Description
The purpose of this paper was to construct a canonical correlation analysis (CCA) model for
the Zimbabwe stock exchange (ZSE). This paper analyses the impact of macroeconomic variables on
stock returns for the Zimbabwe Stock Exchange using the canonical correlation analysis (CCA).
Journal of Financial Economic Policy
Canonical correlation analysis: Macroeconomic variables versus stock returns
Peter Mazuruse
Article information:
To cite this document:
Peter Mazuruse , (2014),"Canonical correlation analysis", J ournal of Financial Economic Policy, Vol. 6 Iss 2
pp. 179 - 196
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Canonical correlation analysis
Macroeconomic variables versus stock returns
Peter Mazuruse
Harare Institute of Technology, Harare, Zimbabwe
Abstract
Purpose – The purpose of this paper was to construct a canonical correlation analysis (CCA) model for
the Zimbabwe stock exchange (ZSE). This paper analyses the impact of macroeconomic variables on
stock returns for the Zimbabwe Stock Exchange using the canonical correlation analysis (CCA).
Design/methodology/approach – Data for the independent (macroeconomic) variables and
dependent variables (stock returns) were extracted fromsecondary sources for the period fromJanuary
1990 to December 2008. For each variable, 132 sets of data were collected. Eight top trading companies
at the ZSE were selected, and their monthly stock returns were calculated using monthly stock prices.
The independent variables include: consumer price index, money supply, treasury bills, exchange rate,
unemployment, mining and industrial index. The CCA was used to construct the CCA model for the
ZSE.
Findings – Maximization of stock returns at the ZSEis mostly infuenced by the changes in consumer
price index, money supply, exchange rate and treasury bills. The four macroeconomic variables greatly
affect the movement of stock prices which, in turn, affect stock returns. The stock returns for Hwange,
Barclays, Falcon, Ariston, Border, Caps and Bindura were signifcant in forming the CCA model.
Research limitations/implications – During the research period, some companies delisted due to
economic hardships, and this reduced the sample size for stock returns for respective companies.
Practical implications – The results from this research can be used by policymakers, stock market
regulators and the government to make informed decisions when crafting economic policies for the
country. The CCA model enables the stakeholders to identify the macroeconomic variables that play a
pivotal role in maximizing the strength of the relationship with stock returns.
Social implications – Macroeconomic variables, such as consumer price index, infation, etc.,
directly affect the livelihoods of the general populace. They also impact on the performance of
companies. The society can monitor economic trends and make the right decisions based on the current
trends of economic performance.
Originality/value – This research opens a new dimension to the study of macroeconomic variables
and stock returns. Most studies carried out so far in Zimbabwe zeroed in on multiple regression as the
central methodology. No study has been done using the CCA as the main methodology.
Keywords Stock returns, Macroeconomic variables, Canonical correlation, Canonical variates,
Canonical loadings, Redundancy index
Paper type Research paper
1. Introduction
The study of macroeconomic variables and their impact on stock returns has been an
area of intense interest among academics, investors and stock market regulators since
the 1970s. The relationship between macroeconomic variables and stock returns affects
the valuation of securities and risk management. Because the inter-dependence of the
JEL classifcation – C51, C61
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1757-6385.htm
Canonical
correlation
analysis
179
Journal of Financial Economic Policy
Vol. 6 No. 2, 2014
pp. 179-196
©Emerald Group Publishing Limited
1757-6385
DOI 10.1108/JFEP-09-2013-0047
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stock market and the real economy is very strong, it means that any movement in stock
prices directly affects the real economy. It is vital to determine what drives the volatility
of both the fxed-income securities and stock prices, as this can assist in predicting the
path of economic growth. For investors, volatility implies holding more securities in
portfolios so as to achieve diversifcation. That is why macroeconomists and fnance
specialists are increasingly focusing on the relationship between macroeconomic
variables and stock returns. Researchers, such as Aftab (2000), Fazal etal (2001), Nishat
(2004), Shahaz (2006) and Sharma (2007), have extensively researched on the
relationship between macroeconomic variables and stock returns. The fundamental
differences fromtheir fndings are that they selected different types of variables in their
research. The number of variables they used also differed. The researchers concentrated
mostly on developed and emerging markets, ignoring developing markets like
Zimbabwe. Methodologies that were used were different, and conclusions that were
drawn somehow differed. This research seeks to broaden the subject to developing
markets like Zimbabwe. It seeks to consider those macroeconomic variables that
directly affect stock returns at the Zimbabwe Stock Exchange (ZSE). The canonical
correlation analysis (CCA) was used to analyse the data. This methodology is a complete
deviation from the multiple regression which is usually applied. Thus, the motive is to
determine the macroeconomic variable sensitivities to stock price movements at the
ZSE.
Since the formation of the ZSE in 1896 in Bulawayo, policymakers, stock market
regulators, investors and stock market analysts have relied on the supply and demand
model in pricing assets and calculating stock returns, disregarding other determinants
of stock returns like macroeconomic variables. Macroeconomic variables provide a
platformfor a detailed analysis of stock returns because they infuence the performance
of the overall economy. Probably a macroeconomic variable approach to stock market
returns may fll the gap being left by the supply and demand theory approach currently
in use. The results for the research can be used for decision-making.
The research seeks to unveil a working strategy for policymakers, investors and fund
managers on how best to structure investment portfolios for the maximization of stock
returns. It aims to come up with a “prescription” for the “treatment” of macroeconomic
variables so that they positively impact the stock market returns for the beneft of the
Zimbabwean economy. Thus, the research seeks to construct an optimization model for
ZSE using the CCA.
The theoretical and practical motivation for undertaking this research on stock
returns and stock market forces can be discussed as follows.
An effciently operating stock market ensures economic growth and prosperity. It
helps in the diversifcation of domestic funds and their channelling into productive
investments. To achieve this, the stock market should have a signifcant relationship
with macroeconomic variables. Nowadays, the capital market has become a key element
of a market-based economy. It transfers the long-termfunds fromsavers to borrowers of
capital, which is very essential for economic development. After globalization,
international capital markets are being integrated rapidly. This integration positively
affects economic growth by reducing the contagious effect of risk on fnancial assets.
A well-performing stock market positively affects economic activities through
growth, saving and the effcient allocation of investment. Such an economy attracts
foreign direct investment. The stock market can boost the confdence of savers by
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providing domestic households with investment funds. The effcient market hypothesis
(EMH) suggests that all the relevant information to investors about proft maximization
is refected through stock prices and macroeconomic variables (Chong, 2003). If EMHis
implemented in stock markets, the role of stock brokerage frms diminishes gradually.
The stock market plays the vital role of transferring funds from capital borrowers to
capital investors. This is very essential for economic growth. It increases liquidity of
fnancial assets and diversifcation of global risk. This enables investors to make wiser
investment decisions (Agrawalla, 2006).
The research also seeks to broaden research previously carried out on this subject.
Previous researchers concentrated mostly on developed and emerging markets, leaving
out developing markets like Zimbabwe. The methodologies used by previous
researchers concentrated on multiple regression analysis, ignoring other methodologies
such as the CCA. A research of this nature has practical implications on policymakers,
stock market regulators, investors and stock market analysts in Zimbabwe.
2. Literature review
The literature reviewis divided into two categories: the frst reviewfocuses on research
carried out on stock returns versus macroeconomic variables with the CCA as the
modelling tool. The second review focuses on general studies that were carried out on
the relationship between stock returns and macroeconomic variables using other
methodologies.
After 1986, the relationship between macroeconomic factors and stock returns was
extensively investigated. The fndings from literature point to the existence of a
signifcant link between macroeconomic factors and stock returns in the countries
examined. Arnold (1996) applied CCA to explore the relationship between security
returns and economic factors in international settings, namely, the UKand the USA. The
CCA was applied in investigating a set of economic indicators that synthetically
infuence security returns. The results showed that the CCAsuccessfully links the stock
market and the economic factors. Azhar (2006) used the CCA, the principal components
analysis and the generalised least squares to determine the arbitrage pricing factors
in the Malaysian Stock Exchange. The CCA results suggest that macroeconomic
variables and stock market returns are signifcantly correlated at principal
component scores of the macroeconomic variables, namely, FECONI (0.4470),
FECON2 (0.3214) and FECON3 (0.7791). The results suggest that FECON1 represents
the macroeconomic variables such as the US exchange rate, the Singapore exchange rate
and Money supply (M2). The FECON2 represents macroeconomic variables such as
export, import, industrial index and gold production. FECON3 represents
macroeconomic variables such as oil prices (petroleum) and the Japanese exchange rate.
Sabetfar et al. (2011) carried out a study on the Tehran Stock Exchange (1991-1998).
They wanted to fnd out the common risk factors in the returns of non-oil-based stocks
in the Tehran Stock Exchange using the principal component analysis, the
cross-sectional regression and the CCA. The research results showed that the 13
macroeconomic variables which were used failed to fully explain excess returns of the
samples. All the 13 macroeconomic variables did not affect stock market returns.
Fan Fah et al. (2010) used the CCA to test the arbitrage pricing theory on the Tehran
Stock Exchange. Tests conducted using the principal component analysis showed that
at least one to three factors can explain the cross-section of expected returns in this
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market. The results suggest that there are four groups of macroeconomic variables in
the test period (1991-2008) that affect stock returns. But the signifcance of these factors
was found to be inconsistent over time. In general, the fndings document a weak
applicability of the arbitrage pricing theory in this market.
Vuyyuri (2005) investigated co-integration relationship and the causality between
the fnancial and real sectors of the Indian economy using monthly observations from
July 1992 to December 2002. The fnancial variables used were interest rate, infation
rate, stock returns and industrial production (proxy for real economy). The augmented
Dickey – Fuller and Phillips – Perron unit root tests were applied to check for
stationarity. The Granger test showed unidirectional Granger cause between the
fnancial and the real sector of the economy.
Afzal (2011) carried out an empirical analysis of the relationship between
macroeconomic variables and stock prices in Bangladesh using co-integration and the
Granger causality test. The results suggest that co-integration exists between stock
prices with money supply values, M1 and M2, and infation rate. This indicates the
existence of a long-run relationship between them. Unidirectional causality was found to
exist fromthe stock market to the exchange rate, M1 and M2. Cakmarh et al. (2010) found
that macroeconomic variables have useful information for predicting monthly US
excess stock returns and volatility over the period 1980-2005 using linear regression
models.
Chen et al. (1986) tested the multifactor model in the USA by using seven
macroeconomic variables. They found that consumption, oil prices and market index
were not priced by the fnancial market. However, industrial production, changes in risk
premiumand twists in the yield curve were signifcant in explaining stock returns. Chen
(1991) performed the second study covering the USA. Research fndings showed that
future market stock returns could be forecasted by interpreting macroeconomic
variables such as default spread, term spread, one-month treasury-bill rate, industrial
production growth rate and the dividend price ratio. Clare et al. (1994) investigated the
effect of 18 macroeconomic factors on stock returns in the UK. They found that oil prices,
retail price index, bank lending and corporate default risks were important risk factors
for the UK stock returns. Mukherjee and Naka (1995) used the vector error correction
approach to model the relationship between the Japanese stock returns and
macroeconomic variables. A co-integration relation was detected among stock prices
and the six macroeconomic variables, namely, exchange rate, long-term government
bond rate and call money rate. Gjrde and Saettem (1999) examined the causal relation
between stock returns and macroeconomic variables in Norway. Results showed that a
positive link exists between oil price, real activity and stock returns. However, the study
failed to show a signifcant relationship between stock returns and infation.
A recent study by Flannery and Protopapadakis (2002) re-evaluated the effect of
some series announcements on stock returns. Among these series, six macroeconomic
variables, namely, balance of trade, housing starts, employment, consumer price index,
money supply and producer price index affected stock returns. On the other hand, two
popular measures of aggregate economic activity (real gross national product [GNP] and
industrial production) were not related to stock returns.
Bailey and Chung (1996) examined the impact of macroeconomic risks on the equity
market of Philippines. Findings of the study showed that fnancial fuctuations,
exchange rate movements, political changes and equity cannot explain Philippine stock
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returns. Mokerjee and Qiao (1997) investigated the effect of macroeconomic variables on
the Singapore Stock Exchange market. Results suggest that stock prices co-integrated
with both measures of the money supply (M1 and M2) and aggregate foreign exchange
reserves. However, stock prices and exchange rates did not have a long-term
relationship. Ibrahimand Aziz (2003) investigated the relationship between stock prices
and industrial production, money supply, consumer price index and exchange rate in
Malaysia. Stock prices were found to share a positive long-run relationship with
industrial production and consumer price index. On the contrary, stock prices have a
negative association with money supply and the Ringgit exchange rate. Cheung and Ng
(1998) investigated the relationship between stock prices and some macroeconomic
factors, namely, real oil prices, total personal consumption, money supply (M1) and GNP
in Canada, Germany, Italy, Japan and the USA. Results showed that there is a long-run
co-movement between the selected macroeconomic variables and real stock market
prices. Billson and Hooper (2001) used value-weighted world market index and some
macroeconomic variables for explaining stock returns in selected emerging markets.
Findings showed that goods prices and real activity have limited ability to explain the
variation in return. Sharma et al. (2002) investigated the relationship between stock
prices and some macroeconomic factors in fve Asian countries, namely, Indonesia,
Malaysia, Philippines, Singapore and Thailand. Results suggest that in the long run,
stock prices will be positively related to growth and output. In the short run, stock prices
are found to be functions of past current values of macroeconomic variables.
3. Research methodology
3.1 Data collection
The data used in this research were sourced fromthe Central Statistics Offce, ZSE, Old
Mutual Asset Managers Zimbabwe (Private) Lt. and the Reserve Bank of Zimbabwe.
Top trading companies (counters) were selected because there was delisting and listing
of companies during the period under study due to harsh economic conditions. Those
counters which remained listed during the period under study were considered because
they had enough sets of data needed for the research. The counters selected were
categorized into two groups, namely, the industrial and the mining. The industrial
group consists of Ariston Holding Limited (Ariston), Border Timbers Limited (Border),
Barclays Bank of Zimbabwe Limited (Barclay), CAFCA Limited (Cafca) and Caps
Holding Limited (Caps). The mining group consists of Bindura Nickel Corporation
Limited (Bindura), Falcon Gold Zimbabwe Limited (Falcon) and Hwange Colliery
Company Limited (Hwange). These were the top eight movers at the ZSE during the
study period, and they never delisted from ZSE.
The data collected from Central Statistics Offce included: industrial and mining
indices, infation rate, money supply rate and unemployment rate. Prior to 1993, the
Central Statistics Offce produced a Consumer Price Index for high-income urban
families and another for low-income urban families. This was later revised with an
overall Consumer Price Index calculated from 1993 onwards. Because of hyperinfation
between 2005 and 2008, the research omitted data during that period. Data during that
period could distort the research fndings. This was the reason for not considering the
period prior to 1994 in the study because the overall index is diffcult to derive, given
that two indices were available. Data for money supply, exchange rate and interest rate
(91 days Treasury bill rate) were gathered from the Reserve Bank of Zimbabwe. The
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secondary data gathered have data on 12 cases, ? 11 time periods, for a total of 132
observations. The study combined data gathered through discussion, arbitration and
abstraction to shed light on the main theme of the problembeing investigated. The CCA
was used to analyse the data. Data on stock returns were deduced from stock prices for
each counter. The return from buying and holding a stock depends on the price at the
time of purchase, holding period, total dividend payments received and price at the end
of the holding period. Given P
it
, the price for the i
th
asset for month is t and d
it
is the
dividend paid to stock i in the t
th
month, then the rate of return for asset i in the t
th
month
is given by:
R
it
?
P
it
? P
i(t?1)
? d
it
P
i(t?1)
?
P
it
? P
i(t?1)
P
i(t?1)
?
d
it
P
i(t?1)
(3.1)
The equation (3.1) is the sumof the capital gain yield and dividend gain yield. Assuming
that investors have not declared any dividend on the exchange market, the cash
dividend declared is zero, and, as a result, the equation (3.1) is reduced to:
R
it
?
P
it
? P
i(t?1)
P
i(t?1)
(3.2)
Equation (3.2) was used to convert rawasset prices to the rate of return. The mean return
R
?
calculated by 1 / N ?
t?1
N
R
t
,R
?
, subtracted from R
it
to get a zero return, and R
?
was the
adjusted return obtained from the transformation process.
3.2 Model building: the theory behind
3.2.1 CCA (hotelling). This is a multivariate technique of measuring the linear
relationship between two multidimensional variables. It can be thought of as multiple –
multiple regression. It fnds two bases, one for each variable that are optimal with
respect to correlations. At the same time, it fnds the corresponding correlations. In other
words, it fnds the two bases in which the correlation matrix between the variables is
diagonal and correlations on the diagonal are maximized. The dimensionality of these
newbases is equal to or less than the smallest dimensionality of the two sets of variables.
Mathematically:
Consider the following two equations:
W
1
? a
11
X
1
? a
12
X
2
? … ? a
1p
X
p
(3.3)
V
1
? b
11
y
1
? b
12
y
2
? … ? b
1
q
y
q
(3.4)
Where W
1
is a linear combination of the X variables.
V
1
is the linear combination of the y variables.
Let C
1
be the correlation between W
1
and V
1
.
The objective of canonical correlation is to estimate a
11
, a
12
[…] a
1p
and b
11
, b
12
[…]
b
1q
such that C
1
is maximized. Equations (3.3) and (3.4) are canonical equations, W
1
and
V
1
are the canonical variates and C
1
is the canonical correlation. Once W
1
and V
1
have
been estimated, the next step is to identify another set of canonical variates:
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W
2
? a
21
X
1
? a
22
X
2
? … ? a
2p
X
p
V
2
? b
21
y
1
? b
22
y
2
? … ? b
2q
y
p
(3.5)
such that the correlation C
2
between themis maximum. W
2
and V
2
are uncorrelated with
W
1
and V
1
, i.e. the two sets of canonical variates are uncorrelated. This procedure is
continued until the m
th
set of canonical variates;
W
m
? a
m1
? a
m2
X
2
? … ? a
mp
X
p
V
m
? a
m1
y
1
? b
m2
y
2
? … ? b
mq
y
q
(3.6)
In summary, the objective of canonical correlation is to identify the m sets of canonical
variates, (W
1
, V
1
), (W
2
, V
2
) […] (W
m
, V
m
), such that the corresponding canonical
correlations C
1
, C
2
[…], Cm are maximum and Corr (V
j
, V
k
) ?0, for all j k, Corr (W
j
,
Wk) ?0, for all j k.
Cor (W
j
, V
k
) ?0, for all j k. This is clearly an optimization/maximization problem
subject to certain constraints. Fitting the overall model is done using redundancy
analysis.
3.2.2 Redundancy analysis. Relying on the canonical correlations alone on
signifcance testing is quite misleading. Thus, another test called the redundancy
analysis is carried out for both the dependent and the independent variables. This
overcomes the inherent bias and uncertainty in using canonical roots (squared canonical
correlations) as a measure of shared variance. The redundancy index is calculated by
squaring multiple correlation coeffcients between the total independent variable set
and each variable in the dependent variable set, and then averaging these squared
coeffcients to get an average R
2
value.
Redundancy index ? ?average loading?
2
[canonical correlation]
2
Aredundancy index close to one (1) is considered to be the highest. Avalue close to one
shows that the amount of say, the dependent variable’s variance being shared with the
independent variable, is signifcant and vice versa. Those close to zero (0) are considered
to be very low and have no signifcance in the variance being shared.
3.2.3 Interpreting the canonical variates. These interpretations involve examining
the canonical functions. This determines the signifcance of each of the original
variables in deriving the canonical relationships. The three methods for interpretation
include canonical weights (standardized coeffcients), canonical loadings (structure
correlations) and canonical cross-loadings. The three procedures consider a coeffcient
close to 1 (100 per cent) as being very signifcant. The one close to zero is very
insignifcant. The research study uses all the three methods in the interpretation. From
the three procedures, canonical cross-loadings is the best.
4. Data analysis and interpretation
The statistical analysis software (SAS) was used for data analysis. The software was
used to carry out the CCA (multiple – multiple regression) on stock returns and the
macroeconomic variables. The PROC CANCORR command in SAS was used to invoke
the CCA, and a SAS output was produced as shown in the Appendix.
185
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correlation
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4.1 Deriving the canonical functions
The frst statistical signifcance test is for the canonical correlations of each of the seven
canonical correlations functions. The seven canonical functions are:
Maximize ?
1
? Max Corr (V
1
, W1) …Maximize ?
7
? Max Corr (V
7
, W
7
)
Subject to
Var (V
1
) ? 1 … Var (V
7
) ? 1
Var (W
1
) ? 1 …Var (W
7
) ?1
(3.7)
where W
n
is a function of the macroeconomic variables, with n ?1, 2, 0.7.
V
n
is a function of stock returns-counters with n ?1, 2 […] 8.
The fundamental question is: are all the seven canonical functions of any
practical signifcance? The seven canonical functions have canonical correlation
coeffcients of: 0.990, 0.907, 0.760, 0.549, 0.252, 0.134 and 0.061 (SAS output frst
table). The frst four canonical functions are statistically signifcant as depicted by
the high canonical correlations which are closer to one (1). The last three are
insignifcant because their coeffcients are nearer to zero (0). Using the F-values for
multivariate tests, we fail to reject the frst four canonical functions because they
have signifcant F-values either at the 1 or 5 per cent level of signifcance (SAS
output-multivariate statistics and F-approximations). But a use of the R
2
-value leads
to the rejection of the fourth canonical function, and the frst three remain (SAS
output frst Table fourth column). Thus, the last four canonical functions are
rejected in this frst test of signifcance because their R
2
values of 0.302, 0.063, 0.018
and 0.004 are too small. Further tests are carried out using the Wilks’ Lambda,
Pillai’s trace, Hotelling – Lawley’s trace and Roy’s greatest root (SAS output:
multivariate statistics and F-approximations). All except the Wilks’ lambda “agree”
on the overall signifcance of all canonical functions taken collectively.
4.2 Redundancy analysis
Table I shows the redundancy indices for the independent variables for the frst canonical
function. The following information was deduced: ?CR
2
? 4.1199, Average ?
?CR 2 / 7 ?0.5885, Canonical R
2
-value ?0.980576 [fromoutput third table]:
Redundancy index ??Average of CR
2
][Canonical R
2
? value]
? [0.5885??0.9805? ? 0.5771
Fromthe computations, we can deduce that: although an index of 0.58 is not very close to 1,
it is fairly a large index value. The index value shows some relatively high degree of
signifcance. Thus, we can conclude that the amount of the independent variable’s variance
being accounted for or shared with the dependent variable is fairly signifcant. From
Table I.
The independent
variables redundancy
index calculations
Loadings ER M
1
CPI M
3
TB II U
Canonical loadings 0.9265 0.7156 0.3662 0.9931 0.8549 0.0100 0.5446
[Canonical loading]
2
0.8584 0.51208 0.1341 0.9862 0.7308 0.001 0.2965
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Table II, the following can also be deduced: ?CR
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?5.2926, Average ?5.2926/7 ?0.6616
canonical R
2
?0.980576.
Redundancy Index ??0.6616??0.9806? ?0.6487
From the calculations, it can be deduced that a redundancy index of 0.65 is high. This
implies that the amount of dependent variable’s variance being shared between the
independent variables is very signifcant. This redundancy analysis leads to the
conclusion that the frst canonical function:
Max ?
1
? Max Corr ?V
1
, W
1
?
Subject to
Var (V
1
) ? 1
Var (W
1
) ? 1
(3.8)
is signifcant in explaining the relationship between the dependent and independent
variables.
Similar computations were carried out. For the second canonical function, the
redundancy index for the independent variable was 0.031, which is quite insignifcant.
The redundancy index for the dependent variable was 0.103, which again is quite an
insignifcant index. Even the computations for the other canonical functions yielded
insignifcant indices. Thus, all the other six canonical functions were rejected, and
ultimately, the frst canonical function was chosen as the most representative of all.
4.3 Interpreting the canonical variates
4.3.1 Standardized canonical weights. [SAS output: standardised canonical weights for
the VAR variables]
Based on the size of the weights, the order of magnitude in terms of contribution of
the independent variables to the frst variate is: M
3
(1.96), CPI (?0.832), ER(?0.164), TB
(0.065), U(?0.045), MI (0.026) and II (?0.003). For the dependent variables they are in the
order:
Hwange (0.362), Ariston (?0.117), Caps (?0.216), Border (0.157), Cafca (0.156),
Barclays (0.099), Falcon (0.065) and Bindura (?0.017), i.e.
W
1
[M
3
, CPI, ER, TB, U, MI, II] – for the independent variate.
V
1
[Hwange, Ariston, Caps, Border, Cafca, Barclays, Falcon, Bindura] – dependent
variate.
For the second canonical function:
W
2
[CPI, M
3
, ER, U, TB, M1, II].
V
2
[Ariston, Border, Hwange, Caps, Barclays, Bindura, Falcon, Cafca].
Table II.
The dependent variable
redundancy index
calculations
Loadings Hwang Falcon Caps Cafca Barclays Border Bindura Ariston
CL 0.9784 0.9265 0.7156 0.3662 0.9474 0.8520 0.6131 0.9118
CL
2
0.9573 0.8584 0.5121 0.1341 0.8976 0.7259 0.3759 0.8314
Notes: Where CL – canonical loadings; CL
2
– [canonical loadings]
2
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W
1
and W
2
have some similarities in the order in which the factors contribute to the
formation of the variate and, consequently, the canonical function (model). It means that
the frst fve independent variables, namely, CPI, M3, ER, Uand M1 are signifcant in the
formation of the frst canonical variate and the frst canonical model. The dependent set
of variables, namely, Ariston, Border, Hwange, Caps and Barclays are the top movers in
terms of contribution to the variate and the canonical model. Canonical loadings and
cross-loadings are the most appropriate methods in interpreting because they are very
robust in instances of multi-collinearity.
4.3.2 Canonical loadings. [SAS output: correlations between the with variables and
their canonical variables].
The loadings for the independent variables are: money supply (99 per cent),
consumer price index (98 per cent), exchange rate (94 per cent), treasury bills (85 per
cent), unemployment (55 per cent), mining index (51 per cent) and industrial index (1 per
cent). These fgures suggest that the frst four macroeconomic variables greatly
infuence the formation of the frst variate W
1
, resulting in the frst canonical correlation
model. Only the contributions of the mining and industrial indices contributions fall
below the average mark of 54.5 per cent (average cross loadings for the seven
macroeconomic variables). This implies that the two indices are insignifcant. The
loadings for the dependent variables are: Hwange (98 per cent), Barclays (95 per cent),
Falcon (93 per cent), Ariston (91 per cent), Border (85 per cent), Caps (72 per cent),
Bindura (62 per cent) and Cafca (37 per cent). The most infuential ones include: Hwange,
Barclays, Falcon, Ariston and Border because their loadings exceed the 79 per cent
average point. Analysis of other variates is no longer necessary because an overall
pattern has been deduced already.
4.3.3 Canonical cross-loadings. [SAS output: last two tables]. Again similar patterns
appear inthe cross-loadingvalues for M3 (98 per cent), CPI (97 per cent), ER(93 per cent) and
TB (85 per cent). These fgures support conclusions that were drawn earlier. If the
corresponding correlation values are squared, the percentage of the variance being
explained by the canonical variates is obtained. The values are: M3 (97 per cent), CPI (94 per
cent), ER (87 per cent), TB (72 per cent), U (29 per cent), MI (26 per cent) and II (0 per cent).
Hwange (97 per cent), Barclays (94 per cent), Falcon (92 per cent), Ariston (90 per cent),
Border (84 per cent), Caps (71 per cent), Bindura (61 per cent) and Cafca (36 per cent). The
squared values render Bindura and Cafca insignifcant.
4.3.4 Overall model ftting. From the analysis above, W
1
(money supply, consumer
price index, exchange rate and treasury bills) contribute signifcantly in maximizing
stock returns at ZSE. Thus, they should be incorporated into the fnal CCAmodel. When
developing the model, emphasis should be on the macroeconomic variables’ impact on
stock returns rather than the reverse. Thus, for the dependent variables, we include top
movers like V
1
(Hwange, Barclays, Border, Caps, Ariston, Bindura and Falcon).
The specifc model with regards to the research fndings is given by:
Maximize ?
1
? Maximize Corr (V
1
, W
1
)
Subject to
Var (V
1
) ? 1
Var (W
1
) ? 1
(3.9)
where V
1
?stock returns for Hwange, Barclays, Border, Caps, Ariston, Bindura, Falcon.
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W
1
? [money supply, consumer price index, exchange rate, treasury bills].The
general canonical correlation model is of the form:
Maximize ?
n
? Max Corr (V
n
, W
n
)
Subject to
Var (V
n
) ? 1
Var (W
n
) ? 1
(4.0)
4.4.5 Model validation and diagnosis. The last stage involves validating the prescribed
model through sensitivity analysis. The canonical loadings are examined for stability of
the model ftted. Individual values for the independent variables are deleted from the
analysis. The results obtained after omitting one value from each of money supply,
unemployment and exchange rate show little or no signifcant changes of the weights,
loadings and cross-loadings. This implies that the ftted model is very signifcant even
if the population is changed.
5. Conclusion
From the research results, the following were found: CCA identifes dimensions among
stock returns and macroeconomic variables that maximize the relationship between the
dimensions. Through CCA, the research managed to fnd the strength of the relationship
between macroeconomic variables and stock returns for the Zimbabwe economy during
“harsh” economic conditions. A fairly good model was ftted because a redundancy
index of 0.68 is very high (for a comparable multiple regression). This means that the
variance of stock returns that is being accounted for by the macroeconomic variables is
also high.
Five major macroeconomic variables, namely, money supply, treasury bills,
exchange rate, consumer price index and unemployment were found to be signifcant in
maximizing stock returns at ZSE. Mining index and industrial production were rejected.
Thus, a general CCA model was found that can be used by ZSE upon further scrutiny.
The model constructed can be used for prediction purposes. Thus, the model can be used
by traders at ZSE, economic regulators and the government to manage portfolios. The
model gives insight into what macroeconomic factors to closely monitor for portfolios at
ZSE to yield maximum stock returns during an economic meltdown like what the
country experienced between 1990 and 2008.
6. Practical implications and future research guide lines
Investors and stock market regulators can use the canonical correlation model to
monitor macroeconomic trends because they affect stock price movements. New
investors can have an insight of areas of investment. They can monitor the performance
of some companies in similar types of business. The research results can assist the
government in crafting economic policies for the country based on the listed stocks’
reaction to movements of the macroeconomic variables. The CCA results enable
policymakers to simultaneously predict multiple stock returns from multiple
macroeconomic variables.
This research is limited to top trading counters and selected macroeconomic
variables during the research period at ZSE. Future researchers may include all the
listed companies as well as more macroeconomic variables into the model so that a
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broader view of ZSE is envisaged. There may be a need to incorporate other modelling
techniques such as the principal components analysis and factor analysis to
complement the CCA. For example, factor analysis can be considered with the intention
of discussing scale development. The CCA is a form of scale development. The
dependent and independent variates represent dimensions of the variable sets similar to
the relationship between them. But factor analysis maximizes the explanation (shared
variate) of the variable set. The results of the CCAmodel showa set of stock returns and
a set of macroeconomic variables being correlated. These two sets of variables can be
used to develop a multidimensional regression model for the ZSE. This is a matrix of all
regression models for each company and provides another dimension of analyzing the
stock market using multiple models.
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Appendix
Tables on canonical correlation analysis and SAS output
Table AI.
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Table AII.
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Table AIII.
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Table AIV.
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Corresponding author
Peter Mazuruse can be contacted at: [email protected]
To purchase reprints of this article please e-mail: [email protected]
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Table AV.
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doc_534728519.pdf
The purpose of this paper was to construct a canonical correlation analysis (CCA) model for
the Zimbabwe stock exchange (ZSE). This paper analyses the impact of macroeconomic variables on
stock returns for the Zimbabwe Stock Exchange using the canonical correlation analysis (CCA).
Journal of Financial Economic Policy
Canonical correlation analysis: Macroeconomic variables versus stock returns
Peter Mazuruse
Article information:
To cite this document:
Peter Mazuruse , (2014),"Canonical correlation analysis", J ournal of Financial Economic Policy, Vol. 6 Iss 2
pp. 179 - 196
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Canonical correlation analysis
Macroeconomic variables versus stock returns
Peter Mazuruse
Harare Institute of Technology, Harare, Zimbabwe
Abstract
Purpose – The purpose of this paper was to construct a canonical correlation analysis (CCA) model for
the Zimbabwe stock exchange (ZSE). This paper analyses the impact of macroeconomic variables on
stock returns for the Zimbabwe Stock Exchange using the canonical correlation analysis (CCA).
Design/methodology/approach – Data for the independent (macroeconomic) variables and
dependent variables (stock returns) were extracted fromsecondary sources for the period fromJanuary
1990 to December 2008. For each variable, 132 sets of data were collected. Eight top trading companies
at the ZSE were selected, and their monthly stock returns were calculated using monthly stock prices.
The independent variables include: consumer price index, money supply, treasury bills, exchange rate,
unemployment, mining and industrial index. The CCA was used to construct the CCA model for the
ZSE.
Findings – Maximization of stock returns at the ZSEis mostly infuenced by the changes in consumer
price index, money supply, exchange rate and treasury bills. The four macroeconomic variables greatly
affect the movement of stock prices which, in turn, affect stock returns. The stock returns for Hwange,
Barclays, Falcon, Ariston, Border, Caps and Bindura were signifcant in forming the CCA model.
Research limitations/implications – During the research period, some companies delisted due to
economic hardships, and this reduced the sample size for stock returns for respective companies.
Practical implications – The results from this research can be used by policymakers, stock market
regulators and the government to make informed decisions when crafting economic policies for the
country. The CCA model enables the stakeholders to identify the macroeconomic variables that play a
pivotal role in maximizing the strength of the relationship with stock returns.
Social implications – Macroeconomic variables, such as consumer price index, infation, etc.,
directly affect the livelihoods of the general populace. They also impact on the performance of
companies. The society can monitor economic trends and make the right decisions based on the current
trends of economic performance.
Originality/value – This research opens a new dimension to the study of macroeconomic variables
and stock returns. Most studies carried out so far in Zimbabwe zeroed in on multiple regression as the
central methodology. No study has been done using the CCA as the main methodology.
Keywords Stock returns, Macroeconomic variables, Canonical correlation, Canonical variates,
Canonical loadings, Redundancy index
Paper type Research paper
1. Introduction
The study of macroeconomic variables and their impact on stock returns has been an
area of intense interest among academics, investors and stock market regulators since
the 1970s. The relationship between macroeconomic variables and stock returns affects
the valuation of securities and risk management. Because the inter-dependence of the
JEL classifcation – C51, C61
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1757-6385.htm
Canonical
correlation
analysis
179
Journal of Financial Economic Policy
Vol. 6 No. 2, 2014
pp. 179-196
©Emerald Group Publishing Limited
1757-6385
DOI 10.1108/JFEP-09-2013-0047
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stock market and the real economy is very strong, it means that any movement in stock
prices directly affects the real economy. It is vital to determine what drives the volatility
of both the fxed-income securities and stock prices, as this can assist in predicting the
path of economic growth. For investors, volatility implies holding more securities in
portfolios so as to achieve diversifcation. That is why macroeconomists and fnance
specialists are increasingly focusing on the relationship between macroeconomic
variables and stock returns. Researchers, such as Aftab (2000), Fazal etal (2001), Nishat
(2004), Shahaz (2006) and Sharma (2007), have extensively researched on the
relationship between macroeconomic variables and stock returns. The fundamental
differences fromtheir fndings are that they selected different types of variables in their
research. The number of variables they used also differed. The researchers concentrated
mostly on developed and emerging markets, ignoring developing markets like
Zimbabwe. Methodologies that were used were different, and conclusions that were
drawn somehow differed. This research seeks to broaden the subject to developing
markets like Zimbabwe. It seeks to consider those macroeconomic variables that
directly affect stock returns at the Zimbabwe Stock Exchange (ZSE). The canonical
correlation analysis (CCA) was used to analyse the data. This methodology is a complete
deviation from the multiple regression which is usually applied. Thus, the motive is to
determine the macroeconomic variable sensitivities to stock price movements at the
ZSE.
Since the formation of the ZSE in 1896 in Bulawayo, policymakers, stock market
regulators, investors and stock market analysts have relied on the supply and demand
model in pricing assets and calculating stock returns, disregarding other determinants
of stock returns like macroeconomic variables. Macroeconomic variables provide a
platformfor a detailed analysis of stock returns because they infuence the performance
of the overall economy. Probably a macroeconomic variable approach to stock market
returns may fll the gap being left by the supply and demand theory approach currently
in use. The results for the research can be used for decision-making.
The research seeks to unveil a working strategy for policymakers, investors and fund
managers on how best to structure investment portfolios for the maximization of stock
returns. It aims to come up with a “prescription” for the “treatment” of macroeconomic
variables so that they positively impact the stock market returns for the beneft of the
Zimbabwean economy. Thus, the research seeks to construct an optimization model for
ZSE using the CCA.
The theoretical and practical motivation for undertaking this research on stock
returns and stock market forces can be discussed as follows.
An effciently operating stock market ensures economic growth and prosperity. It
helps in the diversifcation of domestic funds and their channelling into productive
investments. To achieve this, the stock market should have a signifcant relationship
with macroeconomic variables. Nowadays, the capital market has become a key element
of a market-based economy. It transfers the long-termfunds fromsavers to borrowers of
capital, which is very essential for economic development. After globalization,
international capital markets are being integrated rapidly. This integration positively
affects economic growth by reducing the contagious effect of risk on fnancial assets.
A well-performing stock market positively affects economic activities through
growth, saving and the effcient allocation of investment. Such an economy attracts
foreign direct investment. The stock market can boost the confdence of savers by
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providing domestic households with investment funds. The effcient market hypothesis
(EMH) suggests that all the relevant information to investors about proft maximization
is refected through stock prices and macroeconomic variables (Chong, 2003). If EMHis
implemented in stock markets, the role of stock brokerage frms diminishes gradually.
The stock market plays the vital role of transferring funds from capital borrowers to
capital investors. This is very essential for economic growth. It increases liquidity of
fnancial assets and diversifcation of global risk. This enables investors to make wiser
investment decisions (Agrawalla, 2006).
The research also seeks to broaden research previously carried out on this subject.
Previous researchers concentrated mostly on developed and emerging markets, leaving
out developing markets like Zimbabwe. The methodologies used by previous
researchers concentrated on multiple regression analysis, ignoring other methodologies
such as the CCA. A research of this nature has practical implications on policymakers,
stock market regulators, investors and stock market analysts in Zimbabwe.
2. Literature review
The literature reviewis divided into two categories: the frst reviewfocuses on research
carried out on stock returns versus macroeconomic variables with the CCA as the
modelling tool. The second review focuses on general studies that were carried out on
the relationship between stock returns and macroeconomic variables using other
methodologies.
After 1986, the relationship between macroeconomic factors and stock returns was
extensively investigated. The fndings from literature point to the existence of a
signifcant link between macroeconomic factors and stock returns in the countries
examined. Arnold (1996) applied CCA to explore the relationship between security
returns and economic factors in international settings, namely, the UKand the USA. The
CCA was applied in investigating a set of economic indicators that synthetically
infuence security returns. The results showed that the CCAsuccessfully links the stock
market and the economic factors. Azhar (2006) used the CCA, the principal components
analysis and the generalised least squares to determine the arbitrage pricing factors
in the Malaysian Stock Exchange. The CCA results suggest that macroeconomic
variables and stock market returns are signifcantly correlated at principal
component scores of the macroeconomic variables, namely, FECONI (0.4470),
FECON2 (0.3214) and FECON3 (0.7791). The results suggest that FECON1 represents
the macroeconomic variables such as the US exchange rate, the Singapore exchange rate
and Money supply (M2). The FECON2 represents macroeconomic variables such as
export, import, industrial index and gold production. FECON3 represents
macroeconomic variables such as oil prices (petroleum) and the Japanese exchange rate.
Sabetfar et al. (2011) carried out a study on the Tehran Stock Exchange (1991-1998).
They wanted to fnd out the common risk factors in the returns of non-oil-based stocks
in the Tehran Stock Exchange using the principal component analysis, the
cross-sectional regression and the CCA. The research results showed that the 13
macroeconomic variables which were used failed to fully explain excess returns of the
samples. All the 13 macroeconomic variables did not affect stock market returns.
Fan Fah et al. (2010) used the CCA to test the arbitrage pricing theory on the Tehran
Stock Exchange. Tests conducted using the principal component analysis showed that
at least one to three factors can explain the cross-section of expected returns in this
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market. The results suggest that there are four groups of macroeconomic variables in
the test period (1991-2008) that affect stock returns. But the signifcance of these factors
was found to be inconsistent over time. In general, the fndings document a weak
applicability of the arbitrage pricing theory in this market.
Vuyyuri (2005) investigated co-integration relationship and the causality between
the fnancial and real sectors of the Indian economy using monthly observations from
July 1992 to December 2002. The fnancial variables used were interest rate, infation
rate, stock returns and industrial production (proxy for real economy). The augmented
Dickey – Fuller and Phillips – Perron unit root tests were applied to check for
stationarity. The Granger test showed unidirectional Granger cause between the
fnancial and the real sector of the economy.
Afzal (2011) carried out an empirical analysis of the relationship between
macroeconomic variables and stock prices in Bangladesh using co-integration and the
Granger causality test. The results suggest that co-integration exists between stock
prices with money supply values, M1 and M2, and infation rate. This indicates the
existence of a long-run relationship between them. Unidirectional causality was found to
exist fromthe stock market to the exchange rate, M1 and M2. Cakmarh et al. (2010) found
that macroeconomic variables have useful information for predicting monthly US
excess stock returns and volatility over the period 1980-2005 using linear regression
models.
Chen et al. (1986) tested the multifactor model in the USA by using seven
macroeconomic variables. They found that consumption, oil prices and market index
were not priced by the fnancial market. However, industrial production, changes in risk
premiumand twists in the yield curve were signifcant in explaining stock returns. Chen
(1991) performed the second study covering the USA. Research fndings showed that
future market stock returns could be forecasted by interpreting macroeconomic
variables such as default spread, term spread, one-month treasury-bill rate, industrial
production growth rate and the dividend price ratio. Clare et al. (1994) investigated the
effect of 18 macroeconomic factors on stock returns in the UK. They found that oil prices,
retail price index, bank lending and corporate default risks were important risk factors
for the UK stock returns. Mukherjee and Naka (1995) used the vector error correction
approach to model the relationship between the Japanese stock returns and
macroeconomic variables. A co-integration relation was detected among stock prices
and the six macroeconomic variables, namely, exchange rate, long-term government
bond rate and call money rate. Gjrde and Saettem (1999) examined the causal relation
between stock returns and macroeconomic variables in Norway. Results showed that a
positive link exists between oil price, real activity and stock returns. However, the study
failed to show a signifcant relationship between stock returns and infation.
A recent study by Flannery and Protopapadakis (2002) re-evaluated the effect of
some series announcements on stock returns. Among these series, six macroeconomic
variables, namely, balance of trade, housing starts, employment, consumer price index,
money supply and producer price index affected stock returns. On the other hand, two
popular measures of aggregate economic activity (real gross national product [GNP] and
industrial production) were not related to stock returns.
Bailey and Chung (1996) examined the impact of macroeconomic risks on the equity
market of Philippines. Findings of the study showed that fnancial fuctuations,
exchange rate movements, political changes and equity cannot explain Philippine stock
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returns. Mokerjee and Qiao (1997) investigated the effect of macroeconomic variables on
the Singapore Stock Exchange market. Results suggest that stock prices co-integrated
with both measures of the money supply (M1 and M2) and aggregate foreign exchange
reserves. However, stock prices and exchange rates did not have a long-term
relationship. Ibrahimand Aziz (2003) investigated the relationship between stock prices
and industrial production, money supply, consumer price index and exchange rate in
Malaysia. Stock prices were found to share a positive long-run relationship with
industrial production and consumer price index. On the contrary, stock prices have a
negative association with money supply and the Ringgit exchange rate. Cheung and Ng
(1998) investigated the relationship between stock prices and some macroeconomic
factors, namely, real oil prices, total personal consumption, money supply (M1) and GNP
in Canada, Germany, Italy, Japan and the USA. Results showed that there is a long-run
co-movement between the selected macroeconomic variables and real stock market
prices. Billson and Hooper (2001) used value-weighted world market index and some
macroeconomic variables for explaining stock returns in selected emerging markets.
Findings showed that goods prices and real activity have limited ability to explain the
variation in return. Sharma et al. (2002) investigated the relationship between stock
prices and some macroeconomic factors in fve Asian countries, namely, Indonesia,
Malaysia, Philippines, Singapore and Thailand. Results suggest that in the long run,
stock prices will be positively related to growth and output. In the short run, stock prices
are found to be functions of past current values of macroeconomic variables.
3. Research methodology
3.1 Data collection
The data used in this research were sourced fromthe Central Statistics Offce, ZSE, Old
Mutual Asset Managers Zimbabwe (Private) Lt. and the Reserve Bank of Zimbabwe.
Top trading companies (counters) were selected because there was delisting and listing
of companies during the period under study due to harsh economic conditions. Those
counters which remained listed during the period under study were considered because
they had enough sets of data needed for the research. The counters selected were
categorized into two groups, namely, the industrial and the mining. The industrial
group consists of Ariston Holding Limited (Ariston), Border Timbers Limited (Border),
Barclays Bank of Zimbabwe Limited (Barclay), CAFCA Limited (Cafca) and Caps
Holding Limited (Caps). The mining group consists of Bindura Nickel Corporation
Limited (Bindura), Falcon Gold Zimbabwe Limited (Falcon) and Hwange Colliery
Company Limited (Hwange). These were the top eight movers at the ZSE during the
study period, and they never delisted from ZSE.
The data collected from Central Statistics Offce included: industrial and mining
indices, infation rate, money supply rate and unemployment rate. Prior to 1993, the
Central Statistics Offce produced a Consumer Price Index for high-income urban
families and another for low-income urban families. This was later revised with an
overall Consumer Price Index calculated from 1993 onwards. Because of hyperinfation
between 2005 and 2008, the research omitted data during that period. Data during that
period could distort the research fndings. This was the reason for not considering the
period prior to 1994 in the study because the overall index is diffcult to derive, given
that two indices were available. Data for money supply, exchange rate and interest rate
(91 days Treasury bill rate) were gathered from the Reserve Bank of Zimbabwe. The
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secondary data gathered have data on 12 cases, ? 11 time periods, for a total of 132
observations. The study combined data gathered through discussion, arbitration and
abstraction to shed light on the main theme of the problembeing investigated. The CCA
was used to analyse the data. Data on stock returns were deduced from stock prices for
each counter. The return from buying and holding a stock depends on the price at the
time of purchase, holding period, total dividend payments received and price at the end
of the holding period. Given P
it
, the price for the i
th
asset for month is t and d
it
is the
dividend paid to stock i in the t
th
month, then the rate of return for asset i in the t
th
month
is given by:
R
it
?
P
it
? P
i(t?1)
? d
it
P
i(t?1)
?
P
it
? P
i(t?1)
P
i(t?1)
?
d
it
P
i(t?1)
(3.1)
The equation (3.1) is the sumof the capital gain yield and dividend gain yield. Assuming
that investors have not declared any dividend on the exchange market, the cash
dividend declared is zero, and, as a result, the equation (3.1) is reduced to:
R
it
?
P
it
? P
i(t?1)
P
i(t?1)
(3.2)
Equation (3.2) was used to convert rawasset prices to the rate of return. The mean return
R
?
calculated by 1 / N ?
t?1
N
R
t
,R
?
, subtracted from R
it
to get a zero return, and R
?
was the
adjusted return obtained from the transformation process.
3.2 Model building: the theory behind
3.2.1 CCA (hotelling). This is a multivariate technique of measuring the linear
relationship between two multidimensional variables. It can be thought of as multiple –
multiple regression. It fnds two bases, one for each variable that are optimal with
respect to correlations. At the same time, it fnds the corresponding correlations. In other
words, it fnds the two bases in which the correlation matrix between the variables is
diagonal and correlations on the diagonal are maximized. The dimensionality of these
newbases is equal to or less than the smallest dimensionality of the two sets of variables.
Mathematically:
Consider the following two equations:
W
1
? a
11
X
1
? a
12
X
2
? … ? a
1p
X
p
(3.3)
V
1
? b
11
y
1
? b
12
y
2
? … ? b
1
q
y
q
(3.4)
Where W
1
is a linear combination of the X variables.
V
1
is the linear combination of the y variables.
Let C
1
be the correlation between W
1
and V
1
.
The objective of canonical correlation is to estimate a
11
, a
12
[…] a
1p
and b
11
, b
12
[…]
b
1q
such that C
1
is maximized. Equations (3.3) and (3.4) are canonical equations, W
1
and
V
1
are the canonical variates and C
1
is the canonical correlation. Once W
1
and V
1
have
been estimated, the next step is to identify another set of canonical variates:
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W
2
? a
21
X
1
? a
22
X
2
? … ? a
2p
X
p
V
2
? b
21
y
1
? b
22
y
2
? … ? b
2q
y
p
(3.5)
such that the correlation C
2
between themis maximum. W
2
and V
2
are uncorrelated with
W
1
and V
1
, i.e. the two sets of canonical variates are uncorrelated. This procedure is
continued until the m
th
set of canonical variates;
W
m
? a
m1
? a
m2
X
2
? … ? a
mp
X
p
V
m
? a
m1
y
1
? b
m2
y
2
? … ? b
mq
y
q
(3.6)
In summary, the objective of canonical correlation is to identify the m sets of canonical
variates, (W
1
, V
1
), (W
2
, V
2
) […] (W
m
, V
m
), such that the corresponding canonical
correlations C
1
, C
2
[…], Cm are maximum and Corr (V
j
, V
k
) ?0, for all j k, Corr (W
j
,
Wk) ?0, for all j k.
Cor (W
j
, V
k
) ?0, for all j k. This is clearly an optimization/maximization problem
subject to certain constraints. Fitting the overall model is done using redundancy
analysis.
3.2.2 Redundancy analysis. Relying on the canonical correlations alone on
signifcance testing is quite misleading. Thus, another test called the redundancy
analysis is carried out for both the dependent and the independent variables. This
overcomes the inherent bias and uncertainty in using canonical roots (squared canonical
correlations) as a measure of shared variance. The redundancy index is calculated by
squaring multiple correlation coeffcients between the total independent variable set
and each variable in the dependent variable set, and then averaging these squared
coeffcients to get an average R
2
value.
Redundancy index ? ?average loading?
2
[canonical correlation]
2
Aredundancy index close to one (1) is considered to be the highest. Avalue close to one
shows that the amount of say, the dependent variable’s variance being shared with the
independent variable, is signifcant and vice versa. Those close to zero (0) are considered
to be very low and have no signifcance in the variance being shared.
3.2.3 Interpreting the canonical variates. These interpretations involve examining
the canonical functions. This determines the signifcance of each of the original
variables in deriving the canonical relationships. The three methods for interpretation
include canonical weights (standardized coeffcients), canonical loadings (structure
correlations) and canonical cross-loadings. The three procedures consider a coeffcient
close to 1 (100 per cent) as being very signifcant. The one close to zero is very
insignifcant. The research study uses all the three methods in the interpretation. From
the three procedures, canonical cross-loadings is the best.
4. Data analysis and interpretation
The statistical analysis software (SAS) was used for data analysis. The software was
used to carry out the CCA (multiple – multiple regression) on stock returns and the
macroeconomic variables. The PROC CANCORR command in SAS was used to invoke
the CCA, and a SAS output was produced as shown in the Appendix.
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4.1 Deriving the canonical functions
The frst statistical signifcance test is for the canonical correlations of each of the seven
canonical correlations functions. The seven canonical functions are:
Maximize ?
1
? Max Corr (V
1
, W1) …Maximize ?
7
? Max Corr (V
7
, W
7
)
Subject to
Var (V
1
) ? 1 … Var (V
7
) ? 1
Var (W
1
) ? 1 …Var (W
7
) ?1
(3.7)
where W
n
is a function of the macroeconomic variables, with n ?1, 2, 0.7.
V
n
is a function of stock returns-counters with n ?1, 2 […] 8.
The fundamental question is: are all the seven canonical functions of any
practical signifcance? The seven canonical functions have canonical correlation
coeffcients of: 0.990, 0.907, 0.760, 0.549, 0.252, 0.134 and 0.061 (SAS output frst
table). The frst four canonical functions are statistically signifcant as depicted by
the high canonical correlations which are closer to one (1). The last three are
insignifcant because their coeffcients are nearer to zero (0). Using the F-values for
multivariate tests, we fail to reject the frst four canonical functions because they
have signifcant F-values either at the 1 or 5 per cent level of signifcance (SAS
output-multivariate statistics and F-approximations). But a use of the R
2
-value leads
to the rejection of the fourth canonical function, and the frst three remain (SAS
output frst Table fourth column). Thus, the last four canonical functions are
rejected in this frst test of signifcance because their R
2
values of 0.302, 0.063, 0.018
and 0.004 are too small. Further tests are carried out using the Wilks’ Lambda,
Pillai’s trace, Hotelling – Lawley’s trace and Roy’s greatest root (SAS output:
multivariate statistics and F-approximations). All except the Wilks’ lambda “agree”
on the overall signifcance of all canonical functions taken collectively.
4.2 Redundancy analysis
Table I shows the redundancy indices for the independent variables for the frst canonical
function. The following information was deduced: ?CR
2
? 4.1199, Average ?
?CR 2 / 7 ?0.5885, Canonical R
2
-value ?0.980576 [fromoutput third table]:
Redundancy index ??Average of CR
2
][Canonical R
2
? value]
? [0.5885??0.9805? ? 0.5771
Fromthe computations, we can deduce that: although an index of 0.58 is not very close to 1,
it is fairly a large index value. The index value shows some relatively high degree of
signifcance. Thus, we can conclude that the amount of the independent variable’s variance
being accounted for or shared with the dependent variable is fairly signifcant. From
Table I.
The independent
variables redundancy
index calculations
Loadings ER M
1
CPI M
3
TB II U
Canonical loadings 0.9265 0.7156 0.3662 0.9931 0.8549 0.0100 0.5446
[Canonical loading]
2
0.8584 0.51208 0.1341 0.9862 0.7308 0.001 0.2965
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Table II, the following can also be deduced: ?CR
2
?5.2926, Average ?5.2926/7 ?0.6616
canonical R
2
?0.980576.
Redundancy Index ??0.6616??0.9806? ?0.6487
From the calculations, it can be deduced that a redundancy index of 0.65 is high. This
implies that the amount of dependent variable’s variance being shared between the
independent variables is very signifcant. This redundancy analysis leads to the
conclusion that the frst canonical function:
Max ?
1
? Max Corr ?V
1
, W
1
?
Subject to
Var (V
1
) ? 1
Var (W
1
) ? 1
(3.8)
is signifcant in explaining the relationship between the dependent and independent
variables.
Similar computations were carried out. For the second canonical function, the
redundancy index for the independent variable was 0.031, which is quite insignifcant.
The redundancy index for the dependent variable was 0.103, which again is quite an
insignifcant index. Even the computations for the other canonical functions yielded
insignifcant indices. Thus, all the other six canonical functions were rejected, and
ultimately, the frst canonical function was chosen as the most representative of all.
4.3 Interpreting the canonical variates
4.3.1 Standardized canonical weights. [SAS output: standardised canonical weights for
the VAR variables]
Based on the size of the weights, the order of magnitude in terms of contribution of
the independent variables to the frst variate is: M
3
(1.96), CPI (?0.832), ER(?0.164), TB
(0.065), U(?0.045), MI (0.026) and II (?0.003). For the dependent variables they are in the
order:
Hwange (0.362), Ariston (?0.117), Caps (?0.216), Border (0.157), Cafca (0.156),
Barclays (0.099), Falcon (0.065) and Bindura (?0.017), i.e.
W
1
[M
3
, CPI, ER, TB, U, MI, II] – for the independent variate.
V
1
[Hwange, Ariston, Caps, Border, Cafca, Barclays, Falcon, Bindura] – dependent
variate.
For the second canonical function:
W
2
[CPI, M
3
, ER, U, TB, M1, II].
V
2
[Ariston, Border, Hwange, Caps, Barclays, Bindura, Falcon, Cafca].
Table II.
The dependent variable
redundancy index
calculations
Loadings Hwang Falcon Caps Cafca Barclays Border Bindura Ariston
CL 0.9784 0.9265 0.7156 0.3662 0.9474 0.8520 0.6131 0.9118
CL
2
0.9573 0.8584 0.5121 0.1341 0.8976 0.7259 0.3759 0.8314
Notes: Where CL – canonical loadings; CL
2
– [canonical loadings]
2
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W
1
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2
have some similarities in the order in which the factors contribute to the
formation of the variate and, consequently, the canonical function (model). It means that
the frst fve independent variables, namely, CPI, M3, ER, Uand M1 are signifcant in the
formation of the frst canonical variate and the frst canonical model. The dependent set
of variables, namely, Ariston, Border, Hwange, Caps and Barclays are the top movers in
terms of contribution to the variate and the canonical model. Canonical loadings and
cross-loadings are the most appropriate methods in interpreting because they are very
robust in instances of multi-collinearity.
4.3.2 Canonical loadings. [SAS output: correlations between the with variables and
their canonical variables].
The loadings for the independent variables are: money supply (99 per cent),
consumer price index (98 per cent), exchange rate (94 per cent), treasury bills (85 per
cent), unemployment (55 per cent), mining index (51 per cent) and industrial index (1 per
cent). These fgures suggest that the frst four macroeconomic variables greatly
infuence the formation of the frst variate W
1
, resulting in the frst canonical correlation
model. Only the contributions of the mining and industrial indices contributions fall
below the average mark of 54.5 per cent (average cross loadings for the seven
macroeconomic variables). This implies that the two indices are insignifcant. The
loadings for the dependent variables are: Hwange (98 per cent), Barclays (95 per cent),
Falcon (93 per cent), Ariston (91 per cent), Border (85 per cent), Caps (72 per cent),
Bindura (62 per cent) and Cafca (37 per cent). The most infuential ones include: Hwange,
Barclays, Falcon, Ariston and Border because their loadings exceed the 79 per cent
average point. Analysis of other variates is no longer necessary because an overall
pattern has been deduced already.
4.3.3 Canonical cross-loadings. [SAS output: last two tables]. Again similar patterns
appear inthe cross-loadingvalues for M3 (98 per cent), CPI (97 per cent), ER(93 per cent) and
TB (85 per cent). These fgures support conclusions that were drawn earlier. If the
corresponding correlation values are squared, the percentage of the variance being
explained by the canonical variates is obtained. The values are: M3 (97 per cent), CPI (94 per
cent), ER (87 per cent), TB (72 per cent), U (29 per cent), MI (26 per cent) and II (0 per cent).
Hwange (97 per cent), Barclays (94 per cent), Falcon (92 per cent), Ariston (90 per cent),
Border (84 per cent), Caps (71 per cent), Bindura (61 per cent) and Cafca (36 per cent). The
squared values render Bindura and Cafca insignifcant.
4.3.4 Overall model ftting. From the analysis above, W
1
(money supply, consumer
price index, exchange rate and treasury bills) contribute signifcantly in maximizing
stock returns at ZSE. Thus, they should be incorporated into the fnal CCAmodel. When
developing the model, emphasis should be on the macroeconomic variables’ impact on
stock returns rather than the reverse. Thus, for the dependent variables, we include top
movers like V
1
(Hwange, Barclays, Border, Caps, Ariston, Bindura and Falcon).
The specifc model with regards to the research fndings is given by:
Maximize ?
1
? Maximize Corr (V
1
, W
1
)
Subject to
Var (V
1
) ? 1
Var (W
1
) ? 1
(3.9)
where V
1
?stock returns for Hwange, Barclays, Border, Caps, Ariston, Bindura, Falcon.
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W
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? [money supply, consumer price index, exchange rate, treasury bills].The
general canonical correlation model is of the form:
Maximize ?
n
? Max Corr (V
n
, W
n
)
Subject to
Var (V
n
) ? 1
Var (W
n
) ? 1
(4.0)
4.4.5 Model validation and diagnosis. The last stage involves validating the prescribed
model through sensitivity analysis. The canonical loadings are examined for stability of
the model ftted. Individual values for the independent variables are deleted from the
analysis. The results obtained after omitting one value from each of money supply,
unemployment and exchange rate show little or no signifcant changes of the weights,
loadings and cross-loadings. This implies that the ftted model is very signifcant even
if the population is changed.
5. Conclusion
From the research results, the following were found: CCA identifes dimensions among
stock returns and macroeconomic variables that maximize the relationship between the
dimensions. Through CCA, the research managed to fnd the strength of the relationship
between macroeconomic variables and stock returns for the Zimbabwe economy during
“harsh” economic conditions. A fairly good model was ftted because a redundancy
index of 0.68 is very high (for a comparable multiple regression). This means that the
variance of stock returns that is being accounted for by the macroeconomic variables is
also high.
Five major macroeconomic variables, namely, money supply, treasury bills,
exchange rate, consumer price index and unemployment were found to be signifcant in
maximizing stock returns at ZSE. Mining index and industrial production were rejected.
Thus, a general CCA model was found that can be used by ZSE upon further scrutiny.
The model constructed can be used for prediction purposes. Thus, the model can be used
by traders at ZSE, economic regulators and the government to manage portfolios. The
model gives insight into what macroeconomic factors to closely monitor for portfolios at
ZSE to yield maximum stock returns during an economic meltdown like what the
country experienced between 1990 and 2008.
6. Practical implications and future research guide lines
Investors and stock market regulators can use the canonical correlation model to
monitor macroeconomic trends because they affect stock price movements. New
investors can have an insight of areas of investment. They can monitor the performance
of some companies in similar types of business. The research results can assist the
government in crafting economic policies for the country based on the listed stocks’
reaction to movements of the macroeconomic variables. The CCA results enable
policymakers to simultaneously predict multiple stock returns from multiple
macroeconomic variables.
This research is limited to top trading counters and selected macroeconomic
variables during the research period at ZSE. Future researchers may include all the
listed companies as well as more macroeconomic variables into the model so that a
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broader view of ZSE is envisaged. There may be a need to incorporate other modelling
techniques such as the principal components analysis and factor analysis to
complement the CCA. For example, factor analysis can be considered with the intention
of discussing scale development. The CCA is a form of scale development. The
dependent and independent variates represent dimensions of the variable sets similar to
the relationship between them. But factor analysis maximizes the explanation (shared
variate) of the variable set. The results of the CCAmodel showa set of stock returns and
a set of macroeconomic variables being correlated. These two sets of variables can be
used to develop a multidimensional regression model for the ZSE. This is a matrix of all
regression models for each company and provides another dimension of analyzing the
stock market using multiple models.
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Appendix
Tables on canonical correlation analysis and SAS output
Table AI.
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Table AII.
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Table AIII.
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Table AIV.
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Corresponding author
Peter Mazuruse can be contacted at: [email protected]
To purchase reprints of this article please e-mail: [email protected]
Or visit our web site for further details: www.emeraldinsight.com/reprints
Table AV.
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