Particular cases in Granger's causality test

Question

I'd like to expand a question about Granger's causality test by focusing on some particular cases one may happen to encounter.

I have two time series x and y where the Granger causality test gives:

> grangertest(x,y,1)

Granger causality test

Model 1: y ~ Lags(y, 1:1) + Lags(x, 1:1)
Model 2: y ~ Lags(y, 1:1)
  Res.Df Df      F Pr(>F)
1     23                 
2     24 -1 1.4754 0.2368

and

> grangertest(y,x,1)

Granger causality test

Model 1: x ~ Lags(x, 1:1) + Lags(y, 1:1)
Model 2: x ~ Lags(x, 1:1)
  Res.Df Df      F Pr(>F)
1     23                 
2     24 -1 1.0869  0.308

The fact that both p-values are quite high (say I choose a confidence level 0.05) makes me think that I cannot reject the null hypothesis that x does not cause y for Y = f(X). In other words I can't say neither if x causes y, nor if y causes x. Is this interpretation correct?

Just out of curiosity, then, I have tried to see what happens when I use two synthetic random series.

x <- rnorm(1000,0,1)
y <- rnorm(1000,0,1)

and I get

> grangertest(x,y,1)
Granger causality test

Model 1: y ~ Lags(y, 1:1) + Lags(x, 1:1)
Model 2: y ~ Lags(y, 1:1)
  Res.Df Df      F Pr(>F)
1    996                 
2    997 -1 1.5105 0.2193

and

Granger causality test

Model 1: x ~ Lags(x, 1:1) + Lags(y, 1:1)
Model 2: x ~ Lags(x, 1:1)
  Res.Df Df      F Pr(>F)
1    996                 
2    997 -1 0.1859 0.6664

This is an outcome quite similar to my data and, it seems to me, consistent with the interpretation of the function output I gave above, but please let me know if I'm missing something.

Answer 1

Is this interpretation correct?

Yes, it is. The null hypothesis is absence of causality, and you cannot reject the null since your $p$-values are above your chosen significance level.

This is an outcome quite similar to my data and, it seems to me, consistent with the interpretation of the function output I gave above.

Your logic is fine.