Omitted formula

问题如下：

a. What is the population value of $\beta_1$ in the regression $Y_i = \alpha + \beta_1X_1 + \epsilon_i$ if $\rho_{X_1X_2}= 0.6$ and $\sigma_{X_1}^2 = 1$ and $\sigma_{X_2}^2 = 1.2$?

解释：

Using the omitted variable formula, when X2 is omitted from the regression,

$\beta_1 = 1.2-2.1*(\frac{0.6*\sqrt 1*\sqrt 1.2}{1})=-0.18$

where $\frac{0.6*\sqrt 1*\sqrt 1.2}{1}$ is the regression slope from a regression of X2 on X1.