```Eco 205: Econometrics
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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
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Formula for SE(bˆ1)
5
4 ways to conduct Hypothesis Tests
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Example: Wage vs. Age
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Example: Wage vs. Age
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Regression when X is Binary
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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
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14
Heteroskedasticity and Homoskedasticity
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Homoskedasticity
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Heteroskedasticity
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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?
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The class size data
Heteroskedastic or homoskedastic?
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So far we have (without saying so) assumed
that u might be heteroskedastic.
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What if the errors are in fact homoskedastic?
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Two formulas for the se( bˆ1 )
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Robust standard errors in STATA
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Further Questions
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The Extended LS Assumptions
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Efficiency of OLS
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t-critical values
29
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