# A question about normalization in probit model (binary response model with normal error)

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!