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 )




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