Granger causality is a statistical concept that measures the predictive power of one time series over another. It was proposed by the Nobel laureate economist Clive Granger in 1969, and it has been widely used in various fields such as economics, neuroscience, and climatology.
The idea behind Granger causality is simple: if a variable X can help us forecast the future values of another variable Y, better than using the past values of Y alone, then we say that X Granger-causes Y. For example, if we want to predict the stock price of Apple, and we find that the stock price of Microsoft can improve our prediction accuracy, then we can say that Microsoft Granger-causes Apple.
However, Granger causality does not imply true causality in the sense of cause-and-effect. It only captures the temporal precedence and correlation between two variables, but it does not account for other factors that may influence them. For instance, Microsoft may not have any direct impact on Apple’s performance, but they may both be affected by some common market trends or events. Therefore, Granger causality should be interpreted with caution and supplemented with other methods or evidence.
To test for Granger causality, we need to perform a statistical hypothesis test using a regression model. The null hypothesis is that X does not Granger-cause Y, which means that adding lagged values of X to the model does not improve the prediction of Y. The alternative hypothesis is that X does Granger-cause Y, which means that adding lagged values of X to the model does improve the prediction of Y. We can use an F-test or a t-test to compare the models and reject or accept the null hypothesis based on a significance level.
The choice of lag length is an important issue in Granger causality testing. If we use too few lags, we may miss some relevant information from the past. If we use too many lags, we may introduce noise and multicollinearity into the model. There are various criteria and methods to select the optimal lag length, such as the Akaike information criterion (AIC), the Bayesian information criterion (BIC), or cross-validation.
Granger causality is a useful tool for exploring the relationships between time series variables, but it has some limitations and assumptions that need to be considered. First, it assumes that the time series are stationary, which means that their statistical properties do not change over time. If the time series are non-stationary, they need to be transformed or differenced before applying Granger causality. Second, it assumes that there are no hidden or confounding variables that may affect both X and Y. If there are such variables, they need to be included in the model or controlled for by other methods. Third, it assumes that the relationship between X and Y is linear and additive. If there are non-linear or interactive effects between X and Y, they need to be captured by using more complex models or extensions of Granger causality.
In conclusion, Granger causality is a statistical concept that measures the predictive power of one time series over another. It does not imply true causality in the sense of cause-and-effect, but it can reveal some interesting patterns and associations between variables. It requires some assumptions and conditions to be valid, and it can be extended or modified to deal with more complex situations.