ysem GMM esmaon wh a small sample Marcelo oo July 15, 9 Barcelona Economcs Workng Paper eres Workng Paper nº 395
YTEM GMM ETIMATION WITH A MALL AMPLE Marcelo oo July 9 Properes of GMM esmaors for panel aa, whch have become very popular n he emprcal economc growh leraure, are no well known when he number of nvuals s small. Ths paper analyses hrough Mone Carlo smulaons he properes of varous GMM an oher esmaors when he number of nvuals s he one ypcally avalable n counry growh sues. I s foun ha, prove ha some perssency s presen n he seres, he sysem GMM esmaor has a lower bas an hgher effcency han all he oher esmaors analyse, nclung he sanar frs-fferences GMM esmaor. Keywors: Economc Growh, ysem GMM esmaon, Mone Carlo mulaons JEL classfcaon: C15, C33, O11 Insu Anàls Econòmca, Barcelona. Emal: marcelo.soo@ae.csc.es. I am graeful o Rchar Blunell, Frank Wnmejer an parcpans n he Economerc ocey meengs n Mexco DF an Wellngron for commens an helpful suggesons. The suppor from he pansh Mnsry of cence an Innovaon uner projec ECO8-4837/ECON s graefully acknowlege. The auhor acknowleges he suppor of he Barcelona GE Research Nework an of he Governmen of Caalona.
1. Inroucon The evelopmen an applcaon of Generalse Mehos of Momens (GMM) esmaon for panel aa has been exremely fruful n he las ecae. For nsance, Arellano an Bon (1991), who poneere he apple GMM esmaon for panel aa, have more han 1, caons accorng o II Web of Knowlege as of July 9. In he emprcal growh leraure, GMM esmaon has become parcularly popular. The Arellano an Bon (1991) esmaor n parcular nally benefe from wesprea use n fferen opcs relae o growh 1. ubsequenly he relae Blunell an Bon (1998) esmaor has gane an even graer aenon n he emprcal growh leraure. However, hese GMM esmaors were esgne n he conex of labour an nusral sues. In such sues he number of nvuals N s large, whereas he ypcal number of cross-uns n economc growh samples s much smaller. Inee, avalably of counry aa lms N o a mos 1 an ofen o less han half ha value. The lack of knowlege abou he properes of GMM esmaors when N s small reners hem a sor of a black box. Moreover, a praccal problem no aresse n he earler leraure refers o fac ha he low number of cross-uns may preven he use of he full se of nsrumens avalable. Ths mples ha, n orer o make esmaon possble, he number of nsrumens mus be reuce. The performance of he varous GMM esmaors n panel aa s no well known when only a paral se of nsrumens s use for esmaon. Ths paper analyses hrough Mone Carlo smulaons he performance of he sysem GMM an oher sanar esmaors when he number of nvuals s small. The smulaons follow closely hose mae by Blunell e al () n he sense ha he srucure of he moel smulae s exacly he same as hers. The only fference s ha Blunell e al. chose N=5, whle hs paper repors resuls for N more aape o he acual sample sze of growh regressons n a panel of counres (N=1, 5, 35). A small N consrans he researcher o lm he number of nsrumens use for esmaon, whch may also have a consequence on he properes of he esmaors. The paper sues he behavour of he esmaors for fferen choces on he nsrumens. 1 For nsance Casell e al (1996) use o es he olow moel; Greeanway e al () for analysng he mpac of rae lberalsaon n evelopng counres; an Banerjee an Duflo (3) o nvesgae he effec of ncome nequaly on growh. ome examples are sues on a an growh (Dalgaar e al, 4); eucaon an growh (Cohen an oo, 7); an exchange rae volaly an growh (Aghon e al, 9).
The nex secon epcs he economerc moel uner conseraon. econ 3 presens he esmaon resuls obane by Mone Carlo smulaons. econ 4 conclues.. The economerc moel We wll conser an auoregressve moel wh one aonal regressor: y αy 1 βx η u (1) for = 1,, N an =,, T, wh α 1.The surbances η an u have he sanar properes. Tha s, η, Eu, Eη u E for = 1,, N an =,, T. () Aonally, he me-varyng errors are assume uncorrelae: E u s u for = 1,, N an s. (3) Noe ha no conon s mpose on he varance of u, hence he momen conons use below o no requre homoskeascy. The varable x s also assume o follow an auoregressve process: x x 1 τη θu e (4) for = 1,, N an =,, T, wh 1. The properes of he surbance e are analogous o hose of u. More precsely, e, Eη e E for = 1,, N an =,, T. (5) Two sources of enogeney are presen n he x process. Frs, he fxe-effec componen has an effec on x hrough a parameer mplyng ha y an x have boh a seay sae eermne only by. An secon, he me-varyng surbance u mpacs x wh a parameer. A suaon n whch he aenuaon bas ue o measuremen error preomnaes over he upwar bas ue o smulaney eermnaon may be smulae wh <. For smplcy, s useful o express x an y as evaons from her seay sae values. Uner he aonal hypohess ha (4) s a val represenaon of x for = 1,,, x 3
may be wren as a evaon from s seay sae: x τη 1 ξ (6) where he evaon from seay sae ξ s equal o ξ 1 1 L θu e. In hs las expresson L s he lag operaor an so, for any varable w an parameer, (1 L) 1 w s efne as 1 (1 λl) w w λw λ -1 w -... mlarly, assumng ha (1) s a val represenaon of y for = 1,,, we have, y η 1 α β(1 αl) 1 x (1 αl) 1 u Afer subsung n hs las expresson x by (6), y may be wren as y 1 βτ 1 α1 η ζ (7) wh s evaon from seay sae gven by, ζ 1 θu e (1 αl) u 1 1 β(1 αl) (1 L). Hence, he evaon from he seay sae s he sum of wo nepenen AR() processes an one AR(1) process. 3. Mone Carlo smulaons Ths secon repors Mone Carlo smulaons for he moel escrbe n (1) o (5) an analyses he performance of fferen esmaors. To summarse, he moel specfcaon s: y αy βx η u (8) 1 4
x x τη θu e (9) 1 wh η ~ (; σ η ); u ~ N(; σ u ); e ~ N(; σ N. e ) We wll conser hree fferen cases for he auoregressve processes: no perssency (= = ), moerae perssency (= =.5) an hgh perssency (= =.95). The oher parameers are kep fxe n each smulaon as follows 3 : σ η 1; σ 1; σ.16 = 1; =.5; =.1;. u e The parameer s negave n orer o emulae he effecs of measuremen error n x 4. The hypohess of homoskeascy s roppe n subsequen smulaons. Inally, he sample sze consere s N = 1 an T = 5. In laer smulaons N s se a 5 an 35 wh T=1, n orer o llusrae he effecs of a low number of nvuals (relave o T). Each resul presene below s base on a fferen se of 1 replcaons, wh new nal observaons generae for each replcaon. The appenx A explans wh more eals he generaon of nal observaons. The esmaors analyse are OL, fxe-effecs, fference GMM, level GMM an sysem GMM. One an wo-sep resuls are repore for each GMM esmaon. All he esmaons are performe wh he program DPD for Gauss (Arellano an Bon, 1998). 3.1 Accuracy an effcency resuls The man fnng s ha, prove ha some perssency s presen n he seres, he sysem GMM esmaor yels he resuls wh he lowes bas. Conser Table 1, whch presens resuls for N=1 an T=5. The performance of each esmaor vares accorng o he egree of perssency n he seres. For nsance, when an are boh equal o zero, OL esmaes wrongly assgn a hghly sgnfcan coeffcen o he lagge epenen varable, whereas he Whn esmaor proves a negave an sgnfcan coeffcen 5. However, 3 These values are he same as hose selece by Blunell e al () 4 Hauk an Waczarg (9) carry ou Mone Carlo smulaons for he convergence equaon erve from he olow moel by recly nroucng nose n he varables. 5 Recall ha he OL coeffcen on y 1 s base owars 1 an he Whn groups coeffcen s base ownwars (wh a bas ecreasng wh T). 5
OL proves esmaes for wh he lowes roo mean square error (RME) n he noperssency case 6. The hgh RME on splaye by all GMM esmaes s a consequence of he weakness of he nsrumens for x scusse n appenx B when =. In he moerae perssency case ( an equal o.5), he OL esmaor has agan a srong upwars bas for an a ownwars bas for. The Whn esmaor s srongly base ownwars n boh cases. The fference GMM esmaor resuls n coeffcens beween 6% an 7% he real parameer values an presens he hghes RME for. Ths shows ha lagge levels are weak nsrumens for varables n fferences even n a moerae-perssency envronmen. As o he level an sysem GMM esmaes, hey splay sysemacally he lowes bas for boh an. In aon, hese esmaors resul n he lowes RME for. In he hgh perssency case ( an equal o.95), he sysem GMM esmaor ouperform all he oher esmaors n erms of bas an effcency. Noe however ha he bas n he lagge epenen varable of he OL esmaor s conserably reuce. Ths s ue o he fac ha hs coeffcen s base owars 1. One well known cavea of GMM esmaors refers o her repore wo-sep sanar errors, whch sysemacally uneresmae he real sanar evaon of he esmaes (Blunell e al, ). For nsance, sanar errors of sysem GMM are 6% o 74% lower han he sanar evaon of he esmaes of an 7% o 83% lower n he case of. Ths resul suggess akng he one-sep esmaes for nference purposes, snce accuracy an effcency (measure by he RME) are smlar o hose of he wo-seps. The varance correcon suggese by Wnmejer (5) s mplemene n he curren smulaons. The nex sep s o replcae he Mone Carlo expermens by changng he sample szes. Resuls are now obane by seng N = 5 an T = 1. The reucon of N relave o T preclues he use of he full se of nsrumens erve from conons (1) an (1). Inee, f all hose momen conons were exploe he number of nsrumens woul be (T)(T+1), whch excees N. On he oher han, he opmal weghng marx W efne n (1) has a rank of N a mos. Therefore, f he number of nsrumens excees N, W s sngular an he wo-sep esmaor canno be compue. In orer o make esmaon possble only he mos relevan (.e. he mos recen) nsrumens are use n R 6 The RME on s efne as β j β 1 where R s he oal number of replcaons. R j1 6
each pero. Tha means ha only levels lagge wo peros are use for he equaon n fferences an, as before, fferences lagge one pero are use for he equaon n levels. Ths proceure resuls n 4(T) nsrumens. The resuls are presene n Table. The man conclusons are he same as hose obane from Table 1. Tha s, n he smulaons whou perssency, all he esmaors perform baly. The OL an fxe-effec esmaors presen conserable bases an, alhough he sysem GMM esmaor splays a relavely low bas, has a hgh RME for. As o he moerae an hgh-perssency cases, he sysem GMM oes beer han any oher esmaor overall. ll, he moeraeperssency esmaon suffers from a small sample upwars bas of % for an of 14% for. The nex se of resuls s obane by sranng even more he sample sze, wh N = 35 an T = 1. From he prevous scusson becomes apparen ha he sysem GMM esmaon wh he se of nsrumens use n he prevous smulaons s no feasble snce he number of nsrumens 4 (1)=4 excees he number of nvuals. everal alernaves were consere for reucng he number of nsrumens. Frs, lagge levels of x were ome from he nsrumen se for he equaon n fferences an Z l was kep as before. econ, lagge fferences of x were ome from he nsrumen se for he equaon n levels an Z was kep as before. An hr, Z l was kep as before an Z was mofe as follows, y 1 y 1 T T Z (1) y x x x Uner hs las alernave he oal number of nsrumens s 3 (T)+1=31. Alhough all hree alernaves prove smlar resuls n erms of bas, he hr alernave resule n he lowes RME error n he hgh-perssency case. Table 3 compares he fferen esmaors, wh he nsrumens for he sysem GMM esmaor efne n (1) for he equaon n fferences an n (1) for he equaon n levels. In general he RME are hgher han n he prevous smulaons ue o he smaller sample sze. In aon, he upwar bas for obane by sysem GMM s hgher han before. Bu overall, hs esmaor ouperforms n erms of accuracy an effcency once agan. The las case uner conseraon s when errors are heeroskeasc across nvuals. The 7
resuls presene n ables 1 o 3 are base on resuals wh varances σ u 1 an σ e σ.16. Now heeroskeascy s nrouce by generang resuals u an e such ha an. Ths parcular srucure mples ha he u ~ U(.5 ; 1.5) e σ.16σ expece varances of u an e are he same as n he prevous smulaons an ha he u rao σ / σ u e s consan, hus makng easer he comparson wh he resuls prevously shown. Table 4 repors he smulaon resuls wh N = 35 an T =1. The nsrumens use correspon o hose of Table 3. The man effec of heeroskeascy s o slghly ncrease he RME of n he hgh-perssency case. ll, he level an sysem GMM esmaors splay boh he lowes fne-sample bas an he lowes RME. Fgure 1 presens he srbuon of he esmaes obane wh OL, an one an wosep sysem GMM. The srbuons correspon o he sample wh N = 35, T = 1 an wh heeroskeasc resuals across nvuals. The vercal lne correspons o he parameer values. The fgure shows ha he srbuons of he one an wo-sep sysem GMM are more concenrae aroun parameer han he srbuon obane from OL. However, he GMM esmaors are sysemacally base upwars, hough he bas s conserably lower han he one presen n OL. Regarng, he srbuons of GMM esmaes n he no-perssency case hough cenre on he rgh value have very fa queues. The OL esmaor performs beer n hs parcular case. In he cases wh moerae an hgh perssency he sperson of GMM srbuon s conserably reuce an s bas s sysemacally lower han OLs. One srkng feaure of GMM esmaors s ha he gan n effcency from he wo-sep esmaor s almos nexsen: he one an wo sep srbuons are vrually he same. More work shoul be one n orer fn ou he cases n whch he wo-sep esmaor oes beer han he one-sep esmaor. 3. Type-I error an power of sgnfcance ess. Anoher aspec n whch he varous esmaors can be evaluae s he frequency of wrong rejecons of he hypohess ha s no sgnfcan when s n fac equal o zero (.e. ype-i error) an he power o properly rejec lack of sgnfcance when coeffcens are fferen from zero. Table 5 repors he frequency rejecons a a 5% level of he hypohess ha = an = for he fferen smulaons escrbe above. One srkng feaure whch came ou alreay from fgure 1 s ha he OL esmaor 8
always rejecs lack of sgnfcance, even n he case when = (see he no-perssency case). Ths s an aonal flagran mplcaon of he upwar bas of OL esmaes. A smlar phenomenon occurs wh he Whn esmaor, whch fals o scar sgnfcance of n 43% o 99.5% of he smulaons wh =. As menone before, he sanar error of wo-sep GMM esmaors uneresmae he real varably of he coeffcens. A consequence of hs s he relavely hgh number of wrong rejecons of non-sgnfcance of n he smulaons wh = n wo-sep GMM esmaes (up o 69% n he sysem GMM esmaor n he smulaon wh heeroskeascy). The lowes ype-i errors correspon o one-sep GMM esmaors, alhough hey also en o over-rejec as N becomes smaller. The weakness of he fference GMM esmaor s reflece n s low power o rejec nonsgnfcance when parameers are n fac fferen from zero. For nsance, n he hghperssency case he one-sep fference GMM esmaor rejecs non-sgnfcance of n only 4% o 3% of he smulaons ha s, wrongly smsses he sgnfcance of n 7% o 96% of he smulaons. The sysem esmaor s he mos powerful among GMM esmaors, wh s power ncreasng as seres become more perssen. For nsance, accorng o one-sep esmaes n he heeroskeasc case, he non-sgnfcance of was rejece n 56% of he smulaons whou perssency, 9% of smulaons wh moerae perssency, an 1% of smulaons wh hgh perssency. Overall, he one-sep sysem GMM s he more relable esmaor n erms of power an error ype-i. Among all he esmaors presene n he able, he OL esmaor has he hghes power n absolue erms ( never rejece sgnfcance n he smulaons). Bu he counerpar of hs s ha nference base on OL esmaes s a poor gue when he ecson of rejecng a poenal non-sgnfcan bu enogenous varable comes up. 3.3 Overenfyng resrcons ess One crucal feaure of nsrumenal varables s her exogeney. Frequency rejecons of overenfyng resrcon ess n whch he null hypohess s ha nsrumens are uncorrelae wh u are presene n Table 6. By consrucon, he nsrumens use for esmaon are all exogenous, so one woul expec ha a a 5% level, exogeney woul be rejece n 5% of he smulaons. In samples wh N = 1 an T = 5 here s a slgh enency owars uner-rejec exogeney n he wo exreme cases of perssency. Bu n smulaons wh smaller N an larger T he uner-rejecon s much more accenuae. In fac, he sysem GMM esmaon resuls n overenfyng resrcon ess ha (properly) 9
never rejec exogeney. However, he fac ha he frequency of rejecons s lower han her expece value, suggess ha small sample bas s affecng he ess. In orer o unersan beer he consequences of hs bas, smulaons wh auocorrelae resuals shoul be mae. When resuals u are auocorrelae lagge levels or fferences of he regressors woul be correlae wh he u, hence hey coul no be use as nsrumens. Ths kn of smulaons was no performe n hs paper. 4. Conclusons Ths paper has analyse he properes of recenly evelope GMM an oher sanar esmaors, obane wh Mone Carlo smulaons. Alhough earler sues by Blunell an Bon (1998) an Blunell e al () have alreay shown he superory of he sysem GMM esmaor over oher esmaors, he valy of her resuls when he number of nvuals s small are largely unknown. Unersanng beer he properes of hese esmaors when N s small s mporan gven he populary ha hs meho of esmaon s geng n emprcal growh sues. A small number of nvuals ha s, he number of counres ypcally avalable n panel sues oes no seem o have mporan effecs on he properes prevously oulne abou he sysem GMM esmaor. Namely, when seres are moeraely or hghly perssen, hs esmaor presens he lowes bas an hghes precson. As expece, he OL esmaor has he lowes varance, bu he gan n erms of accuracy ha he sysem GMM esmaor presens, makes a more relable ool n he praccal work. Oher wely use esmaors he fxe effec an he fference GMM are sysemacally ouperforme by he sysem GMM esmaor. Moreover, he properes of hs esmaor are no hnere when N s so small ha s no possble o explo he full se of lnear momen conons. Ths s also an mporan fnng snce he earler Mone Carlo smulaons were carre ou by usng he full se of nsrumens, whereas n praccal work hs may no be always feasble, especally n he conex of counry growh sues. Overall he sysem GMM esmaor splays he bes feaures n erms of small sample bas an precson. Ths, ogeher wh s smple mplemenaon conver no a powerful ool n apple economercs. The nex sep s o conras he resuls prove by hs new esmaor n apple economercs wh hose obane by he sanar esmaors. 1
Appenx A: Consrucon of nal observaons In orer o reuce runcaon error, whch s may be mporan parcularly n smulaons wh hgh perssency, nal observaons are bul as follows (see Kve (1995) for a more general scusson on he generaon of nal observaons). Frs, les om he nex an rewre (6) as: x τη 1 ξ (A.1) where ξ represens he evaon of x from s seay sae an s gven by: ξ 1 L θp q 1 θu e (A.) o ξ s he sum of wo nepenen AR(1) processes, p an q, wh varances σ p σ u an 1- σ q σ e. 1- Base on nepenen ranom varables, u 1 an e 1, s possble o generae nal saonary-conssen observaons x 1 as follows: τη 1. 5 θu 1 x 1 1 e1. mlarly, rewrng equaon (7) where y s represene as a evaon from seay sae, we oban: y 1 βτ 1 α1 η. (A.3) ζ The evaon from seay sae ζ s gven by, ζ 1 1 1 θu e (1 αl) u β(1 αl) (1 L). (A.4) 11
Accorng o he expresson (A.4) ζ s a sum of wo nepenen AR() processes an one AR(1) process, an so can be expresse as: ζ βθr βs v (A.5) where r s v r r u, 1 1-1 -1 - s s e an αv -1 u - The auoregressve parameers are 1 = + an =. The varances of each one of hese processes are gven by: σ σ σ 1 1 r σu 1 1 1 s σe 1 v σ u 1 α Wh hese varances a han s possble o generae for each nvual nal observaons of r 1, s 1 an v 1 ha are conssen wh mean an varance saonary. Then nal observaons of y 1 are generae base on equaons (A.3) an (A.5) as follows: y 1 βτ 1 α1 1 1 1 η βθr1 βs1 v1 nce hs proceure oes no ensure covarance saonary of nal observaons, each seres y was generae for 5 + T peros an he frs 5 peros were elee for esmaon. Appenx B: The GMM esmaor Ths appenx escrbes he fference an sysem GMM esmaors an he problem of weak nsrumens. Much of he maeral of hs appenx s base on Arellano an Bon (1991) an Blunell an Bon (1998). B.1 Frs-fference GMM esmaor The sanar Arellano an Bon (1991) esmaor consss n akng equaon (1) n frs 1
fferences an hen usng y - an x - for = 3,, T as nsrumens for changes n pero. The exogeney of hese nsrumens s a consequence of he assume absence of seral correlaon n he surbances u. Namely, here are z = (T1)(T) momen conons mple by he moel ha may be exploe o oban z fferen nsrumens. The momen conons are: E y Δu an x Δu s E s (B.1) for = 3,, T an s =,, 1, where u = u u -1. Thus for each nvual, resrcons (B.1) may be wren compacly as, Z u Δ E (B.) where Z s he (T )z marx gven by, y 1 x 1 y y x x 1 1 Z (B.3) y 1... y x... x T 1 T an u s he (T )1 vecor (u 3, u 4,,u T ). eng he marx Z = ( Z 1, Z,, Z N ), he marx X forme by he sacke marces X = ( (y x 3 ), (y 3 x 4 ),, (y T-1 x T )) an he vecor Y forme by he sacke vecors Y = ( y 3, y 3,, y T ), he GMM esmaon of B = (, ) base on he momen conons (B.) s gven by, B 1 1 ΔX Z W Z ΔX ΔX Z W 1 Z ΔY (B.4) where W s some z z posve efne marx. From equaon (B.4) can be seen ha he sanar nsrumenal varable esmaor wh nsrumens gven by (B.3) s a parcular case of he GMM esmaor. Inee, he sanar IV esmaor s obane by leng W = Z Z. Hansen (198) shows ha he marx W yelng he opmal (.e., mnmum varance) GMM esmaor base only n momen conons (B.) s, 13
N Δu Δu Z (B.5) 1 W Z nce he acual vecors of errors u are unknown, a frs sep esmaon s neee n orer o make Hansens esmaor operaonal. Alhough no knowlege s requre abou he varance of u, Arellano an Bon (1991) sugges akng no accoun he varance srucure of he fference error erm u ha woul resul uner he assumpon of homoskeascy. In ha case, E(u u ) = u G where G s he (T)(T) marx gven by, 1 G 1 1 1 1 1 Therefore, he one-sep Arellano-Bon esmaor s obane by usng, N GZ (B.6) 1 W Z Then, he wo-sep esmaor s obane by subsungu n (B.5) by he resuals from he one-sep esmaon. B. Weak nsrumens Blunell an Bon (1998) show ha he frs-fference GMM esmaor for a purely auoregressve moel.e. whou aonal regressors has a large fne sample bas an low precson n wo cases. Frs, when he auoregressve parameer ens o uny an secon, when he varance of he specfc-effec ncreases wh respec o he varance of u. In boh cases lagge levels of he epenen varable become weaker nsrumens snce hey are less correlae wh subsequen changes. To see hs clearly, Blunell an Bon conser he smple case of T = 3. In hs case only one observaon per nvual s avalable for esmaon, whch leas o he followng sngle nsrumenal varable equaon: 14
Δy (α 1)y 1 η u, for = 1,, N. (B.7) The leas-squares esmaon of ( 1) n equaon (B.7) yels he followng coeffcen, N Δy y 1 πˆ, N y whch has a probably lm equal o, 1 k Plm πˆ (α 1) k σ η σ, where u 1 α k. 1 α No surprsngly he probably lm of πˆ ens owars zero as y approaches o a ranom walk process. Also, ue o he posve correlaon beween y 1 an, πˆ ens η owars zero as σ ncreases relave o σ. u A smlar resul s obane when aonal regressors are nclue as n he moel (1)-(5). To show hs keepng racably, conser agan he smple case wh T = 3 an wh x 1 beng he only lagge varable use as an nsrumen. In ha case he sngle nsrumenal varable equaon for x 3 s Δx 3 ( 1)x τη υ (B.8) 1 3 for = 1,, N an where υ θu e 1θu e esmae for ( 1) s gven by, N Δx3x1 πˆ, N x whch has a probably lm equal o, 1 3 3 3. The leas square 15
( 1) Plm πˆ 1 γ, where γ τ ση. θ σu σe As n he Blunell-Bon example of a purely auoregressve equaon, lagge levels of an aonal regressor are weak nsrumens as ens o 1 an when he varance of he specfc effec componen of x 1, τ σ η, s large relave o he varance of s emporal surbance, θ σ u σ e. The only fference wh he purely auoregressve case s ha when he auoregressve parameer ens o zero, x 1 s also a weak nsrumen for x 3. Ths s he resul of usng levels of varables lagge wo peros as nsrumens for conemporary changes. As ager an ock (1997) show, when nsrumens are weakly correlae wh he regressors, nsrumenal varable mehos have a srong bas an he sanar errors uneresmae he real varably of he esmaors. Therefore, n he conex of growh regressons where even conserng a me ren varables lke he ncome level or human an physcal capal socks have a srong auoregressve componen, he sanar fference GMM esmaor s no lkely o perform sasfacorly. Hence he nee n growh regressons of GMM esmaors ha eal wh he problem of hgh perssency n he seres. B.3 ysem GMM esmaor Arellano an Bover (1995) an Blunell an Bon (1998) sugges o use lagge fferences as nsrumens for esmang equaons n levels. The valy of hese nsrumens requre only a ml conon on nal values, whch s E u η Δx an u η Δy 3 E 3 (B.9) Conons (B.9) ogeher wh he moel se ou n (1) o (5) mply he followng momen conons: E η Δx an η Δy u -1 E u -1 (B.1) for = 1,, N an = 3,,T. o, f conons (B.9) are rue, (B.1) resuls n he exsence of z l = (T) nsrumens for he equaon n levels. In fac conons (B.9) are always rue uner he hypohess of mean saonary mplc n 16
he ervaon of expressons (6) an (7) an he absence of seral correlaon n u. Inee, uner such hypohess we have: an E u η Δx Eu η Δξ 3 u η Δy Eu η Δζ E 3 3. 3 The avanage of esmaon n levels s ha lagge fferences are nformave abou curren levels of varables even when he auoregressve coeffcens approach uny. To see hs, conser agan he smple case of T = 3 an where x s he only varable use as nsrumen for x 3. Then he frs-sage nsrumenal varable equaon s x 3 Δx τη υ 3 where now 3 s a funcon of u an e for = 3, 1 an earler aes. The leas square esmaor for has a probably lm equal o Plm πˆ. Therefore, lagge fferences reman nformave abou curren levels even when he auoregressve parameer ens o one. On he oher han, when he auoregressve parameer ens o, lagge fferences lose her explanaory power. Blunell an Bon (1998) propose o explo conons (B.1) by esmang a sysem of equaons forme by he equaon n frs-fferences an he equaon n levels. The nsrumens use for he equaon n frs-fferences are hose escrbe above, whle he nsrumens for he equaon n levels for each nvual are gven by he (T ) z l marx y x y 3 x3 Z l (B.11) y T1 x T1 Thus, leng he marx Z l = ( Z l1, Z l,, Z ln ), he whole se of nsrumens use n he 17
sysem GMM esmaor s gven by he N(T )(z + z l ) marx Z s Z Z l The one-sep sysem GMM esmaor s obane wh he weghng marx W efne as: W s W, Wl (B.1) N where W l Z l Zl an W s efne n ( B.6). o he one-sep esmaor s: 1 B 1 1 X Z W Z X X Z W where X s s a sacke marx of regressors n fferences an levels an where Y s s a sacke vecor of he epenen varable n fferences an levels. Fnally, he heeroskeascy-robus wo-sep esmaor s obane as explane above. I s sraghforwar o show ha he sysem GMM esmaor s a weghe average of he fference an level coeffcens. Inee, efnng he marces, 1 Z Y -1 Pm ZmWm Zm an Qm X mpm X m where m =, l or s (.e., marces wh varables n fferences, levels or sysem), an nong ha Q Q Q l he sysem esmaor B s may be wren as: B Q -1 X P Y l X P Y l l Q -1 Q B I Q -1 Q B l 18
In hs las expresson an l are he fference an level esmaors, respecvely, an I s he eny marx. The wegh on s: Q -1 Q 1 π Z Z π π Z Z π π Z Z π l l l l where s a z marx of coeffcens obane by leas square n he unerlyng frssage regresson of X on Z, an l s efne analogously. Therefore, as he explanaory power of he nsrumens for he equaon n fferences ecreases an ens owars zero, he sysem GMM esmaor ens owars he level GMM esmaor. References Aghon, P., Bacchea, P., Rancere, R., Rogoff, K., 9. Exchange rae volaly an proucvy growh: The role of fnancal evelopmen. Journal of Moneary Economcs 56(4), 494-513. Arellano, M., Bon,., 1991. ome ess of specfcaon for panel aa: Mone Carlo evence an an applcaon o employmen equaons. Revew of Economc ues 58, 77-97. Arellano, M., Bon,., 1998. Dynamc panel aa esmaon usng DPD98 for Gauss. Arellano, M., Bover, O., 1995. Anoher look a he nsrumenal varable esmaon of error-componens moels. Journal of Economercs 68, 9-51. Banerjee, A.., Duflo, E., 3. Inequaly an growh: Wha can he aa say? Journal of Economc Growh, 8, 67-99. Blunell, R., Bon,., 1998. Inal conons an momen resrcons n ynamc panel aa moels. Journal of Economercs 87, 115-143. Blunell, R., Bon,., Wnmejer, F.,. Esmaon n ynamc panel aa moels: Improvng on he performance of he sanar GMM esmaor, n Balag, B. (e.), Nonsaonary Panels, Panel Conegraon, an Dynamc Panels. Avances n Economercs 15. JAI Press, Elsever cence, Amseram. Casell F., Esquvel, G., Lefor, F., 1996. Reopenng he convergence ebae: a new look a cross-counry growh emprcs. Journal of Economc Growh 1, 363-389. Cohen, D., oo, M., 7. Growh An Human Capal: Goo Daa, Goo Resuls. Journal of Economc Growh 1(1), 51-76. Dalgaar, C., Hansen, H., Tarp F., 4. On he emprcs of foregn a an growh. Economc Journal 114(496), F191-F16. Greenaway, D., Morgan, W., Wrgh, P.,. Trae lberalsaon an growh n evelopng counres. Journal of Developmen economcs 67(1), 9-44. 19
Hansen, L., 198. Large sample properes of generalze meho of momens esmaors. Economerca 5, 19-154. Hauk, W., Waczarg, R., 9. A Mone Carlo suy of growh regressons. Journal of Economc Growh 14, 13-147. Kve, J., 1995. On bas, nconssency an effcency of varous esmaors n ynamc panel aa moels. Journal of Economercs 68, 53-78. ager, D., ock, H., 1997. Insrumenal varables regresson wh weak nsrumens. Economerca 65(3), 557-586. Wnmejer, F., 5. A fne sample correcon for he varance of lnear effcen wosep GMM esmaors. Journal of Economercs 16(1), 5-51.
Table 1: mulaons wh N = 1 an T = 5 Esmaor Mean RME. Dev.. Err. /. Dev. Mean RME Dev. Err. /. Dev. OL.493.496.45 1..977.137.135.988 Whn -.4.47.5.965.394.61.136.961 DIF GMM - 1 -.7.1.96 1.4.395 1.18.819.98 No DIF GMM - -.7.17.13.868.39 1.59.865.848 Perssency LEV GMM - 1.38.118.11 1.13 1.57 1.78 1.685 1.59 LEV GMM -.9.11.117 1.1 1.544 1.83 1.75.968 Y GMM - 1.19.89.87 1.4.886.786.778 1.14 Y GMM -.1.89.86.74.844.811.796.694 OL.8.31..98.773.49.13.98 Whn.136.368.55 1..388.63.146.995 DIF GMM - 1.368.1.166.993.653.633.59.969 Moerae DIF GMM -.363.7.181.89.63.686.579.85 Perssency LEV GMM - 1.577.133.19 1.17 1.174.581.555 1. LEV GMM -.566.135.118.894 1.165.66.583.91 Y GMM - 1.55.113.1.966 1.67.413.48.981 Y GMM -.556.117.13.653 1.3.416.414.7 OL.963.13. 1.1.886.14.49 1.14 Whn.749.6.41.994.574.453.154.98 DIF GMM - 1.895.1.84.993.85 1.8.974 1.1 Hgh DIF GMM -.891.19.9.831.54 1.83 1.44.86 Perssency LEV GMM - 1.958.11.7 1.1.991.17.17 1.17 LEV GMM -.958.11.8 1.1.988.133.13 1.35 Y GMM - 1.958.11.7 1.87.99.113.113 1.158 Y GMM -.958.11.8.65 1..111.111.88 1
Table : mulaons wh N = 5 an T = 1 Esmaor Mean RME. Dev.. Err. /. Dev. Mean RME Dev. Err. /. Dev. OL.493.495.47.956.968.1.118.987 Whn -.84.94.4.97.46.6.1 1.8 DIF GMM - 1 -.5.65.6.987.4.73.43.96 No DIF GMM - -..67.64.418.399.738.48.418 Perssency LEV GMM - 1.74.15.74.984 1.565.9.77.949 LEV GMM -.65.15.8.646 1.58.967.8.69 Y GMM - 1.41.77.65.98 1.13.546.536.954 Y GMM -.43.8.69.364 1.74.545.54.89 OL.818.319.19.967.781.35.86.98 Whn.393.114.39.99.53.48.11 1.15 DIF GMM - 1.41.131.95.896.74.41.85.97 Moerae DIF GMM -.411.136.13.368.7.431.31.415 Perssency LEV GMM - 1.638.153.66.993 1.5.378.318.989 LEV GMM -.631.151.76.638 1.195.43.35.67 Y GMM - 1.61.1.64.979 1.14.319.87 1.8 Y GMM -.61.1.69.338 1.16.35.3.357 OL.963.13..95.883.16.46.947 Whn.9.3.16.96.815.3.84.959 DIF GMM - 1.913.54.4.93.475.69.451.974 Hgh DIF GMM -.91.59.44.377.457.736.497.387 Perssency LEV GMM - 1.959.9.4 1.74.988.83.8 1.6 LEV GMM -.959.1.4.714.983.93.9.71 Y GMM - 1.958.9.4 1.55.99.83.83 1.37 Y GMM -.958.9.4.373.993.89.88.38
Table 3: mulaons wh N = 35 an T = 1 Esmaor Mean RME. Dev.. Err. /. Dev. Mean RME Dev. Err. /. Dev. OL.495.498.57.943.966.15.148.99 Whn -.87.11.51.956.41.61.16.967 DIF GMM - 1 -.8.84.79.977.387.841.576.945 No DIF GMM - -.3.91.88.546.398.893.66.5 Perssency LEV GMM - 1.1.134.89.954 1.51.84.669.997 LEV GMM -.94.136.98.537 1.466.87.737.554 Y GMM - 1.6.11.81.954 1.31.616.571.98 Y GMM -.6.14.84.91 1.19.61.571.5 OL.818.319.3.93.774.5.17.938 Whn.391.119.47.974.59.488.16.958 DIF GMM - 1.43.139.116.945.73.487.4.969 Moerae DIF GMM -.4.15.18.5.715.519.434.555 Perssency LEV GMM - 1.661.177.74.966 1.155.383.35.975 LEV GMM -.656.177.84.539 1.149.47.378.576 Y GMM - 1.64.145.74.948 1.148.367.336.966 Y GMM -.63.145.76.77 1.17.365.34.94 OL.963.13..935.887.16.55.93 Whn.9.35.19.936.817.11.15.94 DIF GMM - 1.91.56.48.987.619.671.553.999 Hgh DIF GMM -.917.67.58.475.63.745.631.47 Perssency LEV GMM - 1.959.1.5 1.6.986.99.98 1.3 LEV GMM -.959.1.5.58.983.11.19.585 Y GMM - 1.958.1.5 1.8.99.98.97 1.11 Y GMM -.959.1.5.34.987.1.11.33 3
Table 4: mulaons wh N = 35 an T = 1 an heeroskeascy across nvuals Esmaor Mean RME. Dev.. Err. /. Dev. Mean RME Dev. Err. /. Dev. OL.49.495.59.938.971.149.146.984 Whn -.89.13.53.96.44.61.133.96 DIF GMM - 1 -.33.9.83.961.41.8.564.969 No DIF GMM - -.7.94.9.51.394.878.635.5 Perssency LEV GMM - 1.98.135.94.935 1.45.81.7.959 LEV GMM -.9.135.11.491 1.398.879.784.493 Y GMM - 1.56.13.86.933 1.185.637.61.936 Y GMM -.56.14.88.49 1.158.63.61.14 OL.818.319.3.988.776.48.16.969 Whn.394.116.48.975.55.491.16.994 DIF GMM - 1.49.136.116.965.78.54.411.958 Moerae DIF GMM -.49.145.16.514.78.54.454.51 Perssency LEV GMM - 1.664.181.75.98 1.146.386.358.977 LEV GMM -.66.18.83.517 1.15.41.383.538 Y GMM - 1.69.149.75.958 1.134.361.335.991 Y GMM -.67.149.77.54 1.117.359.34.73 OL.963.13..95.88.131.57.96 Whn.9.36..93.81.15.14.946 DIF GMM - 1.9.6.5.947.59.716.588.961 Hgh DIF GMM -.915.71.6.444.585.76.637.466 Perssency LEV GMM - 1.959.1.5.95.983.19.18.991 LEV GMM -.959.11.5.59.98.116.115.545 Y GMM - 1.959.1.5.953.985.16.15.989 Y GMM -.959.1.5.77.983.11.19.89 4
Table 5: Frequency rejecons of he null hypohess ha coeffcens are no sgnfcan (a a 5% level) N=35; T=1; N=1; T=5 N=5; T=1 N=35; T=1 Heeroskeascy Esmaor OL 1 1 1 1 1 1 1 1 Whn 995 853 538 975 456 895 46 87 DIF GMM - 1 63 64 8 93 13 98 1 No DIF GMM - 19 117 448 599 316 397 351 398 Perssency LEV GMM - 1 59 8 189 643 47 65 19 586 LEV GMM - 74 35 33 769 469 819 497 79 Y GMM - 1 65 33 111 587 14 63 139 559 Y GMM - 164 41 531 97 63 96 687 946 OL 1 1 1 1 1 1 1 1 Whn 714 764 1 997 1 986 1 987 DIF GMM - 1 651 8 975 733 939 499 933 48 Moerae DIF GMM - 695 333 998 914 98 73 985 731 Perssency LEV GMM - 1 97 63 1 964 1 91 1 97 LEV GMM - 97 636 1 981 1 967 1 973 Y GMM - 1 994 769 1 971 1 94 1 91 Y GMM - 999 86 1 999 1 994 1 995 OL 1 1 1 1 1 1 1 1 Whn 1 96 1 1 1 1 1 1 DIF GMM - 1 999 43 1 81 1 38 1 36 Hgh DIF GMM - 999 97 1 613 1 61 1 576 Perssency LEV GMM - 1 1 99 1 1 1 1 1 1 LEV GMM - 1 99 1 1 1 1 1 1 Y GMM - 1 1 995 1 1 1 1 1 1 Y GMM - 1 997 1 1 1 1 1 1 5
Table 6: Frequency rejecons of he null hypohess ha nsrumens are exogenous (argan es) Esmaor N=1; T=5 N=5; T=1 N=35; T=1 N=35; T=1; Heerosk. No DIF GMM 4 1 Perssency LEV GMM 39 9 Y GMM 9 Moerae DIF GMM 54 1 1 Perssency LEV GMM 74 31 Y GMM 49 Hgh DIF GMM 1 1 1 Perssency LEV GMM 5 1 1 Y GMM 34 6
Fgure 1: Dsrbuons of esmaes n moel wh N=35, T=1 an heeroskeascy across nvuals No perssency ( = ; = 1) 16 3 14 5 1 1 Frequency 8 Frequency 15 6 1 4 5 -. -.16 -.1 -.8 -.4..4.8.1.16..4.8.3.36.4.44.48.5.56.6.64.68 -.6 -.5 -.4 -.3 -. -.1..1..3.4.5.6.7.8.9 1. 1.1 1. 1.3 1.4 1.5 1.6 1.7 1.8 1.9..1..3.4.5.6.7.8.9 3. 3.1 3. OL Y 1 Y OL Y 1 Y Moerae perssency ( =.5; = 1) 5 5 15 15 1 5.37.39.4.44.46.49.51.54.56.58.61.63.66.68.7.73.75.78.8.8.85.87.9 Frequency 1 5 -.8.4.16.8.4.5.64.76.88 1. 1.1 1.4 1.36 1.48 1.6 1.7 1.84 1.96.8..3 Frequency OL Y 1 Y OL Y 1 Y Hgh perssency ( =.95; = 1) 18 14 16 1 14 1 1 Frequency 1 8 Frequency 8 6 6 4 4.938.94.94.944.946.948.95.95.954.956.958.96.96.964.966.968.97.97.974.976.978.98 More.64.676.71.748.784.8.856.89.98.964 1. 1.36 1.7 1.18 1.144 1.18 1.16 1.5 1.88 1.34 1.36 OL Y 1 Y OL Y 1 Y 7