I've got a linear model: $y_i=β_1x_{i1}+β_2x_{i2}+ε_i$ where E($ε_i$)=0 and Var($ε_i$)= $σ^2I_n$ for i=1,...,n
Supposed we don't have the data for $x_{i2}$ and we estimate: $y_i=β_1x_{i1}+ε_i$ for i=1,...,n
So far I've shown that $\hat{\beta}_1$ is biased and that $s^2= {ε_i'ε_i}/(n-1)$ is also biased and that such bias is positive.
I now have to explain why I'd expect that the OLS estimator of $\hat{\beta}_1$ is inconsistent and then suggest a suitable estimator giving conditions where it'd be consistent.
Please could someone give me a helping hand, I'm not sure why $\hat{\beta}_1$ is inconsistent?
Thanks
$\hat{\beta} = (x_1'x_1)^{-1}x_1'y$
Now substitute the true model $y=X\beta + \epsilon$ into $y$
$\hat{\beta} = (x_1'x_1)^{-1}x_1'y = \beta_1 + (x_1'x_1)^{-1}x_1'x_2\beta_2+ (x_1'x_1)^{-1}x_1'\epsilon_i$
Taking the limit sends the last term to $0$, which leaves
$\hat{\beta} \rightarrow \beta_1 + \frac{cov(x_1,x_2)}{var(x_1)}\beta_2$
Which means all you need to do if figure out when $\frac{cov(x_1,x_2)}{var(x_1)}\beta_2$=0