Demand Estimation
Chapter 6 OVERVIEW
Demand Estimation
Chapter 6
1
2
Chapter 6 KEY CONCEPTS
simultaneous relation identification problem consumer interview market experiments regression analysis deterministic relation statistical relation time series cross section scatter diagram linear model multiplicative model simple regression model
Demand Curve Estimation
multiple regression model standard error of the estimate (SEE) correlation coefficient coefficient of determination degrees of freedom corrected coefficient of determination F statistic t statistic two-tail t tests one-tail t tests
3
Simple Linear Demand Curves
The best estimation method balances marginal costs and marginal benefits. Simple linear relations are useful for demand estimation.
Using Simple Linear Demand Curves
Straight-line relations give useful approximations.
4
Price/Quantity Price/Quantity Plot Plot for for Product Product X X
Identification Problem
Changing Nature of Demand Relations
Demand relations are dynamic.
Interplay of Supply and Demand
Economic conditions affect demand and supply.
Shifts in Demand and Supply
Simultaneous Relations
5
Demand Curve Estimation Identification Problem Interview and Experimental Methods Regression Analysis Measuring Regression Model Significance Measures of Individual Variable Significance
Curve shifts can be estimated. Figure 6.1
6
1
Shifting Shifting Supply Supply Curve Curve Tracing Tracing Out Out Stable Demand Curve Stable Demand Curve
Supply Supply and and Demand Demand Curves Curves Incorrectly Incorrectly interpreting interpreting AB AB as as aa demand demand curve curve could could lead lead to to poor poor managerial managerial decisions decisions
IfIf the the demand demand curve curve has has not not shifted shifted and and only only the the supply supply curve curve has has shifted… shifted… Figure 6.3
Figure 6.2
7
8
Interview and Experimental Methods
Consumer Interviews
Demand Demand Elasticities Elasticities for for California California and and Valencia Valencia Oranges Oranges –– Market Market Experiment Experiment
Interviews can solicit useful information when market data is scarce. Interview opinions often differ from actual market transaction data.
Market Experiments
Controlled experiments can generate useful insight. Experiments can become expensive.
9
Percentage Change in the Sales of A 1% Change in the Price of
Florida Interior
Florida Indian River
-3.07
Florida Interior
+1.16
-3.01
California
+0.18
+0.09
California
-2.76
Grand GrandRapids, Rapids,Michigan MichiganExperiment Experiment 10
Regression Analysis
A statistical relation exists when averages are related. A deterministic relation is true by definition.
Specifying the Regression Model
Scatter ScatterDiagrams Diagramsof ofVarious Various Unit UnitCost/Output Cost/OutputRelations Relations
What Is a Statistical Relation?
11
Florida Indian River
Dependent variable Y is caused by X. X variables are independently determined from Y.
Least Squares Method
Minimize sum of squared residuals.
Scatter ScatterDiagrams Diagramscan canimpart impartand andinstinctive instinctive“feel” “feel”for forthe thedata. data. Figure 6.4 12
2
The Statistical Tool We Use To Determine Demand Relationships
Regression • A mathematical model to represent the relationship between a dependent variable (y) and an independent variable (x).
AA deterministic deterministic relation relation isis an an association association between between variables variables that that isis known known with with certainty. certainty.
AA statistical statistical relation relation exists exists between between two two economic economic variables variables ifif the the average average of of one one isis related related to to another. another.
Regression Regression isis used used to to determine determine Demand Demand Relationships. Relationships.
13
14
Regression
Regression
• A mathematical model to represent the relationship between a dependent variable (y) and an independent variable (x). • Can be used to answer questions such as...does Y tend to increase when X increases?
15
A mathematical model to represent the relationship between a dependent variable (y) and an independent variable (x).
Can be used to answer questions such as...Does y tend to increase when x increases?
Describes the way in which one variable is related to another (e.g. Sales and Price).
16
Beware the Software
Many sophisticated software packages are available -- but there is a danger in using canned packages unless you are familiar with the underlying concepts. ForecastX is easy to use, but learn the underlying concepts.
Who Uses ForecastX? FedEx United Parcel Services The Gap Levis Sears LensCrafters Ibbotson Associates Pillsbury Shoney’s Restaurants US Navy Starbucks Wallmart
Visa PWC Keebler Company BP Amoco GTE MCI US West U.S. Cellular Motorola Dell Corporation America Online Ryder
Microsoft Maytag Harley Davidson Ernst & Young Hasbro Lockheed Martin Accenture AT&T 3Com Nintendo Sprint John Deere
ForecastX is the Demand Planning Module in PeopleSoft® 17
18
3
The Regression Model
The Regression Model
• A regression Model is a simplified or ideal representation of the real world.
19
A regression model is a simplified or ideal representation of the real world.
All scientific inquiry is based to some extent on models - that is the set of simplifying assumptions - on which regression is based.
20
Origin of Regression
Origin of Regression
The term "regression analysis" comes from studies carried out by the English statistician Francis Galton in about 1875.
The term "regression analysis" comes from studies carried out by the English statistician Francis Galton in about 1875. Galton compared the heights of parents with the heights of their offspring and found: that very tall parents tended to have offspring shorter than their parents while very short parents had offspring taller than their parents.
21
22
The Retail Sales Function Date 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
23
RS 1037.88 1067.46 1167.42 1283.75 1373.83 1449.21 1538.78 1648.56 1758.45 1845.13 1856.09 1944.61 2072.61 2232.52 2348.67
DPIPC 9565 10107.5 10763.5 11886.5 12585.8 13243.5 13848.3 14856.3 15741.8 16669.3 17190 18061.5 18551.3 19251.8 20169.3
Retail Auto Sales City City
Auto AutoSales Sales Population Population
Seattle Seattle Portland Portland
2658 2658 1795 1795
1758 1758 1159 1159
Tacoma Tacoma Spokane Spokane
718 718 509 509
Yakima Yakima Boise Boise
218 218 208 208 190 190
537 537 358 358 184 184 202 202 123 123
157 157 136 136
147 147 116 116
106 106
81 81
Billings Billings Olympia Olympia
Time Series Data
Bellingham Bellingham Great GreatFalls Falls
Cross-Sectional Data
24
4
Ordinary LeastLeast-squares Regression Model Y = a + bX
Linear Least Squares Regression
Y = Dependent Variable X = Independent Variable a = Intercept of the line b = Slope of the line 25
26
We Actually Estimate
Linear Least Squares Regression
Yˆ = ˆa + bˆ X i • The “Hats” indicate estimated numbers.
27
28
We Actually Estimate Yˆ = ˆa + bˆ X
Rewriting the Equation
or
Yi − Yˆi = ei Y = Yˆ + e
or
Yi = aˆ + bˆX i + ei
i
The “Hats” Hats” indicate estimated numbers.
Y = ˆa + bˆ X i + e
29
The “e” indicates the error (or residual)
i
i
i
30
5
The Minimization Problem Reduces To Two Equations
The Problem
Minimize the sum of the squared deviations (or errors).
bˆ =
∑e
2 i
=
∑ (Y
i
ˆ −Y i
i
i
) (∑ X
− nY X /
2 i
− nX 2
)
aˆ = Y − ˆbX
These squared deviations or errors are sometimes called residuals.
Minimize:
(∑ Y X
)
2
These are called the “Normal Equations.”
31
32
SPECIFYING THE Regression Model
The Intercept and Slope
Linear Model
•The intercept (or "constant term")indicates where the regression line intercepts the vertical axis. Some people call this a "shift parameter" because it "shifts" the regression line up or down on the graph.
Q = b0 + bP P + bA A + bI I
Multiplicative Model
Q = b0 P bP AbA I bI Pg. Pg.172 172
33
34
Retail Sales as a Function of DPIPC
The Intercept and Slope The intercept (or "constant term") indicates where the regression line intercepts the vertical axis. Some people call this a "shift parameter" because it "shifts" the regression line up or down on the graph. The slope indicates how Y changes as X changes (e.g., if the slope is positive, as X increases, Y also increases -- if the slope is negative, as X increases, Y decreases).
( RS) 35
36
6
Regression RegressionRelation RelationBetween BetweenUnits UnitsSold Soldand and Personal PersonalSelling SellingExpenditures ExpendituresFor ForElectronic Electronic Data Processing (EDP), Inc. (Table 6.5) Data Processing (EDP), Inc. (Table 6.5)
Measuring Regression Model Significance
Standard Error of the Estimate (SEE) increases with scatter about the regression line.
Mistake Mistake Pg. Pg.174 174
37
Pg. Pg.177 177
38
Goodness of Fit, r and R2
r = 1 means perfect correlation; r = 0 means no correlation. R2 = 1 means perfect fit; R2 = 0 means no relation. Corrected Coefficient of Determination, R2
Adjusts R2 downward for small samples. Pg. Pg.178 178
39
40
Measures of Individual Variable Significance
F statistic
t statistics
Tells if R2 is statistically significant.
Two-tail t Tests
One-Tail t Tests
Pg. Pg.181 181
41
t statistics compare a sample characteristic to the standard deviation of that characteristic. A calculated t statistic more than two suggests a strong effect of X on Y (95 % confidence). A calculated t statistic more than three suggests a very strong effect of X on Y (99 % confidence).
Tests of effect. Tests of magnitude or direction.
42
7
Case Study
Demand Estimation for Mrs. Smyth’ Smyth’s Pies
Data Data page page 102 102
Pg. Pg.182 182
43
44
Problem 6.10 QY = 10 PY
−1.10
0.5
3 .8
2 .5
PX AY AX I 1.85
⎛ ∂Q ⎞⎛ P ⎞
ε P = ⎜⎜ Y ⎟⎟⎜⎜ Y ⎟⎟ ⎝ ∂PY ⎠⎝ QY ⎠
(
(
))⎛ QP
ε P = (− 1.10) 10 PY −2.10 PX 0.5 AY 3.8 AX 2.5 I 1.85 ⎜⎜
(
⎝
(
Y Y
⎞ ⎟⎟ ⎠
))⎛ 10P
ε P = (− 1.10) 10 PY −2.10 PX 0.5 AY 3.8 AX 2.5 I 1.85 ⎜⎜ εP =
((− 1.10)(10P
Y
⎝
−1.10
0 .5
3 .8
2 .5
PX AY AX I 1.85 1 .0 PY
)) ⎛⎜
Y
−1.10
⎞ PY ⎟ 0 .5 3 .8 2 .5 PX AY AX I 1.85 ⎟⎠
⎞ PY ⎟ ⎜ 10 P −1.10 P 0.5 A 3.8 A 2.5 I 1.85 ⎟ Y X Y X ⎝ ⎠
Chapter Chapter 77
ε P = −1.10 45
46
8
Demand Estimation
Chapter 6
1
2
Chapter 6 KEY CONCEPTS
simultaneous relation identification problem consumer interview market experiments regression analysis deterministic relation statistical relation time series cross section scatter diagram linear model multiplicative model simple regression model
Demand Curve Estimation
multiple regression model standard error of the estimate (SEE) correlation coefficient coefficient of determination degrees of freedom corrected coefficient of determination F statistic t statistic two-tail t tests one-tail t tests
3
Simple Linear Demand Curves
The best estimation method balances marginal costs and marginal benefits. Simple linear relations are useful for demand estimation.
Using Simple Linear Demand Curves
Straight-line relations give useful approximations.
4
Price/Quantity Price/Quantity Plot Plot for for Product Product X X
Identification Problem
Changing Nature of Demand Relations
Demand relations are dynamic.
Interplay of Supply and Demand
Economic conditions affect demand and supply.
Shifts in Demand and Supply
Simultaneous Relations
5
Demand Curve Estimation Identification Problem Interview and Experimental Methods Regression Analysis Measuring Regression Model Significance Measures of Individual Variable Significance
Curve shifts can be estimated. Figure 6.1
6
1
Shifting Shifting Supply Supply Curve Curve Tracing Tracing Out Out Stable Demand Curve Stable Demand Curve
Supply Supply and and Demand Demand Curves Curves Incorrectly Incorrectly interpreting interpreting AB AB as as aa demand demand curve curve could could lead lead to to poor poor managerial managerial decisions decisions
IfIf the the demand demand curve curve has has not not shifted shifted and and only only the the supply supply curve curve has has shifted… shifted… Figure 6.3
Figure 6.2
7
8
Interview and Experimental Methods
Consumer Interviews
Demand Demand Elasticities Elasticities for for California California and and Valencia Valencia Oranges Oranges –– Market Market Experiment Experiment
Interviews can solicit useful information when market data is scarce. Interview opinions often differ from actual market transaction data.
Market Experiments
Controlled experiments can generate useful insight. Experiments can become expensive.
9
Percentage Change in the Sales of A 1% Change in the Price of
Florida Interior
Florida Indian River
-3.07
Florida Interior
+1.16
-3.01
California
+0.18
+0.09
California
-2.76
Grand GrandRapids, Rapids,Michigan MichiganExperiment Experiment 10
Regression Analysis
A statistical relation exists when averages are related. A deterministic relation is true by definition.
Specifying the Regression Model
Scatter ScatterDiagrams Diagramsof ofVarious Various Unit UnitCost/Output Cost/OutputRelations Relations
What Is a Statistical Relation?
11
Florida Indian River
Dependent variable Y is caused by X. X variables are independently determined from Y.
Least Squares Method
Minimize sum of squared residuals.
Scatter ScatterDiagrams Diagramscan canimpart impartand andinstinctive instinctive“feel” “feel”for forthe thedata. data. Figure 6.4 12
2
The Statistical Tool We Use To Determine Demand Relationships
Regression • A mathematical model to represent the relationship between a dependent variable (y) and an independent variable (x).
AA deterministic deterministic relation relation isis an an association association between between variables variables that that isis known known with with certainty. certainty.
AA statistical statistical relation relation exists exists between between two two economic economic variables variables ifif the the average average of of one one isis related related to to another. another.
Regression Regression isis used used to to determine determine Demand Demand Relationships. Relationships.
13
14
Regression
Regression
• A mathematical model to represent the relationship between a dependent variable (y) and an independent variable (x). • Can be used to answer questions such as...does Y tend to increase when X increases?
15
A mathematical model to represent the relationship between a dependent variable (y) and an independent variable (x).
Can be used to answer questions such as...Does y tend to increase when x increases?
Describes the way in which one variable is related to another (e.g. Sales and Price).
16
Beware the Software
Many sophisticated software packages are available -- but there is a danger in using canned packages unless you are familiar with the underlying concepts. ForecastX is easy to use, but learn the underlying concepts.
Who Uses ForecastX? FedEx United Parcel Services The Gap Levis Sears LensCrafters Ibbotson Associates Pillsbury Shoney’s Restaurants US Navy Starbucks Wallmart
Visa PWC Keebler Company BP Amoco GTE MCI US West U.S. Cellular Motorola Dell Corporation America Online Ryder
Microsoft Maytag Harley Davidson Ernst & Young Hasbro Lockheed Martin Accenture AT&T 3Com Nintendo Sprint John Deere
ForecastX is the Demand Planning Module in PeopleSoft® 17
18
3
The Regression Model
The Regression Model
• A regression Model is a simplified or ideal representation of the real world.
19
A regression model is a simplified or ideal representation of the real world.
All scientific inquiry is based to some extent on models - that is the set of simplifying assumptions - on which regression is based.
20
Origin of Regression
Origin of Regression
The term "regression analysis" comes from studies carried out by the English statistician Francis Galton in about 1875.
The term "regression analysis" comes from studies carried out by the English statistician Francis Galton in about 1875. Galton compared the heights of parents with the heights of their offspring and found: that very tall parents tended to have offspring shorter than their parents while very short parents had offspring taller than their parents.
21
22
The Retail Sales Function Date 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
23
RS 1037.88 1067.46 1167.42 1283.75 1373.83 1449.21 1538.78 1648.56 1758.45 1845.13 1856.09 1944.61 2072.61 2232.52 2348.67
DPIPC 9565 10107.5 10763.5 11886.5 12585.8 13243.5 13848.3 14856.3 15741.8 16669.3 17190 18061.5 18551.3 19251.8 20169.3
Retail Auto Sales City City
Auto AutoSales Sales Population Population
Seattle Seattle Portland Portland
2658 2658 1795 1795
1758 1758 1159 1159
Tacoma Tacoma Spokane Spokane
718 718 509 509
Yakima Yakima Boise Boise
218 218 208 208 190 190
537 537 358 358 184 184 202 202 123 123
157 157 136 136
147 147 116 116
106 106
81 81
Billings Billings Olympia Olympia
Time Series Data
Bellingham Bellingham Great GreatFalls Falls
Cross-Sectional Data
24
4
Ordinary LeastLeast-squares Regression Model Y = a + bX
Linear Least Squares Regression
Y = Dependent Variable X = Independent Variable a = Intercept of the line b = Slope of the line 25
26
We Actually Estimate
Linear Least Squares Regression
Yˆ = ˆa + bˆ X i • The “Hats” indicate estimated numbers.
27
28
We Actually Estimate Yˆ = ˆa + bˆ X
Rewriting the Equation
or
Yi − Yˆi = ei Y = Yˆ + e
or
Yi = aˆ + bˆX i + ei
i
The “Hats” Hats” indicate estimated numbers.
Y = ˆa + bˆ X i + e
29
The “e” indicates the error (or residual)
i
i
i
30
5
The Minimization Problem Reduces To Two Equations
The Problem
Minimize the sum of the squared deviations (or errors).
bˆ =
∑e
2 i
=
∑ (Y
i
ˆ −Y i
i
i
) (∑ X
− nY X /
2 i
− nX 2
)
aˆ = Y − ˆbX
These squared deviations or errors are sometimes called residuals.
Minimize:
(∑ Y X
)
2
These are called the “Normal Equations.”
31
32
SPECIFYING THE Regression Model
The Intercept and Slope
Linear Model
•The intercept (or "constant term")indicates where the regression line intercepts the vertical axis. Some people call this a "shift parameter" because it "shifts" the regression line up or down on the graph.
Q = b0 + bP P + bA A + bI I
Multiplicative Model
Q = b0 P bP AbA I bI Pg. Pg.172 172
33
34
Retail Sales as a Function of DPIPC
The Intercept and Slope The intercept (or "constant term") indicates where the regression line intercepts the vertical axis. Some people call this a "shift parameter" because it "shifts" the regression line up or down on the graph. The slope indicates how Y changes as X changes (e.g., if the slope is positive, as X increases, Y also increases -- if the slope is negative, as X increases, Y decreases).
( RS) 35
36
6
Regression RegressionRelation RelationBetween BetweenUnits UnitsSold Soldand and Personal PersonalSelling SellingExpenditures ExpendituresFor ForElectronic Electronic Data Processing (EDP), Inc. (Table 6.5) Data Processing (EDP), Inc. (Table 6.5)
Measuring Regression Model Significance
Standard Error of the Estimate (SEE) increases with scatter about the regression line.
Mistake Mistake Pg. Pg.174 174
37
Pg. Pg.177 177
38
Goodness of Fit, r and R2
r = 1 means perfect correlation; r = 0 means no correlation. R2 = 1 means perfect fit; R2 = 0 means no relation. Corrected Coefficient of Determination, R2
Adjusts R2 downward for small samples. Pg. Pg.178 178
39
40
Measures of Individual Variable Significance
F statistic
t statistics
Tells if R2 is statistically significant.
Two-tail t Tests
One-Tail t Tests
Pg. Pg.181 181
41
t statistics compare a sample characteristic to the standard deviation of that characteristic. A calculated t statistic more than two suggests a strong effect of X on Y (95 % confidence). A calculated t statistic more than three suggests a very strong effect of X on Y (99 % confidence).
Tests of effect. Tests of magnitude or direction.
42
7
Case Study
Demand Estimation for Mrs. Smyth’ Smyth’s Pies
Data Data page page 102 102
Pg. Pg.182 182
43
44
Problem 6.10 QY = 10 PY
−1.10
0.5
3 .8
2 .5
PX AY AX I 1.85
⎛ ∂Q ⎞⎛ P ⎞
ε P = ⎜⎜ Y ⎟⎟⎜⎜ Y ⎟⎟ ⎝ ∂PY ⎠⎝ QY ⎠
(
(
))⎛ QP
ε P = (− 1.10) 10 PY −2.10 PX 0.5 AY 3.8 AX 2.5 I 1.85 ⎜⎜
(
⎝
(
Y Y
⎞ ⎟⎟ ⎠
))⎛ 10P
ε P = (− 1.10) 10 PY −2.10 PX 0.5 AY 3.8 AX 2.5 I 1.85 ⎜⎜ εP =
((− 1.10)(10P
Y
⎝
−1.10
0 .5
3 .8
2 .5
PX AY AX I 1.85 1 .0 PY
)) ⎛⎜
Y
−1.10
⎞ PY ⎟ 0 .5 3 .8 2 .5 PX AY AX I 1.85 ⎟⎠
⎞ PY ⎟ ⎜ 10 P −1.10 P 0.5 A 3.8 A 2.5 I 1.85 ⎟ Y X Y X ⎝ ⎠
Chapter Chapter 77
ε P = −1.10 45
46
8
