# A practical method to test the validity of the standard Gumbel distribution in logit-based multinomial choice models of travel behavior

## Abstract

Most multinomial choice models (e.g., the multinomial logit model) adopted in practice assume an extreme-value Gumbel distribution for the random components (error terms) of utility functions. This distributional assumption offers a closed-form likelihood expression when the utility maximization principle is applied to model choice behaviors. As a result, model coefficients can be easily estimated using the standard maximum likelihood estimation method. However, maximum likelihood estimators are consistent and efficient only if distributional assumptions on the random error terms are valid. It is therefore critical to test the validity of underlying distributional assumptions on the error terms that form the basis of parameter estimation and policy evaluation. In this paper, a practical yet statistically rigorous method is proposed to test the validity of the distributional assumption on the random components of utility functions in both the multinomial logit (MNL) model and multiple discrete-continuous extreme value (MDCEV) model. Based on a semi-nonparametric approach, a closed-form likelihood function that nests the MNL or MDCEV model being tested is derived. The proposed method allows traditional likelihood ratio tests to be used to test violations of the standard Gumbel distribution assumption. Simulation experiments are conducted to demonstrate that the proposed test yields acceptable Type-I andmore »

- Authors:

- Tongji Univ., Shanghai (China)
- National Renewable Energy Lab. (NREL), Golden, CO (United States)
- Maricopa Association of Governments, Phoenix, AZ (United States)
- Arizona State Univ., Tempe, AZ (United States)

- Publication Date:

- Research Org.:
- National Renewable Energy Lab. (NREL), Golden, CO (United States)

- Sponsoring Org.:
- USDOE Office of Energy Efficiency and Renewable Energy (EERE)

- OSTI Identifier:
- 1413182

- Report Number(s):
- NREL/JA-5400-70645

Journal ID: ISSN 0191-2615

- Grant/Contract Number:
- AC36-08GO28308

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Transportation Research, Part B: Methodological

- Additional Journal Information:
- Journal Volume: 106; Journal Issue: C; Journal ID: ISSN 0191-2615

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 30 DIRECT ENERGY CONVERSION; travel behavior models; discrete choice models; violations of distributional assumptions; test of validity of distributional assumption; multinomial logit model; multiple discrete-continuous extreme value model

### Citation Formats

```
Ye, Xin, Garikapati, Venu M., You, Daehyun, and Pendyala, Ram M..
```*A practical method to test the validity of the standard Gumbel distribution in logit-based multinomial choice models of travel behavior*. United States: N. p., 2017.
Web. doi:10.1016/j.trb.2017.10.009.

```
Ye, Xin, Garikapati, Venu M., You, Daehyun, & Pendyala, Ram M..
```*A practical method to test the validity of the standard Gumbel distribution in logit-based multinomial choice models of travel behavior*. United States. doi:10.1016/j.trb.2017.10.009.

```
Ye, Xin, Garikapati, Venu M., You, Daehyun, and Pendyala, Ram M.. Wed .
"A practical method to test the validity of the standard Gumbel distribution in logit-based multinomial choice models of travel behavior". United States.
doi:10.1016/j.trb.2017.10.009.
```

```
@article{osti_1413182,
```

title = {A practical method to test the validity of the standard Gumbel distribution in logit-based multinomial choice models of travel behavior},

author = {Ye, Xin and Garikapati, Venu M. and You, Daehyun and Pendyala, Ram M.},

abstractNote = {Most multinomial choice models (e.g., the multinomial logit model) adopted in practice assume an extreme-value Gumbel distribution for the random components (error terms) of utility functions. This distributional assumption offers a closed-form likelihood expression when the utility maximization principle is applied to model choice behaviors. As a result, model coefficients can be easily estimated using the standard maximum likelihood estimation method. However, maximum likelihood estimators are consistent and efficient only if distributional assumptions on the random error terms are valid. It is therefore critical to test the validity of underlying distributional assumptions on the error terms that form the basis of parameter estimation and policy evaluation. In this paper, a practical yet statistically rigorous method is proposed to test the validity of the distributional assumption on the random components of utility functions in both the multinomial logit (MNL) model and multiple discrete-continuous extreme value (MDCEV) model. Based on a semi-nonparametric approach, a closed-form likelihood function that nests the MNL or MDCEV model being tested is derived. The proposed method allows traditional likelihood ratio tests to be used to test violations of the standard Gumbel distribution assumption. Simulation experiments are conducted to demonstrate that the proposed test yields acceptable Type-I and Type-II error probabilities at commonly available sample sizes. The test is then applied to three real-world discrete and discrete-continuous choice models. For all three models, the proposed test rejects the validity of the standard Gumbel distribution in most utility functions, calling for the development of robust choice models that overcome adverse effects of violations of distributional assumptions on the error terms in random utility functions.},

doi = {10.1016/j.trb.2017.10.009},

journal = {Transportation Research, Part B: Methodological},

number = C,

volume = 106,

place = {United States},

year = {Wed Nov 08 00:00:00 EST 2017},

month = {Wed Nov 08 00:00:00 EST 2017}

}