Simple estimators for the parameters of discrete dynamic games (with entry/exit examples).

We estimate parameters from data on discrete dynamic games, using entry/exit games to illustrate. Semiparametric first-stage estimates of entry and continuation values are computed from sample averages of the realized continuation values of entrants and incumbents. Under certain assumptions, these values are easy-to-compute analytic functions of the parameters of interest. The entry and continuation values are used to determine the model's predictions for entry and exit conditional on the parameter vector, and the estimates compare these predictions with the data on entry and exit rates. Small-sample properties are discussed and lead to the simplest of estimators.

1. Introduction

* This paper uses the structure of dynamic games to develop estimators for their parameters. We concentrate on games with discrete controls and, for ease of exposition, provide our results in the context of a dynamic game of entry and exit. In addition to its importance to industrial organization, the entry/exit example illustrates rather well just why we need these estimation strategies and the major problems that arise in developing them.

Though the costs of entry and the sell-off values (or costs) associated with exit are key determinants of the dynamics of market adjustments, data on these "sunk costs" are much harder to find than data on the determinants of current profits. As a result, we often have to infer the extent of sunk costs from other variables whose behavior depends on them. The variable that is most directly related to the costs of entry is entry itself. To use the connection between actual entry and the costs of entry in estimation, we need to be able to compute the value of entering. Similarly, to make use of the relationship between sell-off values and exit, we need to be able to calculate the value of continuing. Though algorithms for computing these values have been available for some time (e.g., Pakes and McGuire, 1994), they cannot be used in estimation without encountering substantial computational problems.

Consequently, the entry/exit models that have been taken to data have all been two-period models that assume away sunk costs; see Bresnahan and Reiss (1987, 1991), Berry (1992) and, more recently, Mazzeo (2002) and Seim (2005). The lack of estimates of sunk costs induced these papers to focus on characterizing differences in the number of active firms across a cross-section of markets rather than on the likely impacts of policy or environmental changes on the structure of an industry (e.g., the impact of mergers on entry or of pension and/or health-care provisions on exit).

The early entry/exit papers did explicitly consider the estimation issues that arise when the model used to structure the data does not generate a unique equilibrium. The uniqueness issue had been emphasized in the theoretical literature on entry, and both Bresnahan and Reiss (1991) and Berry (1992) considered its impact on estimation in models where fixed costs could vary among agents. When models do not have a unique solution, it is generally not possible to determine the probability of a given outcome conditional on observables and the value of a parameter vector. This rules out many standard estimators. The uniqueness issue became even more important once we allowed for the realism of continuation values that differed across agents, for then the number of possible equilibria increased markedly. The original analysis here, due to Mazzeo (2002) and Seim (2005), allowed continuation values to differ with "location" and began investigating extensions that are crucial to the study of many retail and service sectors.

Our goal here is to make the transition from the two-period setting to truly dynamic models of entry and exit. To do so, we will provide a set of assumptions under which there is only one set of equilibrium policies consistent with the data-generating process. We will then show how some simple ideas (similar to those in Muth, 1961), can be used to deliver estimators that are both easy to compute and grounded in what actually happened.

[] The underlying idea. To determine whether a potential entrant (an incumbent) should enter (continue), we need the expected discounted value of future net cash flows should the firm enter (continue). The potential entrant (incumbent) will enter (continue) if this entry value (continuation value) is greater than the entry fee (the sell-off value). Our measure of the entry values from a particular state is an average of the discounted value of net cash flows actually earned by entrants who did enter at that state. Similarly, our measure of the continuation values from that state is the actual discounted value of net cash flows earned by incumbents who did continue from that state. These measures of entry and continuation values make the relationship between the model and the data transparent, which, together with the estimator's computational ease, simplifies robustness analysis greatly.

We construct the probability of entry conditional on the parameters of the entry-cost distribution as the probability of an entrant drawing an entry fee less than the estimated entry value. Similarly, the probability of an incumbent exiting conditional on the parameters of the sell-off distribution is the probability of drawing a sell-off value greater than the estimated continuation value. The parameters of these distributions are estimated to make the model's predictions for entry and exit rates "as close as possible" to the rates observed in the data. Alternative metrics for closeness produce alternative (root-n) consistent and asymptotically normal estimators, and we provide an extensive discussion of the differences in their computational and statistical properties.

All estimators are semiparametric. There is a first stage that provides a nonparametric estimate of the entry and continuation values and a second stage that treats these estimates as true values in a parametric estimation routine. We provide assumptions under which the first stage need only be done once. That is, the estimation algorithm does not need to compute a complicated fixed point or matrix inverse each time it evaluates its objective function. As a result, the computational burden of our estimator is comparable with that of the estimators for the simple static entry models.

The paper begins with the simplest entry/exit model, a model with one entry location and a fixed number of potential entrants in every period. We then show how to generalize to allow for multiple entry locations and a random number of potential entrants. Once conceptual issues are clarified, a number of alternative estimators suggest themselves. The alternatives have different computational and distributional properties, and so we provide fairly detailed Monte Carlo results. Finally, we investigate the robustness of our estimators to the presence of serially correlated unobservables.

The Monte Carlo results, when combined with a discussion of why they occur, end up being quite informative. Among the alternatives we consider, only one, perhaps two, should be used, and the best performing alternatives are also the least computationally burdensome. The computational burden of these estimators is small enough to think that the effective limitation to estimation will be the richness of the data rather than computational feasibility. Also, provided independent measures of profits are available, there is a sense in which our estimators are robust to the presence of serially correlated unobservables.

[] Limitations of the analysis. First, a conceptual point. We want to stress that, though our assumptions are sufficient to use the data to pick out the equilibrium that was played in the past, they do not allow us to pick out the equilibrium that would follow the introduction of a new policy. On the other hand, the sunk-costs estimates should give the researcher the ability to examine what could happen after a policy change (say, by examining all possible post-policy-change equilibria) and would seem to be a necessary ingredient of any more detailed analysis of post-policy-change behavior (see Pakes, 2005).

Second, we ignore identification issues. Many of the parameters determining behavior in dynamic games can often be estimated without ever computing continuation or entry values. So though we could try and use the entry/exit analysis to estimate all of the underlying primitives, the substantive identification issues are likely to vary with the problem of interest. In particular, measures of variable profits are often available from reported sales and costs information or can be derived from a Nash equilibrium assumption and estimates of demand and cost functions. These estimates are likely to be more reliable than profit estimates obtained using only entry/exit data, and their availability enables more robust estimation strategies for dynamic parameters (see below).

In contrast, it is typically quite difficult to obtain direct measures of sell-off values, as these are determined by such poorly measured objects as "goodwill" the value of the firm's buildings and equipment in their second-best employment and clean-up costs. Similarly, measures of entry costs require estimates of the costs of formulating ideas, testing markets, and accessing both start-up capital and the requisite permissions. As a result, we have focused our discussion on estimation of the parameters of the sunk-cost distributions.