Regression analysis in managerial economics

Marketing Research Approaches to Demand Estimation

Consumer Surveys
data from survey questions
Observational Research
data from observed behavior
Consumer Clinics
data from laboratory experiments
Market Experiments
data from real market tests
Regression Analysis

Scatter Diagram

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
Ordinary Least Squares (OLS)

Model:

Ordinary Least Squares (OLS)

Objective: Determine the slope and intercept that minimize the sum of the squared errors.

Ordinary Least Squares (OLS)

Estimation Procedure

Ordinary Least Squares (OLS)

Estimation Example

Ordinary Least Squares (OLS)

Estimation Example

Tests of Significance

Standard Error of the Slope Estimate

Tests of Significance

Example Calculation

Tests of Significance

Example Calculation

Tests of Significance

Calculation of the t Statistic

Degrees of Freedom = (n-k) = (10-2) = 8

Critical Value at 5% level =2.306

Tests of Significance

Decomposition of Sum of Squares

Total Variation = Explained Variation + Unexplained Variation

Tests of Significance

Coefficient of Determination

Tests of Significance

Coefficient of Correlation

Multiple Regression Analysis

Model:

Multiple Regression Analysis

Adjusted Coefficient of Determination

Multiple Regression Analysis

Analysis of Variance and F Statistic

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.
Durbin-Watson Statistic

Test for Autocorrelation

If d = 2, autocorrelation is absent.

Steps in Demand Estimation

Model Specification: Identify Variables
Collect Data
Specify Functional Form
Estimate Function
Test the Results
Functional Form Specifications

Linear Function:

Power Function:

Estimation Format:

Chapter 5 Appendix

Getting Started

Install the Analysis ToolPak add-in from the Excel installation media if it has not already been installed
Attach the Analysis ToolPak add-in
From the menu, select Tools and then Add-Ins...
When the Add-Ins dialog appears, select Analysis ToolPak and then click OK.
Entering Data

Data on each variable must be entered in a separate column
Label the top of each column with a symbol or brief description to identify the variable
Multiple regression analysis requires that all data on independent variables be in adjacent columns
Example Data

Running the Regression

Select the Regression tool from the Analysis ToolPak dialog
From the menu, select Tools and then Data Analysis...
On the Data Analysis dialog, scroll down the list of Analysis Tools, select Regression, and then click OK
The Regression tool dialog will then be displayed
Select the Data Ranges

Type in the data range for the Y variable or select the range on the worksheet
Type in the data range for the X variable(s) or select the range on the worksheet
If your ranges include the data labels (recommended) then check the labels option
Select an Output Option

Output to a selected range
Selection is the upper left corner of the output range
Output to a new worksheet
Optionally enter a name for the worksheet
Output to a new workbook
And then click OK
Regression Output

Multiple Regression Data

Regression Output

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

ttt

YabXe

=++

ˆ

ˆ

ˆ

tt

YabX

=+

ˆ

ttt

eYY

=-

222

111

ˆ

ˆ

ˆ

()()

nnn

ttttt

ttt

eYYYabX

===

=-=--

ååå

1

2

1

()()

ˆ

()

n

tt

t

n

t

t

XXYY

b

XX

=

=

--

=

-

å

å

ˆ

ˆ

aYbX

=-

10

n

=

1

120

12

10

n

t

t

X

X

n

=

===

å

1

500

50

10

n

t

t

Y

Y

n

=

===

å

1

120

n

t

t

X

=

=

å

1

500

n

t

t

Y

=

=

å

2

1

()30

n

t

t

XX

=

-=

å

1

()()106

n

tt

t

XXYY

=

--=

å

106

ˆ

3.533

30

b

==

ˆ

50(3.533)(12)7.60

a

=-=

Time

t

X

t

Y

t

XX

-

t

YY

-

()()

tt

XXYY

--

2

()

t

XX

-

1

10

44

-2

-6

12

2

9

40

-3

-10

30

3

11

42

-1

-8

8

4

12

46

0

-4

0

5

11

48

-1

-2

2

6

12

52

0

2

0

7

13

54

1

4

4

8

13

58

1

8

8

9

14

56

2

6

12

10

15

60

3

10

30

120

500

106

4

9

1

0

1

0

1

1

4

9

30

22

ˆ

22

ˆ

()

()()()()

tt

b

tt

YYe

s

nkXXnkXX

-

==

----

åå

åå

22

11

ˆ

()65.4830

nn

ttt

tt

eYY

==

=-=

åå

2

ˆ

2

ˆ

()

65.4830

0.52

()()(102)(30)

t

b

t

YY

s

nkXX

-

===

---

å

å

Time

t

X

t

Y

ˆ

t

Y

ˆ

ttt

eYY

=-

22

ˆ

()

ttt

eYY

=-

2

()

t

XX

-

1

10

44

42.90

2

9

40

39.37

3

11

42

46.43

4

12

46

49.96

5

11

48

46.43

6

12

52

49.96

7

13

54

53.49

8

13

58

53.49

9

14

56

57.02

10

15

60

60.55

1.10

1.2100

4

0.63

0.3969

9

-4.43

19.6249

1

-3.96

15.6816

0

1.57

2.4649

1

2.04

4.1616

0

0.51

0.2601

1

4.51

20.3401

1

-1.02

1.0404

4

-0.55

0.3025

9

65.4830

30

ˆ

ˆ

3.53

6.79

0.52

b

b

t

s

===

222

ˆˆ

()()()

ttt

YYYYYY

-=-+-

ååå

2

2

2

ˆ

()

()

t

YY

ExplainedVariation

R

TotalVariationYY

-

==

-

å

å

2

373.84

0.85

440.00

R

==

2

ˆ

rRwiththesignofb

=

0.850.92

r

==

11

r

-££

1122''

kk

YabXbXbX

=++++

L

22

(1)

1(1)

()

n

RR

nk

-

=--

-

/(1)

/()

ExplainedVariationk

F

UnexplainedVariationnk

-

=

-

2

2

/(1)

(1)/()

Rk

F

Rnk

-

=

--

2

1

2

2

1

()

n

tt

t

n

t

t

ee

d

e

-

=

=

-

=

å

å

01234

XXY

QaaPaIaNaPe

=++++++

L

12

()()

bb

XXY

QaPP

=

12

lnlnlnln

XXY

QabPbP

=++