3.4 Estimation techniques
3.4.3 Panel cointegration test
43
44 techniques that are used for testing whether a long run relationship exists or not (Pedroni, 1995). According to Dunis & Ho (2005), the cointegration concept provides a sound methodology for modelling both the long run and short run dynamics in a system.
Furthermore, Alexiou et al. (2016) state that the cointegration methodology is primarily used when one wants to determine if spurious estimation results are evident. It was hypothesised by Johansen & Juselius (1990) that this test examines the null hypothesis of no cointegration in the variables against the alternative that there exists cointegration.
It has to also be verified first that all variables used in the study are integrated of a specific order in levels before the long run relationship can be identified (Christopoulos
& Tsiona, 2004).
The linear stationarity combination is sometimes referred to as the cointegrating vector, which indicates the long run relationship between variables (Gujarati, 2003). According to Costantini & Martini (2009), the Johansen’s VAR procedure and Pedroni’s heterogeneous panel cointegration are only capable of showing whether or not there is cointegration between the variables and also if there is a long run relationship.
The Kao test has the same approach as the Pedroni test as they are both based on the Engel-Granger (1987) two-step that is residual based cointegration test (Dritsakis, 2012;
Ahmad, 2015). Ahmad (2015) further stipulates that the Pedroni and Kao cointegration tests are commonly used to determine the long run association between respective variables used in the study. Such tests can be sensitive to the correct lag length selection in the VAR. Pedroni (1995) also states that many of these tests have inherently low power and it has been confirmed by Shiller & Perron (1985) that the frequency of data does not matter but the duration of the data does for the power of these tests.
3.4.3.1 Panel Johansen cointegration test
The panel Johansen’s cointegration test is proven suitable when dealing with multivariate time series data (Brooks, 2008; Gujarati & Porter, 2009). The cointegration test is performed after the order of integration between the variables has been identified through the stationarity test and after the optimum lag length has been determined. The
45 test is employed as it has been proven suitable also when dealing with multivariate time series data. A linear combination of two or more series can be stationary, despite being individually non-stationary. If there is cointegration of two or more-time series, it means that there is a long-run or equilibrium relationship between the two (Johansen, 1988).
There are two test statistics under the Johansen approach for cointegration, namely the trace equation and the max-eigenvalue equation, which are formulated as:
) 1 ln(
) (
1
g
r i
i
trace r
(3.13) )
1 ln(
) 1 ,
( 1
max
r
r
r
(3.14)
Where r is the number of cointegrating vectors under the null hypothesis,
i is the estimated value for the ith ordered eigenvalue from the matrix and Tis the number of usable observations. The number of cointegration vectors (r) is always smaller or equal to the number of endogenous variables (n). And the larger is the i
, the more large and negative will be ln(1 i)
. The trace test is a joint test that test the null hypothesis of r cointegrating vectors against an unspecified or general alternative hypothesis of r cointegrating vectors. On the other hand, the maximum eigenvalue test conducts separate tests on each eigenvalue and tests the null hypothesis of r cointegrating vectors against an alternative hypothesis of r1 cointegrating vectors (Brooks, 2008).
3.4.3.2 Pedroni panel cointegration test
The Pedroni panel cointegration test usually makes use of the long run variance, both the parametric and non-parametric kernel estimations (Dritsakis, 2012; Ahmad, 2015).
Pedroni proposed various tests for cointegration that allowed heterogeneous intercepts and trend coefficients across cross sections. Pedroni also described various methods that involved establishing statistics for testing the null hypothesis of no cointegration.
The regression is given as follows:
46
t i t mi mi t
i i t i t i i
it t X X X
Y 1 1, 2 2, ... , ,
(3.15)
WhereYand X are assumed to be I(1), t 1,...,Nand m1,...,M . i and iare parameters which can also be set as zero, are the individual and trend effects. When there is no cointegration under the null hypothesis, the residuals i,twill always tend to be I(1) (Gutierrez, 2003).
There are two kind of alternative hypothesis, namely the heterogeneous alternative where 1for all i and homogenous alternative (i )1for alli. According to Gutierrez (2003), Pedroni (1999) introduced seven residual based tests that allowed for heterogeneity among individual units of the panel and where no exogeneity requirements on the regressors in the cointegrating regressions are imposed. The residual based tests dealt with multiple regressors for the null of no cointegration in dynamic panels. Four of which are based on a within-dimension and three on the between dimension (Alexiou et al., 2016). All the seven tests can be formulated as:
it t i n n t
i t
i i
it X X X
Y 1 1,, 2 2,, ... ,,
(3.16) whereXi,tare referred to as the regressors for ncross sections. A regression can also be conducted using the above formula, equation (3.16) and can be written as:
t i t i i t
i, ,1Z,
(3.17)
Seven different statistics can be generated from the preceding estimation process, namely the panel , panel,and panel non-parametric t. Under the within- dimension framework, the null of no cointegration and the alternative of cointegration are usually tested as follows:
1
0 : i
H for all i (3.18)
1
1:i
H for all i (3.19)
47 The group of non-parametric t and group of parametric t, which fall in the between dimension framework of the panel, the alternative hypothesis states that H1 :i 1for at least one i, as the between dimension test allows for heterogeneity as it is less restrictive (Alexiou et al., 2016).
3.4.3.3 Kao panel cointegration test
The Kao and Pedroni cointegration tests follow the same approach, but the Kao test usually specifies the cross-section homogeneous coefficients and intercepts usually of the regressors first stage. Kao (1999) described the bivariate case as:
it it i
it X
Y for Yit Yit1Ui,t and Xit Xit1 it (3.20) Wheret 1,....Tand i 1,...N. And i is referred to as individual constant terms, being the slope of the parameter, it is the stationary disturbance terms, Yit and Xit are the integrated process of order one for all i (Kao, 1999; Gutierrez, 2003). Unlike the Pedroni auxiliary regression, Kao pooled auxiliary regression is given as:
it t i
it ,1
(3.21)
According to Gutierrez (2003), Kao derived the DF and the ADF two type tests as part of the cointegration tests in 1999. Which can be estimated from the following formula:
j it it P
j it j
it u u
U
1 1
(3.22) the residuals of uit
can be obtained from the equation below:
it it i
it X u
Y (3.23)
In which the specifications are used in the Kao cointegration test for the null and alternative hypothesis:
1
0 :
H (3.24)
1 :
HA (3.25)
48 Kao also introduced four kinds of DF type statistics and ADF test statistics, where two DF statistics are based on the assumption of strict exogeneity of the regressors and the other two do allow for the endogeneity of the regressors in respect to the errors used in equations. With the ADF test statistic, from the long-run conditional variances some of the nuisance parameters can be estimated (Kao, 1999; Gutierrez, 2003). All the tests, both the DF type test statistics and ADF type test statistic that Kao proposed, have asymptotic distributions that tend to converge to standard normal distribution N(0.1)as the T and also N (Gutierrez, 2003).
3.4.3.4 Johansen-Fisher panel cointegration test
In 1932, Fisher derived a combined test that used individual independent test results. In the Johansen-Fisher panel cointegration test the number of cointegrating vectors is determined by the trace statistics and the maximum-eigenvalue statistics (Ahmad, 2015).