Goals. PSCI6000 Maximum Likelihood Estimation Multiple Response Model 2. Recap: MNL. Recap: MNL

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1 Goals PSCI6000 Maximum Likelihood Estimation Multiple Response Model 2 Tetsuya Matsubayashi University of North Texas November 9, 2010 Learn multiple responses models that do not require the assumption of IIA Nested logit model and Generalized extreme value (GEV) models Multinomial probit model Mixed logit model 1 / 35 2 / 35 Recap: MNL Recap: MNL Suppose that individual i chooses one of J alternatives. The probability of choice m is P(y i = m) = Pr(U im > U in ) = Pr(ɛ in ɛ im < V im V in ) n m Random components are independently and identically distributed (IID). Random components are distributed according to type I extreme value. The difference between two extreme value variables is distributed logistic. 3 / 35 4 / 35

2 Recap: MNL Recap: MNL The choice probability is: P im = (ɛ in ɛ im < V im V in )f (ɛ im )dɛ im Some algebraic manipulation of this integral results in a succinct, closed form expression: P im = Pr(y i = m) = = which is the logit choice probability. e V im J J=1 ev ij e x i β m J J=1 ex i β J In the multinomial logit model, the equation for the odds of m versus n is P(y i = m) P(y i = n) = exi βm e x i β n = evim e V in This equation indicates that the odds are determined without reference to the other outcomes that might be available. This property is called as the independence of irrelevant alternatives or IIA. This is a consequence of assuming independence of ɛ ij in the random utility model. If the IIA assumption is violated, it means correlation between the random component terms. 5 / 35 6 / 35 Recap: MNL Limitations of MNL and CL Models The IIA is violated when alternatives share unobserved attributes that influence choice. These unobserved attributes cause correlation in the random components of the utilities across alternatives. The IIA assumption can be unrealistic in many settings. If violated, estimates will be misleading. In addition, the IIA assumption results in the following limitations. Taste variation Panel data analysis 7 / 35 8 / 35

3 Taste Variation Panel Data The MNL and CL models can represent systematic taste variation (that is, taste variation that relates to observed characteristics of the decision maker), but not random taste variation (differences in tastes that cannot be linked to observed characteristics.) Two people who have the same issue positions may make different choices because they place different weights on the issue positions of parties or candidates. This also leads to correlations across the utility of alternatives. If unobserved factors are independent over time in repeated choice situations, then the MNL or CL models can capture the dynamics of repeated choice. However, the MNL or CL models cannot handle situations where unobserved factors are correlated over time. Unobservable traits are, for example, correlated over time. 9 / / 35 Generalized Extreme Value Models Generalized extreme value (GEV) models constitute a large class of models that substitute the MNL or CL models when the IIA does not hold. Generalized extreme value distribution allows for correlations over alternatives and is a generalization of the type 1 extreme value distribution that is used for the MNL and CL models. GEV models relax restrictive substitution patterns due to the IIA property, but cannot handle random taste variation and are nor applicable to panel data with temporarily correlated errors. The most widely used member of the GEV family is called a nested logit model. A nested logit model is appropriate when the set of alternatives faced by a decision maker can be partitioned into subsets, called nests in such a way that the following properties hold: 1 For any two alternatives that are in the same nest, the ratio of probabilities is independent of the attributes or existence of all other alternatives. IIA holds within each nest. 2 For any two alternatives in different nests, the ratio of probabilities can depend on the attributes of other alternatives in the two nests. IIA does not hold in general for alternatives in different nests. 11 / / 35

4 Suppose the following set of alternatives is available to a voter in a national election. Suppose that A and B are in the same nest, while C and D are in the same nest. New Probabilities With Alternative Removed Alt. Orig. A B C D A.40.45(+12.5%).52 (+30%).48 (+20%) B (+100%).13 (+30%).12 (+20%) C (+60%).33(+10%).40 (+33%) D (+60%).22(+10%).35 (+70%) IIA holds between A and B and between C and D. We can illustrate the substitution pattern using a tree diagram. Liberal Conservative A B C D Each branch denotes a subset of alternatives within which IIA holds. IIA holds within each nest but not across nests. 13 / / 35 Nested Logit is appropriate when we think that we have groups of alternatives that are similar to each other in an unobserved way. Suppose a two-level nesting structure. Let the probability of choosing alternative m B k be expressed as the product of two probabilities, namely, the probability that an alternative within nest B k is chosen and the probability that the alternative m is chosen given that an alternative in B k is chosen: P im = P im Bk P ibk where P im Bk is the conditional probability of choosing alternative m given that an alternative in nest B k is chosen, and P ibk is the marginal probability of choosing an alternative in nest B k. 15 / / 35

5 The marginal and conditional probabilities can be expressed as: P ibk = P im Bk = e z ikα+λ k I ik k l=1 ez il α+λ l I il e x imβ J j=1 ex ij β where I ik is the inclusive value for category k and defined as: I ik = ln j B k e xij β The probability of choosing B k and then m is P im = ey ikα+λ k I ik k ex imβ l=1 ey il α+λ l I il J j=1 ex ij β The inclusive value for alternative (I k ) captures the expected value for the choices available in the second level if B k is chosen in the first level. 17 / / 35 Estimation and Interpretation The parameter λ k is called a dissimilarity parameter. It corresponds to the degree of dissimilarity between the alternatives within one nest. A higher value of λ k means greater independence and less correlation between alternatives in different nests. Thus, the MNL and CL models follow in the special case where λ K = 1 in all nests. Use nlogit in Stata. See Paolino s application of the nested logit model. You can interpret the results in the same way as the MNL and CL models. 19 / / 35

6 Heteroskedastic GEV Model Multinomial Probit Model We have assumed that the error variance is constant across observations for all logit models. But the error terms in the choice model are heteroskedastic for some applications. You can relax the assumption of homoskedasticity using a heteroskedastic GEV model. See Zeng (2000) for more information. A multinomial probit model resembles the MNL or CL models but replaces the Extreme Value Type I Distribution with a multivariate normal distribution for the utility differences. The multivariate normal distribution allows for a covariance matrix of the dimensions, so in principle this permits each dimension of utility to covary with other choices. MNP obviates the limitations of the MNL and CL models by allowing for random taste variation, unrestricted substitution patterns, and correlation in unobserved factors over time. 21 / / 35 Multinomial Probit Model MNP We begin with the utility function as follows: One limitation in MNP is its reliance on normal distribution. In some situations, normal distribution provides an inadequate representation of the random components. The other limitation is its difficulty in getting the model to converge with many choices because of the computational burden. U im = V im + ɛ im = x i β m + z im γ + ɛ im where x i denotes individual-specific variables and z im denotes choice specific variables. We assume that the disturbance terms have a multivariate normal distribution with a mean vector zero and an arbitrary covariance matrix Σ to be estimated: ɛ im MVN(0, Σ) 23 / / 35

7 MNP MNP If you have three choices, the covariance matrix of the disturbances ɛ i1, ɛ i2, ɛ i3 would be: σ 2 i11 Σ i = σ i12 σi22 2 σ i13 σ i23 σi33 2 In the three-choice case, the choice probability that individual i chooses alternative 1 would be the probability that the utility from alternative 1 exceeds the utilities from alternatives 2 and 3. P(y i = 1) = P[(U i1 > U i2 ) and (U i1 > U i3 )] = P[(V i1 + ɛ i1 > V i2 + ɛ i2 ) and (V i1 + ɛ i1 > V i3 + ɛ i3 )] = P[(ɛ i2 ɛ i1 < V i1 V i2 ) and (ɛ i3 ɛ i1 < V i1 V i3 )] 25 / / 35 MNP MNP Let η i,21 = ɛ i2 ɛ i1 η i,31 = ɛ i3 ɛ i1 Since the difference between two normals is normal, the density of the error difference is also normal with covariance matrix: [ Ω i = σ 2 i,1 + σ2 i,2 2σ i,12 σ 2 i,1 σ i,13 σ i,12 + σ i,23 σ 2 i,1 + σ2 i,3 2σ i,13 ] This allows us to write the probability that voter i will choose alternative 1 in the example with 3 alternatives as: P(y i = 1) = where B i,12 = Bi,12 Bi,13 V i1 V i2 σ 2 i,1 +σ2 i,2 2σ i,12 b 1 (η i,21, η i,31 ; r 1 )dη i,21 dη i,31 and B i,13 = V i1 V i3, b 1 σi,1 2 +σ2 i,3 2σ i,13 denotes the standardized bivariate normal distribution and r 1 denotes the correlation between η i,21 and η i,31. This example can be generalized to more choices. 27 / / 35

8 Estimation Mixed Logit Model MNP is hard to estimate. As shown, the specification requires evaluation of a multivariate normal integral over each of the dimensions of choice. You can use MNP library in R, which relies on Bayesian data augumentation methods, or mprobit that relies on the ML method in Stata. A mixed logit model also obviates the limitations of the MNL and CL models by allowing for random taste variation, unrestricted substitution patterns, and correlation in unobserved factors over time. Unlike MNP, it is not restricted to normal distribution. 29 / / 35 MXL MXL We begin with the utility function as follows: U im = V im + e im = x im β + z im η i + ɛ im where V im = x im β and e im = z im η i + ɛ im. x im and z im are individual and choice specific attributes. x im and z im can have some or all of the same elements in common. ɛ im is IID extreme value. η i is a vector of random terms with mean zero and follows a general distribution, g(η Ω). By considering the impact of z im on the utility, we can specify models that take into account the impacts of unobserved attributes and random taste variation, and avoid the IIA assumption. If IIA holds, η i = 0 for all i, so U im depends only on the systematic portion of utility and an IID stochastic portion of utility. 31 / / 35

9 MXL MXL If the elements of z are also contained in x, this is a random-coefficients model. In this model, the vector β gives the mean values from the random coefficients, while Ω gives the other parameters of the distribution of the random coefficients such as the variance. The random-coefficients model allows for the study of heterogeneity in the impact of independent variables on the dependent variable. If the elements of z are not contained in x, this is an error-component model. The choice probability will depend on β and η. The probability that individual i chooses alternative j is P(y i = m η) = e x imβ+z im η i k J ex ikβ k +z ik η i η is unobserved and drawn from a known joint density function g. To obtain the unconditional choice probability for each individual, the logit probability must get integrated over all values of η weighted by the density of η. [ e x imβ+z im η i ] P(y i = m) = g(η Ω)dη k J ex ikβ k +z ik η i η 33 / / 35 Estimation Simulation techniques are usually applied for estimating MXL models. Use lmer in R or mixlogit in Stata. See Glasgow (2001) as an application. 35 / 35