Estimation of switching models from revealed preferences and

stated intentionsBen-Akiva, Moshe, and Takayuki Morikawa. "Estimation of switching

models from revealed preferences and stated intentions." Transportation Research Part A: General 24.6 (1990): 485-495.

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Claudio Feliciani 37-147293 Fujita Kazuyuki 37-146044

He Le 37-146059Wang Yanjun 37-146920

Paper’s Objective:

Relation with our project:

Find an efficient method to predict switching behavior.In particular analyze the case in which a new metro line is opened and try to predict its share by combining revealed preferences (RP) and stated intentions (SI) data.

This paper deals with the opening of a new metro line in Yokohama in march 1985 and predicting the market share of that line.In our project we are considering the introduction of a bus line during the Tokyo Olympics and we are trying to find out the optimal parameters’ combination (cost, time,…).

1. Introduction

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Revealed Preference (RP) theory (by Paul Samuelson, 1938):

Stated Intentions (or Preferences) (SI):

Someone’s preferences can be revealed by analyzing his/her behavior under different circumstances.

Preferences are directly obtained by asking people what they would prefer and analyzing the results.

Pro: people behavior is analyzed in natural environment.Contra: for behaviors not implicit in the data analyzed, making forecasts may result difficult.

Pro: options to be analyzed are directly used in the survey.Contra: responses may yield significant bias as decision-making protocol generating SI may differ from the one actually used.

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1. Introduction

Revealed Preference model (actual travel behavior – process generating RP data):

𝑈 𝑖=𝛼′𝑤𝑖+𝛽

′ 𝑥 𝑖+𝜖 𝑖=𝑉 𝑖+𝜖 𝑖 ,𝑖=1 ,…, 𝐼

Stated Intentions model (consider alternative not currently available, switch or not?):

Random utility function:

Choice indicator: 𝑑𝑖={1 ,i f 𝑈 𝑖=𝑚𝑎𝑥 𝑗=1 ,… 𝐼 {𝑈 𝑗 }0 , otherwise

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2.1. Theory; combining RP and SI models

~𝑈 𝑖=𝛽

′ 𝑥 𝑖+𝛾′ 𝑧 𝑖+𝑣 𝑖=

~𝑉 𝑖+𝑣 𝑖 , 𝑖=1 ,…, 𝐼Utility function:

Stated intention response:~𝑑={1 ,i f~𝑈 𝑠−𝑚𝑎𝑥 𝑗=1 ,… 𝐼 {𝑈 𝑗 }≥𝛿

0 , otherwise

• s (subway) = index of new alternative.• is a threshold parameter; positive for actual behavior, negative in case of overestimation.• Vector of attributes which appears in RP is omitted in SI; represents attributes which only affect actual

choice but are not taken into account in intentions.• Vector of coefficient is common to both RP and SI models, while and are specific to each model.

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2.1. Theory; combining RP and SI modelsCombining random components:

𝑉 𝑎𝑟 (𝜀 )=𝜇2𝑉𝑎𝑟 (𝑣 ) is a positive parameter expected to be between 0 and 1. represents the “scale” of the SI model relative to RP model.

Log-likelihood function:

𝐿 (𝛼 , 𝛽 ,𝛾 ,𝜇 ,𝛿 )=∑𝑛=1

𝑁

∑𝑖=1

𝐼

𝑑𝑖𝑛 𝑙𝑜𝑔 [𝐹 𝑖 (𝑉 1𝑛 ,… ,𝑉 𝐼𝑛) ]+∑𝑛=1

𝑁 ~𝑑𝑛 𝑙𝑜𝑔 [𝐺(𝜇(~𝑉 𝑠𝑛−

~𝑉 𝑟𝑛−𝛿)) ]

𝐹 𝑖 (𝑉 1𝑛 ,…,𝑉 𝐼𝑛 )=exp (𝑉 𝑖𝑛)

∑𝑗=1

𝐼

exp (𝑉 𝑗𝑛)choice probability of alternative using multinomial logit (MNL) probability function

𝐺 (𝜇(~𝑉 𝑠𝑛−

~𝑉 𝑟𝑛−𝛿))=

1

1+exp (−𝜇 (~𝑉 𝑠𝑛−~𝑉 𝑟𝑛− 𝛿))

binary choice probability (binary logit used here)

• Surveyed in order to investigate the ridership of a new subway line, which opened in March, 1985 in Yokohama, Japan.

• The before survey (3 month before opening); questions:• Regularly used transit route from home to work/school;• Alternative transit route;• Intentions of using the new subway line.

• The after survey (6 months after opening); questions:• Route from home to work using the new subway line;• Route for the same trip without using the new subway line;• Satisfaction level with the service of both routes.

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3.1. Description of the data

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3.2. Before data summary

• Before survey:• 564 respondents• 70% valid

• However, for data analysis only individuals whose principal commuting mode is rail for both regular and alternative route selected:• 107 respondents

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3.2.1. Estimation with RP data – Theory

𝑃𝑛 (1 )=exp (𝜇𝑉 1𝑛)

∑𝑗=1

𝐽 𝑛

exp (𝜇𝑉 𝑗𝑛)

=exp (𝜇𝑉 1𝑛)

exp (𝜇𝑉 1𝑛)+∑𝑗=2

𝐽 𝑛

exp (𝜇𝑉 𝑗𝑛)

=…

¿1

1+exp [𝜇(𝑉 2𝑛−𝑉 1𝑛+1𝜇ln ( 𝐽𝑛−1))]

• We need to take into account the fact that there are several alternatives.• In reality people only choose one regular and alternative route.

• Estimator of the binary choice model will be biased.• individuals and (> 2) alternatives.• 1 = regular route, 2 = alternatives.• … randomly distributed.• Estimation using binary choice set.

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Bus dummy (busdum) =

Bike dummy (bikedum) =

Car dummy (cardum) =

Walk time (walkt) = walking time (in minutes)Access in-vehicle time (accivt) = in-vehicle time of access trip (minutes)Number of transfers (xfern) = number of transfers

{1 ,i f the principal accessmode is bus0 , otherwise

{1 ,i f the principal accessmode is bike0 , otherwise

{1 ,i f the principal accessmode is car0 , otherwise

• “Base” access mode is “walk”.• Regular and alternative routes use rail commuter mode:

• Total travel time or cost had no significant coefficient;• Commuting cost usually covered by employer.

• Socioeconomic variables had non-significant coefficients.

3.2.1. Estimation with RP data – Explanatory variables

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3.2.2. Estimation with SI data

• Stated intentions of using new subway line analyzed by using binary choice model.• Subway specific constant (ASC) introduced to capture:• additional attributes of the subway route not included in the model• response bias toward the new subway route• the threshold value

• Also in this case correction for implicit choice set can be employed.• Utility function of alternative 1 (current route) given by:~𝑈 1𝑛=

~𝑉 1𝑛+

1𝜇ln ( 𝐽𝑛)+𝑣1𝑛

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3.2.3. Estimation with RP + SI data

Revealed Preference Model specifications:

Stated Intentions Model specifications:

• Indexes r, a and s represents regular route, alternative route and subway route.• is the number of alternatives in the full choice set.• set to in as nobody chose that option.• is the alternative specific constant (subway specific constant here).• and cannot be distinguished; set as 0.• Both models described by binary discrete choice model.

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3.2.3. Estimation with RP + SI data

𝑃𝑛 (𝑟 )= 11+exp (𝑈𝑎𝑛−𝑈𝑟𝑛)

=1

1+exp (𝑉 𝑎𝑛+ ln ( 𝐽𝑛−1 )−𝑉 𝑟𝑛)

𝑄𝑛 (𝑟 )= 1

1+exp (~𝑈 𝑠𝑛−~𝑈 𝑟𝑛)

=1

1+exp (𝜇(~𝑉 𝑠𝑛−~𝑉 𝑟𝑛− ln ( 𝐽𝑛)))

• Estimation of discrete choice models requires normalization of scale parameter ( = 1).• For the RP model, choice probability is given by:

• For the SI model, choice probability is given by (scale parameter is used here):

Remember that we assumed: 𝑉 𝑎𝑟 (𝜀 )=𝜇2𝑉𝑎𝑟 (𝑣 )

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3.2.3. Estimation with RP + SI data

• Log-likelihood function for the RP + SI model is given by:

𝐿 (𝛼 , 𝛽 ,𝛾 ,𝜇)=∏𝑛=1

𝑁

𝑃𝑛 (𝑟 )𝑄𝑛(𝑠)~𝑑𝑛𝑄𝑛 (𝑟 )

1−~𝑑𝑛

with being the choice indicator of stated intentions = {1: i f~𝑈 𝑠𝑛−

~𝑈 𝑟𝑛≥0

0 :otherwise

(remember that was set to 0)

RP model SI model RP+SI model

Bus dummy -2.99 (-2.99) -1.57 (-1.78) -2.54 (-2.81)

Bike dummy -5.77 (-4.37) -2.89 (-3.21) -5.73 (-4.55)

Car dummy -8.48 (-5.84) -8.81 (-6.21)

Walk time -0.335 (-3.59) -0.238 (-3.31) -0.365 (-4.27)

Access in-vehicle time 0.00769 (0.16) -0.0856 (-2.22) -0.0630 (-1.63)

Number of transfers -1.12 (-2.73) -1.32 (-2.65) -1.69 (-4.31)

Subway route constant 0.974 (1.67) 1.22 (2.25)

μ 0.559 (3.21)

L(0) -113.51 -97.28 -210.80

L() -74.14 -59.76 -135.77

0.347 0.386 0.356

0.294 0.324 0.318

N 107 107 107

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3.2.4. Before data results• Preferred access mode (in order):

• Walk• Bus• Bike• Car

• In the SI model car not chosen:• Parameter =

• SI model has smaller parameter estimate compared to RP.• Exception: number of transfers.

• In RP+SI model parameters accurately estimated.• Exception: in-vehicle time

• Positive subway constant indicated policy bias toward subway route in SI data (imply negative ).

• Parameters similar to RP model suggesting a RP model replication.

• SI has more random noise.

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3.2.5. Statistical test

• Equality of parameter estimation can be tested by comparing likelihood value of combined model with the one of separate models.• Likelihood ratio test statistic given by: • : log-likelihood for the restricted model• : log-likelihood for the unrestricted model

• Here: • With degrees of freedom , P-value is 0.5217• Null hypothesis cannot be rejected.

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3.3. After data summary

• After survey:• 1201 respondents• 80.7% valid

• However, by selecting only rail users:• 428 observations

• From those, by selecting subway route users:• 254 respondents.

• Ambiguity in the way question was asked may have caused significant errors.

After model Before+After modelBus dummy -0.399 (-1.02) -2.00 (-2.53)

Bike dummy (Before) -5.42 (-4.80)

Bike dummy (After) 0.228 (0.41) 0.0733 (0.04)

Car dummy -1.79 (-1.06) -8.37 (-6.16)

Walk time -0.0768 (-5.09) -0.327 (-4.40)

Access in-vehicle time (Before) -0.0657 (-1.78)

Access in-vehicle time (After) -0.0754 (-3.87) -0.275 (-3.47)

Number of transfers -0.942 (-3.89) -2.03 (-5.41)

Subway route constant (Before) 1.45 (2.94)

Subway route constant (After) -0.112 (-0.36) -0.917 (-1.41)

μ1μ2

0.561 (3.39)0.313 (3.88)

L(0) -296.67 -507.47

L() -257.92 -395.66

0.131 0.220

0.107 0.197

N 428 53517

3.3.1. After data results

• Subway constant not significant.• No bias toward subway in after data.

• Fitting difficult compared to before data.

• Two models combined to test whether after data were generated from same model as before data.• Different scale parameter used.• for before data• for after data

• Bike dummy showing large difference.• Difference in sampling areas.• Therefore separate evaluation.

• After data contains large random error.• Equality of coefficient for uniting before and

after data accepted by likelihood test:• Test statistic: 4.46• P-value: 0.2159

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3.4. Prediction test

• Market share of the subway route given by:

• To obtain the “observed” market share after data used.• Prediction of the market share made by using before data.• Log-likelihood computed to check for goodness-of-fit.

∑𝑛=1

𝑁

�̂�𝑛 (𝑠)

Model Log-likelihood Predicted share of subway route(observed share = 59.35)

After model -260.59 59.35%

RP model -367.64 69.70%

SI model -365.90 81.97%

SI model* -298.63 68.22%

RP+SI combined model -443.85 82.89%

RP+SI combined model* -355.71 68.68%

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3.4.1. Prediction test – Results

• SI bias adjusted (*) obtained by omitting the subway constant.• May lead to an overstatement of the subway route usage.

• Correction of the bias required to reduce overstatement.• After data reveled that captive travelers do not use subway for different reasons.

• Unfamiliarity, disliking or habitual usage of non-subway routes.

• Bias adjusted SI model had closest prediction and best goodness-of-fit.• If over overestimation can be corrected, SI data can lead to good prediction.

• Combining the stated intention data with the RP data increased the accuracy of parameter estimates of the model;• a statistical test showed that the model for the stated intention data, if

properly scaled, had the same coefficients as the RP model;• the stated intention data contained more random noise; and• the utility threshold value for switching routes estimated from the stated

intentions was negative, implying that the respondents overstated their switching to the new alternative.

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4. Conclusions

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End of the presentation.

Thank you for your attention.

Questions?