378
Chapter 15: Multiple Regression Model Building
CHAPTER 15
OBJECTIVES
•
To be able to use quadratic terms in a regression model
•
To be able to use transformed variables in a regression model
•
To be able to measure the correlation among the independent variables
•
To build a regression model using either the stepwise or bestsubsets approach
•
To understand the pitfalls involved in developing a multiple regression model
OVERVIEW AND KEY CONCEPTS
The Quadratic Regression Model
•
In a quadratic regression model, the relationship between the dependent variable and one or
more independent variable is a quadratic polynomial function.
•
The quadratic regression model is useful when the residual plot reveals nonlinear
relationship.
•
The quadratic regression model:
±
2
01
1
2
1
ii
i
i
YX
X
β
ββε
=+
+
+
±
The quadratic relationship can be between
Y
and more than one
X
variable as well.
•
Testing for the overall significance of the quadratic regression model:
±
The test for the overall significance of the quadratic regression model is exactly the
same as testing for the overall significance of any multiple regression model.
±
Test statistic:
MSR
F
MSE
=
.
•
Testing for quadratic effect:
±
The hypotheses:
02
:0
H
=
(no quadratic effect) vs.
12
H
≠
(the quadratic
term is needed)
±
Test statistic:
2
22
b
b
t
S
−
=
.
±
This is a twotail test with a left tail and a righttail rejection region.
Using Transformation in Regression Models
•
The following three transformation models are often used to overcome violations of the
homoscedasticity assumption, as well as to transform a model that is not linear in form into
one that is linear.
•
Squareroot transformation:
±
011
i
βε
+
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±
The dependent variable is
i
Y
and the independent variable is
1
i
X
.
•
Transformed multiplicative model:
±
01
1
log
log
log
log
log
ii
k
k
i
i
YX
X
β
ββ
ε
=+
+
+
+
L
±
The dependent variable is log
i
Y
and the independent variables are
1
log
i
X
,
2
log
i
X
,
etc.
•
Transformed exponential model:
±
1
ln
ln
k
k
i
i
X
+
+
+
L
±
The dependent variable is ln
i
Y
and the independent variables are
1
i
X
,
2
i
X
, etc.
Collinearity
•
Some of the explanatory variables are highly correlated with each other.
•
The collinear variables do not provide new information and it becomes difficult to separate
the effect of such variables on the dependent variable.
Y
X
1
X
2
Large
Overlap
Overlap
in variation of
X
1
and
X
2
is used in explaining
the variation in
Y
but
NOT
in
estimating
and
1
2
Large
Overlap
Overlap
reflects collinearity
between
X
1
and
X
2
•
The values of the regression coefficient for the correlated variables may fluctuate drastically
depending on which independent variables are included in the model.
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 Spring '10
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 Correlation, Regression Analysis, regression model, R Square

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