E ± c iency Com parisonsfor a System G M M
E stim ator inDynam ic P anelData M od els
Frank W ind m eijer
Institute f
or F iscalStud ies
7R id gm ount Street
Lond onW C1E 7AE
IF S W orkingP aper SeriesNo.W 9 8/1
A bstract
T hesystem G M M estimatorin dynamicpaneldatamodels combines mo-mentconditions forthe di®erenced equation with momo-mentconditions for themodelinlevels.A ninitialoptimalweightmatrixunderhomoscedastic-ity and non-serialcorrelation is notknown forthis estimation procedure. Itis common practice touse the inverse ofthe momentmatrix ofthe in-struments as the initialweightmatrix.T his paperassesses the potential e±ciencylossfrom theuseofthisweightmatrixusingthee±ciencybounds as derived byL iu and N eudecker(1 997).
1
In
trod uc tion
A stand ard prac tic e to estim ate the param etersind ynam ic paneld ata m od elsis to take ¯rst d i®erencesto elim inate the c orrelated ind ivid ualspec i¯c e®ec ts,and estim ate the d i®erenced m od elb y G eneralised M ethod ofM om ents(G M M ) us-ingappropriately lagged levelvariab lesasinstrum ents.Asthe inform ationofthe instrum entsfor the d i®erenced m od eld ec reasesasthe seriesb ec om e m ore per-sistent,Arellano and B over (199 5) and B lund elland B ond (199 7) have proposed use ofa system G M M estim ator that c omb inesthe d i®erenced equationw ith the levelequation. T he instrum entsfor the levelequationare lagged d i®erencesof
the variab les, w hic h are valid w henthese d i®erencesare uncorrelated w ith the ind ivid uale®ec ts. B lund elland B ond (19 9 7) show that the system estim ator hassuperior propertiesinterm sofsm allsam ple b iasand R M SE , espec ially for persistent series.
T he G M M estim ator is a tw o-step estim ator. Inthe ¯rst step, aninitial positive sem id e¯nite w eight m atrix isused to ob tainc onsistent estim atesofthe param eters.G iventhese c onsistent estim ates,a w eight m atrixc anb e c onstruc ted that isc onsistent for the e± c ient w eight m atrix, and thisw eight m atrix isused for the asym ptotic ally e± c ient tw o-step estim ates. It is w ellknow n, see e.g. Arellano and B ond (19 9 1), that the tw o-step estim ated stand ard errorshave a sm allsam ple d ow nw ard b iasinthisd ynam ic paneld ata setting, and one-step estim atesw ith rob ust stand ard errorsare oftenpreferred .Although ane± c ient w eight m atrix for the d i®erenced m od elw ith errorsthat are hom osced astic and that are not seriallyc orrelated iseasilyd erived ,thisisnot the c ase for the system estim ator,c omb iningd i®erencesand levelsinform ation.
It isc om m onprac tic e to use the inverse ofthe m om ent m atrixofthe instru-m entsasaninitialw eight instru-m atrix.Inthispaper the potentiale± c iency lossw ill b e c onsid ered ina m od elw ith hom ogeneousand non-serially c orrelated errors. T o d o this, upper b ound sfor the e± c iency lossw illb e c alculated asd erived b y Liuand Neud ec ker (19 97) b ased onthe K antorovic h Inequality(K I).T hese upper b ound sind ic ate that the e± c iency lossc ould potentially b e quite severe.W hen the variance ofthe ind ivid ualunob served heterogeneity issm all, e± c iency c an b e gained b y using a w eight m atrix that isoptim alund er the assum ptionthat the variance ofthe unob served heterogeneity isequalto z ero.
Insec tion2 ,anAR (1) d ynam ic paneld ata m od elisc onsid ered and a d escrip-tionofthe system G M M estim ator isgiven. Insec tion3 the upper b ound sof the e± c iency lossare c alculated for 3 and 4 tim e period srespec tively.Sec tion4 c onclud es.
2
M od elan
d System G M M
E stim ator
Consid er the AR (1) paneld ata spec i¯c ation
yit= ®0yit¡1+ ´i+ "it (1)
for i= 1;:::;N , t= 2 ;:::;T , w ith N large, and T ¯xed .T he error term sfollow the error c om ponentsstruc ture inw hic h
E (´i) = 0 ; E ("it) = 0 ; E³"2 it ´ = ¾2 " ; E ³ ´2 i ´ = ¾2 ´; E (´i"it) = 0 ; E ("it"is) = 0 ; t6= s:
T he yitseriesare assum ed stationary w ith anin¯nite tim e horiz onand therefore
the seriesc analternatively b e w rittenas yit= ´i 1¡®0 + 1 X j= 0 ®j0"it¡j: (2 )
T he O LS and w ithingroupsestim atorsof®0 inm od el(1) are b iased and
inconsistent. A c onsistent estim ator for ®0 isthe system G M M estim ator, as
proposed b y Arellano and B over (19 9 5) and B lund elland B ond (19 9 7),utilising the follow ing(T + 1)(T ¡2 )=2 moment c onditions1
E [(¢ yit¡®0¢ yit¡1)(yit¡2;:::;yi1)] = 0 (3)
E [(yit¡®0yit¡1)¢ yit¡1] = 0 ; (4 )
for t= 3;:::;T . M om ent c ond itions(3) are for the m od elin¯rst d i®erences, utilising appropriately lagged levelsinform ationasinstrum ents, w hereasc ond i-tions(4 ) are for the m od elinlevels, utilising lagged d i®erencesasinstrum ents. AsB lund elland B ond (19 97) show , the system estim ator isc onsid erab ly m ore
1U nderhomoscedasticity there are additionalmomentconditions available toimprove
e± c ient thanthe trad itionalG M M estim ator utilisingthe m om ent c ond itionsof the d i®erenced m od elonly.
De¯ne Z i= 2 6 6 6 66 6 6 6 6 66 6 6 4 yi1 0 0 ::: 0 ::: 0 0 0 ::: 0 0 yi1 yi2 ::: 0 ::: 0 0 0 ::: 0 0 0 0 ::: 0 ::: 0 0 0 ::: 0 0 0 0 ::: yi1 ::: yiT¡2 0 0 ::: 0 0 0 0 ::: 0 ::: 0 ¢ yi2 0 ::: 0 0 0 0 ::: 0 ::: 0 0 ¢ yi3 ::: 0 0 0 0 ::: 0 ::: 0 0 0 ::: 0 0 0 0 ::: 0 ::: 0 0 0 ::: ¢ yT¡1 3 7 7 7 77 7 7 7 7 77 7 7 5 vi= vi(®0) = 2 66 6 6 6 6 66 6 6 6 66 4 ¢ yi3¡®0¢ yi2 ¢ yi4 ¡®0¢ yi3 ::: ¢ yiT ¡®0¢ yiT¡1 yi3¡®0yi2 yi4 ¡®0yi3 ::: yiT ¡®0yiT¡1 3 77 7 7 7 7 77 7 7 7 77 5 ; and fi(®0) = Z i0vi:
M om ent c ond itions(3) and (4 ) im ply that E (fi(®0)) = 0 .T he G M M estim ator2
b ® for ®0 m inim ises " 1 N N X i= 1 fi(® ) #0 WN " 1 N N X i= 1 fi(® ) # ;
w ith respec t to ® ; w here WN is a positive sem id e¯nite w eight m atrix w hic h
satis¯esplimN! 1 WN = W , w ith W a positive d e¯nite m atrix.R egularity c
on-d itionsare inplac e suc h that limN ! 1 N1
PN
i= 1fi(® ) = E (f (® )) and p1N fi(®0)!
N (0 ;ª ):Let F (® ) = E (@fi(® )=@® ) and F0 ´F (®0),then
p
N (®b¡®0) hasa
lim itingnorm ald istrib ution,pN (®b¡®0)! N (0 ;VW),w here
VW = (F00W F0)¡1F00W ª W F0 (F00W F0)¡1: (5)
2SeeH ansen (1 982),O gaki (1 993).H ere,thesamenotation as inL iu andN eudecker(1 997)
3 E ± c ien
c y Com parison
s
Asisc lear from the expressionofthe asym ptotic variance m atrix VW, (5), the
e± c iency ofthe G M M estim ator isa®ec ted b y the c hoic e ofthe w eight m atrix WN .Anoptim alc hoic e isa w eight m atrixfor w hic h W = ª¡1.T he asym ptotic
variance m atrix is thengivenb y (F0
0ª¡1F0)¡1. For any other W the G M M
estim ator islesse± c ient as
³
F00ª¡1F0
´¡1
·(F00W F0)¡1F00W ª W F0 (F00W F0)¡1:
Inpaneld ata m od elsthe e± c ient estim ator isob tained ina tw o-step pro-c ed ure. T he one-step G M M estim ator ® isob tained using anarb itrary w eighte
m atrix WN 1.Let vei= vi(® ).T he e± c ient tw o-step estim ator isthenb ased one
the w eight m atrix WN 2 = à 1 N N X i= 1 Z i0veiveiZ i !¡1 ; plim WN 2 = ª¡1.
Although the e± c ient estim ator iseasily ob tained ,there isa seriousprob lem assoc iated w ith it as the estim ated stand ard errors ofthe tw o-step estim ator c anb e b iased d ow nw ard s quite severely for m od erate sam ple siz es N , as has b eend oc um ented b y Arellano and B ond (19 9 1) and B lund elland B ond (199 7), w ho perform ed M onte Carlo simulationsw ith sam ple siz esN = 2 0 0 .T herefore, inference b ased onthe tw o-step estim ator c anb e very unreliab le. Inc ontrast, the one-step estim ated stand ard errorsb ased onthe asym ptotic variance m atrix (5), using WN 2 asanestim ate for ª and sub stituting® fe or ®0, are found to b e
much lessb iased ,and inference,like W ald tests,muc h m ore reliab le.Inprac tic e therefore,one c anoftenonly rely oninference b ased onthe lesse± c ient one-step estim ator.
For the G M M estim ator that only utilisesthe m om ent c ond itions(3) for the d i®erenced m od el, anoptim alw eight m atrix is³N1 PNi= 1Di0H Di
´¡1
, w here Diis
m aind iagonal,-1'sonthe ¯rst sub d iagonalsand z eroselsew here.SettingWN 1 = ³ 1 N PN i= 1D 0 iH Di ´¡1
resultstherefore inane± c ient one-step estim ator. For the system G M M estim ator suc h ane± c ient one-step w eight m atrix isnot know n, and inprac tic e one usesasaninitialw eight m atrixWN 1 =
³ 1 N PN i= 1Z i0Z i ´¡1 .T o assessthe potentiallossine± c iency from using thisinitialw eight m atrix, the follow ing expressionfor the upper b ound ofthe e± c iency losshasb eend erived b y Liu and Neud ec ker (19 9 7, p.350 ) onthe b asisofthe K antorovic h Inequality (K I): (F00W F0)¡1F00W ª W F0 (F00W F0)¡1 · (¸1+ ¸p)2 4 ¸1¸p ³ F00ª¡1F0 ´¡1 (6) w here ¸1 ¸:::¸¸p are the eigenvaluesofthe m atrixª W .
For T = 3, there is one overid entifying m om ent c ond ition, as the system estim ator utilisesthe follow ingtw o m om ent c ond itions
E [(¢ yi3¡® ¢ yi2)yi1] = 0 E [(yi3¡® yi2)¢ yi2] = 0 ; and ª isgivenb y ª = plim 1 N N X i= 1 " y2 1(¢ "i3)2 yi1¢ yi2 (´i+ "i3)¢ "i3 yi1¢ yi2(´i+ "i3)¢ "i3 (¢ yi2)2 (´i+ "i3)2 # = ¾"2 " 2 ¾2 y ¡(1 ¡® )¾y2 ¡(1 ¡® )¾2 y 2 ¾2 ´+ ¾2" 1+ ® # w here ¾y2 = ¾ 2 ´ (1¡® )2 + ¾2 " 1¡®2: Further, W1 = Ã plim 1 N N X i= 1 Z i0Z i !¡1 = Ã plim 1 N N X i= 1 " y2 i1 0 0 (¢ yi2)2 # !¡1 = " ¾2 y 0 0 2 ¾2" 1+ ® #¡1
and the m atrixG = ª W1 isgivenb y G = " 2 ¾2 " ¡(1 ¡®2)¾y2=2 ¡(1 ¡® )¾2 " ¾2"+ ¾´2 # :
F igure 1 presentsthe plot ofthe functionbK I = (¸G 1+ ¸G p)2 =4 ¸G 1¸G p,w here
the ¸G'sare the eigenvaluesofG , for variousvaluesof®0 and ¾´2=¾2". W hen
¾2
´ = ¾2", bK I isc onstant for d i®erent valuesof® and equalto 4 /3, ind ic ating
that the asym ptotic variance ofthe one-step estim ator c ould potentially b e 33% larger thanthe e± c ient estim ator. W hen¾2
´=¾"2 < 1, bK I isd ec lining w ith ®0,
w hereasit isincreasingw ith ®0 w hen¾´2=¾"2 > 1.T he value ofbK I increasesw ith
¾2
´=¾"2 w hen¾´2=¾"2 > 1.
[F igure 1 ab out here]
W henT = 4 ,there are 4 overid entifyingm om ent c ond itions,and the m atric es ª and W1 are givenb y
ª = ¾2" 2 6 6 66 6 6 6 6 4 2 ¾2 y ¡¾y2 ¡± ¡(1 ¡® )¾2y 21¡®¡®¾ 2 ´ ¡¾2 y 2 ¾2y 2 ± ¾2 " 1+ ® ¡(1 ¡® )± ¡± 2 ± 2 ¾2 y ¡ ¾2 " 1+ ® ¡(1 ¡® )¾y2 ¡(1 ¡® )¾2 y ¾ 2 " 1+ ® ¡ ¾2 " 1+ ® 2 ¾2 ´+ ¾2" 1+ ® ¡1¡®1+ ®¾ 2 ´ 2¡® 1¡®¾2´ ¡(1 ¡® )± ¡(1 ¡® )¾2y ¡11+ ®¡®¾´2 2 ¾2 ´+ ¾2" 1+ ® 3 7 7 77 7 7 7 7 5 W1 = 2 6 6 6 6 6 66 4 ¾2 y 0 0 0 0 0 ¾2 y ± 0 0 0 ± ¾2 y 0 0 0 0 0 2 ¾2" 1+ ® 0 0 0 0 0 2 ¾2" 1+ ® 3 7 7 7 7 7 77 5 ; w here ± = ¾y2 ¡ ¾2 " 1 + ®:
F igure 2 presents the e± c iency b ound s for the one-step system estim ator w henT = 4 . T he valuesfor bK I are larger thanfor the T = 3 c ase. W hen
¾2
´=¾"2 = 1, bK I isno longer c onstant for d i®erent valuesof®0, and takesvalues
b e 3tim esthe variance ofthe e± c ient estim ator.Again,bK I increasesw ith ¾´2=¾"2
w hen¾2
´=¾"2 > 1,and bK I reac hesthe value 6 w hen¾´2=¾"2 = 2 :5.3
[F igure 2 ab out here]
3.
1
AnO ptim alW eight M atrixw hen¾
´2= 0
Anoptim alw eight m atrix for the system G M M estim ator w hen¾2
´ = 0 isgiven b y WN ;¾2 ´= 0 = Ã 1 N N X i= 1 Z i0AZ i !¡1 ; w here A = " H C C0 I T¡2 #
w ith H asd e¯ned ab ove,IT¡2 isthe id entity m atrixoford er (T ¡2 ),and
C = 2 66 6 6 6 6 4 1 0 0 0 ¡1 1 0 0 0 ¡1 1 0 ::: ::: ::: ::: : 0 0 0 ::: 1 3 77 7 7 7 7 5 :
U sing thisw eight m atrix instead ofWN 1 =
³ 1 N PN i= 1Z i0Z i ´¡1 m ay im prove on e± c iency w hen¾2
´ issm all. F igures3 and 4 d isplay the valuesfor bK I w hen
WN ;¾2
´= 0 isused inthe one-step estim ator, for T = 3 and T = 4 respec tively.
Ind eed ,for sm allvaluesof¾2
´ the potentiallossine± c iency isseento b e sm aller
thanw henWN 1 isused .How ever, w hen¾´2=¾2" islarge, the potentiale± c iency
lossgetslarger for WN ;¾2
´= 0,w hic h isw hat one w ould expec t.
[F igures3 and 4 ab out here] O ne w ayto d etec t w hether use ofWN ;¾2
´= 0 c ould b e b ene¯c ial,w ithout ac tually
c alculatingthe variancesofthe c om ponents,isto c alculate the e± c iency b ound s
3T he value ofb
KIincreases with T as the numberofmomentrestrictions increases. For
examplewhen T = 6 and ¾2
bK I for the e± c iency d i®erence b etw eenthe one-step and tw o-step estim ators,
i.e.c alculate bK I from eigenvaluesofthe m atric esWN 2WN ;¾2
´= 0, and WN 2WN 1.
Ifthe form er are c loser to 1 thanthe latter,thisisanind ic ationthat there c ould b e ane± c iency gainfrom usingWN ;¾2
´= 0 instead ofWN 1.
4
Discussion
U pper b ound sfor the e± c iency lossofthe one-step system G M M estim ator for a d ynam ic AR (1) paneld ata m od elasc om pared to the e± c ient tw o-step estim ator, show that the e± c iencylossc ould b e quite severe w henthe w eight m atrixWN 1 =
³ 1 N PN i= 1Z i0Z i ´¡1
isused ,espec ially w henT getslarge.W henthe variance ofthe unob served ind ivid uale®ec ts, ¾2
´, issm all, ane± c iency gainc anb e m ad e b y
usinga w eight m atrixthat isoptim alund er the assum ptionthat ¾2 ´ = 0 .
T he upper b ound sw ere show nto b e quite large, for exam ple w henT = 4 , WN 1 = ³ 1 N PN i= 1Z i0Z i ´¡1 and ¾2
´=¾2" = 2 :5, the ratio ofthe asym ptotic variance
ofthe ine± c ient estim ator to that ofthe e± c ient estim ator c anb e aslarge as 6 for high valuesof®0. InM onte Carlo stud ieshow ever, suc h large d i®erences
ofthe variancesare not found , using norm aland non-norm ald ata generating proc esses.T hisc ould m eanthat the K Iupper b ound s, bK I, are too large to b e
inform ative for these c ases.W henthe bK I are c lose to one, there isevid ence of
ane± c ient one-step estim ator. T he opposite statem ent for large bK I m ay not
b e true.Further researc h isneed ed to assessw hether the bK I are inform ative to
rank d i®erent one-step estim atorsonthe b asisoftheir relative K I-values. Inem piric alsettings,one c aneasily c om pute bK I from the eigenvaluesofthe
m atrixWN 2WN 1,w here WN 2 isthe tw o-step e± c ient w eight m atrix.
Ac know led gm ents
Iw ould like to thank Steve B ond for helpfulc om m entsand for d raw ing my attentionto the use ofWN ;¾2
R ef
eren
c es
[1] Ahn,S.C.and P .Sc hm id t (19 95),E ± c ient E stim ationofM od elsfor Dynam ic P anelData,J ournalofE conometric s,68,5-2 8.
[2 ] Arellano,M .and S.B ond (19 91),Som e T estsofSpec i¯c ationfor P anelData: M onte Carlo E vid ence and anApplic ationto E m ploym ent E quations,R eview ofE conomic Stud ies,58,2 77-9 8.
[3] Arellano,M .and O .B over (199 5),Another Lookat the Instrum ental-Variab le E stim ationofE rror-Com ponentsM od els,J ournalofE conometric s,68,2 9 -51. [4 ] B lund ell,R .and S.B ond (19 97),InitialCond itionsand M om ent R estric tions
inDynam ic P anelData M od els,J ournalofE conom etric s,inpress.
[5] Hansen,L.P .(19 82 ) `Large Sam ple P ropertiesofG eneraliz ed M ethod ofM o-m entsE stio-m ators',E conoo-metrica,50 ,10 2 9-10 54 .
[6] Liu, S.and H.Neud ec ker (19 9 7), K antorovic h Inequalities and E ± c iency Com parisonsfor SeveralClassesofE stim atorsinLinear M od els, Statistica Neerlandica,51,34 5-355.
[7] O gaki, M .(199 3) `G eneraliz ed M ethod ofM om ents: E c onom etric Applic a-tions', in: M ad d ala, G .S., C.R .R ao, and H.D.Vinod (ed s.), `E c onom etric s', HandbookofStatistic s,11,North-Holland ,Am sterd am .
F igure 1: K IE ± c iency B ound s,T = 3;WN 1 = ³1 N P iZ i0Z i ´¡1
F igure 2 : K IE ± c iency B ound s,T = 4 ;WN 1 =
³ 1 N P iZ i0Z i ´¡1
F igure 3: K IE ± c iency B ound s,T = 3;WN 1= WN ;¾2 ´= 0
F igure 4 : K IE ± c iency B ound s,T = 4 ;WN 1= WN ;¾2 ´= 0
