Choice model simulation

Although the concept choice models is widely understood and practiced these days, it is often difficult to acquire hands-on knowledge in simulating choice models. While many stat packages provide useful tools to simulate, researchers attempting to test and simulate new choice models with data often encounter problems from as simple as scaling parameter to misspecification. This article goes beyond simply defining discrete choice models. Rather, it aims at providing a comprehensive overview of how to simulate such models in computer.

Although the concept choice models is widely understood and practiced these days, it is often difficult to acquire hands-on knowledge in simulating choice models. While many stat packages provide useful tools to simulate, researchers attempting to test and simulate new choice models with data often encounter problems from as simple as scaling parameter to misspecification. This article goes beyond simply defining discrete choice models. Rather, it aims at providing a comprehensive overview of how to simulate such models in computer. U_i =\alpha P_i + \beta D_i + \varepsilon_i\, (1) U_a =\alpha P_a + \beta D_a + \varepsilon_a\, (2) where \varepsilon captures unobserved variables that affect consumer utilities. P_i & = \Pr(U_i > U_a) P_i = \Pr( \varepsilon_i - \varepsilon_a > V_a - V_i ) (3) In the end, discrete choice modeling comes down to specifying the distribution of \varepsilon (or \varepsilon_i - \varepsilon_a ) and solving the integral over the range of \varepsilon to calculate P_i . Extending this to more general situations with N consumers (n = 1, 2, ..., N), J choices of consumption (j = 1, 2, ... , J), The choice probability of consumer n choosing j can be written as P_{nj} = \Pr( U_{nj} > U_{ni} ) (4) for all i other than j U_i = C_i + \alpha P_i + \beta D_i + \varepsilon_i\, U_a = C_a + \alpha P_a + \beta D_a + \varepsilon_a\, Then, depending on the value of the estimated alternative-specific constants, the choice probability may change. Also, if we write the choice probability in the format of (3), P_i = \Pr( \varepsilon_i - \varepsilon_a > (C_a - C_i) + \alpha P_a - \alpha P_i + \beta D_a - \beta D_i ) only the difference between C_a and C_i affects the choice probability (i.e. our estimation can only identify the difference). So it's convenient to normalize all the alternative-specific constants to one of the alternatives. If we normalize to C_i , then we estimate the following model: U_i = \alpha P_i + \beta D_i + \varepsilon_i\, U_a = (C_a - C_i ) + \alpha P_a + \beta D_a + \varepsilon_a\, When there are more than 2 choices in the choice set, we can pick any choice i and normalize the alternative-specific constants to that choice by subtracting C_i from all other alternative-specific constants. 3. Sociodemographic variables In deciding between the Irish pub and the American pub, if the researcher has access to additional sociodemographic variables such as income, they can enter the consumer utility equation in various ways. Denote the student's income as Y. If the researcher believes that the income affects the utility linearly, then U_i = \alpha P_i + \beta D_i + \gamma Y +\varepsilon_i\, If the researcher believes that the sociodemographic variable interacts with other variable such as price, then the utility can be written as U_i = \alpha P_i/Y + \beta D_i +\varepsilon_i\, Logistics function has a closed form solution => No simulation necessary. 2. GEV (Generalized extreme value distribution) Allows correlation in unobserved factors across alternatives. iid extreme value unobserved factors The cumulative distribution of difference in extreme values is Logistics function Logistics function has a closed form solution => No simulation necessary. 3. Probit Unobserved factors have a jointly normal distribution. No closed form for the cumulative distribution of normal distribution. Simulation necessary. 4. Mixed logit Allows any distribution in unobserved factors No closed form for the cumulative distribution of normal distribution. Simulation necessary. Colombino, U. (2010). Equilibrium policy simulations with random utility models of labour supply, Carlo Alberto Notebooks 156, Collegio Carlo Alberto. Kenneth E. Train, " Discrete Choice Methods with Simulation", Massachusetts: Cambridge University Press, 2003.

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