Mini Course on Structural Estimation of Static and Dynamic Games

Transcription

1 Mini Course on Structural Estimation of Static and Dynamic Games Junichi Suzuki University of Toronto June 1st,

2 Part : Estimation of Dynamic Games 2

3 ntroduction Firms often compete each other overtime e.g. chain stores trying expand their network One s current decision a ect its own current and future pro t by its decision itself (direct e ect) by a ecting its own future decision (indirect) by a ecting its rivals current and future decisions (indirect) Needs to construct a dynamic game 3

4 Estimation of Dynamic Game Dynamic game contains strategic interaction between agents dynamic optimization Straightforward estimation (nested xed point maximum likelihood) is virtually impossible (don t even think about it!) Need to rely on two-step methods: Nested pseudo likelihood (NPL) by Aguirregabiria and Mira (2007) Another two-step method or BBL by Bajari, Benkard and Levin (2008) 4

5 Recent Research Ryan (2009), MT WP: Studies the impacts of environmental regulation on entry/investment decisions of U.S. Portland cement industry Sweeting (2009), Duke WP: Studies whether radio stations reposition their products to deter future entries Snider (2009), Minnesota WP: Studies whether airline companies are engaged in predatory behavior 5

6 sunk entry cost is random and private 6 Example: Competition between Hotel Chains Suzuki (2009) considers a dynamic entry-exit game between hotel chains to recover the impacts of land use regulation on the cost structure of hotel chains Assume each local market is isolated Hotel chains decide the number of hotels they operate every period Hotel chains incur one-shot sunk entry cost to open a new hotel

7 Empirical Structure The data the number of hotels each chain operates for a certain period each hotel s revenue market characteristics By using these data, want to recover the cost parameters that rationalize both entry/exit behavior and revenue data 7

8 dea of denti cation Suppose you observe a market with one rm overtime This observation suggests this market is large enough to support one hotel but not more Revenue data helps to identify cost level 8

9 The Model: Basic Common state space: s t = (h t, x t ) Choice variable: a it 2 A (h it ) = f 2, 1, 0, 1, 2g \ f0 h it + a it 7g Notations: i 2 f1,..., Ng : hotel chain t 2 f1, 2,..., g : period h it : # of hotels chain i operates at t x t : market characteristics a it : # of hotels chain i opens/closes at t 9

10 The Model: Period Pro t Period pro t function: π i (a it, s t, v it ) = R (s t ) δ i h it (e i + ρ i v it ) 1 (a it > 0) a it = Ψ (a it, s t, v it ) 0 θ i where Ψ (a it, s t, v it ) = θ i = [1, δ i, e i, ρ i ] R i (s t ) h it 1 (a it > 0) a it 1 (a it > 0) a it v it Notations: δ i : operating cost ρ i : the stddev of entry 10 cost e i : the mean of entry cost v it N (0, 1)

11 The Model: Value Function V i (s t ; σ i, σ i ) = E " τ=t β τ t Ψ (σ i (s τ, v iτ ), s τ, v iτ ) 0 θ i σ i # = W i (s t ; σ) 0 θ i where σ i (s t, v i ) : policy function (σ i : S R! A) # W i (s t ; σ) = E β τ t Ψ (σ i (s τ, v iτ ), s τ, v iτ ) σ i " τ=t 11

12 The Model: Markov Perfect Equilibrium n a Markov perfect equilibrium, an equilibrium strategy σ i (s, v) must be the best response to its rivals equilibrium strategy σ i : V i (s; σ i, σ i ) V i for all i, s 2 S and σ 0 i s; σi 0, σ i Exploiting the linearity of the period pro t function, Wi (s; σi, σ i ) W i s; σi 0, σ i θi 0 for all i, s 2 S and σ 0 i 12

13 Nested Fixed Point Algorithm Brute-force method is hopeless in most dynamic games Evaluating the likelihood function even once could take more than a day Plus, multiple equilibria are prevalent 13

14 Using BBL Suzuki (2009) used BBL for estimating structural cost parameters t does not require discretization of continuous state variables calculating the inverse matrix Three continuous variables with 20 intervals would be a matrix 14

15 Estimation Step 1: Revenue Function Estimation ln r ikt = γ i + η 1 + x 0 tη 2 η 3 ln (Σ j h jt ) η 4 ln h it + ɛ ikt Exploit the panel structure Market speci c xed e ects (at least partially) take care of the endogeneity of h it Chain-speci c e ects are also taken into account The transition function (population, establishments and trend) are estimated by AR1 regression 15

16 Estimation Step 2: Policy Function Estimation y it = α i + α m + α 2 ln x t α 3 ln h it α 4 (ln h it ) 2 α 5 (ln (Σ j h jt )) α 6 (ln (Σ j h jt )) 2 + ω it Estimate the policy function by using an ordered logit Use market-dummy to take into account the impact of unobservable market-speci c characteristics on their entry-exit decision Take into account the change in choice set 16

17 Estimation Step 3: Recovering Cost Parameters Recover chain i s structural cost parameters ˆθ i that rationalizes the estimated revenue function and the estimated policy function Each chain s policy function must be the best response to its rivals policy functions The observed policy must beat other possible policies when its rivals play observed policies BBL shows that we can exploit this property to identify structural parameters No data are used once policy functions are estimated 17

18 mplementing BBL Step by Step Focus on chain i s structural parameters in market m Generate chain i s fake policies fσ m i g N perturbing the observed policy function m=1 by slightly How we should make perturbations is an open question Di erent perturbation will a ect the e ciency of estimators Might make a sense to consider a perturbation with clear intuition (e.g., never enter) 18

19 mplementing BBL Step by Step Want to approximate the expected sum of future pro t when they follow certain strategies Assuming other chains follow the observed policy ˆσ j6=i, simulate the model and calculate W i (s t ; σ) for the observed policy ˆσ i as well as σ m i Simulate economy for T periods T should be large enough so that β T is su ciently small The number of simulation should be very large since it needs to capture entry/exit behavior 19

20 Exploiting Linearity Note that linearity assumption allows us to separate simulation from estimation Simulation calculates W i (s t ; σ) No structural parameters are directly involved Otherwise, need to conduct simulations for each θ 20

21 Computational Tips in BBL Under linearity, you can separate simulation from optimization BBL requires running a large number of simulations Good thing is each simulation is independent Can save time by using multiple processors/computers at the same time Matlab has a toolbox for parallel computation 21

22 mplementing BBL Step by Step Try to nd the set of parameters that minimizes the following loss function: θ i = arg min θ N m=1 (min fg m (θ), 0g) 2 g m (θ) = W i (s; σ i, σ i ) W i s; σ 0 i, σ i θi When the observed policy beats a fake policy, it adds zero to the loss function When a fake policy beats the observed policy, it adds squared pro t-di erence to the loss function 22

23 Conducting Counterfactual Experiments A motivation of doing structural estimation is to conduct counterfactual experiments Using structural parameters, the model can calculate an equilibrium under alternative policies Can compare two competing policies by comparing the welfare level they bring 23

24 Conducting Counterfactual Experiments Note that in counterfactual experiments, you need to solve the model explicitly n complicated models, solving an equilibrium might be virtually impossible You might need to make a model simple in simulations Also multiple equilibria are prevalent in dynamic games The impacts of policies might not be pinned down 24

25 Summary Go over the very basics of estimating a dynamics entry-exit model Nested xed point algorithm turns out to be virtually impossible BBL with linearity assumption signi cantly reduces computational burden without hurting consistency 25