## Simple Linear Regression with Heteroskedastic Noise

Introduction The model we consider is $$Y_i = \alpha + \beta x_i + \epsilon_i$$, where $$\epsilon_i$$ are uncorrelated, and $$\mathbb{V}(\epsilon_i)$$ depends on $$i$$. We discuss two solutions to finding estimators of $$\alpha, \beta$$. Weighted least squares regression leads to best linear unbiased estimators (BLUE). Also, with stronger assumptions on $$\epsilon_i$$, maximum likelihood estimators (MLE) can be found. We begin with a discussion of the homoskedastic case with an emphasis on relations between statistical properties of the least squares estimators and assumptions on $$\epsilon_i$$, which is conducive to understanding the heteroskedastic case.