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2 (Nested Logit IIA(Independence from Irrelevant Alternatives [2004] ( [2004] 2

3 2 Spence and Owen[1977] X,Y,Z X Y U 2 U(X, Y, Z X Y X Y Spence and Owen Spence and Owen p X, p Y X Y X Y p Y p X X Y Y L 3

4 2.1 iid (random utility n i U in = V in + ɛ in U in V in ɛ in ɛ Gumbel(0,λ iid(independently and identically distributed i Q i n Q i n = ev in j ev jn (1 X Y X Y X Y X=1(,X=0(,Y=1(,Y=0( (V in = V i (X, Y =(1, 1 V 11 (X, Y =(1, 0, (0, 1, (0, 0 V 10,V 01,V 00 X p X Y p Y ((X, Y =(0, 0, (1, 0, (0, 1, (0, 0 V 11 = u 11 αp X αp Y (2 V 10 = u 10 αp X (3 V 01 = u 01 αp Y (4 V 00 = u 00 (5 u 00,u 10,u 01,u 11 1 Q 00 Q 10 Q 01 Q 11 X Q X (= Q 11 + Q 10 Y Q Y (= Q 11 + Q 01 1 X Y Z Z p Z Z X, Y U(X, Y, Z =u(x, Y +v(z I max X,Y,Z u(x, Y +v(z s.t. I p XX + p Y Y + p ZZ (X, Y =(1, 1 V 11 = max Z U(1, 1,Z=u(1, 1 + v ( I p X X p Y Y p Z V 11 = u 11 αp X αp Y 4

5 1 ɛ in ɛ Gumbel(0,λ iid Q X p Y Q X p X = Prob(Y =1 X =1 Prob(Y =1 X = 0 (6 ɛ in ɛ Gumbel(0,λ iid (1 (2 (5 (X, Y =(1, 1 Q 11 = e u11 αp X αp Y e u11 αp X αp Y + e u 10 αp X + e u 01 αp Y + e u 00 X Q X Q X = e u11 αp X αp Y + e u10 αp X e u11 αp X αp Y + e u 10 αp X + e u 01 αp Y + e u 00 Y p Y Q X p Y αe αp X αp Y (e u00 +u 11 e u10 +u 01 = (e u11 αp X αp Y + e u 10 αp X + e u 01 αp Y + e u 00 2 X Q X X p X Q X p X = α(eu00 + e u01 αp Y (e u11 αp X αp Y + e u10 αp X (e u11 αp X αp Y + e u 10 αp X + e u 01 αp Y + e u 00 2 Q X p Y Q X p X = e αp X αp Y (e u00 +u 11 e u10 +u 01 (e u00 + e u01 αp Y (e u 11 αp X αp Y + e u 10 αp X (7 X Y Prob(Y =1 X =1 = Q11 Q X = Prob(Y =1 X =0 = e u11 αpx αpy e u10 αp X + e u 11 αp X αp Y Q 10 1 Q X = e u01 αpy e u01 αp Y + e u 00 5

6 Prob(Y =1 X =1 Prob(Y =1 X =0= (7 ( η XY e αp X αp Y (e u00 +u 11 e u10 +u 01 (e u00 + e u01 αp Y (e u 11 αp X αp Y + e u 10 αp X αe αp X αp Y (e u00 +u 11 e u10 +u 01 p Y η XY = (e u10 αp Y + e u 11 αp X αp Y (e u 11 αp X αp Y + e u 10 αp X + e u 01 αp Y + e u 00 (6 X Y X Y p X /p Y X Y (6 X Y (X, Y =(1, 1, (0, 0 S = Prob(Y =1 X =1 Prob(Y =1 X = 0 (8 S =1 0=1 (X, Y =(1, 0, (0, 1 S =0 1= 1 - [ 1, 1] (6 (7 e u00 +u 11 e u10 +u 01 (9 X Y (8 (9 e u00 +u 11 e u10 +u 01 6

7 (u 00 + u 11 (u 10 + u 01 (u 11 u 10 (u 01 u 00 (10 u 11 u 10 = u11 u u 01 u 00 = u01 u = Δu ΔY = Δu ΔY X=1 X=0 (10 Δu ΔY Δu X=1 ΔY X= u (6 X Y (6 ɛ in ɛ Gumbel(0,λ iid ɛ in X X Y X X Y X (X, Y =(1, 0or(1, 1 X Y X Y (6 X Y S S S 7

8 2.2 X Y X Y U 00 = V 00 (11 U 10 = V 10 + ɛ x (12 U 01 = V 01 + ɛ y (13 U 11 = V 11 + ɛ x + ɛ y (14 (X, Y =(0, 0, (1, 0, (0, 1, (1, 1 U 00 U 10 U 01 U 11 ɛ x ɛ y f(ɛ F (ɛ exp( x (1 + exp( x 2 (11 X Y ɛ 0 ɛ x ɛ 0 ɛ y ɛ 0 (14 X ɛ x ɛ 0 Y ɛ y (11 (14 ɛ x ɛ y X Y X ɛ x Y ɛ y V 00 V 10 V 01 V 11 8

9 (14 ɛ x + ɛ y U 00 U 10 U 01 U 11 S 2 (11 (14 U 00 + U 11 = U 10 + U 01 S =0 U 00 + U 11 <U 10 + U 01 S U 11 U 00 + U 11 >U 10 + U 01 (U 00 + U 11 (U 10 + U 01 U 00 +U 11 = U 10 +U 01 U 11 U 01 = U 10 U 00 Y X U 11 U 10 = U 01 U 00 X Y S 0 U 00 U 10 U 01 U 11 U 00 + U 11 <U 10 + U 01 S =0 iid iid (6 iid S 3 X Y 0 Prob(Y =1 X =1 Prob(Y =1 X =0 9

10 Prob(Y =1 Z = ez β 1+e Z β β Z X X Prob(X =1 Y =1 Prob(Y =1 X =1= Prob(X =1 Y =1+Prob(X =1 Y =0 Prob(X =0 Y =1 Prob(Y =1 X =0= Prob(X =0 Y =1+Prob(X =0 Y =0 Prob(X =1 Y =1 Prob(X =1 Y =0 Prob(X = 0 Y =1 Prob(X =0 Y =0 (multinominal logit Prob(Y =1 X =1= Prob(X =1 Y =1 Prob(X =1 Prob(X =1 Y =1 Prob(X =1 Z X (multinominal logit X NHK( Y CX( u XY (AGE (MF (INCM (SP u 00 =0(X =0 Y =0 10

11 P C( X =0 AGE Y =1 MF INCM SP C X =1 AGE Y =0 MF INCM SPCAN C X =1 AGE Y =1 MF INCM SP R Log Likelihood n=2614 AGE =41.6, MF =1.53, INCM =5.71, SP =1.70 u 00 =0.000, u 01 =0.532, u 10 =0.347, u 11 =0.034 Q ij e uij =, for i, j =0, 1 ukl k,l=0,1 e Q 00 =0.194, Q 01 =0.330, Q 10 =0.275, Q 11 =0.201 Prob(Y =1 X =1= Prob(Y =1 X =0= Q 11 Q 10 + Q 11 = =0.422 Q 01 Q 00 + Q 01 = =0.630 Prob(Y =1 X =1 Prob(Y =1 X =0= = NHK NTV X NTV( Y CX( u 00 =0(X =0 Y =0 11

12 P C( X =0 AGE Y =1 MF INCM SP C X =1 AGE Y =0 MF INCM SP C X =1 AGE Y =1 MF INCM SP R Log Likelihood n=2614 u 00 =0.000, u 01 = 0.058, u 10 =0.133, u 11 =0.372 Q 00 =0.220, Q 01 =0.208, Q 10 =0.252, Q 11 =0.320 Prob(Y =1 X =1= Prob(Y =1 X =0= Q 11 Q 10 + Q 11 = =0.560 Q 01 Q 00 + Q 01 = =0.485 Prob(Y =1 X =1 Prob(Y =1 X =0= = CX NTV Spence and Owen [1977] 12

13 Spence and Owen Spence and Owen Spence and Owen A NHK NTV CX NTV NHK (X =1 (X =0 NTV (X = X = CX (X = X = A X A A x x x = X A + ɛ ɛ σ A y x S y x S y x 13

14 S y x x NHK CX NTV CX Y NHK Y NHK Y NHK Y NHK NHK CX NHK CX NTV NTV CX CX X A A X B B p A p B A A S B A A B 14

15 A,B A.1 T x 1 = X 1 + ɛ 1 X 1 ɛ 1 N(0,σ 1 15

16 N X i σi 2 j j y j j R j j x i S y j x i (i =1, 2,,N R j j i x i (( P S y j x i max (S y j x k (S y j x i R j k i q j i y j R j (X i,σ i (i =1, 2,,N T i j q j i Po(λj i Tq j λ j i Po(λ A Po(λ B Po(λ A +λ B j T Po( i=1,2,,n λj i λ j i (y j,r j, (x i,σ i i=1,2,,n λj i N N(m, V Po(λ λ N(m, V 2 Po(λ λ x E λ (E x (x; λ = E λ (λ =m E λ (E x ((x m 2 ; λ = E λ (E x ((x λ 2 2(x λ(λ m+(λ m 2 ; λ = E λ (E x ((x λ 2 ; λ 2E λ (E x ((x λ(λ m; λ +E λ (E x ((λ m 2 ; λ = E λ (λ+e λ (λ m 2 = m + V λ ˆM Ŝ m = ˆM V = Ŝ ˆM Po(λ λ N( ˆM,Ŝ ˆM 2 A B j P ( max i=1,2,,n (S yj xi >Rj q j y j R j (X i,σ i(i =1, 2,,N T j Bin(T,q j Tq j m j m j f(m E(m j=m E(m j M 2 = V 16

17 k ( λ k e λ exp 0 k! 2π ˆV (λ ˆM 2 2 ˆV A.1 3 χ ( 24 5 % P = λ j i i λj i ( λ 1/λ 1/2 1/λ 5 dλ 3 4 χ =

18

19 2 NTV 19

20 A.2 5 R W R R w w (R, S, x A,x B,σ A,σ B =( 1, 0, 0, 0, 1, 1 (x A,x B (N(x A,σ A,N(x B,σ B y = W = w 6 R R max(r, S y x A,S y x B t=1 6 R S S R =1 y (y 1,y+1 20

21 (W = w (W = w =0 (W = w =0.4 (x A.x B =(0, 0 (x A.x B = (0.5, 0.5 (W = w

22 (W = w =0.2 ( 5 S y x y x B (X, Y =(0, 0, (1, 0, (0, 1, (1, 1 Q 11 = Prob((U 11 >U 00 (U 11 >U 10 (U 11 >U 01 = Prob((ɛ x + ɛ y >V 00 V 11 (ɛ x >V 10 V 11 (ɛ x >V 01 V 11 ( 1 F (V 01 V 11 ( 1 F (V 10 V 11 V 00 V 10 V 00 V 11 ɛ x = f(ɛ y dɛ y f(ɛ x dɛ x when V 11 + V 00 V 10 + V 01 V ( 01 V 11 V 10 V 11 1 F (V 01 V 11 ( 1 F (V 10 V 11 when V 11 + V 00 <V 10 + V 01 22

23 Q 10 = Prob((U 10 >U 00 (U 10 >U 01 (U 10 >U 11 = Prob((ɛ x >V 00 V 10 (ɛ x ɛ y >V 01 V 10 (ɛ y <V 10 V 11 ( 1 F (V 00 V 10 F (V 10 V 11 when V 11 + V 00 V 10 + V 01 ( = 1 F (V 00 V 10 F (V 10 V 11 V 01 V 11 V 10 V 11 f(ɛ y dɛ y f(ɛ x dɛ x when V 11 + V 00 <V 10 + V 01 V 00 V 10 V 10 V 01 +ɛ x Q 01 = Prob((U 01 >U 00 (U 01 >U 10 (U 01 >U 11 = Prob((ɛ y >V 00 V 01 (ɛ y ɛ x >V 10 V 01 (ɛ x <V 01 V 11 ( 1 F (V 00 V 01 F (V 01 V 11 when V 11 + V 00 V 10 + V 01 ( = 1 F (V 00 V 01 F (V 01 V 11 V 01 V 11 V 10 V 01 +ɛ x f(ɛ y dɛ y f(ɛ x dɛ x when V 11 + V 00 <V 10 + V 01 V 00 V 10 V 00 V 01 Q 00 = Prob((U 00 >U 10 (U 00 >U 01 (U 00 >U 11 = Prob((ɛ x <V 00 V 10 (ɛ y >V 00 V 01 (ɛ x + ɛ y <V 00 V 11 F (V 00 V 01 F (V 00 V 10 V 00 V 10 V 00 V 01 = f(ɛ y dɛ y f(ɛ x dɛ x when V 11 + V 00 V 10 + V 01 V 01 V 11 V 00 V 11 ɛ x F (V 00 V 01 F (V 00 V 10 when V 11 + V 00 <V 10 + V 01 V 11 + V 00 = V 10 + V 01 V 01 V 11 = V 00 V 10 Q 11 Q 11 + Q 10 = = Q 11 = Q 10 = Q 01 = ( ( 1 F (V 01 V 11 1 F (V 10 V 11 ( 1 F (V 00 V 10 F (V 10 V 11 ( 1 F (V 00 V 01 F (V 01 V 11 Q 00 = F (V 00 V 01 F (V 00 V 10 ( 1 F (V 01 V 11 ( 1 F (V 10 V 11 (1 F (V 01 V 11 (1 F (V 10 V 11 + (1 F (V 00 V 10 F (V 10 V 11 ( 1 F (V 01 V 11 ( 1 F (V 10 V 11 (1 F (V 00 V 10 (1 F (V 10 V 11 + (1 F (V 00 V 10 F (V 10 V 11 ( 1 F (V 00 V 10 ( 1 F (V 10 V 11 = (1 F (V 00 V 10 (1 F (V 10 V 11 + (1 F (V 00 V 10 F (V 10 V 11 = 1 F (V 10 V 11 23

24 V 10 V 11 = V 00 V 01 Q 01 ( 1 F (V 00 V 01 F (V 01 V 11 Q 01 + Q 00 = (1 F (V 00 V 01 F (V 01 V 11 +F (V 00 V 01 F (V 00 V 10 ( 1 F (V 00 V 01 F (V 01 V 11 = (1 F (V 00 V 01 F (V 01 V 11 +F (V 00 V 01 F (V 01 V 11 1 F (V 00 V 01 = (1 F (V 00 V 01 + F (V 00 V 01 = 1 F (V 00 V 01 =1 F (V 10 V 11 S = Prob(Y =1 X =1 Prob(Y =1 X =0 = Q11 Q x Q01 1 Q ( x = 1 F (V 10 V 11 ( 1 F (V 10 V 11 =0 Q 11 = Prob((ɛ x + ɛ y >V 00 V 11 (ɛ x >V 10 V 11 (ɛ x >V 01 V 11 V 11 (ɛ x + ɛ y >V 00 V 11 (ɛ x >V 10 V 11 (ɛ x >V 01 V 11 (ɛ x,ɛ y dq11 0 dv 11 Q 10 = Prob((ɛ x >V 00 V 10 (ɛ x ɛ y >V 01 V 10 (ɛ y <V 10 V 11 V 11 (ɛ y <V 10 V 11 dq 10 0 dq01 dv 11 dv 11 0 Q00 = Prob((ɛ x <V 00 V 10 (ɛ y > V 00 V 01 (ɛ x + ɛ y <V 00 V 11 V 11 (ɛ x + ɛ y <V 00 V 11 dq00 dv 11 0 V 11 + V 00 <V 10 + V 01 Q 00 = F (V 00 V 01 F (V 00 V 10 dq00 =0 dv 11 Q 11 + Q 10 + Q 01 + Q 00 =1 dq 11 dq10 + dv 11 dv S V 11 dv dv dq01 dq = d d dv 11 dq 11 (Q 11 + Q 10 Q 11 (dq 11 + dq 10 (Q 11 + Q 10 2 dq01 (Q 01 + Q 00 Q 01 (dq 01 + dq 00 (Q 01 + Q 00 2 dq 11 = dq 10 dq 01 dq 00 1 ( (Q 11 + Q 10 2 (Q 01 + Q 00 2 dq 10 (Q 10 + Q 11 (Q 00 + Q 01 2 dq 01 (Q 10 + Q 11 (Q 00 + Q 01 2 dq 00 Q 10 (Q 00 + Q dq 00 Q 01 (Q 10 + Q

25 dq 10 dq 01 dq 11 V 11 + V 00 <V 10 + V 01 dq 00 =0 V 11 + V 00 V 10 + V 01 (V 11 + V 00 (V 10 + V 01 = (V 00 V 10 (V 01 V 11 dq 00 V 00 V 10 dv 11 = f(v 00 V 11 ɛ x f(ɛ x dɛ x V 01 V 11 dq 00 25

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