Directory UMM :wiley:Public:college:hill:

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PowerPoint Slides

for

Undergraduate Econometrics

Lawrence C. Marsh

To accompany: Undergraduate Econometrics

by R. Carter Hill, William E. Griffiths and George G. Judge Publisher: John Wiley & Sons, 1997

(2)

The Role of

Econometrics

in Economic Analysis

Chapter 1

Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, 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.

(3)

Using Information:

1. Information from

economic theory.

2. Information from economic data.

The Role of Econometrics

(4)

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

(5)

economic theory

economic data

}

economic decisions

To use information effectively:

*Econometrics* helps us combine

economic theory and economic data .

(6)

Consumption, c, is some function of income, i :

c = f(i)

For applied econometric analysis this consumption function must be specified more precisely.

(7)

qd_{= f( p, p}c_{, p}s_{, i )}

supply, qs, of an individual commodity:

qs_{= f( p, p}c_{, p}f₎

p = own price; pc_{= price of complements;}

ps_{= price of substitutes; i = income}

p = own price; pc_{= price of competitive products;}

ps_{= price of substitutes; p}f_{= price of factor inputs}

demand

supply

(8)

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?

(9)

Answering the How Much? question Need to estimate parameters

that are both:

1. unknown

and

2. unobservable

(10)

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

(11)

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

(12)

c = f(i) + e

Need to define f(i) in some way. To make consumption, c,

a linear function of income, i : f(i) = β₁ + β₂ i

The statistical model then becomes: c = β₁ + β₂ i + e

(13)

• Dependent variable, y, is focus of study

(predict or explain changes in dependent variable). • Explanatory variables, X₂ and X₃, help us explain observed changes in the dependent variable.

y = β₁ + β₂ X₂+ β₃ X₃+ e

The Econometric Model

(14)

Statistical Models

Controlled (experimental) vs.

Uncontrolled (observational)

Uncontrolled experiment (econometrics) explaining consump-tion, y : price, X₂, and income, X₃, vary at the same time. Controlled experiment (“pure” science) explaining mass, y : pressure, X₂, held constant when varying temperature, X₃, and vice versa.

(15)

Econometric

model

• economic model

economic variables and parameters. • statistical model

sampling process with its parameters. • data

observed values of the variables.

(16)

• 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

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Note: the textbook uses the following symbol to mark sections with advanced material

:

“Skippy”

(18)

Some Basic

Probability

Concepts

Chapter 2

(19)

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.

(20)

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.

(21)

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

(22)

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

(23)

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).

(24)

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 x f(x)

one dot 1 1/6

two dots 2 1/6

three dots 3 1/6

four dots 4 1/6

five dots 5 1/6

six dots 6 1/6

(25)

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: x₁, x₂, . . . , x_n, then f(x₁) + f(x₂)+. . .+f(x_n) = 1.

Therefore,

(26)

variable, X, can be represented by height:

0 1 2 3 X

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

f(x)

0.2

0.4

0.1

0.3

(27)

A continuous random variable uses area under a curve rather than the

height, f(x), to represent probability:

f(x)

X $34,000 $55,000

. .

per capita income, X, in the United States

0.1324 0.8676

red area green area

(28)

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 ] = 0

Probability is represented by area. Height alone has no area.

An interval for X is needed to get an area under the curve.

(29)

P [ a < X < b ] =

∫

f(x) dx

b a

The area under a curve is the integral of 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.

(30)

Rule 2: Σ ax_i = a Σ x_i

i = 1 i = 1

Rule 1: Σ x_i = x₁ + x₂ + . . . + x_n

i = 1

Rule 3: Σ (x_i + y_i) = Σ x_i + Σ y_i

i = 1 _{i = 1} _{i = 1}

n n n

Note that summation is a linear operator which means it operates term by term.

(31)

Rule 4: Σ (ax_i + by_i) = a Σ x_i + b Σ y_i

i = 1 _{i = 1} _{i = 1}

n n n

Rules of Summation (continued)

Rule 5: x = Σ x_i =

i = 1

n n

1 x₁ + x₂ + . . . + x_n

The definition of x as given in Rule 5 implies the following important fact:

Σ (x_i − x) = 0

i = 1

(32)

Rule 6: Σ f(x_i) = f(x₁) + f(x₂) + . . . + f(x_n)

i = 1

Notation: Σ f(x_i) = Σ f(x_i)= nΣ f(x_i)

x i i = 1

Rule 7: Σ Σ f(x_i,y_j) = Σ [ f(x_i,y₁) + f(x_i,y₂)+. . .+ f(x_i,y_m)]

i = 1 i = 1

n m

j = 1

The order of summation does not matter :

Σ Σ f(x_i,y_j) = Σ Σ f(x_i,y_j)

i = 1

n m

j = 1 j = 1

m n

i = 1

Rules of Summation (continued)

(33)

The

mean

or arithmetic average of a

random variable is its mathematical

expectation or expected value, EX.

The Mean of a Random Variable

(34)

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.

(35)

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] = x₁f(x₁) + x₂f(x₂) + . . . + x_nf(x_n)

In other words:

(36)

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.

(37)

x =

Σ

n x_i

i = 1

where n is the number of sample observations. Empirical (sample) mean:

E[X] =

Σ

x_i f(x_i)

i = 1

where n is the number of possible values of x_i. Analytical mean:

Notice how the meaning of n changes.

(38)

E X =

Σ

x_if(x_i)

i=1

The expected value of X-squared:

E X =

Σ

x_if(x_i)

i=1

2 ₂

It is important to notice that f(x_i) does not change!

The expected value of X-cubed:

E X =

Σ

x_if(x_i)

i=1

3 ₃

(39)

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

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

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

(40)

E[g(X)]

=

Σ

(

x_i

)

f(x_i)

i = 1

g(X) = g₁(X) + g₂(X)

E[g(X)]

=

Σ [

g₁(x_i) + g₂(x_i)] f(x_i)

i = 1

E[g(X)]

=

Σ

g₁(x_i) f(x_i) +

Σ

g₂(x_i) f(x_i)

i = 1

E[g(X)]

=

E[g₁(X)]+ E[g₂(X)]

(41)

Adding

and

Subtracting

Random Variables

E(X-Y) = E(X) - E(Y)

E(X+Y) = E(X) + E(Y)

(42)

E(X+a) = E(X) + a

Adding

a

constant

to a variable will

add a constant to its expected value:

Multiplying by

constant

will multiply

its expected value by that constant:

E(bX) = b E(X)

(43)

var(X) = average squared deviations around the mean of X.

var(X) = expected value of the squared deviations around the expected value of X.

var(X) = E [(X - EX) ] 2

(44)

var(X) = E [(X - EX) ]

= E [X - 2XEX + (EX) ]

= E(X ) - 2 EX EX + E (EX)

= E(X ) - 2 (EX) + (EX) 2 2 2 = E(X ) - (EX) 2 2

var(X) = E [(X - EX) ] 2

var(X) = E(X ) - (EX) 2 2

(45)

variance

of a discrete

random variable, X:

standard deviation is square root of variance

var ( X ) =

(x

- EX )

f(x

)

i = 1 n

∑

(46)

x_i f(x_i) (x_i - EX) (x_i - EX) f(x_i)

2 .1 2 - 4.3 = -2.3 5.29 (.1) = .529 3 .3 3 - 4.3 = -1.3 1.69 (.3) = .507 4 .1 4 - 4.3 = - .3 .09 (.1) = .009 5 .2 5 - 4.3 = .7 .49 (.2) = .098 6 .3 6 - 4.3 = 1.7 2.89 (.3) = .867

Σ x_i f(x_i) = .2 + .9 + .4 + 1.0 + 1.8 = 4.3

Σ (x_i - EX) f(x_i) = .529 + .507 + .009 + .098 + .867 = 2.01

calculate the variance for a discrete random variable, X:

i = 1

n n

i = 1

(47)

Z = a + cX

var(Z) = var(a + cX)

= E [(a+cX) - E(a+cX)]

= c var(X)

var(a + cX) = c var(X)

(48)

A

joint

probability density function,

f(x,y), provides the probabilities

associated with the joint occurrence

of all of the possible pairs of X and Y.

(49)

college grads in household

.15

.05

.45

.35

joint pdf

f(x,y)

Y = 1 Y = 2

vacation homes

owned

X = 0

X = 1

Survey of College City, NY

f

(0,1)

f

(0,2)

f

(1,1)

f

_(1,2)

(50)

E[g(X,Y)] =

Σ

g(x

,y

) f(x

,y

)

i _j

E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=.75

E(XY) =

Σ

x

y

f(x

,y

)

i _j

Calculating the expected value of

functions of two random variables.

(51)

The

marginal

probability density functions,

f(x) and f(y), for discrete random variables,

can be obtained by summing over the f(x,y)

with respect to the values of Y to obtain f(x)

with respect to the values of X to obtain f(y).

f(x

) =

Σ

f(x

,y

)

f(y

) =

Σ

f(x

,y

)

i j

(52)

.15

.05

.45

.35

marginal

Y = 1 Y = 2

X = 0

X = 1

.60

.40

.50

f

(X = 1)

f

(X = 0)

f

(Y = 1)

f

(Y = 2)

marginal pdf for Y:

marginal pdf for X:

(53)

The

conditional

probability density

functions of X given Y=y , f(x

_|

y),

and of Y given X=x , f(y

_|

x),

are obtained by dividing f(x,y) by f(y)

to get f(x

_|

y) and by f(x) to get f(y

_|

x).

f(x

_|

y) =

f(x,y)

_f(y

_|

_{x) =}

f(x,y)

f(y)

_f(x)

(54)

.15

.05

.45

.35

conditonal

Y = 1 _{Y = 2}

X = 0

X = 1

.60

.40

.50

.25 .75 .875 .125 .90 .10 .70 .30

f(Y=2_|X= 0)=.25

f(Y=1_|X = 0)=.75

f(Y=2_|X = 1)=.875

f(X=0_|Y=2)=.30

f(X=1_|Y=2)=.70

f(X=0_|Y=1)=.90

f(X=1_|Y=1)=.10

f(Y=1_|X = 1)=.125

(55)

X and Y are

independent

random

variables if their joint pdf, f(x,y),

is the product of their respective

marginal pdfs, f(x) and f(y) .

f(x

,y

) = f(x

) f(y

)

for independence this must hold for all pairs of i and j

(56)

.15

.05

.45

.35

not independent

Y = 1 Y = 2

X = 0

X = 1

.60

.40

.50

f

(X = 1)

f

(X = 0)

f

(Y = 1)

f

(Y = 2)

marginal pdf for Y:

marginal pdf for X:

.50x.60=.30 .50x.60=.30

.50x.40=.20 .50x.40=.20

The calculations in the boxes show the numbers

required to have

independence.

(57)

The

covariance

between two random

variables, X and Y, measures the

linear association between them.

cov(X,Y) = E[(X - EX)(Y-EY)]

Note that variance is a special case of covariance.

cov(X,X) = var(X) = E[(X - EX) ]

(58)

cov(X,Y) = E [(X - EX)(Y-EY)]

= E [XY - X EY - Y EX + EX EY]

= E(XY) - 2 EX EY + EX EY

= E(XY) - EX EY

cov(X,Y) = E [(X - EX)(Y-EY)]

cov(X,Y) = E(XY) - EX EY

= E(XY) - EX EY - EY EX + EX EY

(59)

.15

.05

.45

.35

Y = 1 Y = 2

X = 0

X = 1

.60

.40

.50

EX=0(.60)+1(.40)=.40 EY=1(.50)+2(.50)=1.50

E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=.75

EX EY = (.40)(1.50) = .60

cov(X,Y) = E(XY) - EX EY

= .75 - (.40)(1.50) = .75 - .60

= .15

covariance

(60)

The

correlation

between two random

variables X and Y is their covariance

divided by the square roots of their

respective variances.

Correlation is a pure number falling between -1 and 1.

cov(X,Y)

ρ

(X,Y) =

var(X) var(Y)

(61)

.15

.05

.45

.35

Y = 1 Y = 2

X = 0

X = 1

.60

.40

.50

EX=.40 EY=1.50

cov(X,Y) = .15

correlation

EX=0(.60)+1(.40)=2 2 2 .40

var(X) = E(X ) - (EX)

= .40 - (.40) = .24

2 2

EY=1(.50)+2(.50) = .50 + 2.0

= 2.50

2 2 2

var(Y) = E(Y ) - (EY)

= 2.50 - (1.50) = .25

2 2

ρ(X,Y) = cov(X,Y)

var(X) var(Y)

ρ(X,Y) = .61

(62)

Independent random variables

have zero covariance and,

therefore, zero correlation.

The converse is not true.

Zero Covariance & Correlation

(63)

The expected value of the weighted sum

of random variables is the sum of the

expectations of the individual terms.

Since expectation is a linear operator, it can be applied term by term.

E[c

₁

X + c

₂

Y] = c

₁

EX + c

₂

EY

E[c

₁

X

₁

+...+ c

X

] = c

₁

EX

₁

+...+ c

EX

In general, for random variables X₁, . . . , X_n :

(64)

variables is the sum of the variances, each times

the square of the weight, plus twice the covariances of all the random variables times the products of

their weights.

var(c₁X + c₂Y)=c₁2var(X)+c₂2var(Y) + 2c₁c₂cov(X,Y)

var(c₁X − c₂Y) = c2₁var(X)+c₂2var(Y) − 2c₁c₂cov(X,Y)

Weighted sum of random variables:

Weighted difference of random variables:

(65)

The Normal Distribution

Y ~ N(

β

,

σ

2 )

f(y) =

2 π

σ

exp

β

y

f(y)

2 σ

(y -

β

)

(66)

The Standardized Normal

Z ~ N(

0 ,

1 )

f(z) =

2 π

exp

-

₂

z

Z = (y -

β

)/

σ

(67)

P [ Y > a ] = P Y - β > = a - β P Z > a - β

σ σ σ

β

_y

f(y)

(68)

P [ a < Y < b ] = P < <

= P < Z < a - β Y - β

σ b - σβ

a - β

b - β

β

_y

f(y)

Y ~ N(

β

,

σ

)

(69)

Y₁ ~ N(β₁,σ₁2), Y₂ ~ N(β₂,σ₂2), . . . , Y_n ~ N(β_n,σ_n2)

W = c

₁

Y

₁

+ c

₂

Y

₂

+ . . . + c

Y

Linear combinations of jointly

normally distributed random variables

are themselves normally distributed.

W ~ N

[

E(W), var(W)

]

(70)

mean: E[V] = E[

χ

_(m)] = m

If Z₁, Z₂, . . . , Z_m denote m independent N(0,1) random variables, and

V = Z2₁+ Z2₂ + . . . + Z2_m, then V ~

χ

_(m)2

V is chi-square with m degrees of freedom.

Chi-Square

variance: var[V] = var[

χ

_(m)] = 2m

If Z₁, Z₂, . . . , Z_m denote m independent N(0,1) random variables, and

V = Z2₁+ Z2₂ + . . . + Z2_m, then V ~

χ

_(m)2

V is chi-square with m degrees of freedom.

(71)

mean: E[

t

] = E[

t

_(m)] = 0 symmetric about zero variance: var[

t

] = var[

t

_(m)] = m / (m

−

If Z ~ N(0,1) and V ~

χ

_(m)and if Z and V are independent then,

~

t

_(m)

t

is student-t with m degrees of freedom.

t =

V _m

(72)

If V₁ ~

χ

_(m

1) and V2 ~

χ

(m2) and if V1 and V2

are independent, then

~

F

_(m

1,m2)

F

is an F statistic with m₁ numerator degrees of freedom and m₂ denominator degrees of freedom.

F =

V_{1 m}

V₂

m₂

(73)

The Simple Linear

Regression

Model

Chapter 3

(74)

1. Estimate a relationship among economic variables, such as y = f(x).

2. Forecast or predict the value of one variable, y, based on the value of

another variable, x.

Purpose of Regression Analysis

(75)

Weekly Food Expenditures

y = dollars spent each week on food items. x = consumer’s weekly income.

The relationship between x and the expected value of y , given x, might be linear:

E(y|x) = β₁+ β₂x

(76)

f(y|x=480)

µ_y|x=480

Figure 3.1a Probability Distribution f(y|x=480) of Food Expenditures if given income x=$480.

(77)

µ_y|x=480 µ_y|x=800

Figure 3.1b Probability Distribution of Food

Expenditures if given income x=$480 and x=$800.

(78)

{

β₁

∆x

∆E(y|x) E(y|x)

Average

Expenditure

x (income) E(y|x)=β₁+β₂x

β₂= ∆E(y|x)

∆x

Figure 3.2 The Economic Model: a linear relationship between avearage expenditure on food and income.

(79)

.

x_t x₁=480 x₂=800

y t

f(y_t)

Figure 3.3. The probability density function for y_t at two levels of household income, x_t

expenditure

Homoskedastic Case

income

(80)

.

x_t x₁ x₂

y t

f(y_t)

Figure 3.3+. The variance of y_t increases as household income, x_t, increases.

expenditure

Heteroskedastic Case

x₃

.

income

(81)

Assumptions of the Simple Linear

Regression Model - I

1. The average value of y, given x, is given by the linear regression:

E(y) = β₁+ β₂x

2. For each value of x, the values of y are

distributed around their mean with variance:

var(y) = σ2

3. The values of y are uncorrelated, having zero covariance and thus no linear relationship:

cov(y_i,y_j) = 0

4. The variable x must take at least two different values, so that x ≠ c, where c is a constant.

(82)

5. (optional) The values of y are normally distributed about their mean for each

value of x:

y ~ N [(β₁+β₂x), σ2 _]

One more assumption that is often used in

practice but is not required for least squares:

(83)

The Error Term

y is a random variable composed of two parts:

I. Systematic component: E(y) = β₁+ β₂x

This is the mean of y.

II. Random component: e = y - E(y) = y - β₁- β₂x

This is called the random error.

Together E(y) and e form the model:

y = β₁+ β₂x + e

(84)

Figure 3.5 The relationship among y, e and the true regression line.

.

y₄ y₁ y₂ y₃

x₁ x₂ _x₃ x₄

}

{

e₁ e₂ e₃

e₄ _{E(y) =}_β

1 + β2x

(85)

}

.

}

.

y₄ y₁

y₂ _y

x₁ x₂ _x₃ x₄

{

e₁ e₂ e₃ e₄ x y

Figure 3.7a The relationship among y, e and the fitted regression line.

y = b₁+ b₂x ^

.

y₁ y₂

y₃ y4

^ ^ ^ ^ ^ ^ ^ ^ 3.13

(86)

{

.

y₄ y₁

y₂ _y 3

x₁ x₂ _x₃ x₄ _x

Figure 3.7b The sum of squared residuals from any other line will be larger.

y = b₁+ b₂x ^

.

y^₁

y₃ ^

y₄

^ y = b^* *1 + b*2x

e₁ ^*

e₂ ^*

y^*₂

e₃ ^*

* ^e4 *

{

(87)

.

) _f(e) _f(y)

Figure 3.4 Probability density function for e and y

0 _β

1+β2x

(88)

The Error Term Assumptions

1. The value of y, for each value of x, is

y = β₁+ β₂x + e

2. The average value of the random error e is:

E(e) = 0

3. The variance of the random error e is:

var(e) = σ2_{= var(y)}

4. The covariance between any pair of e’s is:

cov(e_i,e_j) = cov(y_i,y_j) = 0

5. x must take at least two different values so that

x ≠ c, where c is a constant.

6. e is normally distributed with mean 0, var(e)=σ2 (optional) e ~ N(0,σ2₎

(89)

Unobservable Nature

of the Error Term

1. Unspecified factors / explanatory variables, not in the model, may be in the error term. 2. Approximation error is in the error term if

relationship between y and x is not exactly a perfectly linear relationship.

3. Strictly unpredictable random behavior that may be unique to that observation is in error.

(90)

Population

regression values:

y

=

β

₁

+

β

₂

x

+ e

Population

regression line:

E(y

|x

) =

β

₁

+

β

₂

x

Sample

regression values:

y

= b

₁

+ b

₂

x

+ e

Sample

regression line:

y

= b

₁

+ b

₂

x

(91)

y

=

β

₁

+

β

₂

x

+ e

Minimize error sum of squared deviations:

β

₁

,

β

₂) =

Σ

(

y

-

β

₁

-

β

₂

x

_t )

2_(3.3.4)

t=1 T

e

= y

-

β

₁

-

β

₂

x

(92)

Minimize w.

r.

t.

β

₁

and

β

₂

:

β

₁

,

β

₂) =

Σ

(

y

-

β

₁

-

β

₂

x

_t )

2_(3.3.4)

t =1 T

= - 2

Σ

(

y

-

β

₁

-

β

₂

x

_t )

= - 2

Σ

x

_t(

y

-

β

₁

-

β

₂

x

_t )

∂S(

.

)

∂β

₁

∂S(

.

)

∂β

₂

Set each of these two derivatives equal to zero and

solve these two equations for the two unknowns:

β

₁

β

₂

(93)

S(.)

β_i

b_i

.

Minimize w.

r.

t.

β

₁

and

β

₂

:

.

) =

Σ

(

y

-

β

₁

-

β

₂

x

_t )

t =1 T

∂S(.)

∂β_i < 0

∂S(.)

∂β_i > 0 ∂S(.)

∂β_i = 0

(94)

To minimize S(.), you set the two derivatives equal to zero to get:

= - 2

Σ

(

y

-

b

₁

-

b

₂

x

_t ) = 0

= - 2

Σ

x

_t(

y

-

b

₁

-

b

₂

x

_t ) = 0

∂S(

.

)

∂β

₁

∂S(

.

)

∂β

₂

When these two terms are set to zero,

β

₁ and

β

₂ become

b

₁ and

b

₂ because they no longer represent just any value of

β

₁ and

β

₂ but the special values that correspond to the minimum of S(

.

) .

(95)

- 2

Σ

(

y

-

b

₁

-

b

₂

x

_t ) = 0

- 2

Σ

x

_t(

y

-

b

₁

-

b

₂

x

_t ) = 0

Σ

y

_t -

T

b

₁ -

b

₂

Σ

x

_t = 0

Σ

x

y

_t -

b

₁

Σ

x

_t-

b

₂

Σ

x

_t2 = 0

T

b

₁ +

b

₂

Σ

x

_t =

Σ

y

b

₁

Σ

x

_t+

b

₂

Σ

x

2_t =

Σ

x

y

(96)

Solve for

b

₁ and

b

₂ using definitions of

x

and

y

T

b

₁ +

b

₂

Σ

x

_t =

Σ

y

b

₁

Σ

x

_t+

b

₂

Σ

x

2_t =

Σ

x

y

T

Σ

x

y

_t-

Σ

x

Σ

y

T

Σ

x

_t2

- (

Σ

x

_t)2

b

₂=

b

₁=

y

b

₂

x

(97)

elasticities

percentage change in y percentage change in x

η = ₌

∆x/x

∆y/y

= _∆∆y x

x y

Using calculus, we can get the elasticity at a point:

η = lim ∆y x ₌

∆x y

∂y x

∂x y

∆x→0

(98)

E(y) =

β

₁

+

β

₂

x

∂

E(y)

∂

x

=

β

applying elasticities

∂

E(y)

∂

x

=

β

η

=

E(y)

x

E(y)

x

(99)

estimating elasticities

∂

y

∂

x

=

b

η

=

y

x

y

x

y

^_t

=

b

₁

+ b

₂

x

= 4 + 1.5 x

x

= 8 = average number of years of experience

y

= $10 = average wage rate

= 1.5 = 1.2

8

10 =

b

₂

η

y

x

(100)

Prediction

y

^_t

= 4 + 1.5 x

Estimated regression equation:

x

_t= years of experience

y

^_t = predicted wage rate

x

_t= 2 years, then

y

^ _t =

$7.00

per hour

.

x

_t= 3 years, then

y

^_t =

$8.50

per hour

.

(1)

Data from Surveys

i)

identify the population of interest.

ii)

designing and selecting the sample.

iii)

collecting the information.

iv)

data reduction, estimation and inference.

The survey process has four distinct aspects

:

(2)

Controlled Experiments

1. Labor force participation: negative income tax:

guaranteed minimum income experiment. 2. National cash housing allowance experiment:

impact on demand and supply of housing. 3. Health insurance: medical cost reduction:

sensitivity of income groups to price change. 4. Peak-load pricing and electricity use:

daily use pattern of residential customers.

Controlled experiments were done on these topics:

(3)

Economic Data Problems

I. poor implicit experimental design

(i) collinear explanatory variables.

(ii) measurement errors.

II. inconsistent with theory specification

(i) wrong level of aggregation.

(ii) missing observations or variables.

(iii) unobserved heterogeneity.

(4)

Selecting a Topic

• “What am I interested in?”

• Well-defined, relatively simple topic. • Ask prof for ideas and references.

• Journal of Economic Literature (ECONLIT) • Make sure appropriate data are available.

• Avoid extremely difficult econometrics. • Plan your work and work your plan.

General tips for selecting a research topic

:

ð ð ð ð ð ð ð

(5)

Writing an Abstract

(i) concise statement of the problem.

(ii) key references to available information. (iii) description of research design including: