# Multivariate forecasting

## Vector Autoregression (VAR)

### Vector Autoregression (VAR)

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:

$$y_t$$

$$y_t=c + \sum_{i=1}^pA_i y_{t-i}+\epsilon_t$$

## Structural models

### Autoregressive Distributed Lag (ARDL) model

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??

## ARMAX

### Error Correction Model

#### Static model

Like PAM we start with static estimator.

#### The ECM

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)

### 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.