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Unit Three:
Linear Regression Analysis
Linear regression analysis is a widelyused technique for describing the relationship between a
quantitative
variable Y (the dependent variable) and one or more other variables X
1
, X
2
, …, X
p
(the
independent or predictor variables), at least one of which is quantitative.
In a simple linear regression
analysis, there is exactly one independent variable; in a multiple linear regression analysis, there are
two or more independent variables.
Linear regression analysis has various purposes, including:
(1)
to make predictions
(example:
A large retail department chain wants to assess whether it could predict, with
reasonable certainty and precision, the firstyear sales (Y) of a proposed retail store based on the proposed
location (X
1
= 1 if mall, 0 if shopping center) and measures of population (X
2
), per capita income (X
3
), and
competition (X
4
) within the service area.)
(2)
to estimate impacts
(example:
A manufacturer of ceramic vases wants to estimate the impact of increasing
the temperature (X
1
) within the kiln, or increasing the pressure (X
2
) applied when clay is injected into the
mold, on the breaking strength (Y) of its ceramic vases.)
(3)
to test hypotheses
(example:
A researcher wants to test the hypothesis that the endofyear price (Y) of the
S&P500 would be positively related to the midyear percentage of employed persons in the 45to64 year old
group (X
1
) after controlling for bond yields (X
2
), GDP (X
3
4
), and the
maximum capital gains tax rate (X
5
).)
(4)
to set standards
(example: The manager of a landscaping company wants to estimate the relationship between
the time (Y) it would take an employee to perform a landscaping job and the number of years of experience
(X
1
) of the employee as well as various characteristics of the job or site, including the number of trees to be
planted (X
2
), the number of bushes to be planted (X
3
), and the condition of the terrain (X
4
= 1 if rocky, 0
otherwise) at the site.)
Simple Linear Regression
The (classical normal) simple linear regression model is expressed in the form
Y =
ß
0
+ ß
1
X +
ε
,
where:
•
Y and X are quantitative variables
•
ß
0
and ß
1
are real numbers.
•
ε is called the error (or stochastic or random error) term.
It represents the combined influence on Y
of all factors other than X.
Note
:
Examples of quantitative variables include such “plain” variables as Sales and Age and such transformed
variables as ln(Assets) and
Income
.
Assumptions of the model
include (with x denoting any value in the presumed domain of X):
Across all potential entities* with X = x
:
1.
E(
ε
) = 0 (
⇔
E(Y) = ß
0
+ ß
1
x).
We will call this the
linearity assumption
.
It follows from this assumption that an entity with
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 Fall '08
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