\begin{eqnarray}
\Delta \log       \mathbf{c} _{t+1,i} & = & \alpha_{0} + \alpha_{1} \Delta \log       \mathbf{c} _{t,i} + \alpha_{2} \mathbf{e} _{t,i}[\Delta \log             \mathbf{y} _{t+1,i}] + \alpha_{3} \underline{     a }_{t,i} +    \label{eq:CGrowCross} \\
\Delta \log  \bar{\mathbf{c}}_{t+1,i} & = & \alpha_{0} + \alpha_{1} \Delta \log \bar{\mathbf{c}}_{t,i} + \alpha_{2} \mathbf{e} _{t,i}[\Delta \log       \bar{\mathbf{y}}_{t+1,i}] + \alpha_{3} \underline{\bar{a}}_{t,i} +    \label{eq:CGrowCrossBar}
\end{eqnarray}
