Lecture 14 Discrete Choice Models

Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example

Zhenghui Sha

ME 597: Decision Making for Engineering Systems Design Design Engineering Lab @ Purdue (DELP) School of Mechanical Engineering Purdue University, West Lafayette, IN http://engineering.purdue.edu/delp

November 19, 2014

c Zhenghui Sha Lecture 14 1 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Lecture Outline

1 Discrete Choice Modeling: An Overview

2 Modeling framework of Discrete Data

3 Discrete Choice Models Probit Model Logit Model

4 Model Estimation and Evaluation

5 An Illustrative Example Zero-Model Constant-only Model MNL Model

c Zhenghui Sha Lecture 14 2 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Overall Objective

Objective of using discrete choice models Given the decisions that have already been made, to elicit the preference structures which lead to the observed decision-making activities.

c Zhenghui Sha Lecture 14 3 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Discrete Choice Analysis

The utility theory for decision-making analysis is forward problem. The Discrete Choice Analysis is inverse problem.

c Zhenghui Sha Lecture 14 4 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Discrete Choice

What is Discrete Choice? A dependent variable that is a categorical, unordered variable. The choices/categories are called alternatives (often coded as 1, 2, 3, 4, ...) and only one alternative can be selected.

c Zhenghui Sha Lecture 14 5 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Examples of Discrete Choice

Behavioral Choice Mode of travel: automobile, bus, rail transit, airplane; Type of class of vehicle owned: sedan, cross-over, SUV; Brand of laptop you owned: Dell, Lenovo, HP, etc.

Physical Event Type of a vehicular accident: run-off-road, rear-end, head-on.

c Zhenghui Sha Lecture 14 6 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Making Discrete Choice

Given the three options, which car do you prefer to buy? List your criteria that affects your decision.

c Zhenghui Sha Lecture 14 7 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Decision-making Process

c Zhenghui Sha Lecture 14 8 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Model Structure

For unordered discrete choices, it is natural to start with a linear function of covariates that inﬂuence speciﬁc choice,

Vin = βi xin

Vin is the observed utility of decision-maker n if choosing alternative i.

βi is a vector of estimable parameters for alternative i.

xin is a vector of the observable characteristics (covariates) that determine discrete choice for observation of decision-maker n.

c Zhenghui Sha Lecture 14 9 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Model Structure - Total utility

Total utility

Uin = Vin + in = βi xin + in

Vin is observed component, and in is unobserved component. βi quantiﬁes the decision-makers’ preferences.

in is a error term that quantify the uncertainty from the researcher’s stand point. The reasons for adding this disturbance is that: Variables have been omitted from the utility function. The function form may be incorrectly speciﬁed. Proxy variables may be used.

Variation in βi that are not accounted for.

c Zhenghui Sha Lecture 14 10 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Utility Theory for Discrete Choice

Traditional approaches from microeconomics theory have decision makers choosing among a set of alternatives such that their utility (satisfaction) is maximized subject to the prices of the alternatives and the income constrains.

Assumption: maximizing utility Uin

Pin = P(Uin > Ujn), ∀j 6= i

= P(Vin + in > Vjn + jn), ∀j 6= i

= P(jn − in < Vin − Vjn), ∀j 6= i

= P(jn − in < βinxin − βjnxjn), ∀j 6= i

c Zhenghui Sha Lecture 14 11 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Probability Theory for Discrete Outcome

The discrete choice model can be applied to any applications with discrete outcomes. For example, in the event of a vehicular accident, possible discrete crash outcomes are rear-end, sideswipe, run-off-road, turning, and other.

Linear probabilistic model

Tin = βi xin + in

Pin = P(Tin > Tjn), ∀j 6= i

= P(jn − in < βinxin − βjnxjn), ∀j 6= i

c Zhenghui Sha Lecture 14 12 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Fundamental Difference

Key point The methodological approach used to statistically model these conceptual perspectives is identical. The underlying theory used to derive these models is often quite different. Discrete models of behavioral choices are derived from economic utility theory, often leading to additional insights in the analysis of the model estimation results Models of physical phenomena are derived from simple probability theory.

c Zhenghui Sha Lecture 14 13 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Probit Model - Error Assumption

Assumption of Error term Identically independent distributed (iid) and follows a Normal distribution:

2 f (in) = N(0, σ )

In a binary case, two choices, denoted 1 or 2. Then, the equation estimates the probability of choice 1 is selected by decision-maker n is:

P1n = P(2n − 1n < β1nx1n − β2nx2n) β x − β x = Φ( 1n 1n 2n 2n ) σ q 2 2 where Φ() is the cdf of Normal distribution. σ = σ1 + σ2 − σ12. Typically, σ = 1 is used.

c Zhenghui Sha Lecture 14 14 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Probit Model - Probability Plot

c Zhenghui Sha Lecture 14 15 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Logit Model - Error Assumption

Assumption of Error term Identically independent distributed (iid) and follows a extremem value type 1 (Gumbel) distribution: −e−η(in−ω) F(in) = e where η is a positive scale parameter, ω is a location parameter (mode).

c Zhenghui Sha Lecture 14 16 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Logit Model - Motivation

Motivation of using extreme value type 1 distribution To extend the model from the binary chase to multiple choice case from a computational convenient perspective

In multiple choice case, if there are I alternatives, 1, 2, ..., I, then

Pin = P(βinxin + in > max(βjnxjn + jn)) ∀j6=i A desirable property of an assumed distribution of error is that the maximums of randomly drawn values from the distribution have the same distribution as the values from which they were drawn.

c Zhenghui Sha Lecture 14 17 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Logit Model

Multinomial Logit Model (MNL)

EXP(βinxin) Pin = P EXP(βinxin) + EXP(LN ∀j6=i (EXP(βj xjn)))

EXP(β x ) P = in in in PI j=1 EXP(βj xjn)

c Zhenghui Sha Lecture 14 18 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Logit Model - Logit Probability Plot

c Zhenghui Sha Lecture 14 19 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Logit Model - Key Points

Key Point 1 Variables deﬁning the utility function can be classiﬁed into two categories. Those that vary across outcome alternatives, i.e., customer desired attributes A. Those that do not across outcome alternatives, i.e., socio-demographic attributes about decision-makers S, such as gender, age, income, etc.

Example: Consider a commuters choice of route from home to work where the choices are to take an arterial, a two-lane road, or a freeway

eVa eVt eVf Pa = Pt = Pf = eVa +eVt +eVf eVa +eVt +eVf eVa +eVt +eVf

c Zhenghui Sha Lecture 14 20 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Logit Model - Key Points

Va = β1a + β2axa + β3az Vt = β1t + β2t xt + β3t z Vf = β1f + β2f xf + β3f z

xa, xt and xf are vectors of variables that vary across arterial, two-lane, and freeway choice outcomes respectively, as experienced by commuter n.

z is a vector of characteristic speciﬁc to commuter n.

β1 are constant terms.

β2 are vectors of estimable parameters corresponding to outcome speciﬁc variable in x vectors, and

β3 are vectors of estimable parameters corresponding to variable that do not vary across outcome alternatives.

c Zhenghui Sha Lecture 14 21 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Probit Model Discrete Choice Models Logit Model Model Estimation and Evaluation An Illustrative Example Logit Model - Key Points

Key Point 2 The MNL model is derived using the difference in utility. Estimable parameters relating to variable that do not vary across outcome alternatives can, at most, be estimated in N − 1 of the function determining the discrete outcome. The parameter of at least one of the discrete outcomes must be normalized to zero to make parameter estimation possible.

c Zhenghui Sha Lecture 14 22 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Model Estimation

Based on the collected discrete choice data (either revealed or stated choice), modeling techniques as introduced can be used to create a choice model that can predict the choices individual customer makes and to forecast the market demand. The preference, β, is readily estimated using maximum likelihood methods.

Existing commercial software that offer logit or probit modeling capabilities GENSTAT (www.vsn-intl.com) LIMDEP (www.limdep.com) SAS (www.sas.com) SPSS (www.spss.com) STATA (www.stata.com) SYSTAT (www.systa.com)

c Zhenghui Sha Lecture 14 23 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Model Evaluation - Individual Parameter

t Test The statistical signiﬁcance of individual parameters is approximated using a one tailed t test:

β − 0 t ∗ = S.E.(β) where S.E.(β) is the standard error of the parameter

c Zhenghui Sha Lecture 14 24 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Model Evaluation - A General Method

Likelihood Ratio Test

2 X = −2[LL(βR ) − LL(βU )]

where LL(βR ) is the log-likelihood at convergence of the ”restricted” model and LL(βU ) is the log-likelihood at convergence of the ”unrestricted” model. X 2 statistic is χ2 distributed with degrees of freedom equal to the difference in the numbers of the parameters between the restricted and unrestricted model.

Assessing the signiﬁcance of individual parameter Evaluating overall signiﬁcance of the model examining the appropriateness of of estimating separate parameter for the same variable in different choice function examining the transferability of results over time and space.

c Zhenghui Sha Lecture 14 25 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Discrete Choice Models Model Estimation and Evaluation An Illustrative Example Model Evaluation - Overall Model Fit

ρ2 Statistic LL(β) ρ2 = 1 − LL(0) where LL(β) is the log-likelihood at convergence with parameter β and LL(0) is the initial log-likelihood with all parameter set to zero.

Corrected ρ2 Statistic LL(β) − K ρ2 = 1 − LL(0) where K is the number of the parameter estimated in the model.

c Zhenghui Sha Lecture 14 26 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example An Illustrative Example - Background

Demand estimation model for an academic power saw design

Speed Maintenance Frequency Price Saw 1 High High High Saw 2 Medium Low Medium Saw 3 Low Medium Low

c Zhenghui Sha Lecture 14 27 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Conducting Choice Set

Sample data representing the revealed preference of 15 customers who buy these saws from different vendor. Normalized data is used for convenience computation and interpretation. Socio-demographic attribute is included.

c Zhenghui Sha Lecture 14 28 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Sample Data

Table 3.5 on page 64 (Wei Chen, Christopher Hoyle and Henk J. Wassenaar)

c Zhenghui Sha Lecture 14 29 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Zero-Model

Zero-Model For 1 ≤ n ≤ 15,

Pn(1)[1, 2, 3] = 1/3

Pn(2)[1, 2, 3] = 1/3

Pn(3)[1, 2, 3] = 1/3

where Pn(1)[1, 2, 3] is the probability of alternative i is chosen by customer n.

Reference model to compare the goodness of ﬁt with other models. Not for prediction purpose. The market share predictions for alternative 1, 2, 3 are {1/3, 1/3, 1/3} which do not match well with the observed market share, i.e., {0.4, 0.2, 0.4}

c Zhenghui Sha Lecture 14 30 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Constant-only Model

Utility Function

Uin = βi xin + in = β0i + β1i x1in + ... + βKi xKin + in

β0i - alternative speciﬁc constant (ASC): to represent preferences that are inherent and independent of speciﬁc attribute values.

c Zhenghui Sha Lecture 14 31 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Constant-only Model

Constant-only Model For 1 ≤ n ≤ 15,

U1n = β01

U2n = β02

U3n = β03(= 0)

The ASC corresponding to alternative 3 (i.e., β03) is set to zero, and the ASCs corresponding to other alternatives are evaluated with respect to the reference alternative.

c Zhenghui Sha Lecture 14 32 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Constant-only Model - Output

Figure 3.9 on page 68 (Wei Chen, Christopher Hoyle and Henk J. Wassenaar)

c Zhenghui Sha Lecture 14 33 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Constant-only Model - Analysis

For 1 ≤ n ≤ 15,

U1n = β01 = 0

U2n = β02 = −0.6932

U3n = β03 = 0

eU1n 1 Pn(1)[1, 2, 3] = = = 0.4 eU1n + eU2n + eU3n 1 + e−0.6932 + 1 eU2n e−0.6932 Pn(2)[1, 2, 3] = = = 0.2 eU1n + eU2n + eU3n 1 + e−0.6932 + 1 eU3n 1 Pn(3)[1, 2, 3] = = = 0.4 eU1n + eU2n + eU3n 1 + e−0.6932 + 1

c Zhenghui Sha Lecture 14 34 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Constant-only Model - Key Points

Key Points for Constant-only Model The choice probabilities are identical across all customers. The predicted market shares from the model are identical to the individual choice probabilities. The predicted market share values match exactly with the observed market shares. The model has no purposeful role in guiding engineering design.

c Zhenghui Sha Lecture 14 35 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example MNL Model

For 1 ≤ n ≤ 15,

U1n = βspeed xspeed (1) + βpricexprice(1) + βmaintenancexmaintenance(1)

U2n = βspeed xspeed (2) + βpricexprice(2) + βmaintenancexmaintenance(2)

+ βincome(2)xincome(n,2)

U3n = βspeed xspeed (3) + βpricexprice(3) + βmaintenancexmaintenance(3)

+ βincome(3)xincome(n,3)

Note: the β-coefficient of the alternative specific attributes are identical across all alternatives and all customers. However, the coefficient for the decision-maker specific attribute do vary across alternatives.

c Zhenghui Sha Lecture 14 36 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example MNL Model - Output

Figure 3.9 on page 69 (Wei Chen, Christopher Hoyle and Henk J. Wassenaar)

c Zhenghui Sha Lecture 14 37 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example MNL Model - Utility Results

For n = 3,

U13 = 47.09 × 1 − 55.95 × 0.95 + 28.01 × 0.64 = 11.86

U23 = 47.09 × 0.71 − 55.95 × 0.75 + 28.01 × 1 − 13.67 × 0.44 = 8.42

U33 = 47.09 × 0.67 − 55.95 × 0.60 + 28.01 × 0.89 − 19.66 × 0.44 = 6.98

c Zhenghui Sha Lecture 14 38 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example MNL Model - Predicted Choice Probability

For n = 3,

eU13 e11.86 P3(1)[1, 2, 3] = = = 0.96 eU13 + eU23 + eU33 e11.86 + e8.42 + e6.98 eU23 e8.42 P3(2)[1, 2, 3] = = = 0.03 eU13 + eU23 + eU33 e11.86 + e8.42 + e6.98 eU33 e6.98 P3(3)[1, 2, 3] = = = 0.01 eU13 + eU23 + eU33 e11.86 + e8.42 + e6.98

c Zhenghui Sha Lecture 14 39 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example Summary

1 Discrete Choice Modeling: An Overview

2 Modeling framework of Discrete Data

3 Discrete Choice Models Probit Model Logit Model

4 Model Estimation and Evaluation

5 An Illustrative Example Zero-Model Constant-only Model MNL Model

c Zhenghui Sha Lecture 14 40 / 41 Discrete Choice Modeling: An Overview Modeling framework of Discrete Data Zero-Model Discrete Choice Models Constant-only Model Model Estimation and Evaluation MNL Model An Illustrative Example References

1 W. Chen, C. Hoyle, and H. J. Wassenaar (2013). Decision-Based Design: Integrating Consumer Preferences into Engineering Design. Springer. 2 K. Train (1993). Discrete Choice Methods with Simulation, 2nd Edition. New York, NY, Cambridge University Press.

c Zhenghui Sha Lecture 14 41 / 41 Model Analysis

THANK YOU!

c Zhenghui Sha Lecture 14 1 / 13 Model Analysis

Backup Slides

c Zhenghui Sha Lecture 14 2 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Probit Model - Property of normal distribution

Proposition 2 2 2 2 If 1n ∼ N(0, σ1 ), and 2n ∼ N(0, σ2 ). Then, (1n − 2n) ∼ N(0, σ1 + σ2 − σ12).

where σ12 is the covariance between 1n and 2n

Z (β1nx1n−β2nx2n) 1 σ 1 2 (β1nx1n − β2nx2n) P1n = √ EXP(− ω )dω = Φ( ) 2π −∞ 2 σ

q 2 2 where σ = σ1 + σ2 − σ12. Typically, σ = 1 is used.

c Zhenghui Sha Lecture 14 3 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Probit Model - Estimation

The β is readily estimated using standard maximum likelihood methods.

N I Y Y δ L = P(i) in n=1 i=1 where N is the total number of observations, and I is the total number of alternatives. δin is deﬁned as ( 1, if decision-maker n’s choice is alternative i δin = 0, otherwise

Log-likelihood function for binary case with i = 1 or 2

N X (β1nx1n − β2nx2n) (β1nx1n − β2nx2n) LL = (δ LNΦ( ) + (1 − δ )LNΦ( )) in σ in σ n=1

c Zhenghui Sha Lecture 14 4 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Logit Model - Extreme value type 1 distribution

Deﬁnition of extreme value distributions (Gumbel 1958) Distributions of the maximums of randomly drawn values from some underlying distribution

Property 1

If all jn are iid random variables with mode ωjn and a common scale parameter η (which implies equal variance), then max∀j6=i (βjnxjn + jn) is also 1 P extreme value Type 1 distributed with mode η LN ∀j6=i (EXP(ηβj xjn)) and scale parameter η.

c Zhenghui Sha Lecture 14 5 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Logit Model - Extreme value type 1 distribution

Property 2 For extreme value Type 1 distributed variables, the addition of a positive constant, say a, changes the mode from ω to ω + a without affecting the scale parameter η.

0 0 0 Construct a random variable n + β xn that has the same distribution as 0 max∀j6=i (βjnxjn + jn), where n is a error term associated with the maximum of all possible discrete choice 6= 1 with mode equal to 0, and scale parameter η. 0 0 β xn is the associated product of parameter and covariate.

With Property 1 and 2, we have

0 0 1 X β x = LN (EXP(ηβ x )) n η j jn ∀j6=i

c Zhenghui Sha Lecture 14 6 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Logit Model - Derivation

Property 3 Difference between two independent distributed extreme value Type 1 variables with common scale parameter η is logistic distributed.

Pin = P(βinxin + in > max(βjnxjn + jn)) ∀j6=i 0 0 0 = P(βinxin + in > β xn + n) 0 0 0 = P(n − in < βinxin − β xn) 1 = 0 0 1 + EXP(η(βinxin − β xn))

EXP(η(βinxin)) = 0 0 EXP(η(βinxin)) + EXP(η(β xn))

c Zhenghui Sha Lecture 14 7 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Model Interpretation - Coefﬁcient interpretation

Coefﬁcient interpretation for alternative j: in comparison to the base alternative, an increase in the independent variable makes the selection of alternative j more or less likely.

c Zhenghui Sha Lecture 14 8 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Model Interpretation - Elasticity

Deﬁnition The magnitude of the impact of speciﬁc variables on the outcome probabilities.

P(i) ∂P(i) xki Exki = × ∂xki P(i)

where P(i) is probability of alternative i and xki is the value of variable k for alternative i.

For MNL model, by taking the partial derivative, the elasticity becomes:

P(i) Exki = [1 − P(i)]βki xki

c Zhenghui Sha Lecture 14 9 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Model Interpretation - Elasticity

Interpretation: a 1% change in xki has on the outcome probability P(i)

Inelastic: If the computed elasticity value is less than one. 1% change in xki will have less than a 1% change in alternative i’s selection. Elastic: If the computed elasticity value is greater than one. 1% change in xki will have more than a 1% change in alternative i’s selection.

Note: Elasticities are not applicable to indicator variable.

c Zhenghui Sha Lecture 14 10 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Model Interpretation - Cross-elasticity

Deﬁnition The magnitude of the impact of variables j may have on the probability of alternative i being selected.

P(i) Exkj = −P(j)βkj xkj

Note: This equation implies that there is one cross-elasticity for all i(i 6= j). This property of uniform cross-elasticity is an artifact of the error-term independence assumed in deriving the MNL model.

c Zhenghui Sha Lecture 14 11 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Model Interpretation - Marginal Rates

Deﬁnition The relative magnitude of any two parameters estimated in the model. In MNL, the rate is simply the ratio of parameter for any two variables:

βia MRS(i)ba = βib

c Zhenghui Sha Lecture 14 12 / 13 Model Evaluation Model Analysis Model Interpretation Speciﬁcation Errors

Speciﬁcation Errors

Independence of irrelevant alternatives (IIA) assumption fails. Omitted variables. Presence of an irrelevant variable. Disturbance that are not IID. Random parameter variation. Correlation between explanatory variable and disturbances and endogenous variable. Erroneous data. State dependence and heterogeneity.

c Zhenghui Sha Lecture 14 13 / 13