Compute the Bias of the estimates

Question

If the true expection of response is \begin{align} E(y)=\beta_0+\beta_1x+\beta_2x^2 \end{align} but you assume $E(y)=\beta_0+\beta_1x$ insted. With $x=-2,-1,0,1,2$. How can i compute the Bias of the estimates of $\beta_0$ and $\beta_1$?. I did sth wrong because my solution is 0. Pls support me. Thx a lot

Answer 1

If you simply ignore the $\beta_2$ term but kept the correct $\beta_0$ and $\beta_1$ terms then you would be $\beta_2\times 4$ too low for $x=-2$, $\beta_2\times 1$ too low for $x=-1$, $\beta_2\times 0$ too low for $x=0$, $\beta_2\times 1$ too low for $x=1$, and $\beta_2\times 4$ too low for $x=2$.

Your linear regression will attempt to minimise these errors. The best fit to $(-2,4\beta_2), (-1,\beta_2), (0,0), (1,\beta_2), (2,4\beta_2)$ would be $\hat y=2\beta_2 +0x$

so the expected bias in the estimate of $\beta_0$ is $2\beta_2$, and the expected bias in the estimate of $\beta_1$ is $0$.