#### Estimating a static model

We start by estimating a static model.

$$y_t=\alpha + \theta x_t + \gamma_t$$

#### Equilibrium

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.