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I am trying to optimize a panel regression $G=\beta G+e$. $G \in R^{N\times T}$. $\beta\in R^{N\times N}$ is unknown coefficient, constrained to $diag(\beta)=0$, and reduced rank $rank(\beta)\leq r$. Formally, \begin{eqnarray} \begin{aligned} & \underset{\beta}{\text{min}} \left\Vert G-\beta G \right\Vert_F^2 \\ s.t. & \ diag(\beta)=0, \\ &rank(\beta)\leq r \end{aligned} \end{eqnarray}

Can someone give me some advice on this optimization problem? Thanks in advance!

kangyin ye
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  • A $N\times N$ matrix of rank $r$ can be written as a product $WW^\top$ where $W$ is $N\times r$. The $i$-th element of the diagonal is the squared norm of the $i$-th row of or $W$. If you want the diagonal to be zero, then the only solution is $W=0$ meaning that $\beta=0$. – amoeba Nov 14 '18 at 09:47
  • Thanks for answer @amoeba You are correct. It's impossible to estimate the $\beta$ without modifying the regression model. But i haven't have any idea. – kangyin ye Nov 15 '18 at 08:47

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