Whats is Cointegration?

-In 1987 shortly proceeding the inspirational publishing of Spurious Regression by Paul Newbold and Granger, Nobel laureates Robert Engle and Clive Granger introduced the concept of cointegration. A cointegration test ascertains whether a correlation exists among an excess of two long term time series. Cointegration tests identify events where two or more non-stationary or dynamic time series integrate, so that they cannot delineate from the long-term mean..  These tests, such as the Engle-Granger Two-Step Method, Johansen Test, Trace test, and Maximum Eigenvalue test measure the  sensitivity of two variables to a common average during a certain period of time.

-The Engle-Granger Two-step method entails firstly modeling residuals predicated on static regression and subsequently testing them for any semblance of unit roots. This method uses the Augmented Dickey-Fuller Test and others like it to test for stationary units in the time series. The Engle-Granger method exposes the stationarity of residuals in the eventuality that the time series confers cointegration. Unfortunately, this method is limited in capacity to identify cointegration within exclusively two variables as it may otherwise display an excess of two cointegrating relationships. Additionally, it is a single equation model and thus further restricted in its capacity and application due to a lack of a second equation for alteration or to account for variability and other parameters. The Engle-Granger Two-step method is applied  using softwares such as MATLAB and STAT to determine the presence of cointegration within a set of variables.

-The Johansen Test is simply an additional improved adaptation of the Engle-Gangeer Test that avoids the complication of designating a dependent variable and other errors that ensue during the transition from the initial step which preclude the detection of multiple cointegrating vectors. This test The Johansen Test confers two forms: Trace tests which evaluate the number of linear combinations in time series data, and Maximum Eiganvalue tests which are non-zero vector that change by a scalar factor upon the application of a linear transformation. This test is subject to asymptotic properties, or a large sample size; thus,  Auto Regressive Distributed Lags are often used to supplant a small sample size as it may be insufficient or inaccurate.

-The Philips-Ouliaris Test is an improvement upon the Engle-Gangeer test in that it accounts for supplementary variability derived from the approximate precision of residuals that were conflated with parameter in previous Cointegration Tests published preceding 1987. It is essentially invariant to the normalization of the cointegration relationship, or which variable is implemented as the dependent in the test. This test incorporates asymptotic distributions dependent upon the number of deterministic trends and variables that cointegration tests. Data Scientist frequently substitute these Phillips-Ouliaris distributions and critical values with critical values generated from simulations.

Downloadable reference: https://www.bauer.uh.edu/rsusmel/phd/ec2-7.pdf