# How to obtain bivariate regression coefficients from two univariate regression coefficients? [closed]

Let's assume that we want to obtain the coefficients of the following bivariate regression: $$Y=\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$$

However, we don't have access to the data $$(X_1,X_2,Y)$$. The only information available are:

• Coefficients of the univariate regression $$Y=\gamma_0 + \gamma_1 X_1 + \varepsilon$$
• Coefficients of the univariate regression $$Y=\lambda_0 + \lambda_1 X_2 + \nu$$
• Variances of $$X_1$$ and $$X_2$$, plus correlation between these two
• Variance of $$Y$$

How can we obtain $$\beta_1$$ and $$\beta_2$$ from this information?

• I’m voting to close this question because it is not specific to Quantitative Finance and belongs on Cross Validated Stack Exchange instead. Apr 5 at 8:10

Betas are $$(x'x)^{-1}(x'y)$$. You know $$(x'x)^{-1}$$ from variances and correlation. You know $$x'y$$ from the correlation between each x and y (imply from regression coefficient).
• Hi. The guy that claimed incorrect definitely read the question and answer and has a gigantic rating on cross validated. So, that's why I asked. I got stuck calculating $x^{\prime } y$. Thanks for any details. Apr 7 at 22:29