further inference in the multiple regression model prepared by vera tabakova, east carolina...
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Chapter 6
Further Inference in the Multiple Regression Model
Prepared by Vera Tabakova, East Carolina University
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Chapter 6: Further Inference in the Multiple Regression Model
6.1 The F-Test
6.2 Testing the Significance of the Model
6.3 An Extended Model
6.4 Testing Some Economic Hypotheses
6.5 The Use of Nonsample Information
6.6 Model Specification
6.7 Poor Data, Collinearity and Insignificance
6.8 Prediction Slide 6-2Principles of Econometrics, 3rd Edition
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6.1 The F-Test
Slide 6-3Principles of Econometrics, 3rd Edition
(6.1)
(6.2)
1 2 3i i i iS P A e
0 2: 0H
1 2: 0H
1 3i i iS A e
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6.1 The F-Test
If the null hypothesis is not true, then the difference between SSER and SSEU
becomes large, implying that the constraints placed on the model by the null
hypothesis have a large effect on the ability of the model to fit the data.
Slide 6-4Principles of Econometrics, 3rd Edition
(6.3)
R U
U
SSE SSE JF
SSE N K
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6.1 The F-Test
Hypothesis testing steps:
1. Specify the null and alternative hypotheses:
2. Specify the test statistic and its distribution if the null hypothesis is true
3. Set and determine the rejection region
Using α=.05, the critical value from the -distribution is .
Thus, H0 is rejected if .
Slide 6-5Principles of Econometrics, 3rd Edition
0 2: 0H 1 2: 0H
(1, 72)
1
75 3R U
U
SSE SSEF F
SSE
(.95,1,72) 3.97cF F
3.97F
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6.1 The F-Test
4. Calculate the sample value of the test statistic and, if desired, the p-
value
5. State your conclusion
Since , we reject the null hypothesis and conclude that price does
have a significant effect on sales revenue. Alternatively, we reject H0 because
.
Slide 6-6Principles of Econometrics, 3rd Edition
2961.827 1718.943 152.06
1718.943 75 3R U
U
SSE SSE JF
SSE N K
(1,72)[ 52.06] .0000p P F
52.06 cF F
.0000 .05p
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6.1.1 The Relationship Between t- and F-Tests
The elements of an F-test :
1. The null hypothesis consists of one or more equality restrictions J. The null
hypothesis may not include any ‘greater than or equal to’ or ‘less than or equal
to’ hypotheses.
2. The alternative hypothesis states that one or more of the equalities in the null
hypothesis is not true. The alternative hypothesis may not include any ‘greater
than’ or ‘less than’ options.
3. The test statistic is the F-statistic
Slide 6-7Principles of Econometrics, 3rd Edition
R U
U
SSE SSE JF
SSE N K
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6.1.1 The Relationship Between t- and F-Tests
4. If the null hypothesis is true, F has the F-distribution with J numerator
degrees of freedom and N-K denominator degrees of freedom. The null
hypothesis is rejected if .
5. When testing a single equality null hypothesis it is perfectly correct to use
either the t- or F-test procedure; they are equivalent.
Slide 6-8Principles of Econometrics, 3rd Edition
(1 , , ), where c c J N KF F F F
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6.2 Testing the Significance of the Model
Slide 6-9Principles of Econometrics, 3rd Edition
(6.4)
(6.5)
(6.6)
1 2 2 3 3i i i iK K iy x x x e
0 2 3
1
: 0, 0, , 0
: of the is nonzero
for 2,3,
K
k
H
H At least one
k K
1i iy e
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6.2 Testing the Significance of the Model
Slide 6-10Principles of Econometrics, 3rd Edition
(6.7)
* 2 21
1 1
( ) ( )N N
R i ii i
SSE y b y y SST
( ) / ( 1)
/ ( )
SST SSE KF
SSE N K
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6.2 Testing the Significance of the Model
Example: Big Andy’s sales revenue
1.
2. If the null is true
3. H0 is rejected if
4.
5. Since 29.95>3.12 we reject the null and conclude that price or advertising expenditure or both have an influence on sales.
Slide 6-11Principles of Econometrics, 3rd Edition
1 2 3i i i iS P A e
0 2 3: 0, 0H
1 2 3: 0 or 0, or both are nonzeroH
(2,72)
( ) / (3 1)
/ (75 3)
SST SSEF F
SSE
3.12F
( ) / ( 1) (3115.485 1718.943) / 229.25
/ ( ) 1718.943 / (75 3)
SST SSE KF
SSE N K
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6.3 An Extended Model
Figure 6.1 A Model Where Sales Exhibits Diminishing Returns to Advertising Expenditure
Slide 6-12Principles of Econometrics, 3rd Edition
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6.3 An Extended Model
Slide 6-13Principles of Econometrics, 3rd Edition
(6.8)
(6.9)
(6.10)
1 2 3S P A e
21 2 3 4S P A A e
3 4( held constant)
( ) ( )2
P
E S E SA
A A
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6.3 An Extended Model
Slide 6-14Principles of Econometrics, 3rd Edition
(6.11)
2ˆ 109.72 7.640 12.151 2.768
(se) (6.80) (1.046) (3.556) (.941)i i i iS P A A
1 2 2 3 3 4 4i i i i iy x x x e
22 3 4, , ,i i i i i i i iy S x P x A x A
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6.4 Testing Some Economic Hypotheses
6.4.1 The Significance of Advertising
Slide 6-15Principles of Econometrics, 3rd Edition
(6.12)
(6.13)
21 2 3 4i i i i iS P A A e
1 2i i iS P e
0 3 4: 0, 0H
1 3 4: 0 or 0H
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6.4 Testing Some Economic Hypotheses
Since , we reject the null hypothesis and conclude that advertising does have a significant effect upon sales revenue.
Slide 6-16Principles of Econometrics, 3rd Edition
(2,71)
2
75 4R U
U
SSE SSEF F
SSE
(.95,2,71) 3.126cF F
(2,71)[ 8.44] .0005P F
8.44 3.126cF F
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6.4.2 The Optimal Level of Advertising
Economic theory tells us that we should undertake all those actions
for which the marginal benefit is greater than the marginal cost. This
optimizing principle applies to Big Andy’s Burger Barn as it attempts
to choose the optimal level of advertising expenditure.
Slide 6-17Principles of Econometrics, 3rd Edition
3 4( held constant)
( )2
P
E SA
A
3 42 1oA
ˆ12.1512 2 ( 2.76796) 1oA
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6.4.2 The Optimal Level of Advertising
Big Andy has been spending $1,900 per month on advertising. He
wants to know whether this amount could be optimal.
The null and alternative hypotheses for this test are
Slide 6-18Principles of Econometrics, 3rd Edition
0 3 4 1 3 4: 2 1.9 1 : 2 1.9 1H H
0 3 4 1 3 4: 3.8 1 : 3.8 1H H
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6.4.2 The Optimal Level of Advertising
Slide 6-19Principles of Econometrics, 3rd Edition
3 4
3 4
( 3.8 ) 1
se( 3.8 )
b bt
b b
23 4 3 4 3 4
2
var( 3.8 ) var( ) 3.8 var( ) 2 3.8 cov( , )
12.6463 3.8 .884774 2 3.8 3.288746
.427967
b b b b b b
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6.4.2 The Optimal Level of Advertising
Because , we cannot reject the null
hypothesis that the optimal level of advertising is $1,900 per month.
There is insufficient evidence to suggest Andy should change his
advertising strategy.
Slide 6-20Principles of Econometrics, 3rd Edition
1.6330 1 .633.9676
.65419.427967t
1.994 .9676 1.994
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6.4.2 The Optimal Level of Advertising
Slide 6-21Principles of Econometrics, 3rd Edition
21 2 4 4(1 3.8 )i i i i iS P A A e
21 2 4( ) ( 3.8 )i i i i i iS A P A A e
(1552.286 1532.084) /1.9362
1532.084 / 71F
2 23.976 (1.994)c cF t
(1,71)
(71) (71)
-value [ .9362]
[ .9676] [ .9676] .3365
p P F
P t P t
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6.4.2a A One-Tailed Test with More than One Parameter
Reject H0 if t ≥ 1.667.
t = .9676
Because .9676 < 1.667, we do not reject H0.
There is not enough evidence in the data to suggest the optimal level
of advertising expenditure is greater than $1900.
Slide 6-22Principles of Econometrics, 3rd Edition
0 3 4
1 3 4
: 3.8 1
: 3.8 1
H
H
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6.4.2 Using Computer Software
Slide 6-23Principles of Econometrics, 3rd Edition
21 2 3 4
21 2 3 4
( )
6 1.9 1.9
80
E S P A A
0 3 4
1 2 3 4
: 3.8 1 and
6 1.9 3.61 80
H
5.74 and the -value = .0049F p
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6.5 The Use of Nonsample Information
Slide 6-24Principles of Econometrics, 3rd Edition
(6.14)
(6.15)
1 2 3 4 5ln( ) ln( ) ln( ) ln( ) ln( )Q PB PL PR I
1 2 3 4 5
1 2 3 4 5
2 3 4 5
ln( ) ln ln ln ln
ln( ) ln( ) ln( ) ln( )
ln( )
Q PB PL PR I
PB PL PR I
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6.5 The Use of Nonsample Information
Slide 6-25Principles of Econometrics, 3rd Edition
(6.16)
(6.17)
2 3 4 5 0
1 2 3 4 5ln( ) ln( ) ln( ) ln( ) ln( )t t t t t tQ PB PL PR I e
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6.5 The Use of Nonsample Information
Slide 6-26Principles of Econometrics, 3rd Edition
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6.5 The Use of Nonsample Information
Slide 6-27Principles of Econometrics, 3rd Edition
(6.18)
4 2 3 5
1 2 3
2 3 5 5
1 2
3 5
1 2 3 5
ln( ) ln( ) ln( )
ln( ) ln( )
ln( ) ln( )
ln( ) ln( ) ln( ) ln( )
ln ln ln
t t t
t t t
t t
t t t t t
t t tt
t t t
Q PB PL
PR I e
PB PR
PL PR I PR e
PB PL Ie
PR PR PR
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6.5 The Use of Nonsample Information
Slide 6-28Principles of Econometrics, 3rd Edition
(6.19)ln( ) 4.798 1.2994ln 0.1868ln 0.9458ln
(se) (0.166) (0.284) (0.427)
t t tt
t t t
PB PL IQ
PR PR PR
* * * *4 2 3 5
( 1.2994) 0.1868 0.9458
0.1668
b b b b
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6.6 Model Specification
6.6.1 Omitted Variables
Slide 6-29Principles of Econometrics, 3rd Edition
(6.20)
5534 3132 4523
(se) (11230) (803) (1066)
( -value) (.622) (.000) (.000)
i i iFAMINC HEDU WEDU
p
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6.6.1 Omitted Variables
Slide 6-30Principles of Econometrics, 3rd Edition
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6.6.1 Omitted Variables
Slide 6-31Principles of Econometrics, 3rd Edition
(6.21)
(6.22)
26191 5155
(se) (8541) (658)
( -value) (.002) (.000)
i iFAMINC HEDU
p
1 2 2 3 3i i i iy x x e
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6.6.1 Omitted Variables
Slide 6-32Principles of Econometrics, 3rd Edition
(6.23)
(6.24)
* * 2 32 2 2 3
2
cov( , )bias( ) ( )
var( )
x xb E b
x
7755 3211 4777 14311 6
(se) (11163) (797) (1061) (5004)
( -value) (.488) (.000) (.000) (.004)
i i i iFAMINC HEDU WEDU KL
p
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6.6.2 Irrelevant Variables
Slide 6-33Principles of Econometrics, 3rd Edition
5 67759 3340 5869 14200 6 889 1067
(se) (11195) (1250) (2278) (5044) (2242) (1982)
( -value) (.500) (.008) (.010) (.005) (.692) (.591)
i i i i i iFAMINC HEDU WEDU KL X X
p
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6.6.3 Choosing the Model
1. Choose variables and a functional form on the basis of your
theoretical and general understanding of the relationship.
2. If an estimated equation has coefficients with unexpected signs, or
unrealistic magnitudes, they could be caused by a misspecification
such as the omission of an important variable.
Slide 6-34Principles of Econometrics, 3rd Edition
* *2 2where and c x x c
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6.6.3 Choosing the Model
3. One method for assessing whether a variable or a group of variables
should be included in an equation is to perform significance tests. That
is, t-tests for hypotheses such as
or F-tests for hypotheses such as .
Failure to reject hypotheses such as these can be an indication that the
variable(s) are irrelevant.
4. The adequacy of a model can be tested using a general specification test
known as RESET.
Slide 6-35Principles of Econometrics, 3rd Edition
0 3: 0H
0 3 4: 0H
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6.6.3a The RESET Test
Slide 6-36Principles of Econometrics, 3rd Edition
(6.25)
1 2 2 3 3i i i iy x x e
1 2 2 3 3ˆi i iy b b x b x
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6.6.3a The RESET Test
Slide 6-37Principles of Econometrics, 3rd Edition
(6.26)
(6.27)
21 2 2 3 3 1 ˆi i i i iy x x y e
2 31 2 2 3 3 1 2ˆ ˆi i i i i iy x x y y e
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6.6.3a The RESET Test
Slide 6-38Principles of Econometrics, 3rd Edition
0 1
0 1 2
: 0 5.984 -value .015
: 0 3.123 -value .045
H F p
H F p
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6.7 Poor data, Collinearity and Insignificance
6.7.1 The Consequences of Collinearity
Slide 6-39Principles of Econometrics, 3rd Edition
(6.28)
1 2 2 3 3i i i iy x x e
2
22 2
23 2 21
var( )(1 ) ( )
N
ii
br x x
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6.7.1 The Consequences of Collinearity
The effects of imprecise information:
1. When estimator standard errors are large, it is likely that the usual t-
tests will lead to the conclusion that parameter estimates are not
significantly different from zero. This outcome occurs despite
possibly high or F-values indicating significant explanatory power
of the model as a whole.
Slide 6-40Principles of Econometrics, 3rd Edition
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6.7.1 The Consequences of Collinearity
2. The estimators may be very sensitive to the addition or deletion of a
few observations, or the deletion of an apparently insignificant
variable.
3. Despite the difficulties in isolating the effects of individual
variables from such a sample, accurate forecasts may still be
possible if the nature of the collinear relationship remains the same
within the new (future) sample observations.
Slide 6-41Principles of Econometrics, 3rd Edition
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6.7.2 An Example
MPG = miles per gallon
CYL = number of cylinders
ENG = engine displacement in cubic inches
WGT = vehicle weight in pounds
Slide 6-42Principles of Econometrics, 3rd Edition
42.9 3.558
(se) (.83) (.146)
( -value) (.000) (.000)
i iMPG CYL
p
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6.7.2 An Example
Slide 6-43Principles of Econometrics, 3rd Edition
44.4 .268 .0127 .00571
(se) (1.5) (.413) (.0083) (.0071)
( -value) (.000) (.517) (.125) (.000)
i i i iMPG CYL ENG WGT
p
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6.7.3 Identifying and Mitigating Collinearity
Identifying Collinearity
1. Examining pairwise correlations.
2. Using auxiliary regression
If the R2 from this artificial model is high, above .80 say, the implication is
that a large portion of the variation in is explained by variation in the
other explanatory variables.
Slide 6-44Principles of Econometrics, 3rd Edition
2 1 1 3 3i i i K iKx a x a x a x error
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6.7.3 Identifying and Mitigating Collinearity
Mitigating Collinearity
1. Obtain more information and include it in the analysis.
2. Introduce nonsample information in the form of restrictions on the
parameters.
Slide 6-45Principles of Econometrics, 3rd Edition
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6.8 Prediction
Slide 6-46Principles of Econometrics, 3rd Edition
(6.29)1 2 2 3 3i i i iy x x e
0 1 02 2 03 3 0y x x e
0 1 02 2 03 3y b x b x b
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6.8 Prediction
Slide 6-47Principles of Econometrics, 3rd Edition
1 2 02 3 03 0 1 2 02 3 03
0 1 2 02 3 03
2 20 1 02 2 03 3
02 1 2 03 1 3 02 03 2 3
var( ) var[( ) ( )]
var( )
var( ) var( ) var( ) var( )
2 cov( , ) 2 cov( , ) 2 cov( , )
f x x e b b x b x
e b b x b x
e b x b x b
x b b x b b x x b b
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Keywords
Slide 6-48Principles of Econometrics, 3rd Edition
a single null hypothesis with more than one parameter
auxiliary regressions collinearity F-test irrelevant variable nonsample information omitted variable omitted variable bias overall significance of a regression
model regression specification error test
(RESET) restricted least squares
restricted sum of squared errors single and joint null hypotheses unrestricted sum of squared errors
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Chapter 6 Appendices
Slide 6-49Principles of Econometrics, 3rd Edition
Appendix 6A Chi-Square and F-tests: More Details
Appendix 6B Omitted Variable Bias: A Proof
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Appendix 6A Chi-Square and F-tests: More Details
Slide 6-50Principles of Econometrics, 3rd Edition
(6A.1)
(6A.3)
(6A.2)
( ) /
/ ( )R U
U
SSE SSE JF
SSE N K
21 ( )2
( )R UJ
SSE SSEV
21 ( )2
( )ˆˆ
R UJ
SSE SSEV
(6A.4)
22
2 ( )2
ˆ( )N K
N KV
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Appendix 6A Chi-Square and F-tests: More Details
Slide 6-51Principles of Econometrics, 3rd Edition
(6A.5)
1 11 2
2 2
/( , )
/
V mF F m m
V m
2
2 ,2
2
ˆ ˆ
R U
R UJ N K
SSE SSEJ SSE SSE J
FN K
N K
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Appendix 6A Chi-Square and F-tests: More Details
Slide 6-52Principles of Econometrics, 3rd Edition
1VF
J
0 3 4: 0H
21 2 3 4i i i i iS P A A e
2
8.44 -value .0005
16.88 -value .0002
F p
p
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Appendix 6A Chi-Square and F-tests: More Details
Slide 6-53Principles of Econometrics, 3rd Edition
0 3 4: 3.8 1H
2
.936 -value .3365
.936 -value .3333
F p
p
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Appendix 6B Omitted Variable Bias: A Proof
Slide 6-54Principles of Econometrics, 3rd Edition
1 2 2i i iy x v
1 2 2 3 3i i i iy x x e
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Appendix 6B Omitted Variable Bias: A Proof
Slide 6-55Principles of Econometrics, 3rd Edition
(6B.1)* 2 22 22
2 2
( )( )
( )i i
i i
i
x x y yb w v
x x
2 22
2 2
( )
( )i
ii
x xw
x x
*2 2 3 3i i i ib w x w e
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Appendix 6B Omitted Variable Bias: A Proof
Slide 6-56Principles of Econometrics, 3rd Edition
*2 2 3 3
2 2 32 3 2
2 2
2 2 3 32 3 2
2 2
2 32 3 2
2
( )
( )
( )
( )( )
( )
cov( , )
var( )
i i
i i
i
i i
i
E b w x
x x x
x x
x x x x
x x
x x
x