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What are ways that can be used to estimate correlation matrix $V$ when $cov(\epsilon)=\sigma^2 V$, $y = X \beta+ \epsilon$ and $\epsilon \text{~} N(0, \sigma^2 V)$

Where $V$ is a non-singular positive definite matrix.


Using $V$ for example $\hat{\beta}=(X' V^{-1}X)^{-1}X'V^{-1}y$. And

$$f(y, \beta, \sigma^2)=(2 \pi \sigma)^{-n/2} |V|^{-1/2} \exp(-\frac{1}{2}(y-X \beta)^T V^{-1} (y-X \beta))$$

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  • $\begingroup$ What is your question precisely? It sounds like you are partly providing your answer. Are you asking how to estimate $V$ when neither $V$ nor $\beta$ nor $\sigma$ are known? $\endgroup$ Dec 13, 2016 at 16:38
  • $\begingroup$ @Superpronker I think my answer lies in maybe maximizing $f(y,\beta, \sigma^2)$. But there should be multiple ways to estimate $V$ I believe. $\endgroup$
    – mavavilj
    Dec 13, 2016 at 16:51
  • $\begingroup$ So you mean: how do I estimate $\beta$ and $\sigma$ when $V$ is unknown? In that case, I guess (feasible) GLS is one answer, if you want a linear model. $\endgroup$ Dec 13, 2016 at 17:05
  • $\begingroup$ @Superpronker I think here (en.wikipedia.org/wiki/Estimation_of_covariance_matrices) are some derivations. $V$ is denoted $\Sigma$ there. $\endgroup$
    – mavavilj
    Dec 13, 2016 at 17:07
  • $\begingroup$ If you are saying that you have $N$ observations and $V$ is $N \times N$, then this will not work. If you are saying that you have panel data with $N$ observations of individuals and $T$ time periods, and $V$ is $T \times T$, then you can use feasible GLS. $\endgroup$ Dec 13, 2016 at 17:18

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I believe this is what Aitken estimators address. The challenge is that there are so many unknowns one is trying to estimate for the general case (for symmetric V, this would be $n(n+1)/2$). One technique I've seen is to place extra constraints on V (like a fixed V times a variable).

Searching on "Aitken estimator" brings up many good resources.

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