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The Binary Logit Model

The Binary Logit Model. Definition Characteristics Estimation. Formulating the model. The mathematical form is determined by the assumptions made regarding the error component. First example: The Linear Model. The Linear Model. The assumption leading to the logit model:.

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The Binary Logit Model

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  1. The Binary Logit Model • Definition • Characteristics • Estimation

  2. Formulating the model • The mathematical form is determined by the assumptions made regarding the error component. • First example: The Linear Model

  3. The Linear Model

  4. The assumption leading to the logit model: • The error components are extreme-value (or Gumbel) distributed • The error components are identically and independently distributed (iid) across alternatives • The error components are iid across observatins/indivudals

  5. Probit and Logit • Most common assumption for error distribution is the normal distribution. • Such assumptions lead to the Probit model. • However this lead to some numerical difficulties. • The Gumbel distribution closely approximates the normal distribution and has computational advantages.

  6. The Gumbel cumulative distribution and probability density function

  7. The Binomial Logit Model Pr(1) decreases monotonically with V2 Pr(1) increases monotonically with V1

  8. P1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 5 4 2 3 -6 -5 -4 -3 -2 -1 0 1 V1-V2 The Binomial Logit Model

  9. Specification of the Utility Function Policy variables/level of service variables Va=-Ta-5Ca/y Vb=-Tb-5Cb/y Demographic/socioeconomic variables Example Ta=0.5hr, T6=1.0hr Ca = $1.50, Cb=$0.50, $1.00 Probability of choosing bus Low: DPb= -7% High: DPb= -5%

  10. Example – Effects of Omitting a Variable That is Correlated with a Policy Variable Va=-Ta + 0.5 A Vb=-Tb

  11. Pauto versus Tb-Ta Pauto 1.0 0.9 0.8 0.7 0.6 0.5 0.4 A=1 0.3 A=2 A=3 0.2 Model 2 0.1 1.25 0.25 0.75 1.50 0.50 -0.75 -0.50 -0.25 0 1 Tb-Ta(hr)

  12. Alternative Specific Constant Va = 0.5-Ta Vb = -Ta

  13. Pauto versus Tb-Ta Pauto 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Equation 5.11 0.2 Equation 5.12 0.1 1.25 0.25 0.75 1.50 0.50 -0.75 -0.50 -0.25 0 1 Tb-Ta(hr)

  14. Components of Travel Time Model 1 UA = -TA UB = -TB Model 2 UA = -0.48TIA-1.21TOA UB = -0.48TIB-1.21TO8 TIB = 0.5hr TIA = 0.4hr TOB = 0.3hr TOA = 0.05hr TB = 0.8hr TA = 0.45hr P1A = 0.59 P2A = 0.59 Same effect Different effect

  15. Generic vs. Mode Specific Variables Model 1 UA = -TA UB = -TB Model 2 UA = -4.2TA UB = -2.8TB TA = 0.45hr TB = 0.80hr P1A= 0.59 P2A = 0.59 Same effect Different effect

  16. Travel TimeFunctional Form • Travel cost and income • Auto ownership variables

  17. Travel TimeFunctional Form Pauto 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Model 1 0.1 Model 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 T2

  18. Estimation of the Logit Model • Acquisition of data • Model specification • Model estimation

  19. The Maximum Likelihood Method • Developing a joint probability density function of the observed sample, called the likelihood function • Estimate parameters that maximize the likelihood function for a sample of T individuals with J alternatives

  20. The Maximum Likelihood Example

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