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Econometrics
Instructor: Dr. Jian-fa Li
Ext：04-23323000-4283
Office：T2–1026
E-mail：[email protected]
Required：
Introductory Econometrics：A Modern Approach, 2nd edition,
J.F. Wooldridge (東華) –elementary level.
Reference：
1. Undergraduate economics：R.C.Hill, W.E.Griffiths,G.G.Judge,2nd edition (雙
葉) –elementary level.
2. Econometric Analysis：W.H.Greene,5th edition (雙葉)-Advanced.
Ch1 Warming up
*

Why Study Econometrics?
Econometrics is a set of research tools also employed in the business disciplines
of accounting, finance, marketing and management.
It is also used by social
scientists, specifically researchers in history, political science and sociology.
Econometrics plays an important role in such diverse fields as forestry, and in
agricultural economics.

Studying econometrics fills a gap between being “a student of economics” and
being “a practicing economist.”

By taking this introduction to econometrics you will gain an overview of what
econometrics is about, and develop some “intuition” about how things work.
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* What is Econometrics?

In economics we express our ideas about relationships between economic
variables using the mathematical concept of a function.

For example, to express a relationship between income i and consumption c, we
may write
c  f (i )

The demand for an individual commodity, say the Honda Accord, might be
expressed as
qd  f ( p, p s , pc , i)

The
quantity
of
f ( p, p s , pc , i)
Honda
Accords
demanded,
qd,
is
a
function
of the price of Honda Accords p, the price of cars that are
substitutes ps, the price of items that are complements pc, like gasoline, and the
level of income i.

The supply of an agricultural commodity such as beef might be written as
q s  f ( p, pc , p f )

qs is the quantity supplied, p is the price of beef, pc is the price of competitive
products in production (for example, the price of hogs), and pf is the price of
factors or inputs (for example, the price of corn) used in the production process.
Econometrics is about how we can use economic, business or social science theory
and data, along with tools from statistics, to answer “how much” type questions.
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*Some Examples

A question facing Alan Greenspan is “How much should we increase the discount
rate to slow inflation, and yet maintain a stable and growing economy?” The
answer will depend on the responsiveness of firms and individuals to increases in
the interest rates and to the effects of reduced investment on Gross National
Product. The key elasticities and multipliers are called parameters. The values
of economic parameters are unknown and must be estimated using a sample of
economic data when formulating economic policies.

Econometrics is about how to best estimate economic parameters given the data
we have. “Good” econometrics is important, since errors in the estimates used
by policy makers such as the FRB may lead to interest rate corrections that are too
large or too small, which has consequences for all of us.
Other examples
include:

A city council ponders the question of how much violent crime will be reduced if
an additional million dollars is spent putting uniformed police on the street.

U.S. Presidential candidate Gore questions how many additional California voters
will support him if he spends an additional million dollars in advertising in that
state.

The owner of a local Pizza Hut franchise must decide how much advertising space
to purchase in the local newspaper, and thus must estimate the relationship
between advertising and sales.

Louisiana State University must estimate how much enrollment will fall if tuition
is raised by $100 per semester, and thus whether its revenue from tuition will rise
or fall.
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
The CEO of Proctor & Gamble must estimate how much demand there will be in
ten years for the detergent Tide, as she decides how much to invest in new plant
and equipment.

A real estate developer must predict by how much population and income will
increase to the south of Baton Rouge, Louisiana, over the next few years, and if it
will be profitable to begin construction of a new strip-mall.

You must decide how much of your savings will go into a stock fund and how
much into the money market. This requires you to make predictions of the level of
economic activity, the rate of inflation and interest rates over your planning
horizon.
Writing an Empirical Research Report
* Choosing a Topic
* Writing an Abstract

After you have selected a specific topic, it is a good idea to write up a brief
abstract.

The abstract should be short, usually no more than 500 words, and should include:
1)
a concise statement of the problem;
2)
comments on the information that is available with one or two key
references;
3)
a description of the research design that includes:
(a)
the economic model,
(b)
the econometric estimation and inference methods,
(c)
data sources,
(d)
estimation, hypothesis testing and prediction procedures; and
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4)
the contribution of the work
* A Format for Writing a Research Report
Economics research reports have a standard format in which the various steps of the
research project are discussed and the results interpreted.
The following outline is
typical:
1. Statement of the Problem: The place to start your report is with a summary of the
questions you have investigated, why they are important and who should be
interested in the results.
Identify the contents of each section of the report.
This section should be non-technical and it should motivate the reader to continue
reading the paper.
2. Review of the Literature: Briefly summarize the relevant literature in the research
area you have chosen, so that you may make clear how your work extends our
knowledge.
By all means, cite the works of others that have motivated your
research, but keep it brief. You do not have to survey everything that has been
written on the topic.
3. The Economic or Financial Model: Specify the economic model that you use, and
define the economic variables.
State the model’s assumptions and identify
hypotheses you wish to test. Economic models can get complicated.
Your task
is to explain the model as clearly, but as briefly and simply, as possible.
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Don’t
use unnecessary technical jargon. Use simple terms instead of complicated ones
when possible. Your objective is to display the quality of your thinking, not the
extent of your vocabulary.
4. The Econometric Model: Discuss the econometric model that corresponds to the
economic model. Make sure you include a discussion of the variables in the
model, the functional form, the error assumptions and any other assumptions that
you make. Use notation that is as simple as possible, and do not clutter the body
of the paper with long proofs or derivations.
These can go in a technical
appendix.
5. The Data: Describe the data you used, the source of the data, and any reservations
you have about their appropriateness.
6. The Estimation and Inference Procedures: Describe the estimation methods you
used and why they were chosen. Explain hypothesis testing procedures and their
use.
7. The Empirical Results and Conclusions: Report the parameter estimates, their
interpretation, and values of test statistics.
Comment on their statistical
significance, their relation to previous estimates, and their economic implications.
8. Possible Extensions and Limitations of the Study: Your research will raise
questions about the economic model, data and estimation techniques.
What
future research is suggested by your findings and how might you go about it?
9. Acknowledgments: It is appropriate to recognize those who have commented on
and contributed to your research. This may include your instructor, a librarian
who helped you find data, a fellow student who read and commented on your
paper.
10. References: An alphabetical list of the literature you cite in your study, as well as
references to the data sources you used.
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* Some Basic Probability Concepts:
1. Experiments, Outcomes and Random Variables

A random variable is a variable whose value is unknown until it is observed.
The value of a random variable results from an experiment; it is not perfectly
predictable.

A discrete random variable can take only a finite number of values, which can be
counted by using the positive integers.

A continuous random variable can take any real value (not just whole numbers) in
an interval on the real number line.
2. The Probability Distribution of a Random Variable

When the values of a discrete random variable are listed with their chances of
occurring, the resulting table of outcomes is called a probability function or a
probability density function.

For a discrete random variable X the value of the probability density function f(x)
is the probability that the random variable X takes the value x, f(x)=P(X=x).

Therefore, 0  f(x)  1 and, if X takes n values x1, .., xn, then
f ( x1 )  f ( x2 ) 

 f ( xn )  1.
For the continuous random variable Y the probability density function f(y) can be
represented by an equation, which can be described graphically by a curve. For
continuous random variables the area under the probability density function
corresponds to probability.
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3. Expected Values Involving a Single Random Variable
The Rules of Summation
(1) If X takes n values x1, ..., xn then their sum is
n
x
i 1
 x1  x2 
i
 xn
(2) If a is a constant, then
n
 a  na
i 1
(3) If a is a constant then
n
 ax
i 1
i
n
a  xi
i 1
(4) If X and Y are two variables, then
n
n
n
 ( x  y )  x   y
i 1
i
i
i 1
i
i 1
i
(5) If X and Y are two variables, then
n
n
n
 (ax  by )  a x  b y
i
i 1
i
i 1
i
i
i 1
(6)The arithmetic mean (average) of n values of X is
n
x
x
i 1
n
i

x1  x2 
n
 xn
.
Also,
8
n
(x  x )  0
i
i 1
(7)We often use an abbreviated form of the summation notation.
For
example, if f(x) is a function of the values of X,
n
 f (x )  f (x )  f (x ) 
i
i 1
1
 f ( xn )
2
=  f ( xi ) ("Sum over all values of the index i")
i
  f ( x) ("Sum over all possible values of X ")
x
(7)Several summation signs can be used in one expression. Suppose the
variable Y takes n values and X takes m values, and let f(x,y)=x+y. Then
the double summation of this function is
m
n
m
n
 f ( x , y )   ( x  y )
i 1 j 1
i
j
i 1 j 1
i
j
4. The Mean of a Random Variable

The expected value of a random variable X is the average value of the random
variable in an infinite number of repetitions of the experiment (repeated samples);
it is denoted E[X].

If X is a discrete random variable which can take the values x1, x2,…,xn with
probability density values f(x1), f(x2),…, f(xn), the expected value of X is
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E[ X ]  x1 f ( x1 )  x2 f ( x2 ) 
 xn f ( xn )
n
  xi f ( xi )
i 1
  xf ( x)
x
5. Expectation of a Function of a Random Variable

If X is a discrete random variable and g(X) is a function of it, then
E[ g ( X )]   g ( x) f ( x)
x
However,

E[ g ( X )]  g[ E ( X )]
in general.
If X is a discrete random variable and g(X) = g1(X) + g2(X), where g1(X) and g2(X)
are functions of X, then
E[ g ( X )]  [ g1 ( x)  g 2 ( x)] f ( x)
x
  g1 ( x) f ( x)   g 2 ( x) f ( x)
x
x
 E[ g1 ( x)]  E[ g 2 ( x)]

The expected value of a sum of functions of random variables, or the expected
value of a sum of random variables, is always the sum of the expected values.
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If c is a constant,
E[c]  c
If c is a constant and X is a random variable, then
E[cX ]  cE[ X ]
If a and c are constants then
E[a  cX ]  a  cE[ X ]
6. The Variance of a Random Variable
var( X )  2  E[ g ( X )]  E  X  E ( X )  E[ X 2 ]  [ E ( X )]2
2
Let a and c be constants, and let Z = a + cX. Then Z is a random variable and its
variance is
var(a  cX )  E[(a  cX )  E (a  cX )]2  c 2 var( X )

If X and Y are independent random variables, then
f ( x, y)  f ( x) f ( y)
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for each and every pair of values x and y. The converse is also true.

If X1, …, Xn
are statistically independent the joint probability density function
can be factored and written as
f ( x1, x2 , , xn )  f1 ( x1 )  f 2 ( x2 ) 

 f n ( xn )
If X and Y are independent random variables, then the conditional probability
density function of X given that Y=y is
f ( x | y) 
f ( x, y ) f ( x ) f ( y )

 f ( x)
f ( y)
f ( y)
for each and every pair of values x and y. The converse is also true.
The Expected Value of a Function of Several Random Variables:
Covariance and Correlation

If X and Y are random variables, then their covariance is
cov( X , Y )  E[( X  E[ X ])(Y  E[Y ])]

If X and Y are discrete random variables, f(x,y) is their joint probability density
function, and g(X,Y) is a function of them, then
E[ g ( X , Y )]   g ( x, y ) f ( x, y )
x
y
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
If X and Y are discrete random variables and f(x,y) is their joint probability
density function, then
cov( X , Y )  E[( X  E[ X ])(Y  E[Y ])]
 [ x  E ( X )][ y  E (Y )] f ( x, y )
x

y
If X and Y are random variables then their correlation is
=
cov( X , Y )
var( X ) var(Y )
The Mean of a Weighted Sum of Random Variables
E[aX  bY ]  aE ( X )  bE (Y )

If X and Y are random variables, then


E  X  Y   E  X   E Y 
The Variance of a Weighted Sum of Random Variables

If X, Y, and Z are random variables and a, b, and c are constants, then
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var  aX  bY  cZ   a 2 var  X   b 2 var Y   c 2 var  Z 
 2ab cov  X , Y   2ac cov  X , Z   2bc cov Y , Z 

If X, Y, and Z are independent, or uncorrelated, random variables, then the
covariance terms are zero and:
var  aX  bY  cZ   a 2 var  X   b 2 var Y   c 2 var  Z 

If X, Y, and Z are independent, or uncorrelated, random variables, and if a = b =
c = 1, then
var  X  Y  Z   var  X   var Y   var  Z 
Some Distributions:
* The Normal Distribution
 ( x  )2 
f ( x) 
exp 
,

2
2
2
 2

1
Z
X 
~ N (0,1)

14
  x 

 2 ) and a is a constant, then
If X ~ N(,
a 
 X  a 

P[ X  a ]  P 


P
Z


 
 
 

 2 ) and a and b are constants, then
If X ~ N(,
b 
a  X  b 
a 
P[a  X  b]  P 



P

Z

 

 
 
 

If
X 1 ~ N  1 , 12  , X 2 ~ N  2 , 22  , X 3 ~ N  3 , 32 
c1 , c2 , c3
are constants, then
Z  c1 X1  c2 X 2  c3 X 3 ~ N  E  Z  , var  Z 
The Chi–square random variable
If Z1, Z 2 ,, Z m denote m independent N (0,1) random variable, then
V  Z 12  Z 22    Zm2 ~  2( m)
The notation V ~  2( m) is read as：
the random variable V has a chi-square distribution with m degrees of freedom.


EV   E  2( m)  m


var V   var  2( m)  2m
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and
 m
 V ~ N
The t-Distribution
A “ t ” random variable (no upper case) is formed by dividing a standard normal,
Z ~ N (0,1), random variable by the square root of an independent chi-square random
variable, V ~  2( m) , that has been divided by its degrees of freedom, m.
If Z ~ N (0,1) and V ~  2( m) , and if Z and V are independent, then
t
Z
V
~ t( m )
m
t-distribution’s shape is completely determined by the degrees of freedom parameter,
m, and the distribution is symbolized by t(m).
1. The t-distribution is less “peaked,” and more spread out than the N(0,1).
2. The t-distribution is symmetric, with mean E[t(m)]=0 and variance
var[t(m)]=m/(m2).
3. As the degrees of freedom parameter m   the t(m) distribution approaches the
standard normal N(0,1).
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The F-Distribution
An F random variable is formed by the ratio of two independent chi-square
random variables that have been divided by their degrees of freedom.
If
V1 ~ 2m1 
and
V2 ~ 2m2 
and if
V1
and
V2
are independent,
then
V1
F

V2
m1
~ F m1 .m2 
m2
The F-distribution is said to have
m1
numerator degrees of freedom and
denominator degrees of freedom. The values of
m1
and
m2
m2
determine the
shape of the distribution. The range of the random variable is (0, ) and it has a
long tail to the right.
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