Statistical Assumptions for SLR The assumptions for the simple linear regression model are: 1) The simple linear regression model of the form Yi = β0 + β1Xi + εi where i = 1, …, n is appropriate. 2) E(εi)=0 2) Var(εi) = σ2 3) εi’s are uncorrelated. STA302/1001 week 2

Properties of Least Squares Estimates Estimate of β0 and β1 – functions of data that can be calculated numerically for a given data set. Estimator of β0 and β1 – functions of the underlying random variables. Recall: the least-square estimators are… Claim: The least squares estimators are unbiased estimators for β0 and β1. Proof:… STA302/1001 week 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 S2 is called the MSE – Mean Square Error it is an unbiased estimator of σ2 (proof later on). STA302/1001 week 2

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 β0 and β1 are equivalent to their maximum likelihood estimators. This results in additional nice properties of MLE’s: they are consistent, sufficient and MVUE. STA302/1001 week 2