New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment Enrico Infante * – Università degli studi di Napoli Federico

1

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

2

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


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



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



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!!!



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



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



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



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



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



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



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



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



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





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



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



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



QUESTIONS?

Many Thanks!!!

Documents

New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment Enrico Infante * – Università degli studi di Napoli Federico