Is it necessary to calculate VAR before Granger causality test so that we can have the lag length to be used in Granger causality test
Solved – VAR and Granger causality test
rtime seriesvector-autoregression
Related Solutions
The case of no cointegration
This is easy. If you have no lags, then the model looks like \begin{aligned} \Delta x_{1,t} &= \gamma_{0,1} + u_{1,t}, \\ &\dots \\ \Delta x_{k,t} &= \gamma_{0,k} + u_{k,t}, \\ \end{aligned} where $k$ is the number of series in your model, $\gamma_0$s are intercepts (they would be set to zero if there are no time trends in the nondifferenced $x$s) and $u_t$s are error terms. Then clearly the history of series $j$ is not useful in predicting the series $i$ beyond the history of the series $i$ itself. (Actually, the history of series $j$ is not useful in predicting the series $i$, period.) And this holds for any $(i,j)=1,\dots,k$ where $i\neq j$. Therefore, none of the series Granger-causes any other series. (Also, no group of series Granger-causes another group of series.)
The case with cointegration
Consider a bivariate model for simplicity. Suppose
\begin{aligned}
\Delta x_{1,t} &= \gamma_{0,1} + \alpha_1 (x_{1,t-1}+\beta x_{2,t-1}) + u_{1,t}, \\
\Delta x_{2,t} &= \gamma_{0,2} + \alpha_2 (x_{1,t-1}+\beta x_{2,t-1}) + u_{2,t} \\
\end{aligned}
$\beta\neq 0$ and either $\alpha_1\neq 0$ or $\alpha_2\neq 0$ or both. Then
\begin{aligned}
x_{1,t} &= \gamma_{0,1} + (\alpha_1+1) x_{1,t-1} + \alpha_1\ \beta x_{2,t-1} + u_{1,t}, \\
x_{2,t} &= \gamma_{0,2} + \alpha_2 x_{1,t-1} + (\alpha_2 \beta + 1) x_{2,t-1} + u_{2,t}. \\
\end{aligned}
If $\alpha_1\beta\neq 0$ (i.e. if $\alpha_1\neq 0$ because we already know that $\beta\neq 0$) in the equation for $x_{1,t}$, $x_2$ Granger-causes $x_1$.
Also, if $\alpha_2\neq 0$ in the equation for $x_{2,t}$, $x_1$ Granger-causes $x_2$.
We also know that under cointegration there will be Granger causality at least one way (since $\beta\neq 0$ and either $\alpha_1\neq 0$ or $\alpha_2\neq 0$ or both), so either $x_1$ Granger-causes $x_2$ or $x_2$ Granger-causes $x_1$ or both.
Best Answer
Granger Causality can be defined as: "X is said to Granger-cause Y if Y can be better predicted using the histories of both X and Y than it can by using the history of Y alone." 1
or in other words: "A time series X is said to Granger-cause Y if it can be shown, usually through a series of t-tests and F-tests on lagged values of X (and with lagged values of Y also included), that those X values provide statistically significant information about future values of Y." Wikipedia
This is essentially what a vector autoregression model does.
Also wikipedia says: "Multivariate Granger causality analysis is usually performed by fitting a vector autoregressive model (VAR) to the time series."
Granger causality is therefore more a form of structurally summarizing VAR models.
For some more information: here