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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
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
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
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.
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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
Multicollinearity
Lecture Note #8
1. Perfect Multicollinearity
Definition : Existence of exact linear relationship(s) among independent variables
Example :
Yi = 1 + 2 X2i + 3 X3i + ui
where X2i = X3i
f or i = 1, , n
We cannot obtain unique estimates of
Random Variables (Distribution Theory)
This note is based on Prof. Joon Y. Parks lecture notes series.
1. Random Variables
To summarize the information to random experiments conveniently, we need this information
to be numerical. In general, the outcomes
1
Suggested Solutions for Problem Set 1
1. Question 1
(a) Bivariate probability distribution: we have four possible outcomes cfw_HH, HT, T H, T T
and
Y
0 1 2
X 0 14 14 0 12
1 0 14 14 12
1
1
1
1
4
2
4
(b) Cov (X, Y ) = E [(X X ) (Y Y )] = E (XY ) X Y
1
1
Random Variables (Distribution Theory)
This note is based on Prof. Joon Y. Parks lecture notes series.
Joint Probability Distributions
We discussed the joint probability before, and now consider the interaction between two, or
more, possibly related discr
1
SUGGESTED ANSWER TO PROBLEM SET 2
Question 1.
(a)By the first order condition of OLS, it follows that
0=
=
n
X
i=1
n
X
Yi b1 b2 Xi
ebi .
i=1
However it is not true that
Only E(ei ) = 0 is true.
Pn
i=1 ei
= 0 because ei is regression error and a random
1
Suggested Solutions for Problem Set 4
1. Joint Testing : We use F-test for the testing of this joint hypothesis,
Yi = 1 + 2 X2i + 3 X3i + ei ,
will be unrestricted regression model. Under H0 , 2 = 1 + 3
Yi = 1 + (1 + 3 ) X2i + 3 X3i + ei
Yi X2i = 1 + 3
Random Variables (Distribution Theory)
This note is based on Prof. Joon Y. Parks lecture notes series.
1. Random Vectors and Joint Distribution
(1) A Random Vector
Random Vector : We define random variables X1 , . . . , Xn on the probability space
(, F,
Probability Distributions
This note is based on Prof. Joon Y. Parks lecture notes series.
1. Continuous Probability Distributions
In fact, there are lots of continuous probability density functions. That is, any function that
integrates to one and has non
1
Interval Estimation II
5. Confidence 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 the confidence interval of populatio
1
Suggested Solutions for Problem Set 3
1. Sales vs Advertisement
(a) b1 = 5.35 and b2 = 2.08 Estimation results from both by hands and Stata
should be equal.
(b) Error term reflects the effects from omitted variables whch we think affect
the regressand b
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 Distributions
(1) The Bernoulli Dis