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 \(x_t\) onlyII. \(x_t\) causes \(y_t\) only
III. there is no any causality between \(x_t\) and \(y_t\)
IV. there is causality in both directions