Dynamic Discrete Choice Structural Models in Empirical IO

Transcription

1 Dynamic Discrete Choice Structural Models in Empirical IO Lecture 4: Euler Equations and Finite Dependence in Dynamic Discrete Choice Models Victor Aguirregabiria (University of Toronto) Carlos III, Madrid June 29, 2017 Carlos III, Madrid June 29, /

2 / Motivation For the estimation of dynamic structural models with continuous decision variables, Hansen and Singleton (1982) proposed the Euler equation GMM approach. This approach has some very important advantages: [1] It is not subject to the curse of dimensionality in estimation of structural parameters. [2] Robustness: avoids mis-specification of the stochastic process of state variables. [3] It facilitates dealing with unobservables: time invariant, aggregate, and serially correlated. Carlos III, Madrid June 29, /

3 (2) / Motivation Not surprisingly, these important advantages have made the EE-GMM very influential and the "default" method in the estimation of dynamic structural models with continuous decision variables. Many empirical applications to models of: - life-cycle consumption; - portfolio choices; - firm investment; - firm labor demand; - dynamic pricing; - inventories; etc. Carlos III, Madrid June 29, /

4 (3) / Motivation For more than three decades, the common wisdom has been that the Euler equation GMM had two important limitations: [1] Cannot be applied to discrete choice models. [2] Cannot be used for solution / counterfactuals: in general, Euler equation-policy operator is not a contraction (Coleman, 1990). Recent results have challenged this "common wisdom". Carlos III, Madrid June 29, /

5 Outline Outline [1] EE-GMM approach in continuous decision models [2] Deriving Euler Equations in Dynamic Discrete Choice (DDC) Models [3] Estimating DDC Models using Euler Equations [4] Solving DDC Models using Euler Equations Carlos III, Madrid June 29, /

6 EE-GMM in Continuous Models 1. EE-GMM: Continuous Decision Models Carlos III, Madrid June 29, /

7 EE-GMM in Continuous Models Dynamic Continuous Decision Models Every ( period t, an agent chooses ) the value a t A R to maximize E t j=0 T t βj Π t (a t+j, s t+j ). s t follows a controlled Markov process with transition f t (s t+1 a t, s t ). The sequence of value functions, V t (s t ), satisfy the Bellman equation: { } V t (s t ) = max Π t (a t, s t ) + β V t+1 (s t+1 ) f t (s t+1 a t, s t ) ds t+1 a t A Carlos III, Madrid June 29, /

8 EE-GMM in Continuous Models State Variables and Transitions The vector of state variables is s t (y t, z t ), where z t are exogenous, and y t are endogenous. The exogenous z t follow Markov process f z,t (z t+1 z t ). The transition of the endogenous is: y t+1 = f y (a t, s t ) + ξ t+1 where ξ t+1 is indenpendent of (a t, s t ). Key restriction: y t+1 a t and y t+1 y t are known to the agent at period t. Carlos III, Madrid June 29, /

9 EE-GMM in Continuous Models Deriving Euler Equations Suppose that the expected intertemporal payoff is continuously differentiable in a t and y t. First order conditions: Π t a t Vt+1 y t+1 + β f z,t (z t+1 z t ) dz t+1 = 0 y t+1 a t Envelope Theorem condition: V t y t = Π t y t Vt+1 y t+1 + β f z,t (z t+1 z t ) dz t+1 = 0 y t+1 y t Carlos III, Madrid June 29, /

10 EE-GMM in Continuous Models Deriving Euler Equations (2) ( ) Vt+1 Define β E t y t+1 Vt+1 β y t+1 write F.O.C. and E.T.C. as follows: Π t + y ( ) t+1 Vt+1 β E t = 0 a t a t y t+1 V t y t = Π t y t + y t+1 y t f z,t (z t+1 z t ) dz t+1. We can β E t Combining the two equations, we obtain: ( ) Vt+1 y t+1 V t y t = Π t y t y t+1 y t [ ] 1 yt+1 Π t a t a t This expression give us the marginal value function V t as a y t closed-form expression of primitives of the model that we know without solving the model. Carlos III, Madrid June 29, /

11 EE-GMM in Continuous Models Deriving Euler Equations (2) Dating this expression for V t at t + 1, we get: y t ( ) ( Vt+1 Π t+1 β E t = β E t y [ t+2 yt+2 y t+1 y t+1 y t+1 a t+1 ] 1 Π t+1 a t+1 And plugging this expression into the F.O.C., we obtain the Euler equation: [ ] ( Π t yt+1 Π t+1 + β E t y [ ] ) 1 t+2 yt+2 Π t+1 = 0 a t a t y t+1 y t+1 a t+1 a t+1 ) Carlos III, Madrid June 29, /

12 EE-GMM in Continuous Models GMM estimation using Euler equations We can represent the EE in a compact form as: E t [ h(a t, s t, a t+1, s t+1 ; θ) ] = 0 Function h(.) is known to the researcher up to the vector of structural parameters θ that enter in the marginal payoffs Π t a t transitions y t+1 and y t+1, and β. a t y t This EE implies the conditional moment restrictions: E [ h(a t, s t, a t+1, s t+1 ; θ) a t, s t ] = 0 and Π t y t, in the and for any vector of functions w(a t, s t ), the unconditional moment restrictions: E [ w(a t, s t ) h(a t, s t, a t+1, s t+1 ; θ) ] = 0 Carlos III, Madrid June 29, /

13 EE-GMM in Continuous Models GMM - Euler equations (2) Given a panel data from N individuals, we can construct the sample counterpart of these moment conditions: m N (θ) 1 N We can estimate θ by GMM: N w(a it, s it ) h(a it, s it, a i,t+1, s i,t+1 ; θ) i=1 θ = arg min θ m N (θ) W N m N (θ) Carlos III, Madrid June 29, /

14 EE-GMM in Continuous Models Advantages of the EE-GMM No curse of dimensionality. The estimation of θ does not require solving the model or constructing present values. The computational cost of evaluating m N (θ) for a value of θ does not depend on the dimension of the state space: it is the same when s t is one binary state variable and when s t is a vector of, say, 1, 000 continuous state variables. This is because the "integration" is only over the states that we observe in the sample, i.e., m N (θ) 1 N N w(a it, s it ) h(a it, s it, a i,t+1, s i,t+1 ; θ) i=1 Carlos III, Madrid June 29, /

15 EE-GMM in Continuous Models Advantages of the EE-GMM (2) Dealing with non-stationarities In some applications, the vector of exogenous state variables, z t, can include aggregate variables that are trended in if the sample and that may follow non-stationary processes, e.g., business cycle variables, technological progress, regulatory changes, etc. Using a full-solution method (or any method that requires computing present values) require that we specify and estimate the stochastic process followed by these state variables. Problems: we have short-panel but we need a long time-series; assumption of stable process over time; unrealistic assumptions on rational expectations over a long time period. These restrictions are not needed with the EE-GMM approach. Carlos III, Madrid June 29, /

16 EE-GMM in Continuous Models Advantages of the EE-GMM (3) Dealing with unobservables In the previous notation, we implicitly assume that the whole s t is observable to the researcher. We can allow for unobserved state variables: s t = (x t, ε t ). Suppose: Π it a it = π a (a it, x it ) + ε (a) it ; Π it = π y (a it, x it ) + ε (y ) it y it Suppose that ε (a) it and ε (y ) it have the typical components-of-variance structure in panel data models, i.e., where u (a) it ε (a) it = α (a) i follows an AR(1) process. + γ (a) t + u (a) it Then, we can estimate the parameters accounting for these unobservables using Dynamic Panel Data GMM. Carlos III, Madrid June 29, /

17 EE-GMM in Continuous Models Euler equations in DDC Models As mentioned above, the common wisdom has been that this approach cannot be applied to DDC models: the optimality conditions in these models involve inequalities and not marginal conditions. Recent papers have shown that this is not the case: Arcidiacono and Miller (ECMA, 2011; WP, 2016): Finite dependence representation of DDC models. Aguirregabiria and Magesan (AiE, 2013; WP 2015): Euler equations representation of DDC models. Carlos III, Madrid June 29, /

18 Model and Basic Assumptions 2. DDC: Model and Basic Assumptions Carlos III, Madrid June 29, /

19 Model and Basic Assumptions Dynamic DC Single-Agent Models Single-Agent Models Every time period t an agent makes a choice a t A = {0, 1,..., J} to maximize E t ( j=0 β s Π(a t+j, s t+j ) ). s t follows a controlled Markov process with transition f (s t+1 a t, s t ). The value function V (s t ) solves the Bellman equation: { } V (s t ) = max Π(a t, s t ) + β V (s t+1 ) f (s t+1 a t, s t ) ds t+1 a t A Carlos III, Madrid June 29, /

20 Model and Basic Assumptions Single-Agent Models Observable and unobservable state variables s t = (x t, ε t ): x t observable; ε t unobservable to researcher. Additive separability (AS). Π(a t, x t, ε t ) = π(a t, x t ) + ε t (a t ) Conditional independence (CI). f (x t+1, ε t+1 a t, x t, ε t ) = f x (x t+1 a t, x t ) g(ε t+1 ) Optimal decision rule: {α (x t, ε t ) = a} iff {v(a, x t ) + ε t (a) v(j, x t ) + ε t (j) for any j = a} Carlos III, Madrid June 29, /

21 Model and Basic Assumptions Single-Agent Models Decision rules as Conditional Choice Probabilities Given an arbitrary decision rule α(x t, ε t ) from X R J+1 into A, we can defined Conditional Choice Probability (CCP) function: P(a x) Pr (α(x t, ε t ) = a x t = x) = Λ (a, ṽ(x)) PROPOSITION 1 [Hotz-Miller Inversion]. Mapping Λ (.) is invertible such that there is a one-to-one relationship between the vector of value differences ṽ(x) and the vector of optimal choice probabilities P(x), i.e., ṽ(x) = Λ 1 (P(x)). Carlos III, Madrid June 29, /

22 Model and Basic Assumptions Single-Agent Models Dynamic probability-choice problem Given a vector of CCPs P t {P t (a) : a A {0}}, we define the expected payoff function: where Π P (P t, x t ) J a=0 P t(a) [π t (a, x t ) + e t (a, P t )], e t (a, P t ) = E [ ε t (a) Λ 1 (a, P t ) + ε t (a) Λ 1 (j, P t ) + ε t (j) j ] And define the expected transition probability, f P (x t+1 P t, x t ) J a=0 P t(a) f (x t+1 a, x t ). The Bellman equation of the probability-choice problem is: { V P (x t ) = max P t [0,1] J Π P (P t, x t ) + β x t+1 X V P t+1(x t+1 ) f P (x t+1 P t, x t ) } Carlos III, Madrid June 29, /

23 Model and Basic Assumptions Single-Agent Models Equivalence of discrete-choice and probability-choice problems Define: W (P t, x t ) Π P (P t, x t )+ β xt+1 V P (x t+1 ) f P (x t+1 P t, x t ). PROPOSITION 2. (A) the optimal decision in probability-choice problem is equal to the optimal choice probability in discrete choice problem, i.e., P (x t ) = Λ (ṽ(x t )). (B) W t (P t, x t ) is twice continuously differentiable and globally concave in P t, and the optimal decision rule P (x t ) is uniquely characterized by W (P, x t )/ P t = 0; Carlos III, Madrid June 29, /

24 Euler Equations in DDC Models 3. Euler Equations in DDC Models Carlos III, Madrid June 29, /

25 Euler Equations in DDC Models Euler Equations in DDC models Constrained optimization problem: max {P t,p t+1 } subject to: Π P (P t, x t ) + β xt+1 Π P (P t+1, x t+1 ) f P (x t+1 P t, x t ) f P (2) (x t+2 P t, P t+1, x t ) = f P (x t+2 P t, P t+1, x t) for any x t+2 By construction, Pt and Pt+1 is the unique solution to this constrained optimization problem. Lagrange marginal conditions of optimilaty characterize the solution. Carlos III, Madrid June 29, /

26 Euler Equations in DDC Models Necessary and Suffi cient Condition for Euler Equations Let F t+1 be the matrix with elements: f (x t+2 a t+1, x t+1 ) f (x t+2 a t+1, x t+1 ) f (x t+2 0, x t+1 ) where the columns correspond to all the values x t+2 X (2) (x t ) leaving out one, and the rows correspond to all the values (a t+1, x t+1 ) [A {0}] X (1) (x t ). PROPOSITION 4: The model has an Euler equation representation if and only if F t+1 is full column rank. Relationship with Arcidiacono & Miller s finite state dependence. Carlos III, Madrid June 29, /

27 Euler Equations in DDC Models General Form of the Euler Equations The form of the Euler equation: π (a, x t ) Λ 1 (a, P t ) [ ] β π (0, x t+1 ) + e(0, P t+1 ) λ (x t+2 ) f (x t+2 0, x t+1 ) f (x t+1 a, x t x t+1 x t+2 Carlos III, Madrid June 29, /

28 Euler Equations in DDC Models Example: Dynamic Multi-armed bandit problem Endogenous state variable: f (y t+1 a t, y t ) = 1{y t+1 = a t }. The Euler equations, for any choice a: π(a, x t ) + e(a, P(x t )) + β E t [π(0, a, z t+1 ) + e(0, a, P(x t+1 ))] + π(0, x t ) + e(0, P(x t )) + β E t [π(0, 0, z t+1 ) + e(0, 0, P(x t+1 ))] With extreme value unobservables: π(a, x t ) ln P(a x t ) + β E t [π(0, a, z t+1 ) ln P(0 a, z t+1 )] = π(0, x t ) ln P(0 x t ) + β E t [π(0, 0, z t+1 ) ln P(0 0, z t+1 )] Carlos III, Madrid June 29, /

29 EE-GMM in DDC Models 4. EE-GMM in DDC Models Carlos III, Madrid June 29, /

30 EE-GMM in DDC Models EE-GMM in DDC Models We can represent the EE in the DDC model as: E t [ h(a t, x t, P t (x t ), a t+1, x t+1, P t+1 (x t+1 ); θ) ] = 0 Function h(.) is known to the researcher up to the vector of structural parameters θ that enter in the payoff function and β. This EE implies the conditional moment restrictions: E [ h(a t, x t, P t (x t ), a t+1, x t+1, P t+1 (x t+1 ); θ) a t, x t ] = 0 and for any vector of functions w(a t, x t ), the unconditional moment restrictions: E [ w(a t, s t ) h(a t, x t, P t (x t ), a t+1, x t+1, P t+1 (x t+1 ); θ) ] = 0 Carlos III, Madrid June 29, /

31 EE-GMM in DDC Models EE-GMM in DDC Models Given a panel data from N individuals, we can construct the sample counterpart of these moment conditions: m N (P, θ) 1 N N w(a it, s it ) h(a it, x it, P t (x it ), a i,t+1, x i.t+1, P t+1 (x i,t+1 ) i=1 The main difference with respect to the EE-DDC with continuous decision variables is that now the moment conditions involve the CCPs P t (x it ) and P t+1 (x it ) which are unknown to the researcher. We can apply a semiparametric two-step GMM: [Step 1] Nonparametric estimation of CCPs P t (x it ) and) P t+1 (x i,t+1 ); [Step 2] We plug-in the estimated CCPs in m N ( P, θ and estimate θ by GMM. The estimator is root-n consistent, asymptotically normal, with simple to obtain variance matrix. Carlos III, Madrid June 29, /

32 Euler Policy Iteration Mapping 5. Euler Fixed Point Operator: Contraction Property Carlos III, Madrid June 29, /

33 Euler operator Euler Policy Iteration Mapping The Euler equation implies a fixed point mapping in the space of the vector of value differences ṽ Γ EE v (ṽ) {Γ EE v (a, x t, ṽ) : (a, x t ) (A {0}) X } where Γ EE v (a, x t, ṽ) π (a, x t ) + β x t+1 [ π (0, x t+1 ) + e(0, Λ (ṽ(x t+1 ))) λ (x t+1, Λ (ṽ)) ] f ( Carlos III, Madrid June 29, /

34 Euler Policy Iteration Mapping Euler operator: Multi-armed bandit The Euler operator Γ EE v (a, x t, ṽ) is: π(a, x t ) + β E zt+1 z t [π (0, a, z t+1 ) π (0, 0, z t+1 )] + β E zt+1 z t [ln ( 1 + J exp{ṽ(j, 0, z t+1 )} j=1 ) ln ( 1 + J exp{ṽ(j, a, z t+ j=1 Carlos III, Madrid June 29, /

35 Euler Policy Iteration Mapping Euler operator: Contraction We show that: 1. Γ EE v is a contraction mapping; its unique fixed point is the solution to the DP; 2. The contraction is stronger than value function iterations; 3. A single evaluation of Γ EE v is cheaper than a single value (or relative value) iteration. Carlos III, Madrid June 29, /

36 Sequential Estimators 6. Sample-based Euler operator: Estimators Carlos III, Madrid June 29, /

37 Sequential Estimators Sample-Based Euler Operator It is the sample counterpart of the Euler operator. We replace the conditional expectation at the population level E {zt+1 z t } with its empirical counterpart E (N ) {z t+1 z t }. For the dynamic logit model,γ (N ) EE v (a, y, z; ṽ) is: [π(a, y, z) π(0, y, z)] + β E (N ) {z z 0 } [π (0, a, z ) π (0, 0, z ) ] β E (N ) {z z} [ ln ( 1 + J exp{ṽ t+1 (j, 0, z )} j=1 ) ln ( 1 + J exp{ṽ t+1 (j, a, j=1 The dimension of the operator is given by the sample size and not by the dimension of the space of exogenous state variables z. Carlos III, Madrid June 29, /

38 Sequential Estimators Sample-Based Euler Operator: Properties 1. it is a contraction mapping; a stronger contraction than sample-based value function operator; 2. its evaluation does not suffer of any curse of dimensionality; 3. A fixed point of Γ (N ) EE v (.;θ) is a N consistent and asymptotically normal estimator of the fixed point of Γ EE v (.;θ) Carlos III, Madrid June 29, /

39 Numerical Examples 7. Numerical Examples Carlos III, Madrid June 29, /

40 Numerical Examples Data Generating Process Binary choice model of market entry-exit. Inactive firms get profit π(0, x t ) + ε t (0), with π(0, x t ) = 0, and active firms earn profit π(1, x t ) + ε t (1), with: π(1, x t ) = VP t FC t EC t (1 y t ) and VP t = [ θ VP 0 + θ VP 1 z 1t + θ VP 2 z 2t ] exp ( ω t ) FC t = θ FC 0 + θ FC 1 z 3t EC t = θ EC 0 + θ EC 1 z 4t Carlos III, Madrid June 29, /

41 Numerical Examples Data Generating Process Table 1 Parameters in the DGP Payoff Parameters: θ VP 0 = 0.5; θ VP 1 = 1.0; θ VP θ FC 0 = 0.5; θ FC θ EC 0 = 1.0; θ EC 1 = = = 1.0 Each z j state variable: z jt is AR(1), γ j 0 = 0.0; γj 1 = 0.6 Productivity : ω t is AR(1), γ ω 0 = 0.2; γω 1 = 0.9 Low persistence model: σ ω e = σ e = 1 High persistence model: σ ω e = σ e = 0.01 Discount factor β = 0.95 Carlos III, Madrid June 29, /

42 Comparing the two PI mappings Comparing degree of contraction of operators Lipschitz constant of a fixed point operator Γ(V) = {Γ(x, V) : x X } is defined as: [ ] Γ(V) Γ(W) L sup V,W R X V W = sup V,W R X [ supx X Γ(x, V) Γ(x, W) sup x X V (x) W (x) ] Carlos III, Madrid June 29, /

43 Comparing the two PI mappings Comparing contraction properties Table 2 Degree of Contraction (Lipschitz) Low Persistence High Persistence X EE-v VF RVF EE-v VF RVF , , , , Carlos III, Madrid June 29, /

44 Comparing the two PI mappings Comparing computing times Table 3(a) Comparison of Solution Methods (Model with low persistence) # states Number iters. Time per iter. (secs) X EE-v PF VF RVF EE-v PF VF RVF < <0.001 < < <0.001 < , , Carlos III, Madrid June 29, /

45 Comparing the two PI mappings Comparing computing times Table 3(b) Comparison of Solution Methods (Model with low persistence) Total Time # states (in seconds) X EE-v PF VF RVF EE-v EE-v Time Ratios PF EE-v VF EE-v RVF EE-v 64 < < , , , , M 23,270 1, , Carlos III, Madrid June 29, /

46 Comparing the two PI mappings Comparing computing times Table 4 Comparison of EE-value and RVF Solution Methods (Model with high persistence) Number of Ratio states Euler Relative value total time X # iter. Time-per-iter. # iter. Time-per-iter. RVF / EE-v < < < < , Carlos III, Madrid June 29, /

47 Comparing the two PI mappings Finite Sample Properties Finite Sample Properties: Parameter estimates Table 5(c) Monte Carlos: Estimation of Parameters N = 1, 000 & T = 2; Monte Carlo rep. = 1, 000 Root Mean Squared Error Parameter 2-step 2-step MLE K-step (True value) Eff. HM EE (NPL) EE Total RMSE RMSE HM Ratio RMSE EE Time (in secs) Time PF Ratio Time EE Carlos III, Madrid June 29, /

48 Comparing the two PI mappings Finite Sample Properties Finite Sample Properties: Counterfactual Table 6 Factual and Counterfactual Scenarios Prob. Entry Exit State Being Active Prob. Prob. Persist. Output (A) Factual DGP (B) Counterfactual DGP Policy Effect: (B) - (A) (Percentage change) (-20.1%) (-42.7%) (-11.7%) (15.1%) (-20.0% Carlos III, Madrid June 29, /

49 Comparing the two PI mappings Finite Sample Properties Finite Sample Properties: Counterfactual Table 8 Monte Carlo: Counterfactual Estimates Prob. Entry Exit State Total Being Active Prob. Prob. Persist. Output True Policy Effect Root Mean Square Error VF iterations 67.5% 17.1% 30.1% 11.8% 82.8% PF iterations 67.5% 17.1% 30.1% 11.8% 82.8% EE-value iterations 37.1% 6.3% 14.8% 7.2% 38.6% Carlos III, Madrid June 29, /

50 Conclusion Conclusion Develop and study an alternative EE fixed point mapping which can be used to: [1] Estimate parameters of a structural model; [2] Obtain a solution of the model; [3] Estimate counterfactual probabilities. Provide evidence that the EE-PI mapping is faster than the ST-PI mapping, and that the EE-PI mapping provides better estimates of counterfactual CPs when compared to alternative approximation methods that impose the same computational burden. Carlos III, Madrid June 29, 2017 /