econometricsnotes2-140407141735-phpapp01.pdf

8/18/2019 econometricsnotes2-140407141735-phpapp01.pdf

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Muhammad Ali Econometrics

Lecturer in Statistics GPGC Mardan. BS Economics

M.sc (Peshawar University)

Mhil(A!"U !slama#ad)

$

Introduction

Definition of Econometrics

literally interreted econometrics means %economic measurement%. Econometrics may #e de&ined as

the social science in which the tools o& economic theory' mathematics' and statistical in&erence are

alied to the analysis o& economic henomena(varia#le).Econometrics can also #e de&ined as

%statistical o#servation o& theoretically &ormed concets " alternatively as mathematical economics

worin* with measured data. so economic theory attemts to de&ined the relationshi amon* di&&erent

economic varia#les.

Methodology of Econometrics

+ollowin* are the main stes in methodolo*y o& econometrics

$.

Seci&y mathematical e,uation to descri#e the relationshi #etween economic varia#les.

-.

esi*n methods and rocedures #ased on statistical theory to o#tain reresentative samle

&rom the real world.

/.

eveloment o& methods &or estimatin* the arameters o& the seci&ied relationshis

0.

eveloment o& methods o& main* economic &orecast &or olicy imlications #ased on

estimated arameters.


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-

What are the goals of econometrics

Econometrics hel us to achieves the &ollowin* three *oals1

$.

2ud*e the validity o& the economic theories.

-.

Suly the numerical estimates o& the co3e&&icient o& the economic relationshis which may #e

then used &or some sound economic olicies.

/.

+orecast the &uture values o& the economic ma*nitude with certain de*ree o& ro#a#ility.

The Nature of Econometrics Approach

4he &irst ste o& every econometrics research is the seci&ication o& the model' a model is simly a set o&

mathematical e,uations. !& the model has only one e,uation it is called a sin*le3e,uation model.

5hereas i& it has more than one e,uation it is called a multi3e,uation model. 6ow let us consider the

&ollowin* model.

Y=β0 + β1X

where

Y= Consumption expenditure

β0 = Intercept

β1= Slope or co-efficient of regression

X= income

This is a deterministic model showing the relationship between consumption and income.

The non-deterministic or stochastic models can be written as:

Yi=β0 + β1Xi+ iε


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/

where " iε

" is known as the disturbance or residual term, and hence it is a random variable. It is also

called probabilistic model. The disturbance or error term represent all those factors that affect

consumption but not taken into account. This equation is an example of an "Econometric Model".

In other words it is an example of a "linear regression Model". In this case the response variable 'Y' is

linearly related to the predictor variable 'X' but the relationship between the two is not exact. The

second step is the estimation of model by appropriate econometric method, this will include the

following steps.

1. Collection of Data for the variables included in the model.

2. Choice of appropriate econometric technique for the estimation of technique used is Regression

Analysis in Statistics).

3. The third step is to develop the suitable criteria to find out whether estimates obtained are

in agreement with the expectations of the theory that has been tested, that is to decide whether

the estimates of the parameters of the theoretically meaning full and statistically significant.

0.

4he &inal ste is to use the estimated model to redict the &uture value o& the resonse varia#le.

Deterministic and Stochastic Models

A relation #etween 7 and 8 is said to #e deterministic i& &or each value o& redictor varia#le 7'

there is one and only one corresondin* values o& resonse varia#le 8. "n the other hand the

relation #etween 7 and 8 is said to #e stochastic or ro#a#ilistic i& &or a redictor value o& 7 0

there is a whole ro#a#ility distri#ution o& values o& 8' that is 989 is a random varia#le and 979 is a

&i:ed mathematical varia#le measured without error.


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0

Types of Econometrics

Econometrics can be divided into two main braches

$.

4heoretical

-.

Alied

1.

Theoretical econometrics:

!t is concerned with develoment o& the aroriate methods &or measurin* the economic relationshi

seci&ied #y the econometrics models. 4his tye o& econometrics deends much on mathematical

statistics' sin*le e,uation and simultaneous e,uations techni,ues' the methods used &or measurin*

economic relationshis.

eoussimulabxbxaY

eousSimulbxbxaY

simplebxaY

tan

tan

21

20

→++=

→++=

→+=

2. Applied econometrics:

Alied econometrics escri#e the ractical value o& economic research. !t deals with the alications o&

econometric methods develoed in the theoretical econometrics to the di&&erent &ields o& economics

such as the consumtion &unctions' demand and suly' &raction etc. 4he alied econometrics has

made it ossi#le to o#tained numerical results &rom these studies which are o& *reat imortance to the

lanners.


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=

Regression Analysis

Definition of egression

e*ression analysis is statistical techni,ue &or investi*atin* and modelin* the relationshi

#etween varia#les. 4he term re*ression was &irst time introduced #y %+rancis Galton%. !n his

aer Galton &ound that the hei*ht o& the children o& unusually tall or unusually short arents

tends to move towards the avera*e hei*ht o& the oulation. Galton law o& universal re*ression

was con&irmed #y his &riend >arl Pearson' more than a thousand records o& hei*hts o& mem#ers

o& &amily *rous. ?e &ound that the avera*e hei*ht o& sons o& a *rou o& tall &athers was less

than their &ather hei*ht and the avera*e hei*ht o& sons o& a *rou o& tall &athers was less than

their &athers hei*ht and the avera*e hei*ht o& sons o& a *rou o& short &athers was *reater

than their &athers hei*ht' thus %re*ressin* tall and short sons alie toward the avera*e hei*ht o&

all men. !n the world o& Galton this was %re*ression to mediocrity%.

Modern !nterpretation of egression

Regression analysis is the study of the dependence of one variable the dependent variable, on

one or more other variables, the explanatory variable.

"#$ecti%e of egression

4he o#


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The Simple &inear egression Model

Simle linear re*ression is the most commonly used techni,ue &or determinin* how one

varia#le o& interest(the resonse varia#le)is a&&ected #y chan*es in another varia#le(the

e:lanatory varia#le)4he terms %resonse% and %e:lanatory% mean the same thin* as

%deendent% and %indeendent%' #ut the &ormer terminolo*y is re&erred #ecause the

%indeendent% varia#le may actually #e interdeendent with many other varia#les as well.

Simle linear re*ression is used &or three main uroses1

$. 4o descri#e the linear deendence o& one varia#le on another.

-. 4o redict values o& one varia#le &rom values o& another' &or which more data are availa#le.

/. 4o correct &or the linear deendence o& one varia#le on another' in order to clari&y other

&eatures o& its varia#ility. Linear re*ression determines the #est3&it line throu*h a scatter lot o&

data' such that the sum o& s,uared residuals is minimied e,uivalently' it minimies the error

variance. 4he &it is %#est% in recisely that sense1 the sum o& s,uared errors is as small as

ossi#le. 4hat is why it is also termed %"rdinary Least S,uares% re*ression. Model o& the simle

linear re*ression is *iven #y1

8iD F D$7iF iε

5here

D !ntercet

D$ Sloe or co3e&&icient o& re*ression

iε random error


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An imortant o#


9/23

Muhammad Ali

Lecturer in Statistics GPGC Mard



6ow su#stitutin* th

^

^

∑

∑

∑

β

β

!t is imortant to now that a

samle rather than the entire

&or and . Let9s call and

econometrics is to analye the ,

estimators and under which co

more varia#les. 4he

4he second is the estim

an.

I

X Y ^^

β α −=

e value o& HJ in e,uation ( ii ) we *et1

( )

( ) n X X

n X Y Y X

X X X

X Y Y X

X X X X Y X

X X X X Y X

X X X Y X

ii

iiii

ii

iii

iiiii

iiiii

iiii

/

/

)(

22

2

2^

2^^

2^^

∑−∑

∑∑−∑

∑−∑

∑−∑

∑−∑=∑−

∑+∑−∑=

∑+∑−=

β

β β

β β

nd are not the same as and #ecause th

ulation. !& you too a di&&erent samle' you wo

the "LS estimators o& and . "ne

uality o& these estimators and see under what c

nditions they are not. "nce we have and

&irst is the &itted values' o

tes o& the error terms' which we

Econometrics

BS Economics

y are #ased on a sin*le

uld *et di&&erent values

& the main *oals o&

nditions these are *ood

' we can construct two

r estimates o& y 1

ill call the residuals1


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$

Assumptions of the *lassical linear egression Model:

+ollowin* are the &ew imortant assumtions o& the classical linear re*ression model1

$. &inearity: 4he re*ression model is linear in the arameters. i.e.

8i D FD$7i F ui

-. Non stochastic +1 Kalues o& the indeendent or reressor varia#le assumed to #e &i:ed in

the reeated samlin*.

/. ,ero mean of the error term1 4he e:ected value or mean o& the random distur#ance

term *iven the value o& 7 is ero. i.e.

E ( Ui7i)

0.

-omoscedasticity1 Kariance o& the ui &or all the o#servations remain the same.i.e.

K ( u$) - K(u-)

- K(u/) - N K(un)

-


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Muhammad Ali




;.

No Autocorrelati

error terms' sym#olicall

/.

No relationship #

7. The num#er of o

parameters to #e

greater than the number

sample must not all be t

an.

$$

n #eteen error term1 4here is no corr

1

COV( iε , jε )=0

teen predictor %aria#le and error

E(ui,Xi)=0

#ser%ations 0n0 must #e greater th

estimated. Alternatively, the number of

of explanatory variables. The values of predicto

e same. Technically Var(X) must be a finite posi

Econometrics

BS Economics

lation #etween any two

term.

n the num#er of

bservations 'n' must be

r variable (X) in a given

tive number.


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Muhammad Ali




8.

The regression m

bias or error in the mod

, there are no perfect lin

!nterpretation of *oe

4he interretation o& the coe

estimated intercet (#) tells us

#ecause many o& our varia#les

income' which rarely i& ever ha

relationshi #etween : and y. !t

roperties of &east S(

$.

4he least s,uare estima

we have

an.

$-

del is correctly specified. Alternatively,

l used in empirical analysis. There is no perfect

ar relationships among the explanatory variables

cients3arameters

cients &rom a linear re*ression model is &ai

the value o& y that is e:ected when : . 4his

onOt have true values (or at least not relevan

ve values). 4he sloe (#$) is more imortan

is interreted as the e:ected chan*e in y &or a

are Estimators:

ors are linear &unction o& the actual o#servation

∑=

−

n

i

X Xi1

)( ( )Y Y − ∑=

−

n

i

X Xi1

2)(

∑=

n

i 1

8i (7i333 ) Xi 33 ∑ − )( X XiYi ∑=

n

i 1

(

∑=

n

i 1

8i (7i333 ) Xi ∑=

−

n

i

X Xi1

2)( as

∑ /∑-

∑

Econometrics

BS Economics

there is no specification

ulticollinearity. That is

.

ly strai*ht&orward. 4he

is o&ten not very use&ul'

t ones3lie education or

' #ecause it tells us the

ne3unit chan*e in :.

on 8.

− X i 2)

∑=

n

i 1

(7i333 ) Xi


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Muhammad Ali




5here

Similarl

?ence #oth

-.

4he least s,uare estima

we have

6ow ∑ ∑= = =

=

n

i

n

i

n

i

xiwi1 1 1

/

∑=

n

i

iw1

2

∑

=

n

i

ii xw1

an.

$/

5i

:i ∑

-

we have

x y β α ˆˆ −=

∑ / 33( ∑ )

[ ]∑=

−

n

i

iw X n1

/ 1 8i

nd ' are e:ressed as linear &unction o& the 89

ors and ' are un#iased estimators o& H and

∑=

=

n

i

iiY W 1

^

β

∑=

++

n

i

i Xiwi1

)( ε β α

∑ ∑ ∑= = =

++

n

i

n

i

n

i

iiiii w xww1 1 1

ε β α 3333333

xi1

- ∑ ∑= =

=−=

n

i

n

i

ii x x x1 1

20 / )( 33333333333( A )

∑ ∑ ∑ ∑∑= = ==

==

=

n

i

n

i

n

i

ii

n

i

i x x x xi1 1 1

222

2

1

2 / 1) /( /

( )∑=

=∑

∑=

∑=

n

i i

i

i

i

i

x

x x

x

x

12

2

21

Econometrics

BS Economics

s.

D.

33333( ! )

i x2

3333333333( B )


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Muhammad Ali




Su#stitutin* these values in e,u

^

β (= α

4ain* e:ectation on #

^ β E

4hen is an un#iased

6ow

By usin* e,uation A ' B '

4ain* e:ectation on #

an.

$0

tion ( ! )

)()0 x xw ii −∑+ β F iiw ε ∑

[ ] iiiii ww x xw ε β ∑+∑+∑

[ ] iwiε β ∑++ 01

iwiε β ∑+ 33333333333( ! )

oth sides we *et

( )ii E w ε β ∑+

0+ β since E 0)( =iε

estimate o& D.

[ ] ii Y xwn −∑ / 1

[ ] )( / 1 ii xw xn ε β α ++−∑

−−++∑ iiii w xw x

n x

nn β α ε β α

111

wi xnn

x

n i α ε β α −∑−∑+

∑+∑

11

( ) ( ) ( ) x xin

xnn

β α ε β α −−∑++ 1011

and C.

iii w x

nε ε α ∑−∑+

1

oth sides1

Econometrics

BS Economics

− iii w x ε

iwi xwixi x ε ∑−∑

iwi x ε ∑


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Muhammad Ali




E( )

4hus is an un#iased e

an.

$;

( ) ( )iii E w x E nε ε α ∑−∑+ 1

00 ++α since E(

α

stimator o& α .

Econometrics

BS Economics

0) =iε


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$=

Multiple Linear Regression Model

Definition

A linear re*ression model that involves more than one redictor varia#le is called

multile linear re*ression model. !n this case the resonse varia#le is a linear &unction o&

two or more than two redictor varia#les. A multile linear re*ression model with %%

redictor varia#les is *iven #y1

ε β β β β +++++= p p X X X oYi ...2211 i

p β β β ,...,, 10 are arameters and to #e estimated &rom the samle data. 4hese

arameters are also called re*ression coe&&icients' the arameters

)...3,2,1( p j j = β reresent the e:ected chan*e in the resonse 8 and ercent chan*e

in 7


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$@

"rdinary &east S(uare *riteria to find Estimates

4he least s,uare &unction

S e∑ i-

∑=

−

n

ii

Y Y 1

2^

2

22

^

11

^

0

^

++−∑ X X Y i β β β

2

22

^

11

^

0

^

2

−−−∑ X X Y i β β β 33333333333333( ! )

4he &unction 9S9 is to #e minimied with resect to

^

2

^

1

^

0

^

,, β β β and . +or this urose we have to

di&&erentiate E,uation ( ! ) with resect to

^

2

^

1

^

0

^

,, β β β and .

∑∑==

=

−−−

∂

∂=

∂

∂=

∂

∂ n

i

i

n

ii

X X Y eS

1

2

22

^

11

^

0

^

0

^1

2

^

00

^0

)(

)( β β β

β β β

3333333( !! )

∑∑==

=

−−−

∂

∂=

∂

∂=

∂

∂ n

i

i

n

i

i X X Y eS

1

2

22

^

11

^

0

^

1

^1

2

^

11

^0

)(

)( β β β

β β β

333333( !!!)

∑∑==

=

−−−

∂

∂=

∂

∂=

∂

∂ n

i

i

n

ii

X X Y eS

1

2

22

^

11

^

0

^

2

^1

2

^

22

^0

)(

)( β β β

β β β

3333( !K )


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$

+rom ( !! )

( ) 0121

^

22

^

11

^

0 =−

−−−∑

=

n

i

i X X Y β β β

∑ ∑∑∑= ===

=−−−

n

i

n

i

n

i

n

i

X XiYi1 1

22

^

1

1

^

0

^

1

0 β β β

∑ ∑∑∑= ===

++=

n

i

n

i

n

i

n

i

X XiYi1 1

22

^

1

1

^

0

^

1

β β β

)(22

^

11

^^

1

V X X nYi o

n

i

−−−−∑+∑+=∑=

β β β

+rom ( !!! )

( ) 02122

^

11

^

0

^

=−

−−−∑ X X X Yi β β β

0212

^2

11

^

10

^

1 =∑−∑−∑−∑ X X X X X Y i β β β

)(212

^2

11

^

10

^

1 VI X X X X X Y i −−−−∑+∑+∑=∑ β β β

( ) 02)( 222^

11

^

0

^

=

−−−∑−−− X X X Y IV From i β β β

02

22

^

211

^

20

^

2 =∑−∑−∑−∑ X X X X X Y i β β β

)(2

22

^

211

^

20

^

2 VII X X X X X Y i −−−−∑+∑+∑=∑ β β β

E,uation ( K )' ( K! )' and ( K!! ) are called normal e,uations.

+rom ( K ) we *et


19/23





$I

n X n X nnnYi o

n

i

/ / / / 22

^

11

^^

1∑+∑+=∑=

β β β

22

^

11

^

0

^

X X Y β β β ++=

22

^

11

^

0

^

X X Y β β β −−=

su#stitutin* value o& D in e,uation ( K!) and ( K!!) we *et

212

^2

11

^

122

^

11

^

1)( X X X X X X Y X Y VI From i ∑+∑+∑

−−=∑−− β β β β

2

22

^

211

^

222

^

11

^

2)( X X X X X X Y X Y VII From i ∑+∑+∑

−−=∑−− β β β β

[ ] [ ] A X X X X X X X X Y X Y i −−−∑−∑+∑−∑=∑−∑ 12212^

112

11

^

11 β β

[ ] [ ] B X X X X X X X X Y X Y i −−−∑−∑+∑−∑=∑−∑ 222

22

^

21211

^

22 β β

6ow

( )( )Y Y X X y x −−∑=∑ 111

( ) ( )Y Y X Y Y X y x −∑−−∑=∑ 111

( ) ( ) 0sin,11 =−∑ → −∑=∑ Y Y ceY Y X y x

111 X Y Y X y x ∑−∑=∑

2

11

2

1 )( X X x −∑=∑

( )1111

2

1 )( X X X X x −−∑=∑


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-

( ) ( )111111

2

1 X X X X X X x −−−∑=∑

0)(sin,

0)(sin

22222

2

2

2

11112

1

2

1

=−∑→∑−∑=∑

=−∑→∑−∑=∑

X X ce X X X x

Similarly

X X ce X X X x

( ) ( )

( )Y Y ce X Y Y X y x

Y Y X Y Y X y x

Y Y X X y x

−∑→∑−∑=∑

−∑−−∑=∑

−−∑=∑

sin;222

222

222

Su#stitutin* these results in e,uation A and B we o#tained the normal e,uations in

deviation &orm as &ollow1$

Solvin* the a#ove normal e,uations &or DQ and D$

Q i.e. multilyin* e,uation ( C ) #y

R:-- an e,uation( d ) #y R:$:- and su#tract it' we will *et the &ollowin* estimates

o& D$.

( )

∑−∑∑

∑∑−∑∑

= 2212

2

2

1

212

2

21

1

^

x x x x

x x y x x y x

β

Similarly multilyin* e,uation ( C ) #y 9 R:$:- 9 and e,uation ( d ) #y R:$- and

su#tractin* we will *et

( )

∑−∑∑

∑∑−∑∑=

2212

2

2

1

211

2

122

^

x x x x

x x y x x y x β


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-$

Standardi4ed coefficients:

!n statistics' standardied

coe&&icients or #eta

coe&&icients are the estimates resultin* &rom an analysis

carried out on indeendent varia#les that have #een standardied so that their variances are. 4here&ore'

standardied coe&&icients re&er to how many standard deviations a deendent varia#le will chan*e'

er standard deviation increase in the redictor varia#le. Standardiation o& the coe&&icient is usually

done to answer the ,uestion o& which o& the indeendent varia#les have a *reater e&&ect on

the deendent varia#le in a multile re*ression analysis' when the varia#les are

measuredindi&&erent unitso& (&ore:amle' income measuredin dollars and &amilysie measuredin num#e

r o& individuals). A re*ression carried out on ori*inal (unstandardied) varia#les roduces unstandardied

coe&&icients. A re*ression carried out on standardied varia#les roduces standardied coe&&icients.

Kalues &or standardied and unstandardied coe&&icients can also #e derived su#se,uent to either tye

o& analysis. Be&ore solvin* a multile re*ression ro#lem' all varia#les (indeendent and deendent) can

#e standardied. Each varia#le can #e standardied #y su#tractin* its mean &rom each o& its values and

then dividin* these new values #y the standard deviation o& the varia#le. Standardiin* all varia#les in a

multile re*ression yields standardied re*ression coe&&icients that show the chan*e in the deendent

varia#le measured in standard deviations.

Ad%antages

Standard coefficients' advocates note that the coefficients ignore the independent variable's scale of units,

which makes comparisons easy.


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--

Disad%antages

Critics voice concerns that such a standardiation can #e misleadin* a chan*e o& one standard deviation

in one varia#le has no reason to #e e,uivalent to a similar chan*e in another redictor. Some

varia#les are easy to a&&ect e:ternally' e.*.' the amount o& time sent on an action. 5ei*ht or

cholesterol level are more di&&icult' and some' lie hei*ht or a*e' are imossi#le to a&&ect e:ternally.

5oodness of 6it '2)

4he Coe&&icient o& etermination' also nown as S,uared' is interreted as the *oodness o& &it o& a

re*ression. 4he hi*her the coe&&icient o& determination' the #etter the variance that the deendent

varia#le is e:lained #y the indeendent varia#le. 4he coe&&icient o& determination is the overall

measure o& the use&ulness o& a re*ression. +or e:amle' i& - is .I;. 4his means that the variation in the

re*ression is I; e:lained #y the indeendent varia#le. 4hat is a *ood re*ression. 6ow' i& the

Coe&&icient o& etermination' or -'is .;. !ts means that the variation in the re*ression is ;

e:lained #y the indeendent varia#le. 4his is not a *ood re*ression. 6ote that - lies #etween 99 and

9$9. !& -$' it means that the &itted model e:lains $ o& the variation in resonse varia#le 989. "n the

other hand i& -' the model does not e:lain any o& the variation o& 989. 4he Coe&&icient o&

etermination can #e calculated as the e*ression sum o& s,uares' SS' divided #y the total sum o&

s,uares'SS4

Coe&&icient o& etermination TSS

RSS

Mathematical &ormula o& the coe&&icient o& etermination is *iven as under1


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-/

ro#lems ith the coefficient of Determination

+irst' let9s consider that the Coe&&icient o& etermination will increase as more indeendent varia#les are

added. !t does not matter i& those indeendent varia#les hel to e:lain the variation o& the deendent

varia#le' the S,uare (Coe&&icient o& etermination) will increase as more indeendent varia#les are

added. 4his #rin*s us to the concet o& Ad

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