Multicollinearity
Lecture Note #8
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
1. Perfect Multicollinearity
Definition : Existence of exact linear relationship(s) among independent variables
Example :
Yi = 1 + 2 X2i + 3 X3i + ei
whe

Families
of
Distributions
We
have
discussed
properties
of
discrete
and
continuous
random
variables.
Now,
we
are
going
to
consider
some
important
examples
of
discrete
and
continuous
random
variables.
Discrete
Probability
Distr

Random
Variables
(Distribution
Theory)
This
note
is
based
on
Prof.
Joon
Y.
Park's
lecture
notes
series.
1.
Random
Vectors
and
Joint
Distribution
(1)
A
Random
Vector
.
Random
Vector
:
We
dene
random
variables
X1;:;Xn
on

Hypothesis
Testing
6.
Tests
of
the
Population
Variance
In
this
section,
we
develop
procedures
for
testing
the
population
variance
2
based
on
a
random
sample
of
n
observations
from
a
normal
population.
It
is
natural
to
b

Inference
in
the
Simple
Regression
Lecture
Note
#4
This
note
is
based
on
Prof.
Yoon-Jae
Whang's
lecture
notes
series.
We
have
learned
how
to
get
OLS
estimates
(point
estimates)
in
the
simple
regression
model,
and
investi

1
Linear
Regression
Model:
Basic
Results
1
The
Linear
Regression
Model
.
Model:
yi
=
1
+
2xi2
+
+
kxik
+
ui
for
i
=1,
2,
:,
n.
In
a
matrix
form,
it
can
be
written
as
y
=
X
+
u,
where
1
010
1
0
1.
y1
1
x12

Linear
Regression
Model:
Inference
Why
Study
Hypothesis
Testing?
.
Examples:
(i)
To
evaluate
the
prediction
of
an
economic
theory,
e.g.
interest
elasticity
of
money
demand
=
0?
(ii)
To
detect
absence
of
structure
(
also

Multiple
Regression
Lecture
Note
#6
This
note
is
based
on
Prof.
Yoon-Jae
Whang's
lecture
notes
series.
.
General
Model
Yi
=
1
+
2X2i
+
+
K
XKi
+
ui
There
may
be
more
than
one
explanatory
variable
that
may
inuenc

Dummy
Variables
Lecture
Note
#7
This
note
is
based
on
Prof.
Yoon-Jae
Whang's
lecture
notes
series.
.
Dummy
Variable
:
Explanatory
variables
take
only
one
of
two
values,
1
or
0.
.
Qualitative
variables
may
be
also
impor

Simple
Regression
2
Lecture
Note
#3
This
note
is
based
on
Prof.
Yoon-Jae
Whang's
lecture
notes
series.
Once
we
get
the
OLS
estimator
in
a
simple
regression
model,
next
step
is
to
investigate
the
properties
of
OLS
estim

Autocorrelation
Lecture
Note
#10
1.
The
Nature
of
Autocorrelation.
.
In
classical
assumptions,
regression
errors
are
assumed
to
be
uncorrelated,
that
is,
Cov
(ui;uj)=
0
for
i
6
=
j
(No
Autocorrelation).
In
this
section,
w

Inference
in
the
Simple
Regression
II
Lecture
Note
#5
This
note
is
based
on
Prof.
Yoon-Jae
Whang's
lecture
notes
series.
In
this
chapter,
we
consider
the
following
problems
in
a
linear
regression
model.
.
How
to
measure

Heteroscedasticity
Lecture
Note
#9
1.
The
Nature
of
Heteroscedasticity
.
Consider
the
following
simple
regression
Yi
=
1
+
2X2i
+
ei
to
explain
household
expenditure
on
food
(Y
)
as
a
linear
function
of
household
income

Generalized
Linear
Models
1.
Model
y
=
X
+
u,
where
Eu
=0
Euu.
=
V
.
6
:
Nonspherical
errors
2
=
2I
2.
Sources
of
nonspherical
errors
Non-spherical
errors
are,
most
notably,
due
to
heteroskedasticity
and
autocorrelati

Multicollinearity
Lecture
Note
#8
1.
Perfect
Multicollinearity
.
Denition
:
Existence
of
exact
linear
relationship(s)
among
independent
variables
.
Example
:
Yi
=
1
+
2X2i
+
3X3i
+
ui
where
X2i
=
X3i
for
i
=1,
;n

Simultaneous
Equation
Model
Lecture
Note
#11
1.
Introduction
.
The
regression
models
we
have
considered
so
far
are
all
single
equation
regression
models
.
A
single
dependent
variable
(Y
)
is
expressed
as
a
function
of
on

1
Matrix
Algebra:
A
Review
1
Basic
Denitions
and
Axioms
3
.
.
A
=
666
.
a11
a12
a1n
a21
a22
a2n
.
.
.
am1
am2
amn
777
.
=
(aij)
is
an
m
n
matrix
(m
rows,
n
columns)
with
aij
as
its
element
in
the
i-th

Sampling
Distribution
\A
sample
should
be
representative
of
the
population":
Random
Sampling
.
Some
denitions
.
We
want
to
know
the
properties
of
a
large
group
of
objects,
given
information
on
a
relatively
small
subset
o

Interval
Estimation
II
5.
Condence
Intervals
for
the
Variance
of
a
Normal
Population
Suppose
that
we
have
a
sample
of
n
observations
from
a
normal
population
with
mean
and
variance
2
.
In
this
section,
we
consider
th

DESCRIPTIVE
STATISTICS
Objective
:
To
present
statistical
data
in
such
a
way
that
the
important
characteristics
of
the
data
could
be
easily
understood.
Our
aim
is
to
survey
some
of
the
methods
employed
in
the
summarizati

Multiple Regression
Lecture Note #6
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
General Model
Yi = 1 + 2 X2i + + K XKi + ei
There may be more than one explanatory variable that may influence the dependent
variable.
Example :
Consum

Inference in the Simple Regression
Lecture Note #4
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
We have learned how to get OLS estimates (point estimates) in the simple regression
model, and investigated some sample properties of OLS

Heteroscedasticity
Lecture Note #9
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
1. The Nature of Heteroscedasticity
Consider the following simple regression
Yi = 1 + 2 X2i + ei
to explain household expenditure on food (Y ) as a linea

Inference in the Simple Regression II
Lecture Note #5
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
In this chapter, we consider the following problems in a linear regression model.
How to measure the variation in yi explained by the

1
Autocorrelation
Lecture Note #10
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
1. The Nature of Autocorrelation.
In classical assumptions, regression errors are assumed to be uncorrelated, that is,
Cov (ei , ej ) = 0 f or i 6= j (No

Simple Regression 1
Lecture Note #2
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
1. The Regression Problem
Let
Y : Dependent Variable (or Explained) Variable
X : Independent (or Explanatory) Variable(s)
The regression analysis tries t

Simple Regression 2
Lecture Note #3
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
Once we get the OLS estimator in a simple regression model, next step is to investigate
the properties of OLS estimators for unknown parameters, 1 , 2 .

Dummy Variables
Lecture Note #7
This note is based on Prof. Yoon-Jae Whangs lecture notes series.
Dummy Variable : Explanatory variables take only one of two values, 1 or 0.
Qualitative variables may be also important in explaining a dependent variable.

What subjects are covered in this course?
Literal Interpretation:
Econo + Metrics = Economic Measurement
Purpose: Econometrics gives empirical content to a priori reasoning in economics.
Economic Theory + Mathematical Tools + Statistical Tools.
Major A

Interval
Estimation
I
1.
Interval
Estimator
In
the
previous
section,
we
have
considered
the
point
estimation
of
unknown
parameters
in
the
population.
The
point
estimation
leads
to
a
single
number
for
the
estimation
of
giv