112 sequential aliment method (a) V in = β 1 x 1in + β 2 x 2in + + β K x Kin (3.120) V in n i x kin n i k β k k (x kin ) (β k ) (3.120)

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1 112 sequential aliment method (a) V in = β 1 x 1in + β 2 x 2in + + β K x Kin (3.120) V in n i x kin n i k β k k (x kin ) (β k ) (3.120)

2 U in = β 1 x 1in + β 2 x 2in + + β K x Kin + ϵ in = V in + ϵ in (3.121) V in ( ) ϵ in ( ) U in (random utility) (2002)[1] ( ) β k n i i U in P n (i) (3.121) P n (i) = P r[u in U jn, for j, j i] (3.122) = P r[v in + ϵ in V jn + ϵ jn, for j, j i] P n (i) 2 (2 ) 2 i,j (3.126)

3 114 P n (i) = P r[u in U jn ] = P r[v in + ϵ in V jn + ϵ jn ] = P r[ϵ jn ϵ in V in V jn ] (3.123) = P r[ϵ n V in V jn ] = F ϵ (V in V jn ) ϵ n ϵ jn ϵ in F ϵ ϵ n (cumulative distribution function) ϵ n (probit model) P n (i) = Φ ϵ (V in V jn) Vin V jn 1 [ = exp 1 2πσ 2 2 = V in V jn σ ( Vin V jn) = Φ σ 1 [ exp 1 2π 2 z2] dz ( ϵ σ ) 2 ] dϵ (3.124) σ ϵ n Φ (logit model) 1 P n (i) = 1 + exp( µ(v in V jn )) exp(µv in ) = exp(µv in ) + exp(µv jn ) (3.125)

4 µ ϵ n ϵ n 2 S (b) (g) (h)?? (b) MNL MNL (multinomial choice model(luce(1959),mcfadden(1973)) (a) 2 3 i i i P n (i) = P r[u in U jn, for j, j i] = P r[u in max j,j i U jn] (3.126) (identically and independently distributed; IID) ( 1 ) F (ϵ) = exp( exp( µ(ϵ η))) (3.127) f(ϵ) = µ exp( µ(ϵ η)) exp( exp( µ(ϵ η))) (3.128) µ ϵ n η ( ) η + γ/µ( γ 0.577) π 2 /6µ 2

5 116 2 (1) ϵ 1 ϵ 2 (η 1, µ),(η 2, µ) ϵ = ϵ 1 ϵ 2 1 F (ϵ) = 1 + exp(µ(η 2 η 1 ϵ)) (3.129) (2) ϵ 1, ϵ 2,..., ϵ i,..., ϵ I (ϵ i, µ) ϵ 1,..., ϵ I max(ϵ 1,..., ϵ I ) ( 1 µ ln I i=1 ) exp (µη i ), µ (3.130) ϵ 1, ϵ 2,..., ϵ I (0,µ) (3.126) max j,j i U jn Un (2) ( 1 µ ln ) exp (µv jn ), µ j i (3.131) U n = V n + ϵ n V n = 1 µ ln j i exp (µv jn) ϵ n (0, µ) (3.126) (1) P n (i) = P r[v in + ϵ in V n + ϵ n ] = P r[ϵ n ϵ in V in V n ] 1 = 1 + exp (µ(vn V i n)) exp (µv in ) = exp (µv in ) + exp (µvn ) (3.132) exp (µv in ) = ( exp (µv in ) + exp ln ) j i exp (µv jn) = exp (µv in) j exp (µv jn)

6 µ 1 IIA (independence from irrelevant alternatives) IIA 2 i, j P in /P jn = exp (V in V jn ) i, j i, j IIA IIA 1/2 IIA 1/3 1/3 1/3 1/2 1/4 1/4 IIA (c) (g) (c) NL IIA (Nested Logit model:nl model(ben-akiva(1973))) ( (c))

7 (NL ) d i 1:(d = i = ) 2:( ) 3:( ) 4:( ) 4 ( ) n U di =V d +V i + V di + ϵ d + ϵ di (3.133) U di di V d, V i d, i V di d, i ϵ d d (max U di µ d ) ϵ di d, i ( µ ) ϵ d di P (d, i) P (i d) P (d) P (d, i) = P (i d)p (d) (3.134)

8 [ P (d) = P r max i [ = P r U di max V d + ϵ d + max i i ] U d i, d d ( ) Vi + V di + ϵ di (3.135) ( ) Vi + V d i + ϵ d i, d d] V d + ϵ d + max i ϵ di max i (V i + V di + ϵ di ) µ V d V d 1 µ ln i exp (µ(v i + V di )) V d (3.135) [ ] P (d) = P r V d + V d + ϵ d + ϵ d V d + V d + ϵ d + ϵ d, d (3.136) d ϵ d ϵ d max i(v i + V di + ϵ di ) V d (3.136) V d + V d ϵ d + ϵ d ϵ d P (d) = exp(µd (V d + V d )) d exp(µ d (V d + V d )) (3.137) P (m d)( d i ) [ ] P (m d) = P r U di U di, i i d (3.138) [ ] = P r V i + V di + ϵ di V i + V di + ϵ di, i i d d V d ϵ d ϵ di IID P (m d) P (i d) = exp(µ(v i + V di )) i exp(µ(v i + V di )) (3.139) (3.134) (3.137) (3.139) P (d, i)

9 120 P (d, i) = P (i d)p (d) = exp(µ(v i + V di )) exp(µd (V d + V d )) i exp(µ(v i + V di )) d exp(µd (V d + V d )) (3.140) (3.140) 2 µ d µ µ 1 µ d µ d /µ µ d µ = 1 V ar(ϵ d +ϵ d ) 1 V ar(ϵ di) = V ar(ϵ di ) V ar(ϵ d ) + V ar(ϵ di ) 1 (3.141) ϵ d µ ϵ di (c) ( ) µ d /µ 1 (d) GEV GEV (General Extreme Value model, McFadden(1978)) GEV (b) (g) GEV GEV GEV C = (1,, i,, n) i P (i C) P (i C) = y G(y1,y2,,yn) i y i µg(y 1, y 2,, y n ) y i = exp(v i ), (U i = V i + ϵ i, i = 1, 2,, n) (3.142) n G µ-gev µ-gev 1. G(y) 0 for all y R J + 2. G(y) µ G(λy) = λ µ G(y), λ > 0 3. lim yi G(y 1,, y i,, y n ) = +, for each i = 1,, n

10 G(y) k D κ (y) ( 1) k D κ (y) 0, y R J + (3.143) κ = (i 1,, i k ), D κ (y) = k G y i1 y ik (y) (3.142) F F (ϵ 1,, ϵ j,, ϵ J ) = exp( G(e ϵ 1,, e ϵ j,, e ϵ (3.144) J )) F 3 F lim G = + then lim ϵ j lim G = 0 then lim F = 1 {ϵ j} + {ϵ j} + F = 0 (3.145) ϵ j 0 1 Q k Q 1 = G 1 = G(y 1, y 2,, y n ) y 1 Q k = Q k 1 G k Q k 1 / y k (3.146) Q k Q k 1 G 1 G k 4 Q k 1 G k Q k 1 / y k Q k / y k+1 Q k = Q k 1 G k + Q k 1 y k+1 y k+1 G k y k+1 2 Q k 1 y k y k+1 (3.147) 4 Q k / y k+1 Q k F 1

11 122 F = exp( G(e ϵ 1,, e ϵ J )) ( G(e ϵ 1,, e ϵ J )) ϵ 1 ϵ 1 = F ( G 1 ) ( e ϵ 1 ) (3.148) = e ϵ 1 Q 1 F 0 (k 1) k k 1 F ϵ 1 ϵ k 1 = e ϵ1 e ϵ k 1 Q k 1 F k F = ( ) e ϵ1 e ϵ k 1 Q k 1 F ϵ 1 ϵ k ϵ k ( )( = e ϵ1 e ϵ Qk 1 F ) k 1 F + Q k 1 ϵ k ϵ k = e ϵ1 e ϵ k Q k F 0 (3.149) (3.146) (3.148) F F i j ϵ j = + F = exp( a i exp( ϵ i )) ( a i = G(0,, 0, i = 1, 0,, 0)) µ 1 η 0 F j U j = V j + ϵ j i P (i)

12 P (i) = = = + exp + F (, ϵ V i + V i 1, ϵ, ϵ V i + V i+1, ) dϵ ϵ i e ϵ G i (, e ϵ V i+v i 1, e ϵ, e ϵ V i+v i+1, ) ( ) G(, e ϵ V i+v i 1, e ϵ, e ϵ V i+v i+1, ) dϵ e ϵ G i (, e V i 1, e V i, e V i+1, ) exp ( e ϵ e V i G(, e V i 1, e V i, e V i+1, )dϵ = e Vi G i (, e V i 1, e V i, e V i+1, ) G(, e V i 1, e V i, e V i+1, ) (3.150) G 2 µ = 1, λ = e ϵ e V 1 (3.150) (3.142) (3.142) µ-gev G(y) P (i C) MNL NL (e) CNL MNL : G(y) = NL : G(y) = J y µ i i=1 ( J i D d=1 i=1 ) µ y µ d µ d i IIA (Cross Nested Logit model:cnl model) CNL NL ( 3.55) i m

13 (CNL ) α im 1 µ-gev G(y 1,, y n ) = M ( n m=1 j C jm y j) µm ) µ µm (α 1/µ (3.151) C i P (i C) α im i m M µ m m µ i 0 < µ µ m (µ m = 1 ) P (i C) = M m=1 ( ) µ j C αµm/µ µm jm eµmvj M ) m =1( j C αµ m /µ jn e µ m V j α µ m/µ im eµ mv i (3.152) j C αµ m/µ jm eµ mv j m P m m i P i m

14 P (i C) = P m = P i m = M m=1 P m P i m (3.153) ( ) µ j C αµ m/µ jm eµ µm mv j M ) (3.154) m =1( j C αµ m /µ jn e µ m Vj α µ m/µ im eµ mv i j C αµ m/µ jm eµ mv j (3.155) α im 0 α im 1, M α im = 1, i (3.156) m α im α im V i (f) GNL GNL CNL PCL (Chu, 1989[2]) µ-gev G(y 1, y 2,, y n ) = ( ) µm (α i my i ) 1/µm (3.157) m i N m N m m µ m m 0 < µ m 1 α im i m α im α im 0, m α im = 1 i P i P i = ( m ) µm 1) (α im e V i ) m( 1/µ i N (α m i me V i ) 1/µ m ( ) i (α µm (3.158) Nm i me V i ) 1/µm m

15 126 P i = m P i m P m (3.159) P m = P i m = ( i Nm (α i me V i ) 1/µ m) µm ( ) m i (α µm (3.160) Nm i me V i ) 1/µm (α im e Vi ) 1/µm i N m (α im e Vi ) 1/µm (3.161) Discrete Choice Analysis. Press, Cambridge, MA The MIT ) m P mp i m ((1 P i ) + ( 1 µ m 1)(1 P i m ) βx i (3.162) P i (g) ( m P i + ( 1 network GEV µ m )P m P i m P i m P i ) βx i (3.163) Daly and Bierlaire(2006) network GEV GEV [3] GEV µ-gev network GEV GEV-network GEV-network GEV GEV-network GEV N A G(N, A) G(N, A) (i, j) associated parameter α ij GEV-network

16 Root Alternatives C 3.56 GEV-network (g) GEV-network C (g) 1 GEV-network v i /C G G i G i (y i ) = α ij G j µ i µ (y) j (3.164) v j S(v i ) G i GEV GEV GEV Daly and Bierlaire(2006)[3] S(v i ) v i GEV (h) MNP (multinomial probit; MNP) [4][5] MNP GEV n i MNP U in = V in + ϵ in, i = 1,..., I (3.165) ϵ n = (ϵ 1n, ϵ 2n,..., ϵ In ) (3.166) 0 Ω i

17 128 P (i) = ϕ(ϵ) = ϵi +V i ϵ 1 ϵ 1= 1 2π I 1 2 σ 1 2 ϵi +V i ϵ J ϕ(ϵ)dϵ J (3.167) ϵ 1 ϵ i= ϵ J = exp( 1 2 ϵσ 1 ϵ ) (3.168) -1 MNP 3 10?? MACML(Maximum Approximate Composite Marginal Likelihood)[6][7] open-form 10 (i) MXL(MMNL) Mixed Logit MNL MNL Mixed Logit Mixed Logit n i U in = V in + η in + ϵ in (3.169)

18 129 V in = β i + X in (3.170) V in β X η Ω ϵ IID Mixed Logit 2 n i P in η in P in (η in ) = exp(v in + η in ) j exp(v jn + η jn ) (3.171) η in Ω f(η in Ω) exp(v in + η in ) P in = j exp(v jn + η jn ) f(η in Ω)dη in (3.172) L MNL N J ln L = δ in P in (3.173) n=1 i=1 δ in n i 1 Mixed Logit ln SL ln SL = 1 R R N J δ in ln P n (i) (3.174) r=1 n=1 i=1 δ in n i 1 0

19 130 [1],,,, :,, [2] Chu, C.:A paired combinational logit model for travel demand analysis, Proceedings of Fifth World Conference on Transportation Research, Vol. 4, pp , [3] Daly, A. and Bierlaire, M.: A general and operational representation of generalised extreme value models, Transportation Research Part B, vol. 40, pp , [4] [5] [6] Bhat, C.R.: The maximum approximate composite marginal likelihood (MACML) estimation of multinomial probit-based unordered response choice models, Transportation Research Part B: Methodological, Vol.45, No.7, pp , [7] Bhat, C.R., Sidharthan, R.: A simulation evaluation of the maximum approximate composite marginal likelihood (MACML) estimator for mixed multinomial probit models, Transportation Research Part B: Methodological, Vol.45, No.7, pp , Tobin(1958) Tobit (2012) (Kuhn-Tucker, KT)

20 y1 c y1 (a) Tobit Tobin(1958) Tobit y n > 0 y n y n 0 yn = βx n + ϵ n yn y n = if y n > 0 (3.175) 0 if yn 0 yn n y y n n y x n n y β Tobit 1 Amemiya(1974) Tobit(1958) Heckman(1974,1979) y n1 y n2 y n1 > 0 y i2

21 132 y n1 = β 1 x n1 + ϵ n1 yn2 = β 2 x n2 + ϵ n2 yn2 if yn1 y i2 = > 0 0 if yn1 0 (3.176) Amemiya(1985) Tobit TypeI TypeV Heckman(1979) TypeII TypeII 2 Tobit Tobit TypeIII V Fang(2008) TypeV (Endogenous Switching Regression Model) Maddala(1983) y n1 y n2 = β 1 x n1 + ϵ n1 = β 2 x n2 + ϵ n2 yn3 = β 3 x n3 + ϵ n3 1 if yn1 y n1 = > 0 0 if yn1 0 yn3 if y n1 = 0 y n3 = 0 if y n1 = 1 (3.177)

22 133 Fang(2008) 2 Tobit Orderd Probit BMOPT(Bayesian Multivariate Ordered Probit and Tobit) y n1 y n2 y n3 = β 1 x n1 + ϵ n1 = β 2 x n2 + ϵ n2 = β 3 x n3 + ϵ n3 yn4 = β 4 x n4 + ϵ n4 0 if y nj < α 1 y nj = 1 if α 1 ynj < α 2, for j = 1, 2 2 if ynj > α 2 yn3 if y n1 = 1 or 2 y n3 = 0 if y n1 = 0 yn4 if y n2 = 1 or 2 y n4 = 0 if y n2 = 0 (3.178) α 1 Orderd Probit Cut point Fang(2008) α 1 = Φ 1 (1/3), α 2 = Φ 1 (1/3) Lee(1983), Bhat and Eluru(2009) Lee(1983) Bhat and Eluru(2009) 2 2 (b)

23 134 n I x i n U n U n = f n (z 1n, z 2n,..., z In ) I i=1 p iz in = E n, z in 0 (3.179) p i E n n KT Dubin and McFadden(1984) i Y in = Y in (p i, E n, x in, s n, ϵ in ) (3.180) x in i s n n ϵ in n i P in = P r[y in (p i, E n, x in, s n, ϵ in ) > Y jn (p j, E n, x jn, s n, ϵ jn ), j I, j i] (3.181) ϵ in Y in = Y in (p i, E n, x in, s n ) + ϵ in (3.182) ϵ in i.i.d. P in z i.

24 135 z in = Y in(p i, E n, x in, s n, ϵ in )/ p i Y in (p i, E n, x in, s n, ϵ in )/ E n (3.183) Y in Dubin and McFadden(1984) Y in = [αe n + βp i + γx in + θs n ] exp( ρp i ) + ϵ in (3.184) z in = β α + ρ α (αe n + βp i + γx in + θs n ) (3.185) KT KT Wales and Woodland(1983) 1 ( I ) L in = U in (z) λ p i z in E n (3.186) λ KT (Kuhn and Tucker, 1951) i=1 U in (z) λp i 0 z in, p T i z in 1 0 λ i = 1,..., M (3.187) U in (x) λ λ = U 1n (z)/p 1 (3.188)

25 p 1 U in (z) p i U 1n (z) 0 z in, i = 2,..., M, p T i z in = 1 (3.189) z in > 0 p 1 U in (x) p i U 1n (x) = 0 U in (x)/u 1n (x) = p i /p 1 i 1 i U in (z, ϵ in ) = V in (z) + ϵ in, i = 1,..., M, (3.190) U in (z) V in (z) + ϵ in (p 1 ϵ in p i ϵ 1n ) + [p 1 V in (z) p i V 1n (z)] 0 z in, i = 2,..., M, p T i z in = 1 (3.191) ϵ in ϵ in z in ϵ in 0 Σ ϵ in y in = p 1 ϵ in p i ϵ 1n 0 Ω z z 1 n y i ȳ i (ẑ) 0 z in, i = 2,..., M, (3.192) ẑ = (z 2,..., z M ) M

26 137 y i = ȳ i (ẑ), f(ẑ) = ϕ(ŷ, Ω)abs[J(ẑ)], ŷ = (y 2,..., y M ) (3.193) ϕ J y z M 1 z = 0 f(0) = ȳm ȳ2... ϕ(ŷ, Ω)dy 2... dy M (3.194) K f(z 2,..., z K, 0,..., 0) = (3.195) ȳm... ȳk+1 ϕ(y 2,..., y K, y K+1,..., y M, Ω) abs[j K (ˆx)]dy K+1... dy M J K (z K+1,..., z M ) = 0 (y 2,..., y M ) (z 2,..., z M ) (y 2,..., y M ) (z 2,..., z M ) M!/K!(M K)! z 1 > 0 z 1 = Wales and Woodland(1983) 1 2 KT MDCEV Bhat(2005,2008) Wales and Woodland(1983) Kim et al. (2002) Multiple Discrete-Continuous Extreme Value (MDCEV)

27 138 MDCEV Nested logit model mixed logit model MDCEV MDCEV MDCEV U(z) = i γ i α i [exp(βx i + ϵ i )] {( zi γ i + 1 ) αi 1} (3.196) exp(βx i + ϵ i ) 0 1 x i β ϵ i α i γ i γ i I p i z i = E (3.197) i=1 Wales and Woodland(1983) KT L = [ I ] [exp(βx i + ϵ i )](z i + γ i ) α i λ z i E i i=1 KT (3.198) [exp (βx i + ε i )] α i ( z i + γ i ) αi 1 λ = 0 if z i > 0, i = 1,..., I ( ) [exp (βx i + ε i )] α i z αi 1 i + γ i λ < 0 if z i = 0, i = 1,..., I (3.199)

28 V i + ϵ i = V 1 + ϵ 1, if zi > 0, i = 2,..., I, V i + ϵ i < V 1 + ϵ 1, if zi = 0, i = 2,..., I, where V i = βx i + ln α i + (ln α i 1) ln(zi + γ i), i = 1,..., I (3.200) ϵ i standard extreme value distribution z i I M z 2 z M P (z 2, z 3,..., z M, 0, 0,..., 0) ϵ 1 M = g(v 1 V j + ϵ 1 ) J j=2 I s=m+1 G(V 1 V s + (3.201) ϵ 1 ) g standard extreme value density function G standard extreme value distribution J J jh = [V 1 V j+1 + ϵ 1 ], i, h = 1, 2,..., M 1 (3.202) zh P (z 1, z 2, z 3,..., z M, 0, 0,..., 0) = ( ) M 1 αj M ( z ) zj + γ j + γ j j 1 α j j=1 j=1 M j=1 ev j ( I ) M k=1 ev k (M (3.203) 1)! M = MNL Dubin and McFadden(1984) single discrete-continuous model

29 140 i.i.d. MNL Bhat mixed MDCEV(MMDCEV) model (Bhat,2005) MMDCEV ϵ i 3 ζ i i.i.d. 2 η w i w i I η 0 ωi 2 ω η 3 µ x i x i H h i 1 0 µ H σ 2 h σ µ P (z 2, z 3,..., z M, 0, 0,..., 0) = ( ) M 1 αj M ( z ) zj + γ j + γ j j 1 α j η µ j=1 j=1 M j=1 evj+η w j+µ x j ( I k=1 evk+η w k+µ x k ) M (M 1)!dF (µ σ)df (η ω) (3.204) F multiple discrete-continuous nested extreme value (MDCNEV) model (Pinjari and Bhat, 2010) GEV (multiple discrete-continuous generalized extreme value (MDCGEV) model, Pinjari, 2011) Wales and Woodland(1983) KT Tobit KT 1 0

30 141 Tobit 1 (c) FIML: full information maximum-likelihood (FIML ) Heckman(1974), TypeII Tobit 2 FIML ( ) (( ) ( )) a1 µ1 Σ11 Σ 12 N, a 2 µ 2 Σ 21 Σ 22 a 1 a 2 = b N (3.205) ( ) µ 1 + Σ 12 Σ 1 22 (b µ 2), Σ 11 Σ 12 Σ 1 22 Σt 12 (3.206) L

31 142 L= n:y n2 0 = n:y n2 0 = n:y n2 0 P r(y n2 0) P r(y n2 0) n:y n2 >0 n:y n2>0 [P r(y n1 y n2 > 0)P r(y n2 > 0)] [P r(y n2 > 0 y n1 )P r(y n1 )] [1 Φ(β 2 x n2 )] (3.207) n:y n2 >0 [ Φ ( 1 1 ρ 2 1 ( )] yn1 β 1 x n1 ϕ σ 1 σ 1 { β 2 x n2 + ρ } ) (y n1 β 1 x n1 ) σ 1 (3.208) Lee(1983) FIML Heckman(1979) y 2 > 0 ( y 1 ) y 1 E(y n1 y n2 > 0) = E(β 1 x n1 y n2 > 0) + E(ϵ n1 y n2 > 0) = β 1 x n1 + E(ϵ n1 ϵ n2 > β 2 x n2 ) = β 1 x n1 + (ρσ 1 ) ϕ(β 2x n2 ) Φ(β 2 x n2 ) (3.209) 1 y 1n β 2 λ n λ n = ϕ(β 2x n2 ) Φ(β 2 x n2 ) (3.210)

32 143 2 y 1n β 1 (ρσ 1 ) N n:y n2 >0 ( y n1 β 1 x n1 + (ρσ 1 ) λ n ) 2 (3.211) FIML: full information maximum-likelihood FIML ϵ i (2008) η = 0 P i i η = { ( ) σ 6 π r Pj ln P } j i + ln P j + ν (3.212) 1 P j i j σ η r j η ϵ j ν 1 i η. σ 6r j = π (2008) WESML weighted exogenous sample maximum likelihood

33 144 ln L(θ) N = w(j n ) {ln P (J n x n, θ) + ln f(x Jn x n, θ)} w(j n ) = Q(J n )/H(J (3.213) n ) n=1 N J n n w(j n ) Q(J n ) H(J n ) x n θ xj n P ( ) f( ) WESML θ Σ Λ θ Ω Σ = 1 N Ω 1 ΛΩ 1 (3.214) KT KT (1) z N M K=1 M!/K!(M K)! 1 L(ẑ 1,..., ẑ N ) = N f(ẑ n ) (3.215) n=1 z n z n U(z, ϵ) Σ MDCEV MDCEV mixed

34 145 MMDCEV [ Q ( ) M 1 αj M ( z ) L(β, θ, γ, σ, ω) = q=1 η µ z j=1 j + γ j + γ j j 1 α j=1 j M ] j=1 ev j+η w j +µ x j ( I ) M (M 1)!dF (µ σ)df (η ω) (3.216) k=1 ev k+η w k +µ x k [1] Amemiya, T.: Advanced Econometrics, Harvard University Press, [2] Bhat, C.R.: A multiple discrete-continuous extreme value model: formulation and application to discretionary time-use decisions, Transportation Research Part B: Methodological 39, pp , [3] Bhat, C.R.: The multiple discrete-continuous extreme value (MDCEV) model: Role of utility function parameters, identification considerations, and model extensions, Transportation Research Part B: Methodological 42, pp , [4] Bhat, C.R., Eluru, N.: A copula-based approach to accommodate residential self-selection effects in travel behavior modeling, Transportation Research Part B: Methodological 43, pp , [5] Dubin, J.A., McFadden, D.L.: An econometric analysis of residential electric appliance holdings and consumption. Econometrica 52 (2), pp , [6] Fang, H.A.: A discrete-continuous model of households vehicle choice and usage, with an application to the effects of residential density, Transportation Research Part B: Methodological 42 (1), pp , [7] Heckman, J.: Sample selection bias as a specification error, Econometrica 47, pp , [8] Lee, L.-F.: Generalized econometric models with selectivity, Econometrica 51, pp , [9] Maddala, G. S.: Limited-Dependent and Qualitative Variables in Econometrics, Cambridge University Press, [10] Pinjari, A.R., Bhat, C.: A multiple discrete-continuous nested extreme value (MDCNEV) model: Formulation and application to non-worker activity timeuse and timing behavior on weekdays, Transportation Research Part B: Methodological 44, pp , 2010.

35 146 [11] Tobin, J.: Estimation of relationships for limited dependent variables, Econometrica 26, pp , [12] Wales, T.J., Woodland, A.D.: Estimation of consumer demand systems with binding non-negativity constraints, Journal of Econometrics 21, pp , [13] [14]? 46(CD-ROM), [15] - - Vol.43, pp , (a) 1 n i n n y n d(y n = i n ) (3.217) N d(y n = i n y n = i n 1

36 147 (b) β L (McFadden, 1974[?]) N J L(β) = P n (i β) d in (3.218) n=1 i=1 d in n i N J ln L(β) = d in ln P n (i β) (3.219) n=1 i=1 β (c) 0 L(0)

37 148 µ µ µ ln L(0) ln L( ˆ) β 0 2 ln L(0) ln L( ˆ) β ρ = ln L(0) ln L(β) 0 β ˆβ L( ˆβ) 0 McFadden(1975) McFadden ρ 2 = ln L( ˆβ) ln L(0) ln 1 ln L(0) = ln L(0) ln L( ˆβ) ln L(0) (3.220) McFadden McFadden Ben-Akiva and Swait(1986) K (AIC) ln 2L( ˆβ) + 2K ρ 2 ρ 2 = ln L(0) (ln L( ˆβ) K) ln L(0) (3.221)

38 149 (d) L β L 0 n max L(β) (3.222) β β = (β 1, β 2,..., β n ) (3.223) β ˆβ Newton-Raphson (BFGS ) 2 Berndt-Hall-Hall-Hausman(BHHH) (Berndt et al., 1974[?]) Nelder-Mead (Nelder and Mead, 1965[?]) Newton-Raphson BFGS BHHH Nelder-Mead 1 β 1

39 150 2 k s k s k = L(β k ) (3.224) 3 α k max α L(β k α k s k ) (3.225) 4 β k. β k+1 = β k α k s k ) (3.226) 5 β k k = k + 1 Step2 Newton-Raphson Newton-Raphson 1 2 β 1 β k 2 L(β k ) = A = 2 L β L β 1β n : : : 2 L β 1 β n 2 L β 2 n (3.227)

40 151 3 k s k s k = ( 2 L(β k )) 1 L(β k ) (3.228) 4 α k max α L(β k α k s k ) (3.229) 5 β k. β k+1 = β k α k s k ) (3.230) 6 β k k = k + 1 Step2 BFGS BFGS 1970 Broyden (1970)[?] Fletcher (1970)[?] Goldfarb (1970)[?] Shanno (1970)[?] 4 BFGS Newton-Raphson B 1 2 β 1 H k α k max α L(β k α k H k L(β k )) (3.231)

41 152 3 β k. β k+1 = β k α k H k L(β k ) (3.232) 4 5 s k = βk + 1 βk y k = L(β k+1 ) L(β k ) H H k+1 = [ I s ky T k s T k y k ] H k [ I y ks T k s T k y k ] + s ks T k s T k y k (3.233) β k k = k + 1 Step2 Mixed Logit β f(β θ) n i J j=1 P n (i) = exp(v jn + η jn ) f(β θ)dβ (3.234) exp(v in + η in ) SP n (i) R (Train, 2003[?]) SP n (i) = 1 R P n (i, β r ) (3.235) R r=1 β r f(β θ) r β

42 153 (a) µ (b) µ 3.58 Bhat (2001) (Bhat, 2001[?]) (0,1) 3 (0,1) 3 1/3 2/3 (0,1/3) (1/3,2/3) (2/3,1) 3 3 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 (0,1) EM (Train, 2008[?]) (Harding and Hausman, 2007[?]) (Cherchi, E. and Guevara, 2011[?]) (e)

43 154 Mixed Logit, 2012 day-to-day Parry and Martin, 2013, 2010 OD, 2010; Li, 2009; Hazelton, 2001; Maher, 1983, 2010;, 2009b, 2009a Park et al., 2010, , 2009, 2009;, 2009, 2008 θ n

44 155 y n P n (y n θ) N L(Y θ) = P n (y n θ) (3.236) n=1 Y = {y 1, y 2,..., y n } θ π(θ) Y θ π(θ x) π(θ Y )L(Y ) = L(Y θ)π(θ) (3.237) L(Y ) Y L(Y ) = L(Y θ)π(θ)dθ (3.238) bayes1, 2010 L(Y θ)π(θ) π(θ Y ) = (3.239) L(Y ) L(Y ) θ L(Y θ) L(Y ) π(θ y t ) L(y t θ)π(θ) (3.240) π(θ y t ) t θ t + 1 π(θ y t ) θ, 2010

45 µ µ σ 2 π(θ y t, y t+1 ) L(y t+1 θ)π(θ y t ) (3.241) t y t 3.3 µ 0 Sigma 0 µ 1 Sigma 1 ˆµ ˆΣ ˆµ = ˆΣ(Σ 1 0 µ 0 + Σ 1 1 µ 1) (3.242) ˆΣ = (Σ Σ 1 1 ) 1 (3.243) (1987) (1993) (MCMC )

46 157 θ = {θ 1, θ 2,..., θ k } i θ (i) = {θ (i),..., θ(i) } 1 2 1, θ(i) 2 k θ (0) i = 1 i θ (i) j θ (i) k f(θ k θ (i),..., θ(i) 1 j 1, θ(i 1) j+1,..., θ(i 1) k ) (3.244) 3 i = i (MH) MCMC Metropolis- Hastings, MH) MH 1 θ (0) i = 1 2 i θ new(i) f(θ new(i) θ (i) ) θ new(i) f(θ new(i) θ (i) ) (3.245)

47 θ new(i) α(θ (i 1), θ new(i) ) α(θ (i 1), θ new(i) ) = min(1, π(θnew(i) )f(θ (i 1) θ new(i) ) π(θ (i 1) )f(θ new(i) θ (i 1) ) ) θ (i) (3.246) θ (i) = { θ new(i) α θ (i 1) 1 α (3.247) (a) 2 Bellman i

48 159 V (s i,t ) = max{u i (a, s i,t ) + β V (s i,t+1 )df (s i,t+1 a, s i,t )} (3.248) β j U i (a, s i,t+j ) dynamic j programming(dp) Bellman (Bellman,1957) ,U i (a, s i,t ) a s i,t t,df (s i,d+1 α, s i,d )} β. s i,t x i,t, ε i,t, i.i.d extreme value,. v(a, x i,t )=u (a, x i,t )+ε i,d (a)+β V (x i,t+1 )df(x i,t+1 a, x i,t+1 ) x i,t+1 (3.249) v(α, x i,d ) u (a, x i,d ) V df t + 1

49 160 P (a x i,d, θ) = exp(v(a, x i,d)) J exp(v(j, x i,d )) j=1 (3.250) V Bellman i U i = α i + β i X i + j i γ j y j + ε i (3.251) α X i ε i y j i j j V j 3.60 i j j i

50 U i = α i + β i X i + γ i j y j N + ε i (3.252) N 3.61 N 3.61 (b)

51 162 NFXP 2 NPL 3 2 i {A, B} m 2 U im U im = αx m + β i z im λy im + ε im (3.253) x m m z im y im NFXP NFXP (Nested Fixed Point Method,Rust,1987) NFXP 3.62 NFXP 4 0: θ (0) = {α (0) m β(0) im γ(0) } i {A, B} m p im p (0) im 1: θ (n) p (k) im p(k+1) im p (k+1) im = Φ(α(n) m x m + β (n) im z im + γ (n) p (k) im ) (3.254) Φ

52 163 0: 1: No 2: 1 Yes 3: No 4: : 1 Yes NFXP p (k) im p(k+1) im ε p (k+1) im p(k) im < ε p (3.255) STEP1 p im = Φ(α(n) m x m + β (n) im z im + γ (n) p im ) (3.256) 3: 2 p im y im

53 164 CCP ln L = m i {A,B} {y im ln(p im ) + (1 y im) ln(1 p im )} (3.257) θ (n+1) = {α (n+1) m β (n+1) im γ(n+1) } 4: 2 θ (n) θ (n+1) θ (n+1) θ (n) < εθ (3.258) CCP (Conditional Choice Probabilities Method, Hotz, V. J., and R. A. Miller, 1993) : 2: 3.63 CCP 2 1: p (1) im x m z im 2

54 165 2: 1 p (1) im p (2) im = α mx m + β im z im + γ im p (1) im (3.259) ln L = m i {A,B} {y im ln(p (2) im ) + (1 y im) ln(1 p (2) im )} (3.260) NPL NPL (Nested Pseudo-Likelihood Algorithm,Aguirregabiria and Mira,2002) ( ) NPL 3.64 NPL 4 0: i {A, B} m p im p (0) im 1: p (n) im i p im Φ p im = Φ(α m x m + β im z im + γp (n) im ) (3.261)

55 166 0: 1: 2: 3: No 4: 3.64 Yes NPL ( ) ln L = m {y im ln(p im ) + (1 y im ) ln(1 p im )} i {A,B} (3.262) 2: 1 α (n) m β(n) im γ(n) 3: 2 p (n) im p (n+1) im p (n+1) im = Φ(α (n) m x m + β (n) im z im + γ (n) p (n) im ) (3.263) 4:

56 167 p (n) im p(n+1) im p (n+1) im p(n) im < ε (3.264) ε p (n+1) im 1 4 [1],, :, Vol.43, pp.14-21, [2] Rust, J., Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher, Econometrica, Vol.55,pp , [3] Hotz, V. J., and R. A. Miller, Conditional Choice Probabilities and the Estimation of Dynamic Models, Review of Economic Studies, Vol.60, pp , 1993 [4] V,Aguirregabiria., P,Mira., Swapping the Nested Fixed Point Algorithm: A Class of Estimators for Discrete Markov Decision Models (a).. TDM ITS.

わが国企業による資金調達方法の選択問題

わが国企業による資金調達方法の選択問題 * takeshi.shimatani@boj.or.jp ** kawai@ml.me.titech.ac.jp *** naohiko.baba@boj.or.jp No.05-J-3 2005 3 103-8660 30 No.05-J-3 2005 3 1990 * E-mailtakeshi.shimatani@boj.or.jp ** E-mailkawai@ml.me.titech.ac.jp