Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games

Estimation of dynamic discrete choice models – particularly dynamic discrete games of incomplete information – is a topic of considerable interest in economics. Broadly, likelihood-based estimation of these models takes the form

	$\displaystyle\max_{(\theta,Y)\in\Theta\times\mathcal{Y}}$	$\displaystyle Q_{N}(\theta,Y)$
	s.t.	$\displaystyle G(\theta,Y)=0,$

where $Q_{N}$ is the log-likelihood function based on $N$ independent markets, $\theta$ is a finite-dimensional vector of parameters, $Y$ is a vector of important auxiliary parameters, and $G(\theta,Y)=0$ is a vector equality constraint that represents equilibrium conditions. The parameters $\theta$ usually consist of the structural parameters of the model. Common examples of auxiliary parameters $Y$ include expected/integrated value functions or conditional choice probabilities, since the equality constraint is often derived from an equilibrium fixed point condition of the form $G(\theta,Y)\equiv Y-\Gamma(\theta,Y)=0$.

One approach to estimating these models is to directly impose the fixed point equation for each trial value of $\theta$ visited by the optimization algorithm by solving for $Y_{\theta}$ such that $G(\theta,Y_{\theta})=0$. This approach was pioneered by Miller (1984), Wolpin (1984), Pakes (1986), and Rust (1987) for single-agent models, where the fixed point is unique and can be computed via standard value function iteration or backwards induction. Solution algorithms are available for dynamic games (Pakes and McGuire, 1994, 2001), but it is often infeasible to nest those within an estimation routine because the computational burden can be quite large. Furthermore, in games the model may be incomplete due to multiple solutions $Y_{\theta}$ (Tamer, 2003).

These issues with the nested fixed-point approach led researchers to extend conditional choice probability (CCP) estimators – first introduced in the seminal work of Hotz and Miller (1993) for single-agent models – to the case of dynamic discrete games. Of particular interest here is the nested pseudo-likelihood approach of Aguirregabiria and Mira (2002; 2007).¹¹1Some other examples of CCP estimators are described in Hotz et al. (1994); Bajari et al. (2007); Pakes et al. (2007); Pesendorfer and Schmidt-Dengler (2008). They suggest using a $k$-step nested pseudo-likelihood ($k$-NPL) approach, which defines a sequence of estimators, as an algorithm for computing the nested pseudo-likelihood (NPL) estimator, a fixed point of the sequence. In single-agent models, Aguirregabiria and Mira (2002) show that the $k$-NPL estimator is efficient for $k\geq 1$ when initialized with a consistent CCP estimate, in the sense that it has the same limiting distribution as the (partial) maximum likelihood estimator. Furthermore, Kasahara and Shimotsu (2008) showed that the sequence converges to the true parameter values with probability approaching one in large samples. Indeed, the Monte Carlo simulations in Aguirregabiria and Mira (2002) show that single-agent $k$-NPL reliably converges to the maximum likelihood estimate. The combination of computational simplicity, efficiency, and convergence stability make $k$-NPL an attractive alternative to other approaches, such as nested fixed point (computationally burdensome) or standard Newton or Fisher scoring steps on the full maximum likelihood problem (often diverges in finite samples in practice).²²2This is a well-known limitation of standard Newton steps, and further regularization is often needed to ensure convergence.

Unfortunately, these attractive properties of $k$-NPL are lost in dynamic games. Aguirregabiria and Mira (2007) show that $k$-NPL estimates are in general not efficient for $k\leq\infty$, although they show that the $\infty$-NPL (taking $k\to\infty$ or until convergence) estimator outperforms the $1$-NPL estimator in efficiency when both are consistent. But Pesendorfer and Schmidt-Dengler (2010) show that the sequence may fail to converge to the equilibrium that generated the data, even with very good starting values, so that $\infty$-NPL may not be consistent (see also, Kasahara and Shimotsu (2012), Egesdal et al. (2015), and Aguirregabiria and Marcoux (2021)). Kasahara and Shimotsu (2012) show that inconsistency occurs when the NPL mapping is unstable at the data-generating equilibrium, which is essentially equivalent to best-response instability of the equilibrium.³³3The NPL mapping is first-order equivalent to the best-response mapping at the equilibrium.

Another type of CCP estimator is the minimum-distance estimator (Altug and Miller (1998); Pesendorfer and Schmidt-Dengler (2008)). This estimator is both consistent and efficient in dynamic games, and Bugni and Bunting (2021) develop a sequential version of this estimator, which we refer to as $k$-MD. However, Monte Carlo simulations show that these econometric properties come at the expense of greatly increased computational burden, taking 12 to 26 times longer per iteration than $k$-NPL in even the simplest small-scale dynamic games.⁴⁴4Bugni and Bunting (2021) perform Monte Carlo simulations with a very small-scale game (two players, two actions, four states), and even in that setting they remark, “… computing the optimally weighted 1-MD, 2-MD, and 3-MD estimators takes us roughly 33%, 75%, and 80% more time than computing the 20-[NPL] estimator, respectively.” These translate to roughly 26, 17.5, and 12 times longer per iteration for each respective case. This difference in computation time is likely to grow with the size of the game, which would be a serious concern for empirical applications. This leaves the researcher with an undesirable tradeoff between the computationally simple $k$-NPL sequence and the more burdensome $k$-MD sequence with better efficiency properties.

But there is another concern shared by the $k$-NPL and $k$-MD sequences: finite sample performance of successive iterations. The $k$-MD estimator uses the NPL mapping to update choice probabilities between iterations, so there is reason to be concerned that it may mimic $k$-NPL’s finite sample properties when the data-generating equilibrium is NPL-unstable. While the first-order asymptotic analysis of Bugni and Bunting (2021) implies that both sequences are consistent for finite $k$, this asymptotic consistency does not necessarily lead to good performance in finite samples. Indeed, the finite-sample performance of $k$-NPL deteriorates with $k$ when the equilibrium is NPL-unstable. For example, Pesendorfer and Schmidt-Dengler (2008) consider a finite number of iterations for $k$-NPL and find that it is severely biased in large-but-finite samples when the equilibrium is NPL-unstable. In our own Monte Carlo simulations presented later, we find that substantial bias appears rather quickly, even for low values of $k$. This issue may also be a concern for $k$-MD, since it is also consistent for finite $k$ but has unknown stability properties as $k\to\infty$, which may lead to deterioration in finite-sample performance even for fixed $k$, similar to $k$-NPL. This concern arises because Kasahara and Shimotsu (2012) show that instability/inconsistency of the $k$-NPL sequence arises from instability of the NPL mapping used to update the choice probabilities between iterations.⁵⁵5However, a rigorous econometric analysis of the the $k$-MD estimator’s behavior as $k\to\infty$ is beyond the scope of this paper.

With these concerns about various sequential methods in mind, an important question arises: is there a CCP-based sequence that achieves a balance between computational simplicity, asymptotic efficiency, and good finite-sample properties with any number of iterations – including as $k\to\infty$ (iterating to convergence) – in dynamic games? The primary contribution of this paper is to provide such a method, which we name the $k$-step Efficient Pseudo-Likelihood ($k$-EPL) estimator. We show that $k$-EPL estimates are first-order asymptotically equivalent to the maximum likelihood estimate for any number of iterations, $k\geq 1$. Thus, every estimate in the sequence is efficient. Furthermore, we also show that higher-order improvements are achieved with iteration, so the $k$-EPL sequence converges to the finite-sample maximum likelihood estimator almost surely. The convergence rate is fast, approaching super-linear as $N\to\infty$. This convergence result for $k$-EPL holds even when the data-generating equilibrium is best-response unstable, rendering $k$-NPL inconsistent.

One key distinction between $k$-EPL and $k$-NPL lies in how we incorporate simultaneous play by multiple agents. While $k$-NPL focuses on single agents’ responses to a combination of an exogenous state transition process and other agents’ equilibrium play, $k$-EPL incorporates the simultaneous nature of the game and is based on the joint behavior of multiple players. Incorporating this additional information yields increased asymptotic precision.

Despite this conceptual modification, our $k$-EPL estimator retains a simple computational structure similar to $k$-NPL. When utility is linear in the parameters of interest, both $k$-EPL and $k$-NPL iterations proceed in two stages: (i) solve a set of linear systems to generate pseudo-regressors; and (ii) use the pseudo-regressors in a static logit/probit maximum likelihood problem. The linear systems only need to be solved once per iteration in the sequence, and the static logit/probit problem is a low-dimensional, strictly concave problem that has a unique solution and is easy to solve with out-of-the-box optimization software. Because $k$-EPL incorporates all players simultaneously, the linear systems in $k$-EPL have larger dimension than those in k-NPL. While this increases the relative computation time in large-dimensional problems, the increase appears much less severe than the computation time required for $k$-MD. We also find that iterating $k$-EPL to convergence can be faster than doing so with $k$-NPL when the game is not too large, since the need for fewer iterations results in lower overall computation time.

One interesting implication of our higher-order analysis is that iterating $k$-EPL can also provide a convenient algorithm for computing the maximum likelihood estimator. We explore this in some of our Monte Carlo simulations and find that it performs quite well, even with random starting values. However, our primary focus is on the entire $k$-EPL sequence, beginning with consistent initial estimates. We find that much of the practical improvement from iteration is achieved with just a few iterations in our Monte Carlo experiments, suggesting that low values of $k$ can be quite effective when the initial CCP estimates are consistent.

In recent related work, Aguirregabiria and Marcoux (2021) studied the finite-sample properties of $\infty$-NPL ($k$-NPL with $k\to\infty$) and introduced a variant of the $k$-NPL algorithm that updates the conditional choice probabilities by applying spectral methods to the CCP updates. The goal of their algorithm is to improve convergence properties of $\infty$-NPL for unstable fixed points.⁶⁶6In the population or in finite samples, the NPL operator may have spectral radius larger than one for some equilibria, rendering it unstable. Conversely, the spectral radius of the EPL operator is zero in the population and near zero in finite samples. However, there are several differences between their work and ours. First, they limit their analysis of the spectral algorithm to computing the best fixed point of the $k$-NPL sequence, whereas we provide analysis for all the iterations in the $k$-EPL sequence. Second, upon convergence, the $k$-NPL and spectral $k$-NPL algorithms do not produce the maximum likelihood estimator, and convergence can require many iterations. In contrast, our $k$-EPL estimator has the same limiting distribution as the MLE at each iteration and usually converges locally to the MLE after few iterations in finite samples. We verify these properties in our simulation studies.

Our work is also related to methods leveraging Neyman orthogonalization, which has played a central role in recent advances in the broader econometrics literature (see, e.g., Chernozhukov et al. (2018); Chernozhukov et al. (2022); Chernozhukov et al. (2022)). $k$-EPL leverages a type of quasi-Newton step in its construction, leading to an important “zero Jacobian” property. Consequently, each estimate in the $k$-EPL sequence is asymptotically Neyman orthogonal to the previous estimate, and which leads to many of $k$-EPL’s attractive econometric properties.

We demonstrate the application of the $k$-EPL estimator through an empirical analysis of the U.S. wholesale club industry, with a specific focus on its three major players: Sam’s Club, Costco, and BJ’s. We construct a structural dynamic model to examine the industry’s competitive landscape. Leveraging data on club stores operating across the United States from 2009 to 2021, we employ $k$-EPL to estimate structural parameters such as fixed costs, entry costs, the effect of market size, and the competitive effect. Additionally, we consider a counterfactual experiment designed to identify the key determinants of market entry behavior and to explore their potential influence on the industry’s structure.

The remainder of the paper proceeds as follows. Section 2 describes a generic dynamic discrete choice game of incomplete information. Section 3 describes the $k$-EPL estimator, its asymptotic and finite-sample properties, and its numerical implementation. Section 4 provides Monte Carlo simulations, and additional simulation results are included in the appendix. Section 5 describes our empirical application to the U.S. wholesale club industry. Section 6 concludes. All proofs appear in the Appendix.

Here we describe a canonical stationary dynamic discrete game of complete information in the style of Aguirregabiria and Mira (2007) and Pesendorfer and Schmidt-Dengler (2008). Time is discrete, indexed by $t=1,2,3.\dots$. In a given market, there are $J$ firms indexed by $j\in\mathcal{J}=\{1,2,\dots,|\mathcal{J}|\}$. Given a vector of state variables observable to all agents and the econometrician, $x_{t}$, and its own private information $\varepsilon_{t}^{j}$, each firm chooses an action, $a_{t}^{j}\in\mathcal{A}=\{0,1,2,\dots,|\mathcal{A}|-1\}$. Action zero is the “outside option” when applicable. All players choose their actions simultaneously.

Agents have flow utilities (profits), $U^{j}(x_{t},a_{t}^{j},a_{t}^{-j},\varepsilon_{t}^{j};\theta_{u})$, where $a_{t}^{-j}$ are the actions of the other players. States transition according to $p(x_{t+1},\varepsilon_{t+1}\mid a_{t},x_{t},\varepsilon_{t};\theta_{f})$, and the discount factor is $\beta\in(0,1)$. Agents choose actions to maximize expected discounted utility,

$$\mathrm{E}\left\{\sum_{s=0}^{\infty}\beta^{s-t}U^{j}(x_{s},a_{s}^{j},a_{s}^{-j},\varepsilon_{s}^{j};\theta_{u})\,\bigg{|}\,x_{t},\varepsilon_{t}^{j}\right\}.$$

The primary parameter of interest is $\theta=(\theta_{u},\theta_{f})$. Furthermore, we impose the following standard assumptions on the primitives.

Assumptions 1–4 here correspond to Assumptions 1–4 in Aguirregabiria and Mira (2007). In most applications in the literature, the private shocks are assumed to be either i.i.d. Type 1 Extreme Value or normal, both of which satisfy the assumptions.

The operative equilibrium concept here will be that of a Markov Perfect Nash equilibrium. We will consider stationary equilibria only, so from here we drop the time subscript. Because moves are simultaneous, the actions of player $j$ do not depend directly on $a^{-j}\in\mathcal{A}^{|\mathcal{J}|-1}$, but rather on $P^{-j}(x)\in\Delta^{|\mathcal{J}|-1}$, where $P^{-j}(x)$ is player $j$’s belief about the other players’ probability of playing the corresponding actions in state $x$ and $\Delta$ is the unit simplex in $\mathbb{R}^{\left|\mathcal{A}\right|-1}$. So, from here on out we will work with the following utility function and transition probabilities:

$$u^{j}(x,a^{j};P^{-j},\theta_{u})=\sum_{a^{-j}\in\mathcal{A}^{|\mathcal{J}|-1}}P^{-j}(x,a^{-j})\bar{u}(x,a^{j},a^{-j};\theta_{u})$$

$$f^{j}(x^{\prime}\mid x,a^{j};P^{-j},\theta_{f})=\sum_{a^{-j}\in\mathcal{A}^{|\mathcal{J}|-1}}P^{-j}(x,a^{-j})f(x^{\prime}\mid x,a^{j},a^{-j};\theta_{f})$$

Now consider the vector of player $j$’s (expected) choice-specific value functions, $v^{j}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{A}|}$, and define the corresponding choice probabilities as $\Lambda^{j}(x,a^{j};v^{j})$, which is the probability agent $j$ chooses action $a^{j}$ in state $x$, conditional on having choice-specific value function $v^{j}$.⁸⁸8We note that the choice-specific value functions, $v^{j}$, are also often referred to as conditional value functions. In equilibrium, the choice probabilities will be $P^{j}(a)=\Lambda^{j}(a;v^{j})$. And let

$$\Lambda^{-j}(v^{-j})=(\Lambda^{1}(v^{1}),\dots,\Lambda^{j-1}(v^{j-1}),\Lambda^{j+1}(v^{j+1}),\dots,\Lambda^{|\mathcal{J}|}(v^{|\mathcal{J}|})),$$

so that in equilibrium $P^{-j}=\Lambda^{-j}(v^{-j})$. Aguirregabiria and Mira (2007) show that in equilibrium, the choice-specific value functions are equal to

$$v^{j}(x,a^{j})=u^{j}(x,a^{j};P^{-j},\theta_{u})+\beta\sum_{x^{\prime}}f^{j}(x^{\prime}\mid x,a^{j};P^{-j},\theta_{f})\Gamma^{j}(x^{\prime};\theta,P),$$

where

$$\Gamma^{j}(\theta,P)=\left(I-\beta F(\theta_{f},P)\right)^{-1}\sum_{a^{j}}P^{j}(a^{j})*\left(u^{j}(a^{j};P^{-j},\theta_{u})+e(a^{j};P^{j})\right)$$

maps into ex-ante (or integrated) value function space. Here, $e\left(a^{j};P^{j}\right)$ stacks the values of $e(x,a^{j};P^{j})\equiv E[\varepsilon^{j}(a^{j})\mid x,a^{j},P^{j}]$, as defined in Aguirregabiria and Mira (2007, Equation 11), and $F(\theta_{f},P)$ is an unconditional state transition matrix with elements $F(\theta_{f},P)\{k,l\}=\sum_{a\in\mathcal{A}^{J}}\left(\Pi_{j=1}^{J}P^{j}(a^{j}\mid x=k)\right)f(x^{\prime}=l\mid x,a;\theta_{f})$.⁹⁹9See footnote 6 in Aguirregabiria and Mira (2007). They then define the NPL operator, $\Psi(\theta,P)$, such that $\Psi^{j}(\theta,P)=\Lambda^{j}(\Gamma^{j}(\theta,P))$ and combining all players yields the following fixed-point condition that describes any Markov perfect equilibrium (Aguirregabiria and Mira 2007, Lemma 1):

$$P=\Psi(\theta,P).$$

While this equilibrium representation based on the NPL operator is often useful, we will ultimately want to work with an alternative representation when implementing our new estimator. This alternative arises due to a change of variables from $P$ space to $v$ space. (See Section 3.3 for a detailed discussion of the importance of this change.) Define the function

$\displaystyle\Phi^{j}(x,a^{j};v^{j},v^{-j},\theta)$

$\displaystyle=u^{j}(x,a;\Lambda^{-j}(v^{-j}),\theta_{u})+\beta\sum_{x^{\prime}}f^{j}(x^{\prime}\mid x,a^{j};\Lambda^{-j}(v^{-j}),\theta_{f})S(v^{j}(x^{\prime})),$

where $\Phi:\Theta\times\mathbb{R}^{|\mathcal{J}|\times|\mathcal{X}|\times|\mathcal{A}|}\to\mathbb{R}^{|\mathcal{J}|\times|\mathcal{X}|\times|\mathcal{A}|}$ and $S(\cdot)$ is McFadden’s social surplus function.¹⁰¹⁰10For example, $S(v^{j}(x))=\ln(\sum_{a^{j}}\exp(v^{j}(x,a^{j}))+\bar{\gamma}$ when the private values are i.i.d. and follow the type 1 extreme value distribution, where $\bar{\gamma}$ is the Euler-Mascheroni constant. This $\Phi(\cdot)$ function allows us to characterize the equilibrium with an alternative fixed-point equation, as described in the following Lemma.

This section describes the $k$-EPL estimator and discusses its asymptotic and finite-sample properties, as well as computational aspects of its implementation in dynamic discrete choice games. We begin by discussing maximum likelihood estimation, subject to an equilibrium constraint based on some nuisance parameter, $Y$. The model is parameterized by a finite-dimensional vector, $\theta\in\Theta\subset\mathbb{R}^{|\Theta|}$, and a constraint $G(\theta,Y)=0$ where $Y\in\mathcal{Y}\subset\mathbb{R}^{|\mathcal{Y}|}$ and $G:\Theta\times\mathcal{Y}\to\mathbb{R}^{|\mathcal{Y}|}$. The true parameter values are $\theta^{*}$ and $Y^{*}$, with $G(\theta^{*},Y^{*})=0$. Note that there may be other values of $Y$ satisfying the constraint at $\theta^{*}$, but we will assume that the data are generated from only one such value, a common assumption in the literature.

The alternative statements of the equilibrium conditions in Section 2 yield two potential choices for $Y$ and the corresponding constraint. Our asymptotic consistency and efficiency results in this section do not depend on the choice of nuisance parameter, but this choice can have serious implications for the computational implementation of the $k$-EPL estimator. So, for computational purposes we ultimately use the choice-specific value functions as the nuisance parameter ($Y\equiv v$), and the constraint comes from the equation in Lemma 1 ($v-\Phi(\theta,v)=0$). We discuss the computational concerns in more detail in Section 3.3. But for now, we present the asymptotic theory with the generic nuisance parameter, $Y$.

Let $w_{i}$ for $i=1,\dots,N$ denote the observations from $N$ independent markets, and define

$\displaystyle Q_{N}(\theta,Y)$

$\displaystyle\equiv N^{-1}\sum_{n=1}^{N}q_{i}(\theta,Y)=N^{-1}\sum_{i=1}^{N}\ln Pr(w_{i}\mid\theta,Y),$

where $q_{i}(\theta,Y)\equiv\ln Pr(w_{i}\mid\theta,Y)$. Furthermore, let $Q^{*}(\theta,Y)\equiv\mathrm{E}[Q_{N}(\theta,Y)]$.

Assumptions 5(a)-(c) echo standard identification assumptions. We note that assuming that $Q^{*}$ has a unique maximum does not rule out games with multiple equilibria. Non-singularity of the Jacobian in (d) is the defining feature of regular Markov perfect equilibria in the sense of Doraszelski and Escobar (2010). Regularity essentially means that the equilibrium is locally isolated and we can apply the implicit function theorem to obtain $Y(\theta)$ locally.¹¹¹¹11Aguirregabiria and Mira (2007) directly assume the local existence of $Y(\theta)$, instead of appealing to the implicit function theorem.

One method to estimate these models is via constrained maximum likelihood:

	$\displaystyle\left(\hat{\theta}_{\text{MLE}},\hat{Y}_{\text{MLE}}\right)=\arg$	$\displaystyle\max_{(\theta,Y)\in\Theta\times\mathcal{{Y}}}Q_{N}(\theta,Y)$
		$\displaystyle\text{s.t. }G(\theta,Y)=0.$

An equivalent statement is $\hat{\theta}_{\text{MLE}}=\textrm{arg max}_{\theta}\quad Q_{N}(\theta,Y(\theta))$, where $G(\theta,Y(\theta))=0$ and $\hat{Y}_{\text{MLE}}=Y(\hat{\theta}_{\text{MLE}})$. Pseudo-likelihood estimation replaces $Y(\theta)$ with some other mapping. Aguirregabiria and Mira (2007) define $Y\equiv P$ and replace $Y(\theta)\equiv P(\theta)$ with their $\Psi(\theta,\hat{P}_{k-1})$ for the $k$-th iteration in the $k$-NPL sequence. However, this procedure suffers from the issues discussed in Section 1.

Our $k$-step Efficient Pseudo-Likelihood ($k$-EPL) sequence instead uses a “Newton-like” step, which provides a good approximation to the full Newton step but uses a fixed value of $\nabla_{Y}G(\theta,Y)^{-1}$; this value varies between steps but does not vary as the optimizer searches over different values $\theta$ within each step. Algorithm 1 below defines our sequential estimation procedure. It uses our Newton-like mapping, $\Upsilon(\cdot)$, which is a function of an initial compound parameter vector, $\gamma=(\theta,Y)$, and an additional (possibly different) value of $\theta$.

This procedure enjoys some nice econometric properties, both asymptotically and in finite samples. These properties arise from some convenient features of the $\Upsilon(\cdot)$ function, which are detailed in the following lemma.

1. 1.

   Roots of $G$ and fixed points of $\Upsilon$ are identical: $\Upsilon(\theta,\gamma_{\theta})=Y(\theta)\iff G(\theta,Y_{\theta})=0$.

2. 2.

   $\nabla_{\theta}\Upsilon(\theta,\gamma_{\theta})=\nabla_{\theta}Y(\theta)$.

3. 3.

   $\nabla_{\gamma}\Upsilon(\theta,\gamma_{\theta})=0$ (Zero Jacobian Property).

Lemma 2 is the key to most of the results in this section, which arise from applying the lemma at $(\theta^{*},\gamma^{*})$ and $(\hat{\theta}^{MLE},\hat{\gamma}^{MLE})$. For now, we note that Result 3 of Lemma 2 is analogous to the “zero Jacobian” property from Proposition 2 of Aguirregabiria and Mira (2002), which was the key to both their efficiency results and the finite-sample convergence results of Kasahara and Shimotsu (2008) for single-agent $k$-NPL. By utilizing Newton-like steps on the equilibrium constraint, the $k$-EPL algorithm restores this zero Jacobian property in dynamic games.

One notable difference between $k$-EPL and some other sequential estimators is that $k$-EPL is initialized from a consistent estimate of both the structural parameter and the nuisance parameter, while other estimators – including $k$-NPL – only require an initially consistent estimate of the (nuisance) CCPs. Other examples include the finite-dependence-based estimators (Hotz and Miller (1993); Arcidiacono and Miller (2011)) and inequality-based estimators (Bajari et al. (2007)). While none of these other estimators offer efficiency or convergence guarantees as $k\to\infty$, they can be used to obtain a consistent parameter estimate for initializing $k$-EPL. If the model exhibits finite-dependence, then finite-dependence-based estimators can be particularly attractive for obtaining initial estimates because of there computational simplicity. However, we note that the example models in our Monte Carlo simulations and empirical application may not have the finite dependence property because exit is not permanent. So, we instead use a $1$-NPL estimate for initialization.¹²¹²12Finite dependence is more difficult to establish in entry/exit games without a permanent exit decision that leads directly to a terminal state property. See, e.g., the discussion in Section 5.2 of Arcidiacono and Ellickson (2011). However, Arcidiacono and Miller (2019) show how to derive finite dependence in a class dynamic games that does not require a terminal state.