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I am reading this PDF:
https://warwick.ac.uk/fac/soc/economics/staff/gboero/personal/hand2_cointeg.pdf
where on pages 4 and 5 it says that if the residuals are stationary, the OLS regression is superconsistent even if the Y and X variables are non-stationary.
My question then is why is it necessary to estimate the Error Correction Model (Second step)? Isn't the stationarity of residuals a sufficient condition?
A rejection of the null of the Engle-Granger test indeed establishes cointegration (up to a type I error, of course).
But estimating the error correction model tells you interesting things about the economic question of interest. E.g., which variable adjusts to deviations from the equilibrium relationship in the previous period so as to restore convergenve towards the long-run cointegration vector.
Why negative $\alpha$? Suppose the error-correction equation is $\Delta y_t=\alpha(y_{t-1}-\beta x_{t-1})+e_t$. When $y_{t-1}-\beta x_{t-1}>0$, it means that the lagged value $y_{t-1}$ is "too large" relative to the cointegrating relationship, so that if $y_t$ is to contribute to restoring the equilibrium relationship, its change in $t$ must be negative, whence $\alpha<0$ is required.
