ECON 4550
Econometrics
Memorial University of Newfoundland
Further Inference in the Multiple Regression
Model
Adapted from Vera Tabakova’s notes

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
Principles of Econometrics, 3rd Edition
Slide 6-2
Si  1  2 Pi  3 Ai  ei
(6.1)
H 0 : 2  0
H1 :  2  0
Si  1  3 Ai  ei
Principles of Econometrics, 3rd Edition
(6.2)
Slide 6-3
SSER  SSEU  J

F
SSEU  N  K 
(6.3)
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.
Principles of Econometrics, 3rd Edition
Slide 6-4
Hypothesis testing steps:
1.
Specify the null and alternative hypotheses: H 0 : 2  0 H1 : 2  0
2.
Specify the test statistic and its distribution if the null hypothesis is true
F
3.
 SSER  SSEU  1
SSEU  75  3
F(1, 72)
Set and determine the rejection region
Using α=.05, the critical value from the -distribution is Fc  F(.95,1,72)  3.97 .
Thus, H0 is rejected if F  3.97 .
Principles of Econometrics, 3rd Edition
Slide 6-5
4.
Calculate the sample value of the test statistic and, if desired, the p-value
F
 SSER  SSEU  J   2961.827  1718.943 1  52.06
SSEU  N  K 
1718.943  75  3
p  P[ F(1,72)  52.06]  .0000
5.
State your conclusion
Since F  52.06  Fc , we reject the null hypothesis and conclude that price does
have a significant effect on sales revenue. Alternatively, we reject H0 because
p  .0000  .05 .
Principles of Econometrics, 3rd Edition
Slide 6-6
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 F 
Principles of Econometrics, 3rd Edition
 SSER  SSEU  J
SSEU  N  K 
Slide 6-7
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 F  Fc , where Fc  F(1, J , N K ) .
5.
When testing a single equality null hypothesis it is perfectly correct to use
either the t- or F-test procedure; they are equivalent.
Principles of Econometrics, 3rd Edition
Slide 6-8
yi  1  xi 22  xi 33 
H 0 : 2  0, 3  0,
 xiK  K  ei
, K  0
H1 : At least one of the k is nonzero
for k  2,3,
(6.5)
K
yi  1  ei
Principles of Econometrics, 3rd Edition
(6.4)
(6.6)
Slide 6-9
N
N
i 1
i 1
SSER   ( yi  b1* )2   ( yi  y ) 2  SST
( SST  SSE ) / ( K  1)
F
SSE / ( N  K )
Principles of Econometrics, 3rd Edition
(6.7)
Slide 6-10
Example: Big Andy’s sales revenue Si  1  2 Pi  3 Ai  ei
1.
H 0 : 2  0, 3  0
H1 : 2  0 or 3  0, or both are nonzero
( SST  SSE ) / (3  1)
SSE / (75  3)
2.
If the null is true F 
3.
H0 is rejected if F  3.12
4.
F
5.
Since 29.95>3.12 we reject the null and conclude that price or advertising
expenditure or both have an influence on sales.
F(2,72)
( SST  SSE ) / ( K  1) (3115.485  1718.943) / 2

 29.25
SSE / ( N  K )
1718.943 / (75  3)
Principles of Econometrics, 3rd Edition
Slide 6-11
Figure 6.1 A Model Where Sales Exhibits Diminishing
Returns to Advertising Expenditure
Principles of Econometrics, 3rd Edition
Slide 6-12
S  1  2 P  3 A  e
(6.8)
S  1  2 P  3 A  4 A2  e
(6.9)
E ( S )
E ( S )

 3  24 A
A ( P held constant)
A
Principles of Econometrics, 3rd Edition
(6.10)
Slide 6-13
yi  1  2 xi 2  3 xi 3  4 xi 4  ei
yi  Si ,
xi 2  Pi ,
xi 3  Ai ,
xi 4  Ai2
Sˆi  109.72  7.640 Pi  12.151 Ai  2.768 Ai2
(se)
(6.80) (1.046)
Principles of Econometrics, 3rd Edition
(3.556)
(.941)
(6.11)
Slide 6-14

6.4.1 The Significance of Advertising
Si  1 2 Pi 3 Ai 4 Ai2  ei
Si  1  2 Pi  ei
(6.12)
(6.13)
H 0 : 3  0, 4  0
H1 : 3  0 or 4  0
Principles of Econometrics, 3rd Edition
Slide 6-15
SSER  SSEU  2

F
SSEU  75  4 
F(2,71)
Fc  F(.95,2,71)  3.126
P [ F(2,71)  8.44]  .0005
Since F  8.44  Fc  3.126 , we reject the null hypothesis and conclude
that advertising does have a significant effect upon sales revenue.
Principles of Econometrics, 3rd Edition
Slide 6-16
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.
E ( S )
 3  24 A
A ( P held constant)
3  24 Ao  1
12.1512  2  (2.76796) Aˆo  1
Principles of Econometrics, 3rd Edition
Slide 6-17
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
H 0 : 3  2 4 1.9  1
H 0 : 3  3.8 4  1
Principles of Econometrics, 3rd Edition
H1 : 3  2 4 1.9  1
H1 : 3  3.8 4  1
Slide 6-18
(b3  3.8b4 )  1
t
se(b3  3.8b4 )
var(b3  3.8b4 )  var(b3 )  3.82  var(b4 )  2  3.8  cov(b3 , b4 )
 12.6463  3.82  .884774  2  3.8  3.288746
 .427967
Principles of Econometrics, 3rd Edition
Slide 6-19
1.6330  1
.633
t

 .9676
.427967 .65419
Because 1.994  .9676  1.994 , 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.
Principles of Econometrics, 3rd Edition
Slide 6-20
Si  1  2 Pi  (1  3.84 ) Ai  4 Ai2  ei
(Si  Ai )  1  2 Pi  4 ( Ai2  3.8 Ai )  ei
(1552.286  1532.084) /1
F
 .9362
1532.084 / 71
Fc  3.976  tc2  (1.994)2
p-value  P[ F(1,71)  .9362]
 P[t(71)  .9676]  P[t(71)  .9676]  .3365
Principles of Econometrics, 3rd Edition
Slide 6-21
H 0 : 3  3.8 4  1
H1 : 3  3.8 4  1
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.
Principles of Econometrics, 3rd Edition
Slide 6-22
E ( S )  1  2 P  3 A  4 A2
 1  62  1.9 3  1.9 2  4
 80
H 0 : 3  3.8 4  1 and
1  6 2  1.9 3  3.614  80
F  5.74 and the p-value = .0049
Principles of Econometrics, 3rd Edition
Slide 6-23
ln(Q)  1  2 ln( PB)  3 ln( PL)  4 ln( PR)  5 ln( I )
(6.14)
ln(Q)  1  2 ln  PB   3 ln  PL   4 ln  PR   5 ln  I 
 1  2 ln( PB)  3 ln( PL)  4 ln( PR)  5 ln( I )
(6.15)
  2  3  4  5  ln( )
Principles of Econometrics, 3rd Edition
Slide 6-24
2  3  4  5  0
(6.16)
ln(Qt )  1  2 ln( PBt )  3 ln( PLt )  4 ln( PRt )  5 ln( I t )  et (6.17)
Principles of Econometrics, 3rd Edition
Slide 6-25
Principles of Econometrics, 3rd Edition
Slide 6-26
4  2  3  5
ln(Qt )  1  2 ln( PBt )  3 ln( PLt )
  2  3  5  ln( PRt )  5 ln( I t )  et
 1  2  ln( PBt )  ln( PRt ) 
(6.18)
3  ln( PLt )  ln( PRt )   5  ln( I t )  ln( PRt )   et
 PBt
 1  2 ln 
 PRt
Principles of Econometrics, 3rd Edition

 PLt
  3 ln 

 PRt

 It
  5 ln 

 PRt

  et

Slide 6-27
 PBt
ln(Qt )  4.798  1.2994ln 
 PRt
(se)
(0.166)

 PLt
  0.1868ln 

 PRt
(0.284)

 It 
  0.9458ln 
 (6.19)
PR

 t
(0.427)
b4*  b2*  b3*  b5*
 (1.2994)  0.1868  0.9458
 0.1668
Principles of Econometrics, 3rd Edition
Slide 6-28

6.6.1 Omitted Variables
FAMINC i  5534  3132 HEDU i  4523WEDU i
(se)
(11230) (803)
( p-value)
Principles of Econometrics, 3rd Edition
(.622) (.000)
(1066)
(6.20)
(.000)
Slide 6-29
Principles of Econometrics, 3rd Edition
Slide 6-30
FAMINC i  26191  5155 HEDU i
(se)
(8541)
(658)
( p-value)
(.002)
(.000)
yi  1  2 xi 2  3 xi 3  ei
Principles of Econometrics, 3rd Edition
(6.21)
(6.22)
Slide 6-31
bias(b )  E (b )  2  3
*
2
*
2
cov( x2 , x3 )
(6.23)
var( x2 )
FAMINC i  7755  3211 HEDU i  4777WEDU i  14311 KL6i
(se)
( p-value)
(11163) (797)
(1061)
(5004)
(.488) (.000)
(.000)
(.004)
Principles of Econometrics, 3rd Edition
(6.24)
Slide 6-32
FAMINC i  7759  3340 HEDU i  5869WEDU i  14200 KL6i  889 X i 5 1067 X i 6
(se)
( p-value)
(11195) (1250)
(2278)
(5044)
(2242)
(1982)
(.500)
(.010)
(.005)
(.692)
(.591)
(.008)
Principles of Econometrics, 3rd Edition
Slide 6-33
1.
Choose variables and a functional form on the basis of your
theoretical and general understanding of the relationship.
where  2*  c 2 and x*  x c
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.
Principles of Econometrics, 3rd Edition
Slide 6-34
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 H 0 : 3  0
or F-tests for hypotheses such as H 0 : 3  4  0 .
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.
Principles of Econometrics, 3rd Edition
Slide 6-35
yi  1  2 xi 2  3 xi 3  ei
yˆi  b1  b2 xi 2  b3 xi 3
Principles of Econometrics, 3rd Edition
(6.25)
Slide 6-36
yi  1 2 xi 2 3 xi 3  1 yˆi2  ei
(6.26)
yi  1 2 xi 2 3 xi 3  1 yˆi2   2 yˆi3  ei
(6.27)
Principles of Econometrics, 3rd Edition
Slide 6-37
H 0 : 1  0
F  5.984
p-value  .015
H 0 : 1   2  0
F  3.123
p-value  .045
Principles of Econometrics, 3rd Edition
Slide 6-38

6.7.1 The Consequences of Collinearity
yi  1  2 xi 2  3 xi 3  ei
var(b2 ) 
2
(1  r ) ( xi 2  x2 )
2
23
Principles of Econometrics, 3rd Edition
N
2
(6.28)
i 1
Slide 6-39
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.
Principles of Econometrics, 3rd Edition
Slide 6-40
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.
Principles of Econometrics, 3rd Edition
Slide 6-41
MPG = miles per gallon
CYL = number of cylinders
ENG = engine displacement in cubic inches
WGT = vehicle weight in pounds
MPG i  42.9  3.558 CYLi
(se)
(.83) (.146)
( p-value) (.000) (.000)
Principles of Econometrics, 3rd Edition
Slide 6-42
MPG i  44.4  .268 CYLi  .0127 ENGi  .00571WGTi
(se)
(1.5) (.413)
( p-value) (.000) (.517)
Principles of Econometrics, 3rd Edition
(.0083)
(.0071)
(.125)
(.000)
Slide 6-43
Identifying Collinearity
1.
Examining pairwise correlations.
2.
Using auxiliary regression
xi 2  a1 xi1  a3 xi 3 
 aK xiK  error
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.
Principles of Econometrics, 3rd Edition
Slide 6-44
Mitigating Collinearity
1.
Obtain more information and include it in the analysis.
2.
Introduce nonsample information in the form of restrictions on the
parameters.
Principles of Econometrics, 3rd Edition
Slide 6-45
yi  1  xi 22  xi 33  ei
(6.29)
y0  1  x022  x033  e0
ŷ0  b1  x02b2  x03b3
Principles of Econometrics, 3rd Edition
Slide 6-46
var( f )  var[(1  2 x02  3 x03  e0 )  (b1  b2 x02  b3 x03 )]
 var(e0  b1  b2 x02  b3 x03 )
2
2
 var(e0 )  var(b1 )  x02
var(b2 )  x03
var(b3 )
 2 x02 cov(b1 , b2 )  2 x03 cov(b1 , b3 )  2 x02 x03 cov(b2 , b3 )
Principles of Econometrics, 3rd Edition
Slide 6-47











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
Principles of Econometrics, 3rd Edition



restricted sum of squared errors
single and joint null hypotheses
unrestricted sum of squared errors
Slide 6-48
Principles of Econometrics, 3rd Edition
Slide 6-49
(SSER  SSEU ) / J
F
SSEU / ( N  K )
(6A.1)
( SSER  SSEU )
V1 
2
(2J )
(6A.2)
( SSER  SSEU )
ˆ
V1 
ˆ 2
(2J )
(6A.3)
( N  K )ˆ 2
V2 
2
Principles of Econometrics, 3rd Edition
(2N  K )
(6A.4)
Slide 6-50
V1 / m1
F
V2 / m2
 SSER  SSEU 

 N  K  ˆ 2
2
2
J
N  K
Principles of Econometrics, 3rd Edition
F (m1 , m2 )
SSER  SSEU 


ˆ
2
J
F J , N  K 
(6A.5)
Slide 6-51
Vˆ1
F
J
H 0 : 3  4  0
Si  1 2 Pi 3 Ai 4 Ai2  ei
F  8.44
2
  16.88
Principles of Econometrics, 3rd Edition
p-value  .0005
p-value  .0002
Slide 6-52
H 0 : 3  3.8 4  1
F  .936
 2  .936
Principles of Econometrics, 3rd Edition
p-value  .3365
p-value  .3333
Slide 6-53
yi  1  2 xi 2  3 xi 3  ei
yi  1  2 xi 2  vi
Principles of Econometrics, 3rd Edition
Slide 6-54
*
2
b
( xi 2  x2 )( yi  y )


 2   wi vi
2
 ( xi 2  x2 )
(6B.1)
( xi 2  x2 )
wi 
2
(
x

x
)
 i2 2
b2*  2  3  wi xi 3   wi ei
Principles of Econometrics, 3rd Edition
Slide 6-55
E (b2* )  2  3  wi xi 3
( xi 2  x2 )xi 3

  2  3
2
(
x

x
)
 i2 2
( xi 2  x2 )( xi 3  x3 )

  2  3
2
(
x

x
)
 i2 2
  2  3
Principles of Econometrics, 3rd Edition
cov( x2 , x3 )
var( x2 )
 2
Slide 6-56