Estimating Sequential Search Models using MPEC

The dynamic programming approach of Weitzman (1979) has recently been used in several papers estimating consumer sequential search (Kim et al., 2010; Ghose et al. 2012; Honka and Chintagunta, 2014; Chen and Yao, 2014). The popularity of this method owes to its ability to exactly characterize the sequence of choices made by consumers. However, in order to characterize this sequence, knowledge of reservation values is required. Reservation values are values that make consumers indifferent between searching and stopping at each stage in their search process and are solutions to structural equations relating consumer utilities and their search costs. Inverting these equations to solve for reservation values for each draw of the structural parameters makes estimation of the model using nested fixed point (NFXP) methods computationally challenging. In this paper, we follow Su and Judd (2012) and show how to recast the optimal sequential search problem as a mathematical program with equilibrium constraints (MPEC). Instead of computing reservation values for each draw, this method augments the likelihood function with these reservation values and imposes the structural equation defining reservation values as a constraint. The main advantage of this method is that it only requires checking this equation once, without solving it, thereby significantly reducing computational time. Method (preliminary) In the optimal sequential search model of Weitzman (1979), Pandora faces a set of closed boxes and must decide whether or not to continue searching, and if so, which box to open next. By opening a box i, Pandora observes a reward ui ∼ Fi(ui) and must pay a search cost ci. Let Pandora’s preferences u(θ) and search costs c(θ) be parameterizes by the vector θ. A fallback ∗University of Chicago, Booth School of Business, [email protected]. †University of Chicago, [email protected].