PanelDataNotes 2 A

7/25/2019 PanelDataNotes 2 A

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Part 2A: Basic Econometrics [ 1/75]

Econometric Analysis of Panel Data

William Greene

Department of Economics

Stern School of Business


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Endogeneity

y= X+, Definition: E[|x]0 Why not?

Omitted variables nobserved hetero!eneity "e#$ivalent to omitted

variables% &eas$rement error on the '() "e#$ivalent to omitted

variables% )tr$*t$ral as+e*ts of the model Endo!eno$s sam+lin! and attrition )im$ltaneity "?%


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Instrumental Variable Estimation One +roblem- variable . the last- one yit= //it1 22it1 3 1 44it1 it E[it|4it] 05 "0 for all others%

6here eists a variable 7its$*h that E[4it| /it, 2it,3, 48/,it,7it] = !"/it, 2it,3, 48/,it,7it%

9n the +resen*e of the other variables, 7 ite+lains- it E[it| /it, 2it,3, 48/,it,7it] = 0 9n the +resen*e of the other variables, 7itand it are

$n*orrelated5 +ro;e*tion inter+retation: 9n the +ro;e*tion 8/48/,it1 4 7it,

4 05


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The First IV Study: Natural Exeriment!Sno"# $%# &n the 'ode of (ommunication of (holera# )*++,

htt:--"""%h%ucla%edu-ei-sno"-sno"boo./%html

ondon @holera e+idemi*, *a /ABC8 @holera = f"Water $rity,$%15

F@a$salG eHe*t of Iater +$rity on *holera? $rity=f"*holera +rone environment "+oor,

!arba!e in streets, rodents, et*5%5 'e!ression doesnot Ior>5

6Io ondon Iater *om+anies

ambeth )o$thIar>

Main sewage discharge

Paul Grootendorst: A Review of Instrumental Variables Estimation of Treatment Effects

http://individualutorontoca/!rootendorst/pdf/IV"Paper"#ept$"%&&'pdf

Rier

!hames


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IV Estimation @holera=f"$rity,$%1 J = Iater *om+any

@ov"@holera,J%=K@ov"$rity,J% J is randomly mied in the +o+$lation

"tIo f$ll sets of +i+es% and $n*orrelatedIith behavioral $nobservables, $%

@holera=L1K$rity1$1 $rity = &ean1random variation1M$ @ov"@holera,J%= K@ov"$rity,J%


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Part 2A: Basic Econometrics [ "/75]

(orn"ell and 0uert Data

Cornwell and Rupert Returns to Schooling Data, 595 Individuals, !ears"aria#les in the $le are

E e+erien*eW4) = Iee>s Ior>edO@@ = o**$+ation, / if bl$e *ollar,

9ND = / if man$fa*t$rin! ind$stry)O6( = / if resides in so$th)&) = / if resides in a *ity ")&)%&) = / if marriedE& = / if femaleN9ON = / if Ia!e set by $nion *ontra*tED = years of ed$*ationWPE = lo! of Ia!e = de+endent variable in re!ressions

6hese data Iere analy7ed in @ornIell, @5 and '$+ert, 5, QER*ient EstimationIith anel Data: n Em+iri*al @om+arison of 9nstr$mental SariableEstimators,Q To$rnal of ++lied E*onometri*s, C, /UAA, ++5 /U8/BB5 )eealta!i, +a!e /22 for f$rther analysis5 6he data Iere doInloaded from theIebsite for alta!iXs tet5


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Part 2A: Basic Econometrics [ #/75]

Secification: 1uadratic Effect of Exerience


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Part 2A: Basic Econometrics [ $/75]

The Effect of Education on 23A4E

= + + + + +1 2 3 4 ...

What is ? ,... + eer!thin" el#$il seit!, %otiation

#$ilit!, %otiation& f' , , , ,...(

%&A'E E()* E+P

E()* 'E,(ER -M-A -.)!/

%&X'


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3hat Influences 23A4E5= +

+ + +

+

1 2

3 4

#$ilit!, %otiation

#$ilit!, %otiat

' , ,...(

...

' (

)ncrease* is associate* ith increases in

ion

#$ilit!

#$ilit!, %otiati' , ,on

%&A'E E()* +

E+P

E()* +

%&X'

2

...( an* ' (

What loos lie an effect *ue to increase in ma!$e an increase in . -he estimate of pics up

the effect of an* the hi**en effect of .

#$ilit!, %otiation

#$ilit!

#$ilit!

E()*

E()*


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An Exogenous Influence

= +

+ + +

+

1 2

3 4

' , , ,...(

...

' (

)ncrease* is asso

#$ili

ciate

t!, %otiation

#$ilit!, %otiation

#$ilit!, %otia

* ith increases in

' , , ,ti .n .o

%&A'E E() 1

1

1

* +

E+P

E()* +

%&X'

2

.( an* not ' (

#n effect *ue to the effect of an increase on ill

onl! $e an increase in . -he estimate of pics up

the effect of onl!.

#$ilit!, %oti

ation

E()*

E()*

E(

1

1

)*

is an 2nstr3menta4 5aria64e


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Instrumental Variables )tr$*t$re

WPE "&D,&X',&X'S(,)*S,CC, S-.,S/S0,1I1%

ED "/S, 2&/%

'ed$*ed orm:

WPE[ &D"/S, 2&/%, &X',&X'S(,)*S,CC, S-.,S/S0,1I1 ]


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T"o Stage 2east S6uares Strategy 'ed$*ed orm:

WPE[ &D"/S, 2&/,X%, &X',&X'S(,)*S,CC,

S-.,S/S0,1I1 ] )trate!y "/% $r!e ED of the inY$en*e of everythin! b$t &), E&

"and the other variables%5 redi*t ED $sin! all eo!eno$s

information in the sam+le "Xand 3%5 "2% 'e!ress WPE on this +redi*tion of ED and

everythin! else5 )tandard errors m$st be ad;$sted for the +redi*ted ED


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&2S


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!he weird rests or thecoeicient on E( ha88ened

6ecase the instrments9

M- and EM are dmm;

aria6es< !here is not

enogh ariation in these

aria6es: y = X 1 , 4 variables in X5 6here eists a set of 4 variables, 3 s$*h that

+lim"36X7n% 8 b$t +lim"36Zn% = 8

6he variables in 3are *alled instr$mentalvariables5

n alternative "to least s#$ares% estimator of is

#9S = "36X%8/36y

We *onsider the folloIin!: Why $se this estimator? What are its +ro+erties *om+ared to least

s#$ares? We Iill also eamine an im+ortant a++li*ation


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IV Estimators

@onsistent

#9S= "36X%8/36y

= "36XZn%8/ "36XZn%1 "36XZn%8/36Zn

= 1 "36XZn%8/36Zn sym+toti*ally normal "same a++roa*h to +roof as for O)%

9neR*ient . to be shoIn5

y *onstr$*tion, the 9S estimator is *onsistent5 We have

an estimator that is *onsistent Ihen least s#$ares is not5


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Part 2A: Basic Econometrics [ 2#/75]

IV Estimation

)hy use an I" esti:ator? )$++ose thatXandare not $n*orrelated5 6hen least s#$ares isneither $nbiased nor *onsistent5

'e*all the +roof of *onsisten*y of least s#$ares:

# = 1 "X6XZn%8/"X6Zn%5

lim #= re#$ires +lim"X6Zn% = 8; 9f this doesnot hold, the estimator is in*onsistent5


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Part 2A: Basic Econometrics [ 2$/75]

A Poular 'isconcetion

9f only one variable in Xis *orrelated Iith , the other*oeR*ients are *onsistently estimated5 alse5

6he +roblem is smeared- over the other *oeR*ients5

=

=

/

///

2/

8/ /

/

/

)$++ose only the \rst variable is *orrelated Iith

0nder the ass$m+tions, +lim" Zn% = 5 6hen

555

5

#

0 #+lim = +lim" Zn% 555 555

5 #

times

K

X


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(onsistency and Asymtotic

Normality of the IV Estimator8/

8/

8/

" %

" %

+lim +lim" N% N

sy5Sar[ sy5Sar[ N]

" N% [+lim N] "/Z

=>

3X

=> =>

3X X3

=> =>

3X X3

? 3


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Asymtotic (o7ariance 'atrix of bIV

/9S

/ 8/9S 9S

2 / 8/9S 9S

" % X

" %" % X " % X X " %

E[" %" % X| ] " % X " %

=

= =

# 3


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Asymtotic Efficiency

sym+toti* eR*ien*y of the 9S estimator5 6hevarian*e is lar!er than that of )5 " lar!esam+le ty+e of Pa$ss8&ar>ov res$lt is at Ior>5%

"/% 9tGs a moot +oint5 ) is in*onsistent5"2% &ean s#$ared error is $n*ertain:

&)E[estimator|]=Sarian*e 1 s#$are of bias5

9S may be better or Iorse5 De+ends on the data


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T"o Stage 2east S6uares(oI to $se an e*ess- of instr$mental variables

"/% Xis 4 variables5 )ome "at least one% of the 4

variables in Xare *orrelated Iith 5

"2% 3is & ^ 4 variables5 )ome of the variables in

3are also in X, some are not5 None of the

variables in 3are *orrelated Iith ;

"C% Whi*h 4 variables to $se to *om+$te 36Xand36yE


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(hoosing the Instruments

@hoose 4 randomly? @hoose the in*l$ded


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Algebraic E6ui7alence

6Io sta!e least s#$ares is e#$ivalent to

"/% ea*h variable in Xthat is also in 3isre+la*ed by itself5

"2% Sariables in Xthat are not in 3are re+la*edby +redi*tions of that X$sin!

ll other variables in Xthat are not *orrelated Iith ll the variables in 3that are not in X5


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Part 2A: Basic Econometrics [ 3"/75]

!he weird rests or thecoeicient on E( ha88ened

6ecase the instrments9

M- and EM are dmm;

aria6es< !here is not

enogh ariation in these

aria6es

%S4S

=>

3 3

3 3 3

%S4S

X 3@3


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#s!mptotic .oariance %atri/ for 2S0S

2 / 8/

9S 9S

2 / 8/2)) 2))

Peneral 'es$lt for 9nstr$mental Sariable Estimation

E[" %" % X| ] " % X " %

)+e*iali7e for 2)), $sin! = " %

E[" %" % X| ] " % X " %

=

=

3

# # X,3 3


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8S2S 9as 2arger Variance than 2S2 8/

2 8/

*om+arison to O)

sy5Sar[2))]= " X %

Ne!le*tin! the in*onsisten*y,

sy5Sar[)] = " X %

"6his is the varian*e of ) aro$nd its mean, not %

sy5Sar[2))] sy5Sar[)] in the matri sense5

@om

X X

X X

8/ 8/ 2

2 2J J

+are inverses:

sy5Sar[)] 8 sy5Sar[2))] "/Z %[ X X ]

"/Z %[ X X" % ]="/Z %[ X ]

6his matri is nonne!ative de\nite5 "Not +ositive de\nite

as it mi!ht have some roIs and *ol$mns

=

=

X X= X X

X X= X I / X X / X

Ihi*h are 7ero5%

9m+li*ation for Q+re*isionQ of 2))5

6he +roblem of QWea> 9nstr$mentsQ


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Estimating 8

2

2 n/i / in

Estimatin! the asym+toti* *ovarian*e matri 8

a *a$tion abo$t estimatin! 5

)in*e the re!ression is *om+$ted by re!ressin! y on ,

one mi!ht $se

"y %

6his is i

=

= %sls

x

x


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0obust estimation of V(

( )8/ 2 8/i,t it it

@o$nter+art to the White estimator alloIs heteros*edasti*ity

Est5:sy5Sar[ ]=" % "y % " % it itX


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8S2S 7s% 0obust Standard Errors+--------------------------------------------------+| Robust Standard Errors |+---------+--------------+----------------+--------+|Variable | Coefficient | Standard Error |b/St.Er.|+---------+--------------+----------------+--------+B_1 45.4848! 4."5#!11 11.#8B_ ."5$54484 ."1%4#$ 4.$$

B_$ -.""1%#%%4 ."""#""% -5.84#B_4 ."1#4854 ."5!5!1!# .5B_5 .$85$!$ ."!"%5%" 5.454B_% .$%!!!4! ."%4!185 5.%8B_! .#55$"115 ."8%81%1 11."""+--------------------------------------------------+| S&S Standard Errors |+---------+--------------+----------------+--------+|Variable | Coefficient | Standard Error |b/St.Er.|

+---------+--------------+----------------+--------+B_1 45.4848! .$%#"8158 1$.$%B_ ."5$54484 ."$1$##"4 1.!"5B_$ -.""1%#%%4 ."""%#1$8 -.454B_4 ."1#4854 .1%%%4$5 ."8"B_5 .$85$!$ .1!%45815 .184B_% .$%!!!4! .1!845!4 .18B_! .#55$"115 ."84%41 4.58$


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Endogeneity Test5 !9ausman,

Eo!eno$s Endo!eno$s

O) @onsistent, ER*ient 9n*onsistent

2)) @onsistent, 9neR*ient @onsistent

ase a test on d = #2))8 #O)

se a Wald statisti*, dG[Sar"d%]8/

d What to $se for the varian*e matri?

(a$sman: "2)) 8 "O)


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9ausman Test


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9ausman Test: &ne at a Time5


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Endogeneity Test: 3u

@onsiderable *om+li*ation in (a$smantest "tet, ++5 2C82C_%

)im+li\*ation: W$ test5 'e!ress yon Xand estimated for the

endo!eno$s +art of X5 6hen $se anordinary Wald test5

?X


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3u Test


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0egression ;ased Endogeneity Test

= + +

it it it

n easy t test5 "Wooldrid!e 20/0, +5 /2_%

y #

= a set of & instr$ments5

Write = 1@an be estimated by ordinary least s#$ares5

Endo!eneity *on*erns *orrelation betIeen v and 5

dd v

itxH

3

3J v

= + + +

it it it it

= # 8 to the e#$ation and $se O)

y # v 1 ` error

)im+le t test on Ihether e#$als 05

Even easier, al!ebrai*ally identi*al, "W$, /U_C%, add

to the e#$ation and do the same tes

it

xH

t5


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Testing Endogeneity of 3-"4 .%"$484>-"4 -1.$5 .185 514.4"5"40CC -.88855# ."15$$ -$.%8 ."""" .5111%44!S023 -.%!#8#1 ."14$#5%1 -18.55 ."""" .#"!%11SS ."$%1%514 ."1$%#!4$ .%4" .""8$ .%5$!8151*S .""$54"8 .""11%45# $."4" .""4 4%.81154%

*S32 .$4#%"141 ."1%484 1.8" ."""" 4%.81154%


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4eneral Test for Endogeneity

2

Etendin! the W$ test

6here are & 4 instr$ments 5

6o *arry o$t the test, *om+$te *ol$mn

by *ol$mn Iith re!ressions on 5

@om+$te by O) the etended model

se

> > % %

%

> > % % %

y? X +X +

3

X

3

y? X +X + X + error

0an test to test ( : 8=


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Alternati7e to 9ausman=s Formula5

( test re#$ires the diHeren*e betIeen aneR*ient and an ineR*ient estimator5

ny Iay to *om+are any tIo *om+etin!estimators even if neither is eR*ient?

ootstra+? "&aybe%


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3ea. Instruments

)ym+tom: 6he relevance condition, +lim 36XZn not 7ero,is *lose to bein! violated5

Dete*tion: )tandard test in the re!ression of > on J5 /0

s$!!ests a +roblem5 statisti* based on 2)) . see tet +5 CB/5

'emedy: Not m$*h . most of the dis*$ssion is abo$t the

*ondition, not Ihat to do abo$t it5 se 9&? 'e#$ires a normality ass$m+tion5 robably

not too restri*tive5


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3ea. Instruments !cont%,

8/+lim = 1 [@ov" %] @ov" %

9f @ov" % is QsmallQ b$t non7ero, small

@ov" % may h$!ely ma!nify the eHe*t5

9S is not only ineR*ient, may be very badly

biased by QIea>Q instr$ments5

)ol$tions

3,X 3,

3,

3,X

? @an one QtestQ for Iea> instr$ments?


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3ea. Instruments

8/ 8/J<


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A study of moral ha>ard

0ihahn# 3ambach# 'illion: ?Incenti7e Effects in

the Demand for 9ealthcare@

$ournal of Alied Econometrics# 8/


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Ali ation: 9ealth (are Panel Data

German Health Care Usage Data, ,!"# Individuals, Varying $umbers o% &eriodsVariables in the %ile are

0ata downloaded from @ournal of Applied Econometrics Archive This is an unbalanced panel with ')%,

individuals The+ can be used for re!ression) count models) binar+ choice) ordered choice) and bivariate binar+

choice 'his is a large data set 'here are altogether !,#! observations 'he number o% observations

ranges %rom * to (+reuen-ies are. */*!, !/!*1, #/1!, 2/"!, /*0*, /*000, /"1) Bote) the

variable BC*(1# below tells how man+ observations there are for each person This variable is repeated in

each row of the data for the person 60ownloaded from the @AE Archive8 D3C'34 / *($umber o% do-tor visits 5 0)

H36&I'78 / *($umber o% hos9ital visits 5 0)

H67' / health satis%a-tion, -oded 0 (lo:) ; *0 (high)

D3CVI6 / number o% do-tor visits in last three months

H36&VI6 / number o% hos9ital visits in last -alendar year

&Us ? *0000

6. observations with income=& were dropped8 HH@ID6 / -hildren under age * in the household / *= other:ise / 0

ADUC / years o% s-hooling

7GA / age in years

B744IAD / marital status

ADUC / years o% edu-ation

P 2A B i E i [ 5#/75]


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E7idence of 'oral 9a>ard5

P t 2A B i E t i [ 5$/75]


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0egression Study

P t 2A B i E t i [ "0/75]


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Part 2A: Basic Econometrics [ "0/75]

Endogenous Dummy Variable

Do*tor Sisits = f"!e, Ed$*, (ealth,resen*e of 9ns$ran*e,

Other $nobservables%

9ns$ran*e = f"E+e*ted Do*tor Sisits,

Other $nobservables%

P t 2A B i E t i [ "1/75]


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Aroaches

"arametri*% @ontrol $n*tion: $ild astr$*t$ral model for the tIo variables"(e*>man%

")emi+arametri*% 9nstr$mental Sariable:@reate an instr$mental variable for thed$mmy variable "arnoIZ@ainZ Poldber!er,

n!rist, @$rrent !eneration of resear*hers% "?% ro+ensity )*ore &at*hin! "(e*>man et

al5, e*>erZ9*hino, &any re*ent resear*hers%



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9ec.man=s (ontrol Function Aroach c = 1 K6 1 E[|6] 1 ` 8 E[|6] M = E[|6] , *om+$ted from a model for Ihether 6 = 0 or /

*a!nitude = ---%&& is nonsensical in this conte3t



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Instrumental Variable Aroach

@onstr$*t a +redi*tion for 6 $sin! only the eo!eno$s information

se 2)) $sin! this instr$mental variable5

*a!nitude = %,&-% is also nonsensical in this conte3t



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Proensity Score 'atching @reate a model for 6 that +rod$*es +robabilities for 6=/: ro+ensity )*ores- ind +eo+le Iith the same +ro+ensity s*ore . some Iith 6=/, some Iith 6=0 @om+are n$mber of do*tor visits of those Iith 6=/ to those Iith 6=05



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Treatment Effect

Earnin!s and Ed$*ation: EHe*t of anadditional year of s*hoolin!

Estimatin! vera!e and o*al vera!e6reatment EHe*ts of Ed$*ation Ihen@om+$lsory )*hoolin! aIs 'eally &atter

hili+ Oreo+o$los E', U,/, 200, /B28/_B

Part 2A: Basic Econometrics [ ""/75]


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Part 2A: Basic Econometrics [ ""/75]

Treatment Effects and

Natural Exeriments



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9o" do anel data fit into this5

We *an $se the $s$al models5 We *an $se far more elaborate models We *an st$dy eHe*ts thro$!h time

Observations are s$rely *orrelated5 6he same individ$al is observed more than on*e nobserved hetero!eneity that a++ears in the

dist$rban*e in a *ross se*tion remains +ersistenta*ross observations "on the same F$nitG%5

ro*ed$res m$st be ad;$sted5 Dynami* eHe*ts are li>ely to be +resent5

Part 2A: Basic Econometrics [ "#/75]


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Part 2A: Basic Econometrics [ "#/75]

Aendix:

Structure and0egression

Part 2A: Basic Econometrics [ "$/75]


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Part 2A: Basic Econometrics [ "$/75]

2east S6uares 0e7isited

8/ 8/

8/

8/ 8/

" % " %

+lim +lim" N% +lim" N%

+lim " % " % = " N% " N%

[ N]

=>

XX

=> =>

XX XX

=> => =>XX XX XX

#? X


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Inference "ith IV Estimators

"/% Wald )tatisti*s:

" % " %

"E5!5, the $s$al Xt8statisti*sX%

"2% ty+e of statisti*:

@om+$te ))=" %X" % Iitho$t restri*tions

@om+$te ))'=" %X" % Iit

=>

u u

R R

R= < M&st;0sy;"arLBN R=

y X y X

y X y X

h restri*tions @om+$te ))=" %X" % Iitho$t restri*tions

"))' ))% Z T = [T,N 4]

))Z"N84%

y X y X



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(omaring &2S and IV

8/


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Testing for Endogeneity!5,

it it it

8/

test for endo!eneity? @onsider one variable:

y # ==^ = 1

# may be endo!eno$s5 = a set of & instr$ments5

"/% O): = " % 9n*onsistent if @ov[#, ] 05

= + +

itxH y X

3

# X


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[ ]

Structure 7s% 0egression

'ed$*ed orm vs5 )t$*t$ral &odel )im$ltaneo$s e#$ations ori!in

g"d% = a01 a/ 1 a29 1 e"d% "demand%g"s% = b01 b/ 1 b2' 1 e"s% "s$++ly%

g"5% = g"d% = g"s%What is the eHe*t of a *han!e in 9 on g"5%?"Not a re!ression%

'ed$*ed form: g = *01 */9 1 *2' 1 v5

"'e!ression% &odern *on*e+ts of str$*t$re vs5 re!ression:

6he sear*h for *a$sal eHe*ts5



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[ ]

Imlications

6he str$*t$re is the theory 6he re!ression is the *onditional mean 6here is alIays a *onditional mean

9t may not e#$al the str$*t$re 9t may be linear in the same variables What is the im+li*ation for least s#$ares

estimation? ) estimates re!ressions ) does not ne*essarily estimate str$*t$res )tr$*t$res may not be estimable . they may not be

identi\ed5



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[ ]

Structure and 0egression

)im$ltaneity? What if E[|x]0 y=1, =Ky1$5 @ov[, ]0

is not the re!ression? What is the re!ression?

'ed$*ed form: ss$me and $ are $n*orrelated5 y = [Z"/8 K%]$ 1 [/Z"/8 K%] = [/Z"/8 K%]$ 1 [K Z"/8 K%] @ov[,y]ZSar[] =

6he re!ression is y = 1 v, Ihere E[v|]=0

2 2 2 2 2

2 2 2 2

[ ] Z[ ]

"/ %"/Z % Ihere I= Z[ ]

u u

u uw w

= + +

= + +



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[ ]

Structure 7s% 0egression-88; a 6CPrice cC*a8acit;

(emand A BCPrice *Cncome

PanelDataNotes 2 A

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