1

Consider the model $$ y_{i}=\beta_{0}+\beta_{1} x_{i 1}+\beta_{2} x_{i 2}+e_{i}, \quad e_{i} \sim N\left(0, \sigma^{2}\right), $$ $ i=1, \ldots, n $. Write down its mean square error (MSE). Prove that the MSE is still an unbiased estimator for $ \sigma^{2} $ when $ \beta_{2} $ is in fact equal to zero, i.e., the true model is $$ y_{i}=\beta_{0}+\beta_{1} x_{i 1}+e_{i} \quad e_{i} \sim N\left(0, \sigma^{2}\right) . $$

I know that $ \frac{\mathrm{SSE}}{\sigma^{2}} \sim \chi_{n-2}^{2} $, but I'm not sure how this is affected by $ \beta_{2} = 0 $.

Dennis
  • 39
  • $SSE/\sigma^2$ has a $\chi^2$ distribution whenever the errors are i.i.d normal. In the first model, the degrees of freedom of $\chi^2$ is $n-3$, while in the second model it is $n-2$ (as you say). So, $MSE$ (i.e. $SSE$ divided by its degrees of freedom) remains unbiased for $\sigma^2$ in any case. – StubbornAtom Dec 11 '21 at 14:10
  • Answered at https://math.stackexchange.com/q/3319241/321264, https://math.stackexchange.com/q/3100557/321264, https://math.stackexchange.com/q/1527021/321264 and in their linked posts. – StubbornAtom Dec 11 '21 at 14:35

0 Answers0