1. The Binary Logit Model Definition Characteristics Estimation 0

2. Formulating the model The mathematical form is determined by the assumptions made regarding the error component. First example: The Linear Model 1

3. The Linear Model 2

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 3

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. 4

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8. The Gumbel cumulative distribution and probability density function 7

9. The Binomial Logit Model 8 P r (1) decreases monotonically with V2 P r (1) increases monotonically with V1

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11. The Binomial Logit Model 10 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 P1P1 -6-5-4-3-201 V 1 -V 2 2 3 4 5

12. Specification of the Utility Function 11 Policy variables/level of service variables V a =-T a -5C a /y V b =-T b -5C b /y Demographic/socioeconomic variables Example T a =0.5hr, T 6 =1.0hr C a = $1.50, C b =$0.50, $1.00 Low:  P b = -7% High:  P b = -5% C b = $0.50 C to = $1.00 Low Income (20K)0.440.41 High Income (40K)0.410.39 Probability of choosing bus

13. Example – Effects of Omitting a Variable That is Correlated with a Policy Variable 12 V a =-T a + 0.5 A V b =-T b Group T b - T a A P auto 10.2510.68 20.6020.83 30.7530.90

14. P auto versus T b -T a 13 -0.75-0.50-0.250 0.75 0.50 0.25 1 1.25 1.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 P auto T b -T a (hr) A=1 A=2 A=3 Model 2

15. Alternative Specific Constant 14 V a = 0.5-T a V b = -T a

16. P auto versus T b -T a 15 -0.75-0.50-0.250 0.75 0.50 0.25 1 1.25 1.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T b -T a (hr) Equation 5.11 Equation 5.12 P auto

17. Components of Travel Time 16 P auto According to Change from Base Case CaseModel 1Model 2Model 1Model 2 Base0.59 0.00 Increase TI B 0.610.600.020.01 Increase TO B 0.610.620.020.03 Model 1 U A = -T A U B = -T B Model 2 U A = -0.48TI A -1.21TO A U B = -0.48TI B -1.21TO 8 TI B = 0.5hrTI A = 0.4hr TO B = 0.3hr TO A = 0.05hr T B = 0.8hrT A = 0.45hr P 1 A = 0.59P 2 A = 0.59 Same effect Different effect

18. Generic vs. Mode Specific Variables 17 Model 1 UA = -T A UB = -T B Model 2 U A = -4.2T A U B = -2.8T B T A = 0.45hrT B = 0.80hr P 1 A = 0.59P 2 A = 0.59 P auto According to Change from Base Case CaseModel 1Model 2Model 1Model 2 Base0.59 0.00 Increase T B 0.610.650.020.06 Decrease T A 0.610.680.020.09 Same effect Different effect

19. Travel Time Functional Form Travel cost and income Auto ownership variables 18

20. Travel Time Functional Form 19 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.0 Model 1 Model 2 P auto T2T2

21. Estimation of the Logit Model Acquisition of data Model specification Model estimation 20

22. 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 21

23. The Maximum Likelihood Example 22

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