Two variables: Which moves first?
"Who moves first" can be conveniently detached from any causal inference, since there may be some third variable influencing both. Certainly one could build a logical and reasonable argument that links increases in investment to increases in GDP (or vice versa), but it is not necessary.  

Then, examining cross-correlation coefficients on the levels or on an appropriate transformation (like first differences, or first log-differences) is a simple and valid way to examine the issue, as long as the chosen transformation will provide stationary series.  

Specifically, one can examine (sample) ${\rm Corr} (X_t, Y_{t})$, ${\rm Corr} (X_t, Y_{t-k})$ and ${\rm Corr} (X_{t}, Y_{t+k})$, where, again, $X$ and $Y$ can represent the chosen transformation and not necessarily the levels of the variables. Also, what calendar time length will $t$ represent should also be decided according to the phenomenon under study. In turn, the distance represented by $k$ is a matter of experimentation, but it is advisable to start with $k=1$.

If there is a temporal order, then the contemporaneous cross-correlation ${\rm Corr} (X_t, Y_{t})$ should be negligible. Then, the magnitude and sign of the other two cross-correlations should provide statistical evidence as to "who moves first".