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PowerPoint Slides for

Undergraduate Econometrics

Lawrence C. Marsh

To accompany: Undergraduate Econometrics

by Carter Hill, William Griffiths and George Judge

Publisher: John Wiley & Sons, 1997

1

Copyright 1996 Lawrence C. Marsh

The Role of

Econometrics

in Economic Analysis

Chapter 1

that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the

John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution

or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these

programs or from the use of the information contained herein.

1.1

Using Information:

1. Information from

economic theory.

2. Information from

economic data.

The Role of Econometrics

1.2

Copyright 1996 Lawrence C. Marsh

Understanding Economic Relationships:

federal

budget

Dow-Jones

Stock Index

trade

deficit

Federal Reserve

Discount Rate

capital gains tax

rent

control

laws

short term

treasury bills

power of

labor unions

crime rate

inflation

unemployment

money supply

1.3

Copyright 1996 Lawrence C. Marsh

economic theory

economic data

}

economic

decisions

To use information effectively

*Econometrics* helps us combine

economic theory

and

economic data

Economic Decisions

1.4

Consumption

, c, is some function of

income

, i :

c = f(i)

For applied econometric analysis

this consumption function must be

specified more precisely.

The Consumption Function

1.5

Copyright 1996 Lawrence C. Marsh

demand

q

d

, for an individual commodity:

q

d

= f( p, p

c

, p

, i )

supply

, of an individual commodity:

q

s

= f( p, p

, p

)

p = own price; p

= price of complements;

p

s

= price of substitutes; i = income

p = own price; p

c

= price of competitive products;

p

s

= price of substitutes; p

= price of factor inputs

demand

supply

1.6

2

Copyright 1996 Lawrence C. Marsh

Listing the variables in an economic relationship is not enough.

For effective policy we must know the

amount of change

needed for a policy instrument to bring about the desired

effect:

How much ?

• By

how much

should the Federal Reserve

raise interest rates to prevent inflation?

• By

how much

can the price of football tickets

be increased and still fill the stadium?

1.7

Answering the

How Much?

question

Need to estimate parameters

that are both:

1. unknown

and

2. unobservable

1.8

Copyright 1996 Lawrence C. Marsh

Average or systematic behavior

over many individuals or many firms.

Not

a single individual or single firm.

Economists are concerned with the

unemployment rate and not whether

a particular individual gets a job.

The Statistical Model

1.9

Copyright 1996 Lawrence C. Marsh

The Statistical Model

Actual vs. Predicted Consumption:

Actual = systematic part + random error

Systematic part provides prediction, f(i),

but actual will miss by random error, e.

Consumption, c, is function, f, of income, i, with error, e:

c = f(i) + e

1.10

Copyright 1996 Lawrence C. Marsh

c = f(i) + e

Need to define f(i) in some way.

To make consumption, c,

linear function

of income, i :

f(i) =

The statistical model then becomes:

c =

i + e

The Consumption Function

1.11

Copyright 1996 Lawrence C. Marsh

• Dependent variable , y, is focus of study

(predict or explain changes in dependent variable).

• Explanatory variables, X

and X

, help us explain

observed changes in the dependent variable.

y =

+ e

The Econometric Model

1.12

3

Copyright 1996 Lawrence C. Marsh

Statistical Models

Controlled (experimental)

vs.

Uncontrolled (observational)

Uncontrolled experiment (econometrics) explaining consump-

tion, y : price,

2

, and income,

, vary at the same time.

Controlled experiment (“pure” science) explaining mass, y :

pressure,

2

, held constant when varying temperature,

and vice versa.

1.13

Econometric

model

• economic model

economic variables and parameters.

• statistical model

sampling process with its parameters.

• data

observed values of the variables.

1.14

• Uncertainty regarding an outcome.

• Relationships suggested by economic theory.

• Assumptions and hypotheses to be specified.

• Sampling process including functional form.

• Obtaining data for the analysis.

• Estimation rule with good statistical properties.

• Fit and test model using software package.

• Analyze and evaluate implications of the results.

• Problems suggest approaches for further research.

The Practice of Econometrics

1.15

Copyright 1996 Lawrence C. Marsh

Note: the textbook uses the following symbol

to mark sections with advanced material

“Skippy”

1.16

Copyright 1996 Lawrence C. Marsh

Some Basic

Probability

Concepts

Chapter 2

that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the

John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution

or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these

programs or from the use of the information contained herein.

2.1

Copyright 1996 Lawrence C. Marsh

random variable

A variable whose value is unknown until it is observed.

The value of a random variable results from an experiment.

The term random variable implies the existence of some

known or unknown probability distribution defined over

the set of all possible values of that variable.

In contrast, an arbitrary variable does not have a

probability distribution associated with its values.

Random Variable

2.2

4

Copyright 1996 Lawrence C. Marsh

Controlled experiment

values

of explanatory variables are chosen

with great care in accordance with

an appropriate

experimental design

Uncontrolled experiment

values

of explanatory variables consist of

nonexperimental observations over

which the analyst has

no control

2.3

Copyright 1996 Lawrence C. Marsh

discrete random variable

A discrete random variable can take only a finite

number of values, that can be counted by using

the positive integers.

Example: Prize money from the following

lottery is a discrete random variable:

first prize: $1,000

second prize: $50

third prize: $5.75

since it has only four (a finite number)

(count: 1,2,3,4) of possible outcomes:

$0.00; $5.75; $50.00; $1,000.00

Discrete Random Variable

2.4

Copyright 1996 Lawrence C. Marsh

continuous random variable

A continuous random variable can take

any real value (not just whole numbers)

in at least one interval on the real line.

Examples:

Gross national product (GNP)

money supply

interest rates

price of eggs

household income

expenditure on clothing

Continuous Random Variable

2.5

Copyright 1996 Lawrence C. Marsh

A discrete random variable that is restricted

to two possible values (usually 0 and 1) is

called a

dummy variable

(also, binary or

indicator variable).

Dummy variables account for qualitative differences:

gender (0=male, 1=female),

race (0=white, 1=nonwhite),

citizenship (0=U.S., 1=not U.S.),

income class (0=poor, 1=rich).

Dummy Variable

2.6

A list of all of the possible values taken

by a discrete random variable along with

their chances of occurring is called a probability

function or probability density function (pdf).

die

f(x)

one dot

1/6

two dots

1/6

three dots 3

1/6

four dots

1/6

five dots

1/6

six dots

1/6

2.7

Copyright 1996 Lawrence C. Marsh

A discrete random variable X

has pdf, f(x), which is the

probability

that X takes on the value x.

f(x) = P(X=x)

0 < f(x) < 1

If X takes on the n values: x

, x

, . . . , x

,

then f(x

) + f(x

)+. . .+f(x

) = 1.

Therefore,

2.8

5

Copyright 1996 Lawrence C. Marsh

Probability, f(x), for a discrete random

variable, X, can be represented by

height

:

number, X, on Dean’s List of three roommates

f(x)

0.2

0.4

0.1

0.3

2.9

Copyright 1996 Lawrence C. Marsh

A continuous random variable uses

area

under a curve rather than the

height, f(x), to represent probability:

f(x)

$34,000

$55,000

.

.

per capita income, X, in the United States

0.1324

0.8676

red area

green area

2.10

Since a continuous random variable has an

uncountably infinite

number of values,

the probability of one occurring is

zero

P [ X = a ] = P [ a < X < a ] =

Probability is represented by

area

Height alone has no

area

An interval for X is needed to get

area

under the curve.

2.11

Copyright 1996 Lawrence C. Marsh

P [ a < X < b ] =

∫

f(x) dx

The

area

under a curve is the

integral

the equation that generates the curve:

For continuous random variables it is the

integral of f(x)

, and not f(x) itself, which

defines the area and, therefore, the

probability

2.12

Copyright 1996 Lawrence C. Marsh

n

Rule 2:

= a

i = 1

i = 1

n

Rule 1:

= x

+ x

+ . . . + x

i = 1

Rule 3:

(

x

)

i = 1

n

n

n

Note that summation is a linear operator

which means it operates term by term.

Rules of Summation

2.13

Copyright 1996 Lawrence C. Marsh

Rule 4:

(

ax

)

= a

+ b

i = 1

n

n

n

Rules of Summation (continued)

Rule 5: x =

i = 1

n

n

1

+ x

+ . . . + x

n

n

The definition of x as given in Rule 5 implies

the following important fact:

(

−

x) = 0

i = 1

n

2.14

6

Copyright 1996 Lawrence C. Marsh

Rule 6:

f(x

) = f(x

) + f(x

) + . . . + f(x

)

i = 1

Notation:

f(x

) =

f(x

)

f(x

)

n

x

i = 1

Rule 7:

f(x

) =

[ f(x

) + f(x

)+. . .+ f(x

,y

m

)]

i = 1

i = 1

n m

j = 1

The order of summation does not matter :

f(x

) =

f(x

)

i = 1

n m

j = 1

j = 1

m n

i = 1

Rules of Summation (continued)

2.15

The

mean

or arithmetic average of a

random variable is its mathematical

expectation or expected value, EX.

The Mean of a Random Variable

2.16

Expected Value

There are two entirely different, but mathematically

equivalent, ways of determining the expected value:

1. Empirically:

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.

In other words, draw an infinite number of samples,

and average the values of X that you get.

2.17

Copyright 1996 Lawrence C. Marsh

Expected Value

2. Analytically:

The

expected value

of a discrete random

variable, X, is determined by weighting all

the possible values of X by the corresponding

probability density function values, f(x), and

summing them up.

E[X] = x

f(x

) + x

f(x

) + . . . + x

f(x

)

In other words:

2.18

Copyright 1996 Lawrence C. Marsh

In the empirical case when the

sample goes to infinity the values

of X occur with a frequency

equal to the corresponding f(x)

in the analytical expression.

As sample size goes to infinity, the

empirical and analytical methods

will produce the same value.

Empirical vs. Analytical

2.19

Copyright 1996 Lawrence C. Marsh

x =

i

n

i = 1

where n is the number of sample observations.

Empirical

(sample) mean:

E[X] =

f(x

)

i = 1

n

where n is the number of possible values of x

.

Analytical

mean:

Notice how the meaning of n changes.

2.20

7

Copyright 1996 Lawrence C. Marsh

E X =

x

i

f(x

i

)

i=1

n

The expected value of

X-squared

:

E X =

x

i

f(x

i

)

i=1

n

2

2

It is important to notice that

f(x

i

) does not change!

The expected value of

X-cubed

:

E X =

x

i

f(x

i

)

i=1

n

3

3

The expected value of

X

:

2.21

Copyright 1996 Lawrence C. Marsh

EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) + 4 (.1)

2

EX

= 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) + 4 (.1)

2

2

2

2

2

= 1.9

= 0 + .3 + 1.2 + 1.8 + 1.6

= 4.9

3

EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) +4 (.1)

3

3

3

3

3

= 0 + .3 + 2.4 + 5.4 + 6.4

= 14.5

2.22

[g(X)]