# How would I find the maximum likelihood estimator for $\sigma^2$ in simple linear regression.

I have a simple linear regression equation $$y_i=\beta_1x_i+\epsilon_i$$ and I need to find the maximum likelihood estimator of $$\sigma^2$$. I believe the log likelihood function for this model is l$$(\beta_1,\sigma^2)=-\frac{n}{2}\ln{(2\pi\sigma^2)}-\frac{1}{2\sigma^2}\sum_{i=1}^n(y_i-\beta_1x_i)$$ but when I derive it with respect to $$\sigma^2$$ , set it equal to 0, and attempt to solve for $$\sigma^2$$ I get a result that is undefined. The way I got to this undefined solution is: dl/d$$\beta_1$$=$$-\frac{n\pi}{\sigma}+\frac{1}{\sigma}\sum_{i=1}^n(y_i\beta_1)^2=^{set}=0$$. When I solve for $$\sigma$$ I get $$0=\frac{1}{\sigma}$$ which is an undefined result. Could anyone help me out?

Thanks for sharing your attempt. You should be writing $$\frac{dl}{d\sigma}$$ instead of $$\frac{dl}{d\beta_1}$$, but you seem to be trying to differentiate with respect to $$\sigma$$ anyway.
However, you seem to not have taken the derivative w.r.t $$\sigma$$ correctly. I think the second term should be $$\frac{1}{\sigma^3}\sum_i (y_i - \beta_1 x_i)$$