How to perform Johansen cointegration test?

By Divya Dhuria & Priya Chetty on September 18, 2018

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).

Testing cointegration

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:

  1. Eigen value statistic and
  2. 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.

Priya is the co-founder and Managing Partner of Project Guru, a research and analytics firm based in Gurgaon. She is responsible for the human resource planning and operations functions. Her expertise in analytics has been used in a number of service-based industries like education and financial services.

Her foundational educational is from St. Xaviers High School (Mumbai). She also holds MBA degree in Marketing and Finance from the Indian Institute of Planning and Management, Delhi (2008).

Some of the notable projects she has worked on include:

  • Using systems thinking to improve sustainability in operations: A study carried out in Malaysia in partnership with Universiti Kuala Lumpur.
  • Assessing customer satisfaction with in-house doctors of Jiva Ayurveda (a project executed for the company)
  • Predicting the potential impact of green hydrogen microgirds (A project executed for the Government of South Africa)

She is a key contributor to the in-house research platform Knowledge Tank.

She currently holds over 300 citations from her contributions to the platform.

She has also been a guest speaker at various institutes such as JIMS (Delhi), BPIT (Delhi), and SVU (Tirupati).



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