ISSN 1977-3331 EWP 2011/030
Malaysian Economic Time Series DataNon-Fixed Seasonal Effect: A Case Study of
Norhayati Shuja´Mohd.Alias LazimYap Bee Wah
Improving Trend-Cycle Forecast by Eliminating
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Improving Trend-Cycle Forecast by Eliminating Non-Fixed Seasonal Effect: A Case Study of Malaysian Economic Time Series Data
Norhayati Shuja’1
[email protected]. Department of Statistics, Malaysia.
Mohd. Alias Lazim. Universiti Teknologi MARA, Malaysia. [email protected] Yap Bee Wah. Universiti Teknologi MARA, Malaysia. [email protected]
Abstract
The major festival celebrations in Malaysia are usually related to religious activities and as such, the
dates are determined based on the lunar calendar. As a result, the dates of these holidays do not
move in accordance to the Gregorian calendar. When these holidays take place, they tend to affect the
economic activities for the periods in the vicinity of the holiday dates and may mask the underlying
trend movements and thus provide inaccurate signals for decision making. The impact of non-fixed
holidays on the time series need to be taken into account when performing seasonal adjustment so as
to avoid misleading interpretations on the seasonally adjusted and trend estimates. In this study, two
approaches, the SEAM (Seasonal Adjustment for Malaysia) and Regression-ARIMA (Reg-ARIMA)
were developed for eliminating the non-fixed holiday effects on five selected Malaysian economic time
series. The results show that the SEAM and Reg-ARIMA procedure were able to remove non-fixed
holiday effect. Furthermore, the forecast performances of these models were subsequently evaluated
based on out-of-sample forecasts procedure using two error measures, the RMSE and MAPE. Though
both methods gave equally good results, however, the overall results show that the SEAM method out-
performs the Reg-ARIMA procedure.
Keywords: Non-fixed Seasonal Effect, Seasonal Adjustment, Trend-Cycle Forecast JEL codes: C22, E32, E37
1 Correspondence Address: Norhayati Shuja’. Economic Indicators Division, Department of Statistics, Malaysia.
2
1. Introduction
The desire for time series data that accurately portray economic growth and decline is usually
being complicated by factors unrelated to the trend of the data but nevertheless do influence
the level of data, thus obscuring accurate interpretation. Seasonal variation is the most
common and important source of noise influencing the monthly or quarterly economic data
series. Seasonal variation includes the recurring calendar-related effects caused by economic
and non-economic factors, such as weather conditions, school holidays, festival holidays,
trading days and etc. Findley and Soukup (2000) pointed out that certain kinds of economic
activities and their associated time series are affected significantly by holidays. When these
holidays take place they tend to affect or influence the economic activities for the periods in
the vicinity of the holiday dates. Incidentally, there are some holidays whose event dates are
not fixed at any specific location within a year period but move from one point to the next,
some within a certain time intervals whilst others are not. For such moving holiday effects,
they need to be taken into account in the seasonal process to avoid biased seasonally adjusted
and trend estimates which can lead to wrong decision making by policy makers (Zhang et al.,
2001).
Many researchers have shown interest in undertaking such studies with the aim of achieving
more reliable methods of seasonal adjustment (Hilmer et al, 1983; McKenzie, 1984; Dagum
et al, 1988; Bell & Hilmer, 2002; Matas-Mir et al, 2008; Koopman & Lee, 2009). For
example, Findley et al. (1998) developed a method to remove Easter holiday effect, Labour
Day effect and Thankgiving effect based on a North American holiday period. While, for
Australian Easter, Zhang et al. (2001) developed a method to remove the Easter effect on the
Australian data series. Lin & Liu (2002) also studied the impact of Chinese New Year, the
Dragon boat festival and Mid Autumn holiday on Taiwan time series data. Similarly for
3
Turkey, the Islamic festivals occur according to the Hegirian calendar (Lunar calendar), that
is the Holy month of Ramadhan, the Feast of Ramadhan (Eid-ul Fitr) and the Feast of
Sacrifice (Eid-ul Adha). Therefore, Alper and Aruoba (2001) proposed the method which
eliminates the festivals’ effect by using dummy variables for religious events which are tied
to the lunar calendar in the regression method. These are some of the major works that have
been performed by researchers.
In the context of Malaysia, Malaysian economic time series data are affected by the major
religious festivals such as the Eid-ul Fitr of the Muslims, the Chinese New Year of the
Chinese and the Deepavali of the Indian. Since the major festivals in this country are usually
related to the religious activities and as such, the dates are determined by the respective
religious calendar. The Eid-ul Fitr is based on the Islamic calendar in which the date is
determined upon the sighting of the new moon. The difference in the method of determining
the date has caused the Eid-ul Fitr to move forward from one period to the next in the interval
of eleven or twelve days earlier each year. The date of Chinese New Year is determined on
the first full moon of the Chinese lunar calendar. It usually moves between 21st January and
21st February, whilst the date of Deepavali is determined by the Indian lunar calendar which
usually moves between 15th October and 15th November. The date of the three major
festivals in Malaysia, i.e. the Eid-ul Fitr, Chinese New Year and Deepavali is shown in
Appendix 1.
As a result, the dates of these holidays are not in alignment with the Gregorian calendar.
Hence, they tend to move along the Gregorian calendar and along the way discharge strong
seasonal influence on many economic time series data. Since these non-fixed holiday
impacts on the time series data are large, therefore, they need to be taken into account when
performing seasonal adjustment process so as to avoid misleading seasonally adjusted and
4
trend estimates. Furthermore, the presence of non-fixed holiday effects may complicate the
interpretation of the data, for example when comparing over a short time frame such as
month to month or quarter to quarter (Department of Statistics, Singapore, 1992). Ashley
(2001) concluded that by removing the non-fixed seasonality effect, the important features of
economic series such as direction, turning points and consistency between other economic
indicators can be easily identified. Deutsche Bundesbank (1999) had earlier stated that
seasonally adjusted data are also better suited for the analysis of current business cycle whilst
(Burck et al., 2004) considered it as important for early detection of turning points and
directional change of the socio-economic activities.
In view of the importance of eliminating the non-fixed holiday effects on time series data,
this paper explores a different procedure for eliminating non-fixed seasonal effect with the
aim of achieving more reliable methods in improving trend-cycle forecast. The paper
comprised of four parts. Section 2 explains the methodology for Seasonal Adjustment for
Malaysia which has been named SEAM, the RegARIMA (adjusted for Malaysia) and the
regressors for measuring the holiday effects. Section 3 presents the findings and Section 4
concludes this paper.
2. Methodology
There are many different approaches to seasonal adjustment used by practitioners and
academicians such as the X-11 family or TRAMO/SEATS. The X-11 family is the most
common method used by many statistical agencies throughout the world, such as the U.S.
Census Bureau and Statistics Canada. In this study, two approaches, the Seasonal
Adjustment for Malaysia (SEAM) and Regression-ARIMA (Reg-ARIMA) were developed
for eliminating the non-fixed holiday effects on selected Malaysian economic time series
5
data. This is done by using two different regressors called REG1 and REG3 to measure the
Eid-ul Fitr, Chinese New Year and Deepavali effect.
2.1 Regressors
In the proposed procedures, the numbers of holidays before, during and after the festivals
were taken into account in the construction of the regressors. This is because the change in
activities caused by the non-fixed holiday may affect the prior or the following month’s level.
This type of effect is referred to as proximity effect and it has different impact both before and
after the date of the festival. In the conduct of these procedures, the number of holidays
taken before, during and after the Eid-ul Fitr, Chinese New Year and Deepavali festivals were
used to construct the regressors. To determine the numbers of holidays a term called
‘Window Length’ will be used as explanatory variable. A sample survey was conducted on
350 individuals primarily to collect the information pertaining to the behavioural pattern of
the Malaysia public with respect to the number of holidays they normally take to celebrate
these festivals (Norhayati et al, 2007). The holiday weights used in constructing the
regressors can also vary depending on the characteristics of the time series data. Table 1
shows the various ‘Window Length’ for each of the festivals.
Table 1: The Number of Window Length
Festival Before During & After Total
1w 2w w Eid-ul Fitr
Chinese New Year
Deepavali
2
2
1
5
6
3
7
8
4
REG1 uses the weight comprising of the combination of the three festivals, namely the
Chinese New Year, Eid-ul Fitr and Deepavali. For example, the Eid-ul Fitr has a total
6
window length of seven-days, two-days to model the effect prior and five-days during & after
the holiday. Similar procedure was also used for the Chinese New Year and Deepavali.
However, if two festivals fall in the same month, then the number of holidays for the two
festivals is combined. The weight variable is defined as follows;
Case 1 : When the festival fall in the beginning of the month (1st-15th)
wn1 in the respective festive month
wn2 before the respective festive month
0 Otherwise
Case 2 : When the festival fall at the end of the month (16th-31st)
wn1 in the respective festive month
wn2 during & after the respective festive month
1 otherwise
where, w is the total number of window lengths, w=8 for Chinese New Year, w=7
for Eid-ul Fitr, w=4 for Deepavali,
1n is the number of holidays fall in the festive month
2n is the number of holidays fall before or after the festive month.
The REG3 on the other hand, uses three separate weight variable, each representing the Eid-
ul Fitr, Chinese New Year and Deepavali festivals. In this method, the weight variables used
are similar to that of REG1, except in this case the value of (-1) is assigned so that the weight
variable for any one year sums up to zero. This is defined as follows:
Reg1 =
Reg1 =
7
Case 1 : When the festival fall in the beginning of the month (1st-15th)
wn1 in the respective festive month
wn2 before the respective festive month
-1 after the respective festive month
0 otherwise
Case 2 : when the festival fall at the end of the month (16th-31st)
wn1 in the respective festive month
wn2 after the respective festive month
-1 before the respective festive month
0 otherwise
where, w is the total number of window lengths, w=8 for Chinese New Year, w=7
for Eid-ul Fitr, w=4 for Deepavali,
1n is the number of holidays fall in the festive month
2n is the number of holidays fall before or after the festive month.
2.2 Seasonal Adjustment for Malaysia (SEAM)
The application of the proposed SEAM procedure requires the support of X-12 ARIMA
program. The SEAM procedure is based on the use of irregular values obtained after
performing an initial seasonal adjustment with the assumption that the moving holiday effect
resides in the irregular series and this series is used to derive correction factors. This is done
in two stages.
In stage one, the irregular ( tI ) estimate is obtained by running the X-12 ARIMA program
which is expected to also contain the irregular series. This series is then used to estimate the
Reg3 =
Reg3 =
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‘true irregular’, ′tI which is free from moving holiday effect. This is done by using two
different regressors called REG1 and REG3, used to measure the Eid-ul Fitr, Chinese New
Year and Deepavali effect. Hence, the moving holiday effect correction is made after the
first seasonal adjustment run of the X-12 ARIMA. The seasonally adjusted time series data
without moving holiday effect is then obtained.
Initially, run the X-12 ARIMA program to obtain the two different components of the time
series called the ‘Final Trend-Cycle’, tT and the ‘Final Irregular’ , tI .
Consider a time series Yt , t=1,2,3,…,n which is represented (based on the multiplicative
assumption) as follows,
tttt ISTY ××=
[1]
where, tT = trend-cycle, tS = seasonal and tI = irregular.
tI is assumed to comprise of three distinct components, i.e.
′××= tttt IHEI [2]
where, tE = the extreme value which falls outside the sigma limit of 2.5, tH = the
moving holiday effect and ′tI = the true irregular assumed free of moving holiday
effect.
However, when the X-12 ARIMA was first run, the component tE [equation 2] was
automatically removed and thus leaving components tH and ′tI . To estimate the moving
holiday effect, fit a regression model to the irregular component ( tI ).
ttt hI εββ ++= 10 [3]
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where, tI = irregular in time t, β 0 = the intercept term value, β1 = parameter for
festival holidays ( th ), th = the weight variable for the month t with the holiday effects
and tε iid with mean of tε is 0)( =tE ε and variance 2)( εσε =t
The estimated function of [3] is then,
tot hI 1ˆˆˆ ββ += [4]
where I tˆ = estimated irregular in time t, 0β = the estimated intercept term value, 1β =
estimated parameter for festival holidays ( th ), th = the weight variable for the month
t with the holiday effects for t= 1, 2, …, 12. The weight variable th is assigned using
REG1 and REG3.
The ‘moving holiday effect’, tH is then removed by dividing the value of the irregular
components ( tI ) obtained from X-12 ARIMA procedure by the estimated value of irregular
components ( tI ) as given in equation [4],
″= tt
t IIIˆ [5]
The resulting value, ″tI in equation 5 is therefore the “irregular” component which is
assumed to be free from the influence of ‘moving holiday’ component ( tH ).
A new set of time series data ′tY (seasonally adjusted for moving holiday effect), is then
generated as the product of tT and ″tI where tT is the “Final trend cycle” obtained from X-
12 ARIMA procedure. Thus,
″×=′ ttt ITY [6]
which is now a new seasonally adjusted series with the seasonal and moving holiday effects
removed.
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2.3 RegARIMA (Adjusted for Malaysia)
The RegARIMA procedure (Findley et al.,(1998) employs the RegARIMA model which is
part of X12-ARIMA modeling capabilities. This method uses a regression model with
ARIMA error term to derive the correction factors and adjust the original data. A correction
to the original data is done before performing any seasonal adjustment. A regressor is used to
estimate the moving holiday effect and is a predictor variable in the regression model. It is
assumed that the festival holidays have an effect of decreasing or increasing the activities
before, during and after a festival. This study proposed two types of regressors, REG1 and
REG3 that are being used in the X-12 ARIMA program as available from SEASABS
(SEASonal Analysis, Australia Bureau of Statistics Standards). SEASABS is a “knowledge-
based” seasonal analysis and adjustment system used by the Australian Bureau of Statistics to
seasonally adjust time series data.
2.4 Forecast Evaluation
A common criteria used to compare the superiority of one model against the other is to
evaluate their out-of-sample forecasting performance. For this study the Box-Jenkins
methodology was used as the basis of comparison to determine which method, SEAM or
RegARIMA perform better at deseasonalising the data series. The accuracy of model’s
forecast performance is determined by measuring the size of the forecast errors, for both the
within-sample and out-of-sample data. To identify whether the SEAM or RegARIMA gives
a better result in forecasting the trend, a fair comparison is made using the same ARIMA
model. Hence, the levels of adequacy for some models may be lower and some may be
higher. The best ARIMA models that have been identified were applied to the five economic
series which was seasonally adjusted using the two seasonal adjustment methods, the SEAM
and RegARIMA with two different regressors, REG1 and REG3. To determine which
11
seasonal adjustment method performs better in removing non-fixed seasonal effect, the
evaluation of forecast accuracy is performed using out-of-sample test rather than within
sample tests. Lettau & Ludvigson (2000), Chatfield (2001) suggested that when assessing
forecast, it is best to look at the model that minimises out-of-sample forecast errors.
3. Results and Findings
The test for the presence of stable (fixed) and moving seasonality (non-fixed) using X-12
ARIMA program was carried out for five selected Malaysian economic time series data
which represent several industries. For analytical purposes these series are assigned special
name as given in brackets.
i) Monthly Total Imports (IMPORT)
ii) Monthly Total Exports (EXPORT)
iii) Monthly Sales Value of Own Manufactured Products (Ex-factory) (OMP)
iv) Monthly Production of Crude Palm Oil (PALM)
v) Monthly Manufacture & Assembly of Motor Vehicle (1600 cc & below)
(VEHICLE)
The Fs-test was observed at the 0.1% and Fm-test was observed at the 5% level of
significance. These two tests are combined to determine whether the seasonality of the series
is ‘identifiable’ or not. If the Fm-test fails, then the two T’s is calculated and if these averages
are less than one, then the seasonality is ‘identifiable’. The results are as given in Table 2,
which indicates that the all the series were found to have significant presence of seasonality
effect.
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Table 2: Test for presence of seasonality results using X-12 ARIMA
ORIGINAL DATA SERIES
STABLE SEASONALITY (test at 0.1%)
MOVING SEASONALITY (test at 1%)
COMBINED TEST
F-value p-value Presence F-value p-value Presence SEASONALITY
PRESENCE
1 EXPORT 16.391 0.00 YES 4.972 0.00 YES YES 2 IMPORT 12.452 0.00 YES 3.719 0.00 YES YES 3 PALM 19.283 0.00 YES 9.341 0.0002 YES YES 4 OMP 3.523 0.00 YES 5.161 0.001 YES YES 5 VEHICLE 11.371 0.99 YES 1.889 0.00 YES YES
Having confirmed that the seasonality effect is present in the data series tested, the next step
is to specifically determine whether moving holiday is also present. The F-test at 5% level of
significance was then performed. The results are summarised in Table 3. For all data series,
the effects of moving holidays were significant at 5% level of significance.
Table 3: Test for Moving Holiday Effects results
REGRESSOR TIME SERIES DATA
TEST FOR MOVING HOLIDAY EFFECT (α =0.05)
F-value p-value Presence
REG1 IMPORT 28.677 0.000 YES EXPORT 44.492 0.000 YES OMP 26.204 0.000 YES PALM 16.724 0.000 YES MACHINE 29.711 0.000 YES REG3 IMPORT 13.881 0.000 YES EXPORT 23.993 0.000 YES OMP 14.356 0.000 YES PALM 12.678 0.000 YES MACHINE 10.823 0.000 YES
To test for the effectiveness of the application of those methodologies, the seasonally
adjusted data series were then tested to determine the presence of seasonality again by
running the X-12 ARIMA program. The results are presented in Table 4. Both methods, the
SEAM and RegARIMA were found to be able to remove the non-fixed seasonal effects.
However, when combined tests were performed, the overall results were much more
conclusive in which all series show absence of any seasonality effect.
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Table 4: Test of seasonality for seasonally adjusted data
SEASONALLY ADJUSTED TIME SERIES DATA M
ETH
OD
REG
RES
SOR
TEST FOR PRESENCE OF SEASONALITY COMBINED
TEST STABLE SEASONALITY
(test at 0.01%) MOVING SEASONALITY (test at 5%) ( * test at 1%) SEASONALITY
F-value p-value Presence F-value p-value Presence Presence
IMPORT
SEAM REG1 0.802 0.64 NO 1.267 0.20 NO NO
REG3 0.740 0.70 NO 0.939 0.54 NO NO
Reg-ARIMA
REG1 0.541 0.87 NO 1.260 0.20 NO NO
REG3 0.531 0.88 NO 1.304 0.17 NO NO
EXPORT
SEAM REG1 0.545 0.87 NO 0.874 0.63 NO NO
REG3 0.355 0.97 NO 0.827 0.69 NO NO
Reg-ARIMA
REG1 0.652 0.78 NO 1.025 0.43 NO NO
REG3 0.720 0.72 NO 1.053 0.40 NO NO
OMP
SEAM REG1 1.000 0.45 NO 1.931 0.01 YES NO
REG3 0.943 0.50 NO 1.960 0.009 * YES NO
Reg-ARIMA
REG1 1.558 0.11 NO 4.883 0.00 * YES NO
REG3 1.482 0.14 NO 4.281 0.00 * YES NO
PALM
SEAM REG1 0.646 0.79 NO 1.045 0.41 NO NO
REG3 0.456 0.93 NO 1.190 0.26 NO NO
Reg-ARIMA
REG1 1.165 0.31 NO 2.527 0.00 * YES NO
REG3 0.354 0.97 NO 1.832 0.02 YES NO
VEHICLE
SEAM REG1 2.230 0.01 NO 1.240 0.22 NO NO
REG3 1.984 0.03 NO 1.119 0.33 NO NO
Reg-ARIMA
REG1 2.127 0.02 NO 2.413 0.001 YES NO
REG3 0.297 0.99 NO 1.596 0.05 NO NO
Since both procedures were effective in removing the moving holiday effect, the next stage is
to determine which procedure performs better. A comparison was made on each of the series
by ranking the probability values of moving seasonality based on the types of regressors
used. For example, IMPORT series, where REG1 for SEAM was compared against REG1
for RegARIMA and rank 1 is given to biggest p-value. When the probability values are equal
as in this case, then the corresponding F-values will be used. Hence, for IMPORT series,
rank 1 is given to REG1 of RegARIMA and rank 2 to REG1 of SEAM. These ranks are then
summed across all data series and the results of ranking are shown in Table 5.
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Table 5: The Results of Ranking the p-value based on Regressors SEASONALLY ADJUSTED TIME SERIES DATA M
ETH
OD
REG
RES
SOR
MOVING SEASONALITY (test at 5%) ( * test at 1%)
RANKING
F-value p-value Presence REG1 REG3
IMPORT
SEAM REG1 1.267 0.20 NO 2
REG3 0.939 0.54 NO 1
Reg-ARIMA
REG1 1.260 0.20 NO 1
REG3 1.304 0.17 NO 2
EXPORT
SEAM REG1 0.874 0.63 NO 1
REG3 0.827 0.69 NO 1
Reg-ARIMA
REG1 1.025 0.43 NO 2
REG3 1.053 0.40 NO 2
OMP
SEAM REG1 1.931 0.01 YES 1
REG3 1.960 0.009 * YES 1
Reg-ARIMA
REG1 4.883 0.00 * YES 2
REG3 4.281 0.00 * YES 2
PALM
SEAM REG1 1.045 0.41 NO 1
REG3 1.190 0.26 NO 1
Reg-ARIMA
REG1 2.527 0.00 * YES 2
REG3 1.832 0.02 YES 2
VEHICLE
SEAM REG1 1.240 0.22 NO 1
REG3 1.119 0.33 NO 1
Reg-ARIMA
REG1 2.413 0.001 YES 2
REG3 1.596 0.05 NO 2
Table 6 summarizes the results of the total rank of p-values associated with each of the
regressor REG1 and REG3 as applied to the two competing methods, SEAM and
RegARIMA. The method that has the smallest total rank values will be considered the more
effective procedure. From the results in Tables 5 and 6, we can conclude that the SEAM
method is more effective than the RegARIMA method when used for removing holiday
effect.
Table 6: Summary of Total Rank for SEAM and RegARIMA
METHOD REG1 REG3 TOTAL
SEAM RegARIMA
6 9
5 10
11 19
15
3.1 Forecast Performance
Although the SEAM method fared better at removing holiday effect than Reg-ARIMA but
more importantly is to determine which procedure performs better in trend forecasting for
seasonally adjusted time series. A comparison of models’ performance, in particular for short
term forecasting of the underlying trend was done for the two types of seasonally adjusted
data. The type 1 was seasonally adjusted using SEAM while the type 2 was seasonally
adjusted using RegARIMA. The ARIMA models were then fitted to the data series. The
models were identified based on the ACF and PACF. The selection of the best model was
carried out using the Akaike Information Criterion (AIC) goodness-of-fit statistics. The
decision rule is to select a model with the smallest AIC which is said to fit the data better. At
the same time the adequacy of the competing models (diagnostic checking) was determined
using the Ljung-Box portmanteau test ( Q* -statistics). Lastly, the models were evaluated
using two error measures, that is, Relative Mean Squared Error (RMSE).
Table 7 : Ljung-Box, AIC, RMSE and Best ARIMA Model
Data Series ARIMA Model Ljung-Box
AIC RMSE Best ARIMA
Model d.f )(2
][ pnK −αχ
p-value
IMPORT (5,1,1)(1,0,0)12 4 7.8 0.101 3989.94 704.30 (5,1,1)(1,0,0)12
(5,1,2)(1,0,0)12 4 4.6 0.329 3991.63 707.24
(5,1,2)(1,0,1)12 3 4.3 0.233 3992.88 704.74
EXPORT (3,1,0)(1,0,0)12 8 7.0 0.533 3942.52 786.05
(4,1,0)(1,0,0)12 7 7.5 0.381 3944.51 787.41
(5,1,0)(1,0,0)12 6 2.5 0.870 3941.88 779.27 (5,1,0)(1,0,0)12
OMP (4,1,0)(1,0,0)12 7 6.5 0.488 7396.24 421688.42
(4,1,0)(1,0,1)12 6 6.5 0.374 7397.83 422466.82
(5,1,0)(1,0,0)12 6 2.3 0.888 7377.45 419533.64 (5,1,0)(1,0,0)12
RPV (3,1,1)(1,0,0)12 7 7.7 0.355 4930.36 3377.04 (3,1,1)(1,0,0)12
(4,1,0)(1,0,0)12 7 7.8 0.355 4931.86 3377.05
(4,1,1)(1,0,0)12 6 7.6 0.270 4931.52 3382.15
VEHICLE (1,1,2)(1,0,0)12 8 16.6 0.034 4624.21 1709.31
(5,1,3)(1,0,0)12 3 6.3 0.099 4624.87 1678.25 (5,1,3)(1,0,0)12
(5,1,3)(1,0,1)12 2 506 0.061 4625.88 1690.17
16
The RMSE within sample for the data series were calculated by using the SAS software and
the results are shown in Table 7. In order to select the best model, the smallest RMSE is
considered. Therefore, among the three models for each series, the best model selected
contains the smallest AIC and RMSE. The best ARIMA models were then applied to the
deseasonalised data which used REG1 and REG3. The test statistics, such as the Ljung –Box
statistics, the AIC and the RMSE were performed and shown in Table 8.
Table 8 : The Summary Statistics for All Seasonally Adjusted Series SEASONALLY ADJUSTED TIME SERIES DATA
ARIMA MODELS
NA
L
AD
JUS
TM
EN
T
PRO
CE
RE
GR
ESS
OR
S Ljung-Box
(Q* statistics) AIC
WITHIN SAMPLE
df 2χ p-value RMSE
IMPORT (5,1,1)(1,0,0)12
SEAM REG1 4 18.9 *0.001 3997.85 697.47
REG3 4 7.8 0.101 3989.94 704.30
Reg- ARIMA
REG1 4 3.7 0.454 3909.40 576.42
REG3 4 5.3 0.260 3914.79 578.49
EXPORT (5,1,0)(1,0,0)12
SEAM REG1 6 5.3 0.504 3951.36 780.27
REG3 6 2.5 0.870 3941.88 779.27
Reg- ARIMA
REG1 6 9.0 0.171 3859.09 577.30
REG3 6 6.4 0.377 3862.64 578.62
OMP (5,1,0)(1,0,0)12
SEAM REG1 6 10.1 0.120 7381.80 444279.51
REG3 6 2.3 0.888 7377.45 419533.64
Reg- ARIMA
REG1 6 13.6 *0.034 7301.18 369081.79
REG3 6 12.9 *0.045 7300.84 363838.17
RPV (3,1,1)(1,0,0)12
SEAM REG1 7 16.7 *0.019 4948.91 3510.90
REG3 7 7.7 0.355 4930.36 3377.04
Reg- ARIMA
REG1 7 10.4 0.165 4915.18 3306.32
REG3 7 10.7 0.150 4900.90 3219.94
VEHICLE (5,1,3)(1,0,0)12
SEAM REG1 3 6.6 0.086 4639.35 1808.87
REG3 3 6.3 0.099 4624.87 1678.25
Reg- ARIMA
REG1 3 5.7 0.126 4655.96 1889.98
REG3 3 6.3 0.099 4589.05 1678.25
* models are not adequate
3.2 Forecast Evaluation
The out-of-sample forecast starts in January 2002 – December 2002 and the first 21 years’
(1981-2001) data are used for model estimation. The results of out-of-sample error measures
17
for the seasonally adjusted time series data are shown in Table 9. Each economic series was
seasonally adjusted using the the SEAM and RegARIMA with two different regressors, that
is, REG1 and REG3. The series were evaluated and two error measures, RMSE and MAPE
were calculated. The results show that both the SEAM and RegARIMA seem to give similar
results. Therefore, it is difficult to identify which method performs better forecasting. For
some series, SEAM procedure is better than RegARIMA whilst for some other series
RegARIMA procedure is better than SEAM.
Therefore, to determine which method performed better forecasting of the trend, the values of
RMSE and MAPE were ranked according to the types of regressors. A comparison is made
on each of the series, e.g., for the import series, REG1 for SEAM is compared against REG1
of RegARIMA. Since the RMSE value of REG1 (SEAM) is smaller than REG1
(RegARIMA), therefore, rank 1 is given to REG1 of SEAM and rank 2 is given to REG1 of
RegARIMA. The same ranking is done for MAPE. Comparing SEAM and RegARIMA
method, the smaller value of MAPE is given rank 1 and subsequently the bigger value is
given rank 2. These ranks are then summed across all data series. The results of ranking are
as shown in Table 9 and the summary of ranking the RMSE and MAPE are shown in Table
10. From the results of ranking the RMSE and MAPE, was found that the SEAM method has
smaller total rank for REG1 and REG3 while the RegARIMA method has smaller total rank
for REG2. Therefore, we can conclude that the SEAM method is more effective in
deseasonalising using REG1 and REG3. As an overall conclusion, the SEAM method
provides better forecasts of the underlying trend.
18
Table 9 : The Results of Out-of-sample Evaluation SEASONALLY ADJUSTED TIME SERIES DATA
ARIMA MODELS
SEA
SON
AL
A
DJU
STM
EN
T
PRO
CE
DU
RE
S
RE
GR
ES
SOR
S
OUT-Of-SAMPLE RANKING
RMSE MAPE RMSE MAPE
IMPORT (5,1,1)(1,0,0)12
SEAM REG1 887.2 2.8 1 1
REG3 1225.6 4.2 1 1
Reg-ARIMA
REG1 956.1 3.2 2 2
REG3 1353.8 4.6 2 2
EXPORT (5,1,0)(1,0,0)12
SEAM REG1 1756.8 5.3 2 2
REG3 2149.7 6.8 2 2
Reg-ARIMA
REG1 1312.7 4.0 1 1
REG3 1545.4 4.7 1 1
OMP (5,1,0)(1,0,0)12
SEAM REG1 1585860.9 5.8 1 1
REG3 1783056.3 6.4 1 1
Reg-ARIMA
REG1 2120293.5 6.9 2 2
REG3 1783283.7 6.4 2 2
PALM (4,1,0)(1,0,0)12
SEAM REG1 46.2 4.1 1 1
REG3 46.0 4.0 1 1
Reg-ARIMA
REG1 58.9 4.5 2 2
REG3 60.3 4.8 2 2
VEHICLE (5,1,3)(1,0,0)12
SEAM REG1 2991.6 10.9 1 1
REG3 3213.5 11.3 1 1
Reg-ARIMA
REG1 4539.9 14.9 2 2
REG3 3364.8 11.8 2 2
Table 10 : Summary of Ranking the RMSE and MAPE
RMSE MAPE TOTAL
SEAM
Reg- ARIMA
SEAM Reg-
ARIMA SEAM
Reg- ARIMA
REG1 6 9 6 9 12 18
REG3 6 9 6 9 12 18
4. Conclusion
This study shows that the festivals such as Eid-ul Fitr, Chinese New Year and Deepavali
celebration that fall on different dates and hence do not follow the Gregorian calendar
significantly affect the eight time series data. These three major festivals have the effect of
either stimulating or reducing activities for the periods in the vicinity of the holiday dates.
19
The application of the seasonal adjustment procedures using SEAM (Seasonal Adjustment
for Malaysia) and RegARIMA (adjusted for Malaysia) can significantly eliminate the
presence of the moving holiday effects. Therefore, the proposed methods, SEAM and
RegARIMA are able to remove moving holiday effects and seasonally adjust the data series.
Although, RegARIMA is a standard program for removing holiday effects, results show that
overall SEAM performs better than the RegARIMA in removing the Malaysian moving
holiday effect.
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Appendix 1: The Dates of Major Festivals in Malaysia, 1981-2010
Year Eid-ul Fitr Chinese New Year Deepavali
1981 1-Aug 5-Feb 26-Oct 1982 22-Jul 25-Jan 14-Nov 1983 12-Jul 13-Feb 4-Nov 1984 30-Jun 2-Feb 23-Oct 1985 20-Jun 20-Feb 11-Nov 1986 9-Jun 9-Feb 1-Nov 1987 29-May 29-Jan 4-Nov 1988 17-May 17-Feb 8-Nov 1989 7-May 6-Feb 29-Oct 1990 26-Apr 27-Jan 17-Oct 1991 16-Apr 15-Feb 5-Nov 1992 4-Apr 4-Feb 26-Oct 1993 25-Mar 23-Jan 13-Nov 1994 14-Mar 10-Feb 3-Nov 1995 3-Mar 31-Jan 23-Oct 1996 20-Feb 19-Feb 10-Nov 1997 9-Feb 7-Feb 30-Oct 1998 30-Jan 28-Jan 19-Oct 1999 19-Jan 16-Feb 7-Nov 2000 8-Jan & 27-Dec 5-Feb 26-Oct 2001 16-Dec 24-Jan 14-Nov 2002 6-Dec 12-Feb 4-Nov 2003 26-Nov 1-Feb 23-Oct 2004 14-Nov 22-Jan 11-Nov 2005 3-Nov 9-Feb 1-Nov 2006 24-Oct 30-Jan 21-Oct 2007 13-Oct 19-Feb 9-Nov 2008 2-Oct 7-Feb 28-Oct 2009 21-Sep 26-Jan 16-Nov 2010 10-Sep 1-Jan 5-Nov
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