Causation: Meaning, Criticisms & Real-World Uses

What Is Granger Causality?

Granger causality is a statistical concept within econometrics and time series analysis that assesses whether one time series is useful in forecasting another. Unlike the philosophical notion of "true causation," Granger causality focuses on predictive ability: if past values of one variable, X, provide statistically significant information for predicting future values of another variable, Y, beyond what past values of Y alone can offer, then X is said to "Granger-cause" Y. This framework helps analysts understand the directional flow of influence between variables in dynamic systems, particularly in financial markets.

History and Origin

The concept of Granger causality was introduced by British econometrician Sir Clive W.J. Granger in a seminal 1969 paper titled "Investigating Causal Relations by Econometric Models and Cross-Spectral Methods," published in Econometrica.¹² Granger's work provided an operational definition of causality that could be tested using empirical data, moving beyond traditional simultaneous equations models.¹⁰, ¹¹ His methodology offered a way to infer a temporal relationship between variables, suggesting that if a past event helps predict a future event, there's a form of predictive causality. For his groundbreaking contributions to time series analysis, including his work on Granger causality and cointegration, Granger was awarded the Nobel Memorial Prize in Economic Sciences in 2003.⁹ Granger himself noted that he adapted a definition of causality proposed by mathematician Norbert Wiener into a practical form.⁸

Key Takeaways

Granger causality is a statistical test determining if one time series is useful in forecasting another.
It is based on the idea that causes must precede their effects in time and provide unique predictive power.
The test involves regression analysis to examine the predictive power of lagged values.
Granger causality does not imply "true" causality in a philosophical sense but rather "predictive causality" or "precedence."
Its application is widespread in economics, finance, and other fields involving multivariate time series data.

Formula and Calculation

The test for Granger causality typically involves performing two regression analysis equations. Consider two stationary time series, (X_t) and (Y_t).

First, regress (Y_t) on its own past values:

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

Next, regress (Y_t) on its own past values and past values of (X_t):

Y_t = \beta_0 + \sum_{i=1}^{p} \beta_i Y_{t-i} + \sum_{j=1}^{q} \gamma_j X_{t-j} + \epsilon_{2t}

Where:

(Y_t) is the current value of the dependent variable.
(X_t) is the current value of the independent variable.
(\alpha_0) and (\beta_0) are constants.
(\alpha_i) and (\beta_i) are coefficients for the lagged values of (Y).
(\gamma_j) are coefficients for the lagged values of (X).
(\epsilon_{1t}) and (\epsilon_{2t}) are the error terms.
(p) and (q) represent the optimal number of lags, often determined by information criteria like AIC or BIC.

To test if X Granger-causes Y, one performs an F-test (a form of hypothesis testing) on the null hypothesis that all (\gamma_j) coefficients in the second equation are jointly equal to zero. If the null hypothesis is rejected, it suggests that past values of X provide significant information for predicting Y, thus X Granger-causes Y. A similar test can be run to check if Y Granger-causes X.

Interpreting Granger Causality

Interpreting Granger causality involves understanding its probabilistic and predictive nature rather than a direct cause-and-effect link. If the test indicates that variable X Granger-causes variable Y, it means that past values of X contribute significantly to the predictive modeling of Y, even after accounting for Y's own past behavior. This can suggest a lead-lag relationship or an informational flow from X to Y.

For instance, if interest rates Granger-cause stock prices, it implies that historical interest rate movements help in predicting future stock price movements. This insight can be valuable for developing investment strategies or informing risk management decisions. However, it does not necessarily mean that changes in interest rates directly "cause" changes in stock prices in a fundamental, mechanistic way; rather, it suggests a strong temporal predictive relationship. When applied in data analysis, it helps identify which variables might serve as leading indicators for others.

Hypothetical Example

Consider a hypothetical scenario involving two financial time series: the Consumer Sentiment Index (CSI) and Retail Sales Growth (RSG). A financial analyst wants to determine if changes in consumer sentiment precede and help predict changes in retail sales.

Collect Data: The analyst gathers monthly data for both CSI and RSG over several years.
Ensure Stationarity: Before applying Granger causality, the analyst verifies that both time series are stationary. If not, they apply differencing or other transformations.
Run Regressions:
- Model 1 (Null Hypothesis): Regress RSG on its own lagged values (e.g., past 3 months of RSG).
  $\text{RSG}_t = c_1 + \alpha_1 \text{RSG}_{t-1} + \alpha_2 \text{RSG}_{t-2} + \alpha_3 \text{RSG}_{t-3} + \epsilon_{1t}$
- Model 2 (Alternative Hypothesis): Regress RSG on its own lagged values AND lagged values of CSI (e.g., past 3 months of CSI).
  $\text{RSG}_t = c_2 + \beta_1 \text{RSG}_{t-1} + \dots + \beta_3 \text{RSG}_{t-3} + \gamma_1 \text{CSI}_{t-1} + \dots + \gamma_3 \text{CSI}_{t-3} + \epsilon_{2t}$
Perform F-test: The analyst compares the explanatory power of Model 2 against Model 1. If the F-test indicates that the lagged CSI terms ((\gamma_1, \gamma_2, \gamma_3)) are jointly statistically significant (i.e., not all equal to zero), then the null hypothesis (that CSI does not Granger-cause RSG) is rejected.

In this example, if the test is significant, it would suggest that past consumer sentiment values contain information useful for predicting future retail sales growth beyond what can be predicted solely from past retail sales growth. This could inform economic forecasts or business strategies.

Practical Applications

Granger causality has numerous practical applications in economics and finance. It is widely used in macroeconomic indicators to study relationships between variables such as inflation, interest rates, GDP growth, and unemployment. For instance, researchers might use it to investigate whether changes in money supply Granger-cause changes in inflation.

In financial markets, analysts employ Granger causality to understand interdependencies between asset prices. It can be applied to assess if volatility in one market (e.g., bond market) Granger-causes volatility in another (e.g., equity market), which is crucial for portfolio management and diversification. Additionally, it helps in identifying leading economic indicators for business cycle analysis. While the test is a statistical tool, its findings can contribute to broader economic discussions on policy and market behavior. The framework has been applied to various domains beyond economics, including neuroscience and genomics, demonstrating its versatility in analyzing time series data.⁷

Limitations and Criticisms

Despite its widespread use, Granger causality has several important limitations and has faced criticism. Primarily, it identifies "predictive causality" rather than true, underlying causal mechanisms. This means that while X might help predict Y, it doesn't necessarily imply that X is the fundamental cause of Y; a third, unobserved variable might be driving both. This issue, known as spurious causality or confounding, can arise if important variables are omitted from the model.⁶

Another criticism is that the original formulation of Granger causality primarily relies on linear relationships.⁵ While extensions for nonlinear and non-Gaussian observations exist, they are often more complex to implement.³, ⁴ The test is also sensitive to the choice of lag length and the stationarity of the time series involved. Furthermore, Granger causality is a bivariate test by default; applying it to high-dimensional systems without accounting for all relevant variables can lead to misleading inferences.² The very philosophical notion of causality in economics is complex and debated, and statistical tests like Granger causality are tools for understanding relationships rather than definitive proofs of cause and effect.

Granger Causality vs. Correlation

Granger causality and correlation are distinct concepts, though both relate to the relationship between variables. Their primary differences lie in the information they convey:

Feature	Granger Causality	Correlation
Concept	Predictive relationship; past values of one variable help forecast another.	Measures the degree to which two variables move together (linearly).
Direction	Implies a temporal direction (X leads Y, or Y leads X, or both).	No inherent direction; merely indicates co-movement.
Time	Explicitly incorporates time lags and the temporal ordering of events.	Typically a contemporaneous measure; does not consider time lags in its basic form.
Implication	Suggests X provides incremental information about Y's future movements.	Suggests a shared tendency to increase or decrease, but not necessarily predictive.
"Causation"	Often described as "predictive causality" or "precedence," not true philosophical causation.	"Correlation does not imply causation" is a fundamental principle.

While a high correlation between two variables might prompt an investigation into a potential Granger causal relationship, correlation alone cannot establish it. Granger causality provides a more nuanced understanding by considering the temporal dimension, indicating whether one variable's past movements add predictive power to another, beyond mere co-movement.

FAQs

What does it mean if X Granger-causes Y?

If X Granger-causes Y, it means that past values of X contain statistically significant information that helps predict future values of Y, even after considering the past values of Y itself. It suggests that X has predictive power over Y.

Can Y Granger-cause X at the same time X Granger-causes Y?

Yes, it is possible for a bidirectional relationship to exist, where X Granger-causes Y, and simultaneously, Y Granger-causes X. This indicates a feedback loop or a dynamic interplay between the two variables.

Does Granger causality prove that one variable "causes" another?

No, Granger causality does not prove "true" or philosophical causation. It only indicates a statistically significant predictive relationship. Other unobserved factors might be influencing both variables, leading to a seemingly causal link. It's often referred to as "predictive causality" or "precedence."

Is Granger causality only used in economics and finance?

While it originated in econometrics, Granger causality is now applied in various fields beyond economics and finance, including neuroscience, climate science, genomics, and social sciences, to analyze the predictive relationships between time series data.

What happens if the time series are not stationary when testing for Granger causality?

If the time series are not stationary, the results of the Granger causality test can be spurious or misleading. Non-stationary data can lead to inflated t-statistics and F-statistics, suggesting a relationship where none truly exists. It is crucial to transform non-stationary series (e.g., through differencing) into stationary ones before conducting the test.¹