LR - Standard Regression Assumptions

each observed response 𝑦^(𝑖) are independent Normal random variables with:
- expectation = 𝐄[𝑌^(𝑖)|𝑋₁^(𝑖), …, 𝑋_𝑘^(𝑖)] = 𝜃₀+ 𝜃₁𝑋₁^(𝑖)+ … + 𝜃_𝑘𝑋_𝑘^(𝑖)
- variance = 𝑉𝑎𝑟(𝑌^(𝑖)|𝑋₁^(𝑖), …, 𝑋_𝑘^(𝑖)) = 𝜎² # variance is same constant for all 𝑋₁^(𝑖), …, 𝑋_𝑘^(𝑖)
predictors {𝑋₁^(𝑖), …, 𝑋_𝑘^(𝑖)} are considered non-random because they are observed
as a consequence:
- 𝜃₀, …, 𝜃_𝑘have Normal Distribution

Desired Properties	Required Assumptions
𝜃ˆ_𝑂𝐿𝑆 is an unbiased estimate of 𝜃	𝐄[𝜖^(𝑖)] = 0, ∀𝑖
𝜃ˆ_𝑂𝐿𝑆 is an unbiased estimate of 𝜃 𝜃ˆ_𝑂𝐿𝑆 is a BLUE estimator	𝐄[𝜖^(𝑖)] = 0, ∀𝑖 𝑉𝑎𝑟(𝜖^(𝑖)) = constant < ∞, ∀𝑖 𝐶𝑜𝑣(𝜖^(𝑖),𝜖^(𝑗)) = 0, ∀𝑖≠𝑗
𝜃ˆ_𝑂𝐿𝑆 is an unbiased estimate of 𝜃 𝜃ˆ_𝑂𝐿𝑆 is a BLUE estimator 𝜃ˆ_𝑂𝐿𝑆 is mathematically equivalent to MLE	𝐄[𝜖^(𝑖)] = 0, ∀𝑖 𝑉𝑎𝑟(𝜖^(𝑖)) = constant < ∞, ∀𝑖 𝐶𝑜𝑣(𝜖^(𝑖),𝜖^(𝑗)) = 0, ∀𝑖≠𝑗 All 𝜖 are Independent and Identically Distributed (IID) from the normal distribution

normality of sampling distribution of means - the distribution of sample means is normally distributed
errors 𝑒^(𝑖)& 𝑒^(𝑗) are independent of each other (where 𝑒^(𝑖) = 𝑦̂^(𝑖) - 𝑦^(𝑖))
absence of outliers - outliers have been removed from the dataset
homogeneity of variance - population variances at different levels of each independent variable {𝑋₁, …, 𝑋_𝑘} are equal

／var／log marcus chiu