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71
Hypothesis Tests and Confidence Intervals
in Multiple Regression
(SW Chapter 7)
Outline
1.
Hypothesis tests and confidence intervals for a single
coefficient
2.
Joint hypothesis tests on multiple coefficients
3.
Other types of hypotheses involving multiple coefficients
4.
How to decide what variables to include in a regression
model?
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Hypothesis Tests and Confidence Intervals for a Single
Coefficient in Multiple Regression
(SW Section 7.1)
•
1
ˆ
(
)
var(
E
β

is approximately distributed
N
(0,1) (CLT).
Thus hypotheses on
can be tested using the usual
t
statistic, and confidence intervals are constructed as {
±
1.96
×
SE
)}.
So too for
2
,…,
k
.
and
are generally not independently distributed – so
neither are their
statistics (more on this later).
73
Example
:
The California class size data
(1)
·
TestScore
= 698.9 – 2.28
×
STR
(10.4)
(0.52)
(2)
= 686.0 – 1.10
– 0.650
PctEL
(8.7)
(0.43)
(0.031)
•
The coefficient on
in (2) is the effect on
TestScores
of a
unit change in
, holding constant the percentage of
English Learners in the district
falls by onehalf
The 95% confidence interval for coefficient on
in (2) is
{–1.10
±
1.96
0.43} = (–1.95, –0.26)
The
t
statistic testing
β
= 0 is
= –1.10/0.43 = –2.54, so
we reject the hypothesis at the 5% significance level
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Standard errors in multiple regression in STATA
reg testscr str pctel
, robust;
Regression with robust standard errors
Number of obs =
420
F(
2,
417) =
223.82
Prob > F
=
0.0000
Rsquared
0.4264
Root MSE
14.464


Robust
testscr 
Coef.
Std. Err.
t
P>t
[95% Conf. Interval]
+
str 
1.101296
.4328472
2.54
0.011
1.95213
.2504616
pctel 
.6497768
.0310318
20.94
0.000
.710775
.5887786
_cons 
686.0322
8.728224
78.60
668.8754
703.189

·
TestScore
= 686.0 – 1.10
×
STR
– 0.650
PctEL
(8.7)
(
0.43
)
0.031)
We use
heteroskedasticityrobust standard errors
– for exactly
the same reason as in the case of a single regressor.
75
Tests of Joint Hypotheses
(SW Section 7.2)
Let
Expn
= expenditures per pupil and consider the
population regression model:
TestScore
i
=
β
0
+
1
STR
2
3
PctEL
u
The null hypothesis that “school resources don’t matter,” and
the alternative that they do, corresponds to:
H
:
= 0
and
vs.
either
≠
0
or
or both
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Tests of joint hypotheses, ctd.
H
0
:
β
1
= 0
and
2
vs.
either
≠
0
or
or both
•
A
joint hypothesis
specifies a value for two or more
coefficients, that is, it imposes a restriction on two or more
coefficients.
In general, a joint hypothesis will involve
q
restrictions.
In
the example above,
= 2, and the two restrictions are
and
= 0.
A “common sense” idea is to reject if either of the
individual
t
statistics exceeds 1.96 in absolute value.
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This note was uploaded on 05/25/2011 for the course ECON 2007 taught by Professor J during the Spring '11 term at UCL.
 Spring '11
 j

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