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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) ?

James
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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.

Richard Hardy
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Ami Tavory
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  • How does this differ from Granger Causality test? – James Oct 19 '16 at 09:57
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    @James There are actually some sources on what's common and what's not. See [here](https://sccn.ucsd.edu/wiki/Chapter_4.5._(Cross-)_correlation_does_not_imply_(Granger-)_causation), for example. – Ami Tavory Oct 19 '16 at 10:47