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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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statistical assumptions for slr
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
Properties of Least Squares Estimates
  • Estimate of β0and β1 – functions of data that can be calculated numerically for a given data set.
  • Estimator of β0and β1 – functions of the underlying random variables.
  • Recall: the least-square estimators are…
  • Claim: The least squares estimators are unbiased estimators for β0and β1.
  • Proof:…

STA302/1001 week 2

estimation of error term variance 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
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 β0and 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

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