Прикладная эконометрика, 2015, № 39 (3)

2015, №39(3)

ISSN 1993-7601

Журнал «Прикладная эконометрика» включен в список периодических изданий ВАК, 
рекомендованных для публикации результатов диссертационных исследований.  
Он также индексирован в международных базах научных журналов по экономике RePEc и EconLit.

Члены редколлегии

Бродский Б. Е. — д-р физ.-мат. наук, ЦЭМИ РАН,
НИУ ВШЭ.

Ван Суст А.  — Ph.D., Тилбургский университет,
Нидерланды.

Вербик М. — Ph.D., школа менеджмента, Роттердам,
Нидерланды.
Денисова И. А. — Ph. D., Центр экономических
и финансовых исследований и разработок (ЦЭФИР),
ЦЭМИ РАН.
Елисеева И. И. — чл.-кор. РАН, д-р экон. наук,
Социологический институт РАН, Санкт-Петербургский университет экономики и финансов.
Канторович Г. Г. — канд. физ.-мат. наук, НИУ ВШЭ.
Карлеваро Ф. — д-р наук, Женевский университет, Швейцария.
Макаров В. Л. — акад. РАН, д-р физ.-мат. наук,
ЦЭМИ РАН, РЭШ.

Максимов А. Г. — канд. физ.-мат. наук, Нижегородский филиал НИУ ВШЭ.
Микушева А. Е. — Ph. D., канд. физ.-мат. наук,
Массачусетский технологический институт, Кэмбридж, США.
Мхитарян В. С. — д-р экон. наук, НИУ ВШЭ.
Рубин Ю. Б. — д-р экон. наук, профессор,
чл.-кор. РАО, ректор МФПУ «Синергия».
Рудзкис Р. — д-р наук, Институт математики и информатики, Каунасский университет, Литва.
Слуцкин Л. Н. — Ph. D., Институт экономики
РАН.
Суслов В. И. — чл.-кор. РАН, д-р экон. наук, Институт экономики и организации промышленного
производства СО РАН.
Харин Ю. С. — чл.-кор. НАН Беларуси, д-р физ.-мат.
наук, Белорусский государственный университет,
НИИ прикладных проблем математики и информатики БГУ, Беларусь.

Главный редактор

Айвазян Сергей Артемьевич — д-р физ.-мат. наук, акад. (иностранный член) НАН Армении, Центральный экономико-математический институт РАН (ЦЭМИ РАН), Московский финансово-промышленный
университет «Синергия», Высшая школа экономики (НИУ ВШЭ), Московская школа экономики МГУ.

Заместитель главного редактора

Пересецкий Анатолий Абрамович — д-р экон. наук, НИУ ВШЭ, ЦЭМИ РАН, Российская экономиче- 
ская школа (РЭШ).

Ответственный секретарь

Сластников Александр Дмитриевич — канд. физ.-мат. наук, ЦЭМИ РАН.

 2015, No. 39(3)

ISSN 1993-7601

Applied

CONOMETRICS
E

Applied Econometrics is indexed in RePEc (Research Papers in Economics)  
and EconLit (The American Economic Association’s electronic bibliography).

Associate Editors

Boris Brodsky — Central Economics and Mathematics Institute (CEMI RAS), Moscow.
Fabrizio Carlevaro — University of Geneva; Geneva 
(Switzerland).
Irina Denisova — Centre for Economic and Financial 
Research (CEFIR); Central Economics and Mathematics Institute (CEMI RAS), Moscow.
Irina Eliseeva — Sociological Institute (SI RAS), SaintPetersburg State University of Economics and Finance,
Saint-Petersburg.
Grigoriy Kantorovich — National Research University Higher School of Economics (NRU HSE), Moscow.
Yury Kharin — Belarusian State University, Research
Institute for Applied Problems of Mathematics and Informatics, Minsk (Belarus).
Valery Makarov — Central Economics and Mathematics Institute (CEMI RAS); New Economic School
(NES), Moscow.

Andrey Maksimov — National Research University
Higher School of Economics (NRU HSE) branch in
Nizhny Novgorod, Nizhny Novgorod.
Anna Mikusheva — Massachusetts Institute of Technology, Cambridge (USA).
Vladimir Mkhitarian — National Research University Higher School of Economics (NRU HSE), Moscow.
Yury Rubin — Moscow Financial-Industrial University (MFIU), Moscow.
Rimantas Rudzkis — Institute of Mathematics and 
Informatics, Vilnius (Lithuania).
Lev Slutskin — Institute of Economics (IE RAS),
Moscow.
Arthur van Soest — Tilburg University, Tilburg (Nether lands).
Viktor Suslov — Institute of Economics and Industrial 
Engineering of the Siberian Branch of RAS, Novosibirsk.
Marno Verbeek — Rotterdam School of Management,
Rotterdam (Netherlands).

Editor-in-Chief

Sergey Aivazian — Central Economics and Mathematics Institute (CEMI RAS), Moscow; Moscow FinancialIndustrial University (MFIU); National Research University Higher School of Economics (NRU HSE); Moscow
School of Economics (MSE), Moscow.

Vice-Editor

Anatoly Peresetsky — National Research University Higher School of Economics (NRU HSE); Central Economics and Mathematics Institute (CEMI RAS); New Economic School (NES), Moscow.

Executive Editor

Alexander Slastnikov — Central Economics and Mathematics Institute (CEMI RAS), Moscow.



3

Applied econometrics / ПРИКЛАДНАЯ ЭКОНОМЕТРИКА

Contents 
Содержание номера

2015, 39(3) 

МакроэконоМика

A. Ndoricimpa
Inflation, output growth and their uncertainties in South Africa: 
Empirical evidence from an asymmetric multivariate GARCH-M model. . . . . . . . . . . . 5

Банки

М. Е. Мамонов
Микроэкономическая модификация общеотраслевого индикатора Буна: 
новые оценки рыночной власти российских банков  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 18

эконоМика спорта

Д. Ю. Орлов
Влияние переговорной силы клубов  
на формирование трансферной стоимости игрока  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 45

страхование

И. А. Тетин
Циклы страховой деятельности в России и макроэкономические показатели .  .  .  .  .  .  . 65

теория и Методология

С. А. Айвазян, А. Н. Березняцкий, Б. Е. Бродский, Б. С. Дарховский
Статистический анализ моделей с переменной структурой .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 84

классические раБоты по эконоМетрике

А. Е. Микушева
Коинтеграция  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Р. Ф. Энгл, К. У. Дж. Грэнджер
Коинтеграция и коррекция ошибок:  
представление, оценивание и тестирование. . . . . . . . . . . . . . . . . . . . . . . . . 107

научная жизнь
XIV Европейская конференция по анализу  
эффективности и производительности (EWEPA 2015) . . . . . . . . . . . . . . . . . . . 136
Пятая международная конференция по финансовому инжинирингу 
и банковскому делу (FEBS 2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Условия публикации статьи  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Содержание номера
Contents

ПРИКЛАДНАЯ ЭКОНОМЕТРИКА / Applied econometrics
2015, 39(3)

MacroeconoMics

Arcade Ndoricimpa
Inflation, output growth and their uncertainties in South Africa: 
Empirical evidence from an asymmetric multivariate GARCH-M model. . . . . . . . . . . . 5

Banking

Mikhail Mamonov
Microeconomic modification of an industry-wide Boone indicator: 
Market power of Russian banks revisited  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

econoMics of sports

Denis Orlov
The effect of clubs’ bargaining power on football player’s transfer value . . . . . . . . . . . . 45

insurance

Ilya Tetin
Underwriting cycle in Russia and macroeconomic indicators .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 65

theory and Methodology

Sergei Aivazian, Alexander Bereznyatskiy, Boris Brodsky, Boris Darkhovsky
Statistical analysis of variable-structure models .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 84

seMinal papers in econoMetrics

Anna Mikusheva
Co-integration  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Robert F. Engle and Claive W. J. Granger
Co-integration and error correction:  
representation, estimation, and testing .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 107

scientific life
14th European Workshop on Efficiency and Productivity Analysis (EWEPA 2015)  . . . . . 136
5th International Conference of the Financial Engineering  
and Banking Society (FEBS 2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Guidelines for authors  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A. Ndoricimpa

5

Applied econometrics / ПРИКЛАДНАЯ ЭКОНОМЕТРИКА

Macroeconomics 
Макроэкономика

2015, 39(3) 

Прикладная эконометрика, 2015, 39(3), с. 5–17.
Applied Econometrics, 2015, 39(3), pp. 5–17.
A. Ndoricimpa1

Inflation, output growth and their uncertainties  
in South Africa: 
Empirical evidence from an asymmetric 
multivariate GARCH-M model

The study examines the relationships between inflation uncertainty and output growth uncertainty, and analyzes their effects on the level of inflation and output growth in South Africa. 
An asymmetric multivariate GARCH-M model suggested by Grier et al. (2004) is applied. 
The findings suggest that while uncertainty about growth is detrimental to output growth, 
inflation uncertainty is not. The findings further reveal that Cukierman and Meltzer (1986) 
hypothesis of a positive impact of inflation uncertainty on the level of inflation is supported, 
and there exists a negative impact of output growth uncertainty on inflation. No association 
could be found between inflation uncertainty and output growth uncertainty. The findings 
imply that output growth and its uncertainty should not be treated separately as it is usually 
suggested by business cycle models. In addition, both output growth uncertainty and inflation 
uncertainty should be considered as part of the determinants of output growth in South Africa.

keywords: inflation; growth; uncertainty; asymmetric multivariate GARCH-M model; South Africa.

Jel classification: E31; E52; C32.

1. introduction
I

n his Nobel Prize lecture, Friedman (1977) pointed out that greater inflation is associated with
greater inflation uncertainty and that inflation uncertainty is harmful to growth since it creates
economic inefficiency through distortions in the effectiveness of the price mechanism thus

hindering the efficient allocation of resources. Although Ball (1992) supports his idea on the link
between inflation and its uncertainty, Cukierman and Meltzer (1986) and Holland (1995) explored
the possibility of a reverse causality. Cukierman and Meltzer (1986) argue that an increase in inflation uncertainty leads to an increase in the level of inflation as policymakers create surprise inflation to stimulate output. Holland (1995) in his stabilization hypothesis points out that an increase
in inflation uncertainty leads to a decrease in inflation as policymakers reduce the growth rate of
money (hence lowering inflation) to attenuate the effects of inflation uncertainty on the economy.
Considering the impact of inflation uncertainty on growth, contrary to Friedman (1977),

Dotsey and Sarte (2000) suggest a positive impact of inflation uncertainty on output growth.
They argue that an increase in the variability of monetary growth, and therefore inflation, makes

1
Arcade Ndoricimpa — Department of Economics, University of Dar es Salaam, Tanzania; 
arcade_ndoricimpa@yahoo.com.

Макроэкономика 
Macroeconomics

ПРИКЛАДНАЯ ЭКОНОМЕТРИКА / Applied econometrics
2015, 39(3)

the return to money balances more uncertain and leads to a fall in the demand for real money 
balances and consumption. Hence, agents increase precautionary savings leading to an increase
in investments which in turn boost output growth.
The impact of real uncertainty (output growth uncertainty) on output growth has also attracted considerable debate among scholars. While Friedman (1968) argues that there is an independence relationship between the two, Black (1987) and Blackburn (1999) suggest a positive impact of output growth uncertainty on output growth by arguing that high volatility will lead to
an increase in savings through precautionary motives which in turn result in an increase in investments. Pindyck (1991) and Ramey and Ramey (1991) on the other hand suggest a negative
impact of output growth uncertainty on output growth. According to them, large fluctuations in
economic activity are more likely to increase the uncertainty regarding the long-run profitability of investments, making the returns of investments riskier and hence reducing the level of investments and therefore output growth.
Scholars have also examined the relationship between inflation uncertainty and output growth

uncertainty. Taylor (1981) and Fuhrer (1997) suggest a trade-off between inflation uncertainty
and output growth uncertainty. According to them, the objective of stabilizing inflation can be
achieved only at the expense of accepting a high volatile output growth and vice-versa. In contrast, Logue and Sweeney (1981) suggest a positive link from inflation volatility to output growth
volatility in contrast to Devereux (1989) who supports a positive link from output growth volatility to inflation volatility.
A number of empirical studies have sought to examine the relationship between inflation

uncertainty and output growth uncertainty and to analyze their effects on the levels of inflation
and output growth, using either one-step approach in GARCH-in-mean models (see, for instance, (Grier et al., 2004; Grier, Grier, 2006; Olayinka, Hassan, 2010; Ndou, Mokoena, 2011;
Omay, 2011; Hachicha, Hooi Hooi, 2013)) or two-step approach where uncertainty measures
for inflation and output growth are initially generated using various types of GARCH models
and then causality tests are conducted to examine the links between the variables (see for instance, (Fountas, Karanasos, 2007; Thornton, 2007; Korap, 2009; Mehrara, Mojab, 2010; Narayan, Narayan, 2013)).
However, to generate uncertainty measures of inflation and output growth, a number of studies use univariate models (see, for instance, (Fountas, Karanasos, 2007; Ndou, Mokoena, 2011;
Hachicha, Hooi Hooi, 2013; Farhan et al., 2012; Mohd et al., 2012)), restrictive models of the
covariance process such as CCC and DCC-GARCH2 models or diagonal BEKK model (see for
instance, (Fountas et al., 2002; Jiranyakul, Opiela, 2011; Korap, 2009; Mehrara, Mojab, 2010;
Türkyılmaz, Ozer, 2010; Conrad, Karanasos, 2008)). As Grier et al. (2004) note, univariate
models do not permit the joint generation of the uncertainty measures for inflation and output
growth nor do they permit one to simultaneously examine the relationships between inflation,
growth and their uncertainties while on the other hand, the restrictive models can lead to misspecification problem.
In addition, when generating uncertainty measures through GARCH models, Grier et al.

(2004) cautioned that imposing diagonality and symmetry restrictions on the variance-covariance
matrix of output growth and inflation might lead to misspecification hence wrong uncertainty

2
CCC and DCC-GARCH models are respectively Constant and Dynamic Conditional Correlation GARCH model

of Bollerslev (1990) and Engle (2002). They both assume that the matrices of ARCH and GARCH terms are diagonal.

A. Ndoricimpa

7

Applied econometrics / ПРИКЛАДНАЯ ЭКОНОМЕТРИКА

Macroeconomics 
Макроэкономика

2015, 39(3) 

measures and faulty conclusions regarding the links between inflation, output growth and their
uncertainties. To address this problem, Grier et al. (2004) proposed a model which allows testing for diagonality and symmetry in the variance-covariance matrix instead of imposing them.
This study therefore follows Grier et al. (2004) and applies an asymmetric multivariate

GARCH-M model to examine the relationships between inflation uncertainty and output growth
uncertainty and to analyze their effects on the level of inflation and output growth in South Africa.
The findings of this study suggest that in South Africa, while uncertainty about growth is

detrimental to output growth, inflation uncertainty is not. Inflation uncertainty promotes output
growth as suggested by Dotsey and Sarte (2000). Friedman’s (1977) hypothesis of a negative
impact of inflation uncertainty on output growth was not supported in this study. The findings
further reveal that Cukierman and Meltzer (1986) hypothesis of a positive impact of inflation
uncertainty on the level of inflation is supported, and there exists a negative impact of output
growth uncertainty on inflation. The results however indicate that there is no association between
inflation uncertainty and output growth uncertainty.
The rest of the paper is organized as follows. Section 2 highlights the methodology used. Section 3 presents and interprets the estimation results and section 4 concludes the study.

2. Methodology

To examine the relationship between inflation uncertainty and output growth uncertainty

and further analyze their effects on the levels of inflation and output growth in South Africa,
this study follows (Grier et al., 2004) and uses an asymmetric BEKK3 GARCH-M model in
which the conditional means of inflation (
)
tp
and output growth (
)
ty
are in form of VARMA

(Vector AutoRegressive Moving Average) GARCH-M model, where the conditional standard
deviations of output growth and inflation are included as explanatory variables in each conditional mean equation. The specification of the conditional means of inflation (
)
tp
and output

growth (
)
ty
is as follows:

1
1

p
q

t
i
t i
t
j
t
j
t

i
j

Y
Y
h

=
=
= m+

+
+
 e
+e


(1)

with
|
(0,
)
t
t
t
N
H
e
 
, where
t
  represents the information set available at time t. In addition,

2
2
,
,
,
,
,
,
,
E(
)
;
E(
)
;
E(
)
y t
y t
t
t
y t
t
y
t
h
h
h
p
p
p
p
e
=
e
=
e e
=
;

( )
( )
,
,
,
,
( )
( )
,
,
,
,

(j)
(j)

(j)
(j)

;
;
;
;
;
;

;
,

i
i

y t
y t
y
y t
y
t
t
yy
y
t
t
t
t
i
i
i

y t
t
y
t
t
t

yy
y
yy
y
j
y
y

h
h
h
y
H
Y
h
h
h
h

p
p

p
p
p
pp
p
p
p

p
p

p
pp
p
pp







e




m






=
=
e =
=
m=
 =














p
e


m

















q
q
 =
 =





q
q





3
BEKK model is a multivariate GARCH model developed by Engle and Kroner (1995) and was named after Baba,

Engle, Kraft and Kroner.

Макроэкономика 
Macroeconomics

ПРИКЛАДНАЯ ЭКОНОМЕТРИКА / Applied econometrics
2015, 39(3)

where
t
H is the conditional variance-covariance matrix,
,y t
h
is the conditional variance of output

growth,
,t
hp is the conditional variance of inflation,
,
y
t
h p  and 
,y t
hp
 are the conditional covariances 
between inflation and output growth,
te is the vector of error terms, m is the matrix of constant

terms,
i is the matrix of Autoregressive coefficients,  is the matrix of in-mean coefficients, and

j
 is the matrix of moving average coefficients. It is important to note that in GARCH models,
uncertainty (volatility) is captured by the conditional variance which is simply the variance of the
one step ahead forecasting error. The conditional variance-covariance matrix of an asymmetric
BEKK model is written as:

1
1
1
1
1
t
t
t
t
t
t
H
C C
A
A
B H
B
D
D






=
+
e
e
+
+
w
w
,
(2)

where

,

,

0

;
;
;
;

yy
yy
y
yy
y
yy
y
y t

y
y
y
y
t

c
C
A
B
D
c
c

p
p
p

p
pp
p
pp
p
pp
p
pp
p











a
a
b
b
d
d
w
=
=
=
=
w=










a
a
b
b
d
d
w











.

In equation (2), C is a lower triangular matrix of constant terms, A is a matrix of ARCH coefficients which captures the ARCH effects, and B is a matrix of GARCH coefficients capturing
the GARCH effects. The diagonal elements in matrix A show the impact of own past shocks on
the current conditional variance, the diagonal elements in Matrix B represent the impact of own
past volatility on the current conditional variance, while the off-diagonal elements in matrices A 
and B represent the volatility spillovers’ effects (Xu, Sun, 2010). Asymmetry in the conditional
variance-covariance matrix is captured by the matrix D which is the matrix of asymmetric coefficients. To introduce asymmetry in the conditional variance-covariance process, Grier et al.
(2004) use the concepts of good and bad news. If inflation is higher than expected, it is bad
news, captured by a positive inflation residual defined as
,
,
max(
,0)
t
t
p
p
w
=
e
. Similarly, if output

growth is lower than expected, it is also bad news, captured by negative output growth innovations defined as
,
,
min(
,0)
y t
y t
w
=
e
. We note that the BEKK model becomes symmetric if asymmetric coefficients are statistically jointly equal to 0, i. e.
ij
d , for all ,
, .
i j
y
=
p  It should also 
be noted that BEKK model is preferred because it ensures the positive definiteness of the conditional variance-covariance matrix unlike the other variants of multivariate GARCH models.
Equation (2) can also be written as follows:

2
2
2
2
2
2
2
2
,
,
1
,
1
,
1
,
1
,
1
,
1
,
1

2
2
2
2
,
1
,
1
,
1
,
1

2
2

             
2
,                                              

y t
yy
y
yy
y t
yy
y
y t
t
y
t
yy
y t
yy
y
y
t
y
t

yy
y t
yy
y
y t
t
y
t

h
c
c
h
h
h
p
p
p p
p p
p p
p 
p
p p
p 
=
+
+a e
+ a a e
e
+a e
+b
+ b b
+b
+

+d w
+ d d w
w
+d w

2
2

,
,
1
,
1
,
1
,
1
,
1

2
,
1
,
1
,
1
,
1
,
1

                     

(
)
(
)

            
(
)
(
)

y
t
y
yy
y
yy
y t
yy
y
y
y t
t
y
t
yy
y
y t

yy
y
y
y
t
y
t
y
yy
y t
yy
y
y
y t
t

h
c
c
c
h

h
h

p
p
pp
p
pp
p
p
p p
pp
p p

pp
p
p
p p
pp
p p
pp
p
p
p 
=
+
+a a e
+ a a
+a a
e
e
+a a
e
+b b
+

+ b b
+b
b
+b
b
+d d w
+ d d
+d d
w
w
+

2
,
1

2
2
2
2
2
2
,
,
1
,
1
,
1
,
1

            
,                                                                                                              

 
2

y
t

t
y
y
y t
y
y t
t
t
y
h
c
c

p
pp
p 
p
pp
p
p
p
pp
p pp
p p

+d d w

=
+
+a e
+ a a
e
e
+a
e
+b2
2

,
1
,
1
,
1

2
2
2
2
,
1
,
1
,
1
,
1

2

           
2
.                                                                   

y t
y
y
t
t

y
y t
y
y t
t
t

h
h
h
p
pp
p pp
p 
p
p
pp
p pp
p 
+ b
b
+b
+

+d w
+ d d w
w
+d w

(3)

From the conditional mean equation (1), we can check how inflation uncertainty and output

growth uncertainty affect the level of inflation and output growth. Assessing the impact of output
growth uncertainty and inflation uncertainty on output growth is done by respectively testing

A. Ndoricimpa

9

Applied econometrics / ПРИКЛАДНАЯ ЭКОНОМЕТРИКА

Macroeconomics 
Макроэкономика

2015, 39(3) 

the null hypotheses that
0
yy

=  and 
0
yp

= . A positive
yy

would mean a positive impact of

output growth uncertainty on output growth, which is Black hypothesis while a negative
yy
  
would imply a negative impact of output growth uncertainty on output growth, supporting the
views of Pindyck (1991) and Ramey and Ramey (1991). A positive
yp

would mean a positive impact of inflation uncertainty on output growth which is Dotsey–Sarte hypothesis while a
negative
yp

would mean a negative impact of inflation uncertainty on output growth, which

is Friedman hypothesis.
Similarly, testing the impact of output growth uncertainty and inflation uncertainty on the

level of inflation is done by respectively testing whether
0
y
p

=  and 
0
pp

= . A positive
y
p

 
would mean a positive impact of output growth uncertainty on inflation which is referred to as
the Devereux hypothesis while a negative
y
p

would imply a negative impact of output growth

uncertainty on inflation. On the other hand, a positive
pp

would mean a positive impact of

inflation uncertainty on inflation, which would support Cukierman–Meltzer hypothesis while
a negative
pp

would mean a negative impact of inflation uncertainty on inflation, which is the

stabilization hypothesis of Holland (1995).
From the conditional variance-covariance matrix (equation (3)), it can be seen that the impact of the conditional variance of one variable on the conditional variance of another variable
is captured by the off-diagonal elements of the matrix B,
y
p
b
 and 
yp
b
. Examining the impact

of inflation uncertainty on output growth uncertainty and vice versa is done respectively by testing
0
y
p
b
=  and 
0
yp
b
= . A negative
y
p
b
 or 
yp
b
would support a trade-off hypothesis between

inflation uncertainty and output growth uncertainty suggested by Taylor (1981). A positive
y
p
b
 
would support Logue and Sweeney (1981) hypothesis while a positive
yp
b
would support Devereux (1989) hypothesis.

3. empirical results and discussion

Monthly data on price and output levels for South Africa are used for the period February 1961 to March 2012. Data were retrieved from International Financial Statistics (IFS) of
the International Monetary Fund (IMF). The price level is captured by Consumer Price Index
(CPI) and output level is captured by Manufacturing Production Index. This is because high
frequency data which are more appropriate with GARCH models are not available for GDP
for most countries including South Africa; proxies for output level such as industrial production index, manufacturing production index, crude oil production index, etc. are therefore
commonly used in this kind of studies (see, for instance, (Korap, 2009; Türkyılmaz, Özer,
2010; Olayinka, Hassan, 2010; Jiranyakul, Opiela, 2011; Bipradas, 2012; Hachicha, Hooi
Hooi, 2013)).
Inflation rate is computed as the monthly difference of the logarithm of CPI,

1
[log(
/
)]
100
t
t
t
CPI
CPI p =

,

and output growth is computed as the monthly difference of the logarithm of the production index
( )
tY ,
1
[log(
/
)]
100
t
t
t
y
Y
Y =

.
Summary statistics in Panel A of Table 1 show that both inflation, p, and output growth, y,

are positively skewed and display platikurtic behavior. In addition, Jarque–Bera (1987) test rejects the null hypothesis of normality in both inflation and output growth series.

Макроэкономика 
Macroeconomics

ПРИКЛАДНАЯ ЭКОНОМЕТРИКА / Applied econometrics
2015, 39(3)

Unit root tests, serial correlation and ARCH tests are preliminarily conducted. Unit root

tests are conducted in order to assess the order of integration of the series, and ARCH test, to
check for evidence of conditional heteroscedasticity in the data, that is, whether the variances
of the series are time-varying. As Grier and Perry (1998) points out, one should be able to reject the null hypothesis of constant variance before estimating a GARCH model and generate
uncertainty measures. An endogenous two-break unit root test of Lee and Strazicich (2003) and
a non-parametric unit root test of Breitung (2002) are used to test for unit root in the series.
Serial correlation is tested using the Ljung–Box (1978) test on the series and on the squared
series, while testing for the presence of ARCH effects in the series is done using LM-ARCH
test of Engle (1982).

Inflation

1961
1964
1967
1970
1973
1976
1979
1982
1985
1988
1991
1994
1997
2000
2003
2006
2009
2012

-1

0

1

2

3

4

5

Output Growth

1961
1964
1967
1970
1973
1976
1979
1982
1985
1988
1991
1994
1997
2000
2003
2006
2009
2012

-10.0

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

10.0

12.5

Fig. 1. Inflation and output growth in South Africa (1961–2012)

Unit root test results reported in Panel B of Table 1 indicate that both tests strongly reject

the null hypothesis of a unit root in inflation and output growth series for South Africa. Inflation
and output growth series are hence integrated of order 0, I (0). This implies that there is no need
to difference them when estimating the Mean equations. In addition, Ljung–Box (1978) test rejects the null hypothesis of no serial correlation in both the series and squared series (Panel C,
Table 1). Serial correlation in the squared data is an evidence of conditional heteroscedasticity and this is confirmed by LM-ARCH test of Engle (1982). ARCH test results in Panel C of
Table 1 indeed suggest that inflation and output growth exhibit significant volatility clustering
for South Africa, implying that the variances of inflation and output growth are not constant but
time-varying. Figure 1 seems to confirm this.

A. Ndoricimpa

11

Applied econometrics / ПРИКЛАДНАЯ ЭКОНОМЕТРИКА

Macroeconomics 
Макроэкономика

2015, 39(3) 

Since the presence of ARCH effects is confirmed, we proceed to estimate our asymmetric

BEKK GARCH-M4 model for South Africa. The estimation results are in Table 2 along with
the diagnostic test results as well as the coefficients restriction tests on the estimated model.

Table 1. Preliminary tests results

Panel A. Summary statistics

Mean
Variance
Skewness
Excess kurtosis
Jarque–Bera test

p
0.6786
0.4021
1.176 [0.000]
2.863 [0.000]
351.308 [0.000]

y
0.2644
8.1814
0.215 [0.000]
0.930 [0.000]
26.895 [0.000]

Panel B. Unit root tests

Lee–Strazicich unit root test
Breitung test

 t Stat
Breaks
B(n)/n
C.V (5%)

p
–13.20*** {2}
1985m12
1998m7
0.00628 [0.009]
0.01046

y
–15.83*** {7}
1980m6
1976m11
0.00052 [0.000]
0.01046

Panel C. Univariate serial correlation and ARCH test

Q(10)
Q(20)
Q2(10)
Q2(20)
ARCH(20)

p
465.9 [0.000]
894.0 [0.000]
125.9 [0.000]
360.0 [0.000]
8.30 [0.000]

y
369.8 [0.000]
568.6 [0.000]
74.81 [0.000]
99.71 [0.000]
4.29 [0.000]

Notes. Lee–Strazicich (LS) test was performed using WinRATS Pro 8.1 while Breitung test was performed using
EasyReg software. Serial correlation and ARCH tests were performed using OxMetrics 6.30. Between {∙} are the optimal
lags used in LS test, selected using the usual criteria and between brackets [∙] are the p-values. For LS test, 1% C.V
is –5.823; 5% C.V is –5.286 and 10% C. V is –4.989 for the model allowing for a shift in intercept and change in trend
slope. P-values reported in brackets [∙] for Breitung test are based on 1000 simulations.

Diagnostic tests, Ljung–Box test, and McLeod–Li test are first used to check for the adequacy of the GARCH model estimated, in other words, to check whether the conditional mean
and the conditional variance-covariance equations are well specified (see Table 2, Panel C).
At 5% level, Ljung–Box test indicates that there is no serial correlation of 5th and 10th order in 
the standardized residuals of inflation and output growth mean equations. Similarly, McLeod–Li
test indicates that the squares of the standardized residuals of inflation and output growth equations are also serially independent at 5% level, implying that there are no remaining ARCH/
GARCH effects. The conditional mean and conditional variance-covariance equations are hence
well specified.
In addition, some coefficient restriction tests (see Table 2, Panel B) are conducted in order

to check whether some of the coefficients in the Mean equation and in the conditional variancecovariance matrix are redundant. The results indicate that the hypotheses of Diagonal VARMA,
no GARCH, no GARCH-M, no Asymmetry and Diagonal GARCH are all rejected at 1% significance level. This suggests that none of the terms included in the conditional mean and variancecovariance equations are redundant. Coefficient restriction tests confirm that the form of the mean

4
In estimating the mean equation, we consider p = q =2 and the diagnostic tests confirm that the mean equation is

well specified with that lag order.

Макроэкономика 
Macroeconomics

ПРИКЛАДНАЯ ЭКОНОМЕТРИКА / Applied econometrics
2015, 39(3)

equation adopted (vector autoregressive moving average, VARMA, plus the in-mean coefficients
included) properly captures the dynamics of inflation and output growth in South Africa and that
the form of the conditional variance-covariance matrix adopted (asymmetry and non-diagonality)
also adequately captures the dynamics of the conditional variance of inflation and conditional
variance of output growth. The conditional standard deviations of inflation and output growth
capturing inflation uncertainty and output growth uncertainty for South Africa are in Figure 2.
Figure 2 indicates that the greatest inflation uncertainty (volatility) is seen in the 2nd half of 
the 1970s, in the 1980s, 1994, 2000, 2004, and 2010, while the greatest output growth uncertainty (volatility) is apparent in 1977 and 2009. Some high output growth uncertainty can be
also seen in the second half of the 1970s, in 1980s, 1999, and 2004. On average, inflation uncertainty seems to have been lower than output growth uncertainty.

Table 2. Estimation results of an asymmetric BEKK GARCH-M model for South Africa

Panel A. Conditional mean equations

1
1

1
2

, where 
(0,
),

2.2727
1.0031
8.5938
0.6469
10.223
(0.232)
(0.008)
(0.075)
(0.0062)
(0.0635)
;
;
0.1037
0.0239
0.7070
0.002
(0.002)
(0.0004)
(0.003)

p
q

t
i
t
i
t
j
t
j
t
t
t
i
j
Y
Y
h
N
H
=
=
= m+

+
+
 e
+e
e









m=
 =
 =

















1
2

;
8
0.2756
(0.0002)
(0.003)

0.4359
0.8442
0.4436
9.0136
0.0154
9.1137
(0.081)
(0.448)
(0.026)
(0.063)
(0.030)
(0.070)
;
;
0.0430
0.0115
0.0453
0.6281
0.
(0.001)
(0.005)
(0.002)
(0.011)




















 =
 =
 =












0301
0.2759
(0.0019)
(0.0086)












Panel B. Conditional variance — covariance

1
1
1
1
1
t
t
t
t
t
t
H
C C
A
A
B H
B
D
D






=
+
e
e
+
+
w
w
,

1.5106
0.2521
0.0193
0.5387
0.0239
0.3674
0
(0.094)
(0.038)
(0.006)
(0.044)
(0.017)
(0.
;
;
;
0.0116
0.00003
0.2072
0.2276
0.0039
0.9487
(0.015)
(0.1317)
(0.279)
(0.026)
(0.226)
(0.004)

C
A
B
D













=
=
=
=



















2
0

2
0

0

0.0149
039)
(0.010)
1.5841
0.2650
(0.284)
(0.039)

Diagonal VARMA :{H :
0,
1,2; 
(8)
207659.3 [0.000]}

No GARCH :
{H :
0,
,
, ;
(12)
219448.3 [0.000]}

No GARCH-M :
{H :

i
i
i
i
y
y
y
y

ij
ij
ij

i

i j
y

p
p
p
p













=
=q
=q
=
=

=

a = b = d =

= p

=


2

2
0

2
0

0,
,
, ;
(4)
2236.9 [0.000]}

No Asymmetry :
{H :
0,
,
, ;
(4)
856.3 [0.000]}

Diagonal GARCH : {H :
0;
(6)
81.1 [0.000]}

ij

ij

y
y
y
y
y
y

i j
y

i j
y

p
p
p
p
p
p

=

= p

=

d =

= p

=

a
= a
= b
= b
= d
= d
=

=

Panel C. Diagnostic tests

Ljung–Box Q (5)
McLeod–Li (5)
Ljung–Box Q (10)
McLeod–Li (10)

zy,t
6.4460 [0.265]
1.117 [0.952]
13.0139 [0.222]
6.1748 [0.800]

zp,t
7.0156 [0.219]
2.2824 [0.808]
17.1519 [0.071]
3.7045 [0.959]

Notes. Results from our estimations using WinRATS Pro 8.1. Between parentheses (∙) are the standard errors and between
brackets [∙] are the p-values. zj,t is the standardized residual defined as
,
,
,
j t
j t
j t
z
h
= e
, where
, .
j
y
=
p