# Multivariate time series

## Multiple time series

### Cointegration

If we have multiple variables, we can explore the order of integration of linear combinations.

If two series have time trends, a linear combination of them could remove this.

### Exogeneity

#### Contemporaneous exogeneity

$$Cov(x_{it},u_{it})=0$$

#### Strict exogeneity

$$Cov (x_{is}, u_{it})=0)$$

This is stronger than contemporeous, all periods.

Shocks don’t affect future outcomes.

#### Sequential exogeneity

Sequential exogeneity: a bit looser than strict exogeneity. only holds when $$s\le t$$.

So shocks can affect, but only in future.

### Introduction

Weak stationary processes can be decomposed to a deterministic and a stochastic component.