\documentclass{article}
\usepackage{amsmath}


\begin{document}


\newlength{\txtonelen}
\newlength{\txttwolen}
\newlength{\txtthreelen}

\settowidth{\txtonelen}{effect}
\settowidth{\txttwolen}{i's effect}
\settowidth{\txtthreelen}{i's effect}


\addtolength{\arraycolsep}{-2pt}
\begin{equation*}
  \begin{array}{*{11}{l}}
    \mathbf{y} & = & \mathbf{X}\boldsymbol{\beta} 
                     &   &                 
                       & + &\mathbf{Z}\boldsymbol{\nu}  
                         &   &       
                           & + & \boldsymbol{\epsilon}\\[10pt]
    Y_{ij}    & = & \beta_0                      
                     & + & \beta_1T_{ij}  
                       & + & 1\nu_{0i} 
                         & + & \nu_{1i}t_{ij} 
                           & + & \epsilon_{ij}\\[10pt]
    \text{HD}  & = & \text{Intercept} 
                     & + & \parbox{\txtonelen}{%
                                   \centering time\\ effect} 
                       & + & \parbox{\txttwolen}{%
                               \centering i's effect\\ on int}  
                         & + & \parbox{\txtthreelen}{%
                                  \centering i's effect\\ on time}
                           & + & \text{Resid}
  \end{array}
\end{equation*}




\end{document}