A short introduc-on to discrete choice models

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1 A short introduc-on to discrete choice models

2 BART Kenneth Train, Discrete Choice Models with Simula-on, Chapter 3.

3 Ques-ons Impact of cost, commu-ng -me, walk -me, transfer -me, number of transfers, distance to first bus, family income, number of drivers in hh, employment density at work loca-on, home loca-on, autos per driver on choice of transporta(on mode? Can we build a model that predicts transporta-on mode choice?

4 OLS is not very useful here The choices are not cardinal. The choices are unordered. Car = 2, Bus = 1, Walking = 0?? The OLS coefficients of such a regression would have no meaning. Cardinal: test score, income, wealth, return. Ordinal: happiness, sa-sfac-on, job market outcomes (for the nerdy). Unordered: transporta-on mode, product choice, loca-on choice.

5 Outline 1. Logit models for mul-ple choice 1. Condi-onal Logit 2. Mul-nomial Logit 2. Independence from irrelevant alterna-ves 3. Es-ma-on Reference: Greene, Chapter Models for Discrete Choice, Sec-on Logit models for mul4ple choice.

6 CONDITIONAL LOGIT

7 Consumer choice U ij = z ij β + ε ij. Consumer i faced with J choices (indexed by j). Linear rela-onship between u-lity and characteris-cs of individual and characteris-cs of the choice. z ij is indexed both by i and j. We assume that eij is extreme value distributed (with fixed variance) so that the difference of two ε ij follows a logit distribu-on. F(ε ij ) = exp( e ε ij ), The βs are unknown and to be es-mated.

Consumer choice (c ted) Prob(U ij > U ik ) for all other k j. Then j is chosen. The consumer chooses the op-on that maximizes his u-lity. Note it is modelled as a sta-c choice.

8 Consumer choice (c ted) Prob(U ij > U ik ) for all other k j. Then j is chosen. The consumer chooses the op-on that maximizes his u-lity. Note it is modelled as a sta-c choice. Because there is unobserved heterogeneity ε ij in the u-lity, two consumers with the same characteris-cs may choose different transporta-on modes. Also the choices are mutually exclusive.

9 Consumer choice (c ted) Prob(Y i = j) = e z ij β J j=1 ez ij β, This is McFadden s simple yet beau-ful formaliza-on of consumer choice. The probability of choosing j over other alterna-ves depends on the observable u-lity derived from the choice j. No-ce that consumer- specific characteris-cs w i (not interacted with choice- specific characteris-cs) do not enter in the probability. Prob(Y i = j) = e β x ij +α w i J j=1 eβ x ij +α w i = e β x ij e α i w i J j=1 eβ x ije α i w i. ot vary across alternatives that is, those specific to the Rule therefore: consumer characteris-cs impact can be es-mated if interacted with choice characteris-cs.

10 Interpreta-on of the coefficients β Prob(Y i = j z i1, z i2,...,z ij ) = e β z ij J j=1 eβ z ij. P j x k = [P j (1( j = k) P k )]β, k = 1,...,J. Marginal effect depends on all probabili-es of choice, as well as the coefficient. The coefficient and the marginal effect don t have the same sign. Changes in the characteris-cs of one choice affect all the choices probabili-es.

11 MULTINOMIAL LOGIT

12 Ques-on Data is only individual specific, i.e. we have informa-on on the individual, not on the choices. 4 occupa-onal choices: 0 = menial, 1= blue collar, 2= crac, 3= white collar, 4 = professional. (J+1 choices here). Choices are not cardinal, and are not ordered. Choice of occupa-on. Prob(Y i = j) = e β j x i 4 k=0 eβ k x i, j = 0, 1,...,4. t of Sections 21.3 and 21.4 is conveniently prod

Indeterminacy As for logit (remember, remember?) we can only iden-fy the effect of xij on the u-lity of a choice rela-ve to another choice. Why? Set β j * = β j + q.

13 Indeterminacy As for logit (remember, remember?) we can only iden-fy the effect of xij on the u-lity of a choice rela-ve to another choice. Why? Set β j * = β j + q. The choice probabili-es are not affected. Which choice? Set β 0 = 0 for instance. The reference choice magers for the interpreta-on of the es-mated coefficients. ln [ Pij P ik ] = x i (β j β k ) = x i β j if k = 0.

14 Interpreta-on: quite tricky Coefficient β j measures the impact of the consumer characteris-c on the rela-ve valua-on of the choice. Rela-ve to choice 0. Marginal effects are useful. An increase in educa-on will increase the probability of choosing occupa-on j by: δ j = P j x i = P j [ β j ] J P k β k = P j [β j β]. k=0

15 INDEPENDENCE FROM IRRELEVANT ALTERNATIVES

Remark The probability of choosing j versus k does not depend on the characteris-cs of choice j. The ra-o P j /P k is independent of the characteris-cs z ij.

16 Remark The probability of choosing j versus k does not depend on the characteris-cs of choice j. The ra-o P j /P k is independent of the characteris-cs z ij. True for the mul-nomial logit: And for the condi-onal logit: P ij P ik = ezij 'β e z ik 'β ompute J log-odds ratios [ ] Pij ln = x i (β j β k ) P ik ew of estimation, it is use

Red bus, Blue bus Traveler has choice of going by car or using a blue bus. Assume for simplicity that there are equal probabili-es of choosing the car over the blue bus P c /P bb = 1.

17 Red bus, Blue bus Traveler has choice of going by car or using a blue bus. Assume for simplicity that there are equal probabili-es of choosing the car over the blue bus P c /P bb = 1. A red bus is introduced, and it is considered iden-cal to the blue bus. Well then P rb /P bb = 1. But the IIA assump-on implies that P c /P bb should not change when the red bus is introduced, so the probability of taking the car goes down to 1/3!!! What about a green bus? Father Amine says: IIA assump-on is not very realis-c if the other choices may be good subs-tutes for the 2 choices.

18 Test for irrelevant alterna-ves Remove one choice (e.g. the red bus) and rees-mate the model. Does it affect the es-mates? χ 2 = ( ˆβ s ˆβ f ) [ ˆV s ˆV f ] 1 ( ˆβ s ˆβ f ), Surprising ( ) : β s, es-mate based on the restricted subset, β f, es-mated based on the full subset, and V s V f the respec-ve covariance matrix. If the removed choice is indeed irrelevant, then bs is more efficient. Converges in distribu-on to a X2 distribu-on with one degree of freedom. Can also test for the removal of K choices (then K degrees of freedom).

19 EXAMPLE: CHOICE OF TRAVEL MODE

20 Ques-on What is the impact of consumers characteris-cs on choice of travel mode?

21 Data TABLE Summary Statistics for Travel Mode Choice Data Number True GC TTME INVC INVT HINC Choosing p prop. Air Train Bus Car Note: The upper figure is the average for all 210 observations. The lower figure is the mean for the observations that made that choice. GC, generalized cost of travel, INVC, wage -mes INVT, TTME terminal -me, INVT -me spent traveling, HINC household income.

22 The model specified is Es-ma-on U ij = α air d i,air + α train d i,train + α bus d i,bus + β G GC ij + β T TTME ij + γ H d i,air HINC i + ε ij. where for each j has the same independent, type 1 extreme value distribution, F ε (ε ij ) = exp( exp( ε ij )) TABLE Parameter Estimates (t Values in Parentheses) Unweighted Sample Choice Based Weighting Estimate t Ratio Estimate t Ratio β G β T γ H α air α train α bus Log likelihood at β = Log likelihood (sample shares) Log likelihood at convergence Dummies for transporta-on choice, and three variables.

23 Predic-ons TABLE Predicted Choices Based on Model Probabilities (Predictions Based on Choice Based Sampling are in Parentheses.) Air Train Bus Car Total (Actual) Air 32 (30) 8 (3) 5 (3) 13 (23) 58 Train 7 (3) 37 (30) 5 (3) 14 (27) 63 Bus 3 (1) 5 (2) 15 (4) 6 (12) 30 Car 16 (5) 13 (5) 6 (3) 25 (45) 59 Total (Predicted) 58 (39) 63 (40) 30 (23) 59 (108) 210 Using: iven in Table m jk = 210 i=1 ˆp ijd ik,

24 Test of the IIA assump-on TABLE Results for IIA Test Full Choice Set Restricted Choice Set β G β T α train α bus β G β T α train α bus Estimate Estimated Asymptotic Covariance Matrix Estimated Asymptotic Covariance Matrix β G 0.194e β T 0.46e α train α bus Note: 0.nnne-p indicates times 10 to the negative p power. H = Critical chi-squared[4] = Excluding choice air to see impact on train/bus and car/bus. IIA assump-on is rejected. What now??

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