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Let $y_1=\alpha_1+e_1,y_2=2\alpha_1-\alpha_2+e_2$ and $y_3=\alpha_1+2\alpha_2+e_3$ where $e_i \sim N(0,\sigma^2)$ iid for $i=1,2,3$. Find the least square estimate of $\alpha_1$ and $\alpha_2$. Derive the test statistic for $H: \alpha_1=\alpha_2$.

So, I have found the least square estimates by minimizing the squares of the errors equating the partial derivatives to $0$ and got the estimate of $\alpha_1$ and $\alpha_2$. But I am struggling to find how to form a test statistic for testing the given hypothesis. Any help would be appreciated. Thank you in advance.

Prof. Shanku
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  • Add the [self-study](https://stats.stackexchange.com/tags/self-study/info) tag. This can be answered using the [general linear hypothesis](https://stats.stackexchange.com/questions/17207/general-linear-hypothesis-test-statistic-equivalence-of-two-expressions). – StubbornAtom Nov 26 '19 at 20:46

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