### Multivariate Regression Coefficient Derivation

 Derive the formula for –from the multivariate regression model with two covariates.The model we hope to estimate, regressing on :       Our goal going forward is to convert this equation into something in terms of that we could solve for. We want to have a nice, clean equation with ,  where " " is in terms of things we know.  Distributing this: Recall lecture notes, So we can add this term (and two other similar terms) to our first order conditions above without issue: Rearranging to get all the beta-one terms on one side: Solving for beta-one  Using alternative notation, that helps us stay organized: And following the same steps with the  first order conditions, we find: Plugging in our equation for into the equation we have for   Now, trying to solve in terms of       Solving this all a slightly different way:Recall from our simple regression model that (using the same notation as above): If we were to estimate the model: . Also, from the model  Now, plugging in those s: Now, plugging in those s  (the coefficients from univariate regressions), we get:Our Solution: Where: is the coefficient estimate of on the multivariate model (from ) is the coefficient estimate from the univariate model: regressed on (from ) is the coefficient estimate from the univariate model: regressed on (from ) is the coefficient estimate regressing on (from ) is the coefficient estimate regressing on (from )