# Relation between regression coefficient and correlation coefficient

For simple linear regression, the regression coefficient's sign and the correlation coefficient's (between independent and dependent variable) sign should be matching or not?

Yes, in the case of a simple linear regression, they should be matching! If $x$ increases and $y$ increases, that means they are positively correlated. In that case, the slope (coefficient of $x$) will be positive. If $x$ increases and $y$ decreases, we always have a negative correlation and a negative slope. Also $cor(y,x) = \sqrt{R^2}$.
• When you said simple linear regression, I assumed you meant with only one independent variable. In that case, it should always match. If it's multiple independent variables, then no, it really doesn't have to match. A correlation is a measure of the linear relationship between 2 variables, while a multiple regression is a measure of the linear relationship between $y$ and multiple $x$s. Jul 27, 2017 at 23:02
• It is not independent. Say you regress $y$ on $x_1$ and $x_2$, and you get significant coefficients $\hat{\beta_1}$ and $\hat{\beta_2}$. Now repeat by omitting $x_2$. Now you have $\hat{\beta_1'}$, which is equal to $\hat{\beta_1}$ plus some bias. The bias is proportional to the correlation between $x_1$ and $x_2$, so the coefficient is not independent of the other variables (to be more precise, the variables are not independent of each other). In any case, none of these are applicable in a simple linear regression setting, was asked in the original question. Jul 27, 2017 at 23:19
• This answer is good, but the final formula is incorrect because $\sqrt{R^2}=|R| \ge 0$ whereas the correlation can be negative.