Two variables: Which moves first? Say I have a time series of two variables, $X$ and $Y$, and want to find out which one of these two moves first. One example is the one of GDP growth and investment growth, where we usually tell the story that at a business cycle frequency, investment moves "earlier" than production or consumption.

I dont want to do eyeballing econometrics ("this thingy here clearly moved first"). What is the standard way of dealing with these things in the literature? What methodology can be used? Answers that allow for more than two variables are very welcome.
"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".