If a series is nonstationary in time series without a constant mean and constant variance, the regression results will be spurious. But regression results can be reliable when a linear combination of non-stationary series (dependent and independent) removes the stochastic trend and produces stationary residuals. Therefore, it is implied that variables are co-integrated. Co-integrated also assumes that there is the occurrence of stochastic non-stationary series, underlying two or more process (p).
To test cointegration, Johansen cointegration test is widely used which determines the number of independent linear combinations (k) for (m) time series variables set that yields a stationary process. The test gives the rank of cointegration.
The order of integration of a series is given by the number of times the series must be differenced in order to produce a stationary series. A series generated by the first difference is integrated of order 1 denoted as I(1). Thus, if a time series, is I(0), it is stationary; if it is I(1) then its change is stationary and its level is non-stationary. The equation for the cointegration test is given below:
Case I: When k=0, then p=m, it means variables in time series are not cointegrated.
Case II: 0 < k < m, 0 < p < m. This case indicates that variables are cointegrated.
Johansen cointegration test
There are many tests to show the long run association between the variables, one of them is the Johansen cointegration test. It is based on the maximum likelihood method and gives two main statistics:
- Eigen value statistic and
- Maximum statistic.
When the rank is zero it means there is no co-integration relationship and if the rank is one it means there is one co-integration equation and so on.
For example in our study, by using Johansen cointegration test, both Eigen value and trace statistics are used to examine the association of long-run relationship between FDI and GDP as well FDI and air pollution and water pollution indicators.