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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?

user608881
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  • Can't you use the same way for $\sigma$ as $\beta_0$ or $\beta_1$? – MathArt Oct 13 '20 at 12:25
  • How exactly do you mean? I rewrote the model that way so that I don't get a parameter next to $\epsilon_t$. Then I minimise the argument $\left( \frac{y_t}{\sigma}-\frac{\beta_0}{\sigma}-\frac{\beta_1}{\sigma}x_t \right)^2$. During minimisation I actually found that $\sigma$ does not affect the parameter estimates. – user608881 Oct 13 '20 at 12:59

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