Lecture_6__Prof._Arkonac's__slides_(Ch_5_and_examples)

Lecture_6__Prof._Arkonac's__slides_(Ch_5_and_examples) -...

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Regression with a single Regressor: Hypothesis Testing and Confidence Intervals Lecture 6 Prof: Seyhan Erden Arkonac, PhD Solutions to Problem Set 1 are posted. Problem Set 2 is due on September 28 th . 1
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TA Information: Naihobe Gonzalez E-mail: ndg2109@columbia.edu Office Hours: Thurs 12-1 (Uris Library), Recitation: Thurs 11-11:50 (PUP 424) TA Information: Ju Hyun Kim E-mail: jk3201@columbia.edu Office Hours: Recitation: Fri 2:10-3PM(PUP 424), Office Hours: Fri 3:10- 4:10PM(Lehman) TA Information: WooRam Park E-mail: wp2135@columbia.edu Office Hours: Office hours : Thurs 2:00~3:00 IAB 1006A Recitation Thurs 3:10~4:00 IAB 403 TA Information: Ran Huo E-mail: rh2346@columbia.edu Office Hours: Recitation: Thursday 12:00-12:50 404IAB; Office Hour: Wednesday 1-2 1006A IAB TA Information: Shreya Agarwal E-mail: sa2628@columbia.edu Office Hours: Mon 12:30pm - 1:30 pm (Uris Library Common Area) Recitation: Fri 11:00am - 11:50am 2
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Stata sessions: Thursday 9/23, 11-11:50am, SCH 558 (Naihobe) Thursday 9/23, 12-12:50pm, SCH 558 (Ran) Thursday 9/23, 3:10-4pm, SCH 558 (WooRam) Friday 9/24, 2:10-3pm, SCH 558 (Ju Hyun) Every week starting 9/24 11-1150am SCH 558 (Shreya)
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Last thing we did last time: When X i is binary (zero or one), what happens to s? Let’s see an example 4
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Let’s work on an example: Random sample of 250 male and 280 female workers ^ Wage = 12.52 + 2.12Male (0.23) (1.06) (a) What is the estimated gender gap? (b) Is the estimated gender gap significantly different from zero? (c) What is the mean wage for women in this sample? what is it for men? 5
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Answers: (a) Estimated gender gap is 2.12 (b) H 0 : β 1 = 0 H 1 : β 1 ≠ 0 Test stat = (2.12-0)/1.06 = 2 Checking from Standard Normal Distr. table p-value = 0.0228x2 = 0.0456 Any α > 0.0456 will reject H 0 meaning β 1 is significant. Any α < 0.0456 will not reject H 0 meaning β 1 is NOT significant 6
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Answers (cont’) (c) Mean wage for women is 12.52 because X=0 when an observation is woman. Mean wage for men is 12.52+2.12=14.64 (since X=1 for men) If we regress Wage on Female using the same data: ^ Wage = 14.64 – 2.12Female where the variable Female=1 for observations that belong to a female and female=0 otherwise. 7
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8 Confidence Intervals for 1 (Section 5.2) Recall that a 95% confidence is, equivalently: The set of points that cannot be rejected at the 5% significance level; A set-valued function of the data (an interval that is a function of the data) that contains the true parameter value 95% of the time in repeated samples. Because the t -statistic for 1 is N(0,1) in large samples, construction of a 95% confidence for 1 is just like the case of the sample mean: 95% confidence interval for 1 = { 1 ˆ 1.96 SE ( 1 ˆ )}
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9 Summary of Statistical Inference about 0 and 1 : Estimation : OLS estimators 0 ˆ and 1 ˆ 0 ˆ and 1 ˆ have approximately normal sampling distributions in large samples Testing : H 0 : 1 = 1,0 v. 1 1,0 ( 1,0 is the value of 1 under H 0 ) t = ( 1 ˆ 1,0 )/ SE ( 1 ˆ ) p -value = area under standard normal outside t act (large n ) Confidence Intervals : 95% confidence interval for 1 is { 1 ˆ 1.96 SE ( 1 ˆ )} This is the set of 1 that is not rejected at the 5% level The 95% CI contains the true 1 in 95% of all samples.
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Earlier example on wage Construct a 95% confidence interval for the gender gap: 2.12 ± (1.96)(1.06) 2.12-(1.96)(1.06)≤ β 1 ≤ 2.12+(1.96)(1.06) approximately, 0.04 ≤ β
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Lecture_6__Prof._Arkonac's__slides_(Ch_5_and_examples) -...

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