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I consider 3 variables $x$, $y$, $z$ and study the relationship of the beta coefficient from OLS regression of two variables connected via a third.

Whilst it is not true that $$\beta_{(x,y)}\times\beta_{(y,z)}\equiv\beta_{(x,z)}$$ (no proof shown - but would be appreciated), does anyone know if there is a simple relationship between $\beta_{(x,y)}$, $\beta_{(y,z)}$ and $\beta_{(x,z)}$?

Another way to look at it is to use the relationship between beta's and correlation coefficient:

1- $\beta_{(x,y)}$ = $\rho_{x,y}$($\sigma_{x}$/ $\sigma_{y}$)

2- $\beta_{(y,z)}$ = $\rho_{y,z}$($\sigma_{y}$/ $\sigma_{z}$)

3- $\beta_{(x,z)}$ = $\rho_{x,z}$($\sigma_{x}$/ $\sigma_{z}$)

1 mutliplied by 2 - we get: $\beta_{(x,y)}$$\times$$\beta_{(y,z)}$ = $\rho_{x,y}$($\sigma_{x}$/ $\sigma_{y}$) $\times$ $\rho_{y,z}$($\sigma_{y}$/ $\sigma_{z}$) = $\rho_{x,y}$ .$\rho_{y,z}$ ($\sigma_{x}$/ $\sigma_{z}$)

therefore this can be proved if we can prove that correlation coeffiencients are compoundable - meaning $\rho_{x,z}$ $\equiv$ $\rho_{x,y}$ $\rho_{y,z}$ ?

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  • $\begingroup$ Please precisely define the meaning of the coefficients. $\endgroup$ Commented May 27, 2020 at 13:22
  • $\begingroup$ I wanted to know if the relationship of the beta coefficient from OLS regression of two variables connected via a third. Beta(x,y) = covariance(x,y)/variance(x) $\endgroup$
    – DGradeci
    Commented May 27, 2020 at 13:29
  • $\begingroup$ - for anyone else the answer comes from the study of path analysis - faculty.cas.usf.edu/mbrannick/regression/Pathan.html $\endgroup$
    – DGradeci
    Commented May 27, 2020 at 16:28
  • $\begingroup$ For a simple, convincing counterexample, let $y$ be a any random variable independent of $x$ and let $z$ be a linear function of $x,$ so that the two betas on the left side of the equation are zero but the beta on the right side is $\pm1.$ $\endgroup$
    – whuber
    Commented Nov 1, 2023 at 22:13

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As you indicate in the comment, for the statement $\beta_{(x,y)}\times\beta_{(y,z)}\equiv\beta_{(x,z)}$ to be true we would need $$ \frac{Cov(X,Y)}{Var(X)}\frac{Cov(Y,Z)}{Var(Y)}\equiv\frac{Cov(X,Z)}{Var(X)} $$ or $$ Cov(X,Y)\frac{Cov(Y,Z)}{Var(Y)}\equiv Cov(X,Z) $$ As a counterexample suffices to show that a statement not true, you might just run

x <- rnorm(100)
y <- rnorm(100)
z <- rnorm(100)

cov(x,y)*cov(y,z)/var(y)
cov(x,z)

As such I am also not aware of a simple relationship.

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