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Suppose we have two censored variables: $$y_1 = \begin{cases} 0, & y_1^*\leq 0\\ y_1^*, & 0< y_1^* < 1000 \\ 1000, & 1000<y_1^* \end{cases}. $$

$$y_2 = \begin{cases} 0, & y_2^*\leq 0\\ y_2^*, & 0< y_2^* < 1000 \\ 1000, & 1000<y_2^* \end{cases}, $$ where $$y_1^*=X\beta_1+\epsilon_1$$ $$y_2^*=X\beta_2+\epsilon_2.$$ Suppose I observe $y=y_1+y_2,$ and $X$. I want to estimate $$y=X\beta_1+\epsilon_1+X\beta_2+\epsilon_2 \equiv X\beta+\epsilon.$$ How should I go about estimating this model to get unbiased and consistent estimates of $\beta$?

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