We consider a vector of observables, not just one
Autoregressive (AR) model for a vector.
VAR(p) looks \(p\) back.
The AR(\(p\)) model is:
\(y_t=\alpha + \sum_{i=1}^p\beta y_{t-i}+\epsilon_t\)
VAR(\(p\)) generalises this to where \(y_t\) is a vector. We define VAR(\(p\)) as:
\(y_t\)
\(y_t=c + \sum_{i=1}^pA_i y_{t-i}+\epsilon_t\)
Include lagged y and lagged x (and current x)
If the processes are stationary, then we can use OLS. THIS IS A BROADER POINT! INTRO??
Like PAM we start with static estimator.
The ECM does a regression with first differences, and includes lagged error terms.
We start with a basic first-difference model.
\(\Delta y_t= \Delta x_t\)
We could also expand this to include laggs for both x and y. Here we don’t.
We know that long term \(y_t=\theta x_t\). We use the error from this in a first difference model.
\(\Delta y_t= \alpha \Delta x_t + \beta (y_{t-1}-\theta x_{t-1})\)
Page on identifying error terms
Also, page on Vector Error Correction Model (VECM)
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.