SUPPLEMENT TO SEQUENTIAL ESTIMATION OF STRUCTURAL MODELS WITH A FIXED POINT CONSTRAINT (Econometrica, Vol. 80, No. 5, September 2012, )

Transcript

1 Econometrca Supplementary Materal SULEMENT TO SEQUENTIAL ESTIMATION OF STRUCTURAL MODELS WITH A FIXED OINT CONSTRAINT (Econometrca, Vol. 80, No. 5, September 2012, ) BY HIROYUKI KASAHARA AND KATSUMI SHIMOTSU THIS SULEMENT contans the followng detals omtted from the man paper due to space constrants: (A) proof of the results n the paper, (B) auxlary results and ther proof, (C) addtonal alternatve sequental algorthms, (D) the convergence propertes of the NL algorthm for models wth unobserved heterogenety, and (E) addtonal Monte Carlo results. Throughout ths Supplement, let a.s. abbrevate almost surely, and let.o. abbrevate nfntely often. C denotes a generc postve and fnte constant that may take dfferent values n dfferent places. For matrx and nonnegatve scalar sequences of random varables {X M M 1} and {Y M M 1}, respectvely, we wrte X M = O(Y M ) (or o(y M ))a.s.f X M AY M for some (or all) A>0a.s.WhenY M belongs to a famly of random varables ndexed by τ T, we say X M = (Y M (τ)) (or o(y M (τ))) a.s.unformly n τ f the constant A>0 can be chosen the same for every τ T. For nstance, n roposton 7 below, we take τ = j 1 and Y M (τ) = j 1 ˆ NL. AENDIX A: ROOFS OF THERESULTS IN THE MAIN TEXT Throughout the proofs, the O( ) terms are unform, but we suppress the reference to ther unformty for brevty. ROOF OF ROOSITION 1: We suppress the subscrpt NL from ˆ NL.Let b>0 be a constant such that ρ(m Ψθ Ψ ) + 2b <1. From Lemma of Horn and Johnson (1985), there s a matrx norm α such that M Ψθ Ψ α ρ(m Ψθ Ψ )+b. Defne a vector norm β for x R L as x β = [x 0 0] α ; then a drect calculaton gves Ax β = A [x 0 0] α A α x β for any matrx A. From the equvalence of vector norms n R L (see, e.g., Corollary of Horn and Johnson (1985)), we can restate roposton 7 n terms of β as follows: there exsts c>0 such that j ˆ = M Ψθ Ψ ( j 1 ˆ) + O(M 1/2 j 1 ˆ β + j 1 ˆ 2 ) a.s. holds unformly n β j 1 { : 0 β <c}. We rewrte ths statement further so that t s amenable to recursve substtuton. Frst, note that M Ψθ Ψ ( j 1 ˆ) β M Ψθ Ψ α j 1 ˆ β (ρ(m Ψθ Ψ )+b) j 1 ˆ β. Second, rewrte the remander term as O(M 1/2 + j 1 ˆ β ) j 1 ˆ β.setc<b; then ths term s smaller than b j 1 ˆ β a.s. Thrd, snce ˆ s consstent, { : ˆ β <c/2} { : 0 β <c} a.s. Consequently, j ˆ β (ρ(m Ψθ Ψ ) + 2b) j 1 ˆ β holds a.s. for all 2012 The Econometrc Socety DOI: /ECTA8291

2 2 H. KASAHARA AND K. SHIMOTSU j 1 n { : ˆ β <c/2}. Because each NL updatng of (θ ) uses the same pseudo lkelhood functon, we may recursvely substtute for the j s, and hence lm k k = ˆ a.s. f 0 ˆ β <c/2. The stated result follows from applyng the equvalence of vector norms n R L to 0 ˆ β and 0 ˆ and usng the consstency of ˆ. Q.E.D. ROOF OF ROOSITION 2: We prove that the stated result holds f j 1 s n a neghborhood N NL c of ˆ NL. The stated result then follows from the strong consstency of ˆ NL. Because roposton 7 holds under the current assumpton, we have (5) j ˆ NL = M Ψθ Ψ 0 ( j 1 ˆ NL ) + f( j 1 ˆ NL ) where f(x) C(x 2 + M 1/2 x ) a.s. Let λ 1 λ 2 λ L be the egenvalues of M Ψθ Ψ 0 such that (6) λ 1 λ r > 1 λ r+1 λ L For any ɛ 0, we may apply a Jordan decomposton to M Ψθ Ψ 0 to obtan H 1 M Ψθ Ψ 0H = D + ɛj, whered = dag(λ 1 λ L ),andj s a matrx wth zeros and ones mmedately above the man dagonal (on the superdagonal) and zeros everywhere else. Defne y j = H 1 ( j ˆ NL ) and g(y) = H 1 f(hy); then multplyng (5) by H 1 gves y j = (D + ɛj)y j 1 + g(y j 1 ),wth g(y) C( y 2 + M 1/2 y ) a.s. Let y 1 j denote the frst r elements of y, and rewrte ths equaton as (7) ( ) ( y 1 j D1 0 = y 2 j 0 D 2 )( y 1 j 1 y 2 j 1 ) ( J1 0 + ɛ 0 J 2 )( y 1 j 1 y 2 j 1 ) + ( ) g1 (y j 1 ) g 2 (y j 1 ) where y 1 and g j 1 1(y j 1 ) are r 1 and D 1 and J 1 are r r wth D 1 = dag(λ 1 λ r ). We frst show that H 1 ( j ˆ NL ) > H 1 ( j 1 ˆ NL ) by provng that y 1 > j y1 j 1 under the stated assumptons. Applyng the trangle nequalty to the frst equaton of (7) gves (8) y 1 j D 1y 1 j 1 ɛj 1y 1 j 1 g 1(y j 1 ) For the frst two terms on the rght hand sde of (8), we have D 1 y 1 = j 1 ( r λ k=1 k 2 (y j 1 k ) 2 ) 1/2 (1 + 3δ) y 1 j 1 for some δ>0 from (6) and ɛj 1 y 1 j 1 δ y1 j 1 by choosng ɛ suffcently small. For the last term of (8), observe that j 1 V(c)f and only f y 1 j 1 c y2 j 1, and hence j 1 / V(c) y j 1 2 = y 1 j y 2 j 1 2 <(1 + c 2 ) y 1 j 1 2

3 ESTIMATION OF STRUCTURAL MODELS 3 Therefore, g 1 (y j 1 ) δ y 1 NL j 1 holds when Nc s suffcently small and M s suffcently large. It then follows from (8) that y 1 (1 + j δ) y1 > j 1 y1. j 1 It remans to show that j / V(c). Applyng the trangle nequalty to the second equaton of (7) gves y 2 D j 2y 2 + ɛj j 1 2y 2 + g j 1 2(y j 1 ). For the frst term on the rght hand sde, D 2 y 2 =( L λ j 1 k=r+1 k 2 (y j 1 k ) 2 ) 1/2 y 2 j 1 from (6). For the second term, smlarly to the updatng of y 1 j, by choosng ɛ and N NL c suffcently small, we have ɛj 2 y 2 + g j 1 2(y j 1 ) c 1 δ y 1 f j 1 j 1 N NL c \ V(c) and M s suffcently large. Therefore, y 2 j y2 + j 1 c 1 δ y 1 j 1 <c 1 (1 + δ) y 1 j 1 <c 1 y 1 j a.s., where the last two nequaltes use y 2 j 1 <c 1 y 1 and j 1 y1 < j 1 y1.thsproves j j / V(c). Q.E.D. ROOF OF ROOSITION 3: Frst, note that j for j 1 satsfes restrcton (2) because t s generated by Ψ(θ ). The restrctons (2) and (3) do not affect the valdty of ropostons 1 and 2 because () the fxed pont constrant n terms of Ψ(θ ) and of Ψ + (θ + ) are equvalent, and () the restrctons (2) and (3) do not affect the order of magntude of the dervatves of Ψ(θ ). For the equvalence of the egenvalues, takng the dervatve of (3) gves ( ) + Ψ Ψ(θ )= + (θ + ) 0 (9) E + Ψ + (θ + = (U ) 0 + Ψ + (θ + ) 0 ) and θ Ψ(θ ) = U θ Ψ + (θ + ). Substtutng ths nto M Ψθ Ψ 0,usngΨ 0 θ = UΨ + θ, and rearrangng terms gves M Ψθ Ψ 0 =[UM+ Ψ θ Ψ + + 0]. Therefore, the updatng formula of + and s gven by + j ˆ + NL = M + Ψ θ Ψ + + ( + j 1 ˆ + NL) + O(M 1/2 + j 1 ˆ + NL + + j 1 ˆ + NL 2 ) a.s. and j ˆ NL = E( + j ˆ NL), + respectvely. Fnally, the equvalence of the egenvalues follows from det(m Ψθ Ψ 0 λi dm()) = det(m + Ψ θ Ψ + λi + dm( + )) det( λi dm( )) and det(ψ 0 λi dm() ) = det(ψ + λi + dm( + )) det( λi dm( )). Q.E.D. ROOF OF EQUATION (4): The notaton follows page 10 of Agurregabra and Mra (2007; henceforth AM07). Let π (a x; θ) = a A (a )Π (a a x; θ) and e (a x; θ) = E[ε (a ) x ],wheree[ε (a ) x ] =E[ε (a ) x ] holds as dscussed on pages 9 10 of AM07. Let V (x) denote the soluton of frm s ntegrated Bellman equaton: (10) V (x) = { max π (a x; θ) + β } V (x )f (x x a ) + ε (a ) a A x X g(dε ; θ)

4 4 H. KASAHARA AND K. SHIMOTSU where f (x x a ) = a A (a )f (x x a a ). Let Π (a a ; θ), π (a ; θ), e (a ; θ), (a ),andv denote the vectors of dmenson X that stack the correspondng state-specfc elements of Π (a a x; θ), π (a x; θ), e (a x; θ), (a x),andv (x), respectvely. Defne the valuaton operator as Γ (θ ) (I βf ) 1 a A (a ) [π (a ; θ) + e (a ; θ)] where F s a matrx wth transton probabltes f (x x), and denotes the Hadamard product. Γ (θ ) gves the soluton of frm s ntegrated Bellman equaton gven θ and. Defne frm s best response mappng gven V and as (cf. equaton (15) of AM07) (11) [Υ (θ V )](a x) ( I a = arg max a A + ε (a) + β x X { π (a x; θ) }) V (x )f (x x a) g(dε ; θ) where f (x x a ) = a A (a )f (x x a a ). Then, the mappng Ψ and ts Jacoban matrx evaluated at (θ 0 0 ) are gven by ( ) ( ) Ψ1 (θ ) Υ1 (θ Γ Ψ(θ )= = 1 (θ ) 2 ) and Ψ 2 (θ ) Υ 2 (θ Γ 2 (θ ) 1 ) ( ) Ψ 0 = 0 2 Ψ 1 (θ 0 0 ) 1 Ψ 2 (θ 0 0 ) 0 where Ψ (θ 0 0 ) = 0 follows from Γ (θ ) = 0 (Agurregabra and Mra (2002), roposton 2). Q.E.D. ROOF OF ROOSITION 4: For part (a), let V denote the soluton of the Bellman equaton (10) gven,andlet be the condtonal choce probabltes assocated wth V.Sncex t = (S t a 1 t 1 a 2 t 1 ) holds and S t follows an exogenous process, we may verfy under Assumpton 3(c) that () V (S t a t 1 a ) = V t 1 (S t a t 1 a ) for t 1 a t 1 a, () V t 1 does not depend on, () Γ (θ ) = Γ (θ ) = V, and (v) Υ (θ V ) = Υ (θ V ) = for any and n the space of s. It follows from () (v) that the model becomes a sngle-agent model for each

5 ESTIMATION OF STRUCTURAL MODELS 5 player and that there exsts a unque Markov perfect equlbrum characterzed by a unque fxed pont = Ψ (θ ) = Υ (θ Γ (θ ) ) for = 1 2, where the fxed pont does not depend on the value of. Defne F(θ ) = Ψ(θ ).Snce F(θ ) = I Ψ(θ ) = I, we may apply the mplct functon theorem to F(θ ) = Ψ(θ ) at (θ ) = (θ ) under Assumpton 2(b), and there exsts an open set N θ contanng θ,anopensetn contanng, and a unque contnuously dfferentable functon (θ): N θ N such that (θ) = Ψ(θ (θ))for any θ N θ. Therefore, a Markov perfect equlbrum exsts n N when the true parameter θ 0 s n N θ. The mappng ρ(m Ψθ Ψ (θ (θ))) s a contnuous functon of θ N θ because (θ) s contnuous n θ N θ, Ψ(θ ) s contnuously dfferentable by Assumpton 2(b), M Ψθ < by Assumpton 2(b), and the spectral radus of a matrx s a contnuous functon of the elements of the matrx. The stated result then follows from Ψ(θ (θ )) = 0 (Remark 1) and the contnuty of ρ(m Ψθ Ψ (θ (θ))). For part (b), under Assumpton 3(d), θ = θ,andβ = 0, the model becomes a sngle-agent model for each player. Therefore, repeatng the argument for part (a) gves the stated result. Q.E.D. ROOF OF ROOSITION 5: Let λ = Re(λ) + Im(λ) = r cos θ + r sn θ be an egenvalue of Ψ 0. Then, the correspondng egenvalue of Λ s λ(α) = αr cos θ + αr sn θ + (1 α). Letf (α) = λ(α) 2 ; then the stated result holds because f(0) = 1and α f(0) = 2(r cos θ 1)<0frcos θ<1and α f(0)>0 f r cos θ>1. Q.E.D. AENDIX B: AUXILIARY RESULTS AND THEIR ROOF roposton 6 strengthens the weak consstency result of roposton 2 of AM07 to strong consstency. roposton 7 descrbeshow an NL step updates θ and. ROOSITION 6: Suppose that Assumpton 1 holds. Then, ( ˆθ NL ˆ NL ) (θ 0 0 ) a.s. ROOF: roposton 2 of AM07 showed weak consstency of ( ˆθ NL ˆ NL ). Therefore, strong consstency ( ˆθ NL ˆ NL ) follows from strengthenng n probablty and wth probablty approachng 1 statements n Steps 2 5 of the proof of roposton 2 of AM07 to almost surely. Frst, observe that AM07 (pp ) showed that Q M (θ ) converges to Q 0 (θ ) a.s. and unformly n (θ ). Thus, the events A M s defned nsteps2,3,and5ofam07satsfyr(a c M o ) = 0. In Step 2, we can strengthen r((θ M ) I) M 1ofAM07to(θ M M ) Ia.s. because AM07 (pp ) showed A M {(θ M ) I}and we have M r(ac M o ) = 0. In

6 6 H. KASAHARA AND K. SHIMOTSU Step 3, an analogous argument strengthens r(sup N( 0 ) θ M () θ 0 () < ε) 1nAM07tosup N( 0 ) θ M () θ 0 () <ε a.s. Smlarly, we can strengthen wth probablty approachng 1 n Steps 4 and 5 to almost surely, and strong consstency of the NL estmator follows. Q.E.D. ROOSITION 7: Suppose that Assumpton 2 holds. Then, there exsts a neghborhood N 1 of 0 such that θ j ˆθ NL = O( j 1 ˆ NL ) a.s. and j ˆ NL = M Ψθ Ψ 0 ( j 1 ˆ NL ) + O(M 1/2 j 1 ˆ NL + j 1 ˆ NL 2 ) a.s. unformly n j 1 N 1. ROOF: We suppress the subscrpt NL from ˆ NL and ˆθ NL.Forɛ>0, defne a neghborhood N (ɛ) ={(θ ) : θ θ <ɛ}. Then, there exsts ɛ 1 > 0 such that N (ɛ 1 ) N and sup (θ ) N (ɛ1 ) θθ Q 0(θ ) 1 < because θθ Q 0 (θ ) s contnuous and θθ Q 0 (θ 0 0 ) s nonsngular. Frst, we assume that ( θ j j 1 ) N (ɛ 1 ) and derve the stated representaton of θ j ˆθ and j ˆ. We later show that ( θ j j 1 ) N (ɛ 1 ) a.s. f N 1 s suffcently small. The frst-order condton for θ j s θ Q M ( θ j j 1 ) = 0. Expandng t around ( ˆθ ˆ) and usng θ Q M ( ˆθ ˆ) = 0 gves (12) 0 = θθ Q M ( θ )( θ j ˆθ) + θ Q M ( θ )( j 1 ˆ) where ( θ ) le between ( θ j j 1 ) and ( ˆθ ˆ). Wrte (12) as θ j ˆθ = θθ Q M ( θ ) 1 θ Q M ( θ )( j 1 ˆ); then the stated unform bound of θ j ˆθ follows because () ( θ ) N (ɛ 1 ) a.s. snce ( θ j j 1 ) N (ɛ 1 ) and ( ˆθ ˆ) s strongly consstent from roposton 6, and () sup (θ ) N (ɛ1 ) θθ Q M(θ ) 1 θ Q M (θ ) = O(1) a.s. snce sup (θ ) N (ɛ1 ) θθ Q 0(θ ) 1 < and sup (θ ) N 2 Q M (θ ) 2 Q 0 (θ ) =o(1) a.s., where the latter follows from Kolmogorov s strong law of large numbers and Theorem 2 and Lemma 1 of Andrews (1992). For the bound of j ˆ, frst we collect the followng results, whch follow from the Taylor expanson around (θ 0 0 ), root-m consstency of ( ˆθ ˆ),and the nformaton matrx equalty: (13) θθ Q M ( ˆθ ˆ) = Ω θθ + O ( M 1/2) a.s. θ Q M ( ˆθ ˆ) = Ω θ + O ( M 1/2) a.s. θ Ψ(ˆθ ˆ) = Ψ 0 + O( θ M 1/2) a.s. Ψ(ˆθ ˆ) = Ψ 0 + O( M 1/2) a.s. Expand the rght hand sde of j = Ψ( θ j j 1 ) twce around ( ˆθ ˆ) and use Ψ(ˆθ ˆ) = ˆ and θ j ˆθ = O( j 1 ˆ ) a.s.; then we obtan j

7 ESTIMATION OF STRUCTURAL MODELS 7 ˆ = θ Ψ(ˆθ ˆ)( θ j ˆθ) + Ψ(ˆθ ˆ)( j 1 ˆ) + O( j 1 ˆ 2 ) a.s. snce sup (θ ) N (ɛ1 ) 3 Ψ(θ )<. Applyng (13)and θ j ˆθ = O( j 1 ˆ ) a.s. to the rght hand sde gves (14) j ˆ = Ψ θ ( θ j ˆθ) + Ψ 0 ( j 1 ˆ) + O( j 1 ˆ 2 ) + O ( M 1/2 j 1 ˆ ) a.s. We proceed to refne (12) towrte θ j ˆθ n terms of j 1 ˆ and substtute t nto (14). Expandng θθ Q M ( θ ) n (12) around( ˆθ ˆ), notng that θ ˆθ θ j ˆθ and ˆ j 1 ˆ,andusng θ j ˆθ = O( j 1 ˆ ) a.s., we obtan θθ Q M ( θ ) = θθ Q M ( ˆθ ˆ)+ O( j 1 ˆ ) a.s. Further, applyng (13) gves θθ Q M ( θ ) = Ω θθ + O(M 1/2 ) + O( j 1 ˆ ) a.s. Smlarly, we obtan θ Q M ( θ ) = Ω θ + O(M 1/2 ) + O( j 1 ˆ ) a.s. Usng these results, refne (12) as θ j ˆθ = Ω 1 θθ Ω θ ( j 1 ˆ) + O(M 1/2 j 1 ˆ + j 1 ˆ 2 ) a.s. Substtutng ths nto (14) n conjuncton wth Ω 1 θθ Ω θ = (Ψ 0 Δ θ Ψ 0 θ ) 1 Ψ 0 Δ θ Ψ 0 gves the stated result. It remans to show ( θ j j 1 ) N (ɛ 1 ) a.s. f N 1 s suffcently small. Let N θ {θ : θ θ 0 <ɛ 1 /2} and defne Δ = Q 0 (θ 0 0 ) sup θ N c Q θ Θ 0(θ 0 )>0, where the last nequalty follows from nformaton nequalty, compactness of N c Θ, and contnuty of Q θ 0(θ ). It follows that { θ j / N θ } {Q 0 (θ 0 0 ) Q 0 ( θ j 0 ) Δ}. Further, observe that Q 0 (θ 0 0 ) Q 0 ( θ j 0 ) Q M (θ 0 j 1 ) Q M ( θ j j 1 ) + 2sup θ Θ Q 0 (θ 0 ) Q 0 (θ j 1 ) + 2sup (θ ) Θ B Q M (θ ) Q 0 (θ ) 2sup θ Θ Q 0 (θ 0 ) Q 0 (θ j 1 ) + 2sup (θ ) Θ B Q M (θ ) Q 0 (θ ), where the second nequalty follows from the defnton of θ j. From contnuty of Q 0 (θ ), there exsts ɛ Δ > 0 such that the frst term on the rght s smaller than Δ/2 f 0 j 1 ɛ Δ. The second term on the rght s o(1) a.s. from Kolmogorov s strong law of large numbers and Theorem 2 and Lemma 1 of Andrews (1992). Hence, r( θ j / N θ o ) = 0 f 0 j 1 ɛ Δ, and settng N 1 ={ : 0 mn{ɛ 1 /2 ɛ Δ }} gves ( θ j j 1 ) N (ɛ 1 ) a.s. Q.E.D. AENDIX C: ADDITIONAL ALTERNATIVE SEQUENTIAL ALGORITHMS C.1. Recursve rojecton Method In ths subsecton, we construct a mappng that has a better local contracton property than Ψ, buldng upon the Recursve rojecton Method (RM) of Shroff and Keller (1993; henceforth SK). Frst, fx θ.let θ denote an element of M θ ={ B : = Ψ(θ )}, so that θ s one of the fxed ponts of Ψ(θ ) when there are multple fxed ponts.

8 8 H. KASAHARA AND K. SHIMOTSU Consder fndng θ by teratng j = Ψ( j 1 θ)startng from a neghborhood of θ. If some egenvalues of Ψ(θ θ ) are outsde the unt crcle, ths teraton does not converge to θ n general. Suppose that, countng multplcty, there are r egenvalues of Ψ(θ θ ) that are larger than δ (0 1) n modulus: (15) λ 1 λ r >δ λ r+1 λ L Defne R L as the maxmum nvarant subspace of Ψ(θ θ ) belongng to {λ k } r k=1,andletq RL be the orthogonal complement of.letπ θ denote the orthogonal projector from R L on. WemaywrteΠ θ = Z θ Z θ,where Z θ R L r s an orthonormal bass of. Then, for each R L, we have the unque decomposton = u + v,whereu Π θ and v (I Π θ ) Q. Now apply Π θ and I Π θ to = Ψ(θ ), and decompose the system as follows: u = f(u v θ) Π θ Ψ(θ u+ v) v = g(u v θ) (I Π θ )Ψ (θ u + v) For a gven j 1, decompose t nto u j 1 = Π θ j 1 and v j 1 = (I Π θ ) j 1. Snce g(u v θ) s contractve n v (see Lemma 2.10 of SK), we can update v j 1 by the recurson v j = g(u v j 1 θ). On the other hand, when the domnant egenvalue of Ψ 0 s outsde the unt crcle, the recurson u j = f(u j 1 v θ)cannot be used to update u j 1 because f(u v θ)s not a contracton n u. Instead, the RM performs a sngle Newton step on the system u = f(u v θ),leadng to the followng updatng procedure: (16) u j = u j 1 + (I Π θ Ψ(θ j 1 )Π θ ) 1 (f (u j 1 v j 1 θ) u j 1 ) h(u j 1 v j 1 θ) v j = g(u j 1 v j 1 θ) Lemma 3.11 of SK shows that the spectral radus of the Jacoban of the stablzed teraton (16) s no larger than δ, and thus the teraton j = h(π θ j 1 (I Π θ ) j 1 θ)+ g(π θ j 1 (I Π θ ) j 1 θ) converges locally. In the followng, we develop a sequental algorthm buldng upon the updatng procedure (16). Let Π(θ ) be the orthogonal projector from R L onto the maxmum nvarant subspace of Ψ(θ )belongng to ts r largest (n modulus) egenvalues, countng multplcty. Defne u, v, h (u v θ),andg (u v θ)by replacng Π θ n u, v, h(u v θ),andg(u v θ) wth Π(θ ), and defne (17) Γ(θ ) h (u v θ)+ g (u v θ) = Ψ(θ )+ [ (I Π(θ ) Ψ (θ )Π(θ )) 1 I ] Π(θ )(Ψ (θ ) )

9 ESTIMATION OF STRUCTURAL MODELS 9 0 s a fxed pont of Γ(θ 0 ), because all the fxed ponts of Ψ(θ ) are also fxed ponts of Γ(θ ). The followng proposton shows two mportant propertes of Γ(θ ): local contracton and the equvalence of fxed ponts of Γ(θ )and Ψ(θ ). ROOSITION 8: (a) Suppose that I Π(θ ) Ψ (θ )Π(θ ) s nonsngular and hence Γ(θ )s well defned. Then Γ(θ )and Ψ(θ ) have the same fxed ponts; that s, Γ(θ ) = f and only f Ψ(θ ) =. (b) ρ( Γ(θ 0 0 )) δ 0, where δ 0 s defned by (15) n terms of the egenvalues of Ψ(θ 0 0 ). Hence, Γ(θ )s locally contractve. Defne Q Γ (θ ) M 1 M T ln Γ (θ )(a M m=1 t=1 mt x mt ). Defne an RM fxed pont as a par ( ˇθ ˇ) that satsfes ˇθ = arg max θ Θ Q Γ (θ M ˇ) and ˇ = Γ(ˇθ ˇ).TheRM estmator, denoted by ( ˆθ RM ˆ RM ), s defned as the RM fxed pont wth the hghest value of the pseudo lkelhood among all the RM fxed ponts. Defne the RM algorthm by the same sequental algorthm as the NL algorthm except that t uses Γ(θ )n place of Ψ(θ ). roposton 9 shows the asymptotc propertes of the RM estmator and the convergence propertes of the RM algorthm. Defne the RM counterparts of θ 0 (), φ 0 (), Ω θθ,andω θ as θ Γ () arg max 0 θ Θ EQΓ (θ ), M φγ () 0 Γ( θ Γ () ), 0 ΩΓ E( θθ θs Γ mt θ sγ mt ),andωγ E( θ θs Γ mt sγ mt ),wheresγ = mt T ln t=1 Γ(θ0 0 )(a mt x mt ).DefneΓ 0 Γ(θ0 0 ) and Γ 0 θ θ Γ(θ0 0 ). We outlne the assumptons frst. ASSUMTION 4: (a) Assumpton 1 holds. (b) Ψ(θ ) s four tmes contnuously dfferentable n N.(c)I Π(θ ) Ψ (θ )Π(θ ) s nonsngular. (d) Γ(θ )>0 for any (a x) A X and (θ ) Θ B.(e)The operator φ Γ 0 () has a nonsngular Jacoban matrx at 0. Assumpton 4(c) s requred for Γ(θ )to be well defned. It would be possble to drop Assumpton 4(d) by consderng a trmmed verson of Γ(θ ), but for brevty we do not pursue t. ROOSITION 9: Suppose that Assumpton 4 holds. Then (a) ˆ RM 0 = O(M 1/2 ) a.s. and M 1/2 ( ˆθ RM θ 0 ) d N(0 V RM ), where V RM =[Ω Γ + θθ Ω Γ (I Γ 0 θ ) 1 Γ 0 θ ] 1 Ω Γ θθ {[ΩΓ + θθ ΩΓ (I Γ 0 θ ) 1 Γ 0 θ ] 1 }. (b) Suppose we obtan ( θ j j ) from j 1 by the RM algorthm. Then, there exsts a neghborhood N 1 of 0 such that θ j ˆθ RM = O( j 1 ˆ RM ) and j ˆ RM = M Γθ Γ 0( j 1 ˆ RM ) + O(M 1/2 j 1 ˆ RM + j 1 ˆ RM 2 ) a.s. unformly n j 1 N 1, where M Γθ I Γ 0 0 θ (Γ Δ θ Γ 0 θ ) 1 Γ 0 Δ θ.

10 10 H. KASAHARA AND K. SHIMOTSU C.2. Approxmate RM Algorthm Implementng the RM algorthm s costly because t requres evaluatng Π(θ ) and Ψ(θ ) for all the tral values of θ. We reduce the computatonal burden by evaluatng Π(θ ) and Ψ(θ ) outsde the optmzaton routne by usng a prelmnary estmate of θ. Ths modfcaton has only a second-order effect on the convergence of the algorthm because the dervatves of Γ(θ )wth respect to Π(θ ) and Ψ(θ ) are zero when evaluated at = Ψ(θ ); see the second term n (17). Let η be a prelmnary estmate of θ. Replacng θ n Π(θ ) and Ψ(θ ) wth η, we defne the followng mappng: Γ(θ η) Ψ(θ )+ [ (I Π(η ) Ψ (η )Π(η )) 1 I ] Π(η )(Ψ (θ ) ) Once Π(η ) and Ψ(η ) are computed, the computatonal cost of evaluatng Γ(θ η)across dfferent values of θ would be smlar to that of evaluatng Ψ(θ ). Let ( θ 0 0 ) be an ntal estmator of (θ 0 0 ). For nstance, θ 0 can be the ML estmator. The approxmate RM algorthm terates the followng steps untl j = k: Step 1.Gven( θ j 1 j 1 ), update θ by θ j = arg max θ Θ j M 1 M m=1 t=1 T ln Γ(θ j 1 θ j 1 )(a mt x mt ) where Θ j {θ Θ : Γ(θ j 1 θ j 1 )(a x) [ξ 1 ξ] for all (a x) A X} for an arbtrary small ξ>0. We mpose ths restrcton to avod computng ln(0). 1 Step 2. Update usng the obtaned estmate θ j by j = Γ( θ j j 1 θ j 1 ). The followng proposton shows that the approxmate RM algorthm acheves the same convergence rate as the orgnal RM algorthm n the frst order. ROOSITION 10: Suppose that Assumpton 4 holds and that we obtan ( θ j j ) from ( θ j 1 j 1 ) by the approxmate RM algorthm. Then, there exsts a neghborhood N 2 of (θ 0 0 ) such that θ j ˆθ RM = O( j 1 ˆ RM + M 1/2 θ j 1 ˆθ RM + θ j 1 ˆθ RM 2 ) a.s. and j ˆ RM = M Γθ Γ 0( j 1 ˆ RM )+O(M 1/2 θ j 1 ˆθ RM + θ j 1 ˆθ RM 2 +M 1/2 j 1 ˆ RM + j 1 ˆ RM 2 ) a.s. unformly n ( θ j 1 j 1 ) N 2. 1 In practce, we may consder a penalzed objectve functon by truncatng Γ(θ j 1 θ j 1 ) so that t takes a value between ξ and 1 ξ, and addng a penalty term that penalzes θ such that Γ(θ j 1 θ j 1 )/ [ξ 1 ξ].

11 ESTIMATION OF STRUCTURAL MODELS 11 By choosng δ suffcently small, the domnant egenvalue of M Γθ Γ les nsde the unt crcle, and the approxmate RM algorthm can converge to a consstent estmator even when the NL algorthm dverges away from the true value. The followng proposton states the local convergence of the approxmate RM algorthm when ρ(m Γθ Γ )<1. ROOSITION 11: Suppose that Assumpton 4 holds, ρ(m Γθ Γ 0 )<1, and { θ k k } s generated by the approxmate RM algorthm startng from ( θ 0 0 ). Then, there exsts a neghborhood N 3 of (θ 0 0 ) such that, for any ntal value ( θ 0 0 ) N 3, we have lm k ( θ k k ) = ( ˆθ RM ˆ RM ) a.s. C.3. Numercal Implementaton of the Approxmate RM Algorthm Implementng the approxmate RM algorthm requres evaluatng (I Π( θ j 1 j 1 ) Ψ( θ j 1 j 1 )Π( θ j 1 j 1 )) 1 as well as computng an orthonormal bass Z( θ j 1 j 1 ) from the egenvectors of Ψ( θ j 1 j 1 ) for j = 1 k. Ths s potentally costly when the analytcal expresson of Ψ(θ ) s not avalable. In ths secton, we dscuss how to reduce the computatonal cost of mplementng the approxmate RM algorthm by updatng (I Π( θ j 1 j 1 ) Ψ( θ j 1 j 1 )Π( θ j 1 j 1 )) 1 and Z( θ j 1 j 1 ) wthout explctly computng Ψ(θ ) n each teraton. Frst, we provde theoretcal underpnnng. The followng corollary shows that, f an alternatve prelmnary consstent estmator (θ ) s used n formng Π(θ ) and Ψ(θ ), t only affects the remander terms n roposton 10. Therefore, f we use a root-m consstent (θ ) to evaluate Π(θ ) and Ψ(θ ) and keep these estmates unchanged throughout teratons, the resultng sequence of estmators s only O(M 1 ) away a.s. from the correspondng estmators generated by the approxmate RM algorthm. COROLLARY 1: Suppose that Assumpton 4 holds. Let (θ ) be a strongly consstent estmator of (θ 0 0 ), and suppose we obtan ( θ j j ) by the approxmate RM algorthm wth Π(θ ) and Ψ(θ ) n place of Π( θ j 1 j 1 ) and Ψ( θ j 1 j 1 ). Then, there exsts a neghborhood N 4 of (θ 0 0 ) such that θ j ˆθ RM = O( j 1 ˆ RM +r Mj ) a.s. and j ˆ RM = M Γθ Γ ( j 1 ˆ RM ) + O(M 1/2 j 1 ˆ RM + j 1 ˆ RM 2 + r Mj ) a.s. unformly n ( θ j 1 j 1 ) N 4, where r Mj = M 1/2 θ j 1 ˆθ RM + θ j 1 ˆθ RM 2 +M 1/2 θ ˆθ RM + θ ˆθ RM 2 + M 1/2 ˆ RM + ˆ RM 2. Usng Corollary 1, n the followng we dscuss how to reduce the computatonal cost of mplementng the RM algorthm by updatng (I Π( θ j 1

12 12 H. KASAHARA AND K. SHIMOTSU j 1 ) Ψ( θ j 1 j 1 )Π( θ j 1 j 1 )) 1 and Z( θ j 1 j 1 ) wthout explctly computng Ψ(θ ) n each teraton. Denote Π j 1 = Π( θ j 1 j 1 ), Z j 1 = Z( θ j 1 j 1 ),and Ψ j 1 = Ψ( θ j 1 j 1 ). Frst, usng Π j 1 = Z j 1 ( Z j 1 ) and ( Z j 1 ) Z j 1 = I, we may verfy that (I Π j 1 Ψ j 1 Π j 1 ) 1 Π j 1 = Z j 1 (I ( Z j 1 ) Ψ j 1 Z j 1 ) 1 ( Z j 1 ) Let Z j 1 =[ z 1 j 1 zr ] and ξ>0. The th column of j 1 Ψ j 1 Z j 1 can be approxmated by Ψ j 1 z (1/ξ)[Ψ( θ j 1 j 1 j 1 + ξ z ) Ψ( θ j 1 j 1 j 1 )], whch requres (r + 1) functon evaluatons of Ψ(θ ). Further, evaluatng (I Π j 1 Ψ j 1 Π j 1 ) 1 only requres the nverson of the r r matrx I ( Z j 1 ) Ψ j 1 Z j 1 nstead of an nverson of an L L matrx. Thus, when r s small, numercally evaluatng (I Π j 1 Ψ j 1 Π j 1 ) 1 s not computatonally dffcult. Second, t s possble to use Ψ j Z j 1 to update an estmate of the orthogonal bass Z. Namely, gven a prelmnary estmate Z j 1,wemayobtan Z j by performng one step of an orthogonal power teraton (see Shroff and Keller (1993), p. 1107, Golub and Van Loan (1996)) by computng Z j = orth( Ψ j Z j 1 ), where orth(b) denotes an orthonormal bass for the columns of B computed by Gram Schmdt orthogonalzaton. Our numercal mplementaton of the RM sequental algorthm s summarzed as follows. Step 0 Intalzaton. (a) Fnd the egenvalues of Ψ 0 Ψ( 0 θ 0 ) for whch the modulus s larger than δ. Let{ λ 0 1 λ 0 r } denote them. 2 (b) Fnd the egenvectors of Ψ 0 assocated wth λ 0 1 λ 0 r. (c) Usng Gram Schmdt orthogonalzaton, compute an orthonormal bass of the space spanned by these egenvectors. Let { z 1 0 zr } denote the bass. (d) Compute 0 Z 0 (I Z Ψ 0 0 Z 0 ) 1 Z and Π 0 0 = Z 0 Z,where 0 Z 0 =[ z 1 0 zr ]. 0 Step 1 Update θ. Gven Z j 1 (I Z Ψ j 1 j 1 Z j 1 ) 1 Z and Π j 1 j 1 = Z j 1 ( Z j 1 ), update θ by θ j = arg max θ Θj M 1 M T ln Γ(θ m=1 t=1 j 1 θ j 1 Z j 1 )(a mt x mt ),whereγ(θ j 1 θ j 1 Z j 1 ) = Π j 1 j 1 + Z j 1 (I Z Ψ j 1 j 1 Z j 1 ) 1 Z (Ψ (θ j 1 j 1 ) j 1 )+(I Π j 1 )Ψ (θ j 1 ) wth Ψ j 1 Ψ( θ j 1 j 1 ). Step 2 Update. Gven( θ j j 1 θ j 1 Z j 1 ), update by j = Γ( θ j j 1 θ j 1 Z j 1 ). 2 Computng the r domnant egenvalues of Ψ 0 s potentally costly. We follow the numercal procedure based on the power teraton method as dscussed n Secton 4.1 of SK.

13 ESTIMATION OF STRUCTURAL MODELS 13 Step 3 Update Z. (a) Update the orthonormal bass Z by Z j = orth( Ψ j Z j 1 ), where the th column of Ψ j Z j 1 s computed by Ψ j z j 1 (1/ξ)[Ψ( θ j j +ξ z j 1 ) Ψ( θ j j )] for small ξ>0, wth Z j 1 =[ z 1 j 1 zr j 1 ]. (b) Compute Π j = Z j ( Z j ) and Z j (I Z Ψ j j Z j ) 1 Z j, where the th row of Ψ j Z j s gven by Ψ j z (1/ξ)[Ψ( θ j j j + ξ z ) Ψ( θ j j j )]. (c) Every J teratons, update the orthonormal bass Z usng the algorthm of Step 0, where ( θ 0 0 ) s replaced wth ( θ j j ). Step 4. Iterate Steps 1 3 k tmes. When an ntal estmate s not precse, the domnant egenspace of Ψ j wll change as teratons proceed. In Step 3(a), the orthonormal bass s updated to mantan the accuracy of the bass wthout changng the sze of the orthonormal bass. If an ntal estmate of the sze of the orthonormal bass s smaller than the true sze, however, the estmated subspace = ΠR L may not contan all the bases for whch egenvalues are outsde the unt crcle. In such a case, the algorthm may not converge. To safeguard aganst such a possblty, the bass sze s updated every J teratons n Step 3(c). In our Monte Carlo experments, we chose J = 10. Corollary 1 mples that ths modfed algorthm wll converge. C.4. Applyng RM to the Example of esendorfer and Schmdt-Dengler (2010) Ths subsecton llustrates how the RM algorthm can be appled to the example of esendorfer and Schmdt-Dengler (2010). We frst derve the relaton between (Γ + θ Γ + ) and (Ψ + + θ Ψ + ). Defne Π + (θ + ) as the + orthogonal projector from R dm(+ ) onto the maxmum nvarant subspace of + Ψ(θ + ) belongng to ts r largest (n modulus) egenvalues, and let Z(θ + ) be an orthonormal bass of the column space of Π + (θ + ), so that Π + (θ + ) = Z(θ + )Z(θ + ). From the proof of roposton 6, we have Γ + (θ + ) + = A(θ + )(Ψ + (θ + ) + ),wherea(θ + ) = Z(θ + )[I Z(θ + ) + Ψ(θ + )Z(θ + )] 1 Z(θ + ) + I Π(θ + ). Consequently, Γ + θ = A(θ 0 0+ )Ψ + θ and Γ + = A(θ 0 0+ )(Ψ + + I) + I. + We proceed to derve M + Γ θ Γ +. Recall + ( ) 0 θ Ψ + = θ p0 1 2 Ψ + 0 = + θ 0 M 0 + Ψ θ = 1 ( ) The egenvectors and egenvalues of Ψ + are gven by + z 1 = 1 ( ) 1 λ = θ 0 z 2 = 1 ( ) λ 2 = θ 0 Because the egenvector z 1 s annhlated by M + Ψ θ,wemaytakez(θ 0 0+ ) = z 2. Suppress (θ 0 0+ ) from Z(θ 0 0+ ), Π(θ 0 0+ ),anda(θ 0 0+ ).SnceZ s the

14 14 H. KASAHARA AND K. SHIMOTSU egenvector of Ψ + + wth egenvalue θ 0,wehaveZ Ψ + + Z = θ 0 Z Z = θ 0 and hence A = Z(1 Z Ψ + + Z) 1 Z + (I Π) = (1 + θ 0 ) 1 Π + (I Π) = I θ 0 (1 + θ 0 ) 1 Π Because Ψ + s symmetrc, we may apply the egenvalue decomposton to t + and wrte Ψ + = θ 0 z + 1 z 1 θ0 z 2 z = 2 θ0 z 1 z 1 θ0 Π. In vew of Az 1 = z 1 and AΠ = (1 + θ 0 ) 1 Π,wehaveΓ + = A(Ψ + + I) + I = θ 0 z + 1 z 1 θ0 (1 + θ 0 ) 1 Π A + I = θ 0 z 1 z. Further, from Ψ + 1 θ = p 0 2z 1,wehaveΓ + θ = AΨ + θ = Ψ + θ and hence M + Γ θ = I z 1 (z 1 z 1 ) 1 z. It follows that 1 M+ Γ θ Γ + = 0, and the local convergence + condton holds. C.5. q-nl Algorthm and Approxmate q-nl Algorthm When the spectral radus of Λ 0 or Ψ 0 s smaller than but close to 1, the convergence of the NL algorthm could be slow and the generated sequence could behave erratcally. Furthermore, n such a case, the effcency loss from usng the NL estmator compared to the MLE s substantal. To overcome these problems, consder a q-fold operator of Λ as Λ q (θ ) Λ ( θ ( Λ ( θ Λ(θ Λ(θ )) ))) }{{} q tmes We may defne Γ q (θ ) and Ψ q (θ ) analogously. Defne the q-nl (q- RM) algorthm by usng a q-fold operator Λ q, Γ q,orψ q n place of Λ, Γ, or Ψ n the orgnal NL (RM) algorthm. In the followng, we focus on Λ q, but the same argument apples to Γ q and Ψ q. If the sequence of estmators generated by the q-nl algorthm converges, ts lmt satsfes ˇθ = arg max θ Θ M 1 M T ln m=1 t=1 Λq (θ ˇ)(a mt x mt ) and ˇθ = Λ q ( ˇθ ˇ). Among the pars ( ˆθ ˆ) that satsfy these two condtons, the one that maxmzes the value of the pseudo lkelhood s called the q-nl estmator and denoted by ( ˆθ qnl ˆ qnl ). Snce the result of roposton 7 also apples here by replacng Ψ wth Λ q, the local convergence property of the q-nl algorthm s prmarly determned by the spectral radus of Λ q Λ q (θ 0 0 ).Whenρ(Λ 0 ) s less than 1, the q-nl algorthm converges faster than the NL algorthm because ρ(λ q ) = (ρ(λ 0 ))q. Moreover, the varance of the q-nl estmator approaches that of the MLE as q. Applyng the q-nl algorthm, as defned above, s computatonally ntensve because the q-nl Step 1 requres evaluatng Λ q at many dfferent values of θ. We reduce the computatonal burden by ntroducng a lnear approxmaton of Λ q (θ ) around (η ), whereη s a prelmnary estmate of θ: Λ q (θ η) Λ q (η ) + θ Λ q (η )(θ η).

15 ESTIMATION OF STRUCTURAL MODELS 15 Gven an ntal estmator ( θ 0 0 ), the approxmate q-nl algorthm terates the followng steps untl j = k: Step 1. Gven ( θ j 1 j 1 ), update θ by θ j = arg max θ Θ q j M 1 M m=1 t=1 T ln Λ q (θ j 1 θ j 1 )(a mt x mt ) where Θ q j {θ Θ : Λ q (θ j 1 θ j 1 )(a x) [ξ 1 ξ] for all (a x) A X} for an arbtrary small ξ>0. Step 2. Gven ( θ j j 1 ), update usng the obtaned estmate θ j by j = Λ q ( θ j j 1 ). Implementng Step 1 requres evaluatng Λ q ( θ j 1 j 1 ) and θ Λ q ( θ j 1 j 1 ) only once outsde of the optmzaton routne for θ and thus nvolves far fewer evaluatons of Λ(θ ) across dfferent values of and θ, compared to the orgnal q-nl algorthm. 3 Defne the q-nl counterparts of θ 0 (), φ 0 (), and Ω θθ as θ q () 0 arg max θ Θ E[ T t=1 ln Λq (θ )(a mt x mt )], φ q 0 () Λq ( θ q 0 () ), and Ωq θθ E( θ s Λ mt θ sλ mt ) wth sλ mt = T t=1 ln Λq (θ 0 0 )(a mt x mt ), respectvely. Defne Ω q θ analogously. ASSUMTION 5: (a) Assumpton 1 holds. (b) Ψ(θ ) s four tmes contnuously dfferentable n N.(c)There s a unque θ 0 such that Λ q (θ 0 0 ) = 0. (d) I (αψ 0 +(1 α)i)q and I Ψ 0 are nonsngular.(e)the operator φq 0 () has a nonsngular Jacoban matrx at 0. Assumpton 5(c) s necessary for dentfyng θ 0 when the condtonal probablty s gven by Λ q (θ ). Ths assumpton rules out θ 1 θ 0 that satsfes Λ q (θ 1 0 ) = 0 even f Λ(θ 1 0 ) 0. Ths occurs, for example, f Λ(θ 1 0 ) = 1 and Λ(θ 1 1 ) = 0 hold for θ 1 θ 0 and 1 0. Assumpton 5(d) s necessary for Ω q θθ to be nonsngular. Snce Λ q = (αψ 0 + (1 α)i)q, the frst condton holds f ρ(λ q )<1 from of Seber (2007). The followng proposton establshes the asymptotcs of the q-nl estmator and the convergence property of the approxmate q-nl algorthm. roposton 12(c) mples that, when q s suffcently large, the q-nl estmator s more effcent than the NL estmator, provded that addtonal condtons n Assumpton 5 hold. roposton 12(d) corresponds to roposton 1. ROOSITION 12: Suppose that Assumpton 5 holds. Then (a) ˆ qnl 0 = O(M 1/2 ) a.s. and M 1/2 ( ˆθ qnl θ 0 ) d N(0 V qnl ), where V qnl = 3 Usng one-sded numercal dervatves, evaluatng θ Λ q ( θ j j ) requres (K + 1)q functon evaluatons of Ψ(θ ).

16 16 H. KASAHARA AND K. SHIMOTSU [Ω q θθ + Ω q θ(i Λ 0 ) 1 Λ q θ] 1 Ω q θθ{[ω q θθ + Ω q θ(i Λ 0 ) 1 Λ q θ] 1 }. (b) Suppose we obtan ( θ j j ) from ( θ j 1 j 1 ) by the approxmate q-nl algorthm. Then, there exsts a neghborhood N 6 of (θ 0 0 ) such that θ j ˆθ qnl = O( j 1 ˆ qnl +M 1/2 θ j 1 ˆθ qnl + θ j 1 ˆθ qnl 2 ) a.s. and j ˆ qnl = M q Λ Λq θ ( j 1 ˆ qnl ) + O(M 1/2 θ j 1 ˆθ qnl + θ j 1 ˆθ qnl 2 + M 1/2 j 1 ˆ qnl + j 1 ˆ qnl 2 ) a.s. unformly n ( θ j 1 j 1 ) N 6, where M q I Λ Λq θ θ(λ q θ Δ Λ q θ) 1 Λ q θ Δ wth Λ q θ θ Λ q (θ 0 0 ).(c)if ρ(λ 0 )<1, then V qnl V MLE as q.(d)suppose { θ k k } s generated by the approxmate q-nl algorthm startng from ( θ 0 0 ) and ρ(m Λ q θ Λq )<1. Then, there exsts a neghborhood N 7 of (θ 0 0 ) such that, for any startng value ( θ 0 0 ) N 7, we have lm k ( θ k k ) = ( ˆθ qnl ˆ qnl ) a.s. C.6. roof of ropostons n Appendx C ROOF OF ROOSITION 8: For part (a), wrte Γ(θ ) as Γ(θ ) = A(θ )(Ψ (θ ) ),wherea(θ ) (I Π(θ ) Ψ (θ )Π(θ )) 1 Π(θ ) + (I Π(θ )).LetZ(θ ) denote an orthonormal bass of the column space of Π(θ ), so that Z(θ )Z(θ ) = Π(θ ) and Z(θ ) Z(θ ) = I r. Suppress (θ ) from Π(θ ), Z(θ ), and Ψ(θ ). A drect calculaton gves (I Π ΨΠ) 1 Π = Z(I Z ΨZ) 1 Z,sowecanwrte A(θ ) as A(θ ) = Z(I Z ΨZ) 1 Z + (I Π). The stated result follows snce A(θ ) s nonsngular because rank[z(i Z ΨZ) 1 Z ]=r, rank(i Π) = L r, andz(i Z ΨZ) 1 Z and I Π are orthogonal to each other. For part (b), defne Γ 0 Γ(θ0 0 ) and Π 0 Π(θ 0 0 ). Defne wth respect to Ψ 0 Ψ(θ0 0 ). Computng Γ(θ ) and notng that Ψ(θ 0 0 ) = 0,wefndΓ 0 = Π0 + (I Π 0 Ψ 0 Π0 ) 1 Π 0 (Ψ 0 I)+ (I Π0 )Ψ 0. Observe that Γ 0 Π0 = (I Π 0 )Ψ 0 Π0 = 0, where the last equalty follows because Ψ 0 Π0 for any R L by the defnton of Π 0. Hence, Γ 0 = Γ 0(I Π0 ).Wealsohave(I Π 0 )Γ 0 = (I Π0 )Ψ 0 because a drect calculaton gves (I Π 0 Ψ 0 Π0 ) 1 Π 0 = Z 0 (I (Z 0 ) Ψ 0 Z0 ) 1 (Z 0 ),wherez 0 = Z(θ 0 0 ), and hence (I Π 0 )(I Π 0 Ψ 0 Π0 ) 1 Π 0 = 0. Then, n conjuncton wth Γ 0 = Γ 0(I Π0 ),weobtan(i Π 0 )Γ 0 = (I Π0 )Ψ 0(I Π0 ).SnceΓ 0(I Π0 ) has the same egenvalues as (I Π 0 )Γ 0 (see Theorem of Horn and Johnson (1985)), we have ρ(γ 0) = ρ(γ 0(I Π0 )) = ρ((i Π 0 )Γ 0) = ρ[(i Π 0 )Ψ 0(I Π0 )] δ 0, where the last nequalty follows from Lemma 2.10 of SK:, Q,andF n SK correspond to our u Π0, I Π 0,andΨ 0. Q.E.D. ROOF OF ROOSITION 9: The stated results follow from roposton 2 of AM07 and our roposton 7 f Assumptons 1(b), 1(c), 1(e) 1(h), and 2(b) and 2(c) hold when Ψ(θ ) s replaced wth Γ(θ ).

17 ESTIMATION OF STRUCTURAL MODELS 17 We check Assumptons 2(b) and 2(c) frst because they are used n showng the other condtons. Frst, note that Chu (1990, Secton4.2;npartcular, lne 17 on p. 1377) proved the followng: f a matrx A(t) s l tmes contnuously dfferentable wth respect to t, andfx(t) spans the nvarant subspace correspondng to a subset of egenvalues of A(t), then X(t) s also l tmes contnuously dfferentable wth respect to t. Consequently, Π(θ ) s three tmes contnuously dfferentable n N (we suppress n N henceforth), snce Ψ(θ ) s three tmes contnuously dfferentable from Assumpton 4(b). Further, I Π(θ ) Ψ (θ )Π(θ ) s nonsngular and three tmes contnuously dfferentable from Assumptons 4(b) and 4(c), and hence Assumpton 2(b) holds for Γ(θ ). For Assumpton 2(c), a drect calculaton gves Ω Γ = Ψ 0 θθ θ A(θ0 0 ) Δ A(θ 0 0 )Ψ 0 θ,wherea(θ ) s defned n the proof of roposton 2 and shown to be nonsngular. Snce rank(ψ 0 θ ) = K from nonsngularty of Ω θθ = Ψ 0 Δ θ Ψ 0, postve defnteness of θ ΩΓ follows. θθ We proceed to confrm that Assumptons 1(b), 1(c), and 1(e) 1(h) hold for Γ(θ ). Assumpton 1(b) for Γ(θ )follows from Assumpton 4(d). Assumpton 1(c) holds because we have already shown that Γ(θ )s three tmes contnuously dfferentable. Assumpton 1(e) holds because Ψ(θ ) and Γ(θ ) have the same fxed ponts by roposton 8. As dscussed on page 21 of AM07, Assumpton 1(f) s mpled by Assumpton 4(e). Assumpton 1(g) for θ Γ () 0 follows from the postve defnteness of Ω Γ θθ and by the mplct functon theorem appled to the frst-order condton for θ. Assumpton 1(h) follows from Assumpton 4(e). Q.E.D. ROOF OF ROOSITION 10: Wrte the objectve functon as Q Γ (θ η) M M 1 M T ln Γ (θ η)(a m=1 t=1 mt x mt ), and defne Q Γ (θ η) 0 EQΓ (θ M η).forɛ>0, defne a neghborhood N 3 (ɛ) ={(θ η):max{ θ θ 0 0 η θ 0 } <ɛ}. Then, there exsts ɛ 1 > 0 such that () Ψ(θ ) s four tmes contnuously dfferentable n (θ ) f (θ η) N 3 (ɛ 1 ), () sup (θ η) N3 (ɛ 1 ) θθ QΓ 0 (θ η) 1 <, and () sup (θ η) N3 (ɛ 1 ) 3 Q Γ (θ 0 η) < because Γ(θ 0 0 θ 0 )(a x) = 0 (a x) > 0, Γ(θ η) s three tmes contnuously dfferentable (see the proof of roposton 9), and θθ Q Γ 0 (θ0 0 θ 0 ) = θθ Q Γ 0 (θ0 0 ) s nonsngular. Frst, we assume ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) and derve the stated representaton of θ j ˆθ and j ˆ. We later show that ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) a.s. f N 2 s taken suffcently small. Henceforth, we suppress the subscrpt RM from ˆθ RM and ˆ RM. Expandng the frst-order condton θ Q Γ ( M θ j j 1 θ j 1 ) = 0 around ( ˆθ j 1 θ j 1 ) gves (18) 0 = θ Q Γ M ( ˆθ j 1 θ j 1 ) + θθ Q Γ M ( θ j 1 θ j 1 )( θ j ˆθ)

18 18 H. KASAHARA AND K. SHIMOTSU where θ [ θ j ˆθ]. Wrtng θ = θ( θ j ),weobtan sup ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) θθ Q Γ M ( θ( θ j ) j 1 θ j 1 ) 1 =O(1) a.s. because () θ( θ j ) θ 0 <ɛ 1 a.s. snce θ j θ 0 <ɛ 1 and ˆθ s strongly consstent, and () sup (θ η) N3 (ɛ 1 ) θθ QΓ M (θ η) 1 = O(1) a.s. snce sup (θ η) N3 (ɛ 1 ) θθ QΓ 0 (θ η) 1 < and sup (θ η) N3 (ɛ 1 ) 2 Q Γ M (θ η) 2 Q Γ 0 (θ η) =o(1) a.s. Therefore, the stated representaton of θ j ˆθ follows f we show that (19) θ Q Γ M ( ˆθ j 1 θ j 1 ) = Ω Γ θ ( j 1 ˆ)+ r Mj where r denotes a generc remander term that s Mj O(M 1/2 θ j 1 ˆθ + θ j 1 ˆθ 2 + M 1/2 j 1 ˆ + j 1 ˆ 2 ) a.s. unformly n ( θ j 1 j 1 ) N 2. We proceed to show (19). Expandng θ Q Γ ( M ˆθ j 1 θ j 1 ) twce around ( ˆθ ˆ ˆθ) gves θ Q Γ ( M ˆθ j 1 θ j 1 ) = θ Q Γ ( M ˆθ ˆ ˆθ) + θ Q Γ ( M ˆθ ˆ ˆθ)( j 1 ˆ)+ θη Q Γ ( M ˆθ ˆ ˆθ)( θ j 1 ˆθ) + O( θ j 1 ˆθ 2 + j 1 ˆ 2 ) a.s. For the frst term on the rght, the RM estmator satsfes θ Q Γ ( M ˆθ ˆ ˆθ) = 0 a.s. because θ Q Γ ( M ˆθ ˆ) = 0 from the frst-order condton, and roposton 8(a) mples Ψ(ˆθ ˆ) = ˆ a.s. and hence θ Γ(ˆθ ˆ ˆθ) = θ Γ(ˆθ ˆ) a.s. For the second and thrd terms on the rght, we have E[ T t=1 θ ln Γ(θ0 0 θ 0 )(a mt x mt )]= Ω Γ and E[ T θ t=1 θη ln Γ(θ0 0 θ 0 )(a mt x mt )]=0 by the nformaton matrx equalty because Γ(θ 0 0 θ 0 ) = Γ(θ 0 0 ), θ Γ(θ 0 0 θ 0 ) = θ Γ(θ 0 0 ), Γ(θ 0 0 θ 0 ) = Γ(θ 0 0 ), and η Γ(θ 0 0 θ 0 ) = 0 from 0 = Ψ(θ 0 0 ). Therefore, (19) follows from the root-m consstency of ( ˆθ ˆ). For the representaton of j ˆ,frstwehave (20) j = ˆ + Γ 0 θ ( θ j ˆθ) + Γ 0 ( j 1 ˆ)+ r Mj by expandng j = Γ( θ j j 1 θ j ) around ( ˆθ ˆ ˆθ) and usng Γ(ˆθ ˆ ˆθ) = ˆ. Next, refne (18) as0= θ Q Γ ( M ˆθ j 1 θ j 1 ) Ω Γ ( θθ θ j ˆθ) + r Mj by expandng θθ Q Γ ( M ˆθ j 1 θ j 1 ) n (18) around( ˆθ ˆ ˆθ) to wrte t as θθ Q Γ ( M ˆθ j 1 θ j 1 ) = Ω Γ + θθ O(M 1/2 ) + O( θ j 1 ˆθ ) + O( j 1 ˆ ) a.s. and usng the bound of θ j ˆθ obtaned above. Substtutng ths nto (19) gves (21) θ j ˆθ = (Ω Γ θθ ) 1 Ω Γ θ ( j 1 ˆ)+ r Mj The stated result follows from substtutng ths nto (20) n conjuncton wth (Ω Γ θθ ) 1 Ω Γ 0 θ = (Γ Δ θ Γ 0 θ ) 1 Γ 0 Δ θ Γ 0.

19 ESTIMATION OF STRUCTURAL MODELS 19 It remans to show that ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) a.s. f N 2 s taken suffcently small. We frst show that (22) sup Q Γ (θ η) (θ η ) Θ M QΓ (θ η) =o(1) 0 j N 2 Q Γ 0 (θ η)s contnuous n (θ η ) Θ j N 2 a.s. Take N 2 suffcently small; then t follows from the strong consstency of ( θ j 1 j 1 ) and the contnuty of Γ(θ η) that Γ (θ η)(a x) [ξ/2 1 ξ/2] for all (a x) A X and (θ η) Θ j N a.s. Observe that () Θ j N s compact because t s an ntersecton of the compact set Θ and A X closed sets, () ln Γ(θ η)s contnuous n (θ η) Θ j N,and () E sup ln Γ (θ η)(a (θ η) Θj N x ) ln(ξ/2) + ln(1 ξ/2) < because of the way we choose N. Therefore, (22) follows from Kolmogorov s strong law of large numbers and Theorem 2 and Lemma 1 of Andrews (1992). Fnally, we show ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) a.s. under (22) by applyng the argument n the proof of roposton 7. Defne Δ = Q Γ 0 (θ0 0 θ 0 ) sup θ Nθ (ɛ 1 ) c Θ QΓ (θ 0 0 θ 0 )>0, where the last nequalty follows from the nformaton nequalty because Q Γ (θ 0 0 θ 0 ) s unquely maxmzed at θ 0 and N θ (ɛ 1 ) c Θ s compact. It follows that { θ j / N θ (ɛ 1 )} {Q Γ 0 (θ0 0 θ 0 ) Q Γ ( 0 θ j 0 θ 0 ) Δ}. roceedng as n the proof of roposton 7, we fnd that, f N 2 s taken suffcently small, then Q Γ 0 (θ0 0 θ 0 ) Q Γ ( 0 θ j 0 θ 0 ) Δ/2+o(1) a.s. and hence ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) a.s. Q.E.D. ROOF OF ROOSITION 11: The proof closely follows the proof of roposton 1. We suppress the subscrpt RM from ˆθ RM and ˆ RM.Defne ( ) 0 (Ω Γ D = θθ ) 1 Ω (23) Γ θ 0 M Γθ Γ 0 Note that ρ(d) = ρ(m Γθ Γ ) and that there exsts a matrx norm α such that D α ρ(d) + b = ρ(m Γθ Γ ) + b. We defne the vector norm for x R k+l as x β = [x 0 0] α ; then Ax β A α x β for any matrx A. From the representaton of j ˆ and θ j ˆθ n roposton 10 and (21), there exsts a neghborhood N ζ of ζ 0 such that ζ j ˆζ = D( ζ j 1 ˆζ) + O(M 1/2 ζ j 1 ˆζ β + ζ j 1 ˆζ 2 ) holds a.s. unformly n β ζ j 1 N ζ.thestated result then follows from repeatng the proof of roposton 1. Q.E.D. ROOF OF COROLLARY 1: The proof closely follows the proof of roposton 10. DefneΓ(θ η Q) Ψ(θ ) +[(I Π(η Q) Ψ (η Q)Π(η Q)) 1 I]Π(η Q)(Ψ (θ ) ), so that the objectve functon n Step 1 s wrtten as Q Γ M (θ j 1 θ ) = M 1 M m=1 T t=1 ln Γ(θ j 1 θ )(a mt x mt ).

20 20 H. KASAHARA AND K. SHIMOTSU For ɛ 1 > 0, defne a neghborhood N 5 (ɛ 1 ) ={(θ η Q):max{ θ θ 0 0 η θ 0 Q 0 } <ɛ 1 }. Then, for any ɛ 1 > 0, we have ( θ j j 1 θ ) N 5 (ɛ 1 ) a.s. f N 4 s chosen suffcently small by the same argument as n the proof of roposton 10. Assumng ( θ j j 1 θ ) N 5 (ɛ 1 ), the stated result follows from startng from the frst order condton θ Q Γ ( M θ j j 1 θ ) = 0, expandng t around ( ˆθ j 1 θ ), and followng the proof of roposton 10 usng Q Γ(θ 0 0 θ 0 0 ) = 0. Q.E.D. ROOF OF ROOSITION 12: art (a) follows from roposton 2 of AM07 f Assumptons 1(b), 1(c), 1(e) 1(h), and 2(b) and 2(c) hold when Ψ(θ ) s replaced wth Λ q (θ ). Smlarly to the proof of roposton 10, we check Assumptons 2(b) and 2(c) frst. Assumpton 2(b) holds for Λ q (θ ) because Ψ(θ ) s three tmes contnuously dfferentable n N from Assumpton 5(b). For Assumpton 2(c), a drect calculaton gves Ω q θθ = ( θ Λ q (θ 0 0 )) Δ θ Λ q (θ 0 0 ) = Λ 0 (I θ (Λ0 )q ) (I Λ 0 ) 1 Δ (I Λ 0 ) 1 (I (Λ 0 )q )Λ θ = Ψ 0 (I (αψ 0 θ +(1 α)i)q ) (I Ψ 0 ) 1 Δ (I Ψ 0 ) 1 (I (αψ 0 +(1 α)i)q )Ψ 0, θ where the second equalty follows from θ Λ q (θ 0 0 ) = ( q 1 j=0 (Λ0 )j )Λ 0 = (I θ Λ 0 ) 1 (I (Λ 0 )q )Λ 0, and the thrd equalty follows from θ Λ0 = αψ 0 and θ θ Λ0 = αψ 0 + (1 α)i. Sncerank(Ψ 0) = K from nonsngularty of Ω θ θθ = Ψ 0 Δ θ Ψ 0, θ postve defnteness of Ω q θθ follows from Assumpton 5(d). The proof of part (a) s completed by confrmng that Assumptons 1(b), 1(c), and 1(e) 1(h) hold for Λ q (θ ). Assumptons 1(b) and 1(c) hold for Λ q (θ ) because Assumptons 1(b) and 1(c) hold for Ψ(θ ). Assumpton 1(e) for Λ q (θ ) follows from Assumpton 5(c). As dscussed on page 21 of AM07, Assumpton 1(f) for Λ q (θ ) s mpled by Assumpton 5(e). Assumpton 1(g) for θ q () follows from the postve defnteness of 0 Ωq θθ and applyng the mplct functon theorem to the frst-order condton for θ. Assumpton 1(h) follows from Assumpton 5(e). Ths completes the proof of part (a). We proceed to prove part (b). Defne the objectve functon and ts lmt as QM(θ η) q M 1 M T ln m=1 t=1 Λq (θ η)(a mt x mt ) and Q q (θ η) 0 EQM(θ η).forɛ>0, q defne a neghborhood N 3 (ɛ) ={(θ η): max{ θ θ 0 0 η θ 0 } <ɛ}. Then, there exsts ɛ 1 > 0 such that () Ψ(θ ) s four tmes contnuously dfferentable n (θ ) f (θ η) N 3 (ɛ 1 ), () sup (θ η) N3 (ɛ 1 ) θθ Qq 0 (θ η) 1 <, and () sup (θ η) N3 (ɛ 1 ) 3 Q q (θ 0 η) < because Λ q (θ 0 0 θ 0 )(a x) = 0 (a x) > 0, Λ q (θ η) s three tmes contnuously dfferentable, and θθ Q q 0 (θ0 0 θ 0 ) = θθ Q q 0 (θ0 0 ) s nonsngular. Frst, we assume that ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) and derve the stated representaton of θ j ˆθ and j ˆ. We later show that ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) a.s. f N 6 s taken suffcently small. Henceforth, we suppress the subscrpt qnl from ˆθ qnl and ˆ qnl. The proof s smlar to the proof of the up-

21 ESTIMATION OF STRUCTURAL MODELS 21 datng formula of roposton 10. For the representaton of θ j ˆθ, expandng the frst-order condton 0 = θ QM( q θ j j 1 θ j 1 ) around ( ˆθ j 1 θ j 1 ) gves 0 = θ QM( q ˆθ j 1 θ j 1 ) + θθ QM( q θ( θ j ) j 1 θ j 1 )( θ j ˆθ), whch corresponds to (18) n the proof of roposton 10. roceedng as n the proof of roposton 10, weobtansup ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) θθ Qq M( θ( θ j ) j 1 θ j 1 ) 1 = O(1) a.s. Therefore, the stated representaton of θ j ˆθ follows f we show θ QM( q ˆθ j 1 θ j 1 ) = Ω q θ( j 1 ˆ) + r Mj,wherer Mj denotes a remander term of O(M 1/2 θ j 1 ˆθ + θ j 1 ˆθ 2 + M 1/2 j 1 ˆ + j 1 ˆ 2 ) a.s. unformly n ( θ j 1 j 1 ) N 6. Ths representaton corresponds to (19) n the proof of roposton 10 and follows from the same argument. Namely, expandng θ QM( q ˆθ j 1 θ j 1 ) twce around ( ˆθ ˆ ˆθ), and notng that () the q-nl estmator satsfes θ QM( q ˆθ ˆ ˆθ) = 0, () Λ q (θ 0 0 θ 0 ) = Λ q (θ 0 0 ), θ Λ q (θ 0 0 θ 0 ) = θ Λ q (θ 0 0 ), Λ q (θ 0 0 θ 0 ) = Λ q (θ 0 0 ), and η Λ q (θ 0 0 θ 0 ) = 0, and usng the nformaton matrx equalty and the root- M consstency of ( ˆθ ˆ), gves the requred result. The proof of the representaton of j ˆ follows from the proof of roposton 10, because () j = ˆ +Λ q θ( θ j ˆθ)+Λ q ( j 1 ˆ)+r Mj, whch corresponds to (20) n the proof of roposton 10, from expandng Λ q ( θ j j 1 ) twce around ( ˆθ ˆ) and usng ˆ = Λ q ( ˆθ ˆ), () θθ QM( q ˆθ j 1 θ j 1 )( θ j ˆθ) = Ω q θθ( θ j ˆθ) + r Mj from expandng θθ QM( q ˆθ j 1 θ j 1 ) around ( ˆθ ˆ ˆθ) and usng the bound of θ j ˆθ obtaned above, and () (Ω q θθ) 1 Ω q θ = (Λ q θ Δ Λ q θ) 1 Λ q θ Δ Λ q. The proof of part (b) s completed by showng that ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) a.s. f N 6 s taken suffcently small. Frst, observe that (22) n the proof of roposton 10 holds wth Q Γ (θ η) and M QΓ (θ η) replacng 0 Qq M(θ η) and Q q (θ η) f we take N 0 6 suffcently small. Therefore, ( θ j j 1 θ j 1 ) N 3 (ɛ 1 ) a.s. follows from repeatng the argument n the last paragraph of the proof of roposton 10 f we show that θ 0 unquely maxmzes Q q (θ 0 0 θ 0 ). Note that (24) Q q (θ 0 0 θ 0 ) Q q 0 (θ0 0 θ 0 ) = TEln( θ Λ q (θ 0 0 )(θ θ 0 ) + 0 )(a mt x mt ) TEln 0 (a mt x mt ) ( ) θ Λ q (θ 0 0 )(a mt x mt )(θ θ 0 ) = TEln (a mt x mt ) Recall that ln(y + 1) y for all y > 1, where the nequalty s strct f y 0. Snce rank( θ Λ q (θ 0 0 )) = K from the postve defnteness of Ω q θθ, t follows that θ Λ q (θ 0 0 )ν 0 for any K-vector ν 0. Therefore, θ Λ q (θ 0 0 )(a mt x mt )(θ θ 0 ) 0 for at least one (a mt x mt ),forall

22 22 H. KASAHARA AND K. SHIMOTSU θ θ 0. Consequently, the rght hand sde of (24) s strctly smaller than TE[ θ Λ q (θ 0 0 )(a mt x mt )(θ θ 0 )/ 0 (a mt x mt )] for all θ θ 0. Because E[ θ Λ q (θ 0 0 )(a mt x mt )/ 0 (a mt x mt )]=0, we have Q q (θ 0 0 θ 0 ) Q q 0 (θ0 0 θ 0 )<0forallθ θ 0. Therefore, θ 0 unquely maxmzes Q q (θ 0 0 θ 0 ),andwe complete the proof of part (b). We prove part (c). From the proof of part (a) n conjuncton wth the relaton Λ 0 = αψ 0 + (1 α)i, wemaywrteωq θθ as Ω q θθ = TΨ 0 (I θ (Λ0 )q ) (I Ψ 0 ) 1 Δ (I Ψ 0 ) 1 (I (Λ 0 )q )Ψ 0. Smlarly, usng the relaton θ Λq (θ 0 0 ) = (Λ 0 )q,weobtanω q θ = TΛ 0 (I θ (Λ0 )q ) (I Λ 0 ) 1 Δ (Λ 0 )q. Therefore, f ρ(λ 0 )<1, then Ωq θθ TΨ 0 0 θ (I Ψ ) 1 Δ (I Ψ 0 ) 1 Ψ 0 and θ Ωq θ 0as q, and t follows that V qnl [TΨ 0 0 θ (I Ψ ) 1 Δ (I Ψ 0 ) 1 Ψ 0 θ ] 1 as q. Ths lmt s the same as V MLE = (T E[ θ ln (θ 0 )(a mt x mt ) θ ln (θ 0 )(a mt x mt )]) 1, where (θ) arg max Mθ E ln (a mt x mt ) wth M θ { B : = Ψ(θ )}, because θ (θ) = (I Ψ (θ (θ))) 1 θ Ψ(θ (θ)) holds n a neghborhood of θ = θ 0. We omt the proof of part (d) because t s dentcal to the proof of roposton 11 except that ˆθ RM, ˆ RM, (Ω Γ θθ ) 1 Ω Γ,andM θ Γ θ Γ are replaced wth ˆθ qnl, ˆ qnl, (Ω q θθ) 1 Ω q θ, andm Λ q θ Λq, respectvely. Q.E.D. AENDIX D: UNOBSERVED HETEROGENEITY Ths secton extends our analyss to models wth unobserved heterogenety. The NL algorthm has an mportant advantage over two-step methods n estmatng models wth unobserved heterogenety because t s dffcult to obtan a relable ntal estmate of n ths context. Suppose that there are K types of agents, where type k s characterzed by a type-specfc parameter θ k, and the probablty of beng type k s π k, wth K k=1 πk = 1. These types capture tme-nvarant state varables that are unobserved by researchers. Wth a slght abuse of notaton, denote θ = (θ 1 θ K ) Θ K and π = (π 1 π K ) Θ π. Then, ζ = (θ π ) s the parameter to be estmated, and let Θ ζ = Θ K Θ π denote the set of possble values of ζ. The true parameter s denoted by ζ 0. Consder a panel data set {{a mt x mt x m t+1 } T t=1 }M such that w m=1 m ={a mt x mt x m t+1 } T t=1 s randomly drawn across m s from the populaton. The condtonal probablty dstrbuton of a mt gven x mt for a type k agent s gven by a fxed pont of θ k = Ψ(θ k θ k). To smplfy our analyss, we assume that the transton probablty functon of x mt s ndependent of types and gven by f x (x m t+1 a mt x mt ) and s known to researchers. 4 4 When the transton probablty functon s ndependent of types, t can be drectly estmated from transton data wthout solvng the fxed pont problem. Kasahara and Shmotsu (2008) analyzed the case n whch the transton probablty functon s also type-dependent n the context of a sngle-agent dynamc programmng model wth unobserved heterogenety.

23 ESTIMATION OF STRUCTURAL MODELS 23 In ths framework, the ntal state x m1 s correlated wth the unobserved type (.e., the ntal condtons problem of Heckman (1981)). We assume that x m1 for type k s randomly drawn from the type k statonary dstrbuton characterzed by a fxed pont of the equaton p (x) = x X p (x )( a A θ k(a x ) f x (x a x )) [T(p θ k)](x). Snce solvng the fxed pont of T( )for gven s often less computatonally ntensve than computng the fxed pont of Ψ(θ ), we assume the full soluton of the fxed pont of T( ) s avalable gven. Let k denote type k s condtonal choce probabltes, stack the k s as = ( 1 K ),andlet 0 denote ts true value. Defne Ψ(θ ) = (Ψ (θ 1 1 ) Ψ(θ K K ) ). Then, for a value of θ, the set of possble condtonal choce probabltes consstent wth the fxed pont constrants s gven by M ={ θ BK : = Ψ(θ )}. The maxmum lkelhood estmator for a model wth unobserved heterogenety s { (25) ˆζ MLE = arg max ζ Θ ζ max M θ M 1 M ln ( [L(π )](w m ) )} m=1 where [L(π )](w m ) = K k=1 πk p (x k m1 ) T t=1 k (a mt x mt )f x (x m t+1 a mt x mt ), and p = T(p k ) s the type k statonary dstrbuton of x when the condtonal choce probablty s k.if 0 s the true condtonal choce probablty k k dstrbuton and π 0 s the true mxng dstrbuton, then L 0 = L(π 0 0 ) represents the true probablty dstrbuton of w. We consder a verson of the NL algorthm for models wth unobserved heterogenety orgnally developed by AM07 as follows. Assume that an ntal consstent estmator 0 = ( 1 0 K 0 ) s avalable. Startng from j = 1, terate the followng steps untl j = l: Step 1. Gven j 1, update ζ = (θ π ) by ζ j = arg max ζ Θ ζ M 1 M ln ([ L(π Ψ (θ j 1 )) ] (w m ) ) m=1 Step 2. Update usng the obtaned estmate θ j by j = Ψ( θ j j 1 ). If teratons converge, the lmt satsfes ˆζ = arg max ζ Θζ M 1 M ln([l(π m=1 Ψ(θ ˆ))](w m )) and ˆ = Ψ(ˆθ ˆ). Among the pars that satsfy these two condtons, the one that maxmzes the pseudo lkelhood s called the NL estmator, whchwedenoteby(ˆζ NL ˆ NL ). Let us ntroduce the assumptons requred for the consstency and asymptotc normalty of the NL estmator. They are analogous to the assumptons used n AM07. Defne ζ 0 () and φ 0 () smlarly to θ 0 () and φ 0 () n the man paper.