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Eco 205: Econometrics • • • • Any questions? GH #2 due Mon in class RAP intro For Tue, read Chapter 6 & pages 112 of Krueger & Whitmore (Economic Journal 2001), Project STAR paper. … • article is in P:\economics\eco205\ space 1 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals (SW Chapter 5) 2 Hypothesis Testing and the Standard Error of bˆ1 The objective is to test a hypothesis, like b1 = 0, using data – to reach a tentative conclusion whether the (null) hypothesis is correct or incorrect. General setup Null hypothesis and two-sided alternative: H0: b1 = b1,0 vs. H1: b1 ¹ b1,0 where b1,0 is the hypothesized value under the null. Null hypothesis and one-sided alternative: H0: b1 = b1,0 vs. H1: b1 < b1,0 3 Hypothesis Testing 4 Formula for SE(bˆ1) 5 4 ways to conduct Hypothesis Tests 6 Example: Wage vs. Age 7 Example: Wage vs. Age 8 Regression when X is Binary 9 Interpretation when X is binary Y i = b0 + b 1Xi + ui , where X is binary ( Xi = 0 or 1): When Xi = 0, Yi = b0 + ui · the mean of Y i is b0 · that is, E(Y i|Xi=0) = b 0 When Xi = 1, Yi = b0 + b1 + ui · the mean of Y i is b0 + b1 · that is, E(Y i|Xi=1) = b 0 + b1 so: b1 = E(Yi |Xi =1) – E(Y i|Xi=0) = population difference in group means 10 11 12 13 14 Heteroskedasticity and Homoskedasticity 15 Homoskedasticity 16 Heteroskedasticity 17 Comparison of Group Means · Standard error when group variances are unequal: ss2 sl2 SE = + ns nl · Standard error when group variances are equal: SE = s p 1 1 + ns nl 2 2 ( n 1) s + ( n 1) s s l l where s 2p = s (SW, Sect 3.6) ns + nl - 2 sp = “pooled estimator of s2” when s l2 = s s2 · Equal group variances = homoskedasticity · Unequal group variances = heteroskedasticity 18 Example from the Current Population Survey Heteroskedastic or homoskedastic? 19 The class size data Heteroskedastic or homoskedastic? 20 So far we have (without saying so) assumed that u might be heteroskedastic. 21 What if the errors are in fact homoskedastic? 22 23 Two formulas for the se( bˆ1 ) 24 Robust standard errors in STATA 25 Further Questions 26 The Extended LS Assumptions 27 Efficiency of OLS 28 t-critical values 29