This document proposes a new 3-way ANOVA test to help determine whether to use a direct, indirect, or mixed approach for seasonally adjusting time series data. The test examines seasonal patterns across time frequencies, years, and individual time series to see if they have common regular seasonal patterns. If seasonal patterns differ, an indirect or mixed approach should be used instead of a direct approach. The document provides an example application to construction industry data from France, Belgium, and Luxembourg, finding common seasonal patterns between Belgium and Luxembourg to allow their mixed adjustment. Future work is proposed to further develop the theoretical underpinnings and test additional applications of the approach.

1. Enrico Infante*
University of Naples Federico II
Dario Buono*
EUROSTAT, Unit B1: Quality, Research and Methodology
Euroindicators PEEI WG – Luxembourg, 11-12 June 2012
*The views and the opinions expressed in this paper are solely of the authors
and do not necessarily reflect those of the institutions for which they work
New innovative 3-way ANOVA a-priori test for
direct vs. indirect approach in Seasonal Adjustment

2. 2
A generic time series Yt can be the result of an aggregation of p series:
( )pthttt XXXfY ,,,,1 =
We focus on the case of the additive function:
∑=
=++++=
p
h
hthptphthtt XXXXY
1
11 ϖϖϖϖ 
Introduction
Enrico Infante, Dario Buono

3. 3
To Seasonally Adjust the aggregate, different approaches can be applied
Direct Approach
Indirect Approach
The Seasonally Adjusted data are
computed directly by Seasonally
Adjusting the aggregate
( ) 





= ∑=
p
h
htht XSAYSA
1
ϖ
The Seasonally Adjusted data are
computed indirectly by Seasonally
Adjusting data per each series
( ) ( )∑=
=
p
h
htht XSAYSA
1
ϖ
Introduction
Enrico Infante, Dario Buono

4. 4
If it is possible to divide the series into groups, then it is possible to
compute the Seasonally Adjusted figures by summing the Seasonally
Adjusted data of these groups
Mixed Approach
Example (two groups):
∑∑ ==
+=
r
u
utu
q
l
ltlt XXY
11
ϖϖ
Group A Group B
prq =+
( ) 





+





= ∑∑ ==
r
u
utu
q
l
ltlt XSAXSAYSA
11
ϖϖ
Introduction
Enrico Infante, Dario Buono

5. 5
To use the Mixed Approach, sub-aggregates must be defined
We would like to find a criterion to divide the series into groups
The series of each group must have
common regular seasonal patterns
How is it possible to decide that two or more series have common
seasonal patterns?
NEW TEST!!!
The basic idea
Enrico Infante, Dario Buono

6. 6
Direct and indirect: there is no consensus on which is the best approach
Direct Indirect
+
-
• Transparency
• Accuracy
• Accounting
Consistency
• No accounting
consistency
• Cancel-out
effect
• Residual
Seasonality
• Calculations
burden
It could be interesting to identify
which series can be aggregated in
groups and decide at which level
the SA procedure should be run
This test gives information about
the approach to follow before
SA of the series
Why a new test?
Enrico Infante, Dario Buono
The presence of residual seasonality should
always be checked in all of the Indirect and
Mixed Seasonally Adjusted aggregates

7. 7
The variable tested is the final estimation of the unmodified Seasonal-
Irregular ratios (or differences) absolute value
ijkSI
1−ijkSI
Additive model
Multiplicative model
It is considered that the decomposition model is the same on all the
series. The series is then considered already Calendar Adjusted
The classic test for moving seasonality is based on a 2-way ANOVA test,
where the two factors are the time frequency (usually months or
quarters) and the years. This test is based on a 3-way ANOVA model,
where the three factors are the time frequency, the years and the series
The test
Enrico Infante, Dario Buono

8. 8
The model is:
ijkkjiijk ecbaSI +++=
Where:
• ai, i=1,…,M, represents the numerical contribution due to the effect of
the i-th time frequency (usually M=12 or M=4)
• bj, j=1,…,N, represents the numerical contribution due to the effect of
the j-th year
• ck, k=1,…,S, represents the numerical contribution due to the effect of
the k-th series of the aggregate
• The residual component term eijk (assumed to be normally distributed
with zero mean, constant variance and zero covariance) represents the
effect on the values of the SI of the whole set of factors not explicitly
taken into account in the model
The test
Enrico Infante, Dario Buono

9. 9
The test is based on the decomposition of the variance of the
observations:
22222
RSNM SSSSS +++=
Sk ,,1 =
Nj ,,1 =
Between time frequencies variance
Between years variance
Between series variance
Residual variance
The test
Enrico Infante, Dario Buono
Mi ,,1 =

10. 10
VAR Mean df
2
MS
2
NS
2
SS
2
RS
∑∑= =
•• =
N
j
S
k
ijki SI
NS
x
1 1
1
∑∑= =
•• =
M
i
S
k
ijkj SI
MS
x
1 1
1
( )∑∑∑= = =
•••••• +−−−
M
i
N
j
S
k
kjiijk xxxxSI
1 1 1
2
2
( )∑=
•• −
M
i
i xxNS
1
2
( )∑=
•• −
N
j
j xxMS
1
2
( )∑=
•• −
S
k
k xxMN
1
2
∑∑= =
•• =
M
i
N
j
ijkk SI
MN
x
1 1
1
1−M
1−N
1−S
( )( )( )111 −−− SNM
The table for the ANOVA test
Sum of Squares
The test
Enrico Infante, Dario Buono

11. 11
The null hypothesis is made taking into consideration that there is no
change in seasonality over the series
( ) ( )( )( )111;12
2
~ −−−−= SNMS
R
S
T F
S
S
F
The test statistic is the ratio of the between series variance and the
residual variance, and follows a Fisher-Snedecor distribution with (S-1)
and (M-1)(N-1)(S-1) degrees of freedom
ScccH === 210 :
Rejecting the null hypothesis is to say that the pure Direct Approach
should be avoided, and an Indirect or a Mixed one should be considered
The test
Enrico Infante, Dario Buono

12. 12
ttt XXY 21 +=
The most simple case: the aggregate is formed of two series, using the
same decomposition model
Do X1t and X2t have the same seasonal patterns?
TEST
Rejecting H0: the two series
have different seasonal patterns
Not rejecting H0: the two series
have common regular seasonal
patterns
Direct Approach
Indirect Approach
Showing the procedure - Example
Enrico Infante, Dario Buono

13. 13
Let’s consider the Construction Production Index of the three French-
speaking European countries: France, Belgium and Luxembourg (data are
available on the EUROSTAT database). The time span is from January
2001 to December 2010
To take an example, a very simple aggregate could be the following:
tttt LUBEFRY ++=
VAR Mean Square df
Months 1.5003 11
Years 0.0226 9
Series 0.1356 2
Residual 0.0117 198
8122.5
0117.0
1356.0
==− ratioF 0035.0=− valueP
There is no evidence of common
seasonal patterns between the
series at 5 per cent level
The Direct Approach
should be avoided
Numerical example
Enrico Infante, Dario Buono

14. 14
If two of them have the same seasonal pattern, a Mixed Approach could
be used. So the test is now used for each couple of series
VAR Mean Square df
Months 2.0403 11
Years 0.0140 9
Series 0.1199 1
Residual 0.0016 99
7591.75=F 0000.0=− valueP 8313.4=F 0303.0=− valueP
VAR Mean Square df
Months 1.0464 11
Years 0.0172 9
Series 0.0793 1
Residual 0.0164 99
LU - FR BE - FR
There is no evidence of common
seasonal patterns between the
series at 5 per cent level
There is no evidence of common
seasonal patterns between the
series at 5 per cent level
Numerical example
Enrico Infante, Dario Buono

15. 15
An excel file with all the calculations is available on request
VAR Mean Square df
Months 0.9579 11
Years 0.0202 9
Series 0.0042 1
Residual 0.0181 99
2314.0=F 6315.0=− valueP
LU - BE
Common seasonal patterns
between the series present
at 5 per cent level
LU and BE have the same seasonal pattern, so it is possible to Seasonally
Adjust them together, using a Mixed Approach
( ) ( ) ( )tttt LUBESAFRSAYSA ++=
Numerical example
Enrico Infante, Dario Buono

16. 16
Theoretical review (F-ratio, trend, co-movements test)
Future research line
Enrico Infante, Dario Buono
• F-ratio: re-building the test upon the ratio of the between months
variance and the residual variance (comments by Kirchner)
Additive and multiplicative
decompositions
Moving Seasonality
+ -
• A-priori estimation of the trend
• Use of the co-movements test as benchmarking

17. 17
Case study (IPC using Demetra+) - ongoing
Simulations (R) - ongoing
Application with a Tukey’s range test
Future research line
Enrico Infante, Dario Buono

18. 18
[1] J. Higginson – An F Test for the Presence of Moving Seasonality When Using
Census Method II-X-11 Variant – Statistics Canada, 1975
[2] R. Astolfi, D. Ladiray, G. L. Mazzi – Seasonal Adjustment of European
Aggregates: Direct versus Indirect Approach – European Communities, 2001
[3] F. Busetti, A. Harvey – Seasonality Tests – Journal of Business and Economic
Statistics, Vol. 21, No. 3, pp. 420-436, Jul. 2003
[4] B. C. Surtradhar, E. B. Dagum – Bartlett-type modified test for moving
seasonality with applications – The Statistician, Vol. 47, Part 1, 1998
[5] M. Centoni, G. Cubbadda – Modelling Comovements of Economic Time Series:
A Selective Survey – CEIS, 2011
[7] A. Maravall – An application of the TRAMO-SEATS automatic procedure; direct
versus indirect approach – Computation Statistics & Data Analysis, 2005
[8] R. Cristadoro, R. Sabbatini - The Seasonal Adjustment of the Harmonised Index
of Consumer Prices for the Euro Area: a Comparison of Direct and Indirect
Method – Banca d’Italia, 2000
[9] B. Cohen – Explaning Psychological Statistics (3rd
ed.), Chapter 22: Three-way
ANOVA - New York: John Wiley & Sons, 2007
[10]I. Hindrayanto - Seasonal adjustment: direct, indirect or multivariate method?
– Aenorm, No. 43, 2004
References
Enrico Infante, Dario Buono

19. 19
Many Thanks!!!
Questions?
Enrico Infante, Dario Buono
We are really grateful for all the comments we already received
(in particular from R. Gatto, R. Kirchner, A. Maravall, G.L. Mazzi, J. Palate)