Time Invariant Regressor in Nonlinear Panel Model with Fixed Effects 1

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1 me Ivarat Regressor Nolear Pael Model wt Fxed ffects Jyog Ha UCLA February 26, 23 I am grateful to Da Ackerberg ad Jerry Hausma for elpful commets.

2 Abstract s paper geeralzes Hausma ad aylor s (98) tuto, ad develop a metod of dealg wt tme-varat regressor olear pael model wt Þxed effects. e metod requres large umber of observatos per dvdual ( ) as well as a large umber of dvduals ().

3 Itroducto Pael data allow te possblty of cotrollg for uobserved dvdual specþc effects, wc may be correlated wt observed explaatory varables. I lear models, suc Þxed effects are usually elmated by dfferecg, wc yelds a model free of suc cdetal parameter. A uteded cosequece of dfferecg s tat t also elmates tme-varat regressor, wc reders te coeffcet of te tmevarat regressor udetþed. Hausma ad aylor (98) used a strumetal varables approac to overcome suc problem. I ts paper, I geeralze ter tuto, ad develop a metod of dealg wt tme-varat regressor te olear framework. s metod requres large umber of observatos per dvdual ( ), so ts applcablty s lmted to te case were s large. Because te strumetal varables estmato requres a large umber of dvduals (), I adopt a asymptotc framework were bot ad grow to Þty at te same rate. s result s made possble by recet teccal progress of pael aalyss uder suc alteratve asymptotcs. See, e.g., Arellao (2), Ha ad Kuersteer (22, 23), Ha ad Newey (22), ad Wouterse (22). 2 Prelmares Suppose tat we are gve a set of momet restrctos [g (y t, γ wδ x tθ )] =, =,...,; t =,..., () for some vector-valued fucto g, were y t, x t,adw deote te depedet varable te tt perod, tme-varyg regressor te tt perod, ad tme-varat regressor. Uobserved dvdual specþc effects are summarzed by te scalar varable γ. Our prmary focus s to estmate te coeffcet δ of te tme-varat regressor w we γ s possbly correlated wt w ad x t. We ow dscuss ow te parameters ca be cosstetly estmated. If bot ad grow to Þty at te same rate, θ ca be -cosstetly estmated. Lettg α γ wδ we ca rewrte te model as [g (y t, α x tθ )] =, to wc we ca apply varats of recetly developed metods dscussed Arellao (2), Ha ad Kuersteer (22, 23), Ha ad Newey (22), ad Wouterse (22). erefore, te coceptual callege s to develop a cosstet estmator of δ. For ts purpose, we assume tat te data are..d. over : Codto ({y,y 2,...},{x,x 2,...},z,w, γ ) s..d. over. Suppose for a momet tat we observe α. Also suppose tat we observe a addtoal varable z wt dm (z )=dm(w ) adsuctat It s easy to geeralze te dscusso to te over-detþed case were dm (z ) > dm (w ). Because te prmary purpose of ts paper s detþcato ad cosstet estmato of δ, I focus o te exactly detþed case.

4 Codto 2 () [z γ ]=;()[z w ] s osgular It s clear tat we ca estmate δ by e δ ( P z w) ( P z α ). It s easy to see tat ts estmator s cosstet for δ as. Hausma ad aylor s (98) tuto was tat e δ would rema cosstet eve f we replace α by a ubased estmate. I te olear cotext, t seems dffcult to come up wt suc a ubased estmator for α. erefore, Hausma ad aylor s (98) metod caot be drectly appled. e basc tuto ts paper s tat, we bot ad grow to Þty at te same rate, we cacomeupwta -cosstet estmator for α,saybα. Because te estmato error becomes very smaller as te sample sze creases, te IV estmator b δ z w z bα wll be cosstet for δ geeral. 3 Cosstet stmato of α Let u (t ; θ, α ) g ( t ; θ, α ) v ( t ; θ, α ) were t (y,...,y,x,...,x ),adweredm (θ) =dm(u) =p ad dm (α) =dm(v) =. We assume tat [g ( t ; θ, α )] =, ad cosder te estmator tat solves = u ³ t ; b θ, bα = t= v ³ t ; bθ, bα t= s dcates tat we are separatg g to two compoets. e Þrst compoet u s used trougout te sample for estmato of θ. e secod compoet v s used oly for te t dvdual to estmate α. s separato was adopted because I wated to explot some recet teccal developmet Ha ad Kuersteer (23) or Ha ad Newey (22). I do ot expect ts separato to be costrag practce. For example, te case of lear models y t = α x tθ ε t (2) wt x t strctly exogeous, we may take u ( t ; θ, α ) = x t (y t α x tθ) v ( t ; θ, α ) = y t α x tθ 2

5 were y P t= y t ad x P t= x t. Note tat ts wll result te usual Þxed effects estmator of θ. We assume followg regularty codtos: Codto 3 () Gve tme-varat varables (α,z,w ), (y t,x t ) s..d. over t; () for every, G () (θ, α )=; () for eac η >, f f {(θ,α): (θ,α) (θ,α ) >η} G () (θ, α) >, were bg () (θ, α ) P t= g ( t; θ, α ), G () (θ, α ) [g ( t ; θ, α )]. Codto 4, suc tat ρ, were < ρ <. Codto 5 () e fucto g ( ; θ, α) s cotuous (θ, α) Υ; () e parameter space Υ s compact; () ere exsts a fucto M ( t ) suc tat mm2 g ( t ; θ, α ) m α m2 M ( t) m m 2,...,6 ad sup M ( t ) Q < for some Q>64. Codto 6 () m v ( t ; θ, α ) 2 > ; () lm P I >, were U ( t ; θ, α ) ³ u ( t ; θ, α ) ρ v ( t ; θ, α ), ρ v(t;θ,α ) α u(t;θ,α ) α,adi U(t;θ,α ). We ca sow tat bα are uformly cosstet for α. eorem Uder Codtos 3, 4, 5, ad 6, we ave (bα α )=λ κ, were µ λ vt /2 α t= v t = O p () ad Pr [ κ ]=o (). Here, v t v t ( t, θ, α ). Proof. See Appedx C. 4 IV stmato of δ Recall tat, f α s kow, we could ave estmate δ by e δ z w z α. (3) Because t s ot kow, we ca replace α by bα, ad cosder b δ z w z bα. (4) 3

6 eorem mples tat we ave ³ bδ δ = z w /2 z γ /2 /2 z λ o p () Note tat, f z λ = P t= z v t does ot ave a zero expectato, t would complcate asymptotc aalyss. We terefore assume tat te strumet z satsþes [z v t ]=. Codto 7 [v t z ]= Codto 7 guaratees tat te error estmato of α does ot complcate te ferece regardg bδ. Suc codto was adopted by Hausma ad aylor (98). We ave v t = ε t te lear model (2), ad Hausma ad aylor s strumet z for suc lear model s requred to satsfy [z ε t ]=. It turs out tat te substtuto of bα for α (3) does ot affect te asymptotc dstrbuto of te resultat estmator (4). It s because /2 /2 P z λ = o p () uder Codtos 3, ad 7. We terefore obta ³ bδ δ = z w /2 z γ o p () Note tat ³ eδ δ = z w /2 z γ o p () as well. s s a tutve result. Note tat b δ s a IV estmator of bα o w.becausebα s a proxy for α, ad because te measuremet error dsappears as, we sould expect tat te asymptotc dstrbuto of ³ bδ δ sould be detcal to tat of ³ eδ δ uder te large asymptotcs. As a cosequece, we obta eorem 2 Assume Codtos, 2, 3, 4, 5, 6, ad 7. Furter assume tat [ z w ] < ad γ 2 z z <. Weteave ³ ³ bδ δ N, ( [z w ]) γ 2 z z ]) ( [w z. 5 Summary ad Future Work I ts paper, I geeralzed Hausma ad aylor s (98) result to olear pael models wt Þxed effects. e result s expected to ave a mplcato for some olear models of socal teractos. ssbecausetemodel() could be uderstood as a model wt some partcular kds of socal effects. Followg te termology troduced by Mask (993), let y t deote te outcome of te tt dvdual of te t group. Also, let x t deote te dvdual level caracterstcs tat operate at te dvdual level oly. If w s equal to te group average [y t ] of y t, te te model cotas edogeous socal effects. If w s equal to te group average [x t ] of x t, te te model cotas exogeous (cotextual) socal effects. Fally γ, wc s ot observed by te ecoometrca, captures te presece of correlated 4

7 group effects. erefore, te result ts paper may be applcable to some partcular class of models of socal teractos. 2 2 IdetÞcato of δ wt varous kds of socal effects s expected to requre some more fuctoal restrcto, suc as te excluso restrcto cosdered by Graam ad Ha (23), or selecto effects as cosdered by Brock ad Durlauf (2). Nolear models of socal teracto are expected to be plagued by multple equlbra geeral. erefore, t s mportat tat te model () s ot uderstood as te Þrst order codto of te log lkelood. After all, textbook style lkelood may ot ext. e model () sould be uderstood as a set of momet restrctos tat are vald regardless of te potetal multplcty of equlbra.wt multple equlbra, te..d. assumpto Codto may soud puzzlg. Suc puzzle would dsappear f te equlbrum selecto s cosdered as a Þxed effect, altoug t does ot appear te model (). 5

8 Appedx A Cosstecy Lemma Assume tat W t are d wt [W t ]=ad Wt 2k <. e, ³P 2k t= W t = C(k) k o( k ) for some costat C(k). Proof. By adoptg a argumet te proof of Lemma 5. Lar (992), we ave ³P 2k t= W P t = 2k P C (α,..., α j ) P j= α I jq s= W α s t s, (5) were for eac Þxed j {,..., 2k}, P α exteds over all j-tuples of postve tegers (α,..., α j ) suc tat α... α j =2k ad P I exteds over all ordered j-tuples (t,..., t j ) of tegers suc tat t j. Also, C(α,..., α j ) stads for a bouded costat. Note, tat f j>kte at least oe of te dces Qj α j =. By depedece ad te fact tat [W t ]=tfollows tat s= W t αs s =weever j>k. ³P s sows tat t= W t 2k C (k) k Wt 2k for some costat C(k). Lemma 2 Suppose tat, for eac, {ξ t,t=, 2,...} s a sequece of zero mea..d. radom varables. We assume tat {ξ t,t=, 2, 3} are depedet across. We also assume tat ξ t 6 <. Fally, we assume tat = O ( ). Weteave Pr for every η >. ξ > η = o 2 t t= Proof. Usg Lemma, weobta 6 ξ C t 8 ξ 6 t, t= were C>s a costat. erefore, we ave 2 Pr ξ > η t 2 C8 6 η 6 ξ 6 t or t= 2 Pr ξ > η C t 6 η t= 6 ξ 6 t = o (). 6

9 Lemma 3 Suppose tat Codtos 4 ad 5 old. We te ave for all η > tat Pr sup bg () (θ, α) G () (θ, α) η = o (θ,α) Proof. Let η > be gve. We ote tat Pr sup bg () (θ, α) G () (θ, α) η (θ,α) Pr sup bg () (θ, α) G () (θ, α) η. (6) Let ε > be cose suc tat 2ε [M ( t )] < η 3. Dvde Υ to subsets Υ, Υ 2,...,Υ M(ε) suc tat (θ, α) θ, α < ε weever (θ, α) ad θ, α are te same subset. Let (θ j, α j ) deote some pot Υ j for eac j. e, sup bg () (θ, α) G () (θ, α) =sup bg () (θ, α) G () (θ, α), (θ,α) j Υ j ad terefore Pr sup bg M(ε) () (θ, α) G () (θ, α) η Pr sup bg () (θ, α) G () (θ, α) η (7) (θ,α) j= Υ j For (θ, α) Υ j,weave bg () (θ, α) G () (θ, α) bg () (θ j, α j ) G () (θ j, α j ) ε (M ( t ) [M ( t )]) 2ε [M ( t)] (θ,α) t= e, Pr sup bg () (θ, α) G () (θ, α) η Υ j Pr bg () (θ j, α j ) G () (θ j, α j ) η 3 = o 2 Pr (M ( t ) [M ( t )]) η 3ε t= (8) by Lemma 2. Combg (6), (7), (8), ad = O ( ), we obta te desred cocluso. eorem 3 Uder Codtos 4, 5, ad 3, Pr b θ θ η = o for every η >. ³ ³ Proof. It s useful to ote bθ, bα,...,bα solves bg () bθ, bα =for every. I oter words, ³ bθ, bα,...,bα solves m P bg () (θ, α ). Letη be gve, ad let ε f f (θ,α) (θ,α ) >η G() (θ, α). Note tat ε >. Wt probablty equal to o,weave m bg () (θ, α ) m θ θ >η,α,...,α (θ,α) (θ bg () (θ, α ),α ) >η > m (θ,α) (θ,α ) >η > 2 ε > = 7 bg () ³ bθ, bα, G() (θ, α ) 2 ε

10 were te secod ad fourt ³ equaltes are based o Lemma 3, ad te last equalty follows from te deþto of te estmators bθ, bα,...,bα. We ca terefore coclude tat Pr b θ θ η = o. eorem 4 Uder Codtos 4, 5, ad 3, Pr [ bα α η] =o for every η >. Proof. We Þrst prove tat Pr bg ³ () bθ, α G () (θ, α) η = o () (9) sup α for every η >. Notetat sup bg ³ () bθ, α G () (θ, α) α sup bg ³ ³ ³ () bθ, α G () bθ, α sup G () bθ, α G () (θ, α) α α sup bg () (θ, α) G () (θ, α) [M ( t)] bθ θ. erefore, (θ,α) Pr Pr sup α sup (θ,α) Pr b θ θ = o () bg ³ () bθ, α G () (θ, α) η bg () (θ, α) G () (θ, α) η 2 η 2( [M ( t )]) by Lemma 3 ad eorem 3. We ow get back to te proof of eorem 4. It suffces to prove tat Pr bα α η = o () for every η >. Let η be gve, ³ ad let ε f f {α: α α >η} G () (θ, α ) >. Codto o te evet sup α bg () bθ, α G () (θ, α) 3 o, ε wc as a probablty equal to o by (9). We te ave m α α >η bg ³ () bθ, α > m α α >η ad terefore, bα α η for every. G () (θ, α ) 2 ε > ³ 2 ε > = G() b bθ, bα, 8

11 B xpaso Lettg U ( t ; θ, α ) u ( t ; θ, α ) ρ v ( t ; θ, α ) µ u (t ; θ, α ) v (t ; θ, α ) ρ α v ( t ; θ, bα (θ)) t= V ( t ; θ, α ) v ( t ; θ, α ) we ca recogze tat b θ s a soluto to = U t= ³ ³ t ; bθ, bα b θ. ³ Let F (F,...,F ) deote te collecto of dstrbuto fuctos. Let bf F b,..., bf, were bf deotes te emprcal dstrbuto fucto for te stratum. DeÞe F (²) F ² ³ F b F for ², /2.ForeacÞxed θ ad ², letα (θ,f (²)) be te soluto to te estmatg equato = V [θ, α (θ,f (²))] df (²), α ad let θ (F (²)) be te soluto to te estmatg equato = U ( t ; θ (F (²)), α (θ (F (²)),F (²))) df (²). By aylor seres expaso, we ave ³ θ F b θ (F )= θ ² () µ 2 θ ²² () µ 3 θ ²²² (e²), 2 6 were θ ² (²) dθ (F (²))/ d², θ ²² (²) d 2 θ (F (²)) ± d² 2,...,ade² s somewere betwee ad /2. We terefore ave ³ ³ θ bf θ (F ) = θ ² () µ 2 θ ²² () r θ ²²² (e²). () 2 6 e last term () ca be sow to be o p () bytesamemetodashaadnewey(22). Let (,²) U ( ; θ (F (²)), α (θ (F (²)),F (²))) () e Þrst order codto may be wrtte as = (,²) df (²) (2) 9

12 Dfferetatg repeatedly wt respect to ², weobta = d (,²) df (²) (,²) d (3) d² = d 2 (,²) d² 2 df (²)2 d (,²) d (4) d² = d 3 (,²) d² 3 df (²)3 d 2 (,²) d² 2 d (5) were ³ F b F. B. θ ² () Because d (,²) d² = (,²) (,²) α α α we may rewrte (3) as = µ (,²) (,²) α valuatg at ² =, ad otg tat U =, we obta α θ ² () = We terefore ave θ ² () = I U d (,²) α α (,²) α I U t= N, lm I lm α df (²) Φ lm (,²) d (6) I (7) B.2 α θ ad α ² I te t stratum, α (θ,f (²)) solves te estmatg equato V ( ; θ, α (θ,f (²))) df (²) = (8) Dfferetatg te LHS wt respect to θ ad ², weobta µ V (, θ,²) V (, θ,²) α (θ,f (²)) = df (²) α df (²), µ V (, θ,²) α (θ,f (²)) = df (²) V (, θ,²) d. α

13 Observe tat α (θ,f (²)) = α (θ,f (²)) = µ V (, θ,²) α µ V (, θ,²) α µ V (, θ,²) df (²) µ df (²) df (²), V (, θ,²) d. quatg tese equatos to zero ad solvg for dervatves of α evaluated at ² =gves µ α θ V V () = α = O (), (9) µ α ² V () = V t = O p (), (2) were α t= α θ α (θ,f ()), α ² α (θ,f ()). B.3 θ ²² () Because d (,²) d² = k= (,²) l= k= (,²) α l α l l= (,²) α l α l we ave d 2 (,²) d² 2 = k= k = l= k= l= k= k = l= l = k= 2 (,²) 2 (,²) α l α l k= l= k= k = l= k= l= l = l= k= 2 (,²) α l 2 (,²) α l α l (,²) α l α l α l 2 α l l= k= k = 2 (,²) α l (,²) 2 θ k 2 α l (,²) α l 2 θ k α l 2 k= 2 (,²) α l α l α l k = (,²) 2 α l α l l= l= l = k= k = l= l= 2 (,²) α l α l l= l = k= (,²) 2 α l α l 2 (,²) α l α l α l 2 (,²) α l α l 2 α l 2 (,²) α l α l α l α l α l α l

14 valuatg (4) at ² =, ad otg tat [U α ]=,weobta = 2 U k 2 k= k = l= k= k = 2 2 U k α l k α l U l= k= k k= 2 U αl α l α l α l l= l = k= k = 2 2 U αl α l α l α l l= l = k= l= 2 µ U d θ k 2 µ U d k α k= l l= k= 2 µ U αl d α l l= Because 2 U αl α l α l α l l= l = α = 2 U () α α α. α 2 U (p) α α α ³ P ³ t= = V t V α ³ P ³ t= V t V α 2 U () α α. 2 U (p) α α ³ ³ V α V α 2 U α l 2 θ k 2 l = ³ ³ k α l 2 U αl α l α l α l αl P t= V t P t= V t 2 l= µ U αl d α l = 2 t= µ U α V α t= V t 2 k= 2 2 k= l= l = k= l= 2 U α l 2 U α l α l 2 k= l= αl αl α l µ U d µ U α l d αl µ µ µ = O p O p = O p µ µ µ = O p O p = O p µ µ µ θ k = O p O p = O p µ µ µ = O p O p = O p 2

15 ad 2 l= k= k = k= k = l= l = k= k = 2 U 2 U αl k α l 2 U αl α l α l α l we may wrte ³ P ³ t= I θ ²² () = V t V α ³ P ³ t= V t V α µ U 2 V α α t= µ µ µ = O () O p O p = O p µ µ µ = O () O p O p = O p µ µ µ = O () O p O p = O p 2 U () α α. 2 U (p) α α t= V t ³ ³ V α V α ³ ³ P t= V t P t= V t or Note tat ad = 2 O p µ µ 2 θ ²² () 2 r r = I I t= ( µ trace t= V t µ V α µ U () α ³ P ³ t= V ³ V α t P ³ t= V t t= 2 U () α l α l V α µ U α 2 U () α l α l V α V α V α µ V t= 2 U () α α. 2 U (p) α α t= ³ ³ µ V α V t = α ³ V α P t= V t ³ V P α t= V t V t o p () t= [V V ] ) o p () ( µ V trace α V t ) U V () α o p () It terefore follows tat 2 µ 2 θ ²² () = 2 r I Υ o p () 3

16 B.4 α θθ, α θ², ad α ²² ³ Secod order dfferetato = α 2 2, 2 2, 2 2 of (8) yelds 2 V (, θ,²) df (²) α µ (θ,f (²)) 2 V (, θ,²) α df (²) µ 2 µ V (, θ,²) α (θ,f (²)) V (, θ,²) 2 α (θ,f (²)) df (²) α df (²) α µ 2 V (, θ,²) α (θ,f (²)) α (θ,f (²)) df (²), ad = = µ 2 µ V (, θ,²) α (θ,f (²)) V (, θ,²) 2 α (θ,f (²)) df (²) df (²) α α µ 2 V (, θ,²) α (θ,f (²)) α (θ,f (²)) α 2 df (²) µ V (, θ,²) V (, θ,²) α (θ,f (²)) d d, α µ V (, θ,²) 2 µ α (θ,f (²)) 2 µ V (, θ,²) α (θ,f (²)) df (²) α 2 α 2 df (²) µ V (, θ,²) α (θ,f (²)) 2 d. α 2 ese tree equaltes caracterzes 2 α (θ,f (²)) Lemma 4 Pr b θ (²) θ η = o () ², 2 α (θ,f (²)),ad 2 α (θ,f (²)). 2 ad Pr ² bα (²) α η = o () for every η >. Proof. Oly te Þrst asserto s proved. e secod asserto ca be proved smlarly. Let η be gve, ad let ε f f {(θ,α): (θ,α) (θ,α ) >η} G() (θ, α) >. Recall tat F (²) F ² ³ bf F, ², We ave ³ g ( ; θ, α (θ)) df (²) = ² G () (θ, α )² G b () (θ, α ) ad ³ g ( ; θ, α (θ)) df (²) G () (θ, α ) ² bg () (θ, α) G () (θ, α) bg () (θ, α) G () (θ, α). 4

17 By Lemma 3, we ave Pr ² sup (θ,α) g ( ; θ, α (θ)) df (²) G () (θ, α) η = o erefore, for every ² wt probablty equal to o,weave m θ θ >η,α,...,α g ( ; θ, α (θ)) df (²) m (θ,α) (θ,α ) >η g ( ; θ, α (θ)) df (²) > m (θ,α) (θ,α ) >η > 2 ε > = We terefore obta tat Pr ² b θ (²) θ η = o. G () (θ, α ) 2 ε g ³ ; b ³ θ (²), α bθ (²) df (²) Lemma 5 Suppose tat, for eac, {ξ t (φ),t=, 2,...} sasequeceofzeromea..d. radom varables dexed by some parameter φ Φ. We assume tat {ξ t (φ),t=, 2, 3} are depedet across. We also assume tat sup φ Φ ξ t (φ) B t for some sequece of radom varables B t tat s..d. across t ad depedet across. Fally, we assume tat B t 64 <, ad = O ( ). We te ave µ Pr ξ t (φ ) > υ = o t= for every υ suc tat υ < 6. Here, {φ } s a arbtrary sequece Φ. Proof. By Markov s equalty, Pr sup ξ φ Φ t (φ ) > υ =Pr sup ξ t= φ Φ t (φ ) > 3 5 υ t= P sup φ Φ t= ξ t (φ ) 64 = υ η 64 were te last equalty s based o domated covergece. By Lemma, weave 64 ξ t (φ ) C 32 ξ t (φ ) C B t 64 t= for some C. erefore, we ave 2 Pr ξ t (φ ) > υ t= 2 C 32 ³ 64υ υ η = O 32 5, 64 P sup φ Φ t= ξ t (φ ) υ η 64, 5

18 ad Pr t= ξ t (φ ) > υ Pr t= ξ t (φ ) > υ = C υ η 64 = o (). Lemma 6 Suppose tat K ( ; θ (²), α (θ (²),²)) s equal to mm2 g ( t ; θ (²), α (θ (²),²)) m α m2 for some m m 2,...,5. e, for ay η >, weave Pr K ( ; θ (²), α (θ (²),²)) df (²) ² [K ( t ; θ, α )] > η = o ad Pr Also, Pr ² ² K ( ; θ (²), α (θ (²),²)) df (²) [K ( t ; θ, α )] > η = o. K ( ; θ (²), α (θ (²),²)) d >C υ = o for some costat C> ad υ < 6. Proof. Note tat we may wrte K ( ; θ (²), α (θ (²),²)) df (²) K ( ; θ (²), α (θ (²),²)) df K ( ; θ (²), α (θ (²),²)) df (²) K ( ; θ (), α (θ (), )) df (²) K ( ; θ (), α (θ (), )) df (²) K ( ; θ (), α (θ (), )) df M( t )(kθ(²) θk α (θ(²),²) α ) d F (²) ² ³ K ( ; θ (), α (θ (), )) d F b F. erefore, we ave K ( ; θ (²), α (θ (²),²)) df (²) [K ( t ; θ, α )] kθ (²) θk [M ( t )] /2 (α (θ (²),²) α ) 2 t= M ( t ) t= [M ( t )] 2 M ( t ) K ( t ; θ (), α (θ (), )) [K ( t ; θ (), α (θ (), ))], 6 t= /2

19 te RHS of wc ca be bouded by usg Lemmas?? ad 4 absolute value by some η > wt probablty o. Because K ( ; θ (²), α (θ (²),²)) df (²) [K ( t ; θ, α )] θ (²) θ [M ( t )] M ( t ) t= α (θ (²),²) α [M ( t )] M ( t ) t= M ( t ) [M ( t )], we ca boud ² t= K ( ; θ (²), α (θ (²),²)) df (²) [K ( t ; θ, α )] absolute value by some η > wt probablty o. Usg Lemma 5, we ca also sow tat K ( ; θ (²), α (θ (²),²)) d ca be bouded by absolute value by C υ for some costat C> ad υ suc tat υ < 6 wt probablty o. Lemma 7 Pr α θ ² (²) >C Pr α ² (²) >C υ ² = o = o for some costat C> ad υ < 6. Proof. From Appedx B.2, we obta µ µ α (θ,f (²)) V (, θ,²) V (, θ,²) = df (²) df (²), α µ µ α (θ,f (²)) V (, θ,²) = df (²) V (, θ,²) d. α ³ R Usg Lemma 6, we ca see tat V (,θ,²) α df (²) s uformly bouded away from zero wt probablty o. We ca also see tat, wt probablty o, R V (,θ,²) df (²) s uformly bouded by some costat C, ad R V (, θ,²) d s uformly bouded by C υ. 7

20 Lemma 8 Pr θ ² (²) >C υ = o ² for some costat C> ad υ < 6. Proof. From (6), we ave θ ² µ (,²) (²) = (,²) α α df (²) µ α (,²) df (²) (,²) d α Usg Lemmas 6, ad 7, we ca boud te deomator of θ ² (²) by some C>, ad te umerator by some C υ wt probablty o. Lemma 9 θ Pr α r θ r ² (²) >C θ Pr α r ² ² (²) >C υ Pr (²) >C ³ υ 2 α ²² ² = o = o = o for some costat C> ad υ < 6. Here, αθ rθ r 2 α r r. We smlarly deþe αθ r². Proof. Note tat 2 V (, θ,²) = df (²) α µ (θ,f (²)) 2 V (, θ,²) α df (²) µ 2 µ V (, θ,²) α (θ,f (²)) V (, θ,²) 2 α (θ,f (²)) df (²) α df (²) α µ 2 V (, θ,²) α (θ,f (²)) α (θ,f (²)) df (²), = α 2 µ 2 V (, θ,²) α (θ,f (²)) df (²) α µ V (, θ,²) 2 α (θ,f (²)) df (²) α µ V (, θ,²) V (, θ,²) d α µ 2 V (, θ,²) α 2 α (θ,f (²)) d α (θ,f (²)) df (²) ad µ V (, θ,²) 2 µ α (θ,f (²)) 2 µ V (, θ,²) α (θ,f (²)) = df (²) α 2 α 2 df (²) µ V (, θ,²) α (θ,f (²)) 2 d. α e result te follows by applyg te same argumet as te proof of Lemma 7. 8, α (θ,f (²)) 2

21 Lemma Pr ² θ ²² (²) >C ³ υ 2 = o for some costat C> ad υ < 6. Proof. e cocluso follows by usg te caracterzato of θ ²² (²) Appedx B.3, ad Lemmas 6, 7, 8, ad 9. Lemma θ Pr α r θ r θ r ² θ Pr α rθ r ² (²) Pr Pr ² θ α r²² (²) >C ² α ²²² ² (²) >C ³ (²) >C >C υ ³ υ 2 υ 3 = o = o = o = o for some costat C> ad υ < 6. Proof. It was see Appedx B.4 tat ad = = µ 2 V (, θ,²) α (θ,f (²)) df (²) r α µ V (, θ,²) 2 µ α (θ,f (²)) 2 V (, θ,²) α (θ,f (²)) α (θ,f (²)) df (²) α r α 2 df (²) r µ V (, θ,²) V (, θ,²) α (θ,f (²)) d d, r α r µ V (, θ,²) 2 µ α (θ,f (²)) 2 µ V (, θ,²) α (θ,f (²)) df (²) α 2 α 2 df (²) µ V (, θ,²) α (θ,f (²)) 2 d. α 2 9

22 We terefore obta 3 µ V (, θ,²) 3 V (, θ,²) α (θ,f (²)) = df (²) df (²) r r r α r r r µ 2 α (θ,f (²)) 2 V (, θ,²) df (²) α µ (θ,f (²)) 3 V (, θ,²) df (²) r r α r r α r r α µ (θ,f (²)) α (θ,f (²)) 3 V (, θ,²) r r α 2 df (²) r µ 3 µ V (, θ,²) α (θ,f (²)) 3 V (, θ,²) α (θ,f (²)) df (²) α r r r α 2 df (²) r r µ 2 V (, θ,²) 2 α (θ,f (²)) df (²) α r r r µ 2 V (, θ,²) 2 µ α (θ,f (²)) 2 V (, θ,²) α (θ,f (²)) df (²) α r r r α 2 df (²) r µ V (, θ,²) 3 α (θ,f (²)) df (²) α r r r µ 3 V (, θ,²) α (θ,f (²)) α (θ,f (²)) α 2 df (²) r r r µ 3 V (, θ,²) α (θ,f (²)) α (θ,f (²)) α (θ,f (²)) α 3 df (²) r r r µ 2 V (, θ,²) 2 α (θ,f (²)) α (θ,f (²)) α 2 df (²) r r r µ 2 V (, θ,²) α (θ,f (²)) 2 α (θ,f (²)) df (²), r r r α 2 α (θ,f (²)) r 2 α (θ,f (²)) r r 2

23 wc caracterzes 3 α (θ,f (²)) r r r, = µ 3 µ V (, θ,²) α (θ,f (²)) 3 V (, θ,²) α (θ,f (²)) α (θ,f (²)) df (²) r r α r α 2 df (²) r µ 2 V (, θ,²) 2 α (θ,f (²)) df (²) r α r µ 2 V (, θ,²) 2 µ α (θ,f (²)) 2 V (, θ,²) α (θ,f (²)) 2 α (θ,f (²)) df (²) r α r α 2 df (²) r r µ V (, θ,²) 3 α (θ,f (²)) df (²) α r r µ 3 V (, θ,²) α (θ,f (²)) α (θ,f (²)) r α 2 df (²) r µ 3 V (, θ,²) α (θ,f (²)) α (θ,f (²)) α (θ,f (²)) α 3 df (²) r r µ 2 V (, θ,²) 2 α (θ,f (²)) α (θ,f (²)) α 2 df (²) r r µ 2 V (, θ,²) α (θ,f (²)) 2 α (θ,f (²)) α 2 df (²) r r 2 µ V (, θ,²) 2 V (, θ,²) α (θ,f (²)) d d r r r α r µ 2 µ V (, θ,²) α (θ,f (²)) 2 V (, θ,²) α (θ,f (²)) α (θ,f (²)) d r α r α 2 d r r µ V (, θ,²) 2 α (θ,f (²)) d, α r r wc caracterzes 3 α (θ,f (²)) r r, = µ 2 V (, θ,²) 2 µ α (θ,f (²)) 2 V (, θ,²) df (²) r α 2 α 2 µ V (, θ,²) 3 α (θ,f (²)) df (²) α r 2 µ 3 µ 2 µ V (, θ,²) α (θ,f (²)) 3 V (, θ,²) r α 2 df (²) α 3 µ 2 V (, θ,²) α (θ,f (²)) 2 α (θ,f (²)) 2 α 2 df (²) r µ 2 µ V (, θ,²) α (θ,f (²)) 2 V (, θ,²) 2 d 2 r α α 2 µ V (, θ,²) 2 α (θ,f (²)) 2 d, α r α (θ,f (²)) 2 α (θ,f (²)) df (²) r 2 α (θ,f (²)) df (²) r µ α (θ,f (²)) α (θ,f (²)) α (θ,f (²)) d r 2 2

24 wc caracterzes 3 α (θ,f (²)) r,ad 2 µ 2 V (, θ,²) α (θ,f (²)) 2 α (θ,f (²)) = α 2 df (²) 2 µ V (, θ,²) 3 α (θ,f (²)) df (²) α 3 µ 3 µ 3 V (, θ,²) α (θ,f (²)) α 3 df (²) µ 2 V (, θ,²) α (θ,f (²)) 2 α (θ,f (²)) 2 df (²) 2 α 2 µ 2 V (, θ,²) 2 α 2 d µ α (θ,f (²)) µ V (, θ,²) µ 2 V (, θ,²) α 2 2 µ V (, θ,²) 2 α α d 2 α (θ,f (²)) 2 µ 2 α (θ,f (²)) d d 2 α (θ,f (²)) 2, wc caracterzes 3 α (θ,f (²)) 3. Ispectg tese dervatves ad applyg Lemmas 6, 7, ad 9, we obta te desred result. Lemma 2 Pr ² θ ²²² (²) >C ³ υ 3 = o for some costat C> ad υ < 6. Proof. From (5), we ave = d 3 (,²) d² 3 df (²)3 d 2 (,²) d² 2 d CombgLemmas6,7,8,9,ad, we ca boud P R ³ d 2 (,²) d² d 2 by C υ 3 wt probablty o. It was see Appedx B.3 tat te r-t compoet d2 (r) (,²) d² of d2 (,²) 2 d² s equal 2 to d 2 (r) (,²) d² 2 = (,²) 2 (r) (,²) (,²) (r) (,²) 2 θ 2 2 (r) (,²) α µ (r) (,²) α 2 (r) (,²) α α α 2 (r) (,²) 2 α α (r) (,²) α 2 (r) µ (,²) α α 2 (r) (,²) α α α 2 (r) µ 2 (,²) α α 2 2 (r) µ (,²) α α α 2 2 α (r) (,²) 2 α α (r) (,²) α 2 θ α 2 2 (r) µ (,²) α α 2 (r) µ 2 (,²) α 2 α 2. α 2 R d 3 (,²) Usg Lemmas 6, 7, 8, 9, ad aga, we ca coclude tat P d² df 3 (²) s equal to ³ P R (,²) df (²) 3 θ plus terms tat ca all be bouded by P R d 2 (,²) 3 d² d 2 by C 22 ³ υ 3

25 wt probablty o ³ P. Because R (,²) df (²) s bouded away from by Lemma 6, we obta te desred cocluso. Lemma 3 ³ bθ θ = θ ² () 2 Proof. Follows from equato () ad Lemma 2. µ 2 θ ²² () o p () C Proof of eorem Let bα (²) be suc tat = V ³ ; b θ (²), bα (²) df (²) (Note slgt dfferece of deþto. I prevous sectos, bα (θ,²) was uderstood to be a soluto to R V ( ; θ, α (θ,f (²))) df (²) =.) Usg te same argumets as earler, we are lookg for te expaso bα (²) α = bα ² () for some ², /2. Let 2 bα²² v (,²) V (θ (F (²)), α (F (²))). ( ²) e Þrst order codto may be wrtte as = v (,²) df (²) Dfferetatg repeatedly wt respect to ², weobta dv (,²) = df (²) v (,²) d (2) d² d 2 v (,²) dv (,²) = d² 2 df (²)2 d (22) d² were ³ bf F. Because dv (,²) d² = v (,²) v (,²) α we may rewrte (2) as µ v (,²) = v (,²) α valuatg at ² =we obta bα ² () = ( [V α ]) µ α α df (²) P t= v (, θ, α ) V θ v (,²) d θ ² () (23) 23

26 were θ ² () s deþed(7). It also follows tat µ µ bα ² v (,²) v (,²) (²) = df (²) α df (²) θ ² (²) Next, cosder d 2 v (,²) d² 2 = ² [,/ ] v (,²) 2 v (,²) ( α ) 2 v (,²) 2 θ () 2 2 v (,²) α µ 2 α v (,²) 2 α α () 2 α v (,²) d. (24) suc tat bα ²² (²) s caracterzed by = θ ² (²) v (,²) df (²) θ ² v (,²) (²) df (²) θ ²² (²) v (,²) 2 df (²) θ ² (²) bα ² 2 α (²) v (,²) ( α ) 2 df (²)(bα ² (²))2 v (,²) df (²) bα ²² v (,²) (²) α d θ ² v (,²) (²) d ² bα (²) (25) α We ow sow tat Pr (bα α ) > υ = o, for ay υ < 6. For ts purpose, t suffces to prove Pr sup bα ² ² [,/ (²) > υ = o, ] Pr bα ² () > υ = o, Pr sup bα ²² (²) > ³ υ 2 = o. I order to prove te Þrst clam, we ote tat Pr sup θ ² (²) υ = o ² [,/ ] from Lemma (8). By Lemma (6), we also ave µ Pr sup v (,²) v (,²) ² [,/ ] df (²) > η = o, Pr sup v (,²) v (,²) df (²) α α > η = o. ² [,/ ] By Lemma (6) aga, t follows tat Pr sup v (,²) d > / υ = o. ² [,/ ] 24

27 s proves te result for bα ² (²) as well as bα (). Forbα ²² (²), te result follows from represetato (25) as well as Lemmas (6), (8), ad (). We ow prove eorem. For ts purpose, we combe bα (²) = α bα ² () 2 bα²² ( ²), bα ² () = ( [V α ]) µ θ ² () = P t= v (, θ, α ) V θ I t= U θ ² () ad obta µ (bα α ) = ( [V α ]) P t= v (, θ, α ) ( [V α ]) V θ I 2 bα²² ( ²), from wc te cocluso follows. t= U 25

28 Refereces [] Arellao, M. (2): Dscrete Coces wt Pael Data, Ivestgacoes coómcas Lecture, V Smposo de Aálss coómco, Bellaterra. [2] Brock, W. ad S. Durlauf (2): Iteractos-based Models, J. Heckma &. Leamer, ds. Hadbook of coometrcs 5, pp Amsterdam: Nort-Hollad. [3] Graam, B. S., ad J. Ha (23): IdetÞcato ad stmato of te Lear--Meas Model of Socal Iteractos, upublsed mauscrpt. [4] Ha, J. ad G. Kuersteer (22): Asymptotcally Ubased Iferece for a Dyamc Pael Model wt Fxed ffects We Bot ad are Large, coometrca, 7, [5] Ha, J. ad G. Kuersteer (23): Bas Reducto for Dyamc Nolear Pael Models wt Fxed ffects, upublsed mauscrpt. [6] Ha, J. ad W.K. Newey (22): Jackkfe ad Aalytcal Bas Reducto for Nolear Pael Models, upublsed mauscrpt. [7] Hausma, J., ad W. aylor (98), Pael Data ad Uobservable Idvdual ffects, coometrca 49, pp [8] Mask, C. (993): IdetÞcato of edogeous socal effects: te reßecto problem, Revew of coomc Studes 6, pp [9] Wouterse,. (22): Robustess agast Icdetal Parameters, upublsed mauscrpt. 26