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week2 - 2 • Recall for random variable Z the estimates of...

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STA302/1001 week 2 1 Statistical Assumptions for SLR The assumptions for the simple linear regression model are: 1) The simple linear regression model of the form Y i = β 0 + β 1 X i + ε i where i = 1, …, n is appropriate. 2) E ( ε i )=0 2) Var( ε i ) = σ 2 3) ε i ’s are uncorrelated.
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STA302/1001 week 2 2 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:
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STA302/1001 week 2 3 Estimation of Error Term Variance σ 2 The variance σ 2 of the error terms ε i ’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 σ
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Unformatted text preview: 2 . • 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 σ 2 is • S 2 is called the MSE – Mean Square Error it is an unbiased estimator of σ 2 (proof later on). ∑ =-= n i i e n s 1 2 2 2 1 STA302/1001 week 2 4 Normal Error Regression Model • In order to make inference we need one more assumption about ε i ’s. • We assume that ε i ’s have a Normal distribution, that is ε i ~ N (0, σ 2 ). • The Normality assumption implies that the errors ε i ’s are independent (since they are uncorrelated). • Under the Normality assumption of the errors, the least squares estimates of β 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....
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week2 - 2 • Recall for random variable Z the estimates of...

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