Lead-and-lag test?

Question

What are the simplest methods to check for the lead and lag relationship between two variables?

I mean.. how to see, between two variables, which leads the other (and which follows) ?

Answer 1

Calculate the cross correlation

$$\frac{E\left[ [f(m) - \mu_f] [g(m + n) - \mu_g] \right]}{\sqrt{E\left[ [f(m) - \mu_f]^2\right]E\left[ [g(m + n) - \mu_g]^2 \right]}}$$.

where $\mu_f$ and $\mu_g$ are the means of $f$ and $g$, respectively.

Note that this is a function of $n$, the relative lag, and that $n$ can be both positive or negative. If $f$ is leading (lagging) over $g$, then this should be significantly positive for some positive (negative) value of $n$.

This is a common first step in econometric lead-lag analysis.