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1 econor Mae our Publcaon Vble. A Servce of Wrchaf Cenre zbwlebnz-informaonzenrum Economc Lechner, Mchael Worng Paper Sequenal Machng Emaon of Dynamc Caual Model IZA Dcuon paper ere, No. 4 Provded n Cooperaon wh: Inue of Labor Economc (IZA) Suggeed Caon: Lechner, Mchael (4) : Sequenal Machng Emaon of Dynamc Caual Model, IZA Dcuon paper ere, No. 4, Inue for he Sudy of Labor (IZA), Bonn Th Veron avalable a: hp://hdl.handle.ne/49/77 Sandard-Nuzungbedngungen: De Doumene auf EconSor dürfen zu egenen wenchaflchen Zwecen und zum Prvagebrauch gepecher und oper werden. Se dürfen de Doumene nch für öffenlche oder ommerzelle Zwece vervelfälgen, öffenlch auellen, öffenlch zugänglch machen, verreben oder anderweg nuzen. Sofern de Verfaer de Doumene uner Open-Conen-Lzenzen (nbeondere CC-Lzenzen) zur Verfügung geell haben ollen, gelen abwechend von deen Nuzungbedngungen de n der dor genannen Lzenz gewähren Nuzungreche. Term of ue: Documen n EconSor may be aved and coped for your peronal and cholarly purpoe. ou are no o copy documen for publc or commercal purpoe, o exhb he documen publcly, o mae hem publcly avalable on he nerne, or o drbue or oherwe ue he documen n publc. If he documen have been made avalable under an Open Conen Lcence (epecally Creave Common Lcence), you may exerce furher uage rgh a pecfed n he ndcaed lcence.

2 DISCUSSION PAPER SERIES IZA DP No. 4 Sequenal Machng Emaon of Dynamc Caual Model Mchael Lechner March 4 Forchungnu zur Zuunf der Arbe Inue for he Sudy of Labor

3 Sequenal Machng Emaon of Dynamc Caual Model Mchael Lechner SIAW, Unvery of S. Gallen, CEPR, ZEW and IZA Bonn Dcuon Paper No. 4 March 4 IZA P.O. Box Bonn Germany Phone: Fax: Emal: za@za.org Any opnon expreed here are hoe of he auhor() and no hoe of he nue. Reearch demnaed by IZA may nclude vew on polcy, bu he nue elf ae no nuonal polcy poon. The Inue for he Sudy of Labor (IZA) n Bonn a local and vrual nernaonal reearch cener and a place of communcaon beween cence, polc and bune. IZA an ndependen nonprof company uppored by Deuche Po World Ne. The cener aocaed wh he Unvery of Bonn and offer a mulang reearch envronmen hrough reearch newor, reearch uppor, and vor and docoral program. IZA engage n () orgnal and nernaonally compeve reearch n all feld of labor economc, () developmen of polcy concep, and () demnaon of reearch reul and concep o he nereed publc. IZA Dcuon Paper ofen repreen prelmnary wor and are crculaed o encourage dcuon. Caon of uch a paper hould accoun for provonal characer. A reved veron may be avalable on he IZA webe ( or drecly from he auhor.

4 IZA Dcuon Paper No. 4 March 4 ABSTRACT Sequenal Machng Emaon of Dynamc Caual Model Th paper propoe equenal machng and nvere elecon probably weghng o emae dynamc caual effec. The equenal machng emaor exend mple, machng emaor baed on propeny core for ac caual analy ha have been frequenly appled n he evaluaon leraure. A Mone Carlo udy how ha he uggeed emaor perform well n mall and medum ze ample. Baed on he applcaon of he equenal machng emaor o an emprcal problem - an evaluaon udy of he Sw acve labour mare polce - ome mplemenaonal ue are dcued and reul are provded. JEL Clafcaon: C4 Keyword: dynamc reamen effec, nonparamerc denfcaon, caual effec, equenal randomaon, programme evaluaon, panel daa Correpondng auhor: Mchael Lechner Sw Inue for Inernaonal Economc and Appled Economc Reearch (SIAW) Unvery of S. Gallen Dufourr S. Gallen Swzerland Emal: Mchael.Lechner@ung.ch Fnancal uppor from he Sw Naonal Scence Foundaon (proec and ) graefully acnowledged. Par of he daa orgnaed from a daabae generaed for he evaluaon of he Sw acve labour mare polcy ogeher wh Mchael Gerfn. I am graeful o he Sw Sae Secreara for Economc Affar (eco; Arbemara) and he Sw Federal Offce for Socal Secury (BSV) for provdng he daa and o Dragana Durdevc, Mchael Gerfn and Hed Seger for her ubanal npu n preparng hem. I han Ruh Mquel for helpful commen on everal ue raed n he paper. The paper ha been preened n emnar a Cambrdge Unvery, Mannhem Unvery and he Tnbergen Inue, Amerdam. I han parcpan for helpful commen. The uual dclamer apple.

5 Inroducon The ample elecon problem n ac veron ha receved conderable aenon n he mcroeconomerc and ac leraure concerned wh uncoverng caual effec of nervenon (e.g. Hecman, 979, Hecman and Robb, 985, Holland, 986, Roy, 95, Rubn, 973, 974, 977). The fundamenal problem ha we dere a comparon of wo (or more) dfferen ae of he world (wh one oucome for each ae) and have o perform h comparon ung he un (ndvdual, frm, ec.) ha acually are oberved n ha ae. If here are facor ha only nfluence elecon no he dfferen ae and he varable ued o meaure he (caual) effec of beng n one ae or he oher, hen a (unadued) comparon of mean of hee oucome varable n he dfferen ae do no emae caual effec. In h cae dfferen mehod of adumen may be ued o recover he caual effec. Whch of hoe mehod approprae depend on he pecfc naure of he connecon of he oucome proce o he elecon proce ha may dffer from one applcaon o he oher. The urvey by Angr and Krueger (999) and Hecman, LaLonde, and Smh (999) gve comprehenve overvew. The good daa cae (all varable ha only nfluence elecon and oucome are obervable) receved conderable aenon n appled a well a mehodologcal ude. Th cae a ue n h paper a well. In parcular machng mehod, whch mplcly or explcly form comparon group o adu for he dfference n obervable characerc relaed o he elecon proce, are popular (e.g. Rubn, 973, Roenbaum and Rubn, 985, Dehea and Wahba, 999,, Hecman, Lalonde, and Smh, 999, Smh and Todd,, Lechner, 999, a). There are recen advance o mprove he underandng of he aympoc propere of varou emaor ued n applcaon (Abade and Imben,, Hahn, 998, Hecman, Ichmura, and Todd, 997, Hecman, Ichmura, Smh, and Todd, 998, Hrano, Imben, and Rdder, 3, Ichmura and Lnon,, among oher). However, he ac model may no be able o addre all elecon ue ha occur n applcaon. Suppoe, for example, one nereed n he effec on female labour upply of gvng brh o wo chldren (equence ) compared o gvng no brh (equence ) n a gven perod of me. Apparenly, elecon 3

6 occur wh repec o he fr and econd ferly decon (gnore wn). However, he econd decon may well depend on he effec of he fr brh on ubequen labour upply (an nermedae oucome). Snce even n he good daa cae elecon canno be conrolled for n he begnnng of he equence (becaue plan may be changed dependng on he nermedae oucome), he ac model no flexble enough o handle elecon bae n uch uaon. Smlar problem occur n oher feld, oo, for example, when evaluang he effec of equence of ranng programme. There are everal way of handlng uch dynamc elecon problem n obervaonal ude. When he oucome varable a duraon varable, le monh of unemploymen, Abbrng and van den Berg (3) ugge modellng he hazard rae no unemploymen wh a elecvy correcon. However, duraon modellng ha he dadvanage ha uually a ey denfyng aumpon he mulplcave proporonal hazard rae condon ha n many cae hard o movae from ubanve conderaon abou he elecon proce. Furhermore, ang accoun of nermedae oucome (oher han hoe comng drecly from duraon dependence) no rval. Oher canddae of paramerc model ha allow nermedae oucome o ome exen are clacal dynamc panel daa model wh ample elecon (e.g. he urvey by Arellano and Honoré, ). Bede he envy wh repec o mpecfcaon, anoher ue ha uch model provde only ndrec emae of he caual effec, and n many cae he connecon beween he emaed coeffcen and he caual parameer of nere no raghforward. Anoher approach o dynamc elecon problem o gnore nermedae oucome and rea he equence parcpaon a beng deermned from he ar. The problem conderably mplfed. Thu, h approach allow ung he emaon mehod avalable for he ac caual model. For example, Arulampalam and Booh () analye he effec of mulple ranng even whn year by modellng hem For he ue of duraon model analyng expermen ee Ham and LaLonde (996). The wor by Hecman and Robb (985) and Hecman and Hoz (989) addre he ue of how o ue panel daa o correc for elecon effec and hu denfy caual effec. However, wh repec o he defnon of he reamen her approach ac. 4

7 a oucome of a coun daa proce. In uch a proce here no role for any nermedae oucome o deermne elecon. Much of he leraure on (ac) caual effec n he good daa cae devoed o reduce he role of funconal form aumpon and o leave he heerogeney of he effec acro un unrerced. To explo hee advanage of he ac evaluaon leraure for he dynamc elecon problem ome paper w he ac caual model by defnng effec n a way ha avod mo of he dynamc elecon problem. Then andard machng echnque are ued for emaon. For example, evaluang programme of he acve labour mare polcy Gerfn and Lechner () crcumven he problem of a ngle ndvdual parcpang n everal programme equenally by emang he effec of he fr programme only. Smlarly, L, Proper, and Roenbaum () and Sane () are emang he effec of a delay of a reamen. Boh paper are baed on machng approache for bnary reamen. Alhough he acual mplemenaon of he machng algorhm ued are farly dfferen, boh paper emae he effec of wang by machng hoe people who a a gven me face he r of reamen bu do no parcpae o hoe onng he reamen, and hen averagng over he drbuon of he ar dae of he reamen. However, wang eenally defne a dynamc reamen equence f leavng he queue allowed durng wang. Bergemann, Fzenberger, and Speceer () analye he effec of programme equence ung a combnaon of a machng and dfference-n-dfference emaor. Agan, her approach con n equenally ung he ac evaluaon model. Snce he caual effec of nere canno be formally defned ung he ac framewor, he condon requred for denfcaon reman unclear. 3 Robn (986) ugge an explcly dynamc caual framewor ha ubequenly ued n applcaon n epdemology and boac. Alhough he focue on pecfc equence and denfyng condon ha would be unconvenonal n economerc (a well a ung a paramerc emaon framewor), he eem o be he fr who explcly formale caual effec of dynamc equence ung poenal ou- 3 Recenly, Mquel (3) dcued denfcaon by IV and dfference-n-dfference n an explcly dynamc model of reamen equence. 5

8 come and allowng for nermedae oucome o deermne he nex ae of he equence. H approach exended by Gll and Robn () o he cae of connuou covarae and reamen. Recenly, Lechner and Mquel (, LM furher on) exend Robn (986) framewor o allow comparon of more general equence and elecon procee. They eablh noaon and aumpon ha are more common n economerc. Focung on he good daa cae (elecon on obervable, ncludng nermedae oucome), LM dcu denfcaon condon, denong hem a he wea and rong dynamc condonal ndependence aumpon (W-DCIA, S-DCIA). They how ha under S-DCIA no pecfc problem are. For W-DCIA he endogeney problem emmng from he endogeney of nermedae oucome lead o a lo of denfcaon even f all elecon varable are obervable, bu nereng caual parameer are ll denfed. Th paper propoe emaor for he model of LM ha rean he flexble propere of he emaor commonly ued n he ac evaluaon leraure, namely ha hey are robu o funconal form aumpon, do no rerc effec heerogeney, and are farly eay o compue. They are a equenal veron of machng-on-he-propeny core emaor. Furhermore, a parcular varan of nvere probably weghed emaor dcued a well. In a Mone Carlo udy ome fne ample propere of hee emaor are exemplfed and compared o oher varan of machng emaor. Furhermore, hey are appled o he evaluaon of he Sw acve labour mare programme. Th emprcal par llurae everal ue concernng he mplemenaon of equenal machng emaor. Secon oulne he dynamc caual framewor uggeed by LM: The noaon nroduced and he bac denfcaon condon are reaed. To focu dea he bac model only preened (an nal perod, wo ubequen perod, and wo reamen each perod). In Secon 3 he emaon problem explaned and equenal machng a well a equenal nvere probably weghed emaor are propoed. The Mone Carlo udy n Secon 4 conan mulaon for wo daa generang procee, one fulfllng S-DCIA and one fulfllng W-DCIA only. Secon 5 preen he applcaon and Secon 6 conclude. Appendx A addree ue ha are wh equenal propeny core mehod wh mulple 6

9 reamen a ued n he emprcal par. Appendx B deal he equenal machng proocol ued n he applcaon and Appendx C dcue ome propere of he nvere-probably weghed emaor. The dynamc caual model Th econ nroduce he dynamc caual model propoed by LM and rephrae her denfcaon condon baed on equenal elecon on obervable. Snce uffcen o ue a hree-perod-woreamen model o dcu all relevan ue ha dnguh he dynamc from he ac model, h econ how he reul only for h bac veron of he model. 4. The varable and he defnon of he effec o be emaed Tme perod ndexed by and τ (, τ {,,}). The vecor of random varable S ( S, S, S) decrbe he reamen receved by a member of he populaon. 5 In perod everybody receve he ame reamen,.e. n he ame ae S. From perod S can ae wo value. A parcular realaon of S denoed by {,}. Furhermore, denoe he hory of varable up o perod by a bar below a varable, e.g. (, ). 6 In perod a member of he populaon can be oberved n exacly one of wo reamen (, ). In perod he parcpae n one of four reamen equence ( (,),(,),(,),(,) ). Therefore, every ndvdual belong o exacly one hor equence defned by and anoher long equence defned by. To um up, n he hree-perod-wo-reamen example LM conder 6 dfferen overlappng poenal oucome correpondng o muually excluve ae For he fne-number-of-perod-fne-number-of-reamen model he reader referred o LM nead. The noaon follow he pr of Rubn (974) and Robn (986). The erm reamen and ae are ued nerchangeably. In he followng member of he populaon are omeme called ndvdual for mplcy. Capal leer uually denoe random varable and mall leer denoe pecfc value of he random varable. To dfferenae beween dfferen equence a leer (e.g. ) may be ued o ndex a equence le. Furhermore, he nal perod gnored when denong dfferen equence. 7

10 defned by reamen au n perod, plu 4 muually excluve ae defned by reamen au n perod and ogeher. Varable ued o meaure he caual effec of he reamen,.e. he poenal oucome, are ndexed by reamen and denoed by (,, ). Poenal oucome are meaured a he end of each perod, reamen au meaured n he begnnng of each perod. For each lengh of a equence, one of he poenal oucome obervable and denoed by. The obervaon rule are defned n equaon (): S + ( S) SS + S( S) + ( S) S + ( S)( S ). (),,,, Nex, varable ha may nfluence reamen elecon and poenal oucome, ofen called arbue or confounder, are defned and denoed by X. Becaue reamen au may nfluence he realaon of hee varable (nroducng ome endogeney), here are poenal value of hee varable a well X ( X, X )). X may conan or funcon of. The K dmenonal vecor X obervable ( a he ame me a (hu oberved only afer he elecon S realed 7 ). The obervaon rule for X analogou o he one for he poenal oucome gven n equaon (). Inere n he emaon of he mean caual effec of a equence of reamen defned up o perod l ( ) compared o anoher equence of he ame lengh ( ) for a parcular populaon and for oucome, of perod (or laer). Th effec denoed by θ. 8, θ may be of nere for everal ubpopulaon: Mean caual effec can be conruced for ubpopulaon defned by varable no nfluenced by he reamen, becaue caual aemen ha are condonal on he effec of he reamen uually do no have ueful nerpreaon. Thu, condonng he effec on X requre exogeney aumpon ha are dcued below. However, condonng on reamen au nereng, becaue allow mlar com- l l 7 8 Therefore, here no role for X n a wo perod model. Caual effec of equence of lengh can be analyed whn he ac model of poenal oucome. 8

11 paron le comparng he well-nown effec of reamen on he reaed (ATET) and reamen on he nonreaed of he ac framewor. The defnon of uch average caual effec gven n equaon ():, l l E S E S τ θ ( ): ( ) ( ), τ, l, l, {,,3,4}, {,..., }. () τ τ τ τ τ l, To conder θ ( ) a caual effec he andard aumpon of he Rubn (974) model are neceary, τ le he Sable Un Treamen Value Aumpon (SUTVA), mplyng ha he effec of reamen on one, peron do no depend on he reamen choce of oher. θ [, θ ( ) ] called he dynamc l l, average reamen effec (DATE). Accordngly, θ ( ) ermed DATE on he reaed (DATET). l l, l Noe ha here are cae n-beween, le θ ( ), when he populaon defned by parcpang n a equence horer han he one evaluaed. Furhermore, he effec are ymmerc when defned for he l,,, ame populaon n he ene of θ ( ) θ ( ), bu ha ( ) Table ummare he noaon nroduced o far. τ l τ l, θ l l θ τ τ ( ), oherwe. Table : Summary of noaon and defnon Symbol Meanng Tmng whn perod,, me perod -- S (, S, S) RV: reamen begnnng, (, ) pecfc equence of reamen unl perod or begnnng {,} excluve reamen n each perod begnnng (, ) RV: poenal oucome end (, ) RV: obervable oucome end X ( X, X ) RV: poenal confounder end X ( X, X) RV: obervable confounder end l, l θ ( τ ) mean caual effec of compared o for hoe parcpang n τ end RV: Random varable. 9

12 . Idenfcaon of he effec of dynamc reamen Suppoe here an nfnely large random ample ({,, x, x, y, y },..., N ) from he populaon S ha defned by he correpondng random varable ( S, S, X, X,, ). LM explore he denfyng power of aumpon ha may be ermed good daa, elecon on obervable or condonal ndependence aumpon. More precely, hey aume ha ou of he varable ha deermne reamen au n each perod he ample conan hoe ha are relaed o he relevan poenal oucome. If hey are no nfluenced by he reamen n a ac model condonng he oberved oucome on hee varable remove all elecon ba (ee Rubn, 977). The dfference beween he ac model and he dynamc caual model ha he laer allow for a econd ype of elecon ba, becaue nermedae oucome mgh nfluence he decon o connue a equence, n oher word, he nermedae oucome may no be condered exogenou anymore. Aume ha n he begnnng of each perod he reearcher ha uffcen nowledge o aume ha agnmen o reamen ndependen of poenal oucome condonal on ha nformaon. 9 Th nformaon may be nfluenced by nermedae oucome. Aumpon W-DCIA of LM formale wha LM call he WEAK DNAMIC CONDITIONAL INDEPENDENCE ASSUMPTION. Aumpon W-DCIA (wea dynamc condonal ndependence aumpon) a),,, S S, X x ;,,,, 9 The followng aumpon relae o denfcaon of all reamen effec defned n Secon. Hence, all poenal oucome are nvolved. If he comparon dered nvolve fewer poenal oucome, or baed on a ubpopulaon defned by a pecfc reamen equence of nere, hen fewer poenal oucome need o appear n he followng aumpon. For he ae of brevy, we do no menon h ue below anymore, bu he requred change wll be obvou (a hey are n he ac framewor). Noe ha he aumpon WDCIA and SDCIA (below) are omewha ronger han neceary, bu have he vrue of beng ealy explaned n erm relang o he elecon proce (ee LM 4 for deal). A B C c mean ha each elemen of he vecor of random varable B ndependen of he random varable A condonal on he random varable C ang a value of c. ( A) B C c mean ha he on drbuon of he elemen of A ndependen of B condonal on C c.

13 b),,, S X x ;,,,, > PS ( X x) >, > PS ( X x, S ) > ; x χ, : {,}. c) Par a) ae ha condonal on he prevou reamen, obervable oucome and confoundng varable, he poenal oucome are ndependen of elecon n perod ( S ). Par b) ae ha condonal on ome exogenou varable X poenal oucome are ndependen of agnmen n perod ( S ). Thee aumpon are vald for all value of x and x n a gven e of nere χ. Par c) he uual common uppor requremen (CSR), bacally ang ha all equence o be evaluaed mu have a pove probably of occurrng n all raa defned by he value of x and x ha are n he e of nere χ. Obvouly, only equence ha are feable can be evaluaed. If W-DCIA combned wh an nal condon for he confoundng varable ( X X X X X,, X X ) and ome regulary, hen,, l THEOREM of LM how ha, θ, (, θ ( ) and, ),(, ) θ (, ) (,, l l,, l l, {,}) are denfed. In plan word, par-we comparon of all equence are denfed, bu only for group of ndvdual defned by her reamen au n perod, or on average n he populaon. The relevan dncon beween he populaon defned by reamen ae n he fr and ubequen perod ha n he fr perod reamen choce random condonal on exogenou varable (he reul of he nal condon for S, X ), wherea n he econd perod, randomaon no hee reamen condonal on varable already nfluenced by he fr par of he reamen. Specfc comparon are denfed for populaon defned by reamen au n boh perod, f populaon and reamen equence are he ame n he fr perod. To denfy he effec of wo dfferen equence defned for a ubpopulaon gven by reamen au n boh perod, addonal exogeney aumpon are requred. The rong condonal ndependence aumpon (S-DCIA) ae ha condonal on X, nowng S doe no help o predc he poenal ou-

14 come gven a value of he oberved X. The drec and eable mplcaon of h aumpon X S X x. Hence, mple ha he confounder are no Granger-caued by prevou reamen. Aumpon S-DCIA (rong dynamc condonal ndependence aumpon) a) b),,, S S, X x ;,,,,,,,,(, X ),(, X ),(, X ),(, X ) S X x ;,,,,,,,, > PS ( X x, S ) >, > PS ( X x) > ; x χ, : {,}. c) ASSUMPTION S-DCIA allow for oucome of prevou reamen (predeermned endogenou varable) o appear n he condonng e, and ll rong enough o denfy all effec. However, hee varable have o fulfl a rong exogeney requremen n he ene ha f hey are nfluenced by he reamen n perod, h nfluence mu no have an mpac on he poenal oucome of nere n perod. 3 Emaon Preumng denfcaon by ASSUMPTIONS W-DCIA or S-DCIA h econ dcue he general rucure of he emaon problem and propoe machng emaor ha are farly cloe o he emaor frequenly ued n ac evaluaon ude baed on model wh elecon on obervable (e.g. Dehea and Wahba, 999,, Hecman, LaLonde, and Smh, 999, Smh and Todd,, Lechner, 999, a). Emaor baed on (equenal) reweghng ung he nvere elecon probable are dcued a well. I hown how he general rucure of adumen o he emaor of he ac model loo le. The focu on hoe clae of emaor for mplcy only. All he uual emaor avalable could be adued o he dynamc conex n he ame way he equenal machng and nvere probably weghed emaor are adued. The o-called regreon mpuaon emaor a dcued for example by Frölch (, 4), Hahn (998), and Hecman, Ichmura, and Todd (998) are effcen bu uffer from he unolved problem of opmally choong he unng parameer, le bandwdh n he nonparamerc regreon ep. The emaor propoed here avod hee problem.

15 Th econ concenrae on he crucal ngreden of he effec o be emaed, namely he repecve counerfacual expecaon and her relaon o he obervable oucome. Gven h connecon, he conrucon of emaor for he effec raghforward. The fr ubecon revew emaon of counerfacual ha can be analyed whn he uual ac framewor, wherea he econd ubecon addree ue concernng counerfacual ha requre a dynamc framewor. 3. Same reamen and condonng equence n perod a ac problem The complexy of he emaon problem depend on he mlary beween he equence ued o ndex he poenal oucome and hoe defnng he populaon of nere. If he equence concde, E( S ) E( S ), E( S, S ) E( S, S )), he ample ( mean n he correpondng ubample a naural nonparamerc emaor. 3 The emaon of counerfacual defned for equence dencal o he condonng e n perod a well-nown emaon problem and exenvely dcued n he leraure abou ac caual model. A hown by LM he ypcal emand have he followng form: E[ S, S ] E[ E ( S, S, X) S, S ], X E [ E( S, S, p ( X ) S, S ], p ( X) p ( x ): P( S X x, S ),, {,}. (3) For h ype of emand wh a bnary reamen many dfferen emaor are uggeed n he leraure and appled n emprcal ude. Such emaor explong he condonal ndependence aumpon (CIA) are ermed machng mehod n he urvey by Hecman, LaLonde, and Smh (999), becaue hey are baed on comparng dfferen obervaon wh dfferen reamen au bu mlar value of 3, Whou lo of generaly we conder he poenal oucome of perod only. However he oucome may be evaluaed n any perod afer he reamen ( > ) a long a hoe poenal oucome do no nfluence S, S. 3

16 ( X, X ) or he repecve condonal elecon probable p ( x ). 4 Roenbaum and Rubn (983) nroduced he frequenly appled prncple o condon on he condonal parcpaon probable (ocalled propeny core) nead of he conrol varable drecly a a way o reduce he dmenon of he emaon problem. Abade and Imben (), Hahn (998), Hecman, Ichmura, and Todd (998), and Ichmura and Lnon () nvegae he aympoc drbuon of dfferen ype of machng emaor. Hrano, Imben, and Rdder (3) and Hernan, Brumbac, and Robn () focu on he aympoc drbuon of parcular varan of nvere probably-weghng emaor. Among oher, Frölch (, 4) and Smh and Todd () dcu praccal and mall ample ue for dfferen ype of uch emaor Dfferen reamen and condonng equence n perod he dynamc problem 3.. Relaon beween poenal and obervable oucome If he equence of nere dffer n he fr perod, hen LM how ha he emand denfed by S- W-DCIA gven by equaon (4): X X E( S ) E{ E[ E( S, X x ) S, X x ] S }, E E E S p x S p X S p ( X) p ( X) { [ (, ( ), ( )] },, p ( x ): P( S X x ), p ( x ): [ p ( x ), p ( x )],,,, {,}. (4) 4 Noe ha he ac leraure ue he erm machng emaon n a more rercve way. I nclude only emaor ha acually form comparon group baed on mlary of reaed and conrol wh repec o ( X, X ), or he correpondng probable (e.g. Roenbaum and Rubn, 983). 5 For more deal ee he excellen urvey by Imben (3). 4

17 The prevou emaon prncple can be appled here a well. However, he reweghng ha o be per-, formed equenally. In a fr age a regreon of on X (or p ( x )) n he ubample of S, performed, leadng o E( S, X x) (or E( S, p ( X )). Whn each raa of X (or p ( x )) n he ubpopulaon S, h regreon funcon averaged accordng o he drbuon of X (or p ( x )) n each uch raum. Thee average are funcon of X only. F- nally, h funcon averaged over he drbuon of X (or p ( x )) n S, leadng o a equenal machng emaor o be dcued n greaer deal laer on. Of coure, f S-DCIA hold, hen (4) hold a well, bu S-DCIA alo mple he followng mpler expreon: X, X E( S ) E [ E( S, X x ) S ], E E S p x S, p ( X) [ (, ( ) },, p ( x ): P( S X x ), p ( x ): [ p ( x ), p ( x )],,,, {,}. (5) Th expreon gve rae o a one machng procedure a n ac model. In prncple, comparng emaor baed eher on (4) or (5) can form he ba of a e for addonal exogeney aumpon mpled by S-DCIA. For he average effec n he populaon he ame prncple apple (equaon (6)): X X E( ) E{ E[ E( S, X x ) S, X x ]}, E E E S p X S p X p ( X) p ( X) { [ (, ( )), ( )]};, {,}. (6) 5

18 Fnally, when S-DCIA hold, E( S ),, denfed a well: E( S ) E { E ( S, X) S } X, X { [, (, {, }] } E E S PS X S S ;,,, {,}. (7) () P Agan, h emaon problem ha he ame rucure a he ypcal ac emaon problem baed on CIA. The only dfference ha of he mulple reamen ype, becaue four dfferen equence are nvolved (cf. Imben,, Lechner, a). Noe ha f S-DCIA vald, hen he reul for he coarer condonng e S and S can eher be obaned by a weghed mean of he effec for popula- on defned by perod reamen (e.g. E( S ) E ( S, S ) PS ( S ) + E S S (, ) [ PS ( S )], or by machng drecly (or weghng he regre- on funcon) accordng o he redefned arge drbuon ( S or S ). Appendx A dcue ome ue ha come up for non-bnary S and S. However, nce he general rucure of he emaon problem doe no change, he man par of he ex focue on he bnary cae. 3.. Sequenal emaor of E( S ) and E( ) under W-DCIA In h econ emaon of he counerfacual ha canno be emaed by he uual ac machng mehod, namely E( S ) and E( ) under W-DCIA (ee equaon (4) and (6)) dcued. 6 The focu on emaor condonng on propeny core, bu he ame prncple apply o emaor ha condon drecly on he repecve conrol varable. The Mone Carlo udy below run boh ype of emaor. All propoed emaor have a mlar rucure, becaue hey are compued a weghed 6 Emaor conen under W-DCIA are conen under he ronger aumpon S-DCIA, bu no converely. 6

19 mean (wh wegh w,, w ) of a funcon of he oucome varable ( g ( p ( x ))) oberved n, ubample S (he reaed populaon):,,,, E S w g p x, w w ( ) ( ( )); ; ; (8), E w g p x, ( ) ( ( )); w ; w. (9),,,, g ( p ( x )) conruced o have mean E [ S, p ( X ) p ( x ))], a lea,, aympocally. The wegh rewegh h funcon oward he arge drbuon f ( x, S ) for equaon (8) and f ( x, S )[ f ( x, )] for equaon (9). Two ype of wegh are condered accordng o wheher hey are emaed by equenal machng or by equenal nvere choce probable., Common choce for g ( p ( x )) n he machng leraure would be eher y ( drec or par, machng) or a (ernel) regreon emae of,, y on p ( x ) and p ( x ). The laer ha he advanage of leadng o an aympocally more effcen emaor f he bandwdh choen appropraely. Th gan hould be parcularly large when many of he wegh are zero ung he drec approach. I ha however he dadvanage ha ubec o he coure of dmenonaly (he number of condonng varable ncreae lnearly wh he number of perod) and ha here appear o be no generally applcable heory of opmally choong he bandwdh. Noe ha here are relaed emaor nown from he machng leraure for he wegh a well a for,, g ( p ( x )) ha can be adued o he dynamc framewor, le blocng or rafcaon emaor (Roenbaum and Rubn, 984, Dehea and Wahba, 999, ee agan Imben, 3, for a comprehenve dcuon). For he ae of brevy hey are gnored. 7

20 3... Sequenal machng emaor (SM) The dea of he machng emaor o equenally adu he covarae drbuon of he reamen populaon oward he arge populaon o a o mmc he equenal condonal expecaon appearng n expreon (8) and (9). The fr ep he ame for boh effec and con n fndng for every mem- ber of S a member of S wh very mlar (he ame) value of p ( x ) and p ( x ).,, Noe ha one obervaon n he reamen populaon may be mached o many or o none of he nermedae arge populaon S (machng wh replacemen). In he econd ep every member of S ((8)) or S ((9)) o be pared wh a member of S wh very mlar (ame) value of, p ( x ). The pove wegh ha are aached o ome or all member of S comng from ep are hen updaed dependng on how ofen an obervaon n S mached o an obervaon of he arge populaon va he nermedae machng ep. Th procedure lead o he followng emaor: w v[ p ( x ), p ( x );] v [ p ( x ), p ( x );] ; S,, SM,,, n, m, m, N n m ; () w v[ p ( x ), p ( x ); ] v [ p ( x ), p ( x ); ] N, SM,,, n, m, m, N n m ; S. () N denoe he number of obervaon for whch S. The funcon v[ p ( x ), p ( x ); ], n, m, m, n defned o be one f p ( x ) cloe o p ( x ) of all obervaon belongng o he ubample de-,, fned by S, and zero oherwe. Smlarly v [ p ( x ), p ( x ); ] one f obervaon, m, cloe o obervaon m n erm of p ( x ) and p ( x ), and zero oherwe. Smlary could for,, example be defned by he Mahalanob merc. Noe ha he wegh of obervaon f no mached o any member of he arge populaon (even f mached n he fr ep). On he oher exreme, f obervaon would be mached o every member of he arge populaon hen wegh would 8

21 be. Parcular ue on how o mplemen uch an emaor (for example on how o nure ha he common uppor condon are afed) are dcued n Secon 4 and 5. A pecfc varan of h emaor ha ued n he emprcal par (Secon 5) dealed n Appendx B. One of he pecal feaure of h emaor ha he number of propeny core o mach on ncreae lnearly wh he number of perod. Abade and Imben () how ha n ac model a machng emaor wh up o wo connuou varable N conen and aympocally normally drbued. However, f baed on a fxed number of mache (no ncreang wh ample ze), hen doe no reach he emparamerc effcency bound derved by Hahn (998). If he number of connuou covarae larger han (a for T > ), hen machng emaor reman conen, bu becaue he ba vanhe a a lower rae han N, hey are no aympocally normally drbued (around he rue value). Abade and Imben () ugge a correcon baed on nonparamerc regreon of he condonal mean funcon of he oucome o elmnae he ba and how ha he ba correced emae aympocally normally drbued ndependen of he number of connuou covarae. 7 In our framewor clear how he ba correcon procedure could be mplemened n he la machng ep. However, no raghforward how o addre bae reulng from mmache n nermedae ep. 8 7 Thee reul are obaned for drec machng emaor whou ung he propeny core. However, one could conecure ha machng on he propeny core approxmaely le machng on a connuou varable f he dmenon of he ndependen varable n he propeny core large enough; even f all ndependen varable are dcree (he problem of machng occur becaue wh connuou varable exac mache have probably zero). 8 The ba adued emaor uffer from he addonal problem ha hey mu be baed on he ame conen emae of he nonparamerc regreon funcon a he regreon mpuaon emaor. However, Abade and Imben () argue ha due o he local naure of he ba correcon when combned wh machng, he reul are no enve o bandwdh choce, wherea he regreon mpuaon emaor are. 9

22 3... Sequenal nvere probably weghng emaor (SIPW) Reweghng by he nvere elecon probable anoher way o oban he approprae wegh. 9 The wegh have he general rucure uch ha every obervaon of he reamen populaon dvded by condonal probably of beng eleced no reamen and mulpled by condonal probably of beng n he counerfacual ae. The wegh for quane denfed under he dfferen DCIA aumpon are gven n equaon () o (5). Noe ha common cale facor - coverng relave ample ze uch he wegh um up o one - are omed. w ; S ; () ( ) ( ), IP pˆ ˆ x, p x, w ; S. (3) ˆ ( ),, IP p x, w ˆ,, IP p ( x, ) pˆ ˆ x, p x, ; S ; (4) ( ) ( ) w ˆ ˆ,, IP p x, p x, pˆ ˆ x, p x, ( ) ( ) ; ; S ; (5) ( ) ( ) The dervaon of he wegh a well a he cale facor are gven n Appendx C. 4 A Mone Carlo udy A a mall ample chec for he above uggeed emaor a Mone Carlo udy conduced. I baed on wo dfferen daa generang procee (DGP) fulfllng he wea and he rong DCIA. The degn 9 See for example Hrano, Imben, and Rdder (3) and Roenbaum (987). Smlar emaor can be found n he leraure on aron and ample elecon (e.g. Horvz and Thomon, 95, Nevo, 3, Robn and Ronzy, 995, Robn, Ronzy, Zhao, 995, and Wooldrdge, 3).

23 am a a very yled mage of ypcal DGP o be found n poenal applcaon whle eepng a mple a poble o eep compuaon me manageable. 4. Daa generang procee The DGP are conruced o ha hey exhb nonlneare, heerogeneou effec, dynamc and elecvy problem o hghlgh he general approach of he dynamc reamen effec leraure. The man ngreden no he daa generang proce (dealed n Table ) are pecfcaon of he poenal oucome and elecon equaon. The poenal oucome n he fr perod are generaed by funcon of an exogenou regreor and an error erm. The poenal oucome n he econd perod are funcon of he poenal oucome of he fr perod plu error erm. The funcon choen dffer for he dfferen poenal oucome. The error erm of he poenal oucome are all muually correlaed, o ha he error erm of he econd perod poenal oucome equaon are correlaed wh he regreor of hee equaon. The elecon equaon n he fr perod gven by an ndcaor funcon of he exogenou confounder ha appear n he poenal oucome equaon for perod plu a whe noe normal error. Table : Specfcaon of he daa generang procee Selecon equaon S ( + X + U7 > ), ~ (,) Oucome equaon.75 [( X) ] + U, X + U Oucome equaon.75 + U, 3 [( / 4) ] + U, 4 ln( ) + U, 5 + U Selecon equaon : DPG (S-DCIA) S (.5 S+.5 X + U8 > ); X ~ N (,) Selecon equaon : DGP (W-DCIA) S ( S+.5+ U8 > ) Drbuon of error erm ( U,..., U8) ~ N(, ), Var( U,..., U 8) (,...,), Corr( U,..., U8) Noe: If no explcly aed oherwe, he pecfcaon relae o boh DGP. All draw of random number (ung Gau 3..3) are ndependen acro obervaon and replcaon. 6 Whle all oher equaon are he ame for boh DGP he elecon equaon n he econd perod dffer. For DGP modelled a a prob ndcaor funcon wh he realed elecon n perod and an ex-

24 ogenou varable a explanaory varable. DGP ubue he exogenou varable by he oberved oucome afer perod. The error erm of he elecon equaon are ndependen of he error erm of he poenal oucome equaon, o ha boh DGP are baed on elecon on obervable. Becaue all varable deermnng elecon are no relaed o oucome of he reamen DGP fulfl W-DCIA and S- DCIA. The crucal dfference beween he wea veron and he rong veron of DCIA ha he rong, l, veron requre F ( X, S, X) F ( X, X ) o hold. I fulflled by DGP, becaue X a regreor ha ndependen of all poenal oucome. In DGP X become he oucome of perod,, whch relaed o poenal oucome of perod by he choen auoregreve pecfcaon of he oucome equaon and he perod elecon rule. Therefore, DGP volae he rong veron of DCIA, becaue of he endogeney of one of he varable deermnng elecon n perod. Neverhele, DGP afe he condon for W-DCIA (all varable nfluencng elecon and poenal oucome are obervable). Funconal form and coeffcen n he oucome equaon of boh DGP have been choen uch ha reamen effec exhb heerogeney and ae dependence. The coeffcen of he elecon equaon are uned o produce margnal parcpaon probable PS ( ) and PS ( ) of abou 5%. Table 3 and 4 gve bac uncondonal decrpve ac for boh DGP. The mean of he poenal oucome dffer ubanally from each oher and hu from he oberved oucome. They exhb conderable ndvdual heerogeney leadng o heerogeney of he effec. Uncondonally, poenal oucome, effec and oberved oucome are hghly correlaed wh boh elecon varable, he correlaon for S beng hgher for DGP due o he dfferen pecfcaon of he elecon equaon n perod. Noe ha alhough X uncorrelaed wh poenal oucome, correlaed wh oberved oucome and hu exhb he clacal feaure of an nrumenal varable.

25 Table 3: Some decrpve ac for he daa generang proce for DGP Mean Sd. Correlaon (x ) X , θ , θ θ ,,,, θ θ θ S S X S S X.. - X.. - Noe: Sample ac baed on one ample of N.. PS (, S ).7; PS (, S ).33; PS (, S ).34; PS (, S ).6. Table 4: Some decrpve ac for he daa generang proce for DGP,,, Mean Sd. Correlaon (x ),,, θ θ θ S S X θ θ θ S S X.. - Noe: Sample ac baed on one ample of N.. PS (, S ).8; PS (, S ).; PS (, S ).; PS (, S ).8. The ample ze condered n he mulaon are N 4, 6, and 64, repecvely. They cover a reaonable range gven recen labour mare applcaon of machng mehod (e.g. Gerfn and Lechner,, or Sane, ). Furhermore, gven he few confoundng varable he ample ze hould be large enough o oberve convergence propere of he emaor. 3

26 4. Emaor Fve dfferen emaor are condered n he mulaon. Four of hem are neare-neghbour ype machng emaor, wo of hem equenal and wo of hem one-ep emaor; wo of he machng emaor ha mach on an emaed propeny core, wo emaor mach drecly on he confoundng varable. When here more han one varable o mach on, he dance merc he Mahalanob dance wh wegh marx compued n he ample of he parcular arge populaon. The remanng emaor ue emaed equenal probable a defned n he prevou econ - o drecly rewegh he reaed obervaon oward he arge populaon. The equenal machng emaor (SM) ung propeny core a well a he nvere probablyweghed emaor (SIPW) are baed on condonal probable emaed by a prob model PS ( S, X), PS ( S, X), PS ( X) ) n he repecve ubample defned by ( reamen au. Boh one-ep-machng emaor are baed on bnary prob for he probably of beng n he parcular ubample condonal on beng eher n he ubample for whch oucome are oberved or n he arge populaon (for example PS (, S X, X) for emang E, or { PS (, S S (,),(,), X, X) for emang } E ( S, S ) ). All varable explanng boh elecon ep are ncluded n hee pecfcaon. Thu, we generally expec he one-ep emaor o be nconen for DGP (wh he excepon of E ( S, S ), E ( S ), ec.), becaue hey have he ame rucure a he propeny core emaor propoed for he ac mulple reamen model ha aume exogenou confounder. Furhermore, o chec he effec of emang he (hgher dmenonal) balancng core n uaon n whch he dmenon of X acually mall E S S (, ) emaed only by one emaor, he (unweghed) ample mean, and hu ha ome deal propere ha we would le he machng emaor o have a well (unbaedne, effcency, conency and N convergence). X denoe he regreor n he prob model ha would eher be (DGP ) or X (DGP ). X need no o be ncluded n he prob emaon becaue no a confoundng varable. Neverhele, ncluded o eep he dmenon of he emaon problem he ame for boh DGP. 4

27 enough o mach drecly on he X varable, he wo drec machng emaor proceed exacly a he propeny core machng emaor, bu nead of ung he core, hey ue he explanaory varable of he core drecly. l l A dealed machng proocol for E( S ) - E( S ) can be found n Appendx B (wh l obvou exenon when oher effec are of nere or he one-ep or drec machng emaor are ued). However, o peed up compuaon chec for common uppor - whch hould no be neceary becaue he varable explanng elecon are connuou - have no been performed for he SM emaor, neher have very mall or very large probable been rmmed for he SIPW emaor. Some remar abou h proocol are warraned: Fr, noe ha machng wh replacemen. Every ep of he machng equence le machng n a ac framewor, o ha all he dfferen emaor dcued n he leraure wh her mer and drawbac are poenal canddae. Noe ha here machng nvolve everal probable o ha here he queon abou how o defne cloene. I eem o be common prace n he propeny core machng leraure o ue he Mahalanob dance. l Nex, ome ue are from he equenal naure of machng for example o oban E( S ): By choong obervaon a mache wh mlar value of he probable nead of he ame value, may happen ha he probable aached o obervaon n early machng ep change over dfferen equenal machng ep due o mprece machng. To preven h happenng every mached compar- on obervaon n perod could be recorded wh he value p of he obervaon mached o, nead of own. Hence he hory of he mach, or n oher word he characerc of he reference drbuon, doe no change when he nex mach occur n he ubequen perod. ˆ l 5

28 l Furhermore, o compue E( S ) he only nformaon ha needed for he N l parcpan n ˆ ˆ l, p,. Smlarly, for parcpan n, all probable of he ype p, are requred. For parc- pan n bu no n only p, needed, and o on. ˆ l l l To emae E( S ) nead of E( S ) (par B n Table B.) he only change n he prevou machng proocol ha he nal machng ep on p, redundan n h cae. When nere n he average effec n he populaon ( E( )), hen he whole populaon play he role of he fr l reference group (nead of ). In h cae n he machng ep baed on p, all parcpan n wll alo be mached o hemelve, a well a eleced parcpan n wll be mached o he parcpan n he remanng reamen n he fr perod. ˆ l ˆ When machng on he propeny core nead of machng drecly on he confoundng varable here he ue of elecng a probably model. I eem ha o far even n he ac model he leraure ha no addreed h horoughly. There are ome reul for pecfc nonparamerc approache a n Hrano, Imben, and Rdder (3), bu he general conenu eem o be ha a flexbly pecfed (and exenvely eed) paramerc model, le a log or a prob model, uffcenly rch and ha he choce of he model doe no really maer (ee for example he Mone Carlo reul by Zhao, ). There one ue, whch ha no been dcued o far becaue no he focu of h paper, bu mporan n prace: emaon of he andard error. In he appled evaluaon leraure here eem o be wo common way o emae andard error: Fr, compue he andard error condonal on he wegh. When he wegh are baed on emaed propeny core, he uncerany of he emaon ep ypcally gnored. Alernavely naïve boorap are ued (ncludng he emaon ep of he propeny core) and andard error (or confdence nerval) are compued from he boorap drbuon. However, here appear o be no proof avalable ha ndeed he condon for he conency of he boorap ( alo no clear ha he andard error he be quany for whch o perform he boorap, 6

29 becaue no aympocally pvoal). In he Mone Carlo udy he andard error are compued ung hee wo mehod. However, due o compung me rercon he boorap ( replcaon) ued for he malle ample only. 4.3 Reul Table 5 (DGP ) and 6 (DGP ) preen he reul of he mulaon. They are baed on replcaon. Snce caual effec are uually emae a he dfference of he emaed (counerfacual) mean of he poenal oucome n he repecve populaon of nere, he able focu on he emaed mean of he counerfacual oucome only. Reporng he reul for all ubpopulaon would requre oo much pace. Therefore, boh able conder only hree populaon, namely he parcpan n he reamen n boh perod ( S (,) ), parcpan n he fr perod ( S ), and he populaon. The reul for oher populaon defned by reamen au are qualavely dencal. Dealed reul are gven for he equenal machng-on-he-emaed-propeny-core emaor ha appear o be of maor nere (rue mean; ba of emaor for mean and andard error; andard devaon, ewne, uro, roo mean quared error and medan abolue error of mean emaor). In boorap veron hown for N4 he repecve lne conan he ame ac for he emaor defned o be he mean of he boorap drbuon, wh one excepon. The ba of he andard error relae o he ba of he andard error for he boorap compared o he Mone Carlo andard error of he non-boorapped equenal machng emaor gven n he lne above he boorap. For he ae of brevy mean quared error only are gven for he oher emaor. 3 For DGP all emaor are almo unbaed. Even for N 4 he ba very mall. The equenal (a well a all oher) machng emaor appear o be farly cloe o he normal drbuon for all ample ze conderng ewne and uro oberved n he Mone Carlo udy. When he ample ze quadruple he andard error are reduced by abou half. In concluon, he machng emaor appear o be N convergen. 7

30 Nex, we compare he varou emaor accordng o her mean quared error, begnnng wh he machng emaor. In many cae he machng emaor baed on he emaed propeny core have lower RMSE han he correpondng machng emaor baed drecly on he varable deermnng elecon. Th reul may eem urprng, becaue here only one regreor n he each prob, o ha he propeny core doe no reduce he dmenon of he condonng varable n h eng. Th reul however n lne wh he fndng of Hrano, Imben, and Rdder (3), alhough he laer (heorecal) reul are obaned for a dfferen cla of emaor. Comparng he equenal and one-ep machng emaor, no yemac dfference appear for DGP (hey wll dffer dracally for DGP ). Table 5: Reul for DGP Sequenal propeny core machng eq. on X one PSM one on X prop. wegh Po. Populaon N rue ba d. ew. ur. ba MAE x RMSE x ouc. d BS BS all BS BS BS all BS Table 5 o be connued. 3 The complee e of reul (all emaor and oher ubpopulaon) avalable on reque from he auhor. 8

31 Table 5: connued Sequenal propeny core machng eq. on X one PSM one on X prop. wegh Po. Populaon N rue ba d. ew. ur. ba MAE x RMSE x ouc. d BS BS all BS BS BS all BS Noe: Ba: Mean of emaed effec - rue effec. Sd.: Sandard devaon oberved n Mone Carlo. Sew.: Sewne. Kur.: Kuro. RMSE: Roo mean quared error. MAE: Medan abolue error. Ba d.: Mean of emaed andard devaon Sd.. Seq. on X.: Sequenal machng ung he approprae conrol varable nead of he propeny core. One PSM: One ep propeny core machng. One on X: One-ep machng ung he approprae conrol varable nead of he propeny core. In cae when he equenal and he one-ep emaor concde, he one-ep emaor are no gven. BS: Boorap emae are baed on boorap ample of N draw (wh replacemen) n he repecve ample. The ba of he boorap andard error doe no relae o he mean of he boorap drbuon, bu hey relae o he equenal machng emaor gven one lne above. All oher number gven n he lne BS relae o he emaor defned a he mean of he boorap drbuon. In many cae he (unrmmed) SIPW emaor ha he lowe RMSE of all emaor condered for N4. When he ample ze ncreae h advanage dappear becaue he emaor converge lower han N. In fac, he uro of he emaor n he mulaon ncreae farly dramacally wh ample ze uggeng ha he hgher order momen (or even he RMSE) of h emaor may no ex. 4 Trmmng very mall and very large probable may be neceary o mprove he convergence propere. Th alo confrmed by conderng he medan abolue error ha how more pronounced reduc- 4 Thee fndng are confrmed by mulaon for N56 whch are compued for he IPW emaor only. 9

32 on wh ncreaed ample ze han he RMSE. A dealed nvegaon no h ue however beyond he cope of h paper. For he malle ample he mean of he boorap drbuon domnae (wh very rare excepon) he non-boorapped emaor. Snce he gan are omeme conderable and hey eem o ex for all emaor, furher reearch on boorappng n he dynamc machng framewor eem o be warraned. If S-DCIA no vald a n DGP, E ( S, S ) and E ( S, S ) are no denfed. Furhermore, he one-ep emaor are nconen for E( ), E( ), E( ), E( ) a well a for E ( S ) and E ( S ). In Table 6 he enre of RMSE for nconen emaor are haded. Table 6: Reul for DGP Sequenal propeny core machng eq. on X one PSM one on X prop. wegh Po. Populaon N rue ba d. ew. ur. ba MAE x RMSE x ouc. mean d BS BS all BS BS BS all BS Table 6 o be connued. 3

33 Table 6: connued Sequenal propeny core machng eq. on X one PSM one on X prop. wegh Po. Populaon N rue ba d. ew. ur. ba MAE x RMSE x ouc. mean d BS BS all BS BS BS all BS Noe: See noe below Table 5. RMSE relang o emaor ha are nconen for he parcular effec are haded. Table 6 how ha all emaor are everely baed for he undenfed effec. For he denfed effec for whch he equenal emaor dffer o he one-ep emaor, he former are eher unbaed or exhb a ba ha dappearng wh ncreang ample ze. Wh one excepon, he RMSE for he nconen one-ep emaor alway larger han for he equenal one. The ngle excepon he emaon of E ( ), for whch he equenal emaor are baed, alhough he ba geng maller he larger he ample. The ba of he one-ep emaor maller for hee ample ze and hu he RMSE maller. However, comparng he yemac developmen of he ba wh ncreang ample ze (.48,.3,.8 for he equenal emaor and.,., -. for he one-ep emaor) may ndcae ha for larger ample ze he one-ep emaor are domnaed by he equenal one (he ba of he equenal machng emaor may evenually dappear, bu peed eem o be below N whch could be an ndcaon of he problem poned ou by Abade and Imben,, dcued n he prevou econ). 3