Granger Causality Analysis

The Granger Causality Analysis (also GCA) is a statistical concept used to determine if a timeseries can predict (Granger-cause) another. It was proposed by Clive Granger in 1969 and has been used widely in economics. However in recent years it has become popular in neuroscience.

Concept

If a Timeseries X₁ Granger-causes another X₂ then information from X₁ help predict future values from X₂ better than information from X₂ alone. For example if housing prices in an area suddenly increase then housing prices in an neighboring area also rise after a little lag, due to the geographically close location of the adjacent area. This means an increase in housing prices in the first area Granger-causes an increase in housing prices in a neighbouring area or you can predict the increase in the neighboring area better by including past housing prices of the first area.

Mathematical Method for multiple timeseries

In general, Granger causality between two stationary timeseries X_t and Y_t can be described by the following autoregressive model:

Y_t = \sum^{p}_{i=1} A_i X_{(t-i)} + \sum^{p}_{i=1} B_i Y_{(t-i)} + u_{YX}(t)

X_t = \sum^{p}_{i=1} A_i'X_{(t-i)} + \sum^{p}_{i=1} B_i'Y_{(t-i)} + u_{XY}(t)

A_i, A_i', B_i, B_i' are therefore autoregressive coefficients, which describe the influence of past data from both timeseries on the current data of one timeseries. p is the order of the autoregressive model, i.e. how many past timesteps are considered. The residuals u_{XY} and u_{YX} are the residual values, which could not be predicted by past data points from either timeseries. Considering the ratio of both residuals one can extract the Granger causality from Y_t to X_t as the following:

GC_{Y\rightarrow X} = ln \big( \frac{var(u_{YX})}{var(u_{XY})} \big)

This results in an n x n matrix for n timeseries.

Resources

• A detailed review of the theory and application of G-causality can be found in Ding et al. (2006)^[1]

References