where µ is a constant and u t is a white noise process. To test for unit roots in our variables, we used
the Augmented Dickey Fuller (ADF) test. Using the
results of ADF, the null hypothesis showed that all
variables have unit roots. Electricity consumption (E)
is non-stationary (therefore the null hypothesis is
rejected).
4.4.2 Test for serial correlation
According to Samson (2015), autocorrelation is the
correlation of a time series with its own past and
future values. We used the Breusch-Godfrey LM test
(Damodar 2004) for both AR(p) and MA (q) error
structures as well as for the presence of lagged regression and explanatory variables. The null hypothesis
(Ho) is that there is no serial correlation of any order. If
the sample size is large enough, Breusch and Godfrey
have shown that:
(n − p)R
2 ∼ X
2
p .
(11)
Implying that asymptotically, n-p times the R
2 follows the chi-square distribution with PDF. If in an
application, (n-p) R
2 exceeds the critical chi-square
value at a chosen level of significance, we reject the
null hypothesis. Thus the null hypothesis is rejected if
p-value is less than 5%, in our case it is 0.00 so we
reject the null hypothesis.
4.4.3 Determining the appropriate lag length for
VECM model
The need for the lags arises because values in the past
affect today’s values for a given variable. This is to
say the variable in question is persistent. There are
various methods to determine how many lags to use.
The AIC was used to determine the appropriate lag
length given the large sample size of 44 observations.
The appropriate lag length is 3.
4.4.4 Cointegration test
This test is used to check if there exists a long-run
relationship between the study variables. Generally, a
set of variables is said to be cointegrated if a linear
combination of the individual series, which are I(d),
is stationary. Intuitively, if xt ∼ I(d) and yt ∼ I(d),
a regression is run, If the residuals, εt, are I(0), then
Et and Y t are cointegrated. We use Johansen’s (1988)
approach, which allows us to estimate and test for
the presence of multiple cointegration relationships.
The choice of lag length is made according to the
AIC criterion. In conclusion there is one cointegration rank (long-run relationship). When determining
lag structures of the data-generating processes (DGP),
we use the Johansen (1988) procedure to test the existence of long-run equilibrium relations using the trace
statistic test for cointegration, because our data are
based on rather small samples, the estimation procedure that we adopt accounts for the Bartlett correction
following Johansen (2000). The Johansen cointegration procedure does not reject the null hypothesis of
one cointegrating equation. The Johansen trace and
max test statistics suggest the existence of at least 1
cointegrating relationship between GDP and electricity consumption. Hence, we estimated the Vector Error
Correction Model (VECM) analysis. Why the VECM?
According to Hendry and Mizon (1978), this choice
of the econometric model is because of existence of
a cointegration problem. Since the error structure in
non-stationary in levels, the problem is estimated in a
first difference formulation. The Dickey Fuller test is
used to determine whether the remainder is stationary
Dickey Fuller, and the Augmented Dickey Fuller test
is applied on the least squares residual to implement
the Engel and Granger procedure.
4.4.5 Test for functional form
We may have a model that is correctly specified, in
terms of including the appropriate explanatory variables, yet commit functional form misspecification.
In this case, the model does not properly account for
the form of the relationship between dependent and
observed explanatory variables.
4.4.6 Test for heteroscedasticity
The error term is found to be homoscedastic using
the Breush Pagan test; this shows the stability of
the parameters using residual diagnostics to minimize
errors (or residuals). The error term is be independently and identically distributed (i.i.d). Using the
correlogram, the error term of the estimated model
is identified. This procedure of log transformation is
important because it stabilises the means, however the
means are also found to be nonstationary.
4.4.7 Test for normality
The Jacque Bera normality test was used to test for
normality, which variable is relevant to be expressed
as linear combination among other variables. Using the
Maximum Likelihood- Autoregressive Conditional
Heteroscedasticity (ML ARCH) the residuals were
normally distributed as shown in Figure 3 in the
appendices.
5 FINDINGS AND DISCUSSION
5.1 Findings
With respect to Eqs. (8a)–(8d), short-run causality is
determined by the statistical significance of the partial F-statistics associated with the corresponding right
hand side variables. The null hypothesis is of no longrun causality in each equation. Equations. (8a)–(8d)
were tested by the statistical significance of the tstatistics for the coefficient on the respective error
correction terms represented by λ. In terms of Equation (8b), both economic growth and real gross fixed
capital formation each have a positive and statistically
significant impact on renewable energy consumption in the short-run, whereas the labour force has a
statistically insignificant impact. For Equation (8c),
economic growth, renewable energy consumption, and
the labour force each have a positive and statistically
280
the Augmented Dickey Fuller (ADF) test. Using the
results of ADF, the null hypothesis showed that all
variables have unit roots. Electricity consumption (E)
is non-stationary (therefore the null hypothesis is
rejected).
4.4.2 Test for serial correlation
According to Samson (2015), autocorrelation is the
correlation of a time series with its own past and
future values. We used the Breusch-Godfrey LM test
(Damodar 2004) for both AR(p) and MA (q) error
structures as well as for the presence of lagged regression and explanatory variables. The null hypothesis
(Ho) is that there is no serial correlation of any order. If
the sample size is large enough, Breusch and Godfrey
have shown that:
(n − p)R
2 ∼ X
2
p .
(11)
Implying that asymptotically, n-p times the R
2 follows the chi-square distribution with PDF. If in an
application, (n-p) R
2 exceeds the critical chi-square
value at a chosen level of significance, we reject the
null hypothesis. Thus the null hypothesis is rejected if
p-value is less than 5%, in our case it is 0.00 so we
reject the null hypothesis.
4.4.3 Determining the appropriate lag length for
VECM model
The need for the lags arises because values in the past
affect today’s values for a given variable. This is to
say the variable in question is persistent. There are
various methods to determine how many lags to use.
The AIC was used to determine the appropriate lag
length given the large sample size of 44 observations.
The appropriate lag length is 3.
4.4.4 Cointegration test
This test is used to check if there exists a long-run
relationship between the study variables. Generally, a
set of variables is said to be cointegrated if a linear
combination of the individual series, which are I(d),
is stationary. Intuitively, if xt ∼ I(d) and yt ∼ I(d),
a regression is run, If the residuals, εt, are I(0), then
Et and Y t are cointegrated. We use Johansen’s (1988)
approach, which allows us to estimate and test for
the presence of multiple cointegration relationships.
The choice of lag length is made according to the
AIC criterion. In conclusion there is one cointegration rank (long-run relationship). When determining
lag structures of the data-generating processes (DGP),
we use the Johansen (1988) procedure to test the existence of long-run equilibrium relations using the trace
statistic test for cointegration, because our data are
based on rather small samples, the estimation procedure that we adopt accounts for the Bartlett correction
following Johansen (2000). The Johansen cointegration procedure does not reject the null hypothesis of
one cointegrating equation. The Johansen trace and
max test statistics suggest the existence of at least 1
cointegrating relationship between GDP and electricity consumption. Hence, we estimated the Vector Error
Correction Model (VECM) analysis. Why the VECM?
According to Hendry and Mizon (1978), this choice
of the econometric model is because of existence of
a cointegration problem. Since the error structure in
non-stationary in levels, the problem is estimated in a
first difference formulation. The Dickey Fuller test is
used to determine whether the remainder is stationary
Dickey Fuller, and the Augmented Dickey Fuller test
is applied on the least squares residual to implement
the Engel and Granger procedure.
4.4.5 Test for functional form
We may have a model that is correctly specified, in
terms of including the appropriate explanatory variables, yet commit functional form misspecification.
In this case, the model does not properly account for
the form of the relationship between dependent and
observed explanatory variables.
4.4.6 Test for heteroscedasticity
The error term is found to be homoscedastic using
the Breush Pagan test; this shows the stability of
the parameters using residual diagnostics to minimize
errors (or residuals). The error term is be independently and identically distributed (i.i.d). Using the
correlogram, the error term of the estimated model
is identified. This procedure of log transformation is
important because it stabilises the means, however the
means are also found to be nonstationary.
4.4.7 Test for normality
The Jacque Bera normality test was used to test for
normality, which variable is relevant to be expressed
as linear combination among other variables. Using the
Maximum Likelihood- Autoregressive Conditional
Heteroscedasticity (ML ARCH) the residuals were
normally distributed as shown in Figure 3 in the
appendices.
5 FINDINGS AND DISCUSSION
5.1 Findings
With respect to Eqs. (8a)–(8d), short-run causality is
determined by the statistical significance of the partial F-statistics associated with the corresponding right
hand side variables. The null hypothesis is of no longrun causality in each equation. Equations. (8a)–(8d)
were tested by the statistical significance of the tstatistics for the coefficient on the respective error
correction terms represented by λ. In terms of Equation (8b), both economic growth and real gross fixed
capital formation each have a positive and statistically
significant impact on renewable energy consumption in the short-run, whereas the labour force has a
statistically insignificant impact. For Equation (8c),
economic growth, renewable energy consumption, and
the labour force each have a positive and statistically
280