## 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.

Read more