Statistical Assumptions for SLR The assumptions for the

Statistical Assumptions for SLR The assumptions for the simple linear regression model are: 1) The simple linear regression model of the form Yi = β 0 + β 1 Xi + εi where i = 1, …, n is appropriate. 2) E(εi)=0 2) Var(εi) = σ2 3) εi’s are uncorrelated. STA 302/1001 week 2 1

Properties of Least Squares Estimates • Estimate of β 0 and β 1 – functions of data that can be calculated numerically for a given data set. • Estimator of β 0 and β 1 – functions of the underlying random variables. • Recall: the least-square estimators are… • Claim: The least squares estimators are unbiased estimators for β 0 and β 1. • Proof: … STA 302/1001 week 2 2

Estimation of Error Term Variance σ2 • The variance σ2 of the error terms εi’s needs to be estimated to obtain indication of the variability of the probability distribution of Y. • Further, a variety of inferences concerning the regression function and the prediction of Y require an estimate of σ2. • Recall, for random variable Z the estimates of the mean and variance of Z based on n realization of Z are…. • Similarly, the estimate of σ2 is • S 2 is called the MSE – Mean Square Error it is an unbiased estimator of σ2 (proof later on). STA 302/1001 week 2 3

Normal Error Regression Model • In order to make inference we need one more assumption about εi’s. • We assume that εi’s have a Normal distribution, that is εi ~ N(0, σ2). • The Normality assumption implies that the errors εi’s are independent (since they are uncorrelated). • Under the Normality assumption of the errors, the least squares estimates of β 0 and β 1 are equivalent to their maximum likelihood estimators. • This results in additional nice properties of MLE’s: they are consistent, sufficient and MVUE. STA 302/1001 week 2 4