# granger causality

## 1. linear regression

- \(y\) and \(x\) are two time sries.
- We want to test whether \(x\) causes \(y\)
- consider the time series \(y_t\) and the time series \(x_t\)
- choose a lag value \(m\), and make an auto-regressive linear regression
- \(y_t = a_0 + a_1 y_{t-1} \cdots a_{m+1} y_{t-m}\)

- Now add lagged values of \(x\) to this regression
- \(y_t = a_0 + a_1 y_{t-1} \cdots a_{m+1} y_{t-m} + b_0 + b_1 x_{t-1} + b_{m+1} x_{t-m}\)
- first check that the regression is a good fit (ex. using an F-test)
- then, check whether the coefficients \(b_i\) are significant according to their t-statistic
- if any of them are, then reject the null hypothesis that \(x\) does not Granger-cause \(y\)