I want to sanity check some results. I've run a TY causality test following the Toda-Yamamoto (TY) procedure as described by Dave Giles in his blog post "Testing for Granger Causality". One variable is stationary at level (I(0)) and one is stationary at its first difference (I(1)). Following the TY procedure, I've constructed a VAR using $p+d$ lags, where $p$ is recommended by AIC and $d=1$ (the max order of integration, as one variable is I(1)). The inputs to the VAR were all variables at level (following Giles procedure, I did not difference the I(1) variable to do a causality test).
I ran a Wald test following the TY procedure for both. Specifically, I excluded the coefficients of the $p+d$th lag term (using only the initial $p$ lags, and not the additional lag added for the I(1) variable as per Giles/TY). The Chi-Square $p$-values are quite low for both directions - 0.005 and 0.0006. The $F$-test $p$-values are 0.097 and 0.063, respectively.
Am I allowed to look at the $F$-test $p$-values using the TY procedure? Does the fact that cointegrations cannot exist between an I(1) and I(0) variable mean I've done something wrong, as it appears mutual causality exists?