A Dynamic Model of Housing Supply

5.3 Estimation - Dynamic Discrete ChoiceUsing the results of the first two stages, the flow profits, π can then be expressed as:π 1nt = ÊP nt − ̂V C nt − δ j − δ t + ɛ nt (14)where ÊP nt is the first-stage estimate of expected prices, ̂V C nt is the second-stage estimateof variable costs and δ j and δ t are the fixed costs parameters that capture the remaining costenvironment. The remaining structural parameters to be estimated are θ d = (σ ɛ , δ j , δ t , β). Oneestimation approach would be to use an estimator similar to Rust (1987), where the valuefunctions are computed by a fixed point iteration for each guess of the parameters to be estimated.Such an approach, while efficient, would be computationally prohibitive in the context of thismodel. Therefore, I use a computationally more simple two-step estimation approach. Tosimplify the problem, I use insights from Hotz and Miller (1993) and Arcidiacono and Miller(2008) to take advantage of the terminal state nature of the dynamic discrete choice problem. 41I can rewrite (9) as:v 0 (Ω) = σ ɛ β( ∫ (log[1 + ev 0 (Ω ′ )/σ ɛ−π 1 (x ′ )/σ ɛ] + π 1 (x ′ )/σ ɛ)q(Ω ′ |Ω, 0)dΩ ′) (15)Each agent chooses the option which yields the higher expected lifetime utility. Given thedistributional assumptions on the error terms, the conditional choice probabilities (CCPs) ofchoosing to not build or to build are given by:P rob(d n = 0|Ω) =P rob(d n = 1|Ω) =e v 0(Ω)/σ ɛ−π 1 (x)/σ ɛ1 + e v (16)0(Ω)/σ ɛ−π 1 (x)/σ ɛ11 + e v 0(Ω)/σ ɛ−π 1 (x)/σ ɛLetting P 1 (Ω) denote the probability that d n = 1 conditional on Ω, I can rewrite (15) as theexpected future per-period profit of choosing to not build and a function of the next period41 The estimation of dynamic discrete choice models with two-step estimators has become increasing common.For examples of other two-step approaches, see Aguirregabiria and Mira (2002), Pakes, Ostrovsky, and Berry(2007), and Bajari, Benkard, and Levin (2007).26