# Question on Stationary & Cointegration Test (Augmented Dickey Fuller & Engle Granger test)

I'm performing the stationary and cointegration test on stock prices.

What I'm confused is

1) the difference between ADF stationary test and ADF cointegration test.

2) Also, in ADF stationary test, I(1) means 'non-statioanry' and thus shows the auto correlation, while I(0) means stationary such that it is more of mean-reverting (rather than moving one way long). am I correct?

3) Is my test design logical?: first test the stationarity in ADF stationary test, and then if so, test cointegration in ADF cointegration test. If certain assets are not stationary in ADF stationary test, then put them through the Engle-Granger test to identify whether the cointegration relationship between assets exists.

Answers will be very much appreciated, and HUGE thanks in advnace :-)

## 1 Answer

I will skip integration of order greater than 1 for simplicity.

1. The standard ADF test considers whether a univariate time series in integrated (denoted $I(1)$) or stationary (denoted $I(0)$). The null hypothesis is that the series is $I(1)$. If you reject the null, that favours the alternative of $I(0)$.
Meanwhile, the Engle-Granger procedure (which makes use of the ADF test on the way) considers whether a number of time series are cointegrated (denoted $CI(1,1)$) or not (denoted $CI(1,0)$). The null hypothesis is that the series are $CI(1,0)$, i.e. not cointegrated. If you reject the null, that favours the alternative of $CI(1,1)$.
An accessible yet thorough treatment of both ADF test and Engle-Granger procedure is given in Zivot & Wang "Modeling Financial Time Series with S-PLUS" sections 4.3.3 and 12.3. (freely available online).

2. Yes, that is about right (I will not pick on details as long as you got the correct intuition).

3. I did not quite get what you were doing, but here is what I suggest. First, test the individual series for stationarity using ADF test. If they are stationary, stop there; if they are integrated, test for cointegration using the Engle-Granger procedure.