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Suppose I have data on $\{Y_i,X_{1i},X_{2i}\}_{i=1}^{N}$ and the data generating process is $Y_i=\mathbf{1}(\beta_1X_{1i}+\beta_2X_{2i}>e_i)$, where $e_i\sim N(0,\sigma^2)$. Usually, we do a normalization using $\sigma$, so that $Y_i=\mathbf{1}(\frac{\beta_1}{\sigma}X_{1i}+\frac{\beta_2}{\sigma}X_{2i}>\frac{e_i}{\sigma})$ and we estimate $(\frac{\beta_1}{\sigma},\frac{\beta_2}{\sigma})$, as now the normalized error is the known standard normal.

But now suppose I already know that $\beta_1>0$, so instead of normalizing using $\sigma$, I normalize using $\beta_1$ : $Y_i=\mathbf{1}(X_{1i}+\frac{\beta_2}{\beta_1}X_{2i}>\frac{e_i}{\beta_1})$. My question is, is it true that in this case, the parameter I need to estimate becomes $(\frac{\beta_2}{\beta_1},\frac{\sigma}{\beta_1})$?

Thanks!

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