Let l be an index for latent variables (l=1,2,3,4). Consider the latent variable z. vector of observed covariates (excluding a constant), α

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1 Olie Supplemet to MaaS i Car-Domiated Cities: Modeli the adoptio, frequecy, ad characteristics of ride-haili trips i Dallas, TX Patrícia S. Lavieri ad Chadra R. Bhat (correspodi author) A Overview of the Geeralized Heteroeeous Data Model Latet Variable Structural Equatio Model Let l be a idex for latet variables (l=1,,3,4). Cosider the latet variable z l ad write it as a liear fuctio of covariates: zl α l w l, (1) where w is a ( D 1) vector of observed covariates (excludi a costat), α l is a correspodi ( D 1) vector of coefficiets, ad l is a radom error term assumed to be stadard ormally distributed for idetificatio purposes (See Bhat, 015). Next, defie the (4 D ) matrix α ( α1, α, α3, α 4), ad the (41) vectors η (,,, )'. To z ( z1, z, z3, z4) ad accommodate iteractios amo the uobserved latet variables, we allow a MVN correlatio structure for η, that is η MVN4[ 04, Γ], whue Γ is (4 4) correlatio matrix. A eeral covariace structure is adopted because i our study there are o coceptual reasos to establish causal relatioships betwee the latet variables. I matrix form, we may write Equatio (1) as: z αw η. () Latet Variable Measuremet Equatio Model Compoets As metioed earlier, we cosider a combiatio of ordial ad omial outcomes explaied by a latet variable vector z ad, whe relevat, a set of other edoeeous ad exoeous variables as well. Cosider N ordial outcomes for the idividual, ad let be the idex for the ordial outcomes ( 1,,..., N ), i our applicatio N=13 for 906 idividuals ad N=1 for the remaider of the sample. Also, let J be the umber of cateories for the th ordial outcome ( J ) ad let the correspodi idex be j j 1,,..., J ). Let y be the latet (

2 uderlyi variable whose horizotal partitioi leads to the observed outcome for the th ordial variable. Assume that the idividual uder cosideratio chooses the a th ordial cateory. The, i the usual ordered respose formulatio, for the idividual, we may write:, ad y γ x d z y, (3), a 1, a where x is a vector of exoeous ad possibly edoeous variables as defied earlier, γ is a correspodi vector of coefficiets to be estimated, d is a (41) vector of latet variable loadis o the th cotiuous outcome, the terms represet thresholds, ad is the stadard ormal radom error for the th ordial outcome. For each ordial outcome,,0,1,..., J 1, J ;,0,, 1 0, ad,j. For later use, let ψ (,..., ) ad ψ ( ψ, ψ,..., ψ ) 1 N. Stack the N uderlyi cotiuous variables,,3, J 1 y ito a ( N 1) vector vector ε. Defie γ ( γ 1, γ,..., γ N ) [( A) ad let y, ad the N error terms N matrix] ad d d, d,..., 1 ito aother ( N 1) [ ( N 4) matrix],, d N IDEN N be the idetity matrix of dimesio N represeti the correlatio matrix of ε (so, ε MVN N 0 N,IDEN N ; aai, this is for idetificatio purposes, ive the presece of the uobserved z vector to eerate covariace. Fially, stack the lower thresholds for the decisiomaker 1,,..., N ito a ( N 1) vector ψ low ad the upper thresholds, a 1 1,,..., N ito aother vector ψ. The, i matrix form, the measuremet equatio for, a up the ordial outcomes (idicators) for the decisio-maker may be writte as: y γx dz ε, ψ y ψ. (4) low up Next, let there be G omial (uordered-respose) variables for a idividual, ad let be the idex for the omial variables, i our applicatio G=. Also, let I be the umber of alteratives correspodi to the th omial variable (I 3) ad let i be the correspodi idex. Both omial outcomes i our applicatio have I=3. Cosider the th omial variable ad assume that the idividual uder cosideratio chooses the alterative usual radom utility structure for each alterative i. U m. Also, assume the b x z, (5) i i i i

3 where x is as defied earlier, b is a ( A 1) colum vector of correspodi coefficiets, ad i i is a ormal error term, ad i is a ( N 1) -colum vector of coefficiets capturi the i effects of latet variables. Let,,... ) ( I 1 vector), ad MVN I ( 0, Λ ). ( 1 I Taki the differece with respect to the first alterative, the oly estimable elemets are foud i the covariace matrix Λ of the error differeces,,,..., ) (where ( 3 I i i 1, i 1). Further, the variace term at the top left diaoal of Λ ( 1, ) is set to 1 to accout for scale ivariace. Λ is costructed from Λ by addi a row o top ad a colum to the left. All elemets of this additioal row ad colum are filled with values of zero. I additio, the usual idetificatio restrictio is imposed such that oe of the alteratives serves as the base whe itroduci alterative-specific costats ad variables that do ot vary across alteratives (that is, wheever a elemet of x is idividual-specific ad ot alterative-specific, the correspodi elemet i b is set to zero for at least oe alterative i ). i To proceed, defie U ( U1, U, U3) ad b ( b 1, b, b 3). Also, defie the I I N i i 1 matrix, which is iitially filled with all zero values. The, positio the 1 ) row vector 1 i the first row to occupy colums 1 to 1 ( N 1 N, positio the 1 ) row vector i the ( N secod row to occupy colums N 1 +1 to N, 1 N ad to occupy colums 3 N G N1N N3. Further, defie G I 1,, 1 +1 to U U U ( G 1 vector),,,... ) ( 1 G ( G 1vector), b ( b 1, b ) ( G A matrix), ad Vech( 1, ) (that is, is a colum vector that icludes all elemets of the matrices 1 ad ). The, i matrix form, we may write Equatio (3) as: U bx z ς (6) where MVN (, Λ). As earlier, to esure idetificatio, we specify Λ as follows: G 0 G Λ1 0 Λ 0 Λ ( GG matrix). (7)

4 G I I the eeral case, this allows the estimatio of ( I 1 1) 1 terms across all the G omial variables, as oriiati from I ( I 1) 1 terms embedded i each Λ matrix. To develop the reduced form equatios, replace the riht side of Equatio () for z i Equatios (4) ad (6) to obtai the followi system: y γx dz ε γx d ( αw η) ε γx dαw dη ε, (8) U bxz ς bx ( αw η) ς bx αw ης. (9) Now, cosider the [( N G) 1)] vector yu y ', U. Defie B 1 yx dαw B ad B bx αw The yu MVN ( B, Ω). NG Ω 1 Ω1 dγd Σ dγ Ω. (10) Ω 1 Ω Γd Γ Λ The model estimatio is performed usi Bhat s (011) MACML. We refer the reader to Bhat (015) for the detailed explaatio as well as iformatio o model idetificatio criteria. REFERENCES Bhat, C.R., 011. The maximum approximate composite marial likelihood (MACML) estimatio of multiomial probit-based uordered respose choice models. Trasportatio Research Part B, 45(7), Bhat, C.R., 015. A ew eeralized heteroeeous data model (GHDM) to joitly model mixed types of depedet variables. Trasportatio Research Part B, 79,

5 Table 1. Thresholds ad costats of idicators ad loadis of latet variables o idicators Attitudial ad lifestyle idicators Privacy-sesitivity I do t mid shari a ride with straers if it reduces my costs (iverse scale) Havi privacy is importat to me whe I make a trip Latet variable Threshold Threshold 3 Threshold 4 Costat loadi Coeff. t-stat Coeff. t-stat Coeff. t-stat Coeff. t-stat Coeff. t-stat I feel ucomfortable sitti close to straers Tech-savviess I frequetly use olie baki services I frequetly purchase products olie Leari how to use ew smartphoe apps is easy for me Variety-seeki lifestyle propesity (VSLP) I thik it is importat to have all sorts of ew experieces ad I am always tryi ew this Looki for advetures ad taki risks is importat to me I love to try ew products before ayoe else Gree lifestyle propesity (GLP) Whe choosi my commute mode, there are may this that are more importat tha bei evirometally friedly (iverse scale) I do t ive much thouht to savi eery at home (iverse scale)