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I have the following likelihood function:

enter image description here

I'm given this information about the $\Omega$ matrix ($\boldsymbol{1}$ is a $T \times 1$ vector of ones): enter image description here

I would like to be able to show that the likelihood function can be rewritten to this: enter image description here

I think there is a misprint in the article, hence the screen images from the article I attached are wrong, meaning that $\Omega^{-1}$ is supposed to be $$\Omega^{-1}=\frac{1}{\sigma^2_{\varepsilon}+T\sigma^2_{c}}\boldsymbol{1}(\boldsymbol{1}'\boldsymbol{1})^{-1}\boldsymbol{1}'+\frac{1}{\sigma^2_{\varepsilon}}(I_t-\boldsymbol{1}(\boldsymbol{1}'\boldsymbol{1})^{-1}\boldsymbol{1}')$$

and not as stated above.

So far I've inserted the expression for $\Omega^{-1}$ in the top log-likelihood function. However, especially the last expression of the rewritten log-likelihood function bothers me, i.e. this part:

\begin{equation*} \begin{split} ...& &+\frac{1}{2}\frac{1}{\sigma^2_{\varepsilon}}\frac{1}{N}\sum_{i=1}^{N} \left(Y_i-\boldsymbol{1}\bar{Y}_i-\left(X_i-\boldsymbol{1}\bar{X}_i\right)\beta\right)'\left(Y_i-\boldsymbol{1}\bar{Y}_i-\left(X_i-\boldsymbol{1}\bar{X}_i\right)\beta\right). \end{split} \end{equation*}

Does anyone see the solution?

I also have this lemma, but not sure if it is to be used in the rewritting of the likelihood function: enter image description here

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@pbr142 is this something you could help me explain? –  Sunv Apr 8 at 14:02
1  
This is basic linear algebra, notably block matrix inversion, and more specifically rank one update. –  nicolas Apr 9 at 8:26
    
@nicolas Thanks for your answer. Could you be more specific as to how I should obtain the rewritten likelihood function? –  Sunv Apr 9 at 8:31
1  
read about block inversion/rank one update, and apply... :) Look at the Horn Johnson or Golub –  nicolas Apr 9 at 10:19
1  
what you added is a special case with block diagonal matrix. I know it's painful, but you should take a few days to get into the comfort zone, this basic linear algebra will help you long time –  nicolas Apr 9 at 11:22

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