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) ?

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#### Best Answer

Calculate the cross correlation

$$frac{Eleft[ [f(m) – mu_f] [g(m + n) – mu_g] right]}{sqrt{Eleft[ [f(m) – mu_f]^2right]Eleft[ [g(m + n) – mu_g]^2 right]}}$$.

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.

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