# MLE for regression linear model

Hi, let $Y_i$ be this regression linear model $$Y_i=\beta_1+\beta_2(x_i-\bar{x})+\beta_3(x_i-\bar{x})^2+\epsilon_i$$

where $(y_i,x_i)\in\mathbb{R}^2$, $\epsilon_i\sim N(0,\sigma^2)$ independent and $(\beta_1, \beta_2, \beta_3,\sigma^2)\in\mathbb{R}^3\times\mathbb{R}^+$ are unknown parameters.

How can I find MLE $\hat\beta$ for $\beta$ and $\hat\sigma$ for $\sigma$ given $(X^TX)^{-1}$, $X^Ty$, $y^Ty$, $\bar{x}$ and $\bar{y}$?

I know that $\hat\beta=(X^TX)^{-1}X^Ty$ and $\hat\sigma=\frac{e^Te}{n}$, but I don't understand how to put all the pieces together.

• Um... you have given us the formula for $\hat\beta$, so what are you trying to ask? – whuber Feb 7 '18 at 22:58
• Good point. I was just wondering why my teacher gave me $y^Ty$, $\bar{x}$ and $\bar{y}$. Maybe I can't find $\hat\sigma$ without those values. – Paul Feb 8 '18 at 16:41
• Are you adding to your question by stating that you also want to find $\hat\sigma$? If so, please edit your post to include that. – whuber Feb 8 '18 at 17:19
• Post edited with your suggestion. – Paul Feb 8 '18 at 21:20