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A practical concern about the relevance of alternative-specific constants for new alternatives in simple logit models
Tran,pn Re~ Vol 15B. %1o fX p p 407 410 Printed in (ircill Bfltitln
I)191 2615 Sl 0 ~ ) ' , (kl ~121~)l) Perpamtln Pre~ Lid
A PRACTICAL C O N C E R N ABOUT THE RELEVANCE OF ALTERNATIVE-SPECIFIC CONSTANTS FOR NEW ALTERNATIVES IN SIMPLE LOGIT MODELS DAVID A. HENSHER School of Economic and Financial Studies, Macquarie University, North Ryde. N.S.W. 2113. Australia (Receired 5 Norember 19791
Abstract-- When a '*new" alternative is introduced, post-estimation, into a legit model, analysts initially exclude an ASC or occasionally assume a correspondence with an existing alternative. Using a recent data set for mode and route choice, the paper highlights the forecasting implications of ignoring the ASC in the utility expression of a "new" alternative, and how sensitive the market shares are to the inclusion' exclusion of the ASC.
I. I N T R O D U C T I O N AND EMPHASIS
In recent years we have witnessed an extensive amount of research and application of multinomiallegit models in the modelling of modal choice. (See Spear. 1978, for a summary of selected studies and Manski and McFadden, 1980, for the state of the art.) In such models, alternative-specific constants (ASCs) are included to capture the mean effect of unobserved factors which influence the selection of alternatives. The importance of including such constants for all but one of the alternatives is well established in the literature, and ways of adjusting their magnitudes in the spatial transferability of models are now well-documented (e.g. Charles River Associates 1978) even if doubt still exists as to the validity of such updating procedures. However, when a "new" alternative is to be introduced, post-estimation, into a legit model (a legitimate excercise given the independence from irrelevant alternative property), no method exists for assigning a magnitude to the new alternative's ASC; invariably we are limited to a utility expression in terms of generic variables. If the actual constants differ from zero, this is clearly a source of bias in the final selection probabilities. Domencich and McFadden (1975) side step the issue by assuming that the representative utility function contains "no pure 'mode" effect but rather evaluates a mode solely in terms of its generic attributes". Hausman and Wise (1978) have also noted this issue, but avoided it. Train (1976) assumed that the ASC associated with post-BART alternatives, "BART with car access" and "BART with bus alternatives" were directly transferable from pre-BART "bus with ear access" and "bus with bus access" respectively. t A self-administered survey procedure was used. with distribution of forms at all crossing points of the Derwent River.
407 I"R{B)Vol. IJB. No.
Charles River Associates (1978} also commented on this issue: "...the problem of what value to give the mode specific constant for a new mode has not been dealt with in the literature and our research failed to turn up a satisfactory solution". (CRA 1978, 132) The aim of this article is to keep this neglected issue alive by drawing on some evidence from a single study, in anticipation that the evidence will generate debate, and assist in resolving this issue. A study in Australia (Hensher, 19791¢ collected individual-specific mode and route data in March 1977 on each of the currently available alternatives
I /~,,,"=~ff/" "%
V I
L )
~- ......
=,,h.o°~
' N~.~, °~.~o,~e('.sr ..........
o .....
I I
, 3
~,
I"I
K C ;*.~
Glenorchl,
I !
.........
4,
/
~I
~
21
Fig. I. Transport facilities for cross-river travel (Hensher. 1979)
408
D a v I D A. HtNSHER
and on the Tasman Bridge route IFig. II, the latter having been an option to all travellers prior to the bridge's collapse in 1975, and which was certainly assured to be an option again after its reopening in September 1977. The mean levels of attributes defining each mode-route alternative (given in Table 1), show the attractiveness of the Tasman Bridge route. This data can be used to obtain some knowledge of the likely bias associated with the absence of an ASC for the "new" alternative. The approach involves the following steps: (I) A comparison of a base choice model from revealed preferences on the existing alternatives to a model based on intentions, both models having alternative-specific constants. (2) Demonstrating that the second model seems to predict realized market shares better than the first even when the mode-specific constant associated with the new alternative is not statistically sig-
"t" Given that each individual ha,d experienced the Tasman Bridge route alternative on repeated occasions up to 1975. :The data did not allow a distinction between modes via the Tasman route. Although each individt,ars level of service data relate to a particular mode, we halve no satisfactory way of identifying the actual mode. Additiomd independent variables were assessed. The results, reported in Flensher 11979) improve the model very marginally. § If the coelticicnts of the generic variables change significantly between the exclusion and inclusion of the Tasm:m route ahernativc, then wc have a problem with the representative utility assumption, placing a great deal of uncertainty on the specilication of the variables defining representative utility. Further diagnosis would be required to test for violation of thc IIA property. Thus even if IIA is not viokttcd in terms of the initial set of alternatives it could be viok, tcd by the application of a "new" ahernativc. A residual test for association {McFadden. Tye and Train. Ig76} could not reject IIA. The test. however, like all tests a,,ailable, generally hits very low power.
nificant. (3) Noting that removing the apparently statistically insignificant mode-specific constant has a major impact on the forecasted market shares.
2. E M P I R I C A L E V I D E N C E
Results are given below for home-based commuter trips. Each individual was assumed to be in a wellinformed position to provide data on the levels of service associated with the Tasman route by the selected mode when it is re-opened.f Since Hobart is a small city with almost no congestion, the likelihood of bias in response is further reduced. Furthermore, each individual was asked to indicate whether she/qae would switch (or return) to the Tasman Bridge route after its re-opening. With this additional information, models were estimated with the Tasman Bridge route as the chosen alternative if the individual indicated an intention to return to this route, otherwise the currently chosen alternative was maintained. The choice set of each individual was expanded by one alternative. Three models were then estimated (Table 2): M O D E L I Btzse Modeh 5 alternatives, which comprise 3 using the Bailey Bridge as a car driver, car passenger or as a bus passenger, and 2 others comprising ferry passenger and "punt" as a car driver. MODEL 2 Base Model with T,tsman Bridffe Alternatire (ASC # 0): 6 alternatives. ! To the 5 in Model 1 is added the Tasman Bridge option. MODEL 3 Base Model with Tasman Bridffe AIternatire (ASC = 0): 6 alternatives. The results in terms of market shares of the second and third models are then compared to the result associated with introducing the Tasman (post-estimation) as a new alternative, delined solely in terms of generic variables. { The results suggest the following:~
Table I. Mean levels of attributes defining each mode-route alternative 8&SI! ~DDEL ALTErnATIVES
Car d r i v e r
(Bailey Bridge
~alk Time (mins)
Wait Time (mins)
lnvehicle Time (mins)
Invehicle Cost (vents)
Out-of-vehicle Cost (cents)
1.09
0
47.9
122.2
8.4
0.88
0
51.9
7.1
0.2
1~.3
2.S
35.2
59.7
0.1
0.3
0.4
66 .b
131.7
2.4
Bus ( B a i l e y B r i d g e )
15.2
2,0
31,9
55.0
0
Hean T o t a l
lO. 39
1,78
39.2
73.-t
2.16
2.50
.95
23.8
b5.9
5.82
Car p a s s e n g e r ( B a i l e y Bridge) F e rry p a s s e n g e r
Car d r i v o r
(print)
NEI~ ALTERNATIVE
Tasman r o u t e (al l modes)
409
New alternatives in simple Iogit models Table Z The alternative mode-route choice models for home-based work trips, 1977 (1324 observations) EX PLA,SATORY VARIABLES
Estimated Coefficient
BASE HODEL ~ I ~ t TASMAN BRIDGE ALTERNATIVE
BASE ~,DDEL WITH T ~ H A N BRIDGE ALTER.NATIVE
BASE ~ D E L
(ASC - O)
(asc ¢ o)
t-statistic
Estimated Coefficient
t-statistic
Estimated Coefficient
t-statistic
Walk t i m e
-.05443
5.2
-.04453
2.6
-.01980
2.S
Wait t i m e
-.12236
6.3
-.04503
1.3
-.033S7
2.4
Invehicle
time
-.01759
3.4
-.01214
1.4
-.01999
S.0
Invehicle
cost
-.00736
S.2
-.01413
4.7
-.00220
2.4
-.01810
3.0
-.01682
2.0
-.00400
1.9
4.05210
8.2
-1.08000
7.6 6.4
Out-of-vehicle Cardriver
cost
- Bailey Bridge ASC
• 22930
O. 7
Ca r p a s s e n g e r B a i l e y B r i d g e ASC
-.64801
0.8
2.44990
4.4
-1.9JO00
Ferry, p a s s e n g e r ASC
1.81650
6.3
5.24120
10.9
-.13300
1.2
-1.57860
3,5
2.31570
3.2
-2.35400
4.9
Car driver
- punt
Bus - B a i l e y
ASC
Bridge
ASC
Tasman Bridge ASC Log-likelihood
at
26.7161 zero
Log-likelihood at ~onv~ r g e l | c e Likelihood ratio ( - 2 log X) o
index
2
.10
-917.7
-1454,6
-1 l S 1 . 6
-5S8.5
-216.0
-1131.8
2477.2
6,15.6
(lOd.f.)
('3d. f.)
718.7 (gd. f. ) • 39
P r o p o r t ion s u c c e s s ft,tl ly predi cted
• 74
(1) The ASCs for 3 of the 4 modes are significantly different across the models, at the 95'~,/, level. (2) The proportion successfully predicted is significantly different. (3) The Tasman Bridge route ASC appears to have a major intluence on all but one variable. (4) The number of switchers to the Tasman Bridge route is least ( = 390) when the new alternative is added postestimation. This contrasts with 824 in model (2) and 793 in model (3). Monitoring after the Bridge reopened shov,'ed a 70",-plus switch for home-based work journeys. Models (2) and (3) predict 62~o and 60~, respectively whereas the post-estimation calculat i o n 'is -9A,. "~ °" However, when the Tasman ASC is introduced all persons who intended to use the Tasman are predicted to choose Tasman ( = 817) whereas when it is excluded 324 are predicted to choose another alternative (5). The inclusion in the model of the Tasman alternative, despite its statistical nonsignificance produces a result closest to the results based on monitoring. In assessing the findings the reader must be aware of the assumption that the comparison between forecasts and realized values implies that service levels of all alternatives remain unchanged.
3. (:ONCI,tDING COMMENT
.85
.22
.91
• IS
bility of the data, the sensitivity of predicted market shares to the inclusion/exclusion of the ASC for a "new" alternative. This problem, however, is confined not only to the new mode case.i" If we accept the interpretation of the ASC as a set of characteristics intrinsic to a given alternative, then it follows that any change in these characteristics implies a change in the ASC. Alternatives may undergo improvement in a way that is not captured by the generic variables but this improvement might fall well short of producing a "new" alternative• Hence the empirical results have wider implications. In a recent paper, Truong and Hensher (1980) argue from theory that ASCs are also central to the derivation of the value of travel time savings; the latter defined by the formula: VOTS =
p-
--=-
+ ;.
where li is the marginal utility of time, 2 is the marginal utility of money and ASC~/t~ is the ratio of the constant to the coefficient of travel time. This additional evaluative dimension adds further support to the importance of ASCs, additional to their predictive relevance•
The empirical probe demonstrates, within the capat I acknowledge the comment of an anonymous referee on this point.
Acknowledgements--The comments of two anonymous referees and Marc Gaudry have been particularly useful in the presentation of the final version of this paper.
410
DAVID A. HENSHER REFERENCES
Charles River Associates (1978) Guidelines for Using the Market Segmentation Technique to Apply Disaggregate Tracel Demand Models with Census Data. (Project 9-13, Phase 2 Report, National Cooperative Highway Research Program), Charles River Associates, Boston. Domencich T. and McFadden D. (1975) Urban Trat~el Demand.. A Beharioural Analysis. North Holland, Amsterdam. Hausman J. and Wise J. (1978) A conditional probit model for qualitative choice: discrete decisions recognising interdependence and heterogeneous preferences. Econometrica, 46~21, 403-26. Hensher D. A, (1979) Individual choice modelling with discrete commodities: theory and application to the reopening of the Tasman Bridge. Economic Rec. 50(150), 243-61. Manski C. F. and McFadden D. (Editorsl (1981) Structural
Analysis of Discrete Data: with Et'onometrk" Application. Massachusetts Institute of Technology Press, Cambridge, Massachusetts. McFadden D, Ty¢ W. and Train K. (1976) Diagnostic tests for the independence from irrelevant alternatives property of the multinomial Iogit model. Urban Trarel Demand Forecasting. Project Working Paper No. 7616, Institute of Transportation Studies, University of California, Berkeley. Spear B. D. (1977) Application of New Trarel Demand Forecasting Techniques to Transportation Planning. U.S. Department of Transportation, Washington, D.C. Train K. (1976) A validation test of disaggregate travel demand models. Urban Trarel Demand Forecasting. Project Working Paper No. 7619, Institute of Transportation Studies, University of California. Berkeley. Truong T. P. and Hensher D. A. ~1980) Valuation of Commodity Time Sacinys and Budget Time. School of Economic and Financial Studies, Macquarie University, Sydney.
I)191 2615 Sl 0 ~ ) ' , (kl ~121~)l) Perpamtln Pre~ Lid
A PRACTICAL C O N C E R N ABOUT THE RELEVANCE OF ALTERNATIVE-SPECIFIC CONSTANTS FOR NEW ALTERNATIVES IN SIMPLE LOGIT MODELS DAVID A. HENSHER School of Economic and Financial Studies, Macquarie University, North Ryde. N.S.W. 2113. Australia (Receired 5 Norember 19791
Abstract-- When a '*new" alternative is introduced, post-estimation, into a legit model, analysts initially exclude an ASC or occasionally assume a correspondence with an existing alternative. Using a recent data set for mode and route choice, the paper highlights the forecasting implications of ignoring the ASC in the utility expression of a "new" alternative, and how sensitive the market shares are to the inclusion' exclusion of the ASC.
I. I N T R O D U C T I O N AND EMPHASIS
In recent years we have witnessed an extensive amount of research and application of multinomiallegit models in the modelling of modal choice. (See Spear. 1978, for a summary of selected studies and Manski and McFadden, 1980, for the state of the art.) In such models, alternative-specific constants (ASCs) are included to capture the mean effect of unobserved factors which influence the selection of alternatives. The importance of including such constants for all but one of the alternatives is well established in the literature, and ways of adjusting their magnitudes in the spatial transferability of models are now well-documented (e.g. Charles River Associates 1978) even if doubt still exists as to the validity of such updating procedures. However, when a "new" alternative is to be introduced, post-estimation, into a legit model (a legitimate excercise given the independence from irrelevant alternative property), no method exists for assigning a magnitude to the new alternative's ASC; invariably we are limited to a utility expression in terms of generic variables. If the actual constants differ from zero, this is clearly a source of bias in the final selection probabilities. Domencich and McFadden (1975) side step the issue by assuming that the representative utility function contains "no pure 'mode" effect but rather evaluates a mode solely in terms of its generic attributes". Hausman and Wise (1978) have also noted this issue, but avoided it. Train (1976) assumed that the ASC associated with post-BART alternatives, "BART with car access" and "BART with bus alternatives" were directly transferable from pre-BART "bus with ear access" and "bus with bus access" respectively. t A self-administered survey procedure was used. with distribution of forms at all crossing points of the Derwent River.
407 I"R{B)Vol. IJB. No.
Charles River Associates (1978} also commented on this issue: "...the problem of what value to give the mode specific constant for a new mode has not been dealt with in the literature and our research failed to turn up a satisfactory solution". (CRA 1978, 132) The aim of this article is to keep this neglected issue alive by drawing on some evidence from a single study, in anticipation that the evidence will generate debate, and assist in resolving this issue. A study in Australia (Hensher, 19791¢ collected individual-specific mode and route data in March 1977 on each of the currently available alternatives
I /~,,,"=~ff/" "%
V I
L )
~- ......
=,,h.o°~
' N~.~, °~.~o,~e('.sr ..........
o .....
I I
, 3
~,
I"I
K C ;*.~
Glenorchl,
I !
.........
4,
/
~I
~
21
Fig. I. Transport facilities for cross-river travel (Hensher. 1979)
408
D a v I D A. HtNSHER
and on the Tasman Bridge route IFig. II, the latter having been an option to all travellers prior to the bridge's collapse in 1975, and which was certainly assured to be an option again after its reopening in September 1977. The mean levels of attributes defining each mode-route alternative (given in Table 1), show the attractiveness of the Tasman Bridge route. This data can be used to obtain some knowledge of the likely bias associated with the absence of an ASC for the "new" alternative. The approach involves the following steps: (I) A comparison of a base choice model from revealed preferences on the existing alternatives to a model based on intentions, both models having alternative-specific constants. (2) Demonstrating that the second model seems to predict realized market shares better than the first even when the mode-specific constant associated with the new alternative is not statistically sig-
"t" Given that each individual ha,d experienced the Tasman Bridge route alternative on repeated occasions up to 1975. :The data did not allow a distinction between modes via the Tasman route. Although each individt,ars level of service data relate to a particular mode, we halve no satisfactory way of identifying the actual mode. Additiomd independent variables were assessed. The results, reported in Flensher 11979) improve the model very marginally. § If the coelticicnts of the generic variables change significantly between the exclusion and inclusion of the Tasm:m route ahernativc, then wc have a problem with the representative utility assumption, placing a great deal of uncertainty on the specilication of the variables defining representative utility. Further diagnosis would be required to test for violation of thc IIA property. Thus even if IIA is not viokttcd in terms of the initial set of alternatives it could be viok, tcd by the application of a "new" ahernativc. A residual test for association {McFadden. Tye and Train. Ig76} could not reject IIA. The test. however, like all tests a,,ailable, generally hits very low power.
nificant. (3) Noting that removing the apparently statistically insignificant mode-specific constant has a major impact on the forecasted market shares.
2. E M P I R I C A L E V I D E N C E
Results are given below for home-based commuter trips. Each individual was assumed to be in a wellinformed position to provide data on the levels of service associated with the Tasman route by the selected mode when it is re-opened.f Since Hobart is a small city with almost no congestion, the likelihood of bias in response is further reduced. Furthermore, each individual was asked to indicate whether she/qae would switch (or return) to the Tasman Bridge route after its re-opening. With this additional information, models were estimated with the Tasman Bridge route as the chosen alternative if the individual indicated an intention to return to this route, otherwise the currently chosen alternative was maintained. The choice set of each individual was expanded by one alternative. Three models were then estimated (Table 2): M O D E L I Btzse Modeh 5 alternatives, which comprise 3 using the Bailey Bridge as a car driver, car passenger or as a bus passenger, and 2 others comprising ferry passenger and "punt" as a car driver. MODEL 2 Base Model with T,tsman Bridffe Alternatire (ASC # 0): 6 alternatives. ! To the 5 in Model 1 is added the Tasman Bridge option. MODEL 3 Base Model with Tasman Bridffe AIternatire (ASC = 0): 6 alternatives. The results in terms of market shares of the second and third models are then compared to the result associated with introducing the Tasman (post-estimation) as a new alternative, delined solely in terms of generic variables. { The results suggest the following:~
Table I. Mean levels of attributes defining each mode-route alternative 8&SI! ~DDEL ALTErnATIVES
Car d r i v e r
(Bailey Bridge
~alk Time (mins)
Wait Time (mins)
lnvehicle Time (mins)
Invehicle Cost (vents)
Out-of-vehicle Cost (cents)
1.09
0
47.9
122.2
8.4
0.88
0
51.9
7.1
0.2
1~.3
2.S
35.2
59.7
0.1
0.3
0.4
66 .b
131.7
2.4
Bus ( B a i l e y B r i d g e )
15.2
2,0
31,9
55.0
0
Hean T o t a l
lO. 39
1,78
39.2
73.-t
2.16
2.50
.95
23.8
b5.9
5.82
Car p a s s e n g e r ( B a i l e y Bridge) F e rry p a s s e n g e r
Car d r i v o r
(print)
NEI~ ALTERNATIVE
Tasman r o u t e (al l modes)
409
New alternatives in simple Iogit models Table Z The alternative mode-route choice models for home-based work trips, 1977 (1324 observations) EX PLA,SATORY VARIABLES
Estimated Coefficient
BASE HODEL ~ I ~ t TASMAN BRIDGE ALTERNATIVE
BASE ~,DDEL WITH T ~ H A N BRIDGE ALTER.NATIVE
BASE ~ D E L
(ASC - O)
(asc ¢ o)
t-statistic
Estimated Coefficient
t-statistic
Estimated Coefficient
t-statistic
Walk t i m e
-.05443
5.2
-.04453
2.6
-.01980
2.S
Wait t i m e
-.12236
6.3
-.04503
1.3
-.033S7
2.4
Invehicle
time
-.01759
3.4
-.01214
1.4
-.01999
S.0
Invehicle
cost
-.00736
S.2
-.01413
4.7
-.00220
2.4
-.01810
3.0
-.01682
2.0
-.00400
1.9
4.05210
8.2
-1.08000
7.6 6.4
Out-of-vehicle Cardriver
cost
- Bailey Bridge ASC
• 22930
O. 7
Ca r p a s s e n g e r B a i l e y B r i d g e ASC
-.64801
0.8
2.44990
4.4
-1.9JO00
Ferry, p a s s e n g e r ASC
1.81650
6.3
5.24120
10.9
-.13300
1.2
-1.57860
3,5
2.31570
3.2
-2.35400
4.9
Car driver
- punt
Bus - B a i l e y
ASC
Bridge
ASC
Tasman Bridge ASC Log-likelihood
at
26.7161 zero
Log-likelihood at ~onv~ r g e l | c e Likelihood ratio ( - 2 log X) o
index
2
.10
-917.7
-1454,6
-1 l S 1 . 6
-5S8.5
-216.0
-1131.8
2477.2
6,15.6
(lOd.f.)
('3d. f.)
718.7 (gd. f. ) • 39
P r o p o r t ion s u c c e s s ft,tl ly predi cted
• 74
(1) The ASCs for 3 of the 4 modes are significantly different across the models, at the 95'~,/, level. (2) The proportion successfully predicted is significantly different. (3) The Tasman Bridge route ASC appears to have a major intluence on all but one variable. (4) The number of switchers to the Tasman Bridge route is least ( = 390) when the new alternative is added postestimation. This contrasts with 824 in model (2) and 793 in model (3). Monitoring after the Bridge reopened shov,'ed a 70",-plus switch for home-based work journeys. Models (2) and (3) predict 62~o and 60~, respectively whereas the post-estimation calculat i o n 'is -9A,. "~ °" However, when the Tasman ASC is introduced all persons who intended to use the Tasman are predicted to choose Tasman ( = 817) whereas when it is excluded 324 are predicted to choose another alternative (5). The inclusion in the model of the Tasman alternative, despite its statistical nonsignificance produces a result closest to the results based on monitoring. In assessing the findings the reader must be aware of the assumption that the comparison between forecasts and realized values implies that service levels of all alternatives remain unchanged.
3. (:ONCI,tDING COMMENT
.85
.22
.91
• IS
bility of the data, the sensitivity of predicted market shares to the inclusion/exclusion of the ASC for a "new" alternative. This problem, however, is confined not only to the new mode case.i" If we accept the interpretation of the ASC as a set of characteristics intrinsic to a given alternative, then it follows that any change in these characteristics implies a change in the ASC. Alternatives may undergo improvement in a way that is not captured by the generic variables but this improvement might fall well short of producing a "new" alternative• Hence the empirical results have wider implications. In a recent paper, Truong and Hensher (1980) argue from theory that ASCs are also central to the derivation of the value of travel time savings; the latter defined by the formula: VOTS =
p-
--=-
+ ;.
where li is the marginal utility of time, 2 is the marginal utility of money and ASC~/t~ is the ratio of the constant to the coefficient of travel time. This additional evaluative dimension adds further support to the importance of ASCs, additional to their predictive relevance•
The empirical probe demonstrates, within the capat I acknowledge the comment of an anonymous referee on this point.
Acknowledgements--The comments of two anonymous referees and Marc Gaudry have been particularly useful in the presentation of the final version of this paper.
410
DAVID A. HENSHER REFERENCES
Charles River Associates (1978) Guidelines for Using the Market Segmentation Technique to Apply Disaggregate Tracel Demand Models with Census Data. (Project 9-13, Phase 2 Report, National Cooperative Highway Research Program), Charles River Associates, Boston. Domencich T. and McFadden D. (1975) Urban Trat~el Demand.. A Beharioural Analysis. North Holland, Amsterdam. Hausman J. and Wise J. (1978) A conditional probit model for qualitative choice: discrete decisions recognising interdependence and heterogeneous preferences. Econometrica, 46~21, 403-26. Hensher D. A, (1979) Individual choice modelling with discrete commodities: theory and application to the reopening of the Tasman Bridge. Economic Rec. 50(150), 243-61. Manski C. F. and McFadden D. (Editorsl (1981) Structural
Analysis of Discrete Data: with Et'onometrk" Application. Massachusetts Institute of Technology Press, Cambridge, Massachusetts. McFadden D, Ty¢ W. and Train K. (1976) Diagnostic tests for the independence from irrelevant alternatives property of the multinomial Iogit model. Urban Trarel Demand Forecasting. Project Working Paper No. 7616, Institute of Transportation Studies, University of California, Berkeley. Spear B. D. (1977) Application of New Trarel Demand Forecasting Techniques to Transportation Planning. U.S. Department of Transportation, Washington, D.C. Train K. (1976) A validation test of disaggregate travel demand models. Urban Trarel Demand Forecasting. Project Working Paper No. 7619, Institute of Transportation Studies, University of California. Berkeley. Truong T. P. and Hensher D. A. ~1980) Valuation of Commodity Time Sacinys and Budget Time. School of Economic and Financial Studies, Macquarie University, Sydney.