*Bounty: 50*

*Bounty: 50*

Assume we are in the OLS setting with $y = Xbeta + epsilon$. When $y$ is a response vector, and $X$ are covariates, we can get two types of covariance estimates:

The homoskedastic covariance

$cov(hat{beta}) = (X’X)^{-1} (e’e)$, and robust covariance

$cov(hat{beta}) = (X’X)^{-1} X’ diag(e^2) X (X’X)^{-1}$.

I’m looking for help on how to derive these covariances when $Y$ is a response matrix, and $E$ is a residual matrix. There is a fairly detailed derivation on slide 49 here, but I think there are some steps missing.

For the homoskedastic case, each column of $E$ is assumed to have a covariance structure of $sigma_{kk} I$, which is the usual structure for a single vector response. Each row of $E$ is also assumed to be i.i.d with covariance $Sigma$.

The derivation starts with collapsing the $Y$ and $E$ matrices back into vectors. In this structure $Var(Vec(E)) = Sigma otimes I$.

First question: I understand the kronecker product produces a block diagonal matrix with $Sigma$ on the block diagonal, but where did $sigma_{kk}$ go to? Is it intentional that the $sigma_{kk}$ values are pooled together so that the covariance is constant on the diagonal, similar to the vector response case?

Using $Sigma otimes I$, the author gives a derivation for $cov(hat{beta})$ on slide 66.

$

begin{align}

cov(hat{beta}) &= ((X’X)^{-1} X’ otimes I) (I otimes Sigma) (X (X’X)^{-1} otimes I) \

&= (X’X)^{-1} otimes Sigma

end{align}

$.

The first line looks like a standard sandwich estimator. The second line is an elegant reduction because of the I matrix and properties of the kronecker product.

Second question: What is the extension for robust covariances?

I imagine we need to revisit the meat of the sandwich estimator, ($I otimes Sigma$), which comes from the homoskedastic assumption per response in the Y matrix. If we use robust covariances, we should say that each column of $E$ has variance $diag(e_k^2)$. We can retain the second assumption that rows in E are i.i.d. Since the different columns in $E$ no longer follow the pattern $scalar * I$, I don’t believe $Var(Vec(E))$ factors into a kronecker product as it did before. Perhaps $Var(Vec(E))$ is some diagonal matrix, $D$?

Revisiting the sandwich-like estimator, is the extension for robust covariance

$

begin{align}

cov(hat{beta}) &= ((X’X)^{-1} X’ otimes I) (D) (X (X’X)^{-1} otimes I) \

&= ?

end{align}

$.

This product doesn’t seem to reduce; we cannot invoke the mixed product property because D does not take the form of a scalar multiplier on I.

The first question is connected to this second question. In the first question on homoskedastic variances, $sigma_{kk}$ disappeared, allowing $Var(Vec(E))$ to take the form $Sigma otimes I$. But if the diagonal of $Var(Vec(E))$ was not constant, it would actually have the same structure as the robust covariance case ($Var(Vec(E))$ is some diagonal matrix $D$). So, what allowed $sigma_{kk}$ to disappear, and is there a similar trick for the robust case that would allow the $D$ matrix to factor?

Thank you for your help.