Asymtotic Prop of Estimators,Plims and Consitency

8/9/2019 Asymtotic Prop of Estimators,Plims and Consitency

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ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

The asymptotic properties of estimators are their

properties as the nm!er of o!ser"ations in a samp#e

!ecomes "ery #ar$e an% ten%s to infinity&

'e sha## !e concerne% (ith the concepts of pro!a!i#ity

#imits an% consistency) an% the centra# #imit theorem&

These topics are sa##y $i"en #itt#e attention in stan%ar%statistics te*ts) $enera##y (ithot an e*p#anation of (hy

they are re#e"ant an% sef#&

+o(e"er) asymptotic properties #ie at the heart of mch

econometric ana#ysis an% so for st%ents of

econometrics they are important&


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The asymptotic properties of estimators are their

properties as the nm!er of o!ser"ations in a samp#e

!ecomes "ery #ar$e an% ten%s to infinity&

'e sha## !e concerne% (ith the concepts of pro!a!i#ity

#imits an% consistency) an% the centra# #imit theorem&

These topics are sa##y $i"en #itt#e attention in stan%ar%statistics te*ts) $enera##y (ithot an e*p#anation of (hy

they are re#e"ant an% sef#&

+o(e"er) asymptotic properties #ie at the heart of mch

econometric ana#ysis an% so for st%ents of

econometrics they are important&



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'e (i## start (ith an a!stract %efinition of a pro!a!i#ity #imit an% then i##strate it (ith a simp#ee*amp#e&

0lim

aXP nn

Pro!a!i#ity #imits



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A se,ence of ran%om "aria!#esXnis sai% to con"er$e in pro!a!i#ity to a constant aif)

$i"en any positi"e ) ho(e"er sma##) the pro!a!i#ity ofXn%e"iatin$ from a!y an amont

$reater than ten%s to -ero as nten%s to infinity&

Pro!a!i#ity #imits

0lim

aXP nn



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The constant ais %escri!e% as the pro!a!i#ity #imit of the se,ence) sa##y a!!re"iate% as

p#im&

0lim

aXP nn

aXn=plim

Pro!a!i#ity #imits



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'e (i## ta.e as or e*amp#e the mean of a samp#e of o!ser"ations)X) $enerate% from a

ran%om "aria!#eX(ith pop#ation mean Xan% "ariance /X& 'e (i## in"esti$ate ho(X

!eha"es as the samp#e si-e n!ecomes #ar$e&

n

0 12

pro!a!i#ity %ensity

fnction ofX

12 022 012 /22

n3 0

2&24

2&25

2&2/

2&26

X



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For con"enience (e sha## assme thatXhas a norma# %istri!tion) !t this %oes not affect

the ana#ysis& IfXhas a norma# %istri!tion (ith mean Xan% "ariance /X)X(i## ha"e a

norma# %istri!tion (ith mean Xan% "ariance

/X7 n&

n

0 12

pro!a!i#ity %ensity

fnction ofX

12 022 012 /22

n3 0

2&24

2&25

2&2/

2&26

X



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For the prposes of this e*amp#e) (e (i## sppose thatXhas pop#ation mean 022 an%

stan%ar% %e"iation 12) as in the %ia$ram&

n

0 12

pro!a!i#ity %ensity

fnction ofX

12 022 012 /22

n3 0

2&24

2&25

2&2/

2&26

X



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n

0 12

9

The samp#e mean (i## ha"e the same pop#ation mean asX) !t its stan%ar% %e"iation (i##

!e 127 ) (here nis the nm!er of o!ser"ations in the samp#e&

12 022 012 /22

n3 0

n

2&24

2&25

2&2/

2&26

pro!a!i#ity %ensity

fnction ofX

X



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n

0 12

10

The #ar$er is the samp#e) the sma##er (i## !e the stan%ar% %e"iation of the samp#e mean&

12 022 012 /22

n3 0

2&24

2&25

2&2/

2&26

pro!a!i#ity %ensity

fnction ofX

X



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n

0 12

11

If nis e,a# to 0) the samp#e consists of a sin$#e o!ser"ation& Xis the same asXan% its

stan%ar% %e"iation is 12&

12 022 012 /22

n3 0

2&24

2&25

2&2/

2&26

pro!a!i#ity %ensity

fnction ofX

X



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n

0 12

5 /1

12

'e (i## see ho( the shape of the %istri!tion chan$es as the samp#e si-e is increase%&

12 022 012 /22

n3 5

2&24

2&25

2&2/

2&26

pro!a!i#ity %ensity

fnction ofX

X



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n

0 12

5 /1

/1 02

13

The %istri!tion !ecomes more concentrate% a!ot the pop#ation mean&

12 022 012 /22

n3 /1

2&24

2&25

2&2/

2&26

pro!a!i#ity %ensity

fnction ofX

X



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n

0 12

5 /1

/1 02

022 1

15

'e ha"e increase% the "ertica# sca#e !y a factor of 02&

12 022 012 /22

n3 022

2&4

2&5

2&/

2&6

pro!a!i#ity %ensity

fnction ofX

X


ASYMPTOTIC PROPERTIES OF ESTIMATORS PLIMS AND CONSISTENCY


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n

0 12

5 /1

/1 02

022 1

0222 0&6

16

The %istri!tion contines to contract a!ot the pop#ation mean&

12 022 012 /22

n3 0222

2&4

2&5

2&/

2&6

pro!a!i#ity %ensity

fnction ofX

X




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n

0 12

5 /1

/1 02

022 1

0222 0&6

1222 2&8

17

In the #imit) the "ariance of the %istri!tion ten%s to -ero& The %istri!tion co##apses to a

spi.e at the tre "a#e& The p#im of the samp#e mean is therefore the pop#ation mean&

12 022 012 /22

n3 1222

2&4

2&5

2&/

2&6

pro!a!i#ity %ensity

fnction ofX

X




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Forma##y) the pro!a!i#ity ofX%ifferin$ from X!y any finite amont) ho(e"er sma##) ten%s to

-ero as n!ecomes #ar$e&

18

0lim Xn XP




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+ence (e can say p#imX3 X&

19

0lim Xn XP

XX

plim




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Consistency

An estimator of a pop#ation characteristic is sai% to !econsistent if it satisfies t(o con%itions:

90 It possesses a pro!a!i#ity #imit) an% so its

%istri!tion co##apses to a spi.e as the samp#e si-e

!ecomes #ar$e) an%

9/ The spi.e is #ocate% at the tre "a#e of the

pop#ation characteristic&

+ence (e can say p#imX3 X&

20




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The samp#e mean in or e*amp#e satisfies !oth con%itions an% so it is a consistent

estimator ofX& Most stan%ar% estimators in simp#e app#ications satisfy the first con%ition

!ecase their "ariances ten% to -ero as the samp#e si-e !ecomes #ar$e&

12 022 012 /22

n3 1222

2&4

2&5

2&/

2&6

pro!a!i#ity %ensity

fnction ofX




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The on#y isse then is (hether the %istri!tion co##apses to a spi.e at the tre "a#e of the

pop#ation characteristic& A sfficient con%ition for consistency is that the estimator

sho#% !e n!iase% an% that its "ariance sho#% ten% to -ero as n!ecomes #ar$e&

12 022 012 /22

n3 1222

2&4

2&5

2&/

2&6

pro!a!i#ity %ensity

fnction ofX




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It is easy to see (hy this is a sfficient con%ition& If the estimator is n!iase% for a finite

samp#e) it mst stay n!iase% as the samp#e si-e !ecomes #ar$e&

12 022 012 /22

n3 1222

2&4

2&5

2&/

2&6

pro!a!i#ity %ensity

fnction ofX




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Mean(hi#e) if the "ariance of its %istri!tion is %ecreasin$) its %istri!tion mst co##apse to

a spi.e& Since the estimator remains n!iase%) this spi.e mst !e #ocate% at the tre "a#e&

The samp#e mean is an e*amp#e of an estimator that satisfies this sfficient con%ition&

12 022 012 /22

n3 1222

2&4

2&5

2&/

2&6

pro!a!i#ity %ensity

fnction ofX




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+o(e"er the con%ition is on#y sfficient) not necessary& It is possi!#e that an estimator may

!e !iase% in a finite samp#e ;

n3 /2

Z

pro!a!i#ity %ensity

fnction of Z




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; !t the !ias !ecomes sma##er as the samp#e si-e increases

n3 022

n3 /2

pro!a!i#ity %ensity

fnction of Z

Z




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; to the point (here the !ias %isappears a#to$ether as the samp#e si-e ten%s to infinity&

Sch an estimator is !iase% for finite samp#es !t ne"erthe#ess consistent !ecase its

%istri!tion co##apses to a spi.e at the tre "a#e&

n3 022

n3 0222

n3 /2

pro!a!i#ity %ensity

fnction of Z

Z

n3 022222



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The estimator is !iase% for finite samp#es !ecase its e*pecte% "a#e is nX79n< 0& =t as

n ten%s to infinity) n79n< 0 ten%s to 0 an% the estimator !ecomes n!iase%&

.

1

1

1

=

=

n

i

iX

n

Z

.1

X

n

nZE

Consistency




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The "ariance of the estimator is $i"en !y the e*pression sho(n& This ten%s to -ero as n

ten%s to infinity& Ths Zis consistent !ecase its %istri!tion co##apses to a spi.e at the

tre "a#e&

.

1

1

1

=

=

n

i

iX

n

Z

.1

X

n

nZE

2

21

)var(X

n

nZ

=

Consistency




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Consistency

In practice (e %ea# (ith finite samp#es) not infinite ones& So (hysho#% (e !e intereste% in (hether an estimator is consistent>

One reason is that sometimes it is impossi!#e to fin% an estimator

that is n!iase% for sma## samp#es& If yo can fin% one that is at

#east consistent) that may !e !etter than ha"in$ no estimate at a##&

A secon% reason is that often (e are na!#e to say anythin$ at a##

a!ot the e*pectation of an estimator& The e*pecte% "a#e r#es are

(ea. ana#ytica# instrments that can !e app#ie% in re#ati"e#y simp#e

conte*ts&

In partic#ar) the m#tip#icati"e r#e E?g9Xh9Y@ 3 E?g9X@E?h9Y@app#ies on#y (henXan% Yare in%epen%ent) an% in most sitations

of interest this (i## not !e the case& =y contrast) (e ha"e a mch

more po(erf# set of r#es for p#ims&




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Consistency








conte*ts&







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Consistency








conte*ts&







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Consistency








conte*ts&







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P#im r#es

Plim rule 1 The p#im of the sm of se"era# "aria!#es is e,a# tothe sm of their p#ims& For e*amp#e) if yo ha"e three

ran%om "aria!#esX) Y) an% Z) each possessin$ a p#im)

p#im 9X< Y< Z 3 p#imX< p#im Y< p#im Z

Plim rule 2 If yo m#tip#y a ran%om "aria!#e possessin$ a p#im !ya constant) yo m#tip#y its p#im !y the same constant&

IfXis a ran%om "aria!#e an% bis a constant)

p#im bX3 bp#imX

Plim rule 3 The p#im of a constant is that constant& For e*amp#e)if bis a constant)

p#im b3 b



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P#im r#es






p#im bX3 bp#imX


p#im b3 b



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P#im r#es






p#im bX3 bp#imX


p#im b3 b



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P#im r#es

Plim rule 4 The p#im of a pro%ct is the pro%ct of the p#ims) ifthey e*ist& For e*amp#e) if Z3XY) an% ifXan% Y!oth

possess p#ims)

p#im Z3 9p#imX9p#im Y

Plim rule 5 The p#im of a ,otient is the ,otient of the p#ims) ifthey e*ist& For e*amp#e) if Z3X/Y) an% ifXan% Y!oth

possess p#ims) an% p#im Yis not e,a# to -ero)

p#im Z =p#imX

p#im Y


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P#im r#es

Plim rule 6 The p#im of a fnction of a "aria!#e is e,a# to thefnction of the p#im of the "aria!#e) pro"i%e% that the

"aria!#e possesses a p#im an% pro"i%e% that the

fnction is continos at that point&

p#im f9X 3 f9p#imX



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E*amp#e se of asymptotic ana#ysis

ZY

To i##strate ho( the p#im r#es can #ea% s to conc#sions (hen the e*pecte% "a#e r#es

%o not) consi%er this e*amp#e& Sppose that yo .no( that a "aria!#e Yis a constant

m#tip#e of another "aria!#e Z



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ZY

Zis $enerate% ran%om#y from a fi*e% %istri!tion (ith pop#ation mean Zan% "ariance /&

is n.no(n an% (e (ish to estimate it& 'e ha"e a samp#e of no!ser"ations&



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ZY

wZX

Yis measre% accrate#y !t Zis measre% (ith ran%om error w(ith pop#ation mean -ero

an% constant "ariance /w& Ths in the samp#e (e ha"e o!ser"ations onX) (hereX3 Z< w)

rather than Z&



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ZY

wZX

One estimator of l9not necessari#y the !est is Yi7 Xi

wZ

w

wZ

w

wZ

Z

wZ

Z

X

Y

ii

i

ii

i

ii

i

i

i

=

=



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ZY

wZX

S!stittin$ from the first t(o e,ations) the estimator can !e re(ritten as sho(n&

wZ

w

wZ

w

wZ

Z

wZ

Z

X

Y

ii

i

ii

i

ii

i

i

i

=

=


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ZY

wZX

=t (e cannot %o this& The ran%om ,antity appears in !oth the nmerator an% the

%enominator an% the e*pecte% "a#e r#es are too (ea. to a##o( s to in"esti$ate the

e*pectation ana#ytica##y&

wZ

w

wZ

w

wZ

Z

wZ

Z

X

Y

ii

i

ii

i

ii

i

i

i

=

=



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ZY

wZX

+o(e"er) (e .no( that a samp#e mean ten%s to a pop#ation mean as the samp#e si-e ten%s

to infinity) an% so p#im w3 2 an% p#im Z3 Z&

wZ

w

wZ

w

wZ

Z

wZ

Z

X

Y

ii

i

ii

i

ii

i

i

i

=

=



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ZY

wZX

Since the p#ims of the nmerator an% the %enominator of the error term !oth e*ist) (e are

a!#e to ta.e the p#im of the error term& Ths (e are a!#e to sho( that the estimator is

consistent) %espite the fact that (e cannot say anythin$ a!ot its finite samp#e properties&

=

00

plimplimplimplim

Zi

i

wZw

X

Y



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Copyri$ht Christopher Do$herty /200&

These s#i%esho(s may !e %o(n#oa%e% !y anyone) any(here for persona# se&

S!Bect to respect for copyri$ht an%) (here appropriate) attri!tion) they may !e

se% as a resorce for teachin$ an econometrics corse& There is no nee% to

refer to the athor&

The content of this s#i%esho( comes from Section R&05 of C& Do$herty)

Introdution to Eonometri!) forth e%ition /200) O*for% ni"ersity Press&

A%%itiona# 9free resorces for !oth st%ents an% instrctors may !e

%o(n#oa%e% from the OP On#ine Resorce Centrehttp:77(((&op&com7.7orc7!in78420168247&

In%i"i%a#s st%yin$ econometrics on their o(n an% (ho fee# that they mi$ht

!enefit from participation in a forma# corse sho#% consi%er the Lon%on Schoo#

of Economics smmer schoo# corse

EC/0/ Intro%ction to Econometrics

http:77(((/se&ac&.7st%y7smmerSchoo#s7smmerSchoo#7+ome&asp*or the ni"ersity of Lon%on Internationa# Pro$rammes %istance #earnin$ corse

/2 E#ements of Econometrics

(((on%oninternationa#&ac&.7#se&
http://www.oup.com/uk/orc/bin/9780199567089/http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspxhttp://g/www.londoninternational.ac.uk/lsehttp://g/www.londoninternational.ac.uk/lsehttp://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspxhttp://www.oup.com/uk/orc/bin/9780199567089/

Asymtotic Prop of Estimators,Plims and Consitency

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