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Is it necessary to create a stable vector autoregressive (VAR) model satisfying all the necessary checks to create it and then and only can we say that Granger causality holds true?

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Yes. In the specific case of VAR models, imagine we want to test the restriction that $x$ does not Granger-cause $y$. We have the regression (in $y$): $$y_t=\alpha + \sum_{l=1}^p \delta_ly_{t-l}+\sum_{l=1}^q \gamma_lx_{t-l}+\epsilon_t$$ The corresponding null to our Granger causality test is $H_0:\gamma_l=0,\,\,l=1,\dots,p$. This is a Wald test, and under the asymptotic normality and consistency of your estimator, it is $\chi^2$ distributed. If your process is not covariance-stationary, the variance of $\epsilon_t$ might change over time so you're estimate will no longer be asymptotically i.i.d. normal and the test statistic may no longer be $\chi^2$ distributed.

In general, Granger causality tests are somewhat model-dependent (they are subject to, for instance, omitted variables bias) so you should make sure you have the "right" model before you run them.

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