**Applied Mathematics
**Vol.5 No.13(2014), Article ID:47884,6 pages
DOI:10.4236/am.2014.513195

A Trivariate Causality Test: A Case Study in Cameroon

Jean Gaston Tamba^{*}, Francis Djanna Koffi, Michel Lissouck, Max Ndame,
Ngwa Engelbert Afuoti

Department of Thermal and Energy Engineering, University Institute of Technology, University of Douala, Douala, Cameroon

Email: tambajeangaston@yahoo.fr

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 7 April 2014; revised 21 May 2014; accepted 1 June 2014

ABSTRACT

In this paper, we examine the causal relationship between diesel consumption, CO_{2}
emissions and GDP in Cameroon during the period 1975-2008. Cointegration and vector
error-correction modelling techniques are used in this study. ADF tests show that
the series, after logarithmic transformation, are non-stationary and integrated
of order one. This study finds the presence of a long-run equilibrium relationship
between the variables. The results of the Granger-causality tests for time series
have been estimated.

**Keywords:**Causality, Coitegration, VECM, Cameroon

1. Introduction

According to Ghosh [1] and Omri
[2] , there have been three streams of research to investigate the causal
relationship between carbon dioxide emissions, energy consumption and economic growth.
Investigating the causal relationship between economic growth and environmental
pollutants constitutes the first stream of research. Here, researchers study the
existence of environmental Kuznets curve [3] [4] . The causal relationship between energy consumption
and economic growth is the second stream of research. This relationship suggests
that energy consumption and economic growth may be jointly determined and the direction
of causality may not be determined a priori. Starting with the work of Kraft and
Kraft [5] , a number of studies examine the causal
relationship between energy consumption and economic growth
[6] [7] . Finally, the last stream of research
has emerged in the recent literature, which combines two approaches earlier by examining
dynamic relationship between CO_{2} emissions, energy consumption and economic
growth. The analysis of the causal relationship between CO_{2} emissions,
energy consumption and economic growth has been subject to many empirical studies
[2] [8]
[9] .

This study has been carried out on various samples of countries. This research shows that there is no clear conclusion about energy consumption, environmental pollutants and economic growth. We note that a wide range of econometric techniques and procedures have been used to test the validity of the relation between environmental pollutants, energy consumption and economic growth. The results and implications of these studies clearly depend on the underlying variables, data frequency, and the development stages of a country [8] . Thus, we remark that conclusion of these studies is various.

To date, no study has been carried out on the causal relationship between CO_{2}
emissions, energy (or diesel) consumption and economic growth in Cameroon. The aims
of this paper are therefore to describe the relationship between total CO_{2}
emissions, diesel consumption and economic growth, and to investigate the long run
and short run causality relationship based on the VECM between total CO_{2}
emissions, diesel consumption and GDP from 1975 to 2008 in Cameroon.

The remainder of this paper is organized as follows: the next section presents an overview of the proposed methodology and data descriptions. In Section 3, the empirical results are reported and the last section concludes the study.

2. Methodology and Data Descriptions

According to Engle and Granger [10] , the series and of a non-stationary linear combination (with the same order of integration) may be stationary. If such a stationary linear combination exists, the variables are considered to be cointegrated. However, the variables may have the property that a particular combination is stationary. The linear combination can be written as follows Equation (1).

(1)

where and are two constants term such that the variable is stationary, and will tend to vary together with time and can be subjected to temporary diversions, but cannot diverge without limit. Equation (1) is a long-run equilibrium relationship and measures the deviation with respect to the equilibrium value.

The first step tests for the order of integration of the natural logarithm of the variables using Augmented Dickey-Fuller (ADF) [11] . For the time series, ADF relationship is expressed as:

. (2)

where is the difference operator, is the auto-regressive lag length and and are the coefficients of interest. When these series are found to be non-stationary, we take first-difference and we apply the ADF tests again on the differenced data and so on.

The second step involves examining cointegration relationship among the variables using vector autoregressive (VAR) approach of Johansen [12] [13] and Johansen and Juselius [14] . We determine the number of longrun equilibrium relationships between integrated variables of the same order. Let be a vector variables integrated of order one of dimension. The representation VAR of order is given by:

. (3)

where are lag coefficient matrices, is a vector of disturbance terms assumed normal and independent with zero mean and non-singular variance-covariance matrix. and are vector of constants terms and trends terms respectively. We can rewrite this structural VAR in error form as:

. (4)

where and.

By examining the matrix, we can detect the existence of cointegrating relationship among the x variables. If the rank of is stationary and all linear combination of are integrated of order one. If, all variables are stationary. The most interesting case is, where r and n denote the rank of and the number of variables constituting the long-run relationship respectively. There exist matrices and, each with rank, such that and is stationary. is a matrix of error correction parameters and is interpreted as a matrix of cointegration vectors. Johansen’s VAR method can be formally tested with two common approaches, namely the maximum eigenvalue test and the trace test. The likelihood ratio statistics for the maximum eigenvalue test and the trace test are Equations (5) and (6) respectively:

. (5)

. (6)

where is the maximum time in the time series. Since the cointegration tests are sensitive to the choice of lag length, we use the Schwartz Information Criteria (SIC) to determine the optimal lag lengths.

Cointegration implies that causality exists between the series, but it does not indicate the direction of the causal relationship [6] . Therefore, we use the VECM to detect the direction of the causality. The causality based on the VECM has the advantage of giving a causal relationship even without any significant estimated coefficients [15] . Granger [16] sustains that if variables are non-stationary, but become stationary after the difference, and cointegrated, it becomes necessary to estimate a VECM for the multivariate causality test. The VECMs for this test can be specified accordingly as Ang [17] , Ang [18] , Odhiambo [19] and Ghosh [1] :

. (7)

(8)

(9)

where,
and
represent CO_{2} emissions, diesel consumption and economic growth in logarithmic
form respectively.
is the difference operator,
s are the parameters to be estimated and
are the serially uncorrelated error terms.
is the error correction term, which is derived from long run
cointegration relationship. In the Wald tests, chi-square
statistics and their probabilities were obtained to determine the short run causalities
in VECM [17] [18]
. The t-statistics on the coefficients of the lagged error-correction term indicates
the significance of the long run causal effect
[1] [19] . The optimum lag
length
is determined on the basis of SC. In our study the Granger causal relationship analysis
is given in Table 1 [19]
.

Our empirical study uses the time data of CO_{2} emissions, diesel consumption
and GDP for the 1975-2008 period in Cameroon. In this paper, CO_{2} emissions
measured by a metric tons CO_{2} and data are obtained from the World Development
Indicators produced by the World Bank [20] . Diesel
consumption and GDP are taken in Tamba et al.
[6] . The logarithm terms of these variables are used because the logarithmic
transformation leads to a more stable variance of data. LNCO2 is the logarithm of
CO_{2} emissions, LNDIE is the logarithm of diesel consumption and LNGDP
is the logarithm of GDP. The GDP is used as a proxy for economic growth. Figure 1 shows logarithmic transformation of the evolution
of total CO_{2} emissions, diesel consumption and development of Cameroon’s
GDP from 1975 to 2008.

3. Empirical Results

The results of the ADF tests on the integration properties of the CO_{2}
emissions, diesel consumption and GDP for Cameroon indicate that the series are
stationary in first difference. This shows that the LNCO2, LNDIE and LNGDP variables
are individually integrated of order one.

Table 1. Granger causal relationship analysis.

The cointegration analysis is typically applied to verify if there exists a long
run relationship between the variables. The results of the Johansen maximum likelihood
cointegration tests are presented in Table2 Table 2 presents maximum eigenvalues and trace statistics
and shows the cointegration relationship among variables. The number of cointegration
test is two according to the trace test and one by the maximum eigenvalue test at
the 5% significance level. In this paper, we are only interested in the first cointegrating
equation because of its ability to determine the impact of the explicative variables
under consideration of the CO_{2} emissions in the long run. Therefore,
we use the maximum eigenvalue tests.

Cointegration implies the existence of Granger-causality but it does not indicate
the direction on the causality relationship. The multivariate Granger-causality
tests based on the VECM are carried out to examine both long run and short run causality.
The results of the Granger-causality tests for time series based on the VECM are
shown in Table3 the results confirm the unidirectional
long run causality and no causality in the short run relationship between total
CO_{2} emissions and diesel consumption, both the bidirectional long run
and short run causality relationship between total CO_{2} emissions and
GDP, and the unidirectional long run causality and no causality in the short run
relationship between diesel consumption and GDP at the 5% level of significance
in Cameroon.

4. Conclusions and Policy Implications

This paper analyses the causal relationship between total CO_{2} emissions,
diesel consumption and GDP in Cameroon over the period 1975-2008. We began by testing
the order of integration of series by the ADF unit root test. Then we tested the
Johansen multivariate cointegration to determine the existence of a long run relationship
between variables. Finally, Granger-causality tests based on the VECM were employed.
VECM tests were used to estimate the direction of Granger-causality for the multivariate
cointegration data.

Figure 1. Logarithmic transformation of the variables from 1975 to 2008.

Table 2. Johansen and Juselius cointegration test.

^{a}denotes rejection of the hypothesis at the 5% significance level; ^{b}MacKinnon-Haug-Michelis
(1999) p-values.

^{a}Denotes significance level at 1%; ^{b}Denotes significance level
at 5%; ^{c}Denotes significance level at 10% respectively.

The results indicate that the time series are in first stationary difference. The Johansen multivariate cointegration tests concluded the existence of a long-run relationship between the variables. The Granger-causality tests based on the VECM show that there exist: (1) unidirectional and bidirectional causality between the variables in the long run and (2) no causality and bidirectional causality between the variables in the short run at the 5% level of significance.

In Cameroon, a government policy aimed at improving energy supply and economic growth
will inevitably have a positive impact on total CO_{2} emissions from an
environmental point of view. That is, “reinforce Cameroon energy demands with low
emission fuel and respect the terms of the UNFCCC by mitigating emissions while
ameliorating economic growth”. Hence, GDP growth and energy consumption will stimulate
total CO_{2} emissions.

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NOTES

^{*}Corresponding author.