### 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-oneUsing 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 )