Rewriting a Likelihood with a Common-Component Covariance Matrix

Summary

The document raises a derivation question about rewriting a maximum-likelihood objective when the covariance matrix has an idiosyncratic variance component and a shared component across observations. The author proposes an inverse covariance expression that separates the common direction, represented by the vector of ones, from deviations around the cross-sectional mean. They are especially unsure how this decomposition produces the centered residual quadratic term in the rewritten likelihood.

The text does not include the displayed likelihoods, the full covariance assumptions, the referenced article figures, or the lemma mentioned by the author. It also contains no proposed derivation or resolution. As a result, it identifies a useful matrix-algebra problem involving projection onto the common component and its orthogonal complement, but does not provide enough information to verify the suspected misprint or reproduce the estimator.

Key ideas

• The question concerns a likelihood with a covariance structure containing shared and idiosyncratic variance components.
• The proposed inverse covariance separates the common direction from deviations around the mean.
• The author is uncertain how the decomposition yields a quadratic term in centered residuals.
• Missing equations and supporting assumptions prevent verification of the proposed correction or completion of the derivation.

Tags

# Derivation of a ML estimator


# Derivation of a ML estimator












I have the following likelihood function:

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

I would like to be able to show that the likelihood function can be rewritten to this:

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:

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.