Granger Causality and Co-integration
Granger causality
Consider a VAR(p) model of 2-d variables, i.e. \(y_t = [y_{i,t}, y_{2,t}]^T\) \(y_t = A_0 + \sum_1^p A_i y_{t-i} + a_t\), and each \(A_i = \begin{bmatrix}\phi_{i,11}&\phi_{i,12}\\\phi_{i,21}&\phi_{i,22}\end{bmatrix}\)
Then, if \(y_{2,t}\) does not Granger cause \(y_{1,t}\), then all \(\phi_{i, 12} = 0\) since it will only be contributing to the equation of \(y_{1,t}\)
Similarly, if \(y_{1,t}\) does not Granger cause \(y_{2,t}\), then all of \(\phi_{i,21}=0\)
Portmanteau test
Let \(X_t, Y_t\) be causal and invertible univariate ARMA processes and be given by
The cross-correlation
If \(\rho(k)=0\) for \(k<0\), then \(X_2\) does not cause \(X_1\)
Cointegration
I(d) process For a time series \(X_t\)
- A stationary and invertible time series is said to be an I(0) process
- A time series is said to be an integrated process of order one, i.e. I(1) process, if \((1-B)X_t\) is stationary and invertible.
- I(d) process, if \((1-B)^d X_t\) is stationary and invertible, \(d>0\) and order \(d\) is referred to as the order of integration or the multiplicity of a unit root.
Consider a multi time series \(X_t\). If \(\forall i, X_{i,t}\) are I(1) but a nontrivial linear combination \(\beta'X_t\) is I(0), then \(X_t\) is said to be cointegrated of order one and the linear combination vector \(\beta\) is called a cointegrating vector.
Granger Representation Theorem
\(X_t, Y_t\) are cointegrated, IFF exists an ECM representation.
Implications
- Vector autoregression on differenced I(1) processes will be a misspecification if the component series are cointegrated
- An equilibrium specification is missing from a VAR representation
- When lagged disequilibrium terms are included as explanatory variables, the model becomes well specified
- Such as model is called an error correction model because the model is structured so that short-rum deviation from the long-run equilibrium will be corrected.