What is the relationship between regression and partial correlation

There is a well-answered question here.

But unfortunately, I don't even understand how the first equation in the answer is derived. Could someone help explain that?

$$\text{Beta:} \quad \beta_{x_1} = \frac{r_{yx_1} - r_{yx_2}r_{x_1x_2} }{1-r_{x_1x_2}^2}$$

What is $r$ here?

• $r_{ab}$ is the correlation between $a$ and $b$. – Glen_b Jun 8 '16 at 8:44
• Thanks @Glen_b but I am still not sure I understand this equation. I think this $\beta$ is the same as $(X_1^TX_1)^{-1}X_1^TY$, how is this related with the right-hand-side? – Haohan Wang Jun 8 '16 at 18:34
• Thanks. @Glen_b I found it here – Haohan Wang Jun 8 '16 at 19:00
• how is this related with the right-hand-side? First, consider the formula (the 1st one here which is homologic with the right-hand one but is for $b$, not for $\beta$. Both this and that are found in nearly any book on linear regression. Second, considering X having just 2 columns, "unwrap" algebraically and in scalar (in place of matrix) notation the expression $(X^TX)^{-1}X^TY$; and I'm sure you will arrive exactly at the formula for $b$. – ttnphns Jun 8 '16 at 20:51

As the comment points out, $r_{ab}$ is the correlation between $a$ and $b$.
This equation: $$\text{Beta:} \quad \beta_{x_1} = \frac{r_{yx_1} - r_{yx_2}r_{x_1x_2} }{1-r_{x_1x_2}^2}$$ can be achieved by solving the following equation:
$$\beta_{x_1} + r_{x_1x_2}\beta_{x_2} = r_{x_1y} \\ r_{x_1x_2}\beta_{x_1} + \beta_{x_2} = r_{x_2y}$$
where $x_1$ and $x_2$ are two predictors and $y$ is the dependent variable. Solving this above equation set by basic linear algebra will lead to the first equation.