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

1
Linear Regression Model: Basic Results
1
The Linear Regression Model
Model:
yi = 1 + 2 xi2 + + k xik + ui for i = 1, 2, ., n.
In a matrix form, it can be written as
y = X + u,
where
y1
y = . , X =
yn
1 x12
.
.
.
.
.
.
1 xn2
x1k
. , =
.
xnk
1
. , u

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 .

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

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

1
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 one or more variables
(independent va

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

1
Linear Regression Model: Inference
1
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 referred to as specification tes

1
Matrix Algebra: A Review
1
Basic Definitions and Axioms
a1n
a2n
A=
= (aij ) is an m n matrix (m rows, n columns) with
am1 am2 amn
aij as its element in the i-th row and j-th column for i = 1, ., m and j = 1, ., n.
a11
a21
.
.
a12
a22
A0 = (aji ) is

1
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 f or i 6= j (No Autocorrelation).
In this section, we relax this assumption by al

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

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 + ui
There may be more than one explanatory variable that may influence the dependent
variable.
Example :
Consum

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

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.

1
Generalized Linear Models
1. Model
y = X + u,
where
Eu = 0
Euu0 = V
2 =
6 2 I : Nonspherical errors
2. Sources of nonspherical errors
Non-spherical errors are, most notably, due to heteroskedasticity and autocorrelation.
(i) Heteroskedasticity: The dia

1
Multivariate (Joint) Normal Distribution
Ex1
x1
and V ar(X) = E[(X
If X = . is a random vector, then E(X) = .
Exn
xn
EX)(X EX)0 ].
Definition 1 (Multivariate Normal): If X Rn has a multivariate normal distribution, i.e.,
X N (, ), iff the joint densit

1
Suggested Solutions to Problem Set 7
1. A simple regression model is given by
Yt = 1 + 2 X2t + et for t = 1, , n
where the regression errors et with V ar (et ) = e2 follow AR(1) model
et = et1 + t , t = 1, . . . , n
(1)
where t s are uncorrelated random

Hypothesis Testing
1. Concepts of Hypothesis Testing
We are often interested in using data in discriminating among economic theories or accessing
the validity of some conjecture, or hypothesis.
Generally, hypotheses are formed about the population paramet

Point Estimation
This note is based on Prof. Joon Y. Parks lecture notes series.
1. Statistical Inference
General Remarks : Suppose that we are given n-numbers x1 , . . . , xn , which are believed to
be generated from some random mechanism in the sense th

1
Suggested Solutions to Problem Set 6
3. Heteroscedasticity :
c = 7.5 + 0.9 N
W
(1)
[ = 0.8 + 7.9 (1/N )
W/N
(2)
(20.5)
(7.2)
(7.5)
(75.5)
(a) The first regression model is just fitted model by OLS without
any transformation, and the second is fitted mod

PROBABILITY
This note is based on Prof. Joon Y. Parks lecture notes series.
Based on the sample information, it is impossible to know exactly the reaction of the
whole population; any information about the population from the sample will inevitably
involv