We start by estimating a static model.
\(y_t=\alpha + \theta x_t + \gamma_t\)
We then use this form an equilibrium for \(y_t\), \(y_t^*\).
\(y_t^*=\hat \alpha + \hat \theta x_t\)
The process depends on the difference from this equilibrium.
\(y_t-y_{t-1}=\beta (y_{t}^*-y_{t-1})+\epsilon_t\)
\(y_t-y_{t-1}=\beta (\hat \alpha + \hat \theta x_t -y_{t-1})+\epsilon_t\)
\(y_t=\beta \hat \alpha + \beta \hat \theta x_t + (1-\beta )y_{t-1}+\epsilon_t\)
\(y_t=\alpha y_{t-1}+(1-\beta )(y_{t}^*-y_{t-1})+\epsilon\)
The higher \(\beta\), the slower the adjustment.
If stationary, can we can use OLS.