LinearRegression2

9252012 p kolm 17 estimating the error variance s 2

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Unformatted text preview: VER. 9/25/2012. © P. KOLM 17 Estimating the Error Variance s 2 One issue: We assumed that the error variance is known. In practice that is not the case. It needs to be estimated We know that s 2 = E (u 2 ) , so if we could observe the true errors, ui , errors we 1n 2 ˆ would estimate s = å ui . But ui are not known n i =1 2 ˆ Instead, we estimate s 2 using the residuals, ui , 1n2 SSR ˆ s= ui = å n - 2 i =1 n -2 2 Fact: This estimator is unbiased (For you: Verify!) s = s 2 is also referred to as the standard error of the regression VER. 9/25/2012. © P. KOLM 18 The Sampling Distribution Finally, if we assume u | x N (0, s 2 ) Then, we have ˆ b1 N (b1, s 2 / SSTx ) n æ ˆ N çb , s 2n -1 å x 2 / SST b0 ç0 i x ç è i =1 ö ÷ ÷ ÷ ÷ ø VER. 9/25/2012. © P. KOLM 19 Statistical Inference Now, as we know the sampling distribution for the classical linear regression model in the two-variable case we can go ahead and perform statistical inference (hypothesis testing, confidence intervals, etc.) However, we will defer this discussion for linear regression when we discuss the multiv...
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