I have the following linear regression model: $y_t=\beta_0+\beta_1x_t+\sigma \epsilon_t$, where $\epsilon_t$ is iid $N(0,1)$.
I am trying to estimate the parameters $\beta_0, \beta_1, \sigma$ using Least-Squares estimation. I am struggling about how to handle the $\sigma$ parameter. I was thinking to rewrite the model as: $$\frac{y_t}{\sigma}=\frac{\beta_0}{\sigma}+\frac{\beta_1}{\sigma}x_t+ \epsilon_t$$ In this case I have the regular linear regression model and can apply the Least-squares method to derive the parameter estimates. Would this be a good approach?