# simplifying to bivariate regression formula

• Jan 17th 2010, 07:26 AM
garymarkhov
simplifying to bivariate regression formula
I want to show that if the number of columns for my matrix of regressors is two and if the first column of my regressors matrix is full of ones then the OLS estimator of the second element of $\displaystyle \beta$ reduces to the bivariate regression formula.

How can I show such a thing? It seems obvious to me that
$\displaystyle Y = \left[ \begin{array}{cc} 1 & x_{10} \\ 1 & x_{20} \\ 1 & x_{30} \\ 1 & x_{40} \end{array} \right] \beta + u$ is the same as something like $\displaystyle y = 1 + Xb + u$ although I'm not sure that's a solid enough "proof". Help!
• Jan 17th 2010, 07:38 PM
matheagle
Though the vector $\displaystyle \beta=(a,b)$
giving you the model $\displaystyle y=a+bx+\epsilon$
Given that $\displaystyle X$ is fixed and a T x k matrix...
If k=2 and the first element of $\displaystyle X$ is a constant, show that the expression for the OLS estimator of the second element of $\displaystyle \beta$ reduces to the familiar bivariate regression formula.