# Prove that System FGLS is Consistent

In the Systems of Equations framework, such as Seemingly Unrelated Regression (SUR), suppose we have $g=1,\ldots,G$ equations. Let $\mathbf{X}_i$ be a $G \times K$ matrix, $\mathbf{y}_i$ be $G \times 1$, and suppose we have collected data on $i=1,\ldots,N$ observations. Let the model be $$\mathbf{y}_i=\mathbf{X}_i\mathbf{\beta}+\mathbf{u}_i$$ Let $\mathbf{\Omega}=E\left(\mathbf{u}_i\mathbf{u}_i^{\top}\right)$ (which is unobserved), and let a consistent estimator be $\hat{\mathbf{\Omega}}=\sum_{i=1}^N \breve{\mathbf{u}}_i\breve{\mathbf{u}}_i$, where $\breve{\mathbf{u}}_i$ are the residuals which come from System OLS (not System GLS). Suppose it is given / already proven that $\hat{\mathbf{\Omega}}\overset{p}{\to}\mathbf{\Omega}$; that is, $\hat{\mathbf{\Omega}}$ is a consistent estimator of $\mathbf{\Omega}$.

Define $$\hat{\mathbf{\beta}}_{SFGLS}=\left(\sum_{i=1}^N \mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{X}_i\right)^{-1}\left(\sum_{i=1}^N \mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{y}_i\right)$$ to be the System FGLS estimator.

The objective is to show that $\hat{\mathbf{\beta}}_{SFGLS}$ is consistent; i.e. $plim \hat{\mathbf{\beta}}_{SFGLS}=\mathbf{\beta}$, under the assumptions

1. $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{O}_{G^2 \times K}$ (this condition implies $E\left(\mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{u}_i\right)=\mathbf{0}$)
2. $E\left(\mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i\right)<\infty$ and $rank E\left(\mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i\right)=K$

My attempt: I wanted to apply the same argument as usual, i.e. write it in error form as $$\hat{\mathbf{\beta}}_{SFGLS}=\mathbf{\beta}+\left(N^{-1}\sum_{i=1}^N \mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{X}_i\right)^{-1}\left(N^{-1}\sum_{i=1}^N \mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{u}_i\right)$$ Then, by the law of large numbers, $$N^{-1}\sum_{i=1}^N \mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{X}_i \overset{p}{\to}E\left(\mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{X}_i\right)$$ and again by the law of numbers, $$N^{-1}\sum_{i=1}^N \mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{u}_i\overset{p}{\to}E\left(\mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{u}_i\right)$$ However, I think what I've written regarding the expectations might be wrong. After all, I know $E\left(\mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{u}_i\right)=\mathbf{0}$, not necessarily $E\left(\mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{u}_i\right)$ (with a hat). Again, I have already shown that $\hat{\mathbf{\Omega}}\overset{p}{\to}\mathbf{\Omega}$, so it seems like it should be true that $N^{-1}\sum_{i=1}^N \mathbf{X}_i^{\top}\hat{\mathbf{\Omega}}^{-1}\mathbf{u}_i\overset{p}{\to}E\left(\mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{u}_i\right)$ (without the hat), however I'm not sure how to piece this into the argument formally.

We know that

$$\text{plim}\left(N^{-1}\sum_{i=1}^N \mathbf{X}_i^{\top}{\mathbf{\Omega}}^{-1}\mathbf{X}_i\right) = E\left(\mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i\right)$$

which is a constant,

and that

$$\text{plim}\left(\mathbf{X}_i^{\top}\hat {\mathbf{\Omega}}^{-1}\mathbf{X}_i\right) = \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i,\;\;\forall i$$

which are random variables.

Combining, it is true that

$$\text{plim}\left(N^{-1}\sum_{i=1}^N \text{plim}\left(\mathbf{X}_i^{\top}\hat {\mathbf{\Omega}}^{-1}\mathbf{X}_i\right)\right) = E\left(\mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i\right)$$

and the question is whether (or under which conditions)

$$\text{plim}\left(N^{-1}\sum_{i=1}^N \text{plim}\left(\mathbf{X}_i^{\top}\hat {\mathbf{\Omega}}^{-1}\mathbf{X}_i\right)\right) =\text{plim}\left(N^{-1}\sum_{i=1}^N \mathbf{X}_i^{\top}\hat {\mathbf{\Omega}}^{-1}\mathbf{X}_i\right)$$

Does this help?