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
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 ) |