103lect5hand

103lect5hand - At the end of the last lecture note, we...

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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 ) A s 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 () 00 ˆˆ SE Var β = ( ) 000 ,Var N ββ Summary • To summarize, we have • This implies that • This will be very useful for hypothesis testing and forming confidence intervals. ( ) 111 N N βββ 11 01 ˆ ˆ 0 ,1 and 0 NN SE SE ∼∼ 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. 1 1 ˆ ˆ STAT c t SE =
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Hypothesis Testing - II • STATA not only computes the estimates β 0 and β 1 , but it also reports values of SE( β 0 ) and SE( β 1 ) (see STATA output). Often (e.g. in the textbook), these results are written in the form of an estimated regression line with SE’s below the respective coefficients. • In our STR-Test Scores example, this is: 01 ˆˆ ˆ (SE( )) (SE( )) ii YX ββ β =+ n Test Scores 698.9 2.28 STR (9.47) (0.48) =− STATA Output regress testscr str Source | SS df MS Number of obs = 420 – -------------+------------------------------ F( 1, 418) = 22.58 Model | 7794.11004 1 7794.11004 Prob > F = 0.0000 Residual | 144315.484 418 345.252353 R-squared = 0.0512 – -------------+------------------------------ Adj R-squared = 0.0490 Total | 152109.594 419 363.030056 Root MSE = 18.581 – ------------------------------------------------------------------------------ testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] – -------------+---------------------------------------------------------------- str | -2.279808 .4798256 -4.75 0.000 -3.22298 -1.336637 _cons | 698.933 9.467491 73.82 0.000 680.3231 717.5428 – ------------------------------------------------------------------------------ Hypothesis Testing - III • So suppose we want to test the hypothesis that class size has no effect on test scores, i.e.
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103lect5hand - At the end of the last lecture note, we...

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