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lect5_103_2010_Compatibility_Mode_

# lect5_103_2010_Compatibility_Mode_ - Properties of 0 At the...

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4/8/2010 1 Properties of β 0 At the end of the last lecture note, we considered the properties of the estimator β 1 (of the “parameter” or “coefficient” β 1 ). We can also consider properties of the estimator β 0 (of β 0 ). It turns out they are similar, i.e. – 1) β 0 is an unbiased estimator of β 0 – 2) β 0 is a consistent estimator of β 0 3) As n increases, the distribution of β 0 is well approximated by a normal distribution, i.e. where Var( β 0 ) is given by a slightly more complicated formula (eq 4.22 in the textbook) that again you will not need to remember. 4) As usual ( ( 0 0 ˆ ˆ SE Var β β = ( ( 0 0 0 ˆ ˆ ,Var N β β β Summary To summarize, we have This implies that This will be very useful for hypothesis testing and forming confidence intervals. ( ( ( 29 ( 29 0 0 0 1 1 1 ˆ ˆ ,Var ˆ ˆ ,Var N N β β β β β β ( 29 ( 29 ( 29 ( 29 0 0 1 1 0 1 ˆ ˆ 0,1 and 0,1 ˆ ˆ N N SE SE β β β β β β - -

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4/8/2010 2 Hypothesis Testing - I Given our results, we can do hypothesis tests just as before, i.e 1) State hypotheses (and select significance level), e.g. H 0 : β 1 = c vs. H A : β 1 c where “c” is the value you are testing that β 1 equals. 2) Compute t STAT 3) Compare t STAT to the appropriate critical value. Reject H 0 if the absolute value of the t STAT is greater than the critical value.
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