Determining causal relationships is a central goal in econometrics. While true causality is difficult to establish definitively with non-experimental data, time series analysis offers tools to investigate predictive relationships. One of the most well-known concepts is Granger causality, introduced by Clive Granger6.
Definition
A time series variable $X_t$ is said to **Granger-cause** another variable $Y_t$ if, given the past history of $Y_t$, past values of $X_t$ are useful for predicting the future values of $Y_t$. In simpler terms, if including lagged values of $X$ in an autoregressive model for $Y$ significantly improves the forecast accuracy compared to a model using only lagged values of $Y$, then $X$ Granger-causes $Y$.
Important Note: This is a test of *predictive* causality, not structural or philosophical causality. It doesn't imply that $X_t$ *structurally causes* $Y_t$, only that it contains useful information for forecasting $Y_t$.
Mathematical Framework: Vector Autoregression (VAR)
Granger causality tests are commonly performed using Vector Autoregression (VAR) models. A VAR model expresses each variable in a system as a linear function of its own past values and the past values of all other variables in the system. For a simple bivariate system with two stationary variables $Y_t$ and $X_t$, a VAR model with $p$ lags, denoted VAR(p), is:
Equation for $Y_t$:
$$ Y_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i Y_{t-i} + \sum_{i=1}^{p} \beta_i X_{t-i} + \epsilon_{1t} \quad \quad (1) $$Equation for $X_t$:
$$ X_t = \gamma_0 + \sum_{i=1}^{p} \gamma_i Y_{t-i} + \sum_{i=1}^{p} \delta_i X_{t-i} + \epsilon_{2t} \quad \quad (2) $$where $\alpha_0, \gamma_0$ are intercepts; $\alpha_i, \beta_i, \gamma_i, \delta_i$ are coefficient matrices/vectors; and $\epsilon_{1t}, \epsilon_{2t}$ are white noise error terms with $E[\epsilon_{jt}] = 0$, $E[\epsilon_{jt}^2] = \sigma_j^2$, and potentially $E[\epsilon_{1t}\epsilon_{2t}] = \sigma_{12} \neq 0$.
Testing for Granger Causality: $X \rightarrow Y$
We want to test whether past $X$ helps predict current $Y$. This corresponds to testing whether the coefficients on lagged $X$ in equation (1) are jointly zero.
Hypotheses Formulation:
The null hypothesis ($H_0$) is that $X$ does *not* Granger-cause $Y$. This means that all $\beta_i$ coefficients associated with lagged $X$ terms in the equation for $Y$ (Equation 1) are simultaneously equal to zero:
$$ H_0: \beta_1 = \beta_2 = \dots = \beta_p = 0 $$The alternative hypothesis ($H_1$) is that $X$ *does* Granger-cause $Y$, meaning at least one $\beta_i \neq 0$.
Testing Procedure using F-test:
The standard approach is to compare the goodness-of-fit between the unrestricted model (Eq. 1) and a restricted model where $H_0$ is imposed.
- Estimate Unrestricted Model: Estimate Equation (1) using Ordinary Least Squares (OLS), including $p$ lags of $Y_t$ and $p$ lags of $X_t$. Record the Sum of Squared Residuals ($SSR_U$). The number of parameters estimated is $k_U = 1 (\text{intercept}) + p (\text{lags of } Y) + p (\text{lags of } X)$.
- Estimate Restricted Model: Estimate Equation (1) *excluding* the $p$ lags of $X_t$, forcing the coefficients $\beta_1, \dots, \beta_p$ to be zero: $$ Y_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i Y_{t-i} + \epsilon_{1t}^* $$ Record the Sum of Squared Residuals from this regression ($SSR_R$). The number of parameters estimated is $k_R = 1 + p$.
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Calculate the F-statistic: The test statistic measures the relative improvement in fit from adding the lagged $X$ terms.
$$ F = \frac{(SSR_R - SSR_U) / q}{SSR_U / (T - k_U)} $$
where:
- $q = p$ is the number of restrictions being tested (the number of $\beta_i$ coefficients set to zero).
- $T$ is the number of observations used in the estimation (after accounting for lags).
- $(T - k_U)$ is the degrees of freedom in the unrestricted model.
- Decision Rule: Compare the calculated $F$ statistic to the critical value $F_{critical}$ from the F-distribution with $q$ numerator degrees of freedom and $(T - k_U)$ denominator degrees of freedom, at a chosen significance level $\alpha$ (e.g., 5%). If $F > F_{critical}$, we reject the null hypothesis $H_0$.
Rejecting $H_0$ provides statistical evidence that $X_t$ Granger-causes $Y_t$. Failure to reject $H_0$ means we do not find evidence that past $X_t$ improves the prediction of $Y_t$ beyond past $Y_t$. Note that testing $Y \rightarrow X$ involves applying the same procedure to Equation (2) and testing $H_0: \gamma_1 = \dots = \gamma_p = 0$.
Important Considerations & Limitations
The validity and interpretation of Granger causality tests rely on several key assumptions7:
- Stationarity: The time series $X_t$ and $Y_t$ are assumed to be (jointly) covariance stationary. Applying the test to non-stationary data without appropriate handling (like differencing or testing for cointegration via VECM) can lead to spurious results.
- Lag Length: The test results can be sensitive to the choice of the lag length $p$. Standard practice involves using information criteria (AIC, BIC, HQIC) to guide the selection of $p$, often testing robustness across different reasonable lag lengths.
- Linearity: The test assumes a linear relationship between the variables. If the true relationship is non-linear, the standard Granger test might fail to detect predictive power.
- No Omitted Variables: The test implicitly assumes that no other relevant past information (from other variables $Z_t$) is omitted from the model. If such variables exist and influence both $X_t$ and $Y_t$, the detected Granger causality might be spurious.
- Parameter Constancy: The coefficients in the VAR model are assumed to be constant over the sample period.
Despite its limitations, the Granger causality test is a valuable tool in the time series econometrician's toolkit for exploring dynamic relationships between variables.