# Testing for Granger causality in a VAR model with zero lags

Is having a maximum lag length of 0 ok after the differencing of variables (variables are not stationary at level). If yes, how can we run the Granger Causality Test through VAR or VECM modelling without having any lags, because it asks for minimum 1 lag to run the Granger Causality Test. The data is annual and I am trying to find the Causality relationship between GDP and Construction Industry.

Should I use VAR modelling or not when the number of lags is 0?
If not, which method should I use?

• umer, what do you think about my answer? Is it clear or do you need further elaboration? (I see you have not accepted it.) – Richard Hardy Sep 7 '17 at 13:10

# 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.