10.1 Granger causality test

• Prior to Granger causality test it is cruical to select the optimal lag length, because the choice of lags $p$ can influence the results (using too few lags may omit relevant information, while too many can introduce noise)

• Lag selection is usually based on information criteria that penalize model complexity, such as Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC) in (6.14)

• Sometimes, AIC and BIC do not indicate the same lag selection and additional criteria can be used: Hannan-Quinn Criterion (HQC) and Final Prediction Error (FPE)

Variable $x_t$ is said to “Granger-cause” $y_t$ if the past values of $x_t$ significantly influence the current value of $y_t$. Assuming that the optimal lag selection $p=1$, Granger causality test is performed simultaneously by testing two null hypothesis

$$\begin{equation} \begin{matrix} H_0:\beta_{1,2}=0~~~~(x_t~~does~~not~~cause~~y_t) \\ \\ H_0:\beta_{2,1}=0~~~~(y_t~~does~~not~~cause~~x_t) \end{matrix} \tag{10.4} \end{equation}$$

• Granger causality test reduces to testing the significance off-diagonal terms of the matrix $A_1$ (so called companion matrix)

• If both null hypothesis (10.4) are rejected then causality in both directions exist

• $F-$statistic or Wald statistic ($\chi^2$ version) can be used to check both null hypothesis

Exercise 44. Conclude about Granger causality direction(s) in the following cases:

I. null hypothesis $H_0: \beta_{1,2}=0$ is not rejected, while null hypothesis $H_0: \beta_{2,1}=0$ is rejected
II. null hypothesis $H_0: \beta_{1,2}=0$ is rejected, while null hypothesis $H_0: \beta_{2,1}=0$ is not rejected
III. both null hypothesis $H_0: \beta_{1,2}=0$ and $H_0: \beta_{2,1}=0$ are rejected
IV. both null hypothesis $H_0: \beta_{1,2}=0$ and $H_0: \beta_{2,1}=0$ are not rejected

Solution

I. $y_t$ causes $x_t$ only
II. $x_t$ causes $y_t$ only
III. there is no any causality between $x_t$ and $y_t$
IV. there is causality in both directions