Micro Economics in a Global
Economy
Chapter 4
The Nature and Scope
of Managerial Economics
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The Identification Problem
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Demand Estimation:
Marketing Research Approaches
•
•
•
•
•
•
Consumer Surveys
Observational Research
Consumer Clinics
Market Experiments
Virtual Shopping
Virtual Management
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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
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Scatter Diagram
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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
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Regression Analysis
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Ordinary Least Squares (OLS)
Model:
Yt  a  bX t  et
€
Y€t  a€ bX
t
et  Yt  Y€t
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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
€ )2
€
€
e

(
Y

Y
)

(
Y

a

bX
t  t t  t
t
PowerPoint Slides by Robert F. Brooker
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Ordinary Least Squares (OLS)
Estimation Procedure
n
b€ 
(X
t 1
t
 X )(Yt  Y )
n
2
(
X

X
)
 t
€
a€  Y  bX
t 1
PowerPoint Slides by Robert F. Brooker
Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.
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 
PowerPoint Slides by Robert F. Brooker
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
b€ 
 3.533
30
t
 X )(Yt  Y )  106
a€  50  (3.533)(12)  7.60
n
(X
( X t  X )(Yt  Y )
Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.
Ordinary Least Squares (OLS)
Estimation Example
n
X 
n  10
n
X
t 1
t
n
(X
t 1
t 1
 120
n
Y
t 1
t
n
 500
Yt 500

 50
10
t 1 n
Y 
t
 X )  30
106
€
b
 3.533
30
t
 X )(Yt  Y )  106
a€  50  (3.533)(12)  7.60
2
n
(X
t 1
X t 120

 12
n
10
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Tests of Significance
Standard Error of the Slope Estimate
sbˆ 
2
ˆ
 (Yt  Y )
(n  k ) ( X t  X )
PowerPoint Slides by Robert F. Brooker
2

e
(n  k ) ( X
2
t
t
 X )2
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Tests of Significance
Example Calculation
Y€t
et  Yt  Y€t
et2  (Yt  Y€t )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  Y€t )2  65.4830
t 1
2
t
t 1
(X
t 1
PowerPoint Slides by Robert F. Brooker
 (Y  Y€)
( n  k ) ( X  X )
2
n
 X )  30
2
t
sb€ 
t
t
2

65.4830
 0.52
(10  2)(30)
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Tests of Significance
Example Calculation
n
n
t 1
t 1
2
2
€
e

(
Y

Y
)
 t  t t  65.4830
n
2
(
X

X
)
 30
 t
t 1
2
€
 (Yt  Y )
65.4830
sb€ 

 0.52
2
( n  k ) ( X t  X )
(10  2)(30)
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Tests of Significance
Calculation of the t Statistic
b€ 3.53
t 
 6.79
sb€ 0.52
Degrees of Freedom = (n-k) = (10-2) = 8
Critical Value at 5% level =2.306
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Tests of Significance
Decomposition of Sum of Squares
Total Variation = Explained Variation + Unexplained Variation
2
2
€
€
 (Yt  Y )   (Y  Y )   (Yt  Yt )
2
PowerPoint Slides by Robert F. Brooker
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Tests of Significance
Decomposition of Sum of Squares
PowerPoint Slides by Robert F. Brooker
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Tests of Significance
Coefficient of Determination
R2 
2
€
(
Y

Y
)

Explained Variation

2
TotalVariation
(
Y

Y
)
 t
373.84
R 
 0.85
440.00
2
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Tests of Significance
Coefficient of Correlation
r  R 2 withthe sign of b€
1  r  1
r  0.85  0.92
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Multiple Regression Analysis
Model:
PowerPoint Slides by Robert F. Brooker
Y  a  b1 X1  b2 X 2 
 bk ' X k '
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Multiple Regression Analysis
Adjusted Coefficient of Determination
(n  1)
R  1  (1  R )
(n  k )
2
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2
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Multiple Regression Analysis
Analysis of Variance and F Statistic
Explained Variation /(k  1)
F
Unexplained Variation /(n  k )
R 2 /(k  1)
F
(1  R 2 ) /(n  k )
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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.
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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.
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Steps in Demand Estimation
•
•
•
•
•
Model Specification: Identify Variables
Collect Data
Specify Functional Form
Estimate Function
Test the Results
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Functional Form Specifications
Linear Function:
QX  a0  a1PX  a2 I  a3 N  a4 PY 
Power Function:
QX  a( PXb1 )( PYb2 )
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e
Estimation Format:
ln QX  ln a  b1 ln PX  b2 ln PY
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