Transcript Managerial Economics in a Global Economy, 5th Edition by Dominick Salvatore Chapter 4 Demand Estimation Prepared by Robert F.

Managerial Economics in a
Global Economy, 5th Edition
by
Dominick Salvatore
Chapter 4
Demand Estimation
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 1
The Identification Problem
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 2
Demand Estimation:
Marketing Research Approaches
•
•
•
•
•
•
Consumer Surveys
Observational Research
Consumer Clinics
Market Experiments
Virtual Shopping
Virtual Management
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 3
Regression Analysis
Year
X
Y
1
10
44
2
9
40
3
11
42
4
12
46
5
11
48
6
12
52
7
13
54
8
13
58
9
14
56
10
15
60
Prepared by Robert F. Brooker, Ph.D.
Scatter Diagram
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 4
Regression Analysis
• Regression Line: Line of Best Fit
• Regression Line: Minimizes the sum of
the squared vertical deviations (et) of
each point from the regression line.
• Ordinary Least Squares (OLS) Method
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 5
Regression Analysis
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 6
Ordinary Least Squares (OLS)
Model:
Yt  a  bX t  et
ˆ
Yˆt  aˆ  bX
t
et  Yt  Yˆt
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 7
Ordinary Least Squares (OLS)
Objective: Determine the slope and
intercept that minimize the sum of
the squared errors.
n
n
n
t 1
t 1
t 1
2
2
ˆ )2
ˆ
ˆ
e

(
Y

Y
)

(
Y

a

bX
t  t t  t
t
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 8
Ordinary Least Squares (OLS)
Estimation Procedure
n
bˆ 
(X
t 1
t
 X )(Yt  Y )
n
(X
t 1
Prepared by Robert F. Brooker, Ph.D.
t
 X)
ˆ
aˆ  Y  bX
2
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 9
Ordinary Least Squares (OLS)
Estimation Example
Time
Xt
1
2
3
4
5
6
7
8
9
10
10
9
11
12
11
12
13
13
14
15
120
n  10
Yt
44
40
42
46
48
52
54
58
56
60
500
n
n
 X t  120
Yt  500
t 1
n
X 
t 1
X t 120

 12
n
10
t 1
n
Yt 500

 50
10
t 1 n
Y 
Prepared by Robert F. Brooker, Ph.D.
Xt  X
Yt  Y
-2
-3
-1
0
-1
0
1
1
2
3
-6
-10
-8
-4
-2
2
4
8
6
10
n
(X
t 1
t 1
( X t  X )2
12
30
8
0
2
0
4
8
12
30
106
4
9
1
0
1
0
1
1
4
9
30
t
 X )2  30
106
bˆ 
 3.533
30
t
 X )(Yt  Y )  106
aˆ  50  (3.533)(12)  7.60
n
(X
( X t  X )(Yt  Y )
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 10
Ordinary Least Squares (OLS)
Estimation Example
n
X 
n  10
n
X
t 1
t
n
(X
t 1
t 1
 120
n
Y  500
t 1
t
n
Yt 500

 50
10
t 1 n
Y 
t
 X )  30
106
ˆ
b
 3.533
30
t
 X )(Yt  Y )  106
aˆ  50  (3.533)(12)  7.60
2
n
(X
t 1
X t 120

 12
n
10
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 11
Tests of Significance
Standard Error of the Slope Estimate
sbˆ 
2
ˆ
 (Yt  Y )
( n  k ) ( X t  X )
Prepared by Robert F. Brooker, Ph.D.
2

e
(n  k ) ( X
2
t
2

X
)
t
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 12
Tests of Significance
Example Calculation
Yˆt
et  Yt  Yˆt
et2  (Yt  Yˆt )2
( X t  X )2
44
42.90
1.10
1.2100
4
9
40
39.37
0.63
0.3969
9
3
11
42
46.43
-4.43
19.6249
1
4
12
46
49.96
-3.96
15.6816
0
5
11
48
46.43
1.57
2.4649
1
6
12
52
49.96
2.04
4.1616
0
7
13
54
53.49
0.51
0.2601
1
8
13
58
53.49
4.51
20.3401
1
9
14
56
57.02
-1.02
1.0404
4
10
15
60
60.55
-0.55
0.3025
9
65.4830
30
Time
Xt
Yt
1
10
2
n
n
 e   (Yt  Yˆt )2  65.4830
t 1
2
t
t 1
Prepared by Robert F. Brooker, Ph.D.
(X
t 1
 (Y  Yˆ )
( n  k ) ( X  X )
2
n
 X )  30
2
t
sbˆ 
t
t
2

65.4830
 0.52
(10  2)(30)
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 13
Tests of Significance
Example Calculation
n
n
t 1
t 1
2
2
ˆ
e

(
Y

Y
)
 t  t t  65.4830
n
2
(
X

X
)
 30
 t
t 1
2
ˆ
 (Yt  Y )
65.4830
sbˆ 

 0.52
2
( n  k ) ( X t  X )
(10  2)(30)
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 14
Tests of Significance
Calculation of the t Statistic
bˆ 3.53
t 
 6.79
sbˆ 0.52
Degrees of Freedom = (n-k) = (10-2) = 8
Critical Value at 5% level =2.306
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 15
Tests of Significance
Decomposition of Sum of Squares
Total Variation = Explained Variation + Unexplained Variation
2
2
ˆ
ˆ
 (Yt  Y )   (Y  Y )   (Yt  Yt )
2
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 16
Tests of Significance
Decomposition of Sum of Squares
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 17
Tests of Significance
Coefficient of Determination
R2 
Explained Variation

TotalVariation
2
ˆ
(
Y

Y
)

2
(
Y

Y
)
 t
373.84
R 
 0.85
440.00
2
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 18
Tests of Significance
Coefficient of Correlation
r  R2 withthe signof bˆ
1  r  1
r  0.85  0.92
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 19
Multiple Regression Analysis
Model:
Prepared by Robert F. Brooker, Ph.D.
Y  a  b1 X1  b2 X 2 
 bk ' X k '
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 20
Multiple Regression Analysis
Adjusted Coefficient of Determination
R 2  1  (1  R 2 )
Prepared by Robert F. Brooker, Ph.D.
(n  1)
(n  k )
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 21
Multiple Regression Analysis
Analysis of Variance and F Statistic
F
Explained Variation /(k  1)
Unexplained Variation /(n  k )
R 2 /(k  1)
F
(1  R 2 ) /(n  k )
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 22
Problems in Regression Analysis
• Multicollinearity: Two or more
explanatory variables are highly
correlated.
• Heteroskedasticity: Variance of error
term is not independent of the Y
variable.
• Autocorrelation: Consecutive error
terms are correlated.
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 23
Durbin-Watson Statistic
Test for Autocorrelation
n
d
2
(
e

e
)
 t t 1
t 2
n
2
e
t
t 1
If d=2, autocorrelation is absent.
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 24
Steps in Demand Estimation
•
•
•
•
•
Model Specification: Identify Variables
Collect Data
Specify Functional Form
Estimate Function
Test the Results
Prepared by Robert F. Brooker, Ph.D.
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 25
Functional Form Specifications
Linear Function:
QX  a0  a1PX  a2 I  a3 N  a4 PY 
Power Function:
QX  a( PXb1 )( PYb2 )
Prepared by Robert F. Brooker, Ph.D.
e
Estimation Format:
ln QX  ln a  b1 ln PX  b2 ln PY
Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved.
Slide 26