Econ 513, USC, Fall 2005Lecture 17. Discrete Response Models :Random Coefficient or Mixed Multinomial Logit ModelsLet us first recall some of the properties of the conditional logit. We consider a case with3 choices, dinner at Spargo, Watergrill, or McDonalds (y ∈ {S, W, M}). There is only onecharacteristic of the choice that matters, price. To make the comparisons simpler, let ussuppose that the prices for the first two are equal and much higher than for the other one,PS= PW>> PM. The coefficient on this characteristic in the utility function is β < 0.(We leave out the intercept in the utility function for simplicity. These would capture tastepreferences for the three restaurants.) So, the utilities for the three choices areUiS= β · PS+ iS,UiW= β · PW+ iW,andUiM= β · PM+ iM.The probability of dinner at Spargo isPr(yi= S) = Pr(UiS= max(UiS, UiW, UiM)) =exp(βPS)exp(βPS) + exp(βPW) + exp(βPM).It follows from the IIA (independence of irrelevant alternatives) property of the conditionallogit thatPr(yi= W |yi6= S) = Pr(UiW> UiM|UiS< max(UiS, UiW, UiM)) =exp(βPW)exp(βPW) + exp(βPM).It is also clear that the probability that UiW> UiMisPr(UiW> UiM) =exp(βPW)exp(βPW) + exp(βPM).Thus it follows thatPr(UiW> UiM|yi= S) = Pr(UiW> UiM|UiS= max(UiS, UiW, UiM)) =exp(βPW)exp(βPW) + exp(βPM).1So an implication of the IIA property is that the probability that the second choice isWatergrill given that the first choice is Spargo is the same as the conditional probabilitythat you choose Watergrill to begin with. Again very unappealing.So, suppose that Spargo raises its prices. Fewer people will go there, and they will goinstead to Watergrill and MacDonalds. The division of this subpopulation of people whowould have gone to Spargo over the two remaining choices is the same as in the p opulationas a whole.Now suppose that there are two types of people in terms of their price sensitivity. Wemodel this as a discrete mixture model to keep it tractable for the time being:βi∈ {β, β},with β < β < 0, and Pr(βi= β) = Pr(βi= β) = 1/2. People with βi= β are more pricesensitive, and thus less likely to go to Spargo and Watergrill than people with βi= β.Now in this mixture/random-coefficients model let us look at the probabilityPr(UiW> UiM|yi= S) (1)and compare this to the marginal probabilityPr(UiW> UiM).The latter isPr(UiW> UiM) = Pr(UiW> UiM|βi= β) · Pr(βi= β) + Pr(UiW> UiM|βi= β) · Pr(βi= β)=exp(βPW)exp(βPW) + exp(βPM)·12+exp(βPW)exp(βPW) + exp(βPM)·12.To study the probability in (1) it is useful to first consider the probabilityPr(βi= β|yi= S) =Pr(yi= S|βi= β) · Pr(βi= β)Pr(yi= S)=exp(βPS)exp(βPS)+exp(βPW)+exp(βPM)·12exp(βPS)exp(βPS)+exp(βPW)+exp(βPM)·12+exp(βPS)exp(βPS)+exp(βPW)+exp(βPM)·12.2With β < β < 0 and PW= PS> PM, it follows thatexp(βPS)exp(βPS) + exp(βPW) + exp(βPM)<exp(βPS)exp(βPS) + exp(βPW) + exp(βPM)and that Pr(βi= β|yi= S) is less than 1/2 = Pr(βi= β). Not surprisingly the probabilitythat someone choosing to eat at Spargo (conditioning on yi= S) is less likely to be a pricesensitive type (a type with βi= β) than a typical person.Now let us go back to the probabilityPr(UiW> UiM|yi= S).Conditional on βi= β we havePr(UiW> UiM|yi= S, βi= β) =exp(βPW)exp(βPW) + exp(βPM).ThusPr(UiW> UiM|yi= S)= Pr(UiW> UiM|yi= S, βi= β) · Pr(βi= β|yi= S)+Pr(UiW> UiM|yi= S, βi= β) · Pr(βi= β|yi= S)=exp(βPW)exp(βPW) + exp(βPM)· Pr(βi= β|yi= S)+exp(βPW)exp(βPW) + exp(βPM)· Pr(βi= β|yi= S)=12·exp(βPW)exp(βPW) + exp(βPM)+exp(βPW)exp(βPW) + exp(βPM)+Pr(βi= β|yi= S) −12·exp(βPW)exp(βPW) + exp(βPM)−exp(βPW)exp(βPW) + exp(βPM)3>12·exp(βPW)exp(βPW) + exp(βPM)+exp(βPW)exp(βPW) + exp(βPM).ThusPr(UiW> UiM|yi= S) > Pr(UiW> UiM).The probability that Watergrill is the second choice given that Spargo is the first choice ishigher than the marginal probability that Watergrill is preferred to MacDonalds. Anotherimplication is that increasing the price of a dinner at Spargo reduces the demand for dinnerat Spargo, with more of that reduction going to Watergrill than to MacDonalds comparedto their original shares. This is much more plausible than the IIA property.The implication of this argument is that allowing for heterogeneity in the coefficients canget us around the undesirable properties of the conditional logit model.Another way of thinking about this approach is to write βi= β0+ ηi. Then we can writethe utilities asUiS= β0· PS+ νiS,UiW= β0· PW+ νiW,andUiM= β0· PM+ νiM,where the three unobserved components (νiS, νiW, νiM) are no longer independent, but in-stead have the structureνiS= PS· ηi+ iS,νiW= PW· ηi+ iW,νiM= PM· ηi+ iM.Unobserved components are now positively correlated, with the strength of the correlationdepending on the closeness of the observed characteristics (the price in this case). At thesame time this correlation structure does not add a lot of additional parameters. In thiscase we can add just a single parameter, one for the variance of βi, and allow for correlations4between all unobserved components. This can be both a advantage and a disadvantage. Sucha structure cannot pick out any correlation structure that exists between the unobservedcomponents in the utility.More generally, we can model the utility for choice j asUij= β0ixij+ εij,with βia random coefficient. We can allow the mixture distribution to partly depend onindividual characteristics:βi= x0iγ + ηi,where the xiare individual specific covariates. Recall that the original conditional logitallowed those to enter additively into the utility function. Now they are allowed to affect boththe slope and the intercept of the utility function. Typically researchers make parametricassumptions on the ηi, e.g., multivariate normal, or a discrete distribution with only a coupleof points of support. Let us denote the parameters of ηiby Ω. Obviously we do not haveto allow all parameters to vary accross individuals. In practice we may only want to do thisfor the most important covariates such as prices.Estimation is difficult for these models. Simply attempting to write down the likelihoodfunction and maximize will only work if there are few parameters. For example, if the ηiare multivariate normal, with dimension K, evaluating the likelihood function will involvesolving a K-dimensional integral. That is computationally difficult. Accurate