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Further Inference in the Multiple Regression Model
Hill et al Chapter 8
The F-TestUsed to test hypotheses on one or more parameters
Unrestricted model:
1 2 3t t t ttr p a e
Restricted model
1 3t t ttr a e
0 2: 0H
1 2: 0H
0R USSE SSE
The F-statisticAre the differences in SSE significant?
R U
U
SSE SSE JF
SSE T K
If the null hypothesis is true, then the statistic F has an F-distribution with J numerator degrees of freedom and T-K denominator degrees of freedom.
Example
USSE = 1805.168 RSSE = 1964.758
1964.758 1805.168 1
1805.168 52 3
4.332
R U
U
SSE SSE JF
SSE T K
p = P[F1,49 4.332] = .0427
Fc= 4.038
Reject the null hypothesis
1 2 3t t t ttr p a e 0 2: 0H 1 2: 0H
1 3t t ttr a e
Testing the significance of a model
1 2 2 3 3t t t tK K ty x x x e
0 2 3
1
: 0, 0, , 0
: of the is nonzeroK
k
H
H at least one
Restricted model 1t ty e *1
tyb y
T
* 2 21( ) ( )R t tSSE y b y y SST
( ) /( 1)
/( )
SST SSE KF
SSE T K
Example1 2 3t t t ttr p a e
0 2 3: 0, 0H 1 2 3: 0, or 0, or both are nonzeroH
ANALYSIS OF VARIANCE
SS DF MS
REGRESSION 11776. 2. 5888.1
ERROR 1805.2 49. 36.840
TOTAL 13581. 51. 266.30
( ) /( 1) (13581.35 1805.168) / 2 5888.09159.83
/( ) 1805.168/(52 3) 36.84
SST SSE KF
SSE T K
Fc = 3.187
An extended model2
1 2 3 4t t t t ttr p a a e
3 4
( held constant)
( ) ( )2t t
tt tp
E tr E tra
a a
2ˆ 104.81 6.582 2.948 0.0017
(6.58) (3.459) (0.786) (0.0361) (s.e.)t t t ttr p a a
2ˆ 110.46 10.198 3.361 0.0268
(3.74) (1.582) (0.422) (0.0159) (s.e.)t t t ttr p a a
The significance of advertising
261.41R U
U
SSE SSE JF
SSE T K
21 2 3 4t t t t ttr p a a e
0 3 4: 0, 0H
1 3 4: 0, or 0, or both are nonzeroH
J=2, T=78, K= 4. SSEU = 2592. SSER = 20907.331
Fc=3.120
The optimal level of advertisingMarginal benefit from advertising:
3 4
( held constant)
( )2t
tt p
E tra
a
Marginal benefit equals marginal cost:
3 42 1ta
ˆ3.361 2( .0268) 1ta ˆ 44.0485ta
Is this significantly different from $40000? T-test
0 3 4: 2 (40) 1H
1 3 4: 2 (40) 1H
3 4
3 4
( 80 ) 1
se( 80 )
b bt
b b
2
3 4 3 4 3 4ˆ ˆ ˆ ˆvar( 80 ) var( ) 80 var( ) 2(80)cov( , ) .76366b b b b b b
1.217 1.248
.76366t
tc = 1.993
Is this significantly different from $40000? F-test
Restricted model obtained by writing the equation under the assumption that the null is true:
0 3 4
0 3 4
: 2 (40) 1
: 1 2 (40)
H
H
21 2 4 4(1 80 )t t t t ttr p a a e
21 2 4( ) ( 80 )t t t t t ttr a p a a e
(2594.533 2592.301) /1.0637
2592.302 / 74R U
U
SSE SSE JF
SSE T K
Fc=3.970
Testing two conjectures
• Optimal advertising is $40000• If advertising is $40000 and price is $2, revenue
will be 175000
0 3 4 1 2 3 4: 2 (40) 1, 2 40 1600 175H Two hypotheses to substitute in to get restricted model
22 4135 2 80 1600t t t t t ttr a p a a e
( ) /1.75
/( )R U
U
SSE SSE JF
SSE T K
Fc=3.120
Incorporating non-sample information
Multiplying each price and income in a demand equation by a constant has no effect on demand
1 2 3 4 5ln ln ln ln lnB L Rq p p p m
1 2 3 4 5
1 2 3 4 5 2 3 4 5
ln ln ln ln ln
ln ln ln ln ln
B L R
B L R
q p p p m
p p p m
2 3 4 5 0
A restricted model
1 2 3 4 5ln ln ln ln lnB L Rq p p p m
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 Bt Lt Rt t t
Bt Rt Lt Rt t Rt t
Bt Lt tt
Rt Rt Rt
q p p p m e
p p p p m p e
p p me
p p p
ˆln 4.798 1.2994ln 0.1868ln 0.9458ln
(3.714) (0.166) (0.284) (0.427)
Bt Lt tt
Rt Rt Rt
p p mq
p p p
Omitted and irrelevant variables
• An omitted variable which is correlated with other variables in the regression will lead to bias.
• The omission of ‘insignificant’ variables may lead to bias (remember all you have done is failed to reject a null)
• Including irrelevant variables will inflate the variances of the estimated parameters.
The RESET test: principle
• If we omit variables and they are correlated with existing variables, including a function of these variables may allow us to pick up some of the effect of the omitted variables.
• If we can artificially improve the model by including powers of the predictions of the model, then a better functional form may exist.
• Overall: if we can improve a model by including powers of the predictions the model is inadequate.
The RESET test: practice
1 2 2 3 3t t t ty x x e 1 2 2 3 3ˆt t ty b b x b x
21 2 2 3 3 1 ˆt t t t ty x x y e
0 1: 0H 1 1: 0H
2 31 2 2 3 3 1 2ˆ ˆt t t t t ty x x y y e
0 1 2: 0H 1 1 2: 0 or 0H
In both cases the null is of no mis-specification
The RESET test: example
1 2 3 4 5ln( ) ln( ) ln( ) ln( ) ln( )t Bt Lt Rt t tq p p p m e
1 2 3 4 5t Bt Lt Rt t tq p p p m e Ramsey RESET Test: LOGLOG Model
F-statistic (1 term) 0.0075 Probability 0.9319
F-statistic (2 terms) 0.3581 Probability 0.7028
Ramsey RESET Test: LINEAR Model
F-statistic (1 term) 8.8377 Probability 0.0066
F-statistic (2 terms) 4.7618 Probability 0.0186
The linear model is mis-specified.
Collinear Economic Variables
• Explanatory variables move together in systematic ways.
• Attribute the increase in TR that is the consequence of two types of advertising.
• Identify the effects of increasing input quantities when technology is of the fixed proportions type.
The consequences of collinearity
2
2 22 2 23
var( )(1 )t
bx x r
• Exact collinearity renders OLS inoperable.• Near exact leads to increased standard errors.• R2 may be high but individual coefficients are
likely to be insignificant.• Estimates will be sensitive to the addition of a few
observations.• Accurate prediction may still be possible.
Identifying and mitigating collinearity
• Identifying:– Large standard errors with high R2.– Pairwise correlation coefficients in excess of
0.8– Auxiliary regressions.
• Mitigating– Additional data.– Parameter restrictions
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