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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 STAFF
 Regression Analysis, sample regression equation

Click to edit the document details