1 Outline. 1. MSL. 2. MSM and Indirect Inference. 3. Example of MSM-Berry(1994) and BLP(1995). 4. Ackerberg s Importance Sampler.

Transcription

1 1 Outline. 1. MSL. 2. MSM and Indirect Inference. 3. Example of MSM-Berry(1994) and BLP(1995). 4. Ackerberg s Importance Sampler.

2 2 Some Examples. In this chapter, we will be concerned with examples of the form: f(y x, θ) = Z f(y x, γ)g(γ θ) We shall discuss some methods for estimating these problems. Then we shall discuss, in detail, two examples. The first is the approach to demand estimation following Berry (1994) and BLP(1995). The second is discrete games using the Ackerberg importance sampler.

3 That is, the density f(y x, γ) is allowed to depend on parameters γ that vary within the population. The population distribution of these parameters is g(γ θ). Examples of this are random effects models of unobserved heterogeneity where an intercept is allowed to vary stochastically. A second is the random coefficient logit. In both cases, integration needs to be performed in order to evaluate the likelihood function. We can evaluate the above integral using either deterministic methods for intergration or stochastic methods based on draw pseudo random numbers.

4 In practice, the latter approach seems to be more common since the asymptotic theory takes account of the approximation error in evaluating the integral. To do Monte Carlo integration, we need to draw s =1,...,S pseudo-random deviaties γ (s) from the density g(γ). Our estimate of the integral is then: bf(y i x i,θ)= 1 S SX s=1 f(y i x i,γ (s) i,θ) Note that we are drawing a seperate set of deviates for each y i and x i in our simulator.

5 Also note that we have assumed that θ does not enter into g(γ). In practice, we may have to carefully parameterize or problems to make sure that this is true. We shall consider an example of this shortly. Note that we could form a confidence interval for the mean in order to evaluate the accuracy of our integral. Also, by doing many monte carlos of size S, we can generate a distribution of b f(y x, θ)toseehow accurate our integral is evaluated. This simulation is unbiased and consistent as S.

6 The MSL (method of simulated likelihood) estimator is: ln L N (θ) = 1 N X i ln b f(y i x i,θ) In some, but not all cases, b f(y i x i,θ)isasmooth and differentiable function of θ. Our estimator b θ MSL is defined as: bθ MSL =argmaxlnl N (θ) Prop 21.1 from Gourieroux and Monfort (stated in the text) demonstrates that if

7 (i)the likelihood satisfies the regularity conditions for asymptotic normality with limit variance Λ 1 (θ 0 ) Λ(θ 0 )= plim N 1 X N i=1 2 ln f(y x, θ) θ θ 0 (ii) the density is simulated with an unbiased simulator (iii)if S,N witih N 1/2 /S 0, then N 1 ( b θ MSL θ 0 ) d N(0, Λ 1 (θ 0 )) There are two approaches one can take to calculating the variance. A first is to bootstrap your standard errors.

8 Asecondisbasedondifferentiating lnf(y b i x i,θ)= P Ss=1 ln f(y i x i,γ (s) i ) 1 S Weshalltalkaboutthemechanicsofimplimenting this estimator in a demand estimation example from Berry (1994) which is closesly related to BLP(1995). 3 MSM and Indirect Inference A MSM estimator starts by specifying a moment equation that depends on the distribution of some random variable. m(y i,x i,θ)= Z h(y i,x i,γ,θ)g(γ)

9 We proceed analogously to the MSL estimator and use Monte Carlo to simulate this integral. We need to draw s =1,...,S pseudo-random deviaties γ (s) from the density g(γ θ). cm(y i,x i,θ)= SX i=1 h(y i,x i,γ (s) i,θ) AsinthecaseofSML,weassumethattheparameters θ do not enter into g(γ). This may require a careful parameterization of our model. If we could perfectly evaluate our integral, in the just identified case, our GMM estimator would be:

10 Q N (θ) = NX i=1 w i m(y i,x i,θ) 0 NX i=1 w i m(y i,x i,θ) where w i corresponds to our weights. In MSM, we plug in the sample analogue- Q N (θ) = NX i=1 w i cm(y i,x i,θ) 0 NX i=1 w i cm(y i,x i,θ) Under the regularity conditions stated in the text, MSM is consistent and asymtotically normal for a fixed S. Unlike MSL, we do not need to let S in order to estimate the model.

11 Given that we have asymptotic normality, this justifies the use of the bootstrap to compute our standard errors. A final method discussed in the text is indirect inference. This method is useful for models that are easy to simulate, but where it is hard to form MSL or MSM estimators. For example, a dynamic stochastic model from Stokey and Lucas might be easy to simulate (assuming it is specified parsimoniously). However, in some cases it might be hard to form moments (e.x. an (S,s) model of inventory behavior).

12 Indirect inference (in the literature, the term EMM, efficient method of moments is used for related estimators) is also common in certain time series problems. Suppose that our model specifies that y i = f(x i,η,θ) where x i is a random variable, η is a stochastic shock and θ is a vector of parameters. Suppose that g(η) is the density for our shock. In indirect inference, we start with an auxiliary model. For example, we might specify ad hoc regressions of the dependent variable y i on the exogenous variables x i.

13 We run this regression on the true data and come up with regression coefficients b β. Given a vector of parameters, θ and we could simulate our model to generate a sequence of psuedo random (ey i, ex i ) i =1,...,N. We could then run our auxilary model on the psuedo random (ey i, ex i )tocomeupwitha b β(θ). The indirect inference estimator is: bθ = ³b β b β(θ) 0 Ω 1 ³b β b β(θ) Where Ω 1 is a weight matrix.

14 Intuitive, indirect inference attempts to match the parameters of the auxiliary model on the real and simulated data. Essentially, it is an extremely convenient way to form moments for a simulation based model. Another attract feature is that often the weight matrix Ω 1 can be formed using the data (y i,x i ) and does not require simulating the model. 4 Example of MSM In Berry (1994) and BLP (1995), consumer preferences can be written as:

15 u(x j,ξ j,p j,v i ; θ d ) where: x j =(x j,1,..., x j,k )isavectorofk characteristics of product j that are observed by both the economist and the consumer. ξ j is a characteristic of product j observed by the consumer but not by the economist. p j is the price of good j v i vector of taste parameters for consumer i θ d vector of demand parameters.

16 One commonly used specification is the logit model with random (normal) coefficients: u ij = x j β i αp j + ξ j + ε ij The K random coefficients are: β i,k = β k + σ k η i,k η i,k N(0, 1), iid Consumer i will purchase good j if and only if it is utility maximizing, just as in the previous lecture. Question: How do we interpret the parameters of this model?

17 It is useful to decompose utility into two parts, the first is a mean level of utility and the second is a heteroskedastic error terms that captures the effect of random tastes parameters: υ ij = X k x jk σ k η i,k + ε ij δ j = x j β αp j + ξ j We can now write utility of person i for product j as: u ij = δ j + υ ij Next, we will write the market shares for aggregate demand in a particularly convenient fashion.

18 First define the set of error terms that make product j utility maximizing given the J dimensional vector δ =(δ j ) A j (δ) = n υ i =(v ij ) δ j + v ij δ j 0 + v ij 0 for all j 0 6= j o The market share of product j can then be written as (assuming a law of large numbers): s j (δ(x, p, ξ),x,θ)= Z A j (δ) f(υ)dυ In this case, the parameter θ is β, α and σ. Given θ and the demand for product j actually observed in the data, es j itmustbethecasethat:

19 es j = s j (δ(x, p, ξ),x,θ) Given θ, this can be expressed as a system of J equations in J unknowns (the ξ j ). To estimate, we find a set of instruments for the ξ j. We must find a set of instruments correlated with the endogenous variable p j, but uncorrelated with the residual ξ j. Commonly used instruments: 1. The product characteristics.

20 2. Prices of products in other markets (interpret ξ j as a demand shifter). 3. Measures of isolation in product space ( P j 0 6=j x j 0,k ) 4. Cost shifters. 4.1 Computation. In this section, I shall outline some of the key steps needed to actually compute Berry (1994). A key step in many programming projects is to doafakedataexperiment/montecarlostudy. Simulate the model using fixed parameter values.

21 Pretend you don t know the parameter values and estimate. This tests the code and sometimes shows you limitations of the models. One of the best ways to really learn the econometricsinapaperistodoafakedataexperiment. We shall consider as an example the random coefficinetlogitmodel. There are basically 4 things we need to do in order to compute the value of the objective function in order to do GMM. 1. For a given value of σ and δ, compute the vector of market shares.

22 2. For a given value of σ, find the vector δ that equates the observed market shares and those predicted by the model using the contraction mapping. 3. Given δ and β, α compute the value of ξ 4. Search for the value of ξ that mimizes the objective function. We shall consider these one at a time. 4.2 Computing Market Shares. In the random coefficient logit model, we can compute the market shares, given δ as follows:

23 s j (δ, σ) = Z exp(δ j + P k x j,k η i,k σ k ) 1+ P j 0 exp(δ j 0 + P k x j 0,k η i,kσ k ) df (η i) In practice, the integral above is computed using simulation. Make a set of S simulation draws for each j and keep them fixed for the whole problem. Denote the draws as η (s) i,s=1,...,s. Let our simulated shares are: bs j (δ, σ) = SX s=1 exp(δ j + P k x j,k η (s) i,k σ k) 1+ P j 0 exp(δ j 0 + P k x j 0,k η(s) i,k σ k)

24 Sometimes importance sampling is useful in order to improve the speed/accuracy of the integration. Imporance sampling is discussed in the text. We can compute confidence intervals using standard methods to see whether the simulated market shares are well estimated. 4.3 The contraction mapping. Next, we wish to find the δ that matches the observed market shares given σ. In Berry and BLP they demonstrate that the following is a contraction:

25 δ (n+1) j = δ (n) j +ln(es j ) ln(bs j (δ, σ)) Berry proves that this is a contraction. Point: Market shares can be inverted very quickly inafairlysimplemanner! 5 Computing the value of ξ The next set is simple. Just let: ξ j = δ j (x j β αp j ) where δ j is computed using the contraction mapping.

26 5.1 Computing the value of the objective function. Let Z be the set of instruments. The objective function is formulated as in all MSM problems assuming E (ξ Z) = 0. The econometrician then chooses β, α, andσ in order to minimize the MSM objective function. Standard mathematical programs (MATLAB, GAUSS, IMSL,NAG) contain software for optimization problems. One standard way to proceed is to do a rough global search first and then use a derivative based method second once you have a very rough sense of the overall shape of the objective function.

27 Multiple starting points commonly used in order to search for multiple local solutions to minimization problem.. See Judd for an overview of numerical minimization. 6 Ackerberg s Importance Sampler. Finally, we consider a method that is useful when the model is difficult to compute but it is possible to reparameterize the model. With the reparameterization, the algorithm is parallel and can be computed more efficiently. Ackerberg (2006) describes this method in detail.

28 As an example, we take the problem of discrete/normal formgamesasstudiedinbajari,hongandryan (2006). Consider static entry game (see Bresnahan and Reiss (1990,1991), Berry (1992), Tamer (2002), Ciliberto and Tamer (2003), and Manuszak and Cohen (2004)). The economist observes a cross section of markets. Theplayersinthegameareafinite set of potential entrants. In each market, the potential entrants simultaneously choose whether to enter.

29 Let a i = 1 denote entry and a i = 0 denote nonentry. In applications, the function f i takes a form such as: f i = n θ 1 x + δ X 1 n a j =1 o if a i =1 j6=i 0ifa i =0 (1) The covariates x are variables which influence the profitability of entering a market. These might include the number of consumers in the market, average income and market specific cost indicators. The term δ measures the influence of j s choice on i s entry decision.

30 The ε i (a) capture shocks to the profitability of entry that are commonly observed by all firms in the market. This is a simultaneous system of logit models! In the paper, we also discuss network effects and peer effects as other examples. 7 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.

31 A = Y i A i. Utility u i : A R, wherer is the real line. Let π i is a mixed strategy. A Nash equilibrium is a set of best responses. Following Bresnahan and Reiss (1990,1991), econometrically a game is a discrete choice model. Except actions of others are right hand size variables. u i (a) =f i (x, a; θ 1 )+ε i (a). (2)

32 Mean utility, f i (x, a; θ 1 ) a, the vector of actions, covariates x, and a parameters θ. ε i (a) preference shocks. ε i (a) g(ε θ 2 )iid. Standard random utility model, except utility depends on actions of others. E(u) set of Nash equilibrium given a vector of utilities u. λ(π; E(u),β) is probability of equilibrium, π E(u) given parameters β.

33 λ(π; E(u),β) corresponds to a finite vector of probabilities. In an application, might let λ depend on 1. Satisfies a particular refinement concept (e.g. trembling hand perfection). 2. The equilibrium is in pure strategies. 3. Maximizes joint payoffs (efficiency). 4. Maximizes profit of incumbent firms (as in airlines examples). In practice, we could create dummy variables for whether a given equilibrium, π E(u) satisfies 1-4 above.

34 Let x(π, u) be this vector of dummies. A straightforward way to model λ is: λ(π; E(u),β) = exp(β x(π, u)) P π 0 E(u) exp(β x(π0,u)) (3) 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. Also not hard to program directly.

35 8 Estimation. P (a x, θ, β) is probability of a given x, θ and β Z X π E(u(x,θ,ε)) P (a x, θ, β) = λ(π; u(x, θ 1,ε),β) ³ Y Ni=1 π(a i ) g(ε θ 2)dε Computation of the above integral is facilitated by the importance sampling procedure of Ackerberg. Make a change of variables to integrate over latent utility u i instead of over ε i. With this change, we won t need to recompute the equilibria to the game during estimation. Estimation not feasible without this insight.

36 Often g(ε θ 2 ) is a simple parametric distribution (e.g. normal, extreme value, etc...) 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) = Y i Y a A φ(ε i (a); f i (θ, x, θ)+μ, σ) 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 cheap. Draw s =1,..., S vectors of vnm utilities, u (s) = (u (s) 1,...,u(s) N )fromanimportancedensityq(u).

37 WecanthensimulateP (a x, θ, β) asfollows: P X Ss=1 π E(u) bp (a x, θ, β) = λ(π; E(u (s) ),β) ³ Y Ni=1 π(a i ) h(u (s) θ,x) q(u (s) ) Precompute E(u (s) ) for a large number of randomly drawn games s =1,...,S. Evaluating b P (a x, θ, β) at new parameters DOES NOT REQUIRE RECOMPUTING E(u (s) )fornew s =1,...,S! Evaluating simulation estimator of P b (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 cheap computation.

38 Normally, the computational expense of structural estimation comes from recomputing the equilibrium many times. Also, note that this approach is naturally parallel. This saves on the computational time by orders of magnitude. Given b P (a x, θ, β) we can simulate the likelihood function or simulate the moments. The asymptotics are standard.

39 9 Application. Study entry decisions by highway paving contractors in CA Focus on decisions by 4 largest contractors in our sample. 414 projects, 271 contractors, $369.2 million dollars awarded. Simultaneous move-antitrust laws forbid communication. Static-bid for the right to complete a single job. f i (a, x) expected profits less markup.

40 Usenonparametricproceduretoestimatemarkups Guerre, Perrigne and Vuong (Emet, 2000). Literature is well developed and seems to give reasonable answers (see Bajari and Hortacsu (JPE, 2005)). Median margin is 2.7% similar to reported margins of publicly traded companies in our sample. Firms compete spatially. Transportation costs significant- closest firm has costs advantage and market power. Only part of f i to be estimated is entry costs (around 1-2% which is reasonable according to industry books, e.g. Park and Chapin).

41 Goal- which equilibrium is most likely? Pure Strategy, efficient, dominated, Nash product?