Maximum-likelihood estimation of the simple-regression model: Deriving the maximum-likelihood estimators of α and β in simple regression is straightforward. Under the assumptions of the model, the Yis...

Maximum-likelihood estimation of the simple-regression model: Deriving the maximum-likelihood estimators of α and β in simple regression is straightforward. Under the assumptions of the model, the Yis are independently and normally distributed random variables with expectations α + βx_i
and common variance σ2 ε . Show that if these assumptions hold, then the least-squares coefficients A and B are the maximum-likelihood estimators of α and β and that σ²
_ε
= PE²
_i
=n is the maximum-likelihood estimator of σ²
ε . Note that the MLE of the error variance is biased. (Hints: Because of the assumption of independence, the joint probability density for the Yis is the product of their marginal probability densities

Find the log-likelihood function; take the partial derivatives of the log likelihood with respect to the parameters α, β, and σ2 ε ; set these partial derivatives to 0; and solve for the maximum likelihood estimators.) A more general result is proved in Section 9.3.3.