# Econ2206 Notes

**Topics:**Normal distribution, Estimator, Scientific method

**Pages:**292 (9594 words)

**Published:**August 21, 2013

Revision

Notes

W2

–

SIMPLE

REGRESSION

MODEL

MOTIVATION

Much

of

applied

econometric

analysis

are

interested

in

“explaining

y

in

terms

of

x”

and

confront

three

issues:

1) Since

there

is

never

an

exact

relationship

between

y

and

x,

how

do

we

account

for

the

“other

unobserved”

variables?

2) What

is

the

function

relationship

between

y

and

x?

3) How

do

we

invoke

a

ceteris

paribus

relationship

or

a

causal

effect

between

y

an

x?

The

simple

linear

regression

model

is:

y

=

β0

+

β1x

+

u

-‐ “u”

is

the

stochastic

error

or

disturbance

term

and

represents

all

those

unobserved

factors

or

other

factors

other

than

x.

-‐ If

all

other

factors

(u)

is

held

fixed,

so

that

change

in

u

is

zero,

we

can

observe

the

function

relationship

between

y

and

x.

-‐ If

we

take

the

expected

value

of

the

model,

(Δu

=

0

and

Δβ0),

then

we

can

see

that

x

has

a

linear

effect

on

y.

We

will

only

get

reliable

estimates

of

β0

and

β1

if

we

make

restricting

assumptions

on

u.

As

long

as

β0

is

included

in

the

model,

nothing

is

lost

by

making

the

assumption

that

the

expected

value

of

u

in

the

population

is

zero;

E(u)

=

0.

ZCM

Our

crucial

assumption

is

by

defining

the

conditional

distribution

of

u

given

any

value

of

x.

This

crucial

assumption

is,

the

average

value

of

u

does

not

depend

on

the

value

of

x.

E(u|x)

=

E(u)

=

0

This

is

the

zero-conditional

mean

assumption

(ZCM)

-‐ The

average

value

of

the

unobserved

factors

is

the

same

across

the

population.

-‐ An

important

implication

of

ZCM

is

that

u

and

x

are

uncorrelated.

OLS

Ordinary

Least

Squares

(OLS)

is

a

method

for

estimating

the

unknown

parameters

in

a

linear

regression

model.

The

estimates

for

β0

and

β1

are

found

by

minimizing

the

sum

of

squared

residuals.

That

is,

the

distance

between

the

observations

in

the

sample

and

the

responses

predicted.

-‐ Fitted

values

and

estimates

are

denoted

by

a

HAT

-‐ The

values

predicted

for

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