A Dynamic Model of Housing Supply

5 EstimationThere are three outcomes associated with the model. The first two are choices made by theparcel owner: the binary decision to build or not in each period, and the housing service provisiondecision made conditional on building. The final outcome is the sales price of all properties thatsell.While all three outcomes are directly related to the full profit function, examining eachoutcome separately helps illustrate what are the sources of identification used to estimate theparameters of the full model. Much can learned by estimating the price outcome in isolation.However, for the housing service and the build/don’t build outcomes, deeper inference can onlybe obtained by also using information from the other outcomes.The price outcomes reveal trends in prices, both across tracts and through time. It isstraightforward what variation in the data identifies the parameters reflecting which tracts arehigh price tracts, or what are the broad time trends or prices. However, the data is also richenough to identify more subtle patterns, such as how the different components of housing arevalued. For example, controlling for lot-size and location, if big houses appreciating at slowerrates than smaller houses, it suggests that the value of land is increasing at a faster rate thanthe value of living space.The raw data reveal how much housing services (or square footage) each parcel owner choseto provide, if building occurred. While this data alone can be used to document cross-sectionaland time series trends in square footage, more can be learned by combining this data with theprice parameter estimates; with data on observed square footage, knowing the marginal benefitof adding a unit of square foot identifies marginal (and variable) costs. Similarly, on their own,the data on when and where building occurred only reveals broad cross-sectional and time-seriespatterns. However, if prices and variable costs are also known, the construction data can thenbe used to estimate fixed costs – controlling for prices and variable costs, different constructionrates imply different fixed costs. These features of the model suggest the appropriateness of amulti-stage estimation strategy, which is more formally outlined below.Let θ p denote (ρ, γ 1 , γ 2 , σ ν ), θ h denote (α 0 , α 1 ), and θ d denote (σ ɛ , δ j , δ t , β). As the errorsacross each of the outcomes are assumed independent, the log-likelihood function can be broken22