Granger Causality — Time Series Causality Testing

Statistics

Testing Whether One Time Series Predicts Another

Granger causality tests whether past values of one series improve predictions of another. It's a statistical notion of predictive causality that reveals lead-lag relationships in temporal data.

Economics — Test whether money supply growth Granger-causes inflation
Finance — Detect lead-lag relationships between stock markets across time zones
Neuroscience — Identify information flow directions between brain regions

If knowing X's past helps predict Y's future, X Granger-causes Y — a powerful test of temporal influence.

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.

DfGranger Causality

A time series $X$ Granger-causes $Y$ if past values of $X$ contain information that improves the prediction of $Y$ beyond what past values of $Y$ alone provide.

Formal Definition

Consider forecasting $Y_t$ using its own past and the past of $X$ :

Restricted Model

Y_t = \alpha_0 + \sum_{i=1}^{p}\alpha_i Y_{t-i} + \varepsilon_t

Here,

$\alpha_i$ =Coefficient on lagged Y
$p$ =Number of lags
$\varepsilon_t$ =Error term with variance $\sigma_1^2$

Unrestricted Model

Y_t = \alpha_0 + \sum_{i=1}^{p}\alpha_i Y_{t-i} + \sum_{j=1}^{p}\beta_j X_{t-j} + \varepsilon_t

Here,

$\beta_j$ =Coefficient on lagged X
$\varepsilon_t$ =Error term with variance $\sigma_2^2$

Testing Granger Causality

If $X$ Granger-causes $Y$ , then $\beta_j \neq 0$ for at least one $j$ , and $\sigma_2^2 < \sigma_1^2$ .

Hypothesis Test

Granger Causality F-Test

F = \frac{(\text{RSS}_r - \text{RSS}_{ur})/p}{\text{RSS}_{ur}/(T-2p-1)}

Here,

$RSS_r$ =Residual sum of squares from restricted model
$RSS_{ur}$ =Residual sum of squares from unrestricted model
$p$ =Number of lags tested
$T$ =Sample size

| Hypothesis | Conclusion |

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

| $H_0$ : $\beta_1 = \cdots = \beta_p = 0$ | $X$ does NOT Granger-cause $Y$ |

| $H_1$ : At least one $\beta_j \neq 0$ | $X$ Granger-causes $Y$ |

VAR Framework

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

VAR(p) Model

\begin{bmatrix} Y_t \ X_t \end{bmatrix} = \mathbf{c} + \sum_{i=1}^{p}\mathbf{A}_i \begin{bmatrix} Y_{t-i} \ X_{t-i} \end{bmatrix} + \begin{bmatrix} \varepsilon_{1t} \ \varepsilon_{2t} \end{bmatrix}

Here,

$\mathbf{A}_i$ =2×2 coefficient matrix at lag i
$\mathbf{c}$ =Constant vector

Important Limitations

Granger Causality ? True Causality

Granger causality only tests predictive dependence, not true causal mechanisms. A significant result means:

$X$ is useful for forecasting $Y$
It does NOT mean $X$ causes $Y$
A third variable $Z$ could cause both $X$ and $Y$
Results are sensitive to the lag length chosen

| 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

Example: GDP and Unemployment

Testing whether GDP growth Granger-causes changes in unemployment:

ADF tests: Both series are non-stationary -> first differences are stationary
VAR lag selection: AIC suggests 2 lags
Granger test ( $H_0$ : GDP does not Granger-cause unemployment):
- F-statistic = 8.32, p-value = 0.0003
- Reject $H_0$ : GDP growth helps predict unemployment changes
Reverse test ( $H_0$ : Unemployment does not Granger-cause GDP):
- F-statistic = 1.45, p-value = 0.236
- Fail to reject $H_0$ : Unemployment does not predict GDP

Conclusion: GDP Granger-causes unemployment, but not vice versa.

Key Takeaways

Summary: Granger Causality

Granger causality tests whether $X$ improves predictions of $Y$ beyond $Y$ 's own past
It is a statistical test, not evidence of true causation
Test within a VAR framework using F-tests or likelihood ratio tests
Results depend on lag selection — always test multiple lag lengths
Both series should be stationary (or cointegrated if non-stationary)
Limitations: linear, predictive only, sensitive to omitted variables

Granger Causality — Time Series Causality Testing

Granger Causality — Time Series Causality Testing

Testing Whether One Time Series Predicts Another

DfGranger Causality

Formal Definition

Restricted Model

Unrestricted Model

Hypothesis Test

Granger Causality F-Test

VAR Framework

VAR(p) Model

Important Limitations

Python Implementation

Worked Example

Example: GDP and Unemployment

Key Takeaways

Summary: Granger Causality

Related Topics

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