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New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal
Adjustment
Enrico Infante * – Università degli studi di Napoli Federico II
Dario Buono * – EUROSTAT, unit B2: Research and Methodology
CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
*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
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INTRODUCTION
A generic time series Yt can be the result of an aggregation of p series:
CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
pthttt XXXfY ,,,,1
We focus on the case of the additive function:
p
hhthptphthtt XXXXY
111
3CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
INTRODUCTION
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
hhtht XSAYSA
1
The Seasonally Adjusted data are computed indirectly by Seasonally Adjusting data per each series
p
hhtht XSAYSA
1
4CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
INTRODUCTION
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
uutu
q
lltlt XXY
11
Group A Group B
prq
r
uutu
q
lltlt XSAXSAYSA
11
5CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
THE BASIC IDEA
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!!!
6CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
WHY A NEW TEST?
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
7CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
THE TEST
The variable tested is the final estimation of the unmodified Seasonal-Irregular differences (or ratios) absolute value
ijkSI
1ijkSI
Additive model
Multiplicative model
If the series are not modelled with the same decomposition model, then the data must be normalized by the column (the time frequency factor). 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
8CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
THE TEST
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
9CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
THE TEST
The test is based on the decomposition of the variance of the observations:
22222RSNM SSSSS
Sk ,,1
Mi ,,1
Between time frequencies variance
Between years variance
Between series variance
Residual variance
Nj ,,1
10CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
THE TEST
VAR Mean df
2MS
2NS
2SS
2RS
N
j
S
kijki SI
NSx
1 1
1
M
i
S
kijkj SI
MSx
1 1
1
M
i
N
j
S
kkjiijk xxxxSI
1 1 1
22
M
ii xxNS
1
2
N
jj xxMS
1
2
S
kk xxMN
1
2
M
i
N
jijkk SI
MNx
1 1
1
1M
1N
1S
111 SNM
The table for the ANOVA test
Sum of Squares
11CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
THE TEST
The null hypothesis is made taking into consideration that there is no change in seasonality over the series
111;12
2
~ SNMSR
ST F
S
SF
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
12CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
SHOWING THE PROCEDURE - EXAMPLE
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
13CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
NUMERICAL EXAMPLE
Let’s consider the Construction Production of the three French speaker European counties: France, Belgium and Luxembourg (data are available on the EUROSTAT database). The time span is from Jan-01 to Dec-10
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
8122.50117.0
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
14CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
NUMERICAL EXAMPLE
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
7591.75F 0000.0 valueP 8313.4F 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
15CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
NUMERICAL EXAMPLE
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
2314.0F 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
16CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
FUTURE RESEARCH LINE
Starting from this idea, there is still work to do!!!
Case study (Demetra+)
Simulations (R)
Application with a Tukey’s range test
Theoretical review
Testing with real data
Create the theoretical base
17CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
REFERENCES
[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] R. Astolfi, D. Ladiray, G. L. Mazzi – Business cycle extraction of Euro-zone GDP: direct versus indirect approach – European Communities, 2001
[7] J. Lothian, M. Morry - A set of Quality Control Statistics for the X-11-ARIMA Seasonal Adjustment Method – Statistics Canada, 1978
[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
18CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
QUESTIONS?
Many Thanks!!!