Development and estimation of a semicompensatory model with a flexible error structure

Transcription

1 Development and estimation of a semicompensatory model with a flexible error structure Sigal Kaplan, Shlomo Bekhor, Yoram Shiftan Transportation Research Institute, Technion Workshop on Discrete Choice Models, Ecole Polytechnique Fédérale de Lausanne, August 27-29, 2009

2 Contents Conclusions Results Case study Model formulation Research objectives Background and motivation

3 Background and motivation When faced with many alternatives, people apply a sequence of a noncompensatory elimination-based heuristic followed by a compensatory utility-based evaluation (Payne, 1976).

4 Background and motivation Semi-compensatory models based on Manski s (1977) formula ( ) ( ) P = P i S P S G iq q q S G The number of theoretical choice sets is 2 J -1 for J alternatives Semi-compensatory models are seldom applied Semi-compensatory models are subject to simplifying assumptions G universal realm Elimination heuristic S - viable choice set Utility maximization i-chosen alternative Latent Observed

5 Background and motivation Assumptions for representing the elimination-based heuristic Study Number of Number of choice alternatives sets Attribute specific thresholds Correlation among thresholds Threshold distribution Thresholds related to individual characteristics Scale for measuring thresholds Representation of utility maximization Borgers et al yes no normal no ordinal MNL Gensch yes no none no ordinal MNL Swait & Ben-Akiva yes no normal irrelevant cardinal MNL Ben-Akiva & Boccara no no logistic yes cardinal MNL Morikawa actually considered yes no logistic no cardinal MNL Başar and Bhat no no logistic yes cardinal MNL Cantillo & Ortúzar yes no normal no cardinal MNL Cantillo et al yes no normal no cardinal MNL Zheng & Guo yes irrelevant orderedresponse yes cardinal MNL

6 Research objective To improve the representation of elimination-based choice set formation To represent correlation patterns and taste variations at the choice stage To increase the number of alternatives and choice sets

7 Developed semi-compensatory model Universal realm of alternatives Choice set formation stage Conjunctive heuristic Overtly specified criteria thresholds No Viable choice set Unmanageable choice set Abort? Yes Choice stage Utility maximization Preference structure Chosen alternative No choice

8 Developed semi-compensatory model Proposed model: P i G P i S P S G ( ) = ( ) ( ) q q q q q Observed choice i Observed choice set S Any GEV or any mixed logit model Observed combination of criteria thresholds that yield the choice set S Multidimensional mixed ordered-response model

9 Developed semi-compensatory model The threshold t of criterion k: t = α Z + ξ + u * ' kq k kq kq kq The selection probability of threshold t of the k-th criterion : ( ) ( m 1) ( m 1) ( ) k The probability to select choice set S: ( ) ( ( )) P θ < t θ =Φ θ α Z + ξ Φ θ α Z + ξ * ' ' kq m k kq kq m k kq kq k k k * * * ( ) = ( 1 ) ( 2 ) K ( ) P S G P t P t P t q q q q kq The correlation matrix of the error terms: Σ 1 = a 1,2 1 L a 2,3 1 a 1, K 1 L O O a a a 1, K 2, K M K 1, K 1

10 Developed semi-compensatory model GEV model: Nested logit P q ( i Sq) = ( ) ' X / ( β ' X j/ λ i r r) β λ e e j Sq, j Br N l= 1 j S, j B q s ( β ' X j/ λs) e λ s λ 1 r Error-component logit model: Nesting logit kernel ( β ' X + FTζ ) i e Pq ( i Sq) = f ( ζ ) ζ ( β ' X j + FTζ ) ζ e j S q Where: F - (J X M) matrix, f jm =1 if alternative j belongs to nest m. f jm =1 otherwise T (M x M) diagonal matrix containing the standard deviation of each factor

11 Developed semi-compensatory model Unconditional log-likelihood function: LL 1 dqj λr ( ' / ) X ( β ' X j/ λ ) β i λ r r e e Q j Sq, j B r, k, k, s = ln λs 1 q= 1 j S N q ( β ' X j/ λs) e l= 1 j Sq, j B s ( βα θ λ) ' ' ( θ( ) ( α )) ( ) 1 1Z 1 1q ξ1 q θm α m 1 1Z1q ξ 1q M Φ + Φ + ξ ( ) k mq q ξkq mk = 1 M k dm q ' '... Φ ( θ( m ) ( α 1 1 )) ( ( 1 )) K ( 1,, ) K K ZKq + ξ q Φ θm α K K ZKq + ξ q ϕk ξ q ξkq dξ1 q dξ K K Kq mk = 1 d The estimation is conducted in a single step by maximum simulated Likelihood. The code was written in GAUSS matrix language.

12 Case study Product: rental apartments Population: university students Geographical scope: Haifa, Israel Survey type: stated preference Survey duration: 1 month Survey method: web-based Incentive: 23 prizes ($1000) Technion campus

13 Case study Respondent s information Database Yes Verification No Questionnaire socio-economic, price perceptions, travel attitudes and study preferences Synthetically generated SQL query apartment dataset 3 < j <100 No Yes Conjunctive choice set formation Criteria thresholds specification (e.g., price, rooms, noise level, parking) Respondent s criteria thresholds and chosen apartment Yes Verification No Utility-based choice stage Rank three most preferred apartments from the choice set

14 Case study

15 Results: data sample Three ranked choice outcomes and their respective thresholds from 631 respondents who searched the database according to a combination of apartment sharing, price and the two most popular neighborhoods. Technion

16 Results: semi-compensatory model Apartment sharing threshold MNL Nested logit EC logit Variables β t-stat β t-stat β t-stat Married Male Age Daily car availability Daily trips to campus Study on-campus (communication efficiency) Monthly expenses $750-$ Current residential accommodation Current residential location $1000-$ Roommates Alone Spouse Suburbs Northern outskirts

17 Results: semi-compensatory model Neighborhood threshold MNL Nested logit EC logit β t-stat β t-stat β t-stat Price-quality ratio consciousness factor Age Daily car availability Study at medical campus Monthly expenses $750-$ > $ Part-time job Student job opportunities Public open space availability Study on-campus (Study efficiency) Daily trips to campus

18 Results: semi-compensatory model Monthly rent price threshold MNL NL EC logit β t-stat β t-stat β t-stat c c c c c c c c c c Married Male Age $500-$ Monthly expenses $750-$ > $

19 Results: semi-compensatory model Monthly rent price threshold (continued) MNL NL EC logit β t-stat β t-stat β t-stat Part-time job Daily car availability Price knowledge factor Apartment search experience (> 3 times) Daily trips to campus current alone/parents Current residential accomodation Current residential location Spouse Roommates Haifa - high income neighborhood Tel-Aviv Preference for non-motorized modes Preference for travel minimization

20 Results: semi-compensatory model Utility-based choice stage MNL NL EC logit β t-stat β t-stat β t-stat Monthly rent price Rooms Roommates Walking time to campus Noise Parking Floor Smoking allowed Bars View Renovated Air condition Solar water boiler λ floor1 (above ground) λ floor2 (ground floor) floor log-likelihood at zero log-likelihood at estimates McFadden's adjusted R^

21 Results: comparison with compensatory models Semi-compensatory Compensatory T-Test of Nested logit Nested logit (βscnl - βnl) βscnl t-stat. βnl t-stat. Dummy for Carmel neighborhood Dummy for shared apartments Monthly rent price Rooms Roommates Walking time to campus Noise Parking Floor Smoking allowed Bars View Renovated Air condition Solar water boiler λ floor1 (above ground) λ floor2 (ground floor) log-likelihood at zero log-likelihood at estimates McFadden's adjusted R^

22 Results: comparison with compensatory models Semi-compensatory Compensatory Nested logit Nested logit βscnl t-stat. βnl t-stat. Dummy for Carmel neighborhood Dummy for shared apartments Monthly rent price Rooms Roommates Walking time to campus Noise Parking Floor Smoking allowed Bars View Renovated Air condition Solar water boiler λ rooms1 ( > 3.0 rooms) λ rooms2 (<= 3.0 rooms) log-likelihood at zero log-likelihood at estimates McFadden's adjusted R^

23 Conclusions Results demonstrate the applicability of representing flexible substitution patterns by the proposed semi-compensatory model The correlations among alternatives at the choice stage can be represented either by a GEV model or a mixed logit model The semi-compensatory model is better than its compensatory counterpart in terms of goodness-of-fit The compensatory model over-estimates the sensitivity to attributes that have a dual role as criteria and as attributes It is impossible to represent a nested structure in the compensatory model for attributes that have a dual role as criteria and as attributes

24 Thank you! Workshop on Discrete Choice Models, Ecole Polytechnique Fédérale de Lausanne, August 27-29, 2009