Lecture 4 - Multinomial Logit

ENCI707: Engineering Demand and Policy Analysis

Outline

• Discrete multinomial choice - Type I EV Distribution
• Multinomial Logit (MNL) - Scale and Systematic Utility Function
• MNL Estimation & Specification
• Elasticity, Marginal Effect, & Substitution
• IIA & IIA violation tests

Discrete Choice Multinomial Choice - Type I Extreme Value Distribution

Binary versus Multinomial Choice

Binary versus Multinomial Choice

Discrete Multinomial Choice

A multinomial choice reduced into an equivalent binary choice problem:

\[𝑈_𝑗 \ge \max_{j,j' \in C_i, j \neq j'} U_{j'}\] \[𝑈_𝑗 \ge 𝑈_𝑘 \text{; } k \neq j \text{ & } j,k \in C_i\]

• where \(C_i\) is the choice set of decision maker \(i\), \(j\) is the best alternative, and \(k\) is the second best alternative

• Probability of choosing alternative j:

\[Pr⁡(𝑗) = Pr⁡((𝑉_𝑗+\epsilon_𝑗) \ge (𝑉_𝑘+\epsilon_𝑘 ))\] \[Pr⁡( 𝑗)=Pr⁡(\epsilon_𝑘 \leq (𝑉_𝑗−𝑉_𝑘+\epsilon_𝑗 )) \text{ (a CDF function)}\]

Discrete Multinomial Choice

A multinomial choice reduced into an equivalen Choice probability in a multinomial choice context: \[Pr⁡(𝑗)=Pr⁡(\epsilon_𝑘 \leq (𝑉_𝑗−𝑉_𝑘+\epsilon_𝑗 ))\]

Probability distribution of random utility (\(\epsilon\)) needs to be fully specified to get the unconditional probability: \[Pr⁡(𝑗)=\int_{\epsilon_𝑗=−\infty}^{+\infty}\int_{\epsilon_𝑘=−\infty}^{𝑉_𝑗−𝑉_𝑘+\epsilon_𝑗} 𝑓(\epsilon_𝑗,\epsilon_𝑘)𝑑 \epsilon_𝑗 𝑑 \epsilon_𝑘\]

• Specification of univariate distribution, \(f(\epsilon_k)\) , and the joint distribution, \(f(\epsilon_j,\epsilon_k)\), are necessary
• Possible distributions:
  • Multivariate normal distribution
  • Type I Extreme Value distribution

Type I Extreme Value Distribution

• Type I (Gumbel) distribution is the most convenient distribution to derive a closed form choice model
  • If is Type I Extreme Value (Gumbel) distributed: \[𝑓(\epsilon)=\mu 𝑒^{−\mu(\epsilon−\eta)} 𝑒^{−𝑒^{−\mu(\epsilon−\eta)}}\] \[F = e^{-e^{-\mu(\epsilon-\eta)}} \text{; } \mu > 0\]
• \(\eta\) is a location parameter and \(\mu\) is a positive scale parameter
• Location parameter is mode of distribution
• Scale parameter defines the dispersion of the PDF shape

Type I Extreme Value Distribution

Basic Properties of Gumbel distribution

1. The mode of the distribution is \(\eta\) (the location of peak of the PDF on x axis from zero).
2. The mean is (\(\eta+\gamma/\mu\)), where \(\gamma\) = Euler’s constant=0.577.
3. The variance is \(\pi^2/6\mu^2\).
4. If \(\epsilon\) is Gumbel distributed with (\(\eta, \mu\)), and if \(\alpha\) is any positive scalar constant, then (\(\alpha \epsilon + V\)) is also Gumbel distributed with parameters (\(\alpha \eta +V\)) and \(\mu/\alpha\).
5. If \(\epsilon_1\) and \(\epsilon_2\) are independent Gumbel variables, then (\(\epsilon_1-\epsilon_2\)) is logistically distributed.
6. If \(\epsilon_1\) and are Identical and Independent Gumbel variables with parameters (\(\eta_1,\mu\)) and (\(\eta_2,\mu\)) respectively then \(\max(\epsilon_1,\epsilon_2)\) is also Gumbel distributed with parameters:
   \[\left(\frac{1}{\mu} \ln⁡(𝑒^{\mu\eta_1}+𝑒^{\mu\eta_2 } ),\mu \right)\]

Basic Properties of Gumbel distribution

7. If \(( \epsilon_1,\epsilon_2,\epsilon_3,\dots ,\epsilon_n )\) are \(n\) independent Gumbel distributed variables with parameters (\(\eta_1,\mu), (\eta_2,\mu),\dots (\eta_n,\mu)\) respectively then \(\max(\epsilon_1,\epsilon_2,\epsilon_3,\dots,\epsilon_n)\) is also Gumbel distributed with parameters (mean, scale): \[\left(\frac{1}{\mu} \ln⁡\sum_{𝑗=1}^𝑛 𝑒^{\mu \eta_𝑗}, \mu \right)\]

Notes:

• Homoskedasticity: Independent Gumbel distributed variables have a common scale parameter, \(\mu\).
• Property 5 is used to derive binomial logit model probability equation.
• Property 7 can be used to derived closed form equation for multinomial logit (MNL) Model.

Multinomial Logit Model - Scale & Systematic Utility Functions

Multinomial Logit Model (MNL)

• In case of a multinomial choice situation: Assume random utility (\(\epsilon\)) is Gumbel distributed with \(\eta = 0\) location parameter and a common scale, \(\mu\).
• Assume that the utility of the second-best alternative is \(U^*\), which is the maximum of all other alternatives except alternative \(j\), the best one. \[𝑈^∗=\max_{k \neq j \text{ & } 𝑗,𝑘 \in 𝐶_𝑖} (𝑉_𝑘+\epsilon_𝑘 )\]
• According to Property 7, \(U^*\) is also Gumbel distributed with parameters (mean, scale)

\[ \left( \frac{1}{\mu} \ln \sum_{k \neq j \text{ & } 𝑗,𝑘 \in 𝐶_𝑖} e^{\mu V_k}, \mu \right)\]

Multinomial Logit Model (MNL)

We can write that the second-best alternative is \[U^∗= V^∗+\epsilon^∗ \text{, } V^* = \frac{1}{\mu} \ln \sum_{k \neq j \text{ & } 𝑗,𝑘 \in 𝐶_𝑖} e^{\mu V_k}\]
Now for the probability of choosing alternative \(j\) (equivalent binary logit form) \[Pr⁡(𝑗)=Pr⁡\left((𝑉_𝑗+\epsilon_𝑗) \ge \max_{k \neq j \text{ & } 𝑗,𝑘 \in 𝐶_𝑖} (V_k + \epsilon_k)\right)\] \[Pr⁡(𝑗)=Pr\left((𝑉_𝑗+\epsilon_𝑗) \ge (𝑉^∗+\epsilon^∗)\right)\] \[Pr⁡(𝑗)=Pr⁡\left((𝑉^∗+\epsilon^∗)−(𝑉_𝑗+\epsilon_𝑗) \leq 0\right)=Pr\left((\epsilon^∗−\epsilon_𝑗) \leq(𝑉_𝑗−𝑉^∗)\right)\]

Multinomial Logit Model (MNL)

• According to Property 5, this function is logistically distributed. So

\[Pr⁡( 𝑗)=\frac{1}{1+𝑒^{\mu(𝑉^∗−𝑉_𝑗)}} = \frac{e^{\mu V_j}}{e^{\mu V_j}+𝑒^{\mu(𝑉^∗−𝑉_𝑗)}}\] \[Pr⁡( 𝑗)=\frac{e^{\mu V_j}}{e^{\mu V_j}+e^{\ln \sum_{k \neq j \text{ & } 𝑗,𝑘 \in 𝐶_𝑖} e^{\mu V_k}}}= \frac{e^{\mu V_j}}{e^{\mu V_j}+\sum_{k \neq j \text{ & } 𝑗,𝑘 \in 𝐶_𝑖} e^{\mu V_k}}\] \[Pr⁡(𝑗)=\frac{e^{\mu V_j}}{\sum_{𝑘 \in 𝐶_𝑖} e^{\mu V_k}}\]

• This is well-known Multinomial Logit Model (MNL) - or softmax function in ML!

Scale of the MNL Model

• Scale (\(\mu\)) is inversely related to the variance of random error term
• Variance is \(\pi^2/6\mu^2\) for a IID Type I EV error
• As \(\mu -> \infty\), Variance \(-> 0\)
  • All information are captured by the deterministic component (\(V\)
  • Choice becomes deterministic (the alternative with maximum V is chosen)
• As \(\mu -> 0\), Variance \(-> \infty\) & \(\exp(\mu V) = 1\)
  • Alternatives are equally likely
  • Choice model does not provide any information
• Constant scale, \(\mu\), is not identified in an MNL using only one dataset
• Separate scale needs to be specified when fusing multiple datasets

Systematic Utility Function

Systematic utility function: Linear-in-parameter function \[𝑉_1=\beta_1+\beta_𝑡 𝑥_{1𝑡}+\beta_{1𝑎} 𝑥_𝑎+\beta_𝑏 𝑥_{1𝑏}^2+\beta_{1𝑎𝑐} 𝑥_𝑎^3+\beta_{1𝑎𝑙} \ln⁡(𝑥_𝑎)+\dots\] \[𝑉_2=\beta_2+\beta_𝑡 𝑥_{2𝑡}+\beta_{2𝑎} 𝑥_𝑎+\beta_𝑏 𝑥_{2𝑏}^2+\beta_{2𝑎𝑐} 𝑥_𝑎^3+\beta_{2𝑎𝑙} + \ln⁡(x_a)+\dots\] \[\vdots\] \[𝑉_J=\beta_J+\beta_𝑡 𝑥_{Jt}+\beta_{J𝑎} 𝑥_𝑎+\beta_𝑏 𝑥_{J𝑏}^2+\beta_{J𝑎𝑐} 𝑥_𝑎^3+\beta_{J𝑎𝑙} + \ln⁡(x_a)+\dots\]

• Alternative Specific Constant - (J-1) are identified
• Permits non-linear transformed variables
• Can be generic if vary across alternatives
• Must be alternative-specific if constant across alternatives (only J-1 are identified)

Systematic Utility Function

• Systematic utility function: Linear-in-parameter function \[𝑉_𝑗=\beta_𝑗+\beta_𝑡 𝑥_{𝑗𝑡}+\beta_𝑘 𝑥_{𝑗𝑘}+\dots \text{ (individual variable space)}\] \[𝑉_𝑗=\beta_𝑗+(\beta_𝑡/\beta_𝑘)𝑥_{𝑗𝑡}+𝑥_{𝑗𝑘}+\dots \text{ (variable ratio space)}\] \[ 𝑉_𝑗=\beta_𝑗+(\beta_𝑡/\beta_𝑐)𝑥_{𝑗−𝑡𝑖𝑚𝑒}+(\beta_𝑟/\beta_𝑐)𝑥_{𝑗−𝑟𝑒𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦}+𝑥_{𝑗−𝐶𝑜𝑠𝑡}+\dots \] \[𝑉_𝑗=\beta_𝑗^{'}+\underset{\begin{array}{c} \text{(Willingness-to-pay} \\ \text{for time savings)} \end{array}}{\beta_{𝑡𝑐}} 𝑥_{𝑗−𝑡𝑖𝑚𝑒}+\underset{\begin{array}{c} \text{(Willingness-to-pay} \\ \text{for improved reliability)} \end{array}}{\beta_{𝑟𝑐}} 𝑥_{𝑗−𝑟𝑒𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦}+𝑥_{𝑗−𝐶𝑜𝑠𝑡}+\dots \]

MNL Estimation & MNL Specification

Estimation of MNL

• Maximum likelihood estimation: \[\text{Likelihood function: } 𝐿_𝑖=\prod_{𝑗=1}^𝐽 (Pr⁡( 𝑗))^{𝑦_𝑖} \] \[\text{Log-likelihood function: } LL_𝑖=\sum_{𝑗 \in C_i} y_j \ln(Pr⁡( 𝑗)) \] \[\text{Sample log-likelihood function: } LL=\sum_{i=1}^N \sum_{𝑗 \in C_i} y_j \ln(Pr⁡( 𝑗)) \]
• Use gradient search algorithms
• Analytical gradient and Hessian functions provide faster estimation than numerical gradient and Hessian functions

Goodness-of-Fit

Model goodness-of-fit is measured by \(\rho^2\) value

- \(k\) = difference in total number of parameters between two models \[\rho_0^2 = \frac{LL(\beta) - LL(0)}{LL(*) - LL(0)} = 1 - \frac{LL(\beta)}{LL(0)} \text{; } adj-\rho_0^2 = 1 - \frac{LL(\beta) - k}{LL(0)}\]

\[\rho_C^2 = \frac{LL(\beta) - LL(C)}{LL(*) - LL(C)} = 1 - \frac{LL(\beta)}{LL(C)} \text{; } adj-\rho_C^2 = 1 - \frac{LL(\beta) - k}{LL(C)}\]

• Adjusted \(\rho^2\) value 0.3 to 0.5 is considered reasonable (but depends on model)
• Alternative: Comparing predicted vs. observed shares

Goodness-of-Fit

• Comparing predicted versus observed share of alternatives:
  • Use holdout sample (20%-30% sample unused for estimation):
    • Problematic for small sample size
    • Holding out may distort market sample share of the alternatives and thereby the alternative specific constants
  • Use bootstrap:
    • Generate large number of samples bootstrapped from the same sample
    • Compare under/over predictions

Validation by Bootstrapping

Validation by Bootstrapping

Validation by 1000 Bootstrap Sample Simulation

Mode	Sample Share (%)	Mean Prediction Error (%)	Max Over-Prediction (%)	Max Under-Prediction (%)
Car Driver	26	0.033	3.26	-2.78
Car Passenger	1	0.015	0.91	-0.71
Transit	59	-0.026	4.05	-3.36
Bicycle	4	0.001	2.35	-1.72
Walk	11	-0.022	2.25	-2.43

Specification Tests

• Expected signs of the coefficients
• Expected substitution patterns - e.g., value of time savings, etc.
• Likelihood ratio test (\(\times\)-2 by Wilks Theorem):\(-2[LL(\beta)-LL(\beta^{'})]\) is \(\chi^2\) distributed with \(k\) (= number of restricted parameters = the difference between number of parameters of two models) degrees of freedom
• Goodness-of-Fit measures: Rho-square, AIC, BIC

Specification of Logit Model

• Logit model works fine if there is no significant random variation of effects of any systematic variables:
  • Unobserved (random) effects of any variable (varying across the alternatives and/or observations) violates the identical error term distribution assumption!
• Example:
  • Effects of driving cost is influenced by income only: systematic effect and logit model will work fine with cost normalized by income
  • Effects of driving cost influenced by income and some unobserved factor: the logit model will be mis-specified

Specification of Logit Model

• Logit model works fine if choice alternatives are perfectly IIA (independent of irrelevant alternatives): proportional substitution exists
  • Imperfect competition or correlated variations among subsets of alternatives cause the violation of identical error term distribution assumption!
• Example:
  • People who prefer private car, will prefer taxi more than they prefer transit - logit model that considers all alternative modes are IIA will be mis-specified

Collinearity of Independent Variables

• If some variables in the model are collinear then the corresponding parameters can change arbitrarily
  • Unstable parameter estimation
  • Hessian matrix may not be definite
• It is possible to have correlated variables in the discrete choice model unless they are highly collinear
  • For high collinearity, a small change in model or data causes drastic changes in parameter estimates
• Example: \(V=\beta_t(Time)+ \beta_c(Cost)\), if \(Cost= \beta_b+ \beta_d(Time)\), then \(V= (\beta_t + \beta_t\beta_d)Time+ \beta_c \beta_b\)
  • Identification issue
• Solution: Create composite variable using the correlated variables (e.g., cost/time)

Income/Budget Effects

• Testing income effects:
  • Use \(Cost\) and \(Cost^2\) as separate variables
  • If \(Cost^2\) has a statistically significant and reasonable parameter value, income effects need to be addressed
• Addressing income effects:
  • Income normalization as (\(Cost/Income\)) or (\(Income-Cost\))
  • Classifying population into different income categories and develop separate models for each
  • Interact income category with other variables

Aggregation/Market Segmentation

Aggregation/Market Segmentation

• Here 60% of vehicle owners are higher income people
  • 60% of low-income vehicle owners are in white-collar group
• So, we should be careful in categorization at the stage of parameter estimation

Aggregation/Market Segmentation

• Multicollinearity & correlation among independent (explanatory) variables may affect model predictions
• Appropriate clustering method can be used to derive finite and exhaustive sets of market segments to develop separate models
• Separate model for each market segment can sometimes help capture heterogeneity in complicated markets

Aside: Correlation, Collinearity, & Multicollinearity

• Correlation: when an independent variable exhibits a strong linear relationship with the dependent variable
• Collinearity: when two (or more) independent variables exhibit a strong linear relationship - may exist between multiple variables
• Multicollinearity: Specific case of collinearity between more than two variables
• Collinearity can occur without linear relationship between any two variables

Elasticity, Marginal Effect, & Substitution in MNL

Marginal Effect/Elasticity: Attributes of Alternatives

• Disaggregate Direct elasticity/marginal effect: with respect to an attribute of the same alternative \[\text{Direct ME} =\frac{\partial Pr⁡(𝑗)}{\partial 𝑥_j}=Pr⁡(𝑗)(1−Pr⁡( 𝑗)) \beta_{𝑥_𝑗}\] \[\text{Direct E}= \frac{\partial Pr⁡(𝑗)}{\partial 𝑥_j}\frac{x_j}{Pr(j)}=x_j(1−Pr⁡( 𝑗)) \beta_{𝑥_𝑗}\]

Marginal Effect/Elasticity: Attributes of Alternatives

• Disaggregate Cross elasticity/marginal effect: with respect to an attribute of different alternative \[\text{Cross ME} =\frac{\partial Pr⁡(𝑗)}{\partial 𝑥_k}=Pr⁡(𝑗)Pr⁡(k) \beta_{𝑥_k}\] \[\text{Cross E}= \frac{\partial Pr⁡(𝑗)}{\partial 𝑥_k}\frac{x_k}{Pr(j)}=-x_kPr⁡(k) \beta_{𝑥_k}\]

• Cross elasticity is the same for all other alternative than alternative \(k\)

• This is proportional substitution: an improvement in one alternative, draws proportionately from all other alternatives!

Fact to check: Attributes of Alternatives

• Summation of all probabilities of a choice model = 1
  • Summation of changes in probabilities (of all alternatives) with respect to a variable (derivative) = 0 \[\sum_j \frac{\partial Pr(j)}{\partial x_{jp}}=\frac{\partial Pr(j)}{\partial x_{jp}} + \sum_{k \neq j} \frac{\partial Pr(k)}{\partial x_{jp}} \text{; (Considering generic coefficients)}\] \[\sum_j \frac{\partial Pr(j)}{\partial x_{jp}}=\beta_p Pr(j) (1-Pr(j)) - \beta_p Pr(j)\sum_{k \neq j}Pr(k)\] \[\sum_j \frac{\partial Pr(j)}{\partial x_{jp}}=\beta_p Pr(j) (1-Pr(j)) - \beta_p Pr(j) (1-Pr(j)) = 0\]
• This is true for any discrete choice model and can be used to verify accuracy of elasticity calculation

Aggregating Elasticity: Attributes of Alternatives

• Disaggregate discrete choice model gives disaggregate values of elasticity for each data point in the sample
• Approaches to meaningfully aggregate:
  • Naïve approach: Consider sample average values of explanatory variables (x) to calculate sample average elasticity
  • Estimate elasticity for each sample respondent and draw distribution of elasticity values
  • Probability weighted sample enumeration (PWSE):

\[𝐸_{𝑥_𝑘}^𝑗=\sum_{i=1}^n \frac{Pr_i(j)}{\sum_{i=1}^n Pr_i(j)}E_{x_k}^{Pr_i(j)}\]

• \(i\) is the individual in the sample of size n
• \(j\) is the specific alternative
• \(x\) is the variable of elasticity
• \(j=k\) indicates direct elasticity
• \(j \neq k\) indicates cross-elasticity

• Same approach is application for marginal effect and for any choice model

Marginal/Elasticity: Attributes of Choice Maker/Choice Context

• Such attributes have alternative-specific coefficients: For example, consider income of choice maker \[\frac{\partial Pr(j)}{\partial Inc_{ij}} = (1-Pr(j))Pr(j)\beta_{inc,ij} \text{; }j,k\in C_i\] \[\frac{\partial Pr(j)}{\partial Inc_{ik}} = Pr(j)Pr(k)\beta_{inc,ik}\]
• Corresponding sum of all alternatives, \(j \neq k\) \[\sum_{j \neq k} \frac{\partial Pr(j)}{\partial Inc_{ik}} = -Pr(j) \sum_{k \neq j} Pr(k) \beta_{inc,ik}\]
• Rate of change of \(Pr(j)\) with respect to income of individual, i: \[\frac{\partial Pr(j)}{\partial Inc_i} = \underset{\text{Direct Elasticity}}{\frac{\partial Pr(j)}{\partial Inc_{ij}}} + \underset{\text{Cross Elasticity}}{\sum_{j \neq k}\frac{\partial Pr(j)}{\partial Inc_{ij}}}\]

Marginal/Elasticity: Attributes of Choice Maker/Choice Context

• Rate of change of \(Pr(j)\) with respect to income: \[\frac{\partial Pr(j)}{\partial Inc_i} = \frac{\partial Pr(j)}{\partial Inc_{ij}} + \sum_{k \neq j} \frac{\partial Pr(j)}{\partial Inc_{ik}} = (1-Pr(j))Pr(j)\beta_{inc,ij}-Pr(j)\sum_{k \neq j} Pr(k) \beta_{inc,ik}\] \[\frac{\partial Pr(j)}{\partial Inc_i} = Pr(j)\beta_{inc,ij}-Pr(j)Pr(j)\beta_{inc,ij}-Pr(j)\sum_{k \neq j}Pr(k)\beta_{inc,ik}\] \[\frac{\partial Pr(j)}{\partial Inc_i} = Pr(j)\left( \beta_{inc,ij} - \overline{\beta_{inc}} \right)\] \[\overline{\beta_{inc,i}} = \sum_{k \in C_i} Pr(k)\beta_{inc,k} \begin{array}{c}\text{; probability weighted average of}\\ \text{alt specific income parameters}\end{array}\]

Marginal/Elasticity: Attributes of Choice Maker/Choice Context

• Elasticity \(Pr(j)\) with respect to income: \[\frac{\partial Pr(j)}{\partial Inc}\frac{Inc}{Pr(j)} = Inc(\beta_{inc,j} - \overline{\beta_{inc}})\] \[\overline{\beta_{inc}} = \sum_{k \in C_i} Pr(k)\beta_{inc,k}\]
• Elastictiy \(Pr(j)\) with respect to age: \[\frac{\partial Pr(j)}{\partial Age}\frac{Age}{Pr(j)} = Age(\beta_{age,j} - \overline{\beta_{age}})\] \[\overline{\beta_{age}} = \sum_{k \in C_i} Pr(k)\beta_{age,k}\]

Aggregating Elasticity: Attributes of Choice Maker/Choice Context

• Use similar approach of aggregating market share from disaggregate choice model predictions (discussed in detail later):
  1. Microsimulation
  2. Sample enumeration
  3. Classification with naïve aggregation
  4. Naïve aggregation
• Same approach is application for marginal effect and for any choice model

Marginal/Elasticity of Categorical Variables

• Choice elasticity with respect to a categorical variable is tricky
  • “1% change in choice probability with respect to 1% change in a categorical variable” - e.g., presence/absence of parking spot at destination has little meaning!
• Arc-elasticity is the best way: Express the change in choice probability for the presence/absence of the categorical variable - i.e., ceteris paribus \[E_{x_i}^{ji} = \frac{Pr(j)_{x_i=1} - Pr(j)_{x_i=0}}{(x_i=1) - (x_i=0)}\frac{((x_i=1)+(x_i=0))/2}{((Pr(j)_{x_i=1})+(Pr(j)_{x_i=0}))/2}\] \[E_{x_i}^{ji} = \frac{Pr(j)_{x_i=1} - Pr(j)_{x_i=0}}{Pr(j)_{x_i=1}+Pr(j)_{x_i=0}}\]
• Arc-elasticity can also be used for continuous variables

Proportional Substitution

• Proportional substitution: an improvement in one alternative, draws proportionately from all other alternatives!

IIA: Independent and Irrelevant Alternatives

\[\frac{Pr(j)}{Pr(k)} = \frac{\exp(V_j)}{\exp(V)k)} = \exp(V_j - V_k)\]

• This property is derived from assumption: the random utilities,\(\epsilon\) are independent and identical in distribution
• Random error are identically distributed, having same scale or variance \[\begin{equation} \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_3 \end{bmatrix} = \begin{bmatrix} \sigma^2 & 0 & 0 \\ 0 & \sigma^2 & 0 \\ 0 & 0 & \sigma^2 \end{bmatrix} = \frac{\pi^2}{6 \mu^2} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{equation}\]

IIA: Independent and Irrelevant Alternatives

• Example of blue bus/red bus: If real alternatives are not following IIA, then MNL would induce serious error
  • Previous: P(car)=1/2 and P(blue bus)=1/2
  • Now a new red bus is introduced and under the IIA assumption that the ratio of choice probabilities between car and blue bus must be constant, so P(car)=1/3, P(blue bus)=1/3 and P(red bus)=1/3, which is unrealistic Actual will be P(car)=1/2, P(blue bus)=1/4 and P(red bus)=1/4; but now the ratio of car and blue bus is not the same as was before introducing red bus

IIA: Independent and Irrelevant Alternatives

• Three alternatives with same utilities 1: car, 2: blue bus and 3: red bus and (V1=V2=V3=V)
  • Now probability of choosing 1: car would be compared against the second highest attractive alternative (\(U^*\)) \[𝑈^∗=\max⁡( 𝑈_2,𝑈_3)=𝑉^∗+\epsilon^∗ \]
• As per Gumbel distribution with 0 mean and unit scale \[𝑉^∗=\ln⁡( 𝑒^{𝑉_2}+𝑒^{𝑉_3})=\ln⁡( 2𝑒^𝑉)=\ln⁡(2)+𝑉 \]
• This step assumed that the random errors of alternative 2(blue bus) and 3(red bus) are independent and in such case, probability of choosing car: \[𝑃(𝑐𝑎𝑟)=\frac{1}{1+𝑒^{𝑉^∗−𝑉_1}} = \frac{1}{1+𝑒^{𝑉+\ln⁡2−𝑉}} = \frac{1}{1+2}=1/3 \]

IIA: Independent and Irrelevant Alternatives

• Actually, 2(blue bus) and 3(red bus) are perfectly correlated and in that case the maximum utility of 2 and 3 are simply \((V+\epsilon)\) that means \(V^*=V\) \[𝑃(𝑐𝑎𝑟) = \frac{1}{1+𝑒^{𝑉^∗−𝑉_1}} = \frac{1}{1+𝑒^{𝑉−𝑉}} = \frac{1}{1+1}=1/2 \]
• That means IID assumption makes \(V^*\) too high by an amount of \(\ln2\)
• IIA is attributed to disaggregate/individual choice situations, not the overall market share

IIA Violation Test

• We can always test IIA by developing alternative nested logit models.
• It is also possible to test statistically whether IIA is violated or not without developing nested logit models.

Steps for testing IIA:

1. Estimate the base MNL model for the full set of alternatives and refer to its parameters as \(\beta_C\)
   
2. Remove one or more alternatives with suspected IIA violation problem to make the constrained choice set \(C^{'}\)
3. This will reduce the number of parameters in the new model and its parameter set is then \(\beta_C^{'}\)

• In each case, in addition to estimating the parameter sets we also produce variance covariance matrices, \(\Sigma\), (Inverse of Hessian Matrix) of the estimated parameters sets

IIA Violation Test

4. Let us refer to the corresponding matrices as \(\Sigma_{\beta_C}\) and \(\Sigma_{\beta_C^{'}}\) of the full model and the restricted model, respectively
5. If the alternatives in the choice set satisfy IIA then the two variance-covariance matrices will have the same values for corresponding cells: \(\Sigma_{\beta_C} = \Sigma_{\beta_C^{'}}\)
   
6. Let us consider the Null Hypothesis, \(H_0: \beta_C = \beta_C^{''}\)
   
7. Conduct tests:

• Hausman and McFadden Test
• Small and Haiso Test

IIA Violation Test

Hausman and McFadden Test (Econometrica, 1984): \[(\hat{\beta}_{C^{'}}−\hat{\beta}_C)^𝑇(\Sigma_{\hat{\beta_C^{'}}}−\Sigma_{\hat{\beta}_C})^{−1}(\hat{\beta}_{C^{'}}-\hat{\beta_C})\] - This is \(\chi^2\) distributed with degrees of freedom equal to the number of alternatives removed in the restricted model

Small and Haiso Test (International Economic Review, 1985):

• Conduct likelihood ratio test between base model and the restricted model \[−2(𝐿𝐿_{\hat{\beta}_𝐶^{'}}−𝐿𝐿_{\hat{\beta}_𝐶})\]
• This likelihood ratio is \(\chi^2\) distributed with degrees of freedom equal to the number of alternatives removed in the restricted model