Frédéric Mirindi

Granger Causality: Predictive Relationships in Time Series

Posted on | Category: Econometrics, Time Series

Illustration of Granger causality in time series
Granger causality helps reveal predictive directionality in time series data.

In economics and data science, we often want to know if changes in one variable can help predict changes in another. Granger causality is a statistical concept that tests whether past values of one time series contain information that helps forecast another series, above and beyond the information contained in its own past. It is a cornerstone for time series analysis in macroeconomics, finance, neuroscience, and many other fields.

What is Granger Causality?

Proposed by Nobel laureate Clive Granger in 1969, Granger causality is not about true causality, but about predictive causality. If including lagged values of $X$ improves the forecast of $Y$, then $X$ is said to "Granger-cause" $Y$. This is tested by comparing models with and without $X$'s lags.

VAR Model for Granger Causality: $$ \begin{cases} Y_t = \alpha_0 + \sum_{i=1}^p \alpha_i Y_{t-i} + \sum_{i=1}^p \beta_i X_{t-i} + \epsilon_t \\ X_t = \gamma_0 + \sum_{i=1}^p \gamma_i X_{t-i} + \sum_{i=1}^p \delta_i Y_{t-i} + \eta_t \end{cases} $$

If the coefficients $\beta_i$ are jointly significant, $X$ Granger-causes $Y$.

Step-by-Step Workflow

  1. 1. Data Collection Gather relevant time series data for all variables of interest.
  2. 2. Stationarity Testing Check each series for stationarity (e.g., using the Augmented Dickey-Fuller test). If non-stationary, difference or transform the data.
  3. 3. Lag Length Selection Determine the optimal lag order (p) using criteria such as AIC or BIC.
  4. 4. Model Estimation Estimate a vector autoregression (VAR) model including all relevant lags.
  5. 5. Granger Causality Test Test whether lagged values of X help predict Y (and/or vice versa), typically using an F-test or Wald test.
  6. 6. Interpretation & Validation Interpret results (statistical significance ≠ true causality) and validate with out-of-sample checks or diagnostics.
This workflow ensures robust and interpretable Granger causality findings.

Example: Consumer Confidence and Retail Sales

Suppose you want to know if changes in consumer confidence help predict retail sales. If adding lagged values of consumer confidence to your model for retail sales significantly improves forecast accuracy, then consumer confidence Granger-causes retail sales. This is a typical application in macroeconomic forecasting.

Best Practices and Pitfalls

  • Ensure stationarity: Use differencing or transformation to stabilize mean/variance.
  • Optimal lag selection: Use AIC/BIC, cross-validation, and domain knowledge.
  • Check for cointegration: If series are non-stationary but cointegrated, use VECM-based causality tests.
  • Interpret with caution: Granger causality is predictive, not proof of true causality—hidden confounders may exist.

Interpreting the Test

F-Statistic p-value Interpretation
> 3.0 < 0.05 Strong evidence of Granger causality
1.5-3.0 0.05-0.1 Marginal evidence
< 1.5 > 0.1 No significant causality

References

  1. Granger, C.W.J. (1969). "Investigating Causal Relations by Econometric Models and Cross-spectral Methods." Econometrica, 37(3), 424–438.
  2. Number Analytics. The Ultimate Guide to Granger Causality Testing (2025).
  3. Wikipedia. Granger causality.
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