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Empirical Framework for Two-Player Repeated Games with Random States

  • Arkadiusz Szydłowski EMAIL logo
Published/Copyright: November 29, 2022
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Journal of Econometric Methods
From the journal Journal of Econometric Methods Volume 12 Issue 1

Abstract

We provide methods for the empirical analysis of a class of two-player repeated games with i.i.d. shocks, allowing for non-Markovian strategies. The number of possible equilibria in these games is large and, usually, theory is silent about which equilibrium will be chosen in practice. Thus, our method remains agnostic about selection among these multiple equilibria, which leads to partial identification of the parameters of the game. We propose a profiled likelihood criterion for building confidence sets for the structural parameters of the game and derive an easily computable upper bound on the critical value. We demonstrate good finite-sample performance of our procedure using a simulation study. We illustrate the usefulness of our method by studying the effect of repealing the Wright Amendment on entry and exit into Dallas airline markets and find that the static game approach overestimates the negative effect of the law on entry into these markets.

Keywords: empirical games; partial identification; parameter on boundary; airline market
JEL Classification: C10; C59; L93
Corresponding author: Arkadiusz Szydłowski, Department of Economics, Finance and Accounting, University of Leicester, University Road, Leicester, LE1 7RH, UK, E-mail:

Funding source: University of Leicester

Award Identifier / Grant number: Unassigned

Acknowledgments

We would like to thank the Associate Editor and two anonymous referees for their suggestions that greatly improved the paper. Also, many thanks to Mike Abito, Wojciech Olszewski, Subir Bose and Andre Stenzel for useful discussions and suggestions and to Arpita Ghosh and Shuai Zhao for providing competent research assistance. We are grateful to participants in the LSE Joint Econometrics and Statistics Workshop, the (Ce)2 Workshop, the UEA Economics seminar and various conferences for useful comments. This research used the ALICE High Performance Computing Facility at the University of Leicester.

Appendix A: Mathematical Proofs

A.1 Proof of Theorem 1

First note that under quasi-concavity:

(6) inf θ 1 ∈ Θ 01 sup θ 2 ∈ Θ 2 ( θ 1 ) L T ( θ 1 , θ 2 ) = min θ 1 ∈ { θ ̲ 1 , θ ̄ 1 } sup θ 2 ∈ Θ 2 ( θ 1 ) L T ( θ 1 , θ 2 )

Further, Assumptions INF2(c)–(g) allow us to apply Proposition 5.1 and Lemma F.1 in Chen, Christensen, and Tamer (2018) in order to obtain quadratic expansion of the likelihood in γ:

sup θ ∈ Θ o T 2 L T ( θ ) = 2 ∑ t = 1 T log ⁡ p ( Y t , γ 0 ) + ‖ V T ‖ 2 − inf θ ∈ Θ o T ‖ T I 0 1 / 2 γ ( θ ) − V T ‖ 2 + o p ( 1 ) = 2 ∑ t = 1 T log ⁡ p ( Y t , γ 0 ) + ‖ V T ‖ 2 + o p ( 1 )

where V T = I 0 − 1 / 2 1 T ∑ t = 1 T ∂ ⁡ log ⁡ p ( Y t , γ 0 ) ∂ γ . Now for the restricted part:

sup θ ∈ Θ o T ( θ 1 ) 2 L T ( θ ) = 2 ∑ t = 1 T log ⁡ p ( Y t , γ 0 ) + ‖ V T ‖ 2 − inf θ = ( θ 1 , θ 2 ) ∈ Θ o T ‖ T I 0 1 / 2 γ ( θ ) − V T ‖ 2 + o p ( 1 ) = 2 ∑ t = 1 T log ⁡ p ( Y t , γ 0 ) + ‖ V T ‖ 2 − inf κ ∈ Γ o T ( θ 1 ) ‖ κ − V T ‖ 2 + o p ( 1 ) = 2 ∑ t = 1 T log ⁡ p ( Y t , γ 0 ) + ‖ V T ‖ 2 − inf κ ∈ K ( θ 1 ) ‖ κ − V T ‖ 2 + o p ( 1 )

This implies:

LR T ( θ 1 ) = 2 sup θ ∈ Θ o T L T ( θ ) − sup θ ∈ Θ o T ( θ 1 ) L T ( θ ) = inf κ ∈ K ( θ 1 ) ‖ κ − V T ‖ 2 + o p ( 1 )

Now use Assumption INF3 and proceed similarly to Shapiro (1985). Note that the cone K ( θ 1 ) can be approximated by:

∂ γ ( θ 0 ) ∂ θ 2 ′ s : s ∈ K 2 ( θ 1 ) , θ 0 = ( θ 1 , θ 2 ) ∈ Θ 0 = Δ θ 1 s : s ∈ K 2 ( θ 1 )

for θ 1 ∈ { θ 1 ̲ , θ 1 ̄ } , which implies that:

LR T ( θ 1 ) = inf s ∈ K 2 ( θ 1 ) ‖ V T − Δ θ 1 s ‖ 2 + o p ( 1 )

and together with (6) concludes the proof.

A.2 Proof of Theorem 2

First we will demonstrate that θ ̲ ̂ 1 → p θ ̲ 1 and θ ̄ ̂ 1 → p θ ̄ 1 as T → ∞. For this purpose we apply Theorem 3.1 in Chernozhukov, Hong, and Tamer (2007). Their condition C.1 is satisfied as follows: part (a) follows from our Assumption INF2(a), lower-semicontinuity of E[log  p(Y t , θ)] in part (b) holds in the neighbourhood N θ of Θ0 by our Assumptions INF2(d) and INF2(e), part (c) follows from our continuity assumptions on p(Y t , θ) and discreteness of Y t , uniform convergence in part (d) can be shown to hold over N θ by applying Jennrich’s ULLN with the help of our Assumptions INF2(a), (d), (e) and noting that 0 ≤ p(Y t , θ) ≤ 1.

Next recall that B k T denotes a ball centred at zero with radius k T → ∞ and define K B k T ( θ 1 ) = { κ : κ = Δ θ 1 s , s ∈ B k T } . If Δ θ 1 = 0 , then trivially inf s ∈ K LF ( θ 1 ) ‖ V T − Δ θ 1 s ‖ 2 = inf s ∈ K LF ( θ 1 ) ∩ B k T ‖ V T − Δ θ 1 s ‖ 2 so we focus on the case when Δ θ 1 ≠ 0 . Note that in this case K B k T ( θ 1 ) is an ellipsoid. For θ 1 in the neighbourhoods of θ ̲ 1 and θ ̄ 1 we have:

P inf s ∈ K LF ( θ 1 ) ‖ V T − Δ θ 1 s ‖ 2 − inf s ∈ K LF ( θ 1 ) ∩ B k T ‖ V T − Δ θ 1 s ‖ 2 ≠ 0 ≤ P ( V T ∉ K B k T ( θ 1 ) )

but as the elliptic radi grow with k T we have that P ( V T ∉ K B k T ( θ 1 ) ) → 0 as T → ∞. Thus we can write:

(7) inf s ∈ K L F ( θ 1 ) ‖ V T − Δ θ 1 s ‖ 2 = inf s ∈ K L F ( θ 1 ) ∩ B k T ‖ V T − Δ θ 1 s ‖ 2 + o p ( 1 )

Assumption INF5 implies that K LF ( θ 1 ) ∩ B k T is a continuous (compact-valued) correspondence around θ ̲ 1 and θ ̄ 1 . Additionally Δ θ 1 is continuous in θ 1 in this neighbourhood by Assumption INF2(e). Thus, we can apply Berge’s maximum theorem to conclude that inf s ∈ K LF ( θ 1 ) ∩ B k T ‖ V T − Δ θ 1 s ‖ 2 is a continuous function of θ 1. Now (7) and continuous mapping theorem imply:

(8) inf s ∈ K LF ( θ ̲ ̂ 1 ) ∩ B k T ‖ V T − Δ θ ̲ ̂ 1 s ‖ 2 = inf s ∈ K LF ( θ ̲ 1 ) ‖ V T − Δ θ ̲ 1 s ‖ 2 + o p ( 1 ) inf s ∈ K LF ( θ ̄ ̂ 1 ) ∩ B k T ‖ V T − Δ θ ̄ ̂ 1 s ‖ 2 = inf s ∈ K LF ( θ ̄ 1 ) ‖ V T − Δ θ ̄ 1 s ‖ 2 + o p ( 1 )

as T → ∞.

Next note that K LF(θ 1) ⊆ K 2(θ 1). This is trivially satisfied when K 2 ( θ 1 ) = R d 2 or K 2(θ 1) is an orthant itself. Consider the remaining case when K 2 ( θ 1 ) = R + d + × R − d − × R d 2 − d + − d − where 0 ≤ d + + d − ≤ d 2 − 1. Now we must have K LF ( θ 1 ) = R + d + × R − d − × R d ̃ + × R d ̃ − with d + ̃ + d ̃ − = d 2 − d + − d − . To see that, without loss of generality, suppose that K LF ( θ 1 ) = R − × R + d + − 1 × R − d − × R d + ̃ × R d ̃ − and let C ̃ be the corner associated with this parameter space and θ 2 = θ 2 * ( θ 1 ) be the profiled-likelihood-minimising value. Now note that the first coordinate of C ̃ , C 1 ̃ , has to be different than the first coordinate of θ 2, θ 2,1, but these are the same for the closest corner, i.e. C θ 1 , 1 = θ 2,1 . We have:

‖ C ̃ − θ 2 ‖ 2 = | C ̃ 1 − θ 2,1 | 2 + ‖ C ̃ − 1 − θ 2 , − 1 ‖ 2 > inf C ∈ C ‖ C − 1 − θ 2 , − 1 ‖ 2 = ‖ C θ 1 − θ 2 ‖ 2

which implies that C ̃ cannot be the closest corner to θ 2.

Finally, we have:

max inf s ∈ K LF ( θ ̲ 1 ) ‖ V T − Δ θ ̲ 1 s ‖ 2 , inf s ∈ K LF ( θ ̄ 1 ) ‖ V T − Δ θ ̄ 1 s ‖ 2 ≥ max inf s ∈ K 2 ( θ ̲ 1 ) ‖ V T − Δ θ ̲ 1 s ‖ 2 , inf s ∈ K 2 ( θ ̄ 1 ) ‖ V T − Δ θ ̄ 1 s ‖ 2

which together with (8) concludes the proof.

A.3 Proof of Theorem 3

For a closed convex cone K let K o denote its polar cone (see e.g. Section 14 in Rockafellar 1970). From the proof of Theorem 1 and Moreau’s decomposition theorem (note that K ( θ 1 ) is a closed convex cone by Assumption INF2(g)):

LR T ( θ 1 ) = ‖ V T ‖ 2 − inf κ ∈ K o ( θ 1 ) ‖ κ − V T ‖ 2 + o p ( 1 )

which implies LR T (θ 1) ≤ ‖V T ‖2 + o p (1) and

sup θ 1 ∈ Θ 01 LR T ( θ 1 ) ≤ max θ 1 ∈ { θ ̲ 1 , θ ̄ 1 } ‖ V T ‖ 2 , ‖ V T ‖ 2 + o p ( 1 ) = ‖ V T ‖ 2 + o p ( 1 )

and the result follows from V T being asymptotically N(0, I).

Appendix B: Alternative Definitions of Market Presence

In Table 4 an airline is present in the market if it operates at least one flight from the market origin to market destination. Here we consider other definitions of market presence. First, we use DB1B ticketing data, which contains 10% sample of airline tickets from reporting carriers, and redefine market presence as selling at least 5 tickets for the specified route (see Table 8). Next, we use T100 Segment data and redefine market as a segment of the trip, for example a flight from ORD to MIA through DCA contains two segments ORD-DCA and DCA-MIA (using previous definition this would only be a single market ORD-MIA).

Table 8:

Share of markets by presence of major (ticketing) carriers over time (in %).

Presence in Q2 1993 – Q2 2002 – Q2 2012 American Delta United US Airways Southwest
In – in – in 43.4 61.1 74.8 42.8 86.8
In – out – in 9 5.9 9.2 12.3 4.8
In – out – out 37 23.1 12.8 27 6
In – in – out 10.6 9.9 3.2 17.9 2.4

  1. DB1B Market data, flights with less than 5 tickets in 1993 dropped. Markets are defined as directional routes between origin and destination airports (irrespective of the number of stops on the way). “In” means that a carrier served at least one flight on the route.

The numbers in Table 8 significantly differ from those in Table 4 as ubiquitous codeshare and interlining agreements drive a wedge between the definitions of operating and ticketing carrier. The differences are smaller between Table 9 and Table 4. Despite these differences the main message remains the same – there is substantial amount of entry and exit across time in the US airline market.

Table 9:

Share of markets by presence of major carriers over time (in %).

Presence in Q2 1990 – Q2 2000 – Q2 2010 American Delta United US Airways Southwest
In – in – in 34 35.8 38.6 16.5 74.6
In – out – in 5 6.8 2.7 3.5 3.1
In – out – out 46.4 33.4 35.8 53.8 14
In – in – out 14.6 24.1 23 26.3 8.3

  1. T100 Segment data, flights with less than 20 passengers in 1990 dropped. Markets are defined as segments of directional routes between origin and destination airports. “In” means that a carrier served at least one flight on the route.

Appendix C: Moment Inequality Characterisation

For a non-empty A ⊊ A let L ( A ; α , V ) denote the probability of observing some a ∈ A in equilibrium in the normal form game under the assumption that in the regions of multiple equilibria an equilibrium in A is always selected. For simplicity let Assumption INF1 hold. Following Galichon and Henry (2011) the marginal identified set for α can be characterised using moment inequalities by:

(9) Θ 01 S = { α : E 0 ( 1 { a ∈ A j } ) ≤ L ( A j ; α , V ) , A j ⊊ A , V ∈ V S ( α ) }

where j = 1, 2, …, J. Now for inference one can implement either the profiled procedure in Kaido, Molinari, and Stoye (2019) (KMS) or Bugni, Canay, and Shi (2017) (BCS). We discuss how the computational burden of these procedures compares to our approach as computation is the main obstacle for a practical inference in our model.[25] For this discussion we employ similar notation as in Section 5 in the paper, namely θ 1 ≡ α, θ 2 ≡ V, θ = (θ 1, θ 2).

C.1 KMS Inference

Let G n , j b be a standardised estimator of E 0 ( 1 { a ∈ A j } ) evaluated on a bootstrap sample scaled by n and let D ̂ n , j ( θ ) denote the gradient of L ( A j ; α , V ) w.r.t. α and V normalised by the sample standard deviation of moment j, σ ̂ n , j . Further, let ξ ̂ n , j ( θ ) denote ( ι n σ ̂ n , j ) − 1 n times the sample estimator of moment j, where ι n → ∞. The KMS critical value is obtained by bootstrapping:

Λ n b ( θ , ρ , c ) = λ ∈ n ( Θ − θ ) ∩ ρ B d : G n , j b + D ̂ n , j ( θ ) λ + ψ j ( ξ ̂ n , j ( θ ) ) ≤ c , j = 1,2 , … , J

where ρB d imposes a technical “box” constraint on the local parameter space and ψ j is a Generalised Moment Selection function of Andrews and Soares (2010), and can be calculated as:

c ̂ ( θ ) = inf { c ∈ R + : P * ( Λ n b ( θ , ρ , c ) ∩ { λ 1 = 0 } ≠ ∅ ) ≥ κ }

where P* denotes the law induced by bootstrap sampling and λ 1 is the first element of λ. Finally, the marginal confidence set is built by finding lowest and highest value of θ 1 for which the sample moment inequalities are satisfied with slackness c ̂ ( θ 1 , θ 2 ) .

Let us now compare our inference method to KMS. Note that the computationally difficult step in our model is the re-evaluation of the continuation value set Θ2(θ 1) for different values of θ 1, which will be embedded in evaluating n ( Θ − θ ) within Λ n b ( θ , ρ , c ) in the KMS procedure. Our procedure in Display 1 controls the number of evaluations of Θ2(θ 1) by controlling the size of the grid for candidate values of θ 1 in the pre-estimation of the identified set in Step 1 and re-using evaluations of the likelihood ratio from Step 1 in building the confidence set in the final Step 5.

Similarly, the first step in the KMS procedure in which candidate values of θ are drawn can be adjusted to include only θ’s on the grid of values for θ 1 to limit number of evaluations of Θ2(θ 1). However, as currently implemented, the KMS procedure proceeds with a smooth iterative algorithm to generate further “good” candidate values of θ (“A-M steps”) and, thus, requires recalculation of Θ2(θ 1) for these newly generated values. As the number of iterations required for this algorithm to converge may differ from application to application, it may be difficult to control the number of evaluations of Θ2(θ 1) in practice without significant changes to this algorithm. Therefore, it seems that using the moment inequality characterisation in (9) and KMS is unlikely to dominate our method in terms of computational convenience.

C.2 BCS Inference

The main BCS critical value is obtained by taking a minimum over two profiled bootstrap statistics, T n DR ( θ 1 ) and T n PR ( θ 1 ) , in order to improve power. As our inference procedure is conservative it seems fair to compare it to T n DR ( θ 1 ) and T n PR ( θ 1 ) separately rather than to the more computationally intensive minimum statistic.

Using the notation from the previous section BCS resampling statistics can be written as:

T n DR ( θ 1 ) = inf θ 2 ∈ Θ ̂ 2 ( θ 1 ) ∑ j = 1 J G n , j b + ψ j ( ξ ̂ n , j ( θ ) ) − T n PR ( θ 1 ) = inf θ 2 ∈ Θ 2 ( θ 1 ) ∑ j = 1 J G n , j b + ξ ̂ n , j ( θ ) −

where Θ ̂ 2 ( θ 1 ) is the set of minimizers of the KMS test statistic (see their paper for details). Note that similarly to our approach we can control the number of evaluations of Θ2(θ 1) by imposing a grid on θ 1. However, note that resampling both T n DR ( θ 1 ) and T n PR ( θ 1 ) requires repeatedly solving a non-linear non-convex constrained optimisation problem. Also, as argued in the main text, solution to this problem will often be reached on the boundary of the set Θ ̂ 2 ( θ 1 ) and Θ2(θ 1).[26] This is much more computationally expensive than repeatedly solving a convex optimisation problem in our simulation procedure in Display 1.

Appendix D: Additional Monte Carlo Simulations

We perform limited number of simulations for δ = 0.95 due to slow convergence of optimization algorithms for this case, which leads to extensive computing times. The results are given in Table 10.

Table 10:

MC simulations: coverage probabilities, δ = 0.95.

Normal shocks
Simulated crit. val. χ 3 2 crit. val.
90% 95% 99% 90% 95% 99%
Θ01 = [0.49, 5.06] 0.979 0.981 0.981 0.981 0.981 0.985
α = 0.1 0 0 0 0 0 0
α = 5.6 0.268 0.270 0.273 0.270 0.270 0.276
Θ01 = [0.49, 5.06] 0.92 0.971 0.98 0.971 0.973 0.996
α = 0.1 0 0 0 0 0 0
α = 5.6 0.131 0.132 0.135 0.133 0.135 0.135
Θ01 = [0.49, 5.06] 0.935 0.961 0.982 0.982 0.982 0.994
α = 0.1 0 0 0 0 0 0
α = 5.6 0.084 0.085 0.088 0.085 0.087 0.088
Θ01 = [0.49, 5.06] 0.942 0.97 0.989 0.971 0.985 0.994
α = 0.1 0 0 0 0 0 0
α = 5.6 0.076 0.077 0.079 0.077 0.077 0.079

  1. 500 Monte Carlo replications.

Appendix E: Inference Without Assumption INF3

In this section we discuss how we can adjust our simulated critical value if θ*(θ 1) is not unique and ∂ γ ( θ 0 ) / ∂ θ 2 ′ contains zero rows for some θ 0. This will happen, for example, if the true probabilities, γ 0, are flat on a set with non-empty interior.

Firstly, the main complication here comes from the fact that now the local parameter space for γ at γ 0 cannot be approximated simply by taking ∂ γ ( θ 0 ) ∂ θ 2 ′ s : s ∈ K 2 ( θ 1 ) , θ 0 = ( θ 1 , θ 2 ) ∈ Θ 0 (cf. proof of Theorem 1) as this set maps to zero when Θ0 is a compact set with non-empty interior. However, given that γ is smooth around γ 0 (see Assumption INF2(e)) a simple expansion Θ 0 η = { θ ∈ Θ : inf θ 0 ∈ Θ 0 ‖ θ − θ 0 ‖ ≤ η } for η > 0 would allow us to bound the asymptotic distribution of our profiled criterion as:

sup θ 1 ∈ Θ 01 LR T ( θ 1 ) ≤ max θ 1 ∈ { θ 1 ̲ , θ 1 ̄ } inf κ ∈ ⋃ ( θ 1 , θ 2 ) ∈ Θ 0 η ∂ γ ( θ 1 , θ 2 ) ∂ θ 2 ′ s : s ∈ K 2 θ 2 ( θ 1 ) V T − κ 2 + o p ( 1 ) = max θ 1 ∈ { θ 1 ̲ , θ 1 ̄ } inf ( θ 1 , θ 2 ) ∈ Θ 0 η inf s ∈ K 2 θ 2 ( θ 1 ) V T − ∂ γ ( θ 1 , θ 2 ) ∂ θ 2 ′ s 2 + o p ( 1 )

where K 2 θ 2 ( θ 1 ) is a cone approximating the local parameter space at θ 2 ∈ θ 2 * ( θ 1 ) .

Now in order to approximate the statistic on the right-hand side above, we can replace θ ̲ 1 and θ ̄ 1 with θ ̲ ̂ 1 and θ ̄ ̂ 1 as before. Next note that the identified set estimator in Chernozhukov, Hong, and Tamer (2007) approximates the identified set from “outside”, thus in practice we can replace Θ0η with their estimator Θ ̂ 0 (note that we only have to estimate a “slice” out of Θ ̂ 0 for θ ̲ ̂ 1 and θ ̄ ̂ 1 ). Finally, note that now the (set of) closest corner(s) to θ 2 ∈ θ 2 * ( θ 1 ) depends on the value of θ 2, C ( θ 1 , θ 2 ) , and Assumption INF5 is too strong in this setup. Thus, letting K LF(C) denote the orthant corresponding to the corner C ∈ C ( θ 1 , θ 2 ) we can replace inf s ∈ K 2 θ 2 ( θ 1 ) V T − ∂ γ ( θ 1 , θ 2 ) ∂ θ 2 ′ s 2 above with max C ∈ C ( θ 1 , θ 2 ) inf s ∈ K LF ( C ) V T − ∂ γ ( θ 1 , θ 2 ) ∂ θ 2 ′ s 2 in order to simulate a conservative critical value.

References

Abito, J. M., and C. Chen. 2021. “How Much Can We Identify from Repeated Games?” Economics Bulletin 41 (3): 1212–22.Search in Google Scholar

Abreu, D., B. Brooks, and Y. Sannikov. 2016. “A ‘Pencil-Sharpening’ Algorithm for Two Player Stochastic Games with Perfect Monitoring.” In Working Paper.10.2139/ssrn.2736574Search in Google Scholar

Abreu, D., B. Brooks, and Y. Sannikov. 2020. “Algorithms for Stochastic Games with Perfect Monitoring.” Econometrica 88 (4): 1661–95. https://doi.org/10.3982/ecta14357.Search in Google Scholar

Abreu, D., D. Pearce, and E. Stacchetti. 1990. “Toward a Theory of Discounted Repeated Games with Imperfect Monitoring.” Econometrica 58 (5): 1041–63. https://doi.org/10.2307/2938299.Search in Google Scholar

Abreu, D., and Y. Sannikov. 2014. “An Algorithm for Two-Player Repeated Games with Perfect Monitoring.” Theoretical Economics 9 (2): 313–38. https://doi.org/10.3982/te1302.Search in Google Scholar

Al Mohamad, D., E. W. Van Zwet, E. Cator, and J. J. Goeman. 2020. “Adaptive Critical Value for Constrained Likelihood Ratio Testing.” Biometrika 107 (3): 677–88. https://doi.org/10.1093/biomet/asaa013.Search in Google Scholar

Amemiya, T. 1986. Advanced Econometrics. Oxford: Blackwell.Search in Google Scholar

Andrews, D. W. K. 2001. “Testing When a Parameter is on the Boundary of the Maintained Hypothesis.” Econometrica 69 (3): 683–734. https://doi.org/10.1111/1468-0262.00210.Search in Google Scholar

Andrews, D. W. K., and G. Soares. 2010. “Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection.” Econometrica 78 (1): 119–57.10.3982/ECTA7502Search in Google Scholar

Bajari, P., C. L. Benkard, and J. Levin. 2007. “Estimating Dynamic Models of Imperfect Competition.” Econometrica 75 (5): 1331–70. https://doi.org/10.1111/j.1468-0262.2007.00796.x.Search in Google Scholar

Bajari, P., H. Hong, and S. P. Ryan. 2010. “Identification and Estimation of a Discrete Game of Complete Information.” Econometrica 78 (5): 1529–68.10.3386/t0301Search in Google Scholar

Beresteanu, A., I. Molchanov, and F. Molinari. 2011. “Sharp Identification Regions in Models with Convex Moment Predictions.” Econometrica 79 (6): 1785–821.10.3982/ECTA8680Search in Google Scholar

Berry, S., and P. Jia. 2010. “Tracing the Woes: An Empirical Analysis of the Airline Industry.” American Economic Journal: Microeconomics 2 (3): 1–43. https://doi.org/10.1257/mic.2.3.1.Search in Google Scholar

Bontemps, C., and R. Kumar. 2020. “A Geometric Approach to Inference in Set-Identified Entry Games.” Journal of Econometrics 218 (2): 373–89. https://doi.org/10.1016/j.jeconom.2020.04.021.Search in Google Scholar

Bugni, F. A., I. A. Canay, and X. Shi. 2017. “Inference for Subvectors and Other Functions of Partially Identified Parameters in Moment Inequality Models.” Quantitative Economics 8 (1): 1–38. https://doi.org/10.3982/qe490.Search in Google Scholar

Chen, X., T. M. Christensen, and E. Tamer. 2018. “Monte Carlo Confidence Sets for Identified Sets.” Econometrica 86 (6): 1965–2018. https://doi.org/10.3982/ecta14525.Search in Google Scholar

Chernozhukov, V., H. Hong, and E. Tamer. 2007. “Estimation and Confidence Regions for Parameter Sets in Econometric Models.” Econometrica 75 (5): 1243–84. https://doi.org/10.1111/j.1468-0262.2007.00794.x.Search in Google Scholar

Ciliberto, F., C. Murry, and E. Tamer. 2021. “Market Structure and Competition in Airline Markets.” Journal of Political Economy 129 (11): 2995–3038. https://doi.org/10.1086/715848.Search in Google Scholar

Ciliberto, F., and E. Tamer. 2009. “Market Structure and Multiple Equilibria in Airline Markets.” Econometrica 77 (6): 1791–828.10.3982/ECTA5368Search in Google Scholar

Galichon, A., and M. Henry. 2011. “Set Identification in Models with Multiple Equilibria.” The Review of Economic Studies 78 (4): 1264–98. https://doi.org/10.1093/restud/rdr008.Search in Google Scholar

Gerardi, K., and A. Shapiro. 2009. “Does Competition Reduce Price Dispersion? New Evidence from the Airline Industry.” Journal of Political Economy 117 (1): 1–37. https://doi.org/10.1086/597328.Search in Google Scholar

Giacomini, R., and T. Kitagawa. 2021. “Robust Bayesian Inference for Set-Identified Models.” Econometrica 89 (4): 1519–56. https://doi.org/10.3982/ecta16773.Search in Google Scholar

Goolsbee, A., and C. Syverson. 2008. “How Do Incumbents Respond to the Threat of Entry? Evidence from the Major Airlines.” Quarterly Journal of Economics 123 (4): 1611–33. https://doi.org/10.1162/qjec.2008.123.4.1611.Search in Google Scholar

Kaido, H., F. Molinari, and J. Stoye. 2019. “Confidence Intervals for Projections of Partially Identified Parameters.” Econometrica 87: 1397–432. https://doi.org/10.3982/ecta14075.Search in Google Scholar

Kaido, H., F. Molinari, and J. Stoye. 2020. “Constraint Qualifications in Partial Identification.” In Working Paper.10.1017/S0266466621000207Search in Google Scholar

Kline, B., and E. Tamer. 2016. “Bayesian Inference in a Class of Partially Identified Models.” Quantitative Economics 7 (2): 329–66. https://doi.org/10.3982/qe399.Search in Google Scholar

Kudo, A. 1963. “A Multivariate Analogue of the One-Sided Test.” Biometrika 50 (3/4): 403–18. https://doi.org/10.2307/2333909.Search in Google Scholar

Lee, B. S., and C. Stewart. 2016. “Identification of Payoffs in Repeated Games.” Games and Economic Behavior 99: 82–8. https://doi.org/10.1016/j.geb.2016.07.004.Search in Google Scholar

Magnac, T., and D. Thesmar. 2002. “Identifying Dynamic Discrete Decision Processes.” Econometrica 70 (2): 801–16. https://doi.org/10.1111/1468-0262.00306.Search in Google Scholar

Mailath, G., and L. Samuelson. 2006. Repeated Games and Reputations: Long-Run Relationships. New York: Oxford University Press.10.1093/acprof:oso/9780195300796.001.0001Search in Google Scholar

Otsu, T., M. Pesendorfer, and Y. Takahashi. 2016. “Pooling Data across Markets in Dynamic Markov Games.” Quantitative Economics 7 (2): 523–59. https://doi.org/10.3982/qe612.Search in Google Scholar

Rockafellar, R. T. 1970. Convex analysis In Princeton Mathematical Series. Princeton University Press.10.1515/9781400873173Search in Google Scholar

Romano, J. P., and A. M. Shaikh. 2008. “Inference for Identifiable Parameters in Partially Identified Econometric Models.” Journal of Statistical Planning and Inference 138 (9): 2786–807. https://doi.org/10.1016/j.jspi.2008.03.015.Search in Google Scholar

Rosen, A. M. 2008. “Confidence Sets for Partially Identified Parameters that Satisfy a Finite Number of Moment Inequalities.” Journal of Econometrics 146 (1): 107–17. https://doi.org/10.1016/j.jeconom.2008.08.001.Search in Google Scholar

Salz, T., and E. Vespa. 2020. “Estimating Dynamic Games of Oligopolistic Competition: An Experimental Investigation.” The RAND Journal of Economics 51 (2): 447–69. https://doi.org/10.1111/1756-2171.12321.Search in Google Scholar

Shapiro, A. 1985. “Asymptotic Distribution of Test Statistics in the Analysis of Moment Structures under Inequality Constraints.” Biometrika 72 (1): 133–44. https://doi.org/10.1093/biomet/72.1.133.Search in Google Scholar

Silvapulle, M. J., and P. K. Sen. 2001. Constrained Statistical Inference. Hoboken, NJ: John Wiley & Sons.10.1002/9781118165614Search in Google Scholar

Tamer, E. 2003. “Incomplete Simultaneous Discrete Response Model with Multiple Equilibria.” The Review of Economic Studies 70 (1): 147–65. https://doi.org/10.1111/1467-937x.00240.Search in Google Scholar
Received: 2022-01-07
Revised: 2022-10-26
Accepted: 2022-11-03
Published Online: 2022-11-29

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Empirical Framework for Two-Player Repeated Games with Random States
  4. The Robustness of Conditional Logit for Binary Response Panel Data Models with Serial Correlation
  5. Density Forecast of Financial Returns Using Decomposition and Maximum Entropy
  6. On the Implementation of Approximate Randomization Tests in Linear Models with a Small Number of Clusters
  7. Quantile Difference in Differences with Time-Varying Qualification in Panel Data
  8. A Random Forest-based Approach to Combining and Ranking Seasonality Tests
  9. Teaching Corner
  10. On the Use of the Helmert Transformation, and its Applications in Panel Data Econometrics
  11. Practitioner's Corner
  12. Linear Rescaling to Accurately Interpret Logarithms
https://doi.org/10.1515/jem-2022-0001
Keywords for this article
empirical games; partial identification; parameter on boundary; airline market

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Empirical Framework for Two-Player Repeated Games with Random States
  4. The Robustness of Conditional Logit for Binary Response Panel Data Models with Serial Correlation
  5. Density Forecast of Financial Returns Using Decomposition and Maximum Entropy
  6. On the Implementation of Approximate Randomization Tests in Linear Models with a Small Number of Clusters
  7. Quantile Difference in Differences with Time-Varying Qualification in Panel Data
  8. A Random Forest-based Approach to Combining and Ranking Seasonality Tests
  9. Teaching Corner
  10. On the Use of the Helmert Transformation, and its Applications in Panel Data Econometrics
  11. Practitioner's Corner
  12. Linear Rescaling to Accurately Interpret Logarithms
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