## Probability convergence with OLS estimators - univariate vs bivariate

Hey guys, this is my first post here. I was hoping you could help point out how I should go about tackling this proof:

Let ($\displaystyle Y_1$, $\displaystyle X_1$, $\displaystyle _1$, $\displaystyle X_1$,$\displaystyle _2$)...,($\displaystyle Y_n$, $\displaystyle X_n$,$\displaystyle _1$, $\displaystyle X_n$,$\displaystyle _2$) be an i.i.d. sample from ($\displaystyle Y$, $\displaystyle X_1$, $\displaystyle X_2$) satisfying:

$\displaystyle Y$ = $\displaystyle \beta_0$ + $\displaystyle \beta_1$$\displaystyle X_1 + \displaystyle \beta_2$$\displaystyle X_2$ + $\displaystyle U$

where (1, $\displaystyle X_1$, $\displaystyle X_2$) is not perfectly collinear, $\displaystyle E[Y^4]$ < $\displaystyle \infty$ and $\displaystyle E[X^4]$ < $\displaystyle \infty$ (** - please note it should be $\displaystyle X_j$ in the above, not just $\displaystyle X$, but my Latex-fu is weak).

Suppose a professor estimates the equation: $\displaystyle Y$ = $\displaystyle \beta_0$* + $\displaystyle \beta_1$*$\displaystyle X_1$ + $\displaystyle U$*

by OLS. Show that:

$\displaystyle \hat\beta_1$* p$\displaystyle \to$ $\displaystyle \beta_1$ + $\displaystyle \beta_2$ Cov[$\displaystyle X_1$ , $\displaystyle X_2$]/Var[$\displaystyle X_1$].

Also, under what conditions will it be true that $\displaystyle \hat\beta_1$* is consistent for $\displaystyle \beta_1$? (Here I'm confused - it's obvious that (X'X)^-1 is well defined since they already explicitly stated that (1, $\displaystyle X_1$, $\displaystyle X_2$) is not perfectly collinear and that $\displaystyle E[Y^4]$ < $\displaystyle \infty$ and $\displaystyle E[X^4]$ < $\displaystyle \infty$).

Any advice to get me started?