Identification and Estimation of Dynamic Games when Players Belief Are Not in Equilibrium

Transcription

1 Identification and Estimation of Dynamic Games when Players Belief Are Not in Equilibrium A Short Review of Aguirregabiria and Magesan (2010) January 25, / 18

2 Dynamics of the game Two players, {i, j} T periods X t is the vector of state variables Y it is player i s choice at time t ɛ it (Y it ) is i s private information; it is distributed by G π i (Y it, Y jt, X t ) is a real value function of i s own action, opponent s action, and state variables; it is the deterministic part of i s payoff payoff function Π it (Y it, Y jt, X t ) = π i (Y it, Y jt, X t ) + ɛ i (Y it ) 2 / 18

3 State variables and transition functions X t = (W t, S 1t, S 2t ) W t are some exogenous, player independent market characteristics S it is player i s specific characteristics f W (W t+1 W t ) is the transition function of W f S (S t+1 Y it, S it ) is the transition function of S; it does not depend on j s action or state 3 / 18

4 Strategies, choice probabilities, and beliefs σ it (X t, ɛ it ) is i s strategy at time t b to jt (X t, ɛ jt ) is player i s belief at period t o about the strategy of player j at time t P it (X t ) = P r(σ it (X t, ɛ it ) = 1 X t ) is i s choice probability B jt (X t ) = P r(b jt (X t, ɛ it ) = 1 X t ) is i s belief of player j s behavior at time t 4 / 18

5 Model assumptions MOD1: Players strategies depend on state variables MOD2: Players maximize expected payoffs MOD3: A player s belief of his own action is consistent with his expectation of his actual actions. equil : Players beliefs about other players actions are unbiased expectations of the actual actions of other players. That is, MOD4: If T <, B (to) jt B jt (X t ) = P jt (X t ) = B jt ; if T <, B (to) jt = B j 5 / 18

6 i s belief of j s behavior is a TxT matrix 6 / 18

7 Best response (1) Given a belief B j, player i best responds by maximizing her expected utility payoff The key optimization criterion is the Bellman equation V B it (X t, ɛ it ) = max Y it where: ( Y it πi B (X t ɛ it ) + β π B i (X t ) = B jt (X t )π i (1, X t ) + (1 B jt (X t ))π i (0, X t ) f B i (X t+1 ) Y it, X t ) =f i (X it+1 Y it, X it ) [B jt (X t )f j (X jt+1 1, X jt )+ (1 B jt (X t ))f j (X jt+1 0, X jt )] ) Vi B (X t+1, ɛ it+1 )f B dg 7 / 18

8 Best response (2) The best response function can be represented by the threshold function {Y it = 1} {ɛ it (0) ɛ it (1) v B it (1, X t ) v B it (0, X t )} where: vit B = πit B (Y it, X t )+β V i (X, ɛ )fi B (X Y it, X t )dg(ɛ ) X,ɛ Denote Λ as the best response function using the explicit distribution function (G) of ɛ, i.e. P r(y it = 1 X t ) = Λ(v B it (1, X t ) v B it (0, X t )) 8 / 18

9 Data There are M markets. The econometrician observes for every market m. {Y imt, Y jmt, X mt } T t=1 We are going to suppress m for our discussion of how to estimate the model. 9 / 18

10 Identification assumptions ID1: X mt = X t, B jmt (X) = B jt (X) ID2: Normalization of the payoff function π( ) ID3: There are two values of player i s opponent s state, Sj L and Sj H, at which player i s beliefs are in equilibrium; that is, B jt (W t, S i, Sj L ) = P jt (W t, S i, Sj L ) B jt (W t, S i, Sj H ) = P jt (W t, S i, Sj H ) 10 / 18

11 Estimation with the assumption equil Suppose T =, 1. Observe the data (Y it, Y jt, X t ); do not observe ɛ it 2. Assume that G (hence Λ) and β are known 3. Estimate ( f t S, f t S, P it, P jt ) non-parametrically 4. Inverts Λ to obtain ṽ it 5. Solve the Bellman equation to obtain Ṽ and π Vit B (X t, ɛ it ) = max{vit B (Y it, X t ) + ɛ it (Y it )} Y it vit B (Y it, X t ) = πit B (Y it, X t ) + β Vit+1(X B t+1, ɛ it+1 )fit B dg note: B it (X t ) = Λ(v B it (1, X t ) v B it (0, X t )) B = P 11 / 18

12 Estimation using backward induction Same as the last slide, but suppose T < Define player i s continuation payoff at time t 1 d it 1 = β X V B it (X )f t 1 (X Y i, Y j, X) Let d it = 0 Solve for π it 1 and V B it 1 Calculate d it 1 Repeat 12 / 18

13 Identification assumptions without the assumption equil Instead of the assumption equil, we assume MOD4 and ID3, which states that there are two values of opponent s state variable, Sj L and Sj H, at which player i s beliefs are in equilibrium. Proposition 2 states that these are sufficient conditions to non-parametrically estimate player i s belief function and payoff function. 13 / 18

14 Estimation without the assumption equil 1. Let d it = 0 2. Calculate ˆB jt by formula (30) in the paper 3. Calculate ˆ V B it by formula (31) 4. Calculate d it 1 of the previous period 5. Repeat 14 / 18

15 Testing unbiased beliefs (1) Under the assumptions MOD1, MOD2, MOD3, MOD4, ID1, and ID2, we can test the null of unbiased belief, i.e. player i s belief of j s behavior is consistent with the j s actual behavior at time t, B jt (X t ) = P jt (X t ). Define q it (X) = Λ 1 (P it (X)) Pick X a, X b, X c, X d s.t. each value has the same value in the component of (S i, W ), but different values of S j. 15 / 18

16 Testing unbiased beliefs (2) Define δ = Further define { qit (X a ) q it (X b ) q it (X c ) q it (X d ) P } jt(x a ) P jt (X b ) P jt (X c ) P jt (X d ) D = H h=1 ( δh i ) 2 se( δ) where δ is the sample mean, then D is asymptotically distributed as Chi-square with H degrees of freedom. H is the number of all possible combinations of four different values of S j with S a j Sb j and Sd j Sd j. 16 / 18

17 Empirically testing the null of unbiased belief 17 / 18

18 Empirically testing the null of unbiased belief 18 / 18