3. Moneymetric and heteroscedastic specification

Although normalizing the scale to 1 is a common practice in random utility models, it is sometimes preferable to normalize another parameter. For instance, normalizing the cost coefficient to -1 sets the units of the utility function as currency units (CHF here), and the estimated coefficients are easily interpreted as willingness to pay. In that case, the scale must be estimated.

We also illustrate here a heteroscedastic specification, where a different scale is associated with different segments of the sample.

Michel Bierlaire, EPFL Wed Jun 18 2025, 11:25:26

from IPython.core.display_functions import display

from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta
from biogeme.models import loglogit
from biogeme.results_processing import (
    EstimationResults,
    get_pandas_estimated_parameters,
)

See the data processing script: Data preparation for Swissmetro.

from swissmetro_data import (
    CAR_AV_SP,
    CAR_CO_SCALED,
    CAR_TT_SCALED,
    CHOICE,
    GROUP,
    SM_AV,
    SM_COST_SCALED,
    SM_TT_SCALED,
    TRAIN_AV_SP,
    TRAIN_COST_SCALED,
    TRAIN_TT_SCALED,
    database,
)

Parameters to be estimated.

asc_car = Beta('asc_car', 0, None, None, 0)
asc_train = Beta('asc_train', 0, None, None, 0)
asc_sm = Beta('asc_sm', 0, None, None, 1)
b_time = Beta('b_time', 0, None, None, 0)
b_cost = Beta('b_cost', -1, None, None, 1)
scale_not_group3 = Beta('scale_not_group3', 1, 0.001, None, 0)
scale_group3 = Beta('scale_group3', 1, 0.001, None, 0)

Definition of the utility functions.

v_train = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED

The scale is defined based on the group membership of each individual.

scale = (GROUP != 3) * scale_not_group3 + (GROUP == 3) * scale_group3

Scale the utility functions, and associate them with the numbering of alternatives.

v = {1: scale * v_train, 2: scale * v_swissmetro, 3: scale * v_car}

Associate the availability conditions with the alternatives.

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

Definition of the model. This is the contribution of each observation to the log likelihood function.

logprob = loglogit(v, av, CHOICE)

These notes will be included as such in the report file.

USER_NOTES = (
    'Illustrates a moneymetric and heteroscedastic specification. A different scale is'
    ' associated with different segments of the sample. The utility function is expressed in CHF.'
)

Create the Biogeme object.

the_biogeme = BIOGEME(database, logprob, user_notes=USER_NOTES)
the_biogeme.model_name = 'b03_scale'

Estimate the parameters.

try:
    results = EstimationResults.from_yaml_file(
        filename=f'saved_results/{the_biogeme.model_name}.yaml'
    )
except FileNotFoundError:
    results = the_biogeme.estimate()

print(results.short_summary())

Results for model b03_scale
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -4976.691
Akaike Information Criterion:   9963.381
Bayesian Information Criterion: 9997.481

Get the results in a pandas table

pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)

               Name     Value  Robust std err.  Robust t-stat.  Robust p-value
0  scale_not_group3  0.357350         0.038417        9.301929        0.000000
1      scale_group3  1.492976         0.075495       19.775879        0.000000
2         asc_train -1.251037         0.119768      -10.445471        0.000000
3            b_time -1.047820         0.080484      -13.018905        0.000000
4           asc_car -0.042907         0.050788       -0.844826        0.398208