This document discusses regression analysis techniques for estimating relationships between variables. It provides examples of using single and multiple regression to model how dependent variables, like income, are impacted by independent variables, such as education levels and population density. Key outputs from regression analyses like the model summary, ANOVA table, and coefficients are also presented to interpret the results and significance of relationships.

1. MMT BATCH 36 1
QUANTITY
DEMAND
ANALYSIS
Joseph Winthrop B. Godoy

2. INTRODUCTION
♦ Shows how a manager can use elasticities of
demand as a quantitative forecasting tool
• Describes regression analysis, which is the
technique economists use to estimate the
parameters of demand functions

3. THE ELASTICITY CONCEPT
Elasticity Analysis
The primary tool used to determine the
magnitude of such a change
Elasticity
Measures the responsiveness of one
variable to changes in another variable

4. THE ELASTICITY CONCEPT
Two aspects of Elasticity
(1) Whether it is positive or negative
(2) Whether it is greater than 1 or less than 1
in absolute value

5. OWN PRICE ELASTICITY OF
DEMAND
 measures the responsiveness of quantity
demanded to a change in price;
 the percentage change in quantity demanded
divided by the percentage change in the price
of the good

6. ELASTIC DEMAND
 demand is said to be elastic if the absolute value
of the own price elasticity is greater than 1:

7. INELASTIC DEMAND
 demand is said to be inelastic if the absolute
value of the own price elasticity is less than 1:

8. UNITARY ELASTIC
 if the absolute value of the own price
elasticity is equal to 1:

9. ELASTICITY & TOTAL
REVENUE
Price of
Software
Quantity of
Software Sold
Own Price
Elasticity
Total
Revenue
A $ 0 80 0.00 $ 0
B 5 70 -0.14 350
C 10 60 -0.33 600
D 15 50 -0.60 750
E 20 40 -1.00 800
F 25 30 -1.67 750
G 30 20 -3.00 600
H 35 10 -7.00 350
I 40 0 0

10. TOTAL REVENUE TEST
♦ If demand is elastic, an increase (decrease)
in price will lead to a decrease (increase) in
total revenue. If demand is inelastic, an
increase (decrease) in price will lead to an
increase (decrease) in total revenue. Finally,
total revenue is maximized at the point
where demand is unitary elastic.

12. THE ELASTICITY CONCEPT
Perfectly Elastic
If the own price elasticity of demand is
indefinite on absolute value
Perfectly Inelastic
If the own price elasticity of demand is
zero

13. FACTORS AFFECTING THE
OWN PRICE ELASTICITY

14. FACTORS AFFECTING THE
OWN PRICE ELASTICITY
1. Available Substitutes
2. Time
3. Expenditure Share

15. FACTORS AFFECTING THE
OWN PRICE ELASTICITY
Available Substitutes
One key determinant of the elasticity of
demand for a good is the number of close
substitutes for the good.
The more substitutes available for the
good, the more elastic the demand for it

16. FACTORS AFFECTING THE
OWN PRICE ELASTICITY
Time
The more time consumers have to react to
a price change, the more elastic the demand
for the good
Time allows the consumer to seek out
available substitutes

17. FACTORS AFFECTING THE
OWN PRICE ELASTICITY
Expenditure Share
Goods that comprise a relatively small
share of consumers’ budgets tend to be more
inelastic than goods for which consumers
spend a sizable portion of their incomes.
When a good comprises only a small
portion of the budget, the consumer can
reduce the consumption of other goods
when the price of the good increases.

18. MARGINAL REVENUE AND THE
OWN PRICE ELASTICITY OF
DEMAND
Marginal Revenue
The change in total revenue due to a
change in output, and that to maximize
profits, a firm should produce where
marginal revenue equals marginal cost.

20. Cross-Price Elasticity

21. Cross-Price Elasticity
♦ A measure of the responsiveness of the
demand for a good to changes in the price
of a related good: the percentage change in
the quantity demanded of one good divided
by the percentage change in the price of a
related good.

23. ♦ Whenever goods X and Y are substitutes,
an increase in the price of Y leads to an
increase in the demand for X.
♦ When goods X and Y are complements, an
increase in the price of Y leads to a
decrease in the demand for X.

24. Example
If the cross-price elasticity of demand between
Corel WordPerfect and Microsoft Word
processing software is 3, a 10% hike in the price
of Word will increase the demand for
WordPerfect by 30 percent, since 30%/10% = 3.
This demand increase for WordPerfect occurs
because consumers substitute away from Word
and toward WordPerfect, due to the price
increase.

25. Cross-Price Elasticity
♦ Cross-price elasticities play an important
role in the pricing decisions of firms that
sell multiple products.

26. Example
In fastfood chains, hamburgers and sodas are
complements. When customers buy
hamburgers, they buy sodas as well. If the
fastfood chain decides to lower the price on
hamburgers, the fastfood chain’s revenues
from both hamburgers and sodas are affected.
In addition, reducing the price of hamburgers
increases the quantity demanded on sodas,
thus increasing soda revenues.

27. Income Elasticity
♦ Income Elasticity is a measure of the
responsiveness of consumer demand to
changes in income.

29. Income Elasticity
♦ When good X is a normal good, an increase
in income leads to an increase in the
consumption of X. When X is an inferior
good, an increase in income leads to a
decrease in the consumption of X.

30. Income Elasticity
The formula for income elasticity is:
Income Elasticity = (% change in quantity
demanded) / (% change in income)

31. Example 1
An example of a product with positive
income elasticity could be Ferraris. Let's say
the economy is booming and everyone's
income rises by 400%. Because people have
extra money, the quantity of Ferraris
demanded increases by 15%.
We can use the formula to figure out the
income elasticity for this Italian sports car:
Income Elasticity = 15% / 400% = 0.0375

32. Example 2
An example of a good with negative income
elasticity could be cheap shoes. Let's again
assume the economy is doing well and
everyone's income rises by 30%. Because
people have extra money and can afford nicer
shoes, the quantity of cheap shoes demanded
decreases by 10%.
The income elasticity of cheap shoes is:
Income Elasticity = -10% / 30% = -0.33

33. Log-Linear Demand
♦ Demand is log-linear if the logarithm of
demand is a linear function of the
logarithms of prices, income, and other
variables.

35. MMT BATCH 36 35
ECONOMETRICS
& REGRESSION
ANALYSIS
Joseph Winthrop B. Godoy

36. MMT BATCH 36 36
Econometrics
Joseph Winthrop B. Godoy

37. MMT BATCH 36 37
Introduction
♦ Managers may obtain estimates of
demand and elasticity from published
studies available in the library or from a
consultant hired to estimate the
demand function based on the specifics
of their company product.
♦ The primary job of a manager is to use
the information to make decisions

38. MMT BATCH 36 38
Introduction
♦ Regardless of how the manager obtains
the estimates, it is useful to have a
general understanding of how demand
functions are estimated and what the
various diagnostic statistics that
accompany the reported output mean.
This entails knowledge of a branch of
economics called econometrics.
♦ Econometrics is simply the statistical
analysis of economic phenomena.

39. Econometrics
♦ Let’s briefly examine the basic ideas
underlying the estimation of the
demand for a product.
♦ Suppose there is some underlying data
on the relation between a dependent
variable, Y, and some explanatory
variable, X.
♦ Suppose that when the values of X and
Y are plotted, they appear as points A,
B, C, D, E, and F in Figure 3–4.
MMT BATCH 36 39

40. Econometrics
MMT BATCH 36 40

41. Econometrics
♦Clearly, the points do not lie on a
straight line, or even a smooth curve
(try alternative ways of connecting the
dots if you are not convinced).
♦The job of the econometrician is to
find a smooth curve or line that does a
“good” job of approximating the
points.
MMT BATCH 36 41

42. Econometrics
♦ For example, suppose the econometrician
believes that, on average, there is a linear
relation between Y and X, but there is also
some random variation in the relationship.
♦ Mathematically, this would imply that the
true relationship between Y and X is
Y = a + bX + e
MMT BATCH 36 42

43. Econometrics
♦ where a and b are unknown parameters
and e is a random variable (an error
term) that has a zero mean.
♦ Because the parameters that determine
the expected relation between Y and X
are unknown, the econometrician must
find out the values of the parameters a
and b.
MMT BATCH 36 43

44. Econometrics
♦ Note that for any line drawn through the
points, there will be some discrepancy
between the actual points and the line.
♦ For example, consider the line in slide 6
or the Figure 3–4, which does a
reasonable job of fitting the data.
MMT BATCH 36 44

45. Econometrics
♦ If a manager used the line to
approximate the true relation, there
would be some discrepancy between
the actual data and the line. For
example, points A and D actually lie
above the line, while points C and E lie
below it.
MMT BATCH 36 45

46. Econometrics
♦ The deviations between the actual
points and the line are given by the
distance of the dashed lines in Figure
3–4, namely êA, êC, êD, and êE.
♦ Since the line represents the expected,
or average, relation between Y and X,
these deviations are analogous to the
deviations from the mean used to
calculate the variance of a random
variable.
MMT BATCH 36 46

47. Econometrics
♦ The econometrician uses a regression
software package to find the values of a and
b that minimize the sum of the squared
deviations between the actual points and the
line. In essence, the regression line is the line
that minimizes the squared deviations
between the line (the expected relation) and
the actual data points. These values of a and
b, which frequently are denoted â and bˆ, are
called parameter estimates, and the
corresponding line is called the least
squares regression.
MMT BATCH 36 47

48. Regression Output in Excel
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.982655
R Square 0.96561
Adjusted R Square 0.959879
Standard Error 26.01378
Observations 15
ANOVA
df SS MS F Significance F
Regression 2 228014.6 114007.3 168.4712 1.65E-09
Residual 12 8120.603 676.7169
Total 14 236135.2
CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%
Intercept 562.151 21.0931 26.65094 4.78E-12 516.1931 608.1089
Temperature -5.436581 0.336216 -16.1699 1.64E-09 -6.169133 -4.704029
Insulation -20.01232 2.342505 -8.543127 1.91E-06 -25.1162 -14.90844
Estimated Heating Oil = 562.15 - 5.436 (Temperature) - 20.012 (Insulation)
Y = B0 + B1 X1 + B2X2 + B3X3 - - - +/- Error
Total = Estimated/Predicted +/- Error

49. MMT BATCH 36 49

50. MMT BATCH 36 50
Regression
Analysis
Joseph Winthrop B. Godoy

51. MMT BATCH 36 51
Introduction
• Many problems in engineering and science
involve exploring the relationships between two
or more variables.
• Regression analysis is a statistical technique
that is very useful for these types of problems.
– For example, in a chemical process, suppose that
the yield of the product is related to the process-
operating temperature.
• Regression analysis can be used to build a
model to predict yield at a given temperature
level.

52. MMT BATCH 36 52
Regression Analysis
♦ Regression Analysis: the study of the
relationship between variables
♦ Regression Analysis: one of the most
commonly used tools for business
analysis
♦ Easy to use and applies to many
situations

53. MMT BATCH 36 53
Regression Modeling Philosophy
♦ Nature of the relationships
♦ Model Building Procedure
– Determine dependent variable (y)
– Determine potential independent variable
(x)
– Collect relevant data
– Hypothesize the model form
– Fitting the model
– Diagnostic check: test for significance

54. Basic idea:
♦Use data to identify
relationships among
variables and use these
relationships to make
predictions
MMT BATCH 36 54

55. Linear Regression
♦Focus:
–Gain some understanding of the
mechanics.
• the regression line
• regression error
– Learn how to interpret and use the
results.
– Learn how to setup a regression
analysis.
MMT BATCH 36 55

56. Linear Regression
Regression is the attempt to explain the
variation in a dependent variable using the
variation in independent variables.
Regression is thus an explanation of causation.
If the independent variable(s) sufficiently
explain the variation in the dependent variable,
the model can be used for prediction.
Independent variable (x)
Dependentvariable

57. Linear Regression
♦Linear dependence: constant rate of
increase of one variable with respect to
another (as opposed to, e.g., diminishing
returns).
♦Regression analysis describes the relationship
between two (or more) variables.
♦Examples:
– Income and educational level
– Demand for electricity and the weather
– Home sales and interest rates
MMT BATCH 36 57

58. MMT BATCH 36 58
Regression Analysis
♦ Simple Regression: single explanatory
variable
♦ Multiple Regression: includes any
number of explanatory variables.

59. SingleSingle
RegressionRegression
Model Summary
.849a .721 .709 2177.791
Model
1
R R Square
Adjusted
R Square
Std. Error of
the Estimate
Predictors: (Constant), Population Per Square Mile,
Percent of Population 25 years and Over with
Bachelor's Degree or More, March 2000 estimates
a.
ANOVAb
5.75E+08 2 287614518.2 60.643 .000a
2.23E+08 47 4742775.141
7.98E+08 49
Regression
Residual
Total
Model
1
Sum of
Squares df Mean Square F Sig.
Predictors: (Constant), Population Per Square Mile, Percent of Population 25 years
and Over with Bachelor's Degree or More, March 2000 estimates
a.
Dependent Variable: Personal Income Per Capita, current dollars, 1999b.
Coefficientsa
13032.847 1902.700 6.850 .000
517.628 78.613 .553 6.584 .000
7.953 1.450 .461 5.486 .000
(Constant)
Percent of Population
25 years and Over
with Bachelor's
Degree or More,
March 2000 estimates
Population Per
Square Mile
Model
1
B Std. Error
Unstandardized
Coefficients
Beta
Standardized
Coefficients
t Sig.
Dependent Variable: Personal Income Per Capita, current dollars, 1999a.
Model Summary
.736a .542 .532 2760.003
Model
1
R R Square
Adjusted
R Square
Std. Error of
the Estimate
Predictors: (Constant), Percent of Population 25 years
and Over with Bachelor's Degree or More, March 2000
estimates
a.
ANOVAb
4.32E+08 1 432493775.8 56.775 .000a
3.66E+08 48 7617618.586
7.98E+08 49
Regression
Residual
Total
Model
1
Sum of
Squares df Mean Square F Sig.
Predictors: (Constant), Percent of Population 25 years and Over with Bachelor's
Degree or More, March 2000 estimates
a.
Dependent Variable: Personal Income Per Capita, current dollars, 1999b.
Coefficientsa
10078.565 2312.771 4.358 .000
688.939 91.433 .736 7.535 .000
(Constant)
Percent of Population
25 years and Over
with Bachelor's
Degree or More,
March 2000 estimates
Model
1
B Std. Error
Unstandardized
Coefficients
Beta
Standardized
Coefficients
t Sig.
Dependent Variable: Personal Income Per Capita, current dollars, 1999a.
MultipleMultiple
RegressionRegression

60. MMT BATCH 36 60
Regression Analysis
♦ Linear Regression: straight-line
relationship
Form: y = mx + b (linear equation)
♦ Non-linear: implies curved relationships,
for example logarithmic or curvilinear
relationships

61. Scatter plots
♦ Regression analysis requires interval
and ratio-level data.
♦ To see if your data fits the models of
regression, it is wise to conduct a
scatter plot analysis.
♦ The reason?
– Regression analysis assumes a linear
relationship. If you have a curvilinear
relationship or no relationship,
regression analysis is of little use.

62. Scatter plot
15.0 20.0 25.0 30.0 35.0
Percent of Population 25 years and Over with Bachelor's Degree or More,
March 2000 estimates
20000
25000
30000
35000
40000
PersonalIncomePerCapita,currentdollars,
1999
Percent of Population with Bachelor's Degree by Personal Income Per Capita
♦This is a linear relationship
♦It is a positive relationship.
♦As population with BA’s increases so does the
personal income per capita.

63. Regression Line
15.0 20.0 25.0 30.0 35.0
Percent of Population 25 years and Over with Bachelor's Degree or More,
March 2000 estimates
20000
25000
30000
35000
40000
PersonalIncomePerCapita,currentdollars,
1999
Percent of Population with Bachelor's Degree by Personal Income Per Capita
R Sq Linear = 0.542
♦Regression line is the best straight line
description of the plotted points and can use it to
describe the association between the variables.
♦If all the lines fall exactly on the line then the
line is 0 and you have a perfect relationship.

64. Types of Lines

65. Scatter Plots of Data with Various
Correlation Coefficients
Y
X
Y
X
Y
X
Y
X
Y
X
r = -1 r = -.6 r = 0
r = +.3r = +1
Y
X
r = 0

66. Y
X
Y
X
Y
Y
X
X
Linear relationships Curvilinear relationships
Linear Correlation

67. Y
X
Y
X
Y
Y
X
X
Strong relationships Weak relationships
Linear Correlation

68. Linear Correlation
Y
X
Y
X
No relationship

69. Things to remember
♦ Regressions are still focuses on
association, not causation.
♦ Association is a necessary
prerequisite for inferring causation, but
also:
1. The independent variable must preceded
the dependent variable in time.
2. The two variables must be plausibly lined
by a theory,
3. Competing independent variables must
be eliminated.

70. Regression Table
♦The regression
coefficient is not a
good indicator for
the strength of the
relationship.
♦Two scatter
plots with very
different
dispersions could
produce the same
regression line.
15.0 20.0 25.0 30.0 35.0
Percent of Population 25 years and Over with Bachelor's Degree or More,
March 2000 estimates
20000
25000
30000
35000
40000
PersonalIncomePerCapita,currentdollars,
1999
Percent of Population with Bachelor's Degree by Personal Income Per Capita
R Sq Linear = 0.542
0.00 200.00 400.00 600.00 800.00 1000.00 1200.00
Population Per Square Mile
20000
25000
30000
35000
40000
PersonalIncomePerCapita,currentdollars,1999
Percent of Population with Bachelor's Degree by Personal Income Per Capita
R Sq Linear = 0.463

71. Simple Linear Regression
Independent variable (x)
Dependentvariable(y)
The output of a regression is a function that predicts the
dependent variable based upon values of the
independent variables.
Simple regression fits a straight line to the data.
y’ = b0 + b1X ± є
b0 (y intercept)
b1 = slope
= ∆y/ ∆x
є

72. Simple Linear Regression
Independent variable (x)
Dependentvariable
The function will make a prediction for each
observed data point.
The observation is denoted by y and the
prediction is denoted by y.
Zero
Prediction: y
Observation: y
^
^

73. Simple Linear Regression
For each observation, the variation can be described as:
y = y + ε
Actual = Explained + Error
Zero
Prediction error: ε
^
Prediction: y^
Observation: y

74. SIMPLE REGRESSION
MMT BATCH 36 74
Relationship
Y X
Dependent Variable Independent Variable
y = mx + b
Linear Equation
ŷ = β0 + β1·x
β0 = y-intercept β1 = slope
β0 = Ῡ − β1•ẋ

75. SIMPLE REGRESSION
VARIABLE DATA
X Y X2
Y2
XY
1 2 1 4 2
2 4 4 16 8
3 5 9 25 15
4 7 16 49 28
5 8 25 64 40
ΣX ΣY ΣX2
ΣY2
ΣXY
15 26 55 158 93
ẋ Ῡ
3 5.2 n =5
MMT BATCH 36 75

76. SIMPLE REGRESSION
MMT BATCH 36 76
`
σ -
σ ሺσ ሺ
= 93 -
ሺ ሺ
ͳͷ
σ -
ሺσ ሺ
= 55 -
ሺ ሺ
ͳͲ
σ -
ሺσ ሺ
= 158 -
ሺ ሺ
ʹ ʹ Ǥͺ
Ԣ= = 1.5
= Ῡ- ·(x) = 5.2 - 1.5(3) = 0.7
{ Ԣ ሺ ሺ
{ Ǥ Ǥ ሺ ሺ

77. SIMPLE REGRESSION
MMT BATCH 36 77

78. Calculating SSE
Independent variable (x)
Dependentvariable
The line that minimizes the sum of squared deviations
between the line and the actual data points is the least
squares regression.
A least squares regression selects the line with the lowest
total sum of squared prediction errors.
This value is called the Sum of Squares of Error, or SSE(σ2
)

79. Calculating SSR
Independent variable (x)
Dependentvariable
The Sum of Squares Regression (SSR) is the
sum of the squared differences between the
prediction for each observation and the
population mean.
Population mean: y

80. Regression Formulas
Calculating SST
The Total Sum of Squares (SST) is equal to SSR + SSE.
Mathematically,
SSR = ∑ ( y – y’ ) (measure of explained variation)
SSE = ∑ ( y – y’ ) (measure of unexplained variation)
SST = SSR + SSE = ∑ ( y – y’ ) (measure of total variation in y)
MMT BATCH 36 80

81. Regression Coefficient
♦ The regression coefficient is the slope
of the regression line and tells you what
the nature of the relationship between
the variables is.
♦ How much change in the independent
variables is associated with how much
change in the dependent variable.
♦ The larger the regression coefficient the
more change.

82. The Coefficient of Determination
The proportion of total variation (SST) that is
explained by the regression (SSR) is known as
the Coefficient of Determination, and is often
referred to as R .
R = =
The value of R can range between 0 and 1, and
the higher its value the more accurate the
regression model is. It is often referred to as a
percentage.
SSR SSR
SST SSR + SSE
2
2
2

83. Standard Error of Regression
The Standard Error of a regression is a measure
of its variability. It can be used in a similar
manner to standard deviation, allowing for
prediction intervals.
y ± 2 standard errors will provide approximately
95% accuracy, and 3 standard errors will provide
a 99% confidence interval.
Standard Error is calculated by taking the square
root of the average prediction error.
Standard Error = SSE
n-k
Where n is the number of observations in the
sample and k is the total number of variables in
the model
√

84. The output of a simple regression is
the coefficient β and the constant A.
The equation is then:
y = A + β * x + ε
where ε is the residual error.
β is the per unit change in the
dependent variable for each unit
change in the independent variable.
Mathematically:
β =
∆ y
∆ x

85. Multiple Linear Regression
More than one independent variable can be
used to explain variance in the dependent
variable, as long as they are not linearly related.
A multiple regression takes the form:
y = A + β X + β X + … + β k Xk + ε
where k is the number of variables, or
parameters.
1 1 2 2

86. Multicollinearity
Multicollinearity is a condition in which at least
2 independent variables are highly linearly
correlated. It will often crash computers.
Example table of
Correlations
Y X1 X2
Y 1.000
X1 0.802 1.000
X2 0.848 0.578 1.000
A correlations table can suggest which
independent variables may be significant.
Generally, an ind. variable that has more than a
.3 correlation with the dependent variable and
less than .7 with any other ind. variable can be
included as a possible predictor.

87. Nonlinear Regression
Nonlinear functions can also be fit as
regressions. Common choices include
Power, Logarithmic, Exponential, and
Logistic, but any continuous function
can be used.

88. Some Aplications
MMT BATCH 36 88

89. House Number Y: Actual Selling
Price ($1,000s)
X: House Size (100s ft2)
1 89.5 20.0
2 79.9 14.8
3 83.1 20.5
4 56.9 12.5
5 66.6 18.0
6 82.5 14.3
7 126.3 27.5
8 79.3 16.5
9 119.9 24.3
10 87.6 20.2
11 112.6 22.0
12 120.8 .019
13 78.5 12.3
14 74.3 14.0
15 74.8 16.7
Averages 88.84 18.17
Sample15housesfromtheregion.

90. MMT BATCH 36 90
Simple Regression Model
♦ y = a + bx + e (Note: y = mx + b)
♦ Coefficients: a and b
♦ Variable a is the y intercept
♦ Variable b is the slope of the line

91. MMT BATCH 36 91
Simple Regression Model
♦ Precision: accepted measure of accuracy is
mean squared error
♦ Average squared difference of actual and
forecast

92. MMT BATCH 36 92
Simple Regression Model
♦ Average squared difference of actual and
forecast
♦ Squaring makes difference positive, and
severity of large errors is emphasized

93. MMT BATCH 36 93
Simple Regression Model
♦ Error (residual) is difference of actual data
point and the forecasted value of dependant
variable y given the explanatory variable x.
Error

94. MMT BATCH 36 94
Simple Regression Model
♦ y = mx + b
♦ Y= a + bX + e
♦ Ŷ = 56,104 + 63.11(Sq ft) + e
♦ If X = 2,500 Square feet, then
♦ $213,879 = 56,104 + 63.11(2,500)

95. MMT BATCH 36 95
Simple Regression Model
♦ Linearity
Square Feet Line Fit Plot
0
50,000
100,000
150,000
200,000
250,000
300,000
350,000
1,500 2,000 2,500 3,000 3,500 4,000
Square Feet
Cost
Cost Predicted Cost

96. MMT BATCH 36 96
Simple Regression Model
♦ Linearity
Square Feet Residual Plot
-100000
-50000
0
50000
100000
1,500 2,000 2,500 3,000 3,500 4,000
Square Feet
Residuals

97. MMT BATCH 36 97
Simple Regression Model
♦ Independence:
– Errors must not correlate
– Trials must be independent

98. MMT BATCH 36 98
Simple Regression Model
♦ Homoscedasticity:
– Constant variance
– Scatter of errors does not change from trial to
trial
– Leads to misspecification of the uncertainty in
the model, specifically with a forecast
– Possible to underestimate the uncertainty
– Try square root, logarithm, or reciprocal of y

99. MMT BATCH 36 99
Simple Regression Model
♦ Normality:
• Errors should be normally distributed
• Plot histogram of residuals

100. MMT BATCH 36 100
Multiple Regression Model
♦ Y = α + β1X1 + … + βkXk + ε

101. Example: An Empirical Model
MMT BATCH 36 101

102. Empirical Model
Figure 1 Scatter Diagram of oxygen purity versus
hydrocarbon level from Table 11-1.

103. Empirical Model
Based on the scatter diagram, it is probably reasonable to
assume that the mean of the random variable Y is related to
x by the following straight-line relationship:
where the slope and intercept of the line are called
regression coefficients.
The simple linear regression model is given by
where ε is the random error
term.

104. Empirical Models
We think of the regression model as an empirical
model.
Suppose that the mean and variance of ε are 0 and σ2
,
respectively, then
The variance of Y given x is

105. Empirical Models
• The true regression model is a line of mean
values:
where β1 can be interpreted as the change in the mean
of Y for a unit change in x.
• Also, the variability of Y at a particular value of x is
determined by the error variance, σ2
.
• This implies there is a distribution of Y-values at
each x and that the variance of this distribution is the
same at each x.

106. Empirical Models
Figure 2 The distribution of Y for a given value of x
for the oxygen purity-hydrocarbon data.

107. Simple Linear Regression
• The case of simple linear regression considers
a single regressor or predictor x and a
dependent or response variable Y.
• The expected value of Y at each level of x is a
random variable:
• We assume that each observation, Y, can be
described by the model

108. Simple Linear Regression
• Suppose that we have n pairs of observations (x1,
y1), (x2, y2), …, (xn, yn).
Figure 3
Deviations of
the data from
the estimated
regression
model.

109. Simple Linear Regression
• The method of least squares is used to estimate the
parameters, β0 and β1 by minimizing the sum of the
squares of the vertical deviations in Figure 3.
Figure 3
Deviations of
the data from
the estimated
regression
model.

110. Simple Linear Regression
• Using the following Equation, the n observations in
the sample can be expressed as
• The sum of the squares of the deviations of the
observations from the true regression line is