# Partial Adjustment Model (PAM)

## Partial Adjustment Model

### Partial Adjustment Model

#### 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.