Consider the model $Y_{ij} = \beta_i + \epsilon_{ij}$ for all $i=1,2,...,k$ and $j=1,2,...,n_i$,
$\epsilon_{ij} \sim \text{i.i.d} \ N(0, \sigma^2)$.
How can I test $H_0 : \beta_i = \alpha \times i\ $ for all $i=1,2,...,k \ \ $ v/s $\ H_1 :$ not $H_0$.
I think we need an F-statistic to test this, and it might be given by $\frac{\frac{SSE(H_0)-SSE}{k}}{\frac{SSE}{n-k}} \sim F_{k,n-k}$. But I am not being able to find $SSE(H_0)$.