Granger Causality — Time Series Causality Testing

Granger causality tests whether past values of one time series help predict future values of another. It is a statistical notion of causality, not true causal inference.

Formal Definition

Consider forecasting using its own past and the past of :

Hypothesis Test

| Hypothesis | Conclusion |

|-----------|-----------|

| : | does NOT Granger-cause |

| : At least one | Granger-causes |

VAR Framework

Granger causality is naturally tested within a Vector Autoregression (VAR) model.

Important Limitations

| Limitation | Explanation |

|-----------|------------|

| Predictive only | Tests statistical predictability, not mechanisms |

| Sensitive to lags | Results can change with different lag lengths |

| Linear only | Standard test assumes linear relationships |

| Stationarity | Series should be stationary or cointegrated |

| Omitted variables | May detect spurious causality if Z is missing |

Python Implementation

import numpy as np

import pandas as pd

from statsmodels.tsa.api import VAR

from statsmodels.tsa.stattools import grangercausalitytests



np.random.seed(42)



# Simulate correlated time series

n = 300

x = np.zeros(n)

y = np.zeros(n)

for t in range(1, n):

    x[t] = 0.5 * x[t-1] + np.random.randn()

    y[t] = 0.3 * x[t-1] + 0.4 * y[t-1] + np.random.randn()



data = pd.DataFrame({'Y': y, 'X': x})



# Granger causality test: X -> Y

print("X Granger-causes Y:")

gc_results = grangercausalitytests(data[['Y', 'X']], maxlag=5, verbose=True)



# VAR approach

model = VAR(data)

lag_order = model.select_order(maxlags=5)

print(f"\nSelected lag order: {lag_order.selected_orders['aic']}")



results = model.fit(maxlags=5)

print(results.summary())

Worked Example

Key Takeaways

Related Topics

• See ARIMA Models for univariate time series modeling

• See Causal Inference for methods that establish true causation

• See Vector Autoregression for multivariate time series models