Theory

General EIV model

\[\begin{equation} \begin{aligned} &\vb*{z}_t~=~ \vb*{z}^*_t + \vb*{q}_t \\ &\vb*{z}^*_t ~=~ \left[ y^*_t \quad \vb*{x}^*_t\right] \\ &\vb*{q}_t ~=~ \left[v_t \quad \vb*{a}_t \right] \\ &y_t^* ~=~ b_0 + \vb*{x}^*_t\vb*{B}^* + \epsilon_t \end{aligned} \end{equation}\]

\[\begin{align} \begin{bmatrix} Z_{11} \\ Z_{21} \\ \vdots \\ Z_{n1}\end{bmatrix} &=~ \begin{bmatrix} 1 & \vb* x_1 \\ 1 & \vb* x_2 \\ \vdots \\ 1 & \vb* x_n \end{bmatrix} \begin{bmatrix} b_0 \\ \vb*{B}^* \end{bmatrix} + \begin{bmatrix}\mathbb{I}_n \quad -(\vb*{B}^*{'} \otimes \mathbb{I}_n) \end{bmatrix} \begin{bmatrix} \vb*{V} \\ vec(\mathbf A) \end{bmatrix} \end{align}\]

Maximum likelihood

\[\begin{align} \vb B_\text{wtls} &=~\left[ (\vb X-\vb A)'\vb W^{-1}\vb X \right] ^{-1}\left[(\vb X-\vb A)\vb W^{-1}\vb Y\right] \\ \Sigma_{B}^\text{wtls} &=~\left[ (\vb X-\vb A)'\vb W^{-1}\vb X \right]^{-1}\mathbf{M}\left[ (\vb X-\vb A)'\vb W^{-1}\vb X \right]^{-1'} \end{align}\]

where $\mathbf{M}=(\vb X-\vb A)'\vb W^{-1}\Sigma \vb W^{-1}(\vb X-\vb A)$.

Bayesian solution

\[\begin{align} \vb B &~=~ \left({\Sigma_{B}^\text{wtls}}^{-1} ~+~ \Sigma_0^{-1}\right)^{-1}\left({\Sigma_{B}^\text{wtls}}^{-1}\vb B_\text{wtls}~+~\Sigma_0^{-1}\vb B_0\right) \\ \Sigma_{B} &~=~\left( {\Sigma_{B}^{\text{wtls}}}^{-1}~+~\Sigma_0^{-1}\right)^{-1} \end{align}\]

Reference

Fischer, M (2023) Measurement Error Proxy System Models: MEPSM v0.2. link to preprint