ECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION

Transcription

1 ECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION Lecture 8: Dynamic Games of Oligopoly Competition Victor Aguirregabiria (University of Toronto) Toronto. Winter 2017 Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

2 Dynamic Games: Outline Outline 1. Structure of empirical dynamic games 2. Data and Identification 3. Estimation 4. Dealing with unobserved heterogeneity 5. Environmental Regulation and Competition: Ryan (2012) 6. Dynamic Product Design: Sweeting (2013) Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

3 Examples of Empirical Applications Examples of Empirical Applications Land use regulation (entry cost) and entry-exit and competition in the hotel industry: Suzuki (IER; 2013); Subsidies to entry in small markets for the dentist industry: Dunne et al. (RAND, 2013); Environmental regulation and entry-exit and capacity choice in cement industry: Ryan (ECMA, 2012); Demand uncertainty and firm investment in the concrete industry: Collard-Wexler (ECMA, 2013); Time-to-build, uncertainty, and dynamic competition in the shipping industry: Kalouptsidi (AER, 2014); Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

4 Examples of Empirical Applications Examples of Empirical Applications [2] Fees for musical performance rights and the choice of format (product design) of radio stations: Sweeting (ECMA, 2013); Hub-and-spoke networks, route entry-exit and competition in the airline industry: Aguirregabiria and Ho (JoE, 2012); Competition in R&D and product innovation between Intel and AMD: Goettler and Gordon (JPE, 2011); Product innovation of incumbents and new entrants in the hard drive industry: Igami (JPE, 2017); Cannibalization and preemption strategies in the Canadian fast-food industry: Igami and Yang (QE, 2016) Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

5 Examples of Empirical Applications Examples of Empirical Applications [3] Release date of a movie: Einav (EI, 2010). Dynamic price competition: Kano (IHIO, 2013); Ellickson, Misra, and Nair (JMR, 2012). Endogenous mergers: Jeziorski (RAND, 2014). Exploitation of a common natural resource (fishing): Huang and Smith (AER, 2014). Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

6 Examples of Empirical Applications When does dynamic oligopoly competition matters empirically? Example: Firms product quality competition a jt quality of firm j s product; z t exogenous market characteristics. Consider the probability (or alternatively, regression function): Pr (a jt z t, a jt, a j,t 1, a j,t 1 ) Dependence with respect to a j,t 1 is evidence of dynamic strategic interactions. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

7 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Dynamic Games: Basic Structure Time is discrete and indexed by t. The game is played by N firms that we index by i. Following the standard structure in the Ericson-Pakes (1995) framework, firms compete in two different dimensions: a static dimension and a dynamic dimension. We denote the dynamic dimension as the "investment decision". Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

8 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Dynamic Games: Basic Structure (2) Let a it be the variable that represents the investment decision of firm i at period t. This investment decision can be an entry/exit decision, a choice of capacity, investment in equipment, R&D, product quality, other product characteristics, etc. Every period, given their capital stocks that can affect demand and/or production costs, firms compete in prices or quantities in a static Cournot or Bertand model. Let p it be the static decision variables (e.g., price) of firm i at period t. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

9 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Dynamic Games: Basic Structure (3) For simplicity and concreteness, I start presenting a simple dynamic game of market entry-exit where every period incumbent firms compete a la Bertrand. In this entry-exit model, the dynamic investment decision a it is a binary indicator of the event "firm i is active in the market at period t". The action is taken to maximize the expected and discounted flow of profits in the market, E t ( r =0 δ r Π it+r ) where δ (0, 1) is the discount factor, and Π it is firm i s profit at period t. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

10 Structure of dynamic games of oligopoly competition Profit function Basic Framework and Assumptions The profits of firm i at time t are given by Π it = VP it FC it EC it IC it + SV it where: VP it represents variable profit; FC it is the fixed cost of operating; EC it is a one time entry cost IC it is an investment cost SV it is the exit value of scrap value Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

11 Structure of dynamic games of oligopoly competition Variable profit function Basic Framework and Assumptions The variable profit VP it is an "indirect" variable profit function that comes from the equilibrium of a static Bertrand game with differentiated product. Consider the simplest version of this type of model. Suppose that firm i has a marginal cost c i (z t ), where z t is the a vector of exogenous state variables, and produces a product with quality v i (z t ). Consumer utility of buying product i is u it = ν i (z t ) α(z t ) p it + ε it, where ν i (.) and α(.) are functions parameters, and ε it is a consumer-specific i.i.d. extreme value type 1 random variable. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

12 Structure of dynamic games of oligopoly competition Variable profit function (2) Basic Framework and Assumptions Under these conditions, the equilibrium variable profit of an active firm depends only on the number of firms active in the market. VP it = (p it c i (z t )) q it where p it and q it represent the price and the quantity sold by firm i at period t, respectively. According this model, the quantity is: q it = H t a it exp{v i (z t ) α(z t ) p it } 1 + N j=1 a jt exp{v j (z t ) α(z t ) p jt } = H t s it where H t is the number of consumers in the market (market size) and s it is the market share of firm i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

13 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Variable profit function (3) Under the Nash-Bertrand assumption the first order conditions for profit maximization are: or q it + (p it c i (z t )) ( α(z t )) q it (1 s it ) = 0 p it = c i (z t ) + 1 α(z t ) (1 s it ) These N equations define a Bertrand equilibrium with prices pt = (p1t, p 2t,..., p Nt ) and market shares: s it = a it exp{v i (z t ) α(z t ) p it } 1 + N j=1 a jt exp{v j (z t ) α(z t ) p jt } Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

14 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Model: Variable profit function (4) It is simple to show that an equilibrium always exists. Equilibrium prices depend on the number and identity of firms active in the market (a t ), and on the exogenous variables z t : p it = p i (a t, z t ). Similarly, the equilibrium market shares s it is a function of (a t, z t ): s it = s i (a t, z t ). Therefore, the indirect or equilibrium variable profit of an active firm is: VP it = a it H t (p i (a t, z t ) c i (z t )) s i (a t, z t ) = a it H t θ VP i (a t, z t ) where θ VP i (.) is a function that represents variable profits per capita. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

15 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Model: Variable profit function (5) For most of the analysis below, I will consider that the researcher does not have access to information on prices and quantities. Therefore, we will treat {θ VP i (a t, z t )}, for each possible value of (a t, z t ), as parameters to estimate from the structural dynamic game. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

16 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Model: Variable profit function (6) We can extend the previous approach to incorporate endogenous product differentiation. Suppose that the decision variable is a it {0, 1,..., A}, where a it = 0 represents that firm i is not active in the market, and a it = a > 0 implies that firm i is active in the market with a product of quality a. Consumers willingness to pay for a product, and marginal cost, now depend on the quality levels: v i (a it, z t ) and c i (a it, z t ) with: v i (1, z t ) < v i (2, z t ) <... < v i (A, z t ) c i (1, z t ) < c i (2, z t ) <... < c i (A, z t ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

17 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Model: Variable profit function (7) Similarly, the demand system is: q it = H t a it exp{v i (a it, z t ) α(z t ) p it } 1 + N j=1 a jt exp{v j (a jt, z t ) α(z t ) p jt } = H t s it It is straightforward to show that, in this model, the equilibrium variable profit of an active firm is: VP it = A 1{a it = a} H t θ VP i (a t, z t ) a=1 where θ VP (.) is the variable profit per capita that now depends on the firm s own quality, and one the number of competitors at each possible level of quality. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

18 Structure of dynamic games of oligopoly competition Fixed cost Basic Framework and Assumptions The fixed cost is paid every period that the firm is active in the market, and it has the following structure (mode of entry-exit): ( ) FC it = a it θ FC i (z t ) + ε FC it θ FC i (z t ) is a function that represents the mean value of the fixed operating cost of firm i. z t is a vector of exogenous state variables that are common knowledge to all the firms. ε FC it is a zero-mean shock that is private information of firm i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

19 Structure of dynamic games of oligopoly competition Fixed cost (2) Basic Framework and Assumptions There are two main reasons why we incorporate these private information shocks in the model. First, as shown in Doraszelski and Satterthwaite (2012), it is a way to guarantee that the dynamic game has at least one equilibrium in pure strategies. Second, they are convenient econometric errors. If private information shocks are independent over time and over players, and unobserved to the researcher, they can explain players heterogeneous behavior without generating endogeneity problems. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

20 Structure of dynamic games of oligopoly competition Fixed cost (3) Basic Framework and Assumptions For the model with endogenous quality choice, we can generalize the structure of fixed costs: FC it = A 1{a it = a} a=1 ( θ FC i ) (a, z t ) + ε FC it (a) where now the mean value of the fixed cost, θ FC i (a, z t ), and the private information shock, ε FC it (a), depend on the level quality. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

21 Structure of dynamic games of oligopoly competition Entry cost Basic Framework and Assumptions The entry cost is paid only if the firm was not active in the market at previous period (entry-exit model): [ ] EC it = a it (1 k it ) θ EC i (z t ) + ε EC it k it a i,t 1 is the binary indicator that is equal to 1 if firm i was active in the market in period t 1. θ EC i (z t ) is a function that represents the entry cost of firm i. ε EC it is a private information shock in the entry cost Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

22 Structure of dynamic games of oligopoly competition Entry cost (2) Basic Framework and Assumptions For the model with endogenous quality, the cost of market entry depends on the firm s quality at the moment of entry. ( A [ ] ) EC it = 1{k it = 0} 1{a it = a} θ EC i (a, z t ) + ε EC it a=1 θ EC i (a, z t ) is the cost of entry with quality a. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

23 Structure of dynamic games of oligopoly competition Investment cost Basic Framework and Assumptions For the model with endogenous quality, we can incorporate also costs of adjusting the level of quality, or repositioning product characteristics. ( ) AC (+) AC ( ) IC it = 1{k it > 0} θi (z t ) 1{a it > x it } + θi (z t ) 1{a it < x it } Now, k it = a i,t 1 also represents the firm s quality at previous period. AC (+) AC ( ) θi (z t ) and θi (z t ) represents the costs of increasing and reducing quality, respectively, once the firm is active. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

24 Structure of dynamic games of oligopoly competition State variables Basic Framework and Assumptions The payoff relevant state variables of this model are: (1) the exogenous state variables affecting demand and costs, z t, and market size H t. For notational simplicity, we represent them in the vector z t (2) the incumbent status (or quality levels) of firms at previous period k t {k it : i = 1, 2,..., N}; (3) the private information shocks {ε it : i = 1, 2,..., N}. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

25 Structure of dynamic games of oligopoly competition State variables (2) Basic Framework and Assumptions The specification of the model is completed with the transition rules of these state variables. (1) Exogenous state variables follow an exogenous Markov process with transition probability function F z (z t+1 z t ). (2) The transition of the incumbent status is trivial: k it+1 = a it. (3) Private information shock ε it is i.i.d. over time and independent across firms with CDF G i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

26 Structure of dynamic games of oligopoly competition Basic Framework and Assumptions Timing of decisions and state variables Note that in this example, I consider that firms dynamic decisions are made at the beginning of period t and they are effective during the same period. An alternative timing that has been considered in many applications is that there is a one-period time-to-build, i.e., the decision is made at period t, and entry costs are paid at period t, but the firm is not active in the market until period t + 1. This is in fact the timing of decisions in Ericson and Pakes (1995). All the results below can be easily generalized to this model with time-to-build. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

27 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium Markov Perfect Equilibrium Most of the recent literature in IO studying industry dynamics focuses on studying a Markov Perfect Equilibrium (MPE), as defined by Maskin and Tirole (Econometrica, 1988). The key assumption in this solution concept is that players strategies are functions of only payoff-relevant state variables. In this model, the payoff-relevant state variables for firm i are (k t, z t, ε it ). We use x t to represent the vector of common knowledge state variables, i.e., x t (k t, z t ). Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

28 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (2) Markov Perfect Equilibrium Let α = {α i (x t, ε it ) : i {1, 2,..., N}} be a set of strategy functions, one for each firm. A MPE is a set of strategy functions α such that every firm is maximizing its value given the strategies of the other players. For given strategies of the other firms, the decision problem of a firm is a single-agent dynamic programming (DP) problem. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

29 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (3) Markov Perfect Equilibrium Let V α i (x t, ε it ) be the value function of the DP problem that describes the best response of firm i to the strategies α i of the other firms. This value function is the unique solution to the Bellman equation: Πi α(a it, x t ) ε it (a it ) Vi α (x t, ε it ) = max a it +δ Vi α (x t+1, ε it+1 ) dg i (ε it+1 ) Fi α (x t+1 a it, x t ) where Π α i (a it, x t ) and F α i (x t+1 a it, x t ) are the expected one-period profit and the expected transition of the state variables, respectively, for firm i given the strategies of the other firms. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

30 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (4) Markov Perfect Equilibrium For the simple entry/exit game, the expected one-period profit Πi α(a it, x t ) is: ] Πi α(a it, x t ) = a it [H t Pr (α i (x t, ε it ) = a it x t ) θ VP i (a it, a it, z t ) a it a it [ θ FC i ] (z t ) + (1 x it ) θ EC i (z t ) And the expected transition of the state variables is: F α i (x t+1 a it, x t ) = 1{x it+1 = a it } F z (z t+1 z t ) [ Pr (α j (x t, ε jt ) = x j,t+1 x t ) j =i ] Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

31 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (5) Markov Perfect Equilibrium A firm s best response function gives his optimal strategy if the other firms behave, now and in the future, according to their respective strategies. In this model, the best response function of player i is: αi (x t, ε it ) = arg max {v a i α (a it, x t ) ε it (a it )} it v α i (a it, x t ) is the conditional choice value function that represents the value of firm i if: (1) the other firms behave according to their strategies in α; and (2) the firm chooses alternative a it today and then behaves optimally forever in the future. vi α (a it, x t ) Πi α (a it, x t ) + δ V α i (x t+1, ε it+1 ) dg i (ε it+1 ) F α i (x t+1 a it, x t ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

32 Structure of dynamic games of oligopoly competition Markov Perfect Equilibrium (6) Markov Perfect Equilibrium A Markov perfect equilibrium (MPE) in this game is a set of strategy functions α such that for any player i and for any (x t, ε it ) we have that: { } αi (x t, ε it ) = arg max v α a i (a it, x t ) ε it (a it ) it Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

33 Structure of dynamic games of oligopoly competition Conditional Choice Probabilities Conditional Choice Probabilities Given a strategy function α i (x t, ε it ), we can define the corresponding Conditional Choice Probability (CCP) function as : P i (a x) Pr (α i (x t, ε it ) = a x t = x) = 1{α i (x t, ε it ) = a} dg i (ε it ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

34 Structure of dynamic games of oligopoly competition Conditional Choice Probabilities Conditional Choice Probabilities (2) Since choice probabilities are integrated over the continuous variables in ε it, they are lower dimensional objects than the strategies α. For instance, when both a it and x t are discrete, CCPs can be described as vectors in a finite dimensional Euclidean space. In our entry-exit model, P i (1 x t ) is the probability that firm i is active in the market given the state x t. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

35 Structure of dynamic games of oligopoly competition Conditional Choice Probabilities Conditional Choice Probabilities (3) Under the additivity and iid assumptions on the ε it, it is possible to show that there is a one-to-one relationship between strategy functions α i (x t, ε it ) and CCP functions P i (a x t ). We also use Π P i and Fi P instead of Πi α and Fi α to represent the expected profit function and the transition probability function, respectively. From now on, we use CCPs to represent players strategies, and use the terms strategy and CCP as interchangeable. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

36 Structure of dynamic games of oligopoly competition MPE in terms of CCPs Conditional Choice Probabilities Based on the concept of CCP, we can represent the equilibrium mapping and a MPE in way that is particularly useful for the econometric analysis. This representation has two main features: (1) a MPE is a vector of CCPs; (2) a player s best response is an optimal response not only to the other players strategies but also to his own strategy in the future. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

37 Structure of dynamic games of oligopoly competition MPE in terms of CCPs (2) Conditional Choice Probabilities A MPE is a vector of CCPs, P {P i (a x) : for any (i, a, x)}, such that: ( { } ) P i (a x) = Pr a = arg max v P a i (a i, x) ε i (a i ) x i vi P (a i, x) is a conditional choice value function, but it has a slightly different definition that before. Now, vi P (a i, x) represents the value of firm i if the firm chooses alternative a i today and all the firms, including firm i, behave according to their respective CCPs in P. The Representation Lemma in Aguirregabiria and Mira (2007) shows that every MPE in this dynamic game can be represented using this mapping. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

38 Structure of dynamic games of oligopoly competition MPE in terms of CCPs (3) Conditional Choice Probabilities The form of this equilibrium mapping depends on the distribution of ε i. For instance, in the entry/exit model, if ε i is N(0, σ 2 ε ): ( v P P i (1 x) = Φ i (1, x) vi P ) (0, x) σ ε In the model with endogenous quality choice, if ε i (a) s are extreme value type 1 distributed: { v P } exp i (a, x) σ P i (a x) = ε { v A P a =0 exp i (a }, x) σ ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

39 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs Since vi P is not based on the optimal behavior of firm i in the future, but just in an arbitrary behavior described by P i (..), calculating vi P does not require solving a DP problem, and it only implies a valuation exercise. By definition: v P i (a i, x) = Π P i (a i, x) + δ x Vi P (x ) Fi P (x a i, x) where Π P i (a i, x) is the expected current profit. In the entry/exit example: [ ] Π P i (a i, x) = a i H Pr (a i x, P i ) θ VP i (a i ) θ FC i (1 x i )θ EC i a i = z P i (a i, x) θ i Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

40 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs where θ i is the vector of parameters: ( θ i = θ VP (a i ) for any a i, θ FC i ), θ EC i and z P i (a i, x) is the vector that depends only on the state x and on the CCPs at state x, but not on structural parameters. z P i (1, x) = (H Pr (a i x,p i ) for any a i, 1, (1 x i )) z P i (0, x) = (0, 0,..., 0) In the dynamic game with endogenous quality choice, we also have Π P i (a i, x) = z P i (a i, x) θ i Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

41 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs The value function Vi P (.) represents the value of firm i if all the firms, including firm i, behave according to their CCPs in P. We can obtain V P i V P i (x) = as the unique solution of the recursive expression: [ ] A P i (a i x) a i =0 z P i (a i, x) θ i + δ Vi P (x ) F P (x a i, x) x When the space X is discrete we can obtain Vi P as the solution of a system of linear equations: ] ] V P i = [ A P i (a i ) z P i (a i ) a i =0 = z P i θ i + δ F P V P i θ i + δ [ A a i =0 P i (a i ) F P i (a i ) V P i Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

42 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs Then, solving for V P i, we have: V P i = ( I δ F P) 1 z P i θ i = W P i θ i ( where Wi P = I δ F P) 1 z P i is a matrix that only depends on CCPs and transition probabilities but not on θ. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

43 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs Solving these expressions into the formula for the conditional choice value function, we have that: v P i (a i, x) = z P i (a i, x) θ i where: z P i (a i, x) = z P i (a i, x) + δ x F P i (x a i, x) W P i Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

44 Structure of dynamic games of oligopoly competition Computing v P i for arbitrary P Computing values and best response probs The best response mapping in the space of CCPs becomes: ( { } ) P i (a x) = Pr a = arg max z P a i (a i, x) θ i ε i (a i ) x i For the entry/exit model with ε i N(0, σ 2 ε ): ( P i (1 x) = Φ [ z i P (1, x) z i P (0, x) ] θi σ ε ) In the model with endogenous quality choice with ε i (a) s extreme value type 1 distributed: { exp z i P (a, x) θ } i σ P i (a x) = { ε A a =0 exp z i P (a, x) θ } i σ ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

45 Data Data, Identification, and Estimation Data The researcher observes a random sample of M markets, indexed by m, over T periods of time, where the observed variables consists of players actions and state variables. For the moment, we consider that the industry and the data are such that: (a) each firm is observed making decisions in every of the M markets; (b) the researcher knows all the payoff relevant market characteristics that are common knowledge to the firms, x. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

46 Data Data, Identification, and Estimation Data With this type of data we can allow for rich firm heterogeneity that is fixed across markets and time by estimating firm-specific structural parameters, θ i. This fixed-effect approach to deal with firm heterogeneity is not feasible in data sets where most of the competitors can be characterized as local players, i.e., firms specialized in operating in a few markets. Condition (b) rules out the existence of unobserved market heterogeneity. Though it is a convenient assumption, it is also unrealistic for most applications in empirical IO. Later I present estimation methods that relax conditions (a) and (b) and deal with unobserved market and firm heterogeneity. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

47 Data Data, Identification, and Estimation Data Suppose that we have a random sample of M local markets, indexed by m, over T periods of time, where we observe: Data = {a mt, x mt : m = 1, 2,..., M; t = 1, 2,..., T } We want to use these data to estimate the model parameters in the population that has generated this data: θ 0 = {θ 0 i : i I }. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

48 Identification Data, Identification, and Estimation Identification A significant part of this literature has considered the following identification assumptions. Assumption (ID 1): Single equilibrium in the data. Every observation in the sample comes from the same Markov Perfect Equilibrium, i.e., for any observation (m, t), P 0 mt = P 0. Assumption (ID 2): No unobserved common-knowledge variables. The only unobservables for the econometrician are the private information shocks ε imt and the structural parameters θ. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

49 Data, Identification, and Estimation Identification (2) Identification First, let s summarize the structure of the dynamic game of oligopoly competition. Let θ be the vector of structural parameters of the model, where θ = {θ i : i = 1, 2,..., N}. Let P(θ) = {P i (a x, θ) : for any (i, a, x)} be a MPE of the model associated with θ. P(θ) is a solution to the following equilibrium mapping: for any (i, a i, x): { exp z i P (a i, x) θ } i σ P i (a i x,θ) = { ε A a =0 exp z i P (a, x) θ } i σ ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

50 Data, Identification, and Estimation Identification (3) Identification where the vector of values z P i (a, x) are z P i (a i, x) = z P i (a i, x) + δ x F P i (x a i, x) W P i ( and Wi P = I δ F P) 1 z P i, and zi P (a i, x) is a vector with the different components of the current expected profit. For instance, in the entry-exit example: z P i (0, x) = (0, 0, 0,...0) z P i (1, x) = (H Pr (a i x,p i ) for any a i, 1, (1 x i )) Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

51 Data, Identification, and Estimation Identification (4) Identification I will use the function Ψ i (a i, x; P, θ) to represent{ the best response or exp z i P (a i, x) θ } i σ ε equilibrium function that in our example is { A a =0 exp z i P (a, x) θ }. i σ ε Then, we can represent in a compact form a MPE as: P = Ψ(P, θ) where Ψ(P, θ) = {Ψ i (a i, x; P, θ) : for any (i, a, x)}. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

52 Data, Identification, and Estimation Identification (5) Identification Under assumptions ID-1 & ID-2, the equilibrium that has generated the data, P 0, can be estimated consistently and nonparametrically from the data. For any (i, a i, x): P 0 i (a i x) = Pr(a imt = a i x mt = x) For instance, we can estimate consistently Pi 0 (a i x) using the following simple kernel estimator: ( ) xmt x m,t 1{a imt = a i } K Pi 0 b (a i x) = ( ) n xmt x m,t K b n Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

53 Data, Identification, and Estimation Identification (6) Identification Second, given that P 0 is identified, we can identify also the expected present values z i P0 (a i, x) at the "true" equilibrium in the population. Third, we know that P 0 is an equilibrium associated to θ 0. Therefore, the following equilibrium conditions should hold: for any (i, a i, x), { } exp z Pi 0 i P0 (a i, x) θ0 i σ (a i x) = 0 ε { } A a =0 exp z i P0 (a, x) θ0 i σ 0 ε If there the components of z i P0 (a i, x) are not collinear, then these equilibrium conditions identify θ0 i σ 0. ε Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

54 Data, Identification, and Estimation Identification (7) Identification For instance, in this logit example, we have that for (i, a i, x), ( P 0 ) ln i (a i x) [ ] θ 0 Pi 0 = z i P0 (a i, x) z i P0 (0, x) i (0 x) Define Y imt ln Then, ( ) P 0 i (a imt x mt ) Pi 0 (0 x mt ) and Z imt z i P0 (a imt, x mt ) z i P0 (0, x mt ). Y imt = Z imt θ 0 i σ 0 ε σ 0 ε And we can also write this system as, E (Z imt Y imt) = E (Z imt Z imt) θ0 i Under assumption ID-3: σ 0 ε. θ 0 i σ 0 ε = E (Z imtz imt ) 1 E (Z imty imt ) θ 0 Victor i Aguirregabiria () Empirical IO Toronto. Winter / 136

55 Data, Identification, and Estimation Identification (7) Identification Note that under the single-equilibrium-in-the-data assumption, the multiplicity of equilibria in the model does not play any role in the identification of the structural parameters. The single-equilibrium-in-the-data assumption is a suffi cient for identification but it is not necessary. Sweeting (2013) and Aguirregabiria and Mira (2012) present conditions for the point-identification of games of incomplete information when there are multiple equilibria in the data. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

56 Estimation Estimation I will describe the following estimators: 1. Two-step estimators with moment equalities 2. Two-step estimators with moment inequalities 3. Recursive K-step and NPL Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

57 Estimation (2) Estimation For the sake of concreteness, we consider the binary choice entry-exit game, where ε it (1) ε it (0) is iid N(0, σ 2 i ). The equilibrium mapping is: P i (a i x, θ i ) = Φ ( ] ) [ z i P (1, x) z i P θi (0, x) σ i For notational simplicity, we use θ i to represent θ i σ i. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

58 Estimation Pseudo Likelihood Function For the description of the different estimators, it is convenient to define the following Pseudo Likelihood function: Q(θ, P) = M N T a imt m=1 i=1 t=1 + (1 a imt ) ln ([ ] ln Φ z i P (1, x mt ) z i P (0, x mt ) [ ([ ] 1 Φ z i P (1, x mt ) z i P (0, x mt ) θ i ) θ i )] This pseudo likelihood function treats firms beliefs P as parameters to estimate together with θ. Note that for given P, the function Q(θ, P) is the likelihood of a Probit model. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

59 Two-step methods Estimation Two-step methods Suppose that we knew the equilibrium in the population, P 0. Given P 0 we can construct the variables z i P0 (1, x mt ) z i P0 (0, x mt ) and then obtain a very simple estimator of θ 0. ^θ = arg max θ Q(θ, P 0 ) This estimator is root-m consistent and asymptotically normal under the standard regularity conditions. It is not effi cient because it does not impose the equilibrium constraints (only asymptotically). While equilibrium probabilities are not unique functions of structural parameters, the best response probabilities that appear in Q(θ, P) are unique functions of structural parameters and players beliefs. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

60 Estimation Two-step methods (2) Two-step methods The previous method is infeasible because P 0 is unknown. However, under the Assumptions "No-unobserved-market-heterogeneity" and "One-MPE-in-the-data" we can estimate P 0 consistently and at with a convergence rate such that the two-step estimator ^θ is root-m consistent and asymptotically normal. For instance, a kernel estimator of P 0 is: ( ) M P i 0 m=1 T xmt x t=1 a imt K b (x) = ( ) M m=1 T xmt x t=1 K b Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

61 Estimation Two-step methods Two-step methods: Finite sample properties (1) The most attractive feature of two-step methods is their relative simplicity. However, they suffer of a potentially important problem of finite sample bias. The finite sample bias of the two-step estimator of θ 0 depends very importantly on the properties of the first-step estimator of P 0. In particular, it depends on the rate of convergence and on the variance and bias of P 0. It is well-known that there is a curse of dimensionality in the NP estimation of a regression function such as P 0. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

62 Estimation Two-step methods Two-step methods: Finite sample properties (2) In particular, the rate of convergence of P 0 declines, and the variance and bias increase, very quickly as the number of conditioning regressors increases. In our simple example, the vector x mt contains only three variables: the binary indicators a im,t 1 and the (continuous) market size S mt. In this case, the NP estimator of P 0 has a relatively high rate of convergence an its variance and bias can be small even with relatively small sample. However, there are applications with more than two (heterogeneous) players and where firm size, capital stock or other predetermined continuos firm-specific characteristics are state variables. Even with binary state variables (a im,t 1 ), when the number of players is relatively large (e.g., more than 10)... Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

63 Estimation Recursive K-step estimator Recursive K-step estimator K-step extension of the 2-step estimator. Given an initial consistent (NP) estimator P 0, the sequence of estimators { θ K, P K : K 1} is defined as: θ K +1 = arg max Q (θ, P ) K θ where: ([ P i K (x) = Φ ] ) z P K 1 i (1, x) z P K 1 i (0, x) θk i Aguirregabiria and Mira (2002, 2007) present Monte Carlo experiments which illustrate how this recursive estimators can have significantly smaller bias than the two-step estimator. Kasahara and Shimotsu (2008) derive a second order approximation to the bias of these K-stage estimators. They show that, if the equilibrium in the population is stable, then this recursive procedure reduces the bias. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

64 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (BBL) Remember that: z P i (a i, x mt ) θ i is the value of choosing alternative a i today, given that firms behave in the future according to the probabilities in P. Then, the value V P imt is: V P imt = (1 P i (x mt )) [ z P i (0, x mt ) θ i ] + Pi (x mt ) [ z P i (1, x mt ) θ i ] = W P imt θ i where W P imt = (1 P i (x mt )) z P i (0, x mt ) + P i (x mt ) z P i (1, x mt ) Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

65 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (2) Let s split the vector of choice probabilities P into the sub-vectors P i and P i, P (P i, P i ) where P i are the probabilities associated to player i and P i contains the probabilities of players other than i. P 0 is an equilibrium associated to θ 0. Therefore, P 0 i is firm i s best response to P 0 i. This implies that for any P i = P 0 i the following inequality should hold: W (P0 i,p0 i) imt θ 0 i W (P i,p 0 i) imt θ 0 i We can define an estimator of θ 0 based on these (moment) inequalities. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

66 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (3) There are infinite alternative policies P i, and therefore there are infinite moment inequalities. For estimation, we should select a finite set of alternative policies. This is a very important decision for this class of estimators (more below). Let H be a (finite) set of alternative policies for each player. Define the following criterion function: R ( θ, P 0, H ) ( { min 0 ; i,m,t P H [ W (P0 i,p0 i) imt ] }) 2 W (P i,p 0 i) imt θ i This criterion function penalizes departures from the inequalities. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

67 Estimation Estimation using Moment Inequalities Estimation using Moment Inequalities (4) Then, given an initial NP estimator of P 0, say ^P 0, we can define the following estimator of θ 0 based on moment inequalities (MI): ^θ = arg min θ i,m,t ( { P H min 0 ; [ W (^P 0 i,^p 0 i) imt ] }) 2 W (P i,^p 0 i) imt θ i There are several relevant comments to make on this MI estimator: 1 Point identification / Set identification 2 Properties (relative to two-step estimators using moment equalities) 3 Continuous dependent variables Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

68 Estimation Estimation using Moment Inequalities MI Estimator: Point / Set identification This estimator is based on exactly the same assumptions as the 2-step moment equalities (ME) estimator. We have seen that θ 0 is point identified by the moment equalities of the ME estimators (e.g., by the pseudo likelihood equations). Therefore, if the set H of alternative policies is large enough, then θ 0 is point identified as the unique maximizer of R ( θ, P 0, H ). However, it is very costly to consider a set H with many alternative policies. For the type of sets H which are considered in practice, R ( θ, P 0, H ) does not have a unique maximizes and therefore θ 0 is set identified. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

69 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML Why to use an estimator that only set-identifies θ 0 when we have alternative estimators which point identified θ 0? The Moment Inequalities (MI) estimator should have other advantages. Let s start examining which ARE NOT the advantages. The MI estimator is not more robust than the 2S-PML estimator. Both estimators are based on exactly the same model and assumptions. Asymptotically, the MI estimator is less effi cient than the 2S-PML estimator. The effi cient 2S-PML has lower asymptotic variance than the MI estimator, even as the set H becomes very large. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

70 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML Computationally, the MI estimator is more costly than the 2S-PML estimator. The main computational cost of implementing the MI and ME estimators comes from obtaining the vectors of values {Wimt P }. The 2S-PML estimators compute {W P imt } only once: at the estimated W ^P 0 imt. Instead, the MI estimator has to calculate also W (P i,^p 0 i) imt at the different alternative policies in H. Furthermore, the MI estimator needs an algorithm for set optimization. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

71 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML There are two potential advantages of MI against 2S-PML estimator: 1 finite sample bias; 2 estimation of models with continuous decision variables. In terms of finite sample bias, the MI estimator may have lower bias than ME estimators. Exploiting the information in the alternative policies might be useful to reduce the bias. However, there is not any formal proof or Monte Carlo evidence on this point. This may depend crucially on the particular choice of the set H. In practice, H contains only a few alternative policies. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

72 Estimation Estimation using Moment Inequalities Properties of MI estimator relative 2S-PML BBL show that, when combined with simulation techniques to approximate the values {Wimt P }, this method can be easily extended to the estimation of dynamic games with continuous decision variables. In fact, the BBL estimator of a model with continuous decision variable is basically the same as with a discrete decision variable. The ME estimator of models with continuous decision variable may be more complicated. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

73 Dealing with the curse of dimensionality Dealing with the curse of dimensionality The solution and estimation of dynamic structural models faces a curse of dimensionality: the computational cost increases very fast with the dimension of the state space. Two-step estimators partly alleviate this problem but they also suffer of this "curse": the cost of computing the present values ( Wi P = I δ F P) 1 z P i is of the order O( X 3 ) where X is the dimension of the state space. This is particularly important in dynamic games with heterogeneous players: if x t = {x it : i = 1, 2,..., N} and X i is the dimension of the space of x it, then the cost of computing Wi P is O( X i 3N ). Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

74 Dealing with the curse of dimensionality Dealing with the curse of dimensionality [2] I briefly discuss here several approaches that have been applied to deal with this curse of dimensionality, their advantages and limitations. 1. No permanent heterogeneity across players / firms. 2. Oblivious equilibrium. 3. Simulation-Based estimation. 4. Euler equations / Finite dependence. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

75 Dealing with the curse of dimensionality No permanent heterogeneity (symmetric MPE) No permanent heterogeneity (symmetric MPE) The vector of firm specific state variables is x it {x (1),..., x (K ) }, and the state in the game is x t = (x 1t, x 2t,..., x Nt ) such that X = K N, e.g., with K = 100 and N = 5, X Suppose that firms do not have permanent heterogeneity: π i = π. Then, we can represent the state variables of firm i as (x it, n ( i)t ), where n ( i)t = (n ( i)xt : x = 1, 2,..., K ) and n ( i)xt = j =i 1{x jt = k}. The possible values of n ( i)t is ( ) [N 1] + [K 1] (N + K 2)! =., e.g., K = 100 and K 1 (K 1)! (N 1)! N = 5, we have X Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

76 Dealing with the curse of dimensionality No permanent heterogeneity (symmetric MPE) No permanent heterogeneity (symmetric MPE) [2] The main limitation of imposing homogeneity is that in many applications firms can be very heterogeneous (in their products, effi ciency, demand and cost shocks, etc) in a way that is quite permanent and cannot be captured by x it. Also, as shown in the example, though the state space can be substantially reduced, it can be still very large, i.e., X goes from O(K N ) to O(( N K )). Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

77 Dealing with the curse of dimensionality Oblivious equilibrium Oblivious equilibrium Again, suppose that the vector of firm specific state variables is x it {x (1),..., x (K ) }, the state in the game is x t = (x 1t, x 2t,..., x Nt ), and firms do not have permanent heterogeneity: π i = π. As explained above, we can represent the state variables of firm i as (x it, n ( i)t ), where n ( i)t = (n ( i)xt : x = 1, 2,..., K ) and n ( i)xt = j =i 1{x jt = k}. The Oblivious Equilibrium (OE) assumption establishes that the strategy function of a player depends on x it but not on n ( i)t, and this strategy function is the same for all players: α(x it, n ( i)t ) = α(x it ). Weintraub, Benkard, and Roy (ECMA, 2008) show that if the distribution of x it satisfies certain light-tail condition, then the OE approximates well the MPE when N becomes large (i.e., N ). Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

78 Dealing with the curse of dimensionality Oblivious equilibrium [2] Oblivious equilibrium If there is not permanent heterogeneity across players (and the tail condition holds), then in the limit as N the state variable n ( i)t /N converges to a vector of constants λ, i.e., there is always the same proportion of firms at each state x it = x (k). It is possible to ignore the vector of constants λ in the strategy function of a player. The OE assumption rules out short-run dynamic strategic interactions between firms [as well as permanent heterogeneity and common industry shocks]. Their conditions are very strong in many IO applications. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

79 Dealing with the curse of dimensionality Simulation-Based Estimation Simulation-based estimation Hotz, Miller, Sanders and Smith (REStud, 1994) proposed an estimator that uses simulation techniques to approximate the values z P imt (a i ) or similarly the vector of values W P imt. In the context of dynamic games, Bajari, Benkard and Levin (BBL) propose a similar simulation method for dynamic games and for models with either discrete or continuous decision and state variables. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

80 Dealing with the curse of dimensionality Simulation-Based Estimation (2) Simulation-based estimation Remember that: z P imt (a i ) z P imt (a i ) + E ( s=1 β s z Pim,t+s (a im,t+s ) x mt, a imt = a i ) The expectations E (.) are taken over all the possible future paths of actions and state variables conditions on (x mt, a imt = a i ) and conditional on future behavior P. The simulator of z P imt (a i ) are obtained by replacing the true expectations E (.) by Monte Carlo approximations to these expectations. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

81 Dealing with the curse of dimensionality Simulation-Based Estimation (3) Simulation-based estimation For every value of x mt in the sample and every choice alternative a i (in the sample or not), we consider (a i, x mt ) as the initial state for player i and then we use the probabilities in P, and the transition probabilities of x, to generate R simulated paths of future actions and state variables from period t + 1 to t + T (i.e., T periods ahead). We index simulated paths by r {1, 2,..., R}. The r th simulated path associated with the initial state (a i, x mt ) is {a (r,a i ) im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T } Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

82 Dealing with the curse of dimensionality Simulation-Based Estimation (4) Simulation-based estimation A simulated path {a (r,a i ) follows. im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T } is obtained as Given (a i, x mt ), we use the transition probability function F x (. a i, x mt ) to obtain a random draw x (r,a i ) m,t+1. Given x (r,a i ) m,t+1, we use the choice probability P i (x (r,a i ) m,t+1 ) to obtain a random draw a (r,a i ) i,m,t+1. Given (a (r,a i ) i,m,t+1, x(r,a i ) m,t+1 ), we use the transition probability function F x (. a (r,a i ) i,m,t+1, x(r,a i ) m,t+1 ) to obtain a random draw x(r,a i ) m,t+2. And so on. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

83 Dealing with the curse of dimensionality Simulation-based estimation Simulation-Based Estimation (5) Then, given the simulated paths {a (r,a i ) im,t+j, x(r,a i ) m,t+j : j = 1, 2,..., T }, we construct the simulator of z P imt (a i ) as: z P,sim imt (a i ) = z P imt (a i )+ 1 R The simulator of W P imt is W P,sim imt [ R T β j z P i r =1 j=1 ( ) ] a (r,a i ) im,t+j, x(r,a i ) m,t+j = (1 P i (x mt )) z P,sim imt (0) + P i (x mt ) z P,sim imt (1) If the DP problem has finite horizon, or if T is large enough such that the approximation error associated with the truncation of paths is negligible, then these simulators are unbiased. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

84 Dealing with the curse of dimensionality Simulation-Based Estimation (6) Simulation-based estimation This simulators can be used either for ME or for MI estimation. The simulation-based ME estimator is the value of θ that solves the system of equations: ( { ( ) }) E H(x mt ) a imt Φ z^p 0,sim imt θ i for any i, t = 0 The simulation-based MI estimator is the value (or set of values) of θ that minimizes the criterion function: ( { [ ] }) 2 min 0 ; W (^P 0 i,^p 0 i),sim imt W (P i,^p 0 i),sim imt θ i i,m,t P H Victor Aguirregabiria () Empirical IO Toronto. Winter / 136

85 Dealing with the curse of dimensionality Simulation-Based Estimation (7) Simulation-based estimation These estimators are consistent only as the number of simulated paths, R, goes to infinity. Note that there are three sources of error in (W ^P 0,sim imt 1 Estimation error: because ^P 0 = P 0 W P0 imt ): 2 Simulation error: because the expectation is not taken over all the possible future histories but only over the simulated paths. 3 Approximation error: because T = (3) can be negligible if T is not too small or β too close to one. (1) can be very important: curse of dimensionality in NP estimation. (2) can be very important. Even with millions of simulated histories we may have a very small proportion of all possible histories. Victor Aguirregabiria () Empirical IO Toronto. Winter / 136