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Discrete Choice Modeling William Greene Stern School of Business IFS at UCL February 11-13, 2004

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Part 3 Modeling Binary Choice

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A Model for Binary Choice Yes or No decision (Buy/Not buy) Example, choose to fly or not to fly to a destination when there are alternatives. Model: Net utility of flying U fly = + 1Cost + 2Time + Income + Choose to fly if net utility is positive Data: X = [1,cost,terminal time] Z = [income] y = 1 if choose fly, U fly > 0, 0 if not.

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What Can Be Learned from the Data? (A Sample of Consumers, i = 1,…,N) Are the attributes “relevant?” Predicting behavior - Individual - Aggregate Analyze changes in behavior when attributes change

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Application 210 Commuters Between Sydney and Melbourne Available modes = Air, Train, Bus, Car Observed: Choice Attributes: Cost, terminal time, other Characteristics: Household income First application: Fly or other

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Binary Choice Data Choose Air Gen.Cost Term Time Income

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An Econometric Model Choose to fly iff U FLY > 0 U fly = + 1Cost + 2Time + Income + U fly > 0 > -( + 1Cost + 2Time + Income) Probability model: For any person observed by the analyst, Prob(fly) = Prob[ > -( + 1Cost + 2Time + Income)] Note the relationship between the unobserved and the outcome

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+ 1Cost + 2TTime + Income

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Econometrics How to estimate , 1, 2, ? It’s not regression The technique of maximum likelihood Prob[y=1] = Prob[ > -( + 1Cost + 2Time + Income)] Prob[y=0] = 1 - Prob[y=1] Requires a model for the probability

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Completing the Model: F( ) The distribution Normal: PROBIT, natural for behavior Logistic: LOGIT, allows “thicker tails” Gompertz: EXTREME VALUE, asymmetric, underlies the basic logit model for multiple choice Does it matter? Yes, large difference in estimates Not much, quantities of interest are more stable.

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Estimated Binary Choice Model | Binomial Probit Model | | Maximum Likelihood Estimates | | Model estimated: Jan 20, 2004 at 04:08:11PM.| | Dependent variable MODE | | Weighting variable None | | Number of observations 210 | | Iterations completed 6 | | Log likelihood function | | Restricted log likelihood | | Chi squared | | Degrees of freedom 3 | | Prob[ChiSqd > value] = | | Hosmer-Lemeshow chi-squared = | | P-value= with deg.fr. = 8 | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| Index function for probability Constant GC TTME HINC

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Estimated Binary Choice Models LOGIT PROBIT EXTREME VALUE Variable Estimate t-ratio Estimate t-ratio Estimate t-ratio Constant GC TTME HINC Log-L Log-L(0)

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+ 1Cost + 2Time + (Income+1) Effect on predicted probability of an increase in income ( is positive)

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How Well Does the Model Fit? There is no R squared “Fit measures” computed from log L “pseudo R squared = 1 – logL0/logL Others… - these do not measure fit. Direct assessment of the effectiveness of the model at predicting the outcome

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Fit Measures for Binary Choice Likelihood Ratio Index Bounded by 0 and 1 Rises when the model is expanded Cramer (and others)

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Fit Measures for the Logit Model | Fit Measures for Binomial Choice Model | | Probit model for variable MODE | | Proportions P0= P1= | | N = 210 N0= 152 N1= 58 | | LogL = LogL0 = | | Estrella = 1-(L/L0)^(-2L0/n) = | | Efron | McFadden | Ben./Lerman | | | | | | Cramer | Veall/Zim. | Rsqrd_ML | | | | | | Information Akaike I.C. Schwarz I.C. | | Criteria | Pseudo – R-squared

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Predicting the Outcome Predicted probabilities P = F(a + b1Cost + b2Time + cIncome) Predicting outcomes Predict y=1 if P is large Use 0.5 for “large” (more likely than not) Count successes and failures

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Individual Predictions from a Logit Model Observation Observed Y Predicted Y Residual x(i)b Pr[Y=1] Note two types of errors and two types of successes.

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Predictions in Binary Choice Predict y = 1 if P > P* Success depends on the assumed P*

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ROC Curve Plot %Y=1 correctly predicted vs. %y=1 incorrectly predicted 45 0 is no fit. Curvature implies fit. Area under the curve compares models

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Aggregate Predictions Frequencies of actual & predicted outcomes Predicted outcome has maximum probability. Threshold value for predicting Y=1 =.5000 Predicted Actual 0 1 | Total | | Total | 210

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Analyzing Predictions Frequencies of actual & predicted outcomes Predicted outcome has maximum probability. Threshold value for predicting Y=1 is P* (This table can be computed with any P*.) Predicted Actual 0 1 | Total N(a0,p0) N(a0,p1) | N(a0) 1 N(a1,p0) N(a1,p1) | N(a1) Total N(p0) N(p1) | N

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Analyzing Predictions - Success Sensitivity = % actual 1s correctly predicted = 100N(a1,p1)/N(a1) % [100(38/58)=65.5%] Specificity = % actual 0s correctly predicted = 100N(a0,p0)/N(a0) % [100(151/152)=99.3%] Positive predictive value = % predicted 1s that were actual 1s = 100N(a1,p1)/N(p1) % [100(38/39)=97.4%] Negative predictive value = % predicted 0s that were actual 0s = 100N(a0,p0)/N(p0) % [100(151/171)=88.3%] Correct prediction = %actual 1s and 0s correctly predicted = 100[N(a1,p1)+N(a0,p0)]/N [100(151+38)/210=90.0%]

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Analyzing Predictions - Failures False positive for true negative = %actual 0s predicted as 1s = 100N(a0,p1)/N(a0) % [100(1/152)=0.668%] False negative for true positive = %actual 1s predicted as 0s = 100N(a1,p0)/N(a1) % [100(20/258)=34.5%] False positive for predicted positive = % predicted 1s that were actual 0s = 100N(a0,p1)/N(p1) % [100(1/39)=2/56%] False negative for predicted negative = % predicted 0s that were actual 1s = 100N(a1,p0)/N(p0) % [100(20/171)=11.7%] False predictions = %actual 1s and 0s incorrectly predicted = 100[N(a0,p1)+N(a1,p0)]/N [100(1+20)/210=10.0%]

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Aggregate Prediction is a Useful Way to Assess the Importance of a Variable Frequencies of actual & predicted outcomes. Predicted outcome has maximum probability. Threshold value for predicting Y=1 =.5000 Predicted Actual 0 1 | Total | | Total | 210 Predicted Actual 0 1 | Total | | Total | 210 Model fit without TTMEModel fit with TTME

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