Dickey fuller/ unit-root test controversy The regression equation of dickey fuller test is defined as:
$$\Delta y_t = py_{t-1}+e_t$$
Trends and constants may also be added.The test statistic can be derived in the following way:
$$y_t=\beta y_{t-1}+ u_t \space \space \space \space\space | -y_{t-1}$$
$$\Delta y_t=(\beta-1)y_{t-1}+u_t$$
Where the $(\beta-1)$ is the p in the first equation. 
Just one question, if you actually run the two different regressions. The coefficient of the first equation won't be the same as the coefficient of the 2nd+1. I am not sure why this is as the equations are exactly the same. Any thoughts? Perhaps the coefficients would be the same if that was actually the true population model. Given this, are there some shortcomings in the test when applying to empirical data?
(Same applies to Johansen test, Engle Granger, and others with the same principle).