11 The Entry-Exit Model
There are a number steps we have to cover before we can talk about identification of this dynamic game with any formality. Namely, we need to cover:
- Identification of discrete choice models.
- Identification of dynamic discrete choice moels (Magnac and Thesmar 2002; Kasahara and Shimotsu 2009)
- Non-parametric instrumental variables (Chernozhukov and Hansen 2005; Chesher 2010)
Instead, the sections below loosely sketch out arguments and complications that arise for this class of models.
11.1 Identification without Unobserved Heterogneity
When there is no persistent unobserved heterogeneity, identification revolves around the fact the probability of entry: \[ p(x,a,a')\] is directly observable and holds the key for identification.
One can show Magnac and Thesmar (2002) that, fixing the discount factor \(\beta\), the transition density, and the distribution of taste shocks, there is a one-to-one mapping between choice probabilities and each \[\tilde{u}(x,a,a') = u_1(x,a,a') - u_0(x,a,a') \] Recall the specification of payoffs in the model: \[ u_{1}(x,a,d^{\prime}) = \phi_{0} + \phi_{1}x - \phi_{2}d^\prime - \phi_{3}(1-a) \] \[{u}_{0}(x,a) = \phi_{4}a \] So fixing \(x\), define: \[ \tilde{u}(x) = [\tilde{u}(x,0,0),\ \tilde{u}(x,1,0),\ \tilde{u}(x,0,1),\ \tilde{u}(x,1,1)]^\prime \] We know that the four terms \((\phi_0+\phi_1x,\phi_2,\phi_3,\phi_4)\) uniquely determine the four-dimensional vector \(\tilde{u}(x)\) and are hence identified. By then observing how choice probabilities vary with \(x\), we identiy \(\phi_0\) and \(\phi_1\) separately.
11.2 Identification with Unobserved Heterogeneity
Suppose that there is a persistent unobserved factor that shifts the returns to entry in each market: \[u_{1,m}(x,a,a') = u_1(x,a,a') + \alpha_{m} \]
Two comments are necessary here:
- We might be worried that \(\alpha\) is correlated with the market characteristic \(X\), which would lead to bias in the estimated parameters if we fail to model and account for this relationship.
- Note that even if \(\alpha\) is independent of \(X\), there is still a problem for identification because of dynamic selection. The incumbency variables \((A,A')\) are enodgenous to this factor \(\alpha\) and hence are mechanically correlated inside the model. A simple instrument that shifts profits from entry would not solve this problem.
In general, three things can assist:
- Panel data: repeated observations of outcomes in each market gives us a way to invert out unobserved market-level heterogeneity.
- Instruments: there is still hope for identification in the cross-section but we would need a richer model that allows covariates to enter flow payoffs in a way that moves \(a\) and \(a'\) independently from each other.
- The steady state assumption imposes a relationship between the choice probability \(p(x,a,a',\alpha)\) and the joint distribution of \((a,a',\alpha)\) that provides additional identifying content that may be employed with either strategy (1) or (2).
Both the IV and panel data approaches for discrete choice models involve technical challenges that go beyond their linear counterparts. We will develop the necessary tools and revisit these identification strategies in later chapters.