Method | Granger Causality Test

Table of contents

• 1. Terminology
• 2. Steps
• 3. References

1. Terminology

Granger causality testing is a statistical method used to assess whether one time series has forcasting or predictive power over another.

The AR model uses past values of Y to forecast its current value. If the predictive power significantly improves after adding past values of X, we say that X Granger-causes Y, which means the following holds in general:

where $I_{t-1}$ represents all past information of both Y and X, and $J_{t-1}$ represents only the past information of Y.

However, it is important to note that Granger causality only indicates a temporal relationship between the variables, not a causal one.

2. Steps

Assume $\{x_t\}$ and $\{y_t\}$ are two stationary time series. Three steps:

1) Determine the order p for the AR model of $\{y_t\}$:

2) Test the joint significance of $x_{t-j}$:

3) Make the conclusion: If we reject H0, it indicates that X is a Granger cause of Y; otherwise, it is not.

Some remarks:

• ⚠️⚠️⚠️The prerequisite of GCT is that all time series are stationary.
• As long as any lagged term of X is significant, X is a Granger cause of Y.
• The choice of lag q for X is not important, but there are some empirical guidelines. For instance, for annual data, 1 to 2 lagged terms may suffice; for quarterly data, 4 or 8; for monthly data, 6 or 12.
• In practice, when testing for Granger causality between X and Y, four possible outcomes may appear: ①only X is a Granger cause of Y, ②only Y is a Granger cause of X, ③both X and Y are Granger causes of each other, or ④there is no Granger causality between X and Y.
• Multivariate GCT = VAR

3. References

• Wooldridge, J. M. (2013). Introductory Econometrics: A Modern Approach (5th ed.). Mason, OH: Cengage Learning. Chapter 18.
• Granger Causality : Time Series Talk - YouTube
• Granger causality - Wikipedia
• https://mp.weixin.qq.com/s/AvJpXnzP4FYMv5vTohC74w