Game Theory and Econometrics: A Survey of Some Recent Research

Transcription

1 Game Theory and Econometrics: A Survey of Some Recent Research Pat Bajari 1 Han Hong 2 Denis Nekipelov 3 1 University of Minnesota and NBER 2 Stanford University 3 University of California at Berkeley Econometric Society World Congress in Shanghai, August 17, / 105

2 Motivation We attempt to survey an emerging literature at the intersection of industrial organization, game theory and econometrics. Researchers specify a set of players, their strategies, information, and payoffs, and use equilibrium concepts to derive positive and normative economic predictions. Researchers specify a set of players, their strategies, information, The predictions of game theoretic models often depend delicately on the specification of the game. Theory often provides little guidance on how to choose between multiple equilibrium generated by a particular game 2 / 105

3 Motivation The empirical literature addresses these issue by letting the data tell us the model specifications that best explain observed behavior. Econometricians observe data from plays of a game and exogenous covariates which influence payoffs or constraints faced by the agent. The payoffs are specified as a parametric or nonparametric function of the actions of other agents and exogenous covariates Use econometric estimators to reverse engineer payoffs to explain the observed behavior. 3 / 105

4 Earlier Literature Bresnahan and Reiss (1990, 1991) Berry (1991) Manski (1993), The Reflection Problem. Brock and Durlauf (2001). 4 / 105

5 Motivation Econometric methods to estimate games for a diverse set of problems. Static games with discrete strategies, e.g. Seim (2006), Sweeting (2009) Dynamic games with discrete strategies, e.g. Aguirregabiria and Mira (2007), Pesendorfer and Schmidt-Dengler (2003, 2008). Dynamic games with possibly continuous strategies, e.g. Pesendorfer and Jofre-Bonet (2003), Bajari, Benkard and Levin (2007). Among many others. We will focus on discrete actions. 5 / 105

6 Literature Early examples, Vuong and Bjorn (1984) and Bresnahan and Reiss (1990,1991). Also, Aradillas-Lopez (2005), Ho(2005), Ishii (2005), Pakes, Porter, Ho and Ishii (2005), Rysman (2004), Seim(2004), Sweeting (2004), Tamer (2003) and Ciliberto and Tamer (2005). Single agent dynamic models: Rust (1987), Hotz and Miller (1993), Magnac and Thesmar (2003), Heckman and Navarro (2005), among many others. Dynamic games: Aguirregabiria and Mira (2003), Bajari, Benkard and Levin (2004), Pakes, Ovstrovsky and Berry (2004), and Pesendorfer and Schmidt-Dengler (2004). 6 / 105

7 Common insight Equilibrium of (dynamic) games can be difficult to compute. Brute force approaches which repeatedly computes the equilibrium for alternative parameter values can be difficult or infeasible. Close relation between information structure and estimation strategy. Private information versus unobserved heterogeneity. With only private information and no unobserved heterogeneity, a flexible two step estimation approach is available. Nonparametric identification and estimation are also feasible under this assumption, where the model resembles a single agent decision problem. 7 / 105

8 Common insight The private information assumption equates the information set of the players (regarding the competition environment) with that of the researchers. In the presence of substantial unobserved heterogeneity, both identification and estimation are more difficult. Computationally intensive but the difficulty of implementation can be reduced with a careful design of estimation strategy. A fascinating broad and general literature. The domain of applicability is not limited to industrial organization. Generalizes to seemingly complicated structural models in other fields such as macro, labor, public finance, or marketing. 8 / 105

9 Self promotion Patrick Bajari, Han Hong, John Krainer and Denis Nekipelov. Estimating Static Models of Strategic Interactions. Patrick Bajari and Victor Chernozhukov and Han Hong and Denis Nekipelov Nonparametric Identification and Semiparametric Estimation of a Dynamic Game. Patrick Bajari, Han Hong and Stephen Ryan. Identification and Estimation of Discrete Games of Complete Information. Ronald Gallant, Han Hong and Ahmed Khwaja. Bayesian Estimation of a Dynamic Oligopolistic Game with Serially Correlated Unobserved Cost Variables 9 / 105

10 A general static model Players, i = 1,..., n. Actions a i {0, 1,..., K}. A = {0, 1,..., K} n and a = (a 1,..., a n ).. s i S i : state for player i. S = Π i S i and s = (s 1,..., s n ) S. For each agent i, K + 1 state variables ɛ i (a i ) ɛ i = (ɛ i (0),..., ɛ i (K)). Density f (ɛ i ), i.i.d. across i = 1,..., n. 10 / 105

11 A general static model Private information model s is common knowledge and also observed by econometrician. ɛ i (a i ): private information to each agent. Private information model with unobserved heterogeneity At least some components of s are not observed by the researchers. Complete information model ɛ i (a i ): public information to all agents. 11 / 105

12 Private information model: Bayesian nash equilibrium Choice probability σ i (a i s) for player i conditional on s, public information: σ i (a i = k s) = 1 {δ i (s, ɛ i ) = k} f (ɛ i )dɛ i. Choice specific expected payoff for player i: Π i (a i, s; θ) = a i Π i (a i, a i, s; θ)σ i (a i s). Expected utility from choosing a i, excluding preference shock. An example from an entry game. 12 / 105

13 Π i (a i, a i, s; θ) is often a linear function, e.g.: { s β + δ 1 {a j = 1} if a i = 1 Π i (a i, a i, s) = j i 0 if a i = 0 Mean utility from not entering normalized to zero. δ measures the influence of j s entry choice on i s profit. If firms compete with each other: δ < 0. β measure the impact of the state variables on profits. ε i (a i ) capture shocks to the profitability of entry. Often ε i (a i ) are assumed to be i.i.d. extreme value distributed: f (ε i (k)) = e ε i (k) e e ε i (k). 13 / 105

14 Equilibrium Choice specific expected payoff under linearity: Π i (a i = 1, s; θ) = s β + δ j i σ j (a j = 1 s). Choice probability under the extreme value distribution σ i (a i = 1 s) = exp(s β + δ j i σ j(a j = 1 s)) 1 + exp(s β + δ j i σ j(a j = 1 s)) Choice probability equations define fixed point mappings for σ j (a j = 1 s). For each θ = (β, δ), solve for for all j = 1,..., n, σ j (a j = 1 s; θ) Maximum likelihood versus two step estimator. Nonparametric estimator also follows from identification arguments. 14 / 105

15 Identification The error term distribution can possibly be identified if a parametric or index structure is imposed on the payoff function. Consider fixing the error term (extreme value without loss of generality) and identifying the payoff nonparametrically. σ i (a i s) only functions of Π i (a i, s) Π i (a i = 0, s). Normalization: For all i and all a i and s, Π i (a i = 0, a i, s) = 0, so that Π i (a i = 0, s) = / 105

16 Hotz and Miller (1993) inversion, for any k, k : log (σ i (k s)) log ( σ i (k s) ) = Π i (k, s) Π i (k, s). More generally let Γ : {0,..., K} S [0, 1]: (σ i (0 s),..., σ i (K s)) = Γ i (Π i (1, s) Π i (0, s),..., Π i (K, s) Π i (0, s)) And the inverse Γ 1 : (Π i (1, s) Π i (0, s),..., Π i (K, s) Π i (0, s)) = Γ 1 i (σ i (0 s),..., σ i (K s)) Invert equilibrium choice probabilities to nonparametrically recover Π i (1, s) Π i (0, s),..., Π i (K, s) Π i (0, s). Π i (a i, s) is known by our inversion and probabilites σ i can be observed by econometrician. 16 / 105

17 Next step: how to recover Π i (a i, a i, s) from Π i (a i, s). Requires inversion of the following system: Π i (a i, s) = a i σ i (a i s)π i (a i, a i, s), i = 1,..., n, a i = 1,..., K.. Given s, n K (K + 1) n 1 unknowns utilities of all agents. Only n (K) known expected utilities. Obvious solution: impose exclusion restrictions. 17 / 105

18 Partition s = (s i, s i ), and suppose Π i (a i, a i, s) = Π i (a i, a i, s i ) depends only on the subvector s i. Π i (a i, s i, s i ) = a i σ i (a i s i, s i )Π i (a i, a i, s i ). Identification: Given each s i, the second moment matrix of the regressors σ i (a i s i, s i ), is nonsingular. Eσ i (a i s i, s i )σ i (a i s i, s i ) Needs at least (K + 1) n 1 points in the support of the conditional distribution of s i given s i. 18 / 105

19 In peer interaction models with a large homogeneous peer group (e.g Brock and Durlauf (2001)), a standard logit regression with an additional equilibrium probability covariate is consistent despite the possibility of multiple equilibria. Still need multiple peer group to identify the peer effect coefficient. In entry games, a single equilibrium is plausible with time series data in a single market (Pesendorfer and Schmidt-Dengler (2003)). In cross section data, multiple markets might play different equilibria. With panel data, the equilibrium can be estimated market by market. 19 / 105

20 Nonparametric Estimation Step 1: Estimation of choice probabilities (e.g. sieve) where and σ i (k s) = T 1(a it = k)q κ(t ) (s t)(q T Q T ) 1 q κ(t ) (s). t=1 Q T = (q κ(t ) (s 1),..., q κ(t ) (s T ), q κ(t ) (s) = (q 1(s),..., q κ(t ) (s)). Step 2: Inversion of expected utilities ( Πi (1, s t) Π i (0, s t),... Π i (K, s t) Π i (0, s t)) In the logit model ˆΠ i (k, s t) ˆΠ i (0, s t) = log ( σ i (k s t)) log ( σ i (0 s t)) = Γ 1 i ( σ i (0 s t),..., σ i (K s t)) 20 / 105

21 Step 3: Recovering structural parameters. Infer Π i (a i, a i, s i ) from ˆΠ i (k, s). Use empirical analog of Π i (a i, s i, s i ) = σ i (a i s i, s i )Π i (a i, a i, s i ). a i For given s i and given a = (a i, a i ), run local regression T ˆΠi (a i, s it, s i ) 2 ˆσ i (a i s it, s i ) Π i (a i, a i, s i ) w (t, s i ). a i t=1 Nonparametric weights can take different forms, e.g. w (t, s i ) = k Asymptotically identified. ( sit s ) i / h T k τ=1 ( siτ s ) i. h 21 / 105

22 Semiparametric Estimation Linear model of deterministic utility Π i (a i, a i, s i ) = Φ i (a i, a i, s i ) θ. Choice specific value function where Π i (a i, s) = E [Π i (a i, a i, s i ) s, a i ] = Φ i (a i, s) θ. Φ i (a i, s) = a i ) σ i (a i s, ˆΦ, θ = Φ i (a i, a i, s i ) j i σ(a j = k j s). Estimated parameter choice probabilities: ) exp (ˆΦ i (a i, s) θ 1 + K k=1 exp (ˆΦ i (k, s) θ Semiparametric method of moment estimator 1 T ( ( )) Â (s t) y t σ s t, ˆΦ, ˆθ = 0. T where ( ) σ s t, ˆΦ, θ = t=1 ( ) ) (σ i k s t, ˆΦ, θ, k = 1,..., K, i = 1,..., n ) 22 / 105

23 Nonparametric vs semiparametric methods Typically σ i (k s) will converge to the true σ i (k s) at a nonparametric rate which is slower than T 1/2. Alternative estimators: kernel smoothing or local polynomial Nonparametric method is more robust against misspecification. May suffer from curse of dimensionality. Semiparametric method more practical in applications. θ can be estimated at parametric rates despite first stage nonparametric estimation. Pseudo MLE can be used in place of Method of moments. 23 / 105

24 Rust Chapter Markov, Discrete dependent variable, Dynamic Programming State space S (can be continuous or discrete) Usually need discretization in value function computation. Action (choice) set D. Time separable utility function: for T, U T (s, d) = T β t u t (s t, d t, θ) t=0 In above s and d are sequences of state variables and actions. δ = (δ 0,..., δ T ), where δ t = δ t (s t ) such as to maximize: max E δu T (s, d). δ=(δ 0,...,δ T ) In above δ generates the DGP for s t. 24 / 105

25 Finite Horizon case: Suppose the continuation value at time T is 0. (Might be questionable in the presence of bequest motives). δ T (s T ) = arg max d D U T (s T, d) For t = 0,..., T 1, { δ t (s t ) = arg max u t (s t, d) + β d D V T (s T ) = max d D U T (s T, d). } V t+1 (s t+1, δ t+1 (s t+1 )) dp (s t+1 s t, d) and { V t (s t ) = max u t (s t, d) + β d D } V t+1 (s t+1, δ t+1 (s t+1 )) dp (s t+1 s t, d) 25 / 105

26 δ = (δ 0,..., δ T ) maximizes [ ) ] V 0 (s) = max E δ U T ( s, d s 0 = s. δ This is called backward induction, can approximate infinite horizon model: ( lim E δ U T s, d ) = lim sup E δ U T ( s, d ) = sup E δ U ( s, d ). T T δ δ 26 / 105

27 Infinite horizon stationary case: time invariance δ t (s) δ (s) V t (s) = V (s). More applicable in IO? δ (s) = arg max d D u (s, d) + β V (s) = arg max u (s, d) + β d D V (s ) dp (s s, d) V (s ) dp (s s, d). 27 / 105

28 Contraction mapping properties V (s) W (s) max u (s, d) + β V (s ) dp (s s, d) d max u (s, d) + β W (s ) dp (s s, d) d max d β V (s ) dp (s s, d) β W (s ) dp (s s, d) max β V (s ) W (s ) dp (s s) d β sup V (s) W (s). s Take sup of left hand side contraction mapping. 28 / 105

29 Define functional mapping: Γ (W ) [s] = max d Bellman equation: [ u (s, d) + β V = Γ (V ). ] W (s ) dp (s s, d). Operationally, set V 0 (s) 0, V 1 (s) = Γ (V 0 ) [s]. This is the same as using T -finite horizon backward induction to approximation the infinite solution. How fast does it converge? How large does T has to be as a function of the sample size? ] V 0 V T 0 (s) = max δ (s) = max δ [ E δ β t u (s t, d t ) s 0 = s t=0 [ T ] E δ β t u (s t, d t ) s 0 = s t=0 lim V 0 T (s) =V0 (s), s S. T This is called successive approximation. 29 / 105

30 Discretization s = 1,..., S, or i u (s, δ (s)) u (i, δ (i)) S S transition probability matrix E δ, i, jth component: Then write Γ (V ) [s] = max d V δ }{{} s 1 [ p [i j, δ (j)]. u (s, d) + β = u (δ) +β E }{{}}{{} δ s 1 s s V δ }{{} s 1 ] S V (s ) p (s s, d) s =1 V δ =G δ (V δ ) = (I βe δ ) 1 u δ =u δ + βe δ u δ + β 2 E 2 δ u δ + 30 / 105

31 This is called policy iterations: δ k (s) = arg max d u (s, d) + β S V δk 1 (s ) p (s s, d) s =1 For each k 1, compute V δk 1 (s ) by the s s matrix inversion, then move on to δ k (s), iterate. Alternative 1: Newton Iteration (see Rust s code) Alternative 2: use a linear (or polynomial) approximation (Judd): V (s) = c 1 χ 1 (s) + c 2 χ 2 (s) + + c N χ N (s). where ĉ = arg max c V (s) Γ (V (s)) 31 / 105

32 Econometric assumptions: stationary markov case δ (s) = d if and only if u (s, d) + β V (s ) dp (s s, d) u (s, d ) + β V (s ) dp (s s, d). Introduce observed and unobserved state variables: s = (x, ɛ). We only observe conditional choice probabilities given observable state variables: P (d x) = 1 (δ (s) = d) dp (s x) = 1 (d = δ (s)) dp (ɛ x) 32 / 105

33 Key assumptions: Additivity: u (s, d) = u (x, d) + ɛ (d). Conditional independence: p (x t+1, ɛ t+1 x t, ɛ t, d t ) =p (x t+1 x t, d t ) p (ɛ t x t, d t ) p (ɛ t x t, d t ) =p (ɛ t ) ɛ t is i.i.d. and independent of everything else. Bellman equation: [ V (s) = V (x, ɛ) = max u (x, d) + ɛ (d) + β V (y, ɛ) dp (ɛ) dp (y x, d) d D = max [V (x, d) + ɛ (d)] d D where V (x, d) is called the choice specific value function. 33 / 105

34 Ex ante value function: (McFadden s social surplus function) V (y) =G (v (y, 1),..., V (y, d)) = V (y, ɛ) dp (ɛ) = max (V (x, d) + ɛ (d)) dp (ɛ). d D Relation between choice probability and social surplus: This is because G (v (x, 1),..., v (x, d)) v (x, i) G (v (x, 1),..., v (x, d)) v (x, i) = p (d = i x). maxd = D (v (x, d ) + ɛ (d )) dq (ɛ) v (x, d) ( ) = 1 d = arg max [u (x, d ) + ɛ (d )] dq (ɛ) d =P (d is chosen x). 34 / 105

35 Under the extreme value assumption: ( ) G (v (x, d), d D) = log exp (v (x, d)). d D Choice specific value function iteration: ( ) v (x, d) = u (x, d) + β log exp (v (y, d )) dp (y x, d). Choice probabilities: d D P (d x) = e v(x,d) e v(x,d ). d D 35 / 105

36 Likelihood estimation: L (θ) = A a=1 t=1 T p (dt a xt a, θ 1, θ 2 ) π ( xt a xt 1, a dt 1, a ) θ 1. Usually, θ 1 can be estimated using the transition process only: ˆθ 1 = arg max θ 1 A a=1 t=1 T π ( xt a xt 1, a dt 1, a ) θ 1. For each θ, compute u θ (x, d), calculate v θ (x, d) either by backward induction or by value function iteration. Then form p (d x, θ) = e vθ(x,d) e v θ(x,d ). d D 36 / 105

37 The utility specifications used in Rust (1987) Bus Engine: { θ1 d = 1 u θ (x, d) = θ 2 x t d = 0 Transition density specification: { g (x π (x t+1 x t, d t ) = t+1, θ 3 ) d t = 1 g (x t+1 x t, θ 3 ) d t = 0 Rust tried both β = 0 and β = / 105

38 Hotz and Hiller (1993, Review of Economic Studies) The idea in this paper predates many recent follow-up works. There is a one-to-one mapping between p (d x) and (x, d) = v (x, d) v (x, 1). In particular, under the extreme value assumption, (x, d) = log p (d x) p (1 x). 38 / 105

39 A two-step estimation procedure: Estimate the transition density ˆπ (x t+1 x t, d t ) first. Flexibly estimate ˆ (x, d), then you can compute the optimal choice rule ˆδ (x, ɛ) = arg max d D [ ˆ (x, d) + ɛ (d) ɛ (1) ] Simulate the entire path (one can take T for stationary models) of x, d and ɛ. Compute ˆV θ (x, d) = T t=0 [ β t u θ ( x t, ˆδ ) )] ( x t, ɛ t ) + ɛ t (ˆδ ( xt, ɛ t ). Use some norm, such as GMM, to match the flexibly estimated ˆ (x, d) to ˆV θ (x, d) ˆV θ (x, 1) through the best choice of θ using the previous step simulation. 39 / 105

40 Identifying Dynamic Discrete Decision Processes Thierry Magnac and David Thesmar Two period model. i I = {1,..., K}. State variable h = (x, ɛ). ɛ = {ɛ 1,..., ɛ K }. Period 1 utility: u i (x, ɛ). Next period state variables: h = ( x, ɛ ) drawn conditional on h = (x, ɛ). 40 / 105

41 Additive separability Assumptions ɛ is independent of x. i I, u i (x, ɛ) = u i (x) + ɛ i, Conditional independence: Random preference shocks at two periods ɛ and ɛ are independent and indepedent of x and d = i. Discrete support: The support of first period state variables x (resp. second period x ) is X (resp. X ). The joint support X = X X is discrete and finite, i.e., X = {x 1,..., x # X } 41 / 105

42 Transition matrix of the (x, ɛ) process P ( h h, d ) = P ( x, ɛ x, d ) = G ( ɛ ) P ( x x, d ) Bellman equation ( ( v i (x, ɛ) = u i (x) + ɛ i + βe max v j x, ɛ ) ) x, d = i j Decompose v i (x, ɛ) = v i (x) + ɛ i where v i (x) = u i (x) + βe ( max j ( ( v j x ) ) + ɛ ) j x, d = i 42 / 105

43 What can be recovered from the data: ( d, d ) I 2, ( x, x ) X X, P ( d, x, d x ) = P ( d x ) P ( x x, d ) P (d x) P (x x, d) is nonparametrically specified and is exactly identified from the data. Can P (d x ) and P (d x) be used to identify the structural parameters: b = {u1 (X ),..., uk (X ), v 1 ( X ),..., vk ( X ), G, β} where f (X ) is a short-cut for {f (x), x X }. For example, u 1 (X ) = {u 1 (x), x X }. 43 / 105

44 The probability that the agent chooses d = i given the structure b and observable state variable x: p i ( x; b) = P ( v Definition of identification ( x, i) X I i ( x; b) + ɛ i = max Observable vector of choice probabilities: p ( x) = (p 1 ( x),..., p K ( x)) j ( v j ( x; b) + ɛ j ) x, b ). Number of observable choice probabilities and number of parameters that can be identified. 44 / 105

45 Mapping between the observable choice probabilities and the vector of value functions: ( x, i) X I, v i ( x) = v K ( x) + q i ( p ( x) ; G) There is no loss of generality in setting vk (X ) = 0. Then for i = 1,..., K 1: vi ( X ) ( = q i ( p X ) ; G ) There is also no loss of generality setting uk (x) = / 105

46 Expected second period value function: for v ( x ) = {v1 ( x ),..., vk ( x ) } Define R ( v ( x ) ; G ) ( ( = E G max v i x ) ) + ɛ i i I Decompose the first period total utility function vi (x) = ui (x) + βe [ R ( v ( x ) ; G ) x, d = i ] Recall Now q i (x) = v i (x) v K (x) v i (x) = u i (x) + βe [ R ( v ( x ) ; G ) x, d = i ] and since uk (x) = 0: v K (x) = βe [ R ( v ( x ) ; G ) x, d = K ] 46 / 105

47 Combine these relations: q i (x) =v i (x) v K (x) =u i (x) + βe [ R ( v ( x ) ; G ) x, d = i ] Therefore u i (x) can be recovered by βe [ R ( v ( x ) ; G ) x, d = K ] u i (x) = q i (x) βe [ R ( v ( x ) ; G ) x, d = i ] + βe [ R ( v ( x ) ; G ) x, d = K ] So given β, G, u K (x) = 0, v K (x ) = 0, the other utility functions u i (x), i = 1,..., K 1 and the other second period value functions v i (x ), i = 1,..., K 1 can be identified. Estimation follows from identification. 47 / 105

48 Exclusion restrictions might identify β: if (x 1, x 2 ) X 2, i I, such that x 1 x 2, u i (x 1 ) = u i (x 2 ) but x 1 and x 2 still generate different transition probabilites P (x x, d), then q i (x 1 ) should be different from q i (x 2 ). 48 / 105

49 Identifying β: q i (x 1 ) βe [ R ( v ( x ) ; G ) x 1, d = i ] + βe [ R ( v ( x ) ; G ) x 1, d = K ] { q i (x 2 ) βe [ R ( v ( x ) ; G ) x 2, d = i ] + βe [ R ( v ( x ) ; G ) x 2, d = K ]} = 0. Parametric restriction: They said it can be used to identify G. Is this true? 49 / 105

50 Unobserved heterogeneity (fixed effect) ( vi (x, θ) = ui (x, θ) + βe max j ( ( v j x, θ ) ) + ɛ ) j d = i, x, θ θ takes two values 0 and 1, with probabilities r(0) and r(1). There are two alternatives only: d = 0, 1. p (x, θ) P (d = 1 x, θ). s (x) = P (d = 1 x) unconditional on θ. State 2 in the second period is absorbing. θ does not affect the transition probabilities P ( x x, d ) = P ( x x, d ) and G (ɛ θ) = G (ɛ) θ is independent of x. 50 / 105

51 Conditional on unobserved heterogeneity u 1 (x, θ) = q (p (x, θ) ; G) βe [ R ( q ( p ( x, θ ) ; G ) ; G ) d = 1, x ] Unconditional (on unobserved heterogeneity) choice probabilities s (x) = p (x, 1) r (1) + p (x, 0) r (0) and for x including x and d, s ( x ) = p ( x, 1 ) p (x; 1) r (1) s (x) + p ( x, 0 ) p (x, 0) r (0) s (x) If you know r(1), hence r(0), and if you know p (x, 1) and p (x, 1), then you can identify p (x, 0) and p (x, 1), and identify u1 (x, θ) in turn. Or you can assume p (x, 1) = p (x, 0) = s (x ) instead. 51 / 105

52 Introducing Dynamics into Incomplete Information Games Players are forward looking. Infinite Horizon, Stationary, Markov Transition Now players maximize expected discounted utility using discount factor β. { W i (s, ɛ i ; σ) = max Π i (a i, s) + ɛ i (a i ) a i A i } + β W i (s, ɛ i ; σ)g(s s, a i, a i )σ i (a i s)f (ɛ i )dɛ i a i Definition: A Markov Perfect Equilibrium is a collection of δ i (s, ɛ i ), i = 1,..., n such that for all i, all s and all ɛ i, δ i (s, ɛ i ) maximizes W i (s, ɛ i ; σ i, σ i ). Equilibrium existence: Pesendorfer and Schmidt-Dengler (2003), Jenkins, Liu, McFadden and Matzkin (2007). 52 / 105

53 Conditional independence: ɛ distributed i.i.d. over time. State variables evolve according to g (s s, a i, a i ). Define choice specific value function V i (a i, s) = Π i (a i, s) + βe [ V i (s ) s, a i ]. Players chooose a i to maximize V i (a i, s) + ɛ i (a i ), Ex ante value function (Social surplus function) V i (s) =E ɛi max a i [V i (a i, s) + ɛ i (a i )] =G (V i (a i, s), a i = 0,..., K) =G (V i (a i, s) V i (0, s), a i = 1,..., K) + V i (0, s) 53 / 105

54 When the error terms are extreme value distributed V i (s) = log K exp (V i (k, s) V i (0, s)) + V i (0, s) k=0 = log σ i (0 s) + V i (0, s) Relationship between Π i (a i, s) and V i (a i, s): V i (a i, s) =Π i (a i, s) + βe [ G ( V i (a i, s ), a i = 0,..., K ) s, a i ] =Π i (a i, s) + βe [ G ( V i (k, s ) V i ( 0, s ), k = 1,..., K ) s, a i ] + βe [ V i ( 0, s ) s, a i ] With extreme value distributed error terms V i (a i, s) =Π i (a i, s) βe [σ i (0 s) s, a i ] + βe [ V i ( 0, s ) s, a i ] 54 / 105

55 Hotz and Miller (1993): one to one mapping between σ i (a i s) and differences in choice specific value functions: (V i (1, s) V i (0, s),...v i (K, s) V i (0, s)) = Ω i (σ i (0 s),..., σ i (K s)) Example: i.i.d extreme value f (ɛ i ): Inverse mapping: σ i (a i s) = exp(v i (a i, s) V i (0, s)) K k=0 exp(v i(k, s) V i (0, s)) log (σ i (k s)) log (σ i (0 s)) = V i (k, s) V i (0, s) Since we can recover V i (k, s) V i (0, s), we only need to know V i (0, s) to recover V i (k, s), k. If we know V i (0, s), V i (a i, s) and Π i (a i, s) is one to one. 55 / 105

56 Identify V i (0, s) first. Set a i = 0: V i (0, s) =Π i (0, s) βe [ σ i (0 s ) s, 0] + βe [V i (0, s ) s, 0] This is a single contraction mapping unique fixed point iteration. Add V i (0, s) to V i (k, s) V i (0, s) to identify all V i (k, s). Then all Π i (k, s) calculated from V i (k, s) through Π i (k, s) = V i (k, s) βe [V i (s ) s, k]. 56 / 105

57 Why normalize Π i (0, s) = 0? Why not V i (0, s) = 0? If a firm stays out of the market in period t, current profit 0, but option value of future entry might depend on market size, number of other firms, etc. These state variables might evolve stochastically. Rest of the identification arguments: identical to the static model. 57 / 105

58 Nonparametric and Semiparametric Estimation Hotz-Miller inversion recovers V i (k, s) V i (0, s) instead of Π i (k, s) Π i (0, s). Nonparametrically compute V i (0, s) using ˆV i (0, s) = βê [σ i (0 s ) s, 0] [ + βê ˆVi (0, s ) s, 0] Obtain and ˆV i (k, s) and forward compute ˆΠ i (k, s). The rest is identical to the static model. 58 / 105

59 Efficient method of estimation Nested fixed point MLE should achieve parametric efficiency. A flexible sieve MLE should also be efficiency when the transition density is nonparametrically specified. Both can be intensive to compute. Alternative: view the value function iteration as a conditional moment model when the transition density is nonparametric. [ [ K ] V i (a i, s) = E Π i (a, s i ; γ) + β log exp (V i (l, s ))] a i, s. l=0 Combine this with the conditional moment: E (1 (a i = k) s m,t ) = σ i (k s m,t ) = exp (V i (k, s)). K exp (V i (l, s)) Make use techniques from the sieve/conditional moment literature (with possibly different conditioning sets), e.g. Newey and Powell, Ai and Chen, Chen and Pouzo. l=0 59 / 105

60 Nonparametric estimation Using kernel or sieve methods to account for continuous state variables in two step estimators are important. When the dimension of continuous state variable is at least 4, discretizing the state space in the first stage of nonparametric estimation will not produce n consistent parameter estimates in the second stage for the payoff functions. Unless an elaborate weighting scheme is used to combine the first stage estimates at different discretized points (similar to higher order kernel), e.g. Hong, Mahajan and Nekipelov (2010). For example, let the distance between two grid points to be h. Bias of order h 2. Need nh 2 0. Also need nh d. Not possible if d / 105

61 Equilibrium Computation Policy simulation still requires computing all equilibria. One approach is to view equilibrium computation as embedded in a constrained optimization program, e.g. Doraszelski, Judd, and Center (2008), Judd and Su (2006). The alternative all-solution homopoty method can be useful. Discussed in, for example, Nekipelov (2010), Borkovsky, Doraszelski, and Kryukov (2009), Besanko and Doraszelski (2004) and Bajari, Hong, Krainer and Nekipelov (2009). 61 / 105

62 Extensions of incomplete information models Nonstationary data and finite horizon Taber (2000) and Magnac and Thesmar (2002). Finite horizon model can be stationary or nonstationary, but typically nonstationary for exact identification with a panel data. Typically assume no continuation value beyond end period. Utility of one of the choices in the last period needs to be normalized to zero. 62 / 105

63 For dynamic models, zero utility for one choice might not be a good assumption for some applications. For static models, only the differences in utilities across choices are identified. For dynamic models, only a linear combination of the utilities under different choices across the state variables can be identified. The exact form of the linear combination depends on the transition probability matrixes. For some applications, parametric models can be more appropriate than normalizing one of the utilities to zero. 63 / 105

64 Identifying discount rates Magnac and Thesmar (2002), Aguirregabiria et al (2007), etc. Insight: need excluded variables that affect transition but not static utility. Intuition: individuals with different values of excluded variables behave similarly if they are myopic but differently if they are forward looking. Formally, find discount rate value so that the nonparametrically recovered static utility is the same for different values of the excluded variable. Identification also applies to hyperbolic discounting models, Fang and Wang (2009). Fernandez-Villaverde et al: difficult to distinguish discount rate heterogeneity from myopic or hyperbolic behavior. 64 / 105

65 Irregular discrete observations The observation time interval can be different from the model time interval, e.g. annual decision but observation only once every two years. Most researchers will assume that they are equal (e.g. using a model with decision once every two years), using a suitable discount rate. But this is not necessarily consistent. Correct approach (in the context of stationary models): First estimate the multi-period transition density from the data. Recover single period transition density from the multi-period one and the estimated choice probability function. Use single period transition density for subsequent estimation steps, e.g. by simulating data from it. 65 / 105

66 Continuous time model Doraszelski and Judd (2007): Players move sequentially, at random times determined by nature. Sequential move continuous time model easier for simulating counterfactual policy outcomes. Bayer, Arcidiacono, Blevins, and Ellickson (2010): two step estimators can also be used for continuous time models. Nonparametric identification argument also goes through. Replace discrete time contraction mapping by, e.g. V i (a i, s) Π i (a i, s) + e ρt E [ ( V i s ) s, a i, t ] f (t) dt. f (t) is nature s distribution of the random arrival time of the next move. Both the conditional expectation above and f (t) can be identified and presumably estimated from the data. 66 / 105

67 Unobserved heterogeneity Some of the state variables that are observed by all the players in the model might not be observed by the econometrician. Partition state variables into s, u, where s is observed by the econometrician but u is not. Focus on the case where the players know more than the econometrician. Also conceivable that the econometrician has more information at the time of analysis, e.g. Aradillas-Lopez (2005). Both period utility Π i (a i, a i, s, u) and the transition process g (s, u s, a, u) can potentially be contaminated by unobserved heterogeneity. The discount rate can also depend on observed and unobserved heterogeneity. 67 / 105

68 Unobserved heterogeneity: identification Kasahara and Shimotsu (2008), Hu and Shum (2008). Insight: identification is a functional mapping, [ β, σi (a s), g ( s s, a )] Π i (a i, a i, s), or [ σi (a s), g ( s s, a )] [β, Π i (a i, a i, s)]. Therefore it suffices to identify the unobserved heterogeneity in the reduced forms of σ i (a s, u) and g (s, u s, a, u). Intuition: unobserved heterogeneity induces serial correlation, and can potentially be identified with panel data. Challenging with possibly continuous unobserved heterogeneity that can be serially correlated. 68 / 105

69 Unobserved heterogeneity: estimation Monte Carlo Markov Chain Bayesian method useful for estimating models with unobserved heterogeneity. Imai, Jain and Ching (2009): update value function once for each parameter iteration along the MCMC chain. Norets (2009): Gibbs sampling to incorporate serially correlated unobserved heterogeneity. Chernozhukov and Hong (2003): MCMC methods are applicable for both frequentist and Bayesian point estimators and inferences. 69 / 105

70 Unobserved heterogeneity: estimation Computational challenges of Bayesian estimators Computing the expected surplus next period given the current state variables and actions. Kernel methods (Imai, Jain and Ching (2009)) Nearest neighborhood and importance sampler (Norets (2009)) Parametric numerical quadrature (Gallant, Hong and Khwaja (2008)). Extending Bayesian methods for DDC models to dynamic games requires a solution algorithm. Both computationally intensive and issues with multiple equilibria. 70 / 105

71 Simulating from the conditional posterior distribution of parameters and latent state variables. Norets (2009) and Imai, Jain and Ching (2009): MCMC for both. Alternative: sequential importance sampler, also called particle filter. Importance sampler does not appear to work well for parameters. But it may work well for unobserved state variables, which has larger dimensions and whose posterior does not concentrate when the sample size increases. Particle filters: Fernandez-Villaverde and Rubio-Ramirez (2007) for DSGE models, Blevins (2009) for incomplete information games in combination with two step estimators, Gallant, Hong and Khwaja (2009) for complete information games. 71 / 105

72 Particle filtering Sequential conditioning versus integration. Particle filter draws from the sequential conditional distribution of latent state variables given data and parameters. Useful for both filtering and prediction. Can be integrated to form the likelihood for each parameter value. The likelihood can be maximized or run through MCMC. Alternative: EM algorithm of Arcidiacono and Miller (2006). Even when parameters are estimated by other methods (GMM etc), particle filtering can still be useful if the latent variable distribution is of interest itself. 72 / 105

73 Unobserved heterogeneity: flexible estimation Based on the insights in Kasahara and Shimotsu (2008), Hu and Shum (2008). One can nonparametrically estimate ˆσ i (a i s, u) and ĝ (s, u a, s, u) for example by a flexible parametric model σ i (a i s, u, θ n ) and g (s, u a, s, u, γ n ). Once the estimates of ˆσ i (a i s, u) and ĝ (s, u a, s, u) are available, they can be mapped into the mean payoff functions, Π i (a s, u) and possibly the discount rate and the error distribution. For example, one can simulate data from ˆσ i (a i s, u) and ĝ (s, u a, s, u), and apply a number of multi-step estimators. The reduced forms ˆσ i (a i s, u) and ĝ (s, u a, s, u) can be estimated using a variety of methods including MCMC, data augmentation, sequential importance sampler, or the EM algorithm. 73 / 105

74 Unobserved heterogeneity Consider single agent dynamic model first. Random coefficient model: Dan Ackerberg A New Use of Importance Sampling to Reduce Computational Burden in Simulation Estimation. Working paper, Bayesian methods: Imai et al Bayesian Estimation of Dynamic Discrete Choice Models. Working paper, Andriy Noretz Dynamic Discrete Choice Models with serially correlated unobservables. Working paper, / 105

75 Ackerberg 2001 Static utility: x i u + ɛ i, ɛ i extreme value. Forward looking agent with discounting Random coefficient: u g (x i, θ). Moments to match: 1 n n [1 (y i = a) P (y i = a x i, θ)] t (x i ). i=1 Conditional choice probabilities: P (y i = a x i, θ) = = e V (a,x i,u) a A ev (a,x i,u) g (u x i, θ) du e V (a,x i,u) a A ev (a,x i,u) g (u x i, θ) q (u x i ) q (u x i) du 75 / 105

76 Using S draws of u s, s = 1,..., S from q (u x i ), the conditional choice probability can be simulated as: ˆP (y i = a x i, θ) = 1 S S s=1 e V (a,x i,u is ) a A ev (a,x i,u is ) g (u is x i, θ) q (u is x i ) Simulated moment conditions: 1 n n i=1 [ 1 (y i = a) ˆP ] (y i = a x i, θ) t (x i ). which is [ 1 n 1 (y i = a) 1 n S i=1 S s=1 e V (a,x i,u is ) a A ev (a,x i,u is ) ] g (u is x i, θ) t (x i ). q (u is x i ) 76 / 105

77 Separation of simulation and value function computation from the estimation step. Draw u is, i = 1,..., n, s = 1,..., S before estimation starts Compute all V (a, x i, u is ) for all i = 1,..., n and s = 1,..., S before hand. No need to recompute V (a, x i, u is ) when estimating θ. θ only reweights the density g (u is x i, θ) q (u is x i ) during optimization estimation. Same logic applies to simulated MLE, and in Bayesian analysis. 77 / 105

78 Noretz 2006 Bayesian method: serially correlated unobserved state variables s t = (y t, x t ). x t observed. y t not observed. Use Gibbs sampler: Given θ, d t, x t, draw y t. Given d t, x t, y t, draw θ Joint likelihood: π (θ) p (y T,i, x T,i, d T,i,..., y 1,i, x 1,i, d 1,i θ) =π (θ) Π T t=1p (d t,i y t,i, x t,i, θ) f (x t,i, y t,i x t 1,i, y t 1,i, d t 1,i ; θ). conditional choice probability can be indicator: p (d i,t y i,t, x t,i, θ) =1 (V (y t,i, x t,i, d t,i ; θ) V (y t,i, x t,i, d; θ), d D). 78 / 105

79 Break unobservables into serially independent and serially dependent components: y t = (ν t, ɛ t ), f (x t+1, ν v+1, ɛ t+1 x t, ν t, ɛ t, d; θ) =p (ν v+1 x t+1, ɛ t+1 ; θ) p (x t+1, ɛ t+1 x t, ɛ t, d; θ). Joint likelihood becomes π (θ) i,t p (d t,i V t,i ) p (V i,t x i,t, ɛ i,t ; θ) p (x t,i, ɛ t,i x t 1,i, ɛ t 1,i, d t 1,i ; θ) V it can be drawn analytically conditional on (x i,t, ɛ i,t ; θ) subject to the constraints specified by the p (d t,i V t,i ) indicators. θ and ɛ i,t are drawn using Metropolis-Hasting steps. 79 / 105

80 Analytic drawing of V t,i = {V t,d,i = V (s t,i, d; θ), d D}, where s = (x, ɛ, ν), requires value function updating. For example: Then u (s t,i, d; θ) = u (x t,i, d; θ) + v t,d,i + ɛ t,d,i. V (s t,i, d; θ) =u (x t,i, d; θ) + v t,d,i + ɛ t,d,i + βe [V (s t+1 ; θ) ɛ t,i, x t,i, d; θ]. At every step θ m, the expected value function E [ V (s t+1 ; θ m ) ɛ m t,i, x m t,i, d; θ m] is updated by averaging over near history point of θ on the mcmc chain and over the importance sampling draws of ɛ s. 80 / 105

81 How to update the approximate value function V m ( s m,j ; θ m) V m (s; θ m ) = max d D {u (s, d; θm ) + βe (m) [ V ( s ; θ m) s, d; θ m] }. At each iteration m, Draw randow states s m,j, j = 1,..., ˆN (m) from an i.i.d. density g ( ) > 0. Each iteration m, only keep track of the history of length N (m): {θ k ; s k,j, V k ( s k,j ; θ k), j = 1,..., ˆN (k)} m 1 k=m N(m). In this history, find i = 1,..., Ñ (m) cloest to θ parameter draws. Only the value functions in importance sampling that correspond to these nearest neighbors are used in the approximation by averaging. 81 / 105

82 Update value function as: E (m) [ V ( s ; θ ) s, d; θ ] Ñ(m) = i=1 Ñ(m) = i=1 ˆN(k i ) j=1 ˆN(k i ) j=1 V k i V k i ( ) s k i,j ; θ k f ( s k i,j s, d; θ ) /g ( s k i,j ) i Ñ(m) ˆN(k r ) r=1 q=1 f (skr,q s, d; θ) /g (s kr,q ). ( ) s k i,j ; θ k i W ki,j,m (s, d, θ) weights simplies with i.i.d. known unobservable components. expected max value function can be integrated out with extreme value errors. 82 / 105

83 Particle filtering Used in macro dynamic models by Fernandez and Rubio. Particle filtering is an importance sampling method. x is latent state, y is observable random variable: We are interested in the posterior distribution of x given y. Want to compute E (t (X ) y) p (y x) p (x) t (x) p (x y) dx = t (x) dx p (y) t (x) p(y x)p(x) g(x) g (x) dx = p(y x)p(x) g(x) g (x) dx 83 / 105

84 If we have r = 1,..., R draws from the density g (x), then we can approximate where E (t (X ) y) 1 R Or one can write where R t (x r ) w (x r y) / 1 R r=1 w (x r y) = p (y x r ) p (x r ). g (x r ) E (t (X ) y) 1 R R w (x r y), r=1 R t (x r ) w (x r y) r=1 w (x r y) = w (x r y) / R w (x r y). r=1 84 / 105

85 In other words, given R draws from g (x), t (X ) is integrated against a discrete distribution that places weights w (x r ) on each of the r = 1,..., R points on the discrete support. Alternatively, one can compute E (t (X ) y) 1 R R t ( x r ) r=1 where x r, r = 1,..., R are R draws from the weighted empirical distribution on the x r, r = 1,..., R, where the weights are w (x r ). When g (x) = p (x), w (x y) = p (y x), and w (x r y) = p (y x r ) / R p (y x r ). r=1 85 / 105

86 In a Bayesian setup, the coefficients θ are the latent state variables x. p (x) is then the prior distribution of θ. A random coefficient model is just like a hierachical Bayesian model where the µ are the hyper-parameters in the prior distribution, and the hyperparameters are to be estimated. The prior for θ can be made dependent on both µ and covariates (state variables) s. Computing the weights p (y θ) involves value function iteration and is computationally difficult. If we use GMM objective function instead of the likelihood to form p (y θ), there might be some computation savings because only one simulation r is needed for each observation. 86 / 105

87 A more interesting case is when there is dynamic unobservable state variables. Suppose s = (x, v), such that x is observed but v is not observed. We might be interested in the posterior distribution of the entire sample path of v in addition to those of θ, the parameters. Particle filtering can be used in this case. How can computation be maximally sped up? Do particles lend themselves to pararrell processing? 87 / 105

88 p (v 0:t y 1:t, x 0:t ) = p (v 0:t, y 1:t, x 0:t ) p (y 1:t, x 0:t ) = p (v 0:t 1, y 1:t 1, x 0:t 1 ) p (v t, y t, x t v 0:t 1, y 1:t 1, x 0:t 1 ) p (y 1:t 1, x 0:t 1 ) p (y t, x t y 1:t 1, x 0:t 1 ) =p (v 0:t 1 y 1:t 1, x 0:t 1 ) p (v t, y t, x t v 0:t 1, y 1:t 1, x 0:t 1 ) p (y t, x t y 1:t 1, x 0:t 1 ) =p (v 0:t 1 y 1:t 1, x 0:t 1 ) p (y t x t, v t ) p (v t, x t v 0:t 1, y 1:t 1, x 0:t 1 ) p (y t, x t y 1:t 1, x 0:t 1 ) =p (v 0:t 1 y 1:t 1, x 0:t 1 ) p (y t x t, v t ) p (v t, x t v t 1, y t 1, x t 1 ) p (y t, x t y 1:t 1, x 0:t 1 ) The fourth equality follows from the conditional independence assumption and the markovian structure, and the fifth equality follows from the markovian structure of the state variable transition. 88 / 105

89 Therefore we only need to make the following small modifications to the filtering algorithm: First, for each particle in period t 1, with its associated value of v t 1, simulate from p (v t x t, v t 1, y t 1, x t 1 ) = p (v t, x t v t 1, y t 1, x t 1 ) p (x t v t 1, y t 1, x t 1 ). If v t and x t are conditionally independent given v t 1, y t 1, x t 1, then we can directly simulate from Then reweight by p (y t x t, v t ). p (v t v t 1, y t 1, x t 1 ). The weights are not easy to compute because they either involve value function iteration orbackward recursion. 89 / 105

90 In summary, three components of recursive particle filtering method: p (v 0:t x 1:t, y 1:t ) p (v 0:t 1 y 1:t 1, x 0:t 1 ) p (v t x t, v t 1, y t 1, x t 1 ) p (y t x t, v t ) The first part p (v 0:t 1 y 1:t 1, x 0:t 1 ), defines recursion. The second part p (v t x t, v t 1, y t 1, x t 1 ), defines the importance sampling density. The third part p (y t x t, v t ), defines the weights on the particles. 90 / 105

91 Relation to the macro model of Jesús Fernández-Villaverde and Juan Rubio-Ramírez. Their paper: Estimating Macroeconomic Models: A Likelihood Approach, forthcoming, Review of Economic Studies. Basically, very similar. They also have feedback from y t to the latent state variables x t+1, v t+1 : p (v t, x t v 0:t 1, y 1:t 1, x 0:t 1 ) = p (v t, x t v t 1, y t 1, x t 1 ) unlike stochastic volatility models, where transitions of v t, x t are automonous. 91 / 105

92 They introduce this dependence by allowing for singularity in the measurement equation: Y t = g (S t, V t ; γ) The states in their transition equation are not necessarily all latent: S t = f (S t 1, W t ; γ) Feedback from Y t to latent states is allowed when Y t is a subcomponent of S t and is directly observed, such that that particular component of g ( ) has no noise V t. 92 / 105

93 In their assumption 2, they assume that both Y and V are continuously distributed and that g ( ) (f ( ) as well) is invertable. Then the conditional density of Y t (which basically is used to calculate the weights) can be determined from the density of V t and the jacobian of the transformation. They do mention though, in one brief sentence, that this assumption can possibly be relaxed as long as the weights can be computed. Our discrete Y t model falls into this extension where there is no invertibility. The weights can still be computed through the logit probability form and the value function iterations (or backward recursion). 93 / 105

94 Calculating value functions by backward induction: Assuming binary choice model. At time T : V T (s) = log [exp (Π (s, 1)) + exp (Π (s, 0))] Suppose V t+1 (s) is known, then at time t: V t (s, 1) =Π (s, 1) + βe [ ( V t+1 s ) s, 1 ] V t (s, 0) =Π (s, 0) + βe [ ( V t+1 s ) s, 0 ] V t (s) = log [exp (V t (s, 1)) + exp (V t (s, 0))]. 94 / 105

95 Complete information Only contains unobserved heterogeneity that are observed by all the players. widely used to model entry behaviors of competing firms. Issues with existence and multiplicity of equilibrium. Important previous work by Bresnahan and Reiss (1990,1991) and Berry (1992) focus on unique features (number of firms) in the data. Recent papers by Tamer (2002) and Ciliberto and Tamer (2003) makes use of bounds. Another approach specifies an equilibrium selection mechanism: Bjorn and Vuong (1984), Berry (1992), Jia (2008). Bajari, Hong and Ryan consider estimating an equilibrium selection mechanism. 95 / 105

96 Complete information Allow for mixed strategies. Equilibrium selection is part of model. Exploit computational tools in McKelvy and McLennan (1994) to find all equilibrium. Reduce computational burden in structural models using importance sampling, e.g. Ackerberg (2008). Nonparametric identification is difficult. 96 / 105

97 The model Simultaneous move game of complete information (normal form game). There are i = 1,..., N players with a finite set of actions A i. Let A = A i. i Utility is u i : A R, where R is the real line. Let π i denote a mixed strategy over A i. A Nash equilibrium is a set of best responses. 97 / 105

98 Following Bresnahan and Reiss (1990,1991), utility is: Mean utility, f i (x, a; θ 1 ) u i (a) = f i (x, a; θ) + ε i (a). a, the vector of actions, covariates x, and a parameters θ. ε i (a) preference shocks. ε i (a) g(ε θ) iid. Standard random utility model, except utility depends on actions of others. E(u) set of Nash equilibrium. λ(π; E(u), β) is probability of equilibrium, π E(u) given parameters β. λ(π; E(u), β) corresponds to a finite vector of probabilities. 98 / 105

99 Example of λ. Theorists have suggested that an equilibrium may be more likely to be played if it: Satisfies a particular refinement concept (e.g. trembling hand perfection). The equilibrium is in pure strategies. The equilibrium is risk dominant. Create dummy variable for whether a given equilibrium, π E(u) satisfies criteria 1-3 above. Let x(π, u) denote the vector of covariates that we generate in this fashion. Then a straightforward way to model λ is: exp(β x(π, u)) λ(π; E(u), β) = π E(u) exp(β x(π, u)) 99 / 105

100 Estimation Computing the set E(u), all of the equilibrium to a normal form game, is a well understood problem. McKelvy and McLennan (1996) survey the available algorithms in detail. Software package Gambit. P(a x, θ, β) is probability of a given x, θ and β P(a x, θ, β) = { λ(π; u(x, θ 1, ε), β) ( N i=1 π(a i ) )} g(ε θ 2)dε π E(u(x,θ,ε)) Computation of the above integral is facilitated by the importance sampling procedure. (cf. Ackerberg (2003)) 100 / 105

101 Often g(ε θ 2 ) is a simple parametric distribution (e.g. normal). For instance, suppose it is normal and let φ( µ, σ) denote the normal density. Then, the density h(u θ, x) for the vnm utilities u is: h(u θ, x) = i φ(ε i (a); f i (θ, x, θ) + µ, σ) a A where for all i and all a, ε i (a) = f i (x, a; θ 1 ) u i (a) Evaluating h(u θ, x) is not difficult. Draw s = 1,..., S vectors of vnm utilities, u (s) = (u (s) 1,..., u(s) ) from an importance density q(u). N 101 / 105

102 We can then simulate P(a x, θ, β) as follows: P(a x, θ, β) = { S s=1 λ(π; E(u (s) ), β) ( N i=1 π(a i ) )} h(u (s) θ,x) q(u (s) ) π E(u) Precompute E(u (s) ) for a large number of randomly drawn games s = 1,..., S. Evaluating P(a x, θ, β) at new parameters does not require recomputing E(u (s) ) for new s = 1,..., S! Evaluating simulation estimator of P(a x, θ, β) of P(a x, θ, β) only requires reweighting of the equilibrium by new λ and h(u (s) θ,x). q(u (s) ) This is a feasible computation. 102 / 105

103 Normally, the computational expense of structural estimation comes from recomputing the equilibrium many times. This can save on the computational time by orders of magnitude. Given P(a x, θ, β) we can simulate the likelihood function or simulate the moments. The asymptotics are standard. Simulated method of moments is unbiased and might have lower number of simulation draws required to converge to the asymptotic distribution at T 1/2 rate. The model we consider is parametric. 103 / 105

104 Importance sampler In private information static or dynamic models, Ackerberg (2009) shows that it works best with a full random coefficient specification. Recomputation of equilibrium or value function typically is needed for the non-random coefficients in each round of their iterations. The importance sampler for the complete information in Bajari et al (2009) does not require random coefficient specification. A dynamic version of the complete information game also requires random coefficient specification. In addition, MCMC appears to be doing a better job than importance sampler for exploring the parameter space. 104 / 105

105 Conclusion Models consistent with Nash equilibrium behavior have many empirical applications. Challenging topics for Nonparametric identification Parametric and semiparametric estimation Computation algorithms for value function iteration and equilibrium solution An exciting area of research for both econometrics and empirical industrial organization. 105 / 105