independence Day are another two important lunar holidays with dates moving 

between April and May, and May and June respectively. A festival falling in a certain month reduces the number of working 

days in that month. For example, during the holidays the industrial production halts. Furthermore, the trading day effect in a 

festival month may totally differ from other months that are not affected by festivals. Festival and trading day effects, 

however, do not necessarily overlap: in many series the occurrence of festivals has an additional effect beyond that of 

reducing the number of working days, such as increasing the demand for workers prior to the festival or increasing custom 

clearance of imported goods to compensate for workdays lost. Sometimes, the festival date effect is dominant and the change 

in the number of working days is of small importance. The importance of pre-adjustments for such effects is clear; it allows 

more detailed decomposition of a time series into its different components and increases the comparability of data within the 

series. 

Since 1992, the Central Bureau of Statistics of Israel uses a special model in order to estimate the combined effects of the 

moving festival dates and the trading day variation while using X11-ARIMA procedure for seasonal adjustment. Section 2 

overviews briefly this model based on postulating a regression relation between the irregular components and the calendar 

effects. The X12-ARIMA method, allows modeling either the irregular series or the original series for pre-adjustment. Being 

an almost uncorrelated series, the irregular series has the appealing simplicity of being a candidate for ordinary least squares 

regression estimation of the above components simultaneously. On the other hand, regression models for time series with 

non-stationary ARIMA residuals, i.e. regARIMA models, enable to model the effects directly rather than as a residual 

component accounting for the correlation structure of the observed series. Hence, in Section 3, the regARIMA modeling is 

elaborated and various diagnostic checking statistics are introduced. Section 4 provides an empirical study illustrating a 

variety of issues concerning pre-adjustment. The first real data set is used to show how the combined moving festival dates 

and trading day effects can be estimated simultaneously by either modeling the irregular series or the original series. 

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Selection of number of regressors and comparison of several regARIMA models for modeling the observed series, are based 

on various statistics such as AICC and out-of-sample forecast performance. The results are further checked by diagnostic 

checking statistics and statistics to assess the quality of seasonal adjustment obtained after adjusting the series for the 

calendar effects. The second real data example demonstrates how the model can be extended to include trend breaks. Finally, 

we show, in brief how a user-defined regressor can be included in the model to account for some of the irregularity in the 

series rather than the automatic outlier detection procedure. Section 5 contains some concluding remarks. 

2. The Currently used Method 

Let {O t T } denote the observed series. First, we assume the multiplicative decomposition model t

: =1,...,

t t t t O = T * S * I (1)

whereTt

is the trend-cycle level, St the seasonal effect and It the irregular factor expressed as percentage points. It is 

assumed that { I

t

:t =1,...,T } are random variables with E(It

) =100 and constant variance σ

2

. In addition they are 

assumed to be uncorrelated. Equation (1) can be extended by assuming that It = Pt * εt

 as 

Ot = Tt * St * Pt * εt 

where Ptis the measurable part of the irregularity that includes calendar effects and ε t

is the remaining irregularity. 

To estimate the overall calendar effect Pij , for month j of year i , assuming an additive relationship with its components we 

write the following model for the estimator of I

ij : 

j ij ij

l

ij ij ij lj lj j ij N I − N = ∑ β D − D γ+ F δ+ H + e

=

6

1

7

*

/100 ) ( )

ˆ

( (2) 

where Nij and N

*

ij are the number of days and average number of days in month j of year i , respectively, D1 j

,..., D7 jare 

the number of Sundays, the number of Mondays,..., the number of Saturdays in month j , Fi

denotes the festival date in year 

i , measured as the number of days between the actual starting date and the earliest possible starting date of the festival, Hij

is the number of intermediate festival days (hol hamoed) in month j of year i and eij is the random error term with 

expectation 0 and constant variance. The coefficient γ j

of the festival effect for month j and is equal 0 for months j not 

affected by festivals. Note that D1k

,..., D7k

are not the traditional daily activity variables; they are recalculated after 

adjustments for festival dates. For example, if the festival eve and the festival dates fall on Wednesday and Thursday 

respectively, the number of Wednesdays and Thursdays are reduced by 1 and the number of Fridays and Saturdays are each 

increased by 1. As a result, we might have months, such as October, with 6 or 7 Fridays and Saturdays. Equation (2) implies 

that the calendar effects should be estimated separately for every month. This is problematic because of the limited number of 

observations available for each month. Our earlier studies have shown that we can group months that share similar calendar 

effects. For example, winter months (November, December, January and February) have similar daily activity effects and are 

unaffected by festivals. For short series, the trading day effects are estimated based on data for all months (one group). In

general, the decision concerning the number of groups of months is based on the length of the series and on statistical testing. 

The above model is estimated separately for each group of months by ordinary least squares using an external program to 

X11-ARIMA. It should be mentioned that the currently used method includes some enhancements mainly dealing with the 

identification and the treatment of the outliers. 

3. Calendar Effect Estimation in X12-ARIMA 

As mentioned above, the X12-ARIMA method allows modeling either the irregular series or the original series for preadjustment. Application of the model for the irregular series defined by equation (2) is straightforward as X12-ARIMA 

allows the provision of used-defined regressors. Modeling the original series is carried out through regARIMA models. 

RegARIMA models: 

It is sometimes useful to transform the series prior to estimating a regARIMA model mainly to stabilize the variance. More 

specifically, transformations that change smoothly in λ maybe used, such as 

Page 145
[ ]















=λ

≠λ

=λ















= = λ + − λ

λ λ

0

1,0

1

log( / ),

( / ) ,/1

/ ,

( )

)( 2

t t

t t

t t

t t

O d

O d

O d

y f O

where dt

is some appropriate sequence of divisors. The regARIMA model for λ =1,0 then becomes: 

t

s

t t q Q

s d D

φ p B)( ΦP(B 1)( − B 1() − B () y − β′X ) θ= B)( Θ (B )a (3) 

where φ p(B) and ΦP(B

s

) are the autoregressive polynomials in B and B

s

of degree p and P, respectively, with roots 

outside the unit circle, θq(B) and Θ (

s

Q B ) are the moving average polynomials of degree q and Q respectively, with roots 

outside the unit circle, d and D are the order of regular and seasonal differencing respectively, s is the seasonal parameter, 

equals to 12 or 4, Xtis the matrix of regressors, and atis white noise with mean 0 and variance σ

2

a

. It follows that: 

1( 1() () )

t t

d D

wt = − B − B y − β′X

is a covariance stationary series that satisfies: 

t

s

t q Q

s

φ p B)( ΦP(B )w θ= B)( Θ (B )a (4) 

Consequently, the model (3) can be rewritten as 

{ } t

r

i

it

d D

t i

d D

1( − B 1() − B) y = ∑β 1( − B 1() − B) x + w

This is a regression model for differenced yt

 with stationary ARMA errors wt. If we assume that at

in (4) are i.i.d. and 

normally distributed with expectation 0 and variance σ

2

, all the model parameters can be estimated by maximizing the 

likelihood function. 

Diagnostics

Checking regARIMA modeling. ACF, PACF and ACF of the squared residuals from a fitted regARIMA model together 

with associated Box-Ljung portmanteau statistics, histogram of the residuals and normality statistics for the residuals are used 

for diagnostically checking regARIMA modeling. The spectrum is used to detect remaining seasonal effects in the residuals. 

Checking the quality of seasonal adjustment. F-test for the presence of seasonality from Table D8A of the X11 output, 

comparison between the standard deviations of the seasonal factors and the irregular factors from Tables D10 and D13, M1-

M11, Q and Q without M2 statistics are used to check the quality of seasonal adjustment. The spectrum is used to detect 

residual seasonality remaining in the seasonally adjusted series. 

Checking the stability of seasonal adjustment. Findley, et al. (1990) describe in detail the sliding span statistics and 

propose their use to analyze the stability of the seasonal adjustment. First, the whole sample is divided into four overlapping

spans. The basic diagnostics are descriptive and reflect how the seasonal factors, trading day factors, seasonally adjusted 

series and their month to month and year to year percent changes vary when the span is altered systematically. Denote by 

S

j

t

and TD j

t

 the seasonal factor and the trading day factor for month t estimated from data in the jth span 

respectively, A

j

t

be the seasonally adjusted value for month t estimated from data in jth span, MM j

t

and YY j

t

denote the 

month to month change and year to year change in the seasonally adjusted series. A month t belonging to at least two spans 

is considered to have unstable seasonal factors, trading day factors and seasonal adjustments if: 

03.

min

max min

>

−

j

tj

j

tj

j

tj

S

S S

, 

02.

min

max min

>

−

j

j t

j

j t

j

j t

TD

TD TD

, 

and 

Page 146
03.

min

max min

>

−

j

tj

j

tj

j

tj

A

A A

, 

respectively. Furthermore, for months t such that both t and t −1 belong to at least two spans, the seasonally adjusted 

month to month percent change is considered as unstable if 

max − min > 03.

j

j t

j

j MMt MM . 

For month t such that both t and t −12 belong to at least two spans the seasonally adjusted year to year percent change is 

considered as unstable if 

max − min > 03.

j

j t

j

j YYt YY . 

S(%), TD(%), A(%), MM(%) and YY(%) are used to denote the percent of unstable months with respect to the number of 

months for which the left hand side of the above inequalities hold. 

Another type of stability diagnostic associated with continuous seasonal adjustment over a period of time is revision histories. 

Let At\uand Tt\u

denote the seasonally adjusted and trend values for month t respectively obtained from data up to time u , 

1≤ t ≤ u ≤ T. The values At\t

and Tt\tobtained from data up to time t are called the concurrent estimates and it is the first 

adjustment obtained for month t. Similarly, At\T and Tt\T

 are the most recent estimates for month t . The revision from the 

concurrent to the most recent adjustment of the seasonally adjusted series expressed as a percentage of the concurrent 

adjustment is calculated by 

tt

A Tt tt

Tt

A

A A

R

\

\ \

\

100 *

−

= . 

Similarly, revisions for month to month percent changes in the seasonally adjusted data R

A%

t\T

as well as their analogous

quantities for trend data, R

T T %

t\T

and Rt\T, are calculated.

Choosing between competing models. Suppose there are competing regARIMA models that differ in the choice of 

regressors, or in the choice of transformations or in the choice of ARMA model (4). For competing regARIMA models 

whose diagnostics seem to be adequate, X12-ARIMA produces likelihood-based model selection criteria: AIC, AICC (Fadjusted AIC), Hannan-Quinn and BIC. One advantage to these criteria over standard t-statistic, χ

2

-statistics and the 

likelihood ratio test is that they may be used to compare nonnested models (one model cannot be obtained simply by 

removing parameters from another model). AICC criterion proposed by Hurvich and Tsai (1989) and the Schwarz BIC 

criterion are defined as: 

AICC = -2*L + 2m / {1 – (m+1)/(T – d –sD)} 

BIC = -2*L + m* log (T- d – sD) 

respectively, where L is the log likelihood and m is the number of estimated parameters. For each of these statistics, the 

model producing a lower value is preferred. 

An alternative approach to model selection is to compare the out-of-sample forecast performance. Assume we are interested 

in h -step ahead forecasting of the time series Ot

. For each t in T0 ≤ t ≤ T − h , let yt+h\t

denote the forecast of 

yt+h

obtained by estimating the regARIMA model using only the data up to time t . The out-of-sample h -step forecast of 

O

−1

t+h

is defined as Ot+h\t = f ( yt+h\t

) and the associated forecast error as et+h\t = Ot+h − Ot+. Consider the sequence of h\t

accumulating sum of squared out-of-sample forecast errors 

= ∑

=

+

M

Tt

Mh tht SS e

0

2

, \

, ,..., . M = T0 T − h

The weighted differences between SS1

h,M and SS 2h,M from two competing models, defined by 

/( )

0

2

,

2

,

1

2,1 ,

,

SS T h T

SS SS

SS

hTh

Mh Mh

Mh

− −

−

=

−

Page 147
can be used for model selection. Let us emphasize that this diagnostic does not require the assumption that any of the models 

being compared is correct. When model selection diagnostics based on the log likelihood, such as AICC, are inapplicable 

because the models are fit to different time series, forecast comparisons are still possible. For example, out-of-sample 

forecast performance can be used to compare the regression models of the irregular component with regARIMA models of 

the observed series. In addition, the smoothness statistic for the series calculated as the square root of the sum of the squared 

first differences can be used to favor one of the competing models. 

 

 

 

4. Empirical Results 

This section contains three real data examples to illustrate a variety of issues concerning pre-adjustment in X12-ARIMA. The 

first example demonstrates how the diagnostics described above can help with decisions about what kind of trading day 

adjustment to use. The second example is an extension of the model to include trend breaks. The third example shows how 

some other user-defined regressors can be included in the regARIMA models, instead of the automatic outlier detection 

procedures, to account for some irregularity in the series. 

Total Industrial Production Index. An important source of the month-to-month fluctuations in the Total Industrial 

Production Index is trading day variation. The model used today for the estimation of these effects together with the moving 

festival effects in X11-ARIMA is given in equation (2). RegARIMA modeling capabilities in X12-ARIMA lead to the 

dilemma whether to model the irregulars or the original series and this is our concern here. The data was processed through 

Table 1: RegARIMA Model Statistics and Statistics Related to Seasonal Adjustment 

Regression on Y Regression on I 

Model 1 Model 5 Model 6 Model 7 Model 2 Model 3 Model 4 

without userdefined 

variables 

one group three groups five groups one group three groups five groups

RegARIMA Related Statistics

Number of parameters 

estimated 3 14 27 39 3 3 3 

Log Likelihood 194.31 284.99 297.16 305.21 279.37 292.68 303.66 

AICC (F-corrected AIC) 776.58 620.88 634.42 663.72 606.47 579.84 557.88 

BIC 784.85 656.51 694.82 736.57 614.73 588.10 566.14 

Residual Normality

Skewness 0.760 0.811 0.771 0.764 0.817 0.804 0.779 

Kurtosis 4.457 2.738 3.780 3.596 2.724 2.754 3.191 

Forecasting Performance 

 

 

 

Average absolute 

percentage error in out-ofsample forecasts (last 3 

years) 

4.62 4.20 3.91 4.65 3.73 3.71 3.24 

Statistics Related to Seasonal Adjustment

D8A F-value - test for 

seasonality 22.19 66.71 73.34 79.67 98.37 110.77 127.23 

std(D10) 4.90 3.63 3.46 3.46 4.88 4.90 4.76 

std(D13) 3.37 1.36 1.19 1.09 1.43 1.23 1.17 

Q 0.66 0.29 0.29 0.25 0.31 0.27 0.22 

Relative Contributions (F2.B)

I (E3) 8.58 2.14 1.14 1.35 2.28 1.77 1.00 

C (D12) 0.70 0.60 0.63 0.57 0.50 0.48 0.43 

S (D10) 90.74 49.36 44.08 40.27 77.09 78.25 75.36 

TD&Holiday (D18) 0.00 47.90 54.15 57.87 20.14 19.50 23.21 

Sliding Spans Statistics

S(%) 0% (0/113) 2.7% (3/113) 5.3% (6/113) 12.4% (14/113) 0% (0/113) 0% (0/113) 0% (0/113) 

TD(%) 0% (0/106) 0% (0/106) 12.3% (13/106) 36.8% (39/106) 0% (0/106) 0% (0/106) 0% (0/106) 

A(%) 0.9% (1/113) 0% (0/113) 0.9% (1/113) 0.9% (1/113) 0% (0/113) 0% (0/113) 0% (0/113) 

MM(%) 4.5% (5/112) 0% (0/112) 2.7% (3/112) 2.7% (3/112) 0% (0/112) 0% (0/112) 0% (0/112) 

YY(%) 0% (0/101) 0% (0/101) 1% (1/101) 7.9% (8/101) 0% (0/101) 0% (0/101) 0% (0/101) 

Page 148
X12-ARIMA program for the time span from January 1990 to May 2001 (125 observations) using multiplicative 

decomposition model. All the values of the likelihood-based model selection criteria, such as AICC, were lower for the 

model with log transformation. Thus, all the models checked hereafter use log data transformations. We systematically 

applied the following models: (1) no regressors for trading day and festival effects. (2) OLS regression model of the 

irregulars for trading day effects for all months (one group), and festival effects. (3) OLS regression model of the irregulars

for trading day effects for three groups of months (three groups), and festival effects. (4) OLS regression model of the 

irregulars for trading day effects for five groups of months (five groups), and festival effects. The analogous regARIMA 

models (2), (3) and (4) will be referred in the following as models (5), (6) and (7).

Table 1 summarizes the statistics related to model selection and quality of the seasonal adjustment. The comparison of the 

regARIMA models when modeling the original series, reveals that model (5) has the lowest AICC and BIC values. The 

seasonal patterns for models (5), (6) and (7) are similar, but the prior adjustment factors have different intensity especially for 

festival months. However, model (6) has the lowest average percent out-of-sample forecast for the last three years and 

outperforms the other two models for one period ahead forecast, based on accumulated forecast errors. The sliding span 

statistics show that the percentage of unstable months is relatively high for model (7). This is probably due to the large 

number of regressors in the model in respect to the number of observations. It is an open question how long a series is needed 

for reliable estimation of trading day and moving festival effects and we will pursue it in the future. When comparing the 

regressions on the irregulars, all the statistics favor model (4). Model (4) has the lowest average percent out-of-sample 

forecast error for the last three years and outperforms all the other models for one period ahead forecast. It should be 

mentioned that this model is almost identical to the currently used model. Figures 1, 2 and 3 present the comparison of the 

prior adjustment factors, seasonal factors and the combined factors for model (4), model (6) and the currently used model, 

respectively. The months that are greatly affected by moving festival dates such as April and October have different 

decompositions in models (4) and (6). Model (6) has produced much larger prior adjustment factors and has compensated for 

this with much smaller seasonal factors. This can also be seen from the high relative contributions of the trading day and 

festival effects to the month-to-month variance in the original series for model (6). As a result, although the seasonal factors 

and the prior adjustment factors differ for models (4) and (6) the combined factors obtained from these models are similar. 

Figure 1: Total Industrial Production - Prior Adjustment Factors

85

90

95

100

105

110

ן

1997

IV VII X ן

1998

IV VII X ן

1999

IV VII X ן

2000

IV VII X ן

2001

IV

Model 6 Currently used model Model 4

Page 149
Figure 2: Total Industrial Production - Seasonal Factors

90

92

94

96

98

100

102

104

106

108

ן

1997

IV VII X ן

1998

IV VII X ן

1999

IV VII X ן

2000

IV VII X ן

2001

IV

Model 6 Currently used model Model 4

Page 150

Figure 3: Total Industrial Production - Combined Factors (Seasonal and Prior Adjustment)

85

90

95

100

105

110

115

ן

1997

IV VII X ן

1998

IV VII X ן

1999

IV VII X ן

2000

IV VII X ן

2001

IV

Model 6 Currently used model Model 4

Total Tourist Arrivals by Air. As of October 2000, due to the security situation in Israel, a change in the level of the series 

of incoming tourism was detected. Therefore, calculation of seasonally adjusted data and trend data since October 2000 was 

carried out after adjusting the data up to September 2000 to the low level observed during the first six months of the crisis. 

The change in level was estimated as the ratio of the average of the original series between October 2000 and March 2001 to 

the average of the ARIMA forecasts for the same period obtained from the X11-ARIMA run on data till September 2000. 

Here, the data was processed through the X12-ARIMA program for the time span from January 1990 to August 2001 (140 
observations), using multiplicative decomposition model. The trading day and the festival effects were estimated from the 

irregulars. The model is slightly different than the one used for the Total Industrial Production series as it includes extra 

variables to account for the Easter effects and has no variables for the effect of the intermediate festival days. First, we 

applied the automatic outlier identification procedure of the program with a critical value 5.00. The program detected one 

level shift outlier for October 2000 with t-value=-6.47. The resulting AICC values for models excluding and including the 

level shift variable were 2305.21 and 2192.2 respectively. Obviously, level shift variable should be included in the model. 

Also, all the other statistics related to the quality of the seasonal adjustment were far better for the model including the level 

shift variable for October 2000. Figure 4 compares the X12-ARIMA adjustment to the X11-ARIMA adjustment with level 

shift estimated as described above. The figure indicates that the size of the change in the level of the series estimated from 

X12-ARIMA seem to be slightly smaller than our current estimation. Also, the seasonally adjusted series from X12-ARIMA 

is smoother. 

Page 151

Figure 4: Comparison of the Seasonally Adjusted Incoming Tourism Series

70000

90000

110000

130000

150000

170000

190000

210000

230000

I II III IV V VIVIIVIIIIX X XIXII I II III IV V VIVIIVIIIIX X XIXII I II III IV V VIVIIVIIIIX X XIXII I II III IV V VIVIIVIII

1998 1999 2000 2001

X12-RegARIMA model Currently used model

Total Electricity Generation. Total Electricity Generation in Israel is highly seasonal as this series is strongly dependent on 

weather conditions. The increasing amount of air-conditioning devices purchased over the past years, has changed the 

seasonal pattern of the series as the peak in January, a cold month, has moved to August, the hottest month. The data was 

processed through the X12-ARIMA program for the time span from January 1990 to April 2003 (148 observations) using 

multiplicative decomposition model. The trading day and the festival effects were estimated from the observed series. The 

sliding span analysis showed that there are not many unstable months, but the maximum percent difference in the seasonal 

factors occur in August, July, November, December and January as expected. The outliers detected by the program occur 

usually on the winter months, November, January and February. For instance, the harsh winter in 1992 lead to the detection 

of January and February 1992 as extreme outliers. We introduced into the model a new variable: the tenth percentile of the 

average temperature measured at 8 p.m. each month. The inclusion of this variable slightly improved most of the statistics 

such as the AICC in the regARIMA models for the observed series, the standard deviation of the irregulars and the relative 

contribution of the irregulars. We are aware that the inclusion of such a variable does not help forecasting but might achieve a 

better decomposition of the time series with the available data. 

5. Concluding Remarks 

In this study, we have addressed various issues concerning pre-adjustment. We have shown how the impact of moving 

festival dates and trading days in Israel can be estimated by either modeling the irregular series or the observed series in X12-
ARIMA. In general, the better inference properties of regARIMA models added to the performance of its estimates lead us 

usually to prefer the regARIMA modeling of the original series for pre-adjustment. However, we have found that some series 

are not modeled well by regARIMA modeling. For example, the Total Industrial Production series, that is greatly affected by 

the moving festival dates and trading day, seem to be better modeled from the irregular series. This conclusion is based on 

out-of-sample forecast performance, accumulated forecast errors and other diagnostic statistics. On the other hand, for the 

Total Tourist Arrivals and the Total Electricity Consumption series the performance of the regARIMA modeling seems to be 

superior or equal, based on the same statistics. Our analysis has also shown that adding other regressors such as those 

associated with trend breaks improves the seasonal decomposition. AICC, accumulated forecast error and sliding-span 

statistics are useful in choosing between different competing methods. 

Page 152

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