Granger causality test in VAR framework: small sample

Question

I am using Granger Causality Test in VAR framework to test the causal relationship between renewable energy consumption, gross domestic product (GDP) and carbon dioxide emissions in one country using annual data from 1982-2012. As I couldn't find data related to renewable energy for that specific country before 1982, I was wondering whether 30 years of time series data is good enough to do such a test?

Answer 1

30 observations seems too few for the central limit theorem (CLT) to really kick in. Therefore, you would need the model residuals to be (roughly) normally distributed and (almost) independent for the $F$-statistic of the Granger causality test to have the $F$-distribution under the null hypothesis of no Granger causality.

If the model residuals are far from normal or there is a strong dependence between them, the $F$-statistic of the Granger causality test will not be $F$-distributed under the null and thus you will not be able to sensibly compare it to the critical values produced by the $F$-distribution. Hence, you will not be able to conclude whether you are able to reject the null hypothesis or not.

The normality of the residuals can be empirically assessed by formally testing for normality (e.g. Shapiro-Wilks test, although there are other tests, too, and I am not sure which of the normality tests is best suited for samples of this size), visually expecting the Q-Q plot etc. The independence of the residuals can be assessed by looking for dependence structures such as autocorrelation (e.g. Breusch-Godfrey test), ARCH patterns (ARCH-LM test) and other.