Existence of Arch Effect in Consumer Goods Deflator the Greek Case 1960-1994

International Research Journal of Finance and Economics
ISSN 1450-2887 Issue 37 (2010)
© EuroJournals Publishing, Inc. 2010
http://www.eurojournals.com/finance.htm

Existence of Arch Effect in Consumer Goods
Deflator the Greek Case 1960-1994

Paraschos Maniatis
Department of Business Administration at Athens University of Economics and Business
76 Patission St., Athens, Greece,GR-104 34
E-mail: pman@sch.gr


Abstract

Heteroscedasticity is a typical phenomenon in economic and financial time series.
The variance of the series changes with time. This effect makes the series ipso facto non-
stationary and deprives the investigator of the mathematical and statistical apparatus
tailored for stationary time series. Given this situation, the investigator is obliged to
consider non-linear stochastic models- the most popular of which are the (G)ARCH ones.
In this study we give the theoretical background of the ARCH model and an econometric
application of the model to the Greek consumer goods deflator. The study gives evidence
that ARCH effects are present in the investigated time series.


Keywords: Arch Effect, Greek Consumer Goods Deflator, Heteroscedasticity.

Introduction
Although there is no general agreement for the meaning of a linear stochastic model there is a strong
tradition to consider a stochastic model as linear if it is linear in the parameters. In this sense an
autoregressive process of p order AR(p) is considered as linear. This conception of linearity results
from the convenience obtained when applying expectations to a stochastic expression, for expectations
are linear transformations. Several families of non-linear models have been introduced in the attempt to
deal with non-linear models as non-linear autoregressive processes, threshold models, bilinear models
(Priestly, 1988) and chaotic models (Mandelbot, 1995). Especially for cases of changing variance
classes of models described as Autoregressive Conditional Heteroscedasticity models (ARCH models)
and Generalized ARCH models (GARCH models) have been introduced. These models do not
generally obtain better point forecasts but they can help to obtain better estimates of the local variance,
which allows construction of more reliable prediction intervals and better risk assessment. This is
important for financial time series, which generally exhibit sudden changes of variance (volatility).
More specifically, the ARCH models are martingale differences and hence knowledge of the variance
in time t σ 2
t does not lead to better point forecast in time t+s. Moreover, as is it is shown in the
appropriate section of this study, even with constant unconditional variance, the conditional variance
can change with time. The last consideration renders the use of (G)ARCH models indispensable even
in cases of stationary time series.
In our study we investigate presence of ARCH effect in the annual rate of change in the
consumer goods deflator. In section 2 we define the data, the variables, their descriptions and symbols.
In section 3 we present the theoretical background of the ARCH modeling, its application to our data
and a discussion of the findings. The section 4- discussion of the findings- contains a summary of some
drawbacks concerning the model the application and the findings.

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International Research Journal of Finance and Economics - Issue 37 (2010)
86
The Data
The data consists of 35 annual measurements of the consumer goods deflator (CGD) from 1960
through 1994 (source: OECD). The data is shown in the column titled D. For all calculations, graphs
and results we have employed the program STATISTICA. For easy reference, we have embedded all
tables and graphs in this text. We have on purpose limited the series up to 1994 since recent economic
data are in the most of times not definite and they involve political considerations and conflict of
interest. In the following tables 1 and 2 we submit the list of all variables and the list of tables and
graphs relevant to this study, accordingly.

Table 1:
List of variables, symbols and descriptions

Symbol Mathematical
definition
Description
D
Dt
Consumer goods deflator in time t
RD
RDt= (Dt –Dt-1)/ Dt-1)
Rate of change of D from time t-1 to time t
RD_1 RDt-1
Rate of change of D from time t-2 to time t-1
U ut
Residual from simple regression of RDt to RDt-1
U2 u 2
t
Squared residual in time t
U2_1 u 2
t-1
Squared residual with lag 1
U2_2 u 2
t-2
Squared residual with lag 2
U2_3 u 2
t-3
Squared residual with lag 3
U2_4 u 2
t-4
Squared residual with lag 4
ACF Autocorrelation
function
Autocorrelation function of the residuals ut
PACF Partial
autocorrelation
function
Partial
autocorrelation function of the residuals ut

Table 2:
List of statistica files for tables and graphs

File serial
File type
File name
File content
number
Table
3
ARCH EFFECT-DATA
Original data, derived series, residuals
Table
4
Regression RD to RD_1
Regression results in model RDt=α0+α1RDt-1
Table
5
ARCH(1)
Regression results in ARCH(1) model
Table
6
ARCH(2)
Regression results in ARCH(2) model
Table
7
ARCH(3)
Regression results in ARCH(3) model
Table
8
ARCH(4)
Regression results in ARCH(4) model
Table 9
-
Regression summary
Graph
1
Graph 1-Deflator (D)
Graph of the deflator
Graph
2
Graph 2-Squared regression residuals
Squared regression residuals in model
RDt=α0+α1RDt-1
Graph
3
Graph 3-Aggregated squared regression
Aggregated squared regression residuals in
residuals
model RDt=α0+α1RDt-1
Graph
4
Graph 4-Regression residuals
Regression residuals in model RDt=α0+α1RDt-
1
Graph
5
Graph 2-Aggregated deflator
Aggregated graph of the deflator
Graph
6
Graph 4-Aggregated RD
Aggregated graph of RD
Graph
7
Graph 3-Change rate in D
Graph of RD
Graph
8
Graph 8-ACF of regression residuals
Autocorrelation function of the regression
residuals in model RDt=α0+α1RDt-1
Graph
9
Graph 9- PACF of regression residuals
Partial autocorrelation function of the
regression residuals in model RDt=α0+α1RDt-1

In the following table 3 are shown the original data (YEAR, D), the derived series and the
residuals

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International Research Journal of Finance and Economics - Issue 37 (2010)
Table 3:
Arch effect-data original data, derived series and residuals

YEAR D
RD RD_1 U
U2 U2_1 U2_2 U2_3 U2_4
1960
0,783

1961
0,792
0,011

1962
0,802
0,013
0,011
-0,020
0,000

1963 0,829 0,034 0,013 0,001 0,000 0,000



1964 0,847 0,022 0,034 -0,029 0,001 0,000 0,000


1965 0,886 0,046 0,022 0,005 0,000 0,001 0,000 0,000

1966 0,917 0,035 0,046 -0,026 0,001 0,000 0,001 0,000 0,000
1967 0,934 0,019 0,035 -0,033 0,001 0,001 0,000 0,001 0,000
1968 0,941 0,007 0,019 -0,031 0,001 0,001 0,001 0,000 0,001
1969 0,970 0,031 0,007 0,002 0,000 0,001 0,001 0,001 0,000
1970 1,000 0,031 0,031 -0,017 0,000 0,000 0,001 0,001 0,001
1971 1,029 0,029 0,031 -0,019 0,000 0,000 0,000 0,001 0,001
1972 1,062 0,032 0,029 -0,015 0,000 0,000 0,000 0,000 0,001
1973 1,222 0,151 0,032 0,101 0,010 0,000 0,000 0,000 0,000
1974 1,509 0,235 0,151 0,087 0,008 0,010 0,000 0,000 0,000
1975 1,701 0,127 0,235 -0,091 0,008 0,008 0,010 0,000 0,000
1976 1,930 0,135 0,127 0,006 0,000 0,008 0,008 0,010 0,000
1977 2,160 0,119 0,135 -0,016 0,000 0,000 0,008 0,008 0,010
1978 2,436 0,128 0,119 0,006 0,000 0,000 0,000 0,008 0,008
1979 2,838 0,165 0,128 0,036 0,001 0,000 0,000 0,000 0,008
1980 3,463 0,220 0,165 0,060 0,004 0,001 0,000 0,000 0,000
1981 4,242 0,225 0,220 0,019 0,000 0,004 0,001 0,000 0,000
1982 5,119 0,207 0,225 -0,003 0,000 0,000 0,004 0,001 0,000
1983 6,047 0,181 0,207 -0,014 0,000 0,000 0,000 0,004 0,001
1984 7,127 0,179 0,181 0,005 0,000 0,000 0,000 0,000 0,004
1985 8,422 0,182 0,179 0,010 0,000 0,000 0,000 0,000 0,000
1986 10,301 0,223 0,182 0,049 0,002 0,000 0,000 0,000 0,000
1987 11,920 0,157 0,223 -0,051 0,003 0,002 0,000 0,000 0,000
1988 13,619 0,143 0,157 -0,011 0,000 0,003 0,002 0,000 0,000
1989 15,672 0,151 0,143 0,009 0,000 0,000 0,003 0,002 0,000
1990 18,776 0,198 0,151 0,050 0,002 0,000 0,000 0,003 0,002
1991 22,231 0,184 0,198 -0,004 0,000 0,002 0,000 0,000 0,003
1992 25,677 0,155 0,184 -0,021 0,000 0,000 0,002 0,000 0,000
1993 29,210 0,138 0,155 -0,014 0,000 0,000 0,000 0,002 0,000
1994 32,335 0,107 0,138 -0,030 0,001 0,000 0,000 0,000 0,002
1995
0,107
0,001
0,000
0,000
0,000
1996

0,001
0,000
0,000
1997

0,001
0,000
1998

0,001
1999










In the following graph 1 is shown the time series of the original data (D, Consumer Goods
Deflator)

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International Research Journal of Finance and Economics - Issue 37 (2010)
88
Graph 1: Deflator (D)

Line Plot (00-ARCH EFFECT-DATA.STA 9v*40c)
35
30
25
20
15
D
10
5
0
-5
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994



Statistical Analysis
The Theoretical Part
Consider the following AR(1) scheme
ut=ρut-1+εt
(1)
which we can, without loss of generality, consider as stationary (|ρ|<1), with εt uncorrelated identically
distributed stochastic variables with zero mean and variance σ 2
ε . The above scheme may be considered
as representing the residuals of a regression, say Xt to Xt-1. Using the symbols E for expectations and V
for variance and exploiting the well-known properties of the expectation and the variance for
uncorrelated variables, we can easily obtain the results:
E(ut)=ρE(ut-1)+E(εt) and V(ut)=ρ2V(ut-1)+V(εt). The last two expressions give
E(ut)=0
(2)
V(u
2
t)=σε /(1-ρ2) (3)
But although the unconditional expectation of ut is constant (zero in this case), its conditional
expectation is not:
E(ut|ut-1)=E(ρut-1+εt)|ut-1)=E(ρut-1|ut-1)+E(εt|ut-1)=ρut-1+0=ρut-1. (4)
However, the conditional variance of the scheme is constant:
V(u
2
2
t|ut-1)=E[(ρut-1+εt)2|ut-1]-(ρut-1)2= (ρut-1)2+V(εt )-(ρut-1)2=σε (5)
The above results indicate that an ordinary linear autoregressive scheme AR(p) cannot help in
situations in which the variance is variable. One way to sort this out is to use for u a model, which
allows for:
• Constant unconditional mean
• Constant conditional mean of
• Uncorrelated u’s
• Constant unconditional variance
• Changing conditional variance
This model exists and is of the (non-linear) form
u
2
t=εt(α0+α1ut-1 )1/2 (6)

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International Research Journal of Finance and Economics - Issue 37 (2010)
α0, α1 non-negative numbers and εt uncorrelated, identically distributed stochastic variables,
uncorrelated with all u’s, with zero mean and unit variance (the last two requirements are not
obligatory since any stochastic variable can be transformed to a new variable with zero mean and unit
variance). The above model is called autoregressive conditional heteroscedasticity model of order 1,
and it is denoted by the symbol ARCH(1). In the general case where u depends on its p previous values
model it is called ARCH of order p and is denoted by
ARCH(p): u
2
2
2
t=εt(α0+α1ut-1 +α2ut-2 +… ....+αput-p )1/2 (7)
Let us prove that the model (6) satisfies the wished propertied 1/ through 4/ above:
For 1.
Ε(u
2
2
2
t)=Ε[εt(α0+α1ut-1 )1/2]=Ε(εt)Ε(α0+α1ut-1 )1/2=0xΕ(α0+α1ut-1 )1/2=0
For 2.
E(u
2
2
2
t|ut-1)=E[εt(α0+α1ut-1 )1/2|ut-1]=E[εt(α0+α1ut-1 )1/2]=E(εt)E(α0+α1ut-1 )1/2]=0
For 3.
Ε(u
2
2
2
2
tut-1)=E{[εt(α0+α1ut-1 )1/2][εt-1(α0+α1ut-2 )1/2]}=E[εtεt-1(α0+α1ut-1 )1/2(α0+α1ut-2 )1/2]=
E(ε
2
2
2
2
tεt-1)E[(α0+α1ut-1 )1/2(α0+α1ut-2 )1/2]=0xE[(α0+α1ut-1 )1/2(α0+α1ut-2 )1/2]=0
For 4.
V(u
2
2
2
2
2
2
t)=V[εt(α0+α1ut-1 )1/2]=E[εt(α0+α1ut-1 )1/2]2=E(εt )E(α0+α1ut-1 )=σε [α0+α1E(ut-1 )]
=σ 2
2
2
2
ε [α0+α1V(ut-1)]=σε [α0+α1V(ut)] => V(ut)=α0σε /(1-α1σε )=α0/(1- α1)
For 5
V(u
2
2
t|ut-1)=E{[εt(α0+α1ut-1 )1/2]2|ut-1}- [E(ut|ut-1)]2 = E{[εt(α0+α1ut-1 )1/2]2|ut-1}-0=
E[ε
2
2
2
2
2
2
t(α0+α1ut-1 )1/2]2=E[εt (α0+α1ut-1 )=(Eεt )(α0+α1ut-1 )=α0+α1ut-1
One can easily show that the conditional variance for an ARCH(p) process is
σ 2
2
2
2
t = α0+α1ut-1 +α2ut-2 +… ....+αput-p (8)
The rationale behind the use of an ARCH model is that if not all parameters α1, α2,.. αp are
simultaneously zeros, then the model can be considered as offering a plausible pattern of changing
variance.
Hence the problem can be formalized as follows:
Step 1- Obtain the residuals ut
Step 2-Proceed to the regressions in the models
u 2
2
2
2
t = α0+α1ut-1 +α2ut-2 +… ....+αput-p p=1,2,...
(9)
Step 3-Test the null hypothesis that all α’s are zero at a given significance level. If the test does
not reject the H0, then the ARCH model cannot be a candidate for explaining the changing variance
(strictly speaking the rejection concerns rejection of ARCH in case that the variance is really
changing). Rejection of H0 advocates for the ARCH model as a good candidate for the description of
changing variance (if it is really changing).
In the following table 4 are shown the regression results in model RDt=α0+α1RDt-1

Table 4:
Regression rd to rd-1 regression results in model rdt=α0+α1rdt-1

Regression Summary
for Dependent
Variable: RD







BETA
ST. ERR.
B
ST. ERR.
T_31_
P_LEVEL
Intercept

0,022623
0,012475062
1,813497671
0,079447128
RD_1 0,860284
0,091565
0,833502
0,088714894
9,39528964
0,00

The graphs of the squared regression residuals, the 5-year aggregated squared regression
residuals and the regression residuals are shown in graphs 2, 3, 4 accordingly.

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International Research Journal of Finance and Economics - Issue 37 (2010)
90
Graph 2: Squared regression residuals

Line Plot (00-ARCH EFFECT-DATA.STA 9v*40c)
0,012
0,010
0,008
0,006
U2
0,004
0,002
0,000
-0,002
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994


Graph 3: Aggregated squared regression residuals

Aggregated Line Plot (00-ARCH EFFECT-DATA.STA 9v*40c)
0,012
0,010
0,008
0,006
U2
0,004
0,002
0,000
-0,002
1960
1965
1970
1975
1980
1985
1990

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International Research Journal of Finance and Economics - Issue 37 (2010)
Graph 4: Regression residuals

Line Plot (00-ARCH EFFECT-DATA.STA 9v*40c)
0,12
0,08
0,04
0,00
U
-0,04
-0,08
-0,12
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999



Application of the Arch Procedure and the Findings
The purpose of this part of the study is to detect ARCH effects in the variable RD. A look at figures 5
and especially 6, in which the data are aggregated in a 5-year basis, clearly shows that the variance
changes.

Graph 5: Aggregated deflator

Aggregated Line Plot (00-ARCH EFFECT-DATA.STA 9v*40c)
35
30
25
20
15
D
10
5
0
-5
1960
1965
1970
1975
1980
1985
1990

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International Research Journal of Finance and Economics - Issue 37 (2010)
92
Graph 6: Aggregated RD

Aggregated Line Plot (00-ARCH EFFECT-DATA.STA 9v*40c)
0,26
0,22
0,18
0,14
RD
0,10
0,06
0,02
-0,02
1960
1965
1970
1975
1980
1985
1990


The following graph 7 shows the annually change rate in D (RD)

Graph 7: Change rate in D (RD)

Line Plot (00-ARCH EFFECT-DATA.STA 9v*40c)
0,26
0,22
0,18
0,14
RD
0,10
0,06
0,02
-0,02
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994


According to the stated theoretical analysis we first proceed to the regression in the model
RDt=α0+α1RDt-1+ut (10)
By this regression the regression residuals ut are obtained. The figures 8 and 9, which exhibit
the autocorrelation and the partial autocorrelation functions accordingly, clearly show that the residuals
are not correlated.

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International Research Journal of Finance and Economics - Issue 37 (2010)
Graph 8: ACF of regression residuals

Autocorrelation Function
U
(Standard errors are white-noise estimates)
Lag Corr.
S.E.
Q
p
1 +,081
,1665
,24
,6256
2 -,223
,1638
2,09
,3526
3 -,152
,1612
2,97
,3961
4 +,017
,1585
2,98
,5607
5 +,042
,1557
3,05
,6916
6 +,112
,1529
3,59
,7322
7 +,079
,1500
3,87
,7950
8 -,022
,1471
3,89
,8670
9 -,075
,1441
4,16
,9006
10 -,027
,1411
4,20
,9381
11 -,035
,1380
4,26
,9616
12 +,078
,1348
4,60
,9701
13 -,057
,1316
4,79
,9796
14 -,267
,1283
9,13
,8227
15 -,140
,1248
10,38
,7950
-1,0
-0,5
0,0
0,5
1,0


Graph 9: PACF of regression residuals

Partial Autocorrelation Function
U
(Standard errors assume AR order of k-1)
Lag Corr.
S.E.
1 +,081
,1741
2 -,231
,1741
3 -,118
,1741
4 -,012
,1741
5 -,020
,1741
6 +,101
,1741
7 +,076
,1741
8 +,017
,1741
9 -,016
,1741
10 -,006
,1741
11 -,065
,1741
12 +,059
,1741
13 -,116
,1741
14 -,272
,1741
15 -,142
,1741
-1,0
-0,5
0,0
0,5
1,0


Therefore, the condition of uncorrelated u’s is fulfilled. We next check if the residuals exhibit
an ARCH effect. For this purpose we apply the model (9) for p=1, 2, 3, 4 and we obtain 4 regression
equations.
The test of the null hypothesis
H0: α0=α1=… ...=0 against the alternative hypothesis
H1: not all α’s are simultaneously zero
is performed by the Fisher’s-F statistic at a 5% significance level

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International Research Journal of Finance and Economics - Issue 37 (2010)
94
In the following tables 5, 6, 7 and 8 are shown the results for each regression from ARCH(1) to
ARCH(4)

Table 5:
Arch(1) regression results in arch(1) model

Regression Summary for
Dependent
Variable:
U2


BETA
ST.
ERR. B ST.
ERR.
T_30_
P_LEVEL
Intercept

0,000789
0,00047
1,678437
0,103651
U2_1
0,454508 0,162627 0,453589 0,162298 2,794795 0,008966

Table 6:
Arch(2) regression results in arch(2) model

Regression Summary for
Dependent
Variable:
U2


BETA
ST.
ERR. B ST.
ERR.
T_28_
P_LEVEL
Intercept

0,000873
0,000516
1,692383
0,101674
U2_1 0,476452
0,188972
0,474291
0,188115
2,52129
0,017664
U2_2
-0,05702 0,188972 -0,05682 0,188308 -0,30173 0,765085

Table 7:
Arch(3) regression results in arch(3) model

Regression Summary for
Dependent Variable: U2







BETA
ST. ERR.
B
ST. ERR.
T_26_
P_LEVEL
Intercept

0,00117
0,000538
2,177281
0,038725
U2_1
0,454713 0,184165 0,454602 0,18412 2,469049 0,020436
U2_2
0,106907 0,203668 0,106732 0,203335 0,524908 0,604094
U2_3
-0,3446 0,184239 -0,34393 0,183881 -1,87041 0,072725

Table 8:
Arch(4) regression results in arch(4) model

Regression Summary for
Dependent Variable: U2







BETA
ST. ERR.
B
ST. ERR.
T_24_
P_LEVEL
Intercept

0,001148
0,000617
1,860927
0,075049
U2_1
0,482397 0,203155 0,480125 0,202198 2,374528 0,025912
U2_2
0,090443 0,211378 0,090306 0,211058 0,427875 0,672557
U2_3
-0,3832 0,211565 -0,38151 0,210636 -1,81125 0,082637
U2_4
0,074465 0,203055 0,07432 0,202661 0,366723 0,717039

Finally, for comparison reasons the results for all regressions are summarized in the following
table 9

Table 9:
Summary of regressions results

Regression results
Regression model
α0
α1
α2
α3
α4
AdjR2
F p-level
RDt=α0+α1RDt-1 0.022623
0.833502


0.7317 88.27 0.00000
ARCH(1) 0.000789
0.453589

0.1801
7.81
0.00087
ARCH(2) 0.000873
0.474300
-0.056800

0.1488
3.62
0.03984
ARCH(3) 0.001170
0.454602
0.106732
-0.343933

0.2174
3.68
0.02461
ARCH(4) 0.001148
0.480125
0.090306
-0.381515
0.074320
0.1886
2.62
0.05953

The p-value corresponding to the F-statistic is less than 0.05 (5%), which leads to the rejection
of the null hypothesis for the parameters of the regression model (10) and the models ARCH(1),

---