Negative coefficient on the error correction term in an ECM Why should $\beta_2$ in the error correction model, $(Y_t – Y_{t−1}) = \beta_0 + \beta_1(X_{t−1} – X_{t−2}) + \beta_2(Y_{t−1} + (–\beta_3)X_{t−1}) + u_t$, be negative? I cannot locate any clear explanations in textbooks or presentation. All simply mention that it is negative.

If the $\beta_2$ term is not negative then there is no "corrective" mechanism, so you don't have a valid Error Correction Model (ECM). What "corrective" mechanism?
When $Y$ is above the level suggested by the cointegration relationship  (call this a case of a positive "error") at time $t-1$ (when $Y_{t−1}>\beta_3X_{t−1}$) , the negative sign of $\beta_2$ when multiplied with a positive number $(Y_{t−1}-\beta_3X_{t−1}>0)$ will give you a negative number. Therefore the contribution of the negative $\beta_2$ term is to push in $Y$ between $t-1$ and $t$ down towards restoring the equilibrium; down towards correcting the "error".
When $Y$ is below the level suggested by the cointegration relationship  (call this a case of a negative "error") at time $t-1$ (when $Y_{t−1}<\beta_3X_{t−1}$), the negative sign of $\beta_2$ when multiplied with a negative number $(Y_{t−1}-\beta_3X_{t−1}<0)$ will give you a positive number. Therefore the contribution of the negative $\beta_2$ term is to push in $Y$ between $t-1$ and $t$ up towards restoring the equilibrium; up towards correcting the "error".
This is why it is called an Error Correction Model.
