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 twovariable 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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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
 Fall '14

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