Constrained Optimization Approaches to Structural Estimation. Che-Lin Su

Transcription

1 Constrained Optimization Approaches to The University of Chicago Booth School of Business ICE 2012 July 16 28, 2012

2 Outline 1. Estimation of Dynamic Programming Models of Individual Behavior

3 Outline 1. Estimation of Dynamic Programming Models of Individual Behavior 2. Estimation of Demand Systems

4 Outline 1. Estimation of Dynamic Programming Models of Individual Behavior 2. Estimation of Demand Systems 3. Estimation of Games of Incomplete Information

5 Optimization Overview Part I Optimization Overview

6 Optimization Overview Unconstrained Optimization: Background min {f(x) : x R n } f : R n R smooth (typically C 2 ) x R n finite dimensional (may be large) Optimality conditions: x local minimizer: f(x ) = 0 Numerical methods: generate a sequence of iterates x k such that the gradient test f(x k ) τ is eventually satisfied; usually τ = 1.e 6 Warning: Any point x that does NOT satisfy f(x) τ should NOT be considered as a solution or a candidate for the solution

7 Optimization Overview Did the solver Find a Solution? First-order Iteration Func-count f(x) Step-size optimality e e e e e Optimization terminated: relative infinity-norm of gradient less than options.tolfun.

8 Optimization Overview Generic Nonlinear Optimization Problem Nonlinear Programming (NLP) problem minimize f(x) objective x subject to c(x) = 0 constraints x 0 variables f : R n R, c : R n R m smooth (typically C 2 ) x R n finite dimensional (may be large) more general l c(x) u possible

9 Optimization Overview Optimality Conditions for NLP Constraint qualification (CQ) Linearizations of c(x) = 0 characterize all feasible perturbations x local minimizer & CQ holds multipliers y, z : f(x ) c(x ) T y z = 0 c(x ) = 0 X z = 0 x 0, z 0 where X = diag(x ), thus X z = 0 x i z i = 0

10 Optimization Overview Solving the FOC for NLP Nonlinear equations: F (w) = 0, where w = (x, y, z) with x, z 0. NLP solvers: generate a sequence of iterates w k such that the test F (w k ) τ with x k 0, z k 0 is eventually satisfied; usually τ = 1.e 6. Same warning applies. Supply exact derivatives: f(x), c(x), 2 L(x, y, z), where is the Lagrangian: L(x, y, z) := f(x) y T c(x) z T x Concerns: NLP is difficult to solve when # of variables and # of constraints are large In many applied models, constraint Jacobian c(x) and Hessian of the Lagragian 2 L(x) are sparse Modern solvers exploit the sparsity structure of c(x) and 2 L(x)

11 Optimization Overview Great interest in estimating models based on economic structure DP models of individual behavior: Rust (1987) Demand Estimation: BLP(1995), Nevo(2000) Nash equilibria of static and dynamic games: AM (2007), BBL (2007), Pakes, Ostrovsky and Berry (2007), Pesendorfer and Schmidt-Dengler (2008) Auctions: Paarsch and Hong (2006), Hubbard and Paarsch (2008) Dynamic stochastic general equilibrium Popularity of structural models in empirical IO and marketing Model sophistication introduces computational difficulties General belief: Estimation is a major computational challenge because it involves solving the model many times Our approach: Formulate structural estimation models as constrained optimization problems and use modern constrained optimization methods and software to solve the models for you

12 Optimization Overview Single-Agent Dynamic Discrete Choice Models Rust (1987): Bus-Engine Replacement Problem Nested-Fixed Point Problem (NFXP) Su and Judd (2012): Constrained Optimization Approach Random-Coefficients Logit Demand Models BLP (1995): Random-Coefficients Demand Estimation Nested-Fixed Point Problem (NFXP) Dubé, Fox and Su (2012): Constrained Optimization Approach Estimating Discrete-Choice Games of Incomplete Information Aguirregabiria and Mira (2007): NPL (Recursive 2-Step) Bajari, Benkard and Levin (2007): 2-Step Pakes, Ostrovsky and Berry (2007): 2-Step Pesendorfer and Schmidt-Dengler (2008): 2-Step Pesendorfer and Schmidt-Dengler (2010): comments on AM (2007) Kasahara and Shimotsu (2012): Modified NPL Su (2012), Egesdal, Lai and Su (2007) : Constrained Optimization

13 Optimization Overview Optimization and Computation in Optimization and computation often perceived as 2nd-order importance to research agenda Typical computational methohd is Nested Fixed-Point procedure: fixed-point calculation embedded in calculation of objective function compute an equilibrium invert a model (e.g. non-linearity in disturbance) compute a value function (i.e. dynamic model) Mis-use of optimization can lead to the wrong answer naively use canned optimization algorithms e.g., Matlab s fminsearch adjust default-settings of solvers to improve speed not accuracy assume there is a unique fixed-point do NOT check or understand solver s output message!! KNITRO: Locally Optimal Solution Found. FilterSQP: Optimal Solution Found. SNOPT: Optimal Solution Found. Matlab Optimization Toolbox: Optimization terminated... does NOT tell you much about what happened at the end

14 Optimization Overview First Step in Solving an Estimation Model Make sure you have a smooth formulation for the model smooth objective function smooth constraints Use the best available NLP solvers! Many free NLP solvers are crappy; they often fail or even worse, can give you wrong solutions Do not attempt to develop numerical algorithms/solvers by yourself You should use solvers developed by professionals, i.e., numerical optimization people Best NLP solvers: SNOPT (Stanford), KNITRO (Northwestern), Filter-SQP (Argonne), IPOPT (IBM), PATH (UW-Madison) Keys to efficient implementation Supply exact 1st and 2nd order derivatives Supply sparsity pattern for constraint Jacobian and Hessian of the Lagragian

15 Part II Estimation of Dynamic Programming Models

16 Estimation of Single-Agent Dynamic Models Rust (1987): Zurcher s Data Bus #: 5297 events year month odometer at replacement 1st engine replacement 1979 June nd engine replacement 1984 August year month odometer reading 1974 Dec Jan Feb Mar Apr May Jun Jul Aug Sep

17 Estimation of Single-Agent Dynamic Models Zurcher s Bus Engine Replacement Problem Each bus comes in for repair once a month Bus manager sees x t : mileage at time t since last engine overhaul ε t = [ε t (d t = 0), ε t (d t = 1)]: other state variable Bus manager chooses between overhaul and ordinary maintenance { 1, replacing the engine; d t = 0, performing regular maintenance. Utility per period u(x t, d t, ε t ; θ c, RC) = ν(x t, d t ; θ c, RC) + ε t (d t ) where { ν(x t, d t, θ c c(xt, θ, RC) = c ) if d t = 0 (RC + c(0, θ c ) if d t = 1 c (x; θ c ): expected operating costs per period at mileage x RC: the expected replacement cost to install a new engine, net of any scrap value of the old engine The mileage x is reset to 0 after the engine replacement

18 Estimation of Single-Agent Dynamic Models Zurcher s Bus Engine Replacement Problem Given (x t, ε t ), the bus manager solves the DP: [ ] max IE β τ t u(x τ, d τ, ε τ ; θ c, RC) {d t,d t+1,d t+2,...} τ=t The expectation IE is taken over the state transition probability p(x t+1, ε t+1 x t, ε t, d t ; θ p ) Value function [ ] V (x t, ε t ) = max IE β τ t u(x τ, d τ, ε τ ; θ c, RC) {d t,d t+1,d t+2,...} τ=t Econometrician Observes mileage x t and decision d t, but not cost Assumes extreme value distribution for ε t (d t ) Structural parameters to be estimated: θ = (θ c, RC, θ p ) Coefficients of operating cost function; e.g., c(x, θ c ) = θ1x c + θ2x c 2 Overhaul cost RC state transition probability p(x t+1, ε t+1 x t, ε t, d t ; θ p )

19 Estimation of Single-Agent Dynamic Models Zurcher s Bus Engine Replacement Problem Bellman equation { } V (x, ε) = max ν(x, d; θ c, RC) + ε(d) + β V (x, ε )p(x, ε x, ε, d; θ p )dx dε d x ɛ Conditional Independence (CI) Assumption: p(x, ε x, ε, d; θ p ) = p 2 (ε x ; θ p 2 )p 3(x x, d; θ p 3 ) Expected value function EV (x) = V (x, ε)p 2 (ε x ; θ p 2 ) ɛ Choice-specific expected value function EV (x, d) = ν(x, d; θ c, RC) + ε(d) + β EV (x)p 3 (x x, d; θ p 3 )dx x

20 Estimation of Single-Agent Dynamic Models Zurcher s Bus Engine Replacement Problem Assume type-1 extreme value distribution for ε = [ε(0), ε(1)] Conditional choice probability P (d x; θ) = exp [ν(x, d; θ c, RC) + βev (x, d)] exp [ ν(x, d ; θ c, RC) + βev (x, d ) ] d {0,1} Choice-specific expected value function EV (x, d) = log x =0 d {0,1} exp [ ν(x, d ; θ c, RC) + βev (x, d ) ] p 3(dx x, d, θ p 3 )

21 Estimation of Single-Agent Dynamic Models Zurcher s Bus Engine Replacement Problem Discretize the mileage state space x into K grid points ˆx = {ˆx 1,..., ˆx K } with ˆx 1 = 0 Mileage transition probability: for j = 1,..., J { Pr{x p 3 (x ˆx k, d, θ p 3 ) = = ˆx k+j θ p 3 }, if d = 0 Pr{x = ˆx 1+j θ p 3 }, if d = 1 Mileage in the next period x can move up at most J grid points Choice-specific expected value function for ˆx ˆx J EV (ˆx, d) = log j=0 d {0,1} exp [ ν(x, d ; θ c, RC) + βev (x, d ) ] p 3(x ˆx, d, θ p 3 )

22 Estimation of Single-Agent Dynamic Models Zurcher s Bus Engine Replacement Problem Data: time series (x t, d t ) T t=1 Likelihood function T L(θ) = P (d t x t, θ c, RC)p 3 (x t x t 1, d t 1, θ p 3 ) t=2 with P (d x, θ c exp{ν(x, d; θ c, RC) + βev θ (x, d)}, RC) = exp{ν(x, d ; θ c, RC) + βev θ (x, d)} d {0,1} EV θ (x, d) = T θ (EV θ )(x, d) J log exp [ ν(x, d ; θ c, RC) + βev (x, d ) ] p3(x ˆx, d, θ3) p j=0 d {0,1}

23 Estimation of Single-Agent Dynamic Models Nested Fixed Point Algo: Rust (1987) Outer loop: Solve likelihood max θ 0 L(θ) = T t=2 P (d t x t, θ c, RC)p 3 (x t x t 1, d t 1, θ p 3 ) Convergence test: θ L(θ) ɛ out Inner loop: Compute expected value function EV θ for a given θ EV θ is the implicit expected value function defined by the Bellman equation or the fixed point function EV θ = T θ (EV θ ) Convergence test: EV k+1 θ EV k θ ɛ in Rust started with contraction iterations and then switched to Newton iterations

24 Estimation of Single-Agent Dynamic Models Smooth Objective Function? Recall first step in solving an estimation problem Is the ML objective function L(θ) smooth (differentiable w.r.t. θ)? T L(θ) = P (d t x t, θ c, RC)p 3 (x t x t 1, d t 1, θ p 3 ) t=2 P (d x, θ c, RC) = exp{ν(x, d; θ c, RC) + βev θ (x, d)} exp{ν(x, d ; θ c, RC) + βev θ (x, d)} d {0,1} Is EV θ differentiable w.r.t. θ? Yes, because T θ (EV θ ) is a contraction mapping(!) Is the approximated ML objective function L(θ, ɛ in ) smooth (differentiable w.r.t. θ)? Is EV θ (ɛ in ) differentiable w.r.t. θ or w.r.t. ɛ in?

25 Estimation of Single-Agent Dynamic Models Concerns with NFXP Dubé Fox and Su (2011) Inner-loop error propagates into outer-loop function and derivatives NFXP needs to solve inner-loop exactly for each vector of parameters to accurately compute the search direction for the outer loop to accurately evaluate derivatives for the outer loop for the outer loop to converge Stopping rules: choosing inner-loop and outer-loop tolerances inner-loop can be slow: contraction mapping is linearly convergent tempting to loosen inner loop tolerance ɛ in used often see ɛ in = 1.e 6 or higher outer loop may not converge with loose inner loop tolerance check solver output message tempting to loosen outer loop tolerance ɛ in to promote convergence often see ɛ out = 1.e 3 or higher Rust s implementation of NFXP was correct ɛ in = 1.e 13 finished the inner-loop with Newton s method

26 Estimation of Single-Agent Dynamic Models Stopping Rules Dubé Fox and Su (2011) Notations: L(θ, ɛ in ): the programmed outer loop objective function with ɛ in Analytic derivatives θ L(θ, ɛ in ) are provided: ɛ out = O( β 1 β ɛ in) Finite-difference derivatives are used: ɛ out = O( β 1 β ɛ in)

27 Estimation of Single-Agent Dynamic Models Constrained Optimization for Solving Zucher Model Form augmented likelihood function for data X = (x t, d t ) T t=1 L (θ, EV ; X) = with P (d x, θ c, RC) = T P (d t x t, θ c, RC)p(x t x t 1, d t 1, θ p ) t=2 exp{ν(x, d; θ c, RC) + βev (x, d)} exp{ν(x, d ; θ c, RC) + βev (x, d )} d {0,1} Rationality and Bellman equation imposes a relationship between θ and EV EV = T (EV, θ) Solve the constrained optimization problem max (θ,ev ) L (θ, EV ; X) subject to EV = T (EV, θ)

28 Estimation of Single-Agent Dynamic Models Equivalent Reformulation? ML-NFXP: max θ 0 L(θ) = T t=2 ML-Constrained Optimization: max (θ,ev ) P (d t x t, θ c, RC)p 3 (x t x t 1, d t 1, θ p 3 ) L (θ, EV ; X) subject to EV = T (EV, θ) Are these two formulations equivalent? Proof? Are the first-order conditions of these two formulations equivalent? Proof?

29 Estimation of Single-Agent Dynamic Models Monte Carlo: Rust s Table X - Group 1,2, 3 Fixed point dimension: 175 Maintenance cost function: c(x, θ c ) = θ c 1 x Mileage transition: stay or move up at most 4 grid points True parameter values: θ1 c = RC = (θ p 30, θp 31, θp 32, θp 33 ) = (0.0937, , , ) Solve for EV at the true parameter values Simulate 250 datasets of monthly data for 10 years and 50 buses Estimation implementations MPEC1: AMPL/Knitro (with 1st- and 2nd-order derivative) MPEC2: Matlab/ktrlink (with 1st-order derivatives) NFXP: Matlab/ktrlink (with 1st-order derivatives) 5 re-start in each of 250 replications

30 Estimation of Single-Agent Dynamic Models Monte Carlo: β = and β Imple. Parameters MSE RC θ c 1 θ p 30 θ p 31 θ p 32 θ p 33 true MPEC (1.613) (0.500) (0.0036) (0.0057) (0.0060) (0.0015) MPEC (1.613) (0.500) (0.0036) (0.0057) (0.0060) (0.0015) NFXP (1.617) (0.500) (0.0036) (0.0057) (0.0060) (0.0015) MPEC (1.570) (0.458) (0.0037) (0.0057) (0.0060) (0.0015) MPEC (1.570) (0.458) (0.0037) (0.0057) (0.0060) (0.0015) NFXP (1.571) (0.459) (0.0037) (0.0057) (0.0060) (0.0015)

31 Estimation of Single-Agent Dynamic Models Monte Carlo: β = and β Imple. Parameters MSE RC θ c 1 θ p 31 θ p 32 θ p 33 θ p 34 true MPEC (1.371) (0.413) (0.0037) (0.0057) (0.0060) (0.0015) MPEC (1.371) (0.413) (0.0037) (0.0057) (0.0060) (0.0015) NFXP (1.368) (0.411) (0.0037) (0.0057) (0.0060) (0.0015) MPEC (1.305) (0.407) (0.0036) (0.0057) (0.0060) (0.0015) MPEC (1.305) (0.407) (0.0036) (0.0057) (0.0060) (0.0015) NFXP (1.305) (0.407) (0.0036) (0.0057) (0.0060) (0.0015)

32 Estimation of Single-Agent Dynamic Models Monte Carlo: β = β Imple. Parameters MSE RC θ c 1 θ p 31 θ p 32 θ p 33 θ p 34 true MPEC (1.308) (0.414) (0.0036) (0.0057) (0.0060) (0.0015) MPEC (1.308) (0.414) (0.0036) (0.0057) (0.0060) (0.0015) NFXP (1.308) (0.414) (0.0036) (0.0057) (0.0060) (0.0015)

33 Estimation of Single-Agent Dynamic Models Monte Carlo: Numerical Performance β Imple. Runs CPU Time # of Major # of Func. # of Contrac. Conv. (in sec.) Iter. Eval. Mapping Iter MPEC MPEC NFXP , MPEC MPEC NFXP , MPEC MPEC NFXP , MPEC MPEC NFXP , MPEC MPEC NFXP ,487

34 Estimation of Single-Agent Dynamic Models Observations MPEC In MPEC/AMPL, problems are solved very quickly. The likelihood function, the constraints, and their first-order and second-order derivatives are evaluated only around 20 times Constraints (Bellman Eqs) are NOT solved exactly in most iterations No need to resolve the fixed-point equations for every guess of structural parameters Quadratic convergence is observed in the last few iterations; in contrast, NFXP is linearly convergent (or super-linear at best) In NFXP, the Bellman equations are solved around 200 times and evaluated between 134,000 and 750,000 times

35 Estimation of Single-Agent Dynamic Models Advantages of Constrained Optimization Newton-based methods are locally quadratic convergent Two key factors in efficient implementations: Provide analytic-derivatives huge improvement in speed Exploit sparsity pattern in constraint Jacobian huge saving in memory requirement nz = 16207

36 Random-Coefficients Demand Estimation Part III Random-Coefficients Demand Estimation

37 Random-Coefficients Demand Estimation Random-Coefficients Logit Demand: BLP (1995) Berry, Levinsohn and Pakes (BLP, 1995) consists of an economic model and a GMM estimator Demand estimation with a large number of differentiated products characteristics approach applicable when only aggregate market share data available flexible substitution patterns / price elasticities control for price endogeneity Computational algorithm to construct moment conditions from a non-linear model Useful for measuring market power, welfare, optimal pricing, etc. Used extensively in empirical IO and marketing: Nevo (2001), Petrin (2002), Dubé ( ), etc.

38 Random-Coefficients Demand Estimation Random-Coefficients Logit Demand Utility of consumer i from purchasing product j in market t u ijt = β 0 i + x jt β x i β p i p jt + ξ jt + ε ijt product characteristics: x jt, p jt, ξ jt x jt, p jt observed; cov(ξ jt, p jt ) 0 ξ jt : not observed not in data β i : random coefficients/individual-specific taste to be estimated Distribution: β i F β (β; θ) BLP s statistical goal: estimate θ in parametric distribution error term ε ijt : Type I E.V. shock (i.e., Logit) Consumer i picks product j if u ijt u ij t, j j

39 Random-Coefficients Demand Estimation Market Share Equations Predicted market shares s j (x t, p t, ξ t, ; θ) = {β i,ε j u ijt u ij t, j j} df β(β; θ)df ε (ε) With logit errors ε s j (x t, p t, ξ t, ; θ) = Simulate numerical integral β ŝ j (x t, p t, ξ t, ; θ) = 1 ns Market share equations exp(β 0 + x jt β x β p p jt + ξ jt ) 1 + J k=1 exp(β0 + x kt β x β p p kt + ξ kt ) df β(β; θ) ns r=1 exp(β 0r + x jt β xr β pr p jt + ξ jt ) 1 + J k=1 exp(β0r + x kt β xr β pr p kt + ξ kt ) ŝ j (x t, p t, ξ t, ; θ) = S jt, j J, t T

40 Random-Coefficients Demand Estimation Random-Coefficients Logit Demand: GMM Estimator Assume E [ξ jt z jt z jt ] = 0 for some vector of instruments z jt Empirical analog g (θ) = 1 T J ξ jt (θ) z jt Data: {(x jt, p jt, S jt, z jt ) j J,t T } t,j Minimize GMM objective function Q(θ) = g (θ) W g (θ) Cannot compute ξ jt (θ) analytically Invert ξ t from system of predicted market shares numerically S t = s (x t, p t, ξ t ; θ) ξ t (θ) = s 1 (x t, p t, S t ; θ) BLP show the inversion of share equations for ξ(θ) is a contraction-mapping

41 Random-Coefficients Demand Estimation BLP/NFXP Estimation Algorithm Outer loop: min θ g (θ) W g (θ) Guess θ parameters to compute g(θ) = 1 T J Stop when θ (g (θ) W g (θ)) ɛ out T t=1 j=1 J ξ jt (θ) zjt

42 Random-Coefficients Demand Estimation BLP/NFXP Estimation Algorithm Outer loop: min θ g (θ) W g (θ) Guess θ parameters to compute g(θ) = 1 T J T t=1 j=1 J ξ jt (θ) zjt Stop when θ (g (θ) W g (θ)) ɛ out Inner loop: compute ξ t (θ) for a given θ Solve s(x t, p t, ξ t ; θ) = S t for ξ by contraction mapping: ξ h+1 t = ξ h t + log S t log s(x t, p t, ξ t ; θ) Stop when ξ h+1 t ξ h t ɛ in Denote the approximated demand shock by ξ(θ, ɛ in ) Stopping rules: need to choose tolerance/stopping criterion for both inner loop (ɛ in ) and outer loop (ɛ out )

43 Random-Coefficients Demand Estimation Smooth Objective Function? Is the GMM objective function Q(ξ(θ)) smooth (differentiable w.r.t. θ)? Q(ξ(θ)) = ξ(θ) ZW Z ξ(θ) Is ξ(θ) differentiable w.r.t. θ? Is the approximated GMM objective function Q(ξ(θ; ɛ in )) smooth (differentiable w.r.t. θ)? Q(ξ(θ, ɛ in )) = ξ(θ, ɛ in ) ZW Z ξ(θ, ɛ in ) Is ξ(θ, ɛ in ) differentiable w.r.t. θ or w.r.t. ɛ in?

44 Random-Coefficients Demand Estimation Our Concerns with NFP/BLP Inefficient amount of computation we only need to know ξ(θ) at the true θ NFP solves inner-loop exactly each stage of parameter search evaluating s(x t, p t, ξ t ; θ) thousands of times in the contraction mapping Stopping rules: choosing inner-loop and outer-loop tolerances inner-loop can be slow (especially for bad guesses of θ): linear convergence at best tempting to loosen inner loop tolerance ɛ in used often see ɛ in = 1.e 6 or higher outer loop may not converge with loose inner loop tolerance check solver output message; see Knittel and Metaxoglou (2008) tempting to loosen outer loop tolerance ɛ in to promote convergence often see ɛ out = 1.e 3 or higher Inner-loop error propagates into outer-loop

45 Random-Coefficients Demand Estimation Knittel and Metaxoglou (2010) Perform extensive numerical studies on BLP/NFXP algorithms with two data sets 10 free solvers and 50 starting points for each solver Find that convergence may occur at a number of local extrema, at saddles and in regions of the ob jective function where the First-Order Conditions are not satisfied. Furthermore, parameter estimates and measures of market performance, such as price elasticities, exhibit notable variation (two orders of magnitude) depending on the combination of the algorithm and starting values in the optimization exercise at hand Recall the optimization output that you saw earlier

46 Random-Coefficients Demand Estimation Analyzing BLP/NFXP Algorithm Let L be the Lipschitz constant of the inner-loop contraction mapping Numerical Errors in GMM function and gradient ( ) Q (ξ(θ, ɛ in )) Q (ξ(θ, 0)) = O L 1 L ɛ in θ Q (ξ (θ)) ξ=ξ(θ,ɛin ) θ Q (ξ (θ)) ( ) ξ=ξ(θ,0) = O L 1 L ɛ in Ensuring convergence: ɛ out = O( L 1 L )ɛ in

47 Random-Coefficients Demand Estimation Errors in Parameter Estimates θ = arg max {Q (ξ(θ, 0))} θ ˆθ = arg max θ {Q (ξ(θ, ɛ in ))} Finite sample error in parameter estimates ( O ˆθ θ 2) ( Q ξ( ˆθ,ɛ ) in ) Q (ξ(θ, 0)) + O ( L ) 1 L ɛ in Large sample error in parameter estimates ˆθ θ 0 ˆθ θ + θ θ 0 Q ( ξ( ˆθ,ɛ ) ( ) ( in ) Q (ξ(θ, 0)) + O L 1 L ɛ in + O 1/ ) T

48 Random-Coefficients Demand Estimation Numerical Experiment: 100 different starting points 1 dataset: 75 markets, 25 products, 10 structural parameters NFP tight: ɛ in = 1.e 10; ɛ out = 1.e 6 NFP loose inner: ɛ in = 1.e 4; ɛ out = 1.e 6 NFP loose both: ɛ in = 1.e 4; ɛ out = 1.e 2 GMM objective values Starting point NFXP tight NFXP loose inner NFXP loose both e 02 Fail e e 02 Fail e e 02 Fail e e 02 Fail e e 02 Fail e e 02 Fail e e 02 Fail e e 02 Fail e e 02 Fail e e 02 Fail e + 03 Main findings: Loosening tolerance leads to non-convergence Check optimization exit flags! Solver does NOT produce a local optimum with loose tolerances!

49 Random-Coefficients Demand Estimation Constrained Optimization Applied to BLP Constrained optimization formulation min (θ,ξ) subject to ξ T ZW Z T ξ s(ξ, θ) = S Advantages: No need to worry about setting up two tolerance levels No inner-loop errors propagated into parameter estimates Easy to code in AMPL and to access good NLP solvers AMPL provides analytic derivatives AMPL analyzes sparsity structure of constraint Jacobian Fewer iterations/function evaluations with first-order and second-order derivatives information Share equations only need to be hold at the solution Bad news: Hessian of the Lagrangian is dense

50 Random-Coefficients Demand Estimation Exploiting Symmetry and Sparsity in the Hessian By adding additional variable g and constraint Z T ξ = g min (θ,ξ,g,) subject to g T W g s(δ; θ 2 ) = S Z T ξ = g Advantages: The Hessian of the objective function is now sparse Increasing the sparsity huge saving on memory

51 Random-Coefficients Demand Estimation Sparsity Pattern of Constraint Jacobian c(x)

52 Random-Coefficients Demand Estimation Sparsity Pattern of Hessian 2 L(x, y, z)

53 Random-Coefficients Demand Estimation AMPL Model: MPEC BLP.mod param ns ; # := 20 ; # number of simulated "individuals" per market param nmkt ; # := 94 ; # number of markets param nbrn ; # := 24 ; # number of brands per market param nbrnplus1 := nbrn+1; # number of products plus outside good param nk1 ; # := 25; # of observable characteristics param nk2 ; # := 4 ; # of observable characteristics param niv ; # := 21 ; # of instrument variables param nz := niv-1 + nk1-1; # of instruments including iv and X1 param nd ; # := 4 ; # of demographic characteristics set S := 1..ns ; # index set of individuals set M := 1..nmkt ; # index set of market set J := 1..nbrn ; # index set of brand (products), including outside good set MJ := 1..nmkt*nbrn; # index of market and brand set K1 := 1..nk1 ; # index set of product observable characteristics set K2 := 1..nk2 ; # index set of product observable characteristics set Demogr := 1..nd; set DS := 1..nd*ns; set K2S := 1..nk2*ns; set H := 1..nz ; # index set of instrument including iv and X1

54 Random-Coefficients Demand Estimation AMPL Model: MPEC BLP.mod ## Define input data format: param X1 {mj in MJ, k in K1} ; param X2 {mj in MJ, k in K2} ; param ActuShare {m in MJ} ; param Z {mj in MJ, h in H} ; param D {m in M, di in DS} ; param v {m in M, k2i in K2S} ; param inva {i in H, j in H} ; # optimal weighting matrix = inv(z Z); param OutShare {m in M} := 1 - sum {mj in (nbrn*(m-1)+1)..(nbrn*m)} ActuShare[mj];

55 Random-Coefficients Demand Estimation AMPL Model: MPEC BLP.mod ## Define variables var theta1 {k in K1}; var SIGMA {k in K2}; var PI {k in K2, d in Demogr}; var delta {mj in MJ} ; var EstShareIndivTop {mj in MJ, i in S} = exp( delta[mj] + sum {k in K2} (X2[mj,k]*SIGMA[k]*v[ceil(mj/nbrn), i+(k-1)*ns]) + sum{k in K2, d in Demogr} (X2[mj,k]*PI[k,d]*D[ceil(mj/nbrn),i+(d-1)*ns]) ); var EstShareIndiv{mj in MJ, i in S} = EstShareIndivTop[mj,i] / (1+ sum{ l in ((ceil(mj/nbrn)-1)*nbrn+1)..(ceil(mj/nbrn)*nbrn)} EstShareIndivTop[l, i]); var EstShare {mj in MJ} = 1/ns * (sum{i in S} EstShareIndiv[mj,i]) ; var w {mj in MJ} = delta[mj] - sum {k in K1} (X1[mj,k]*theta1[k]) ; var Zw {h in H} ; ## Zw{h in H} = sum {mj in MJ} Z[mj,h]*w[mj];

56 Random-Coefficients Demand Estimation AMPL Model: MPEC BLP.mod minimize GMM : sum{h1 in H, h2 in H} Zw[h1]*invA[h1, h2]*zw[h2]; subject to conzw {h in H}: Zw[h] = sum {mj in MJ} Z[mj,h]*w[mj] ; Shares {mj in MJ}: log(estshare[mj]) = log(actushare[mj]) ;

57 Random-Coefficients Demand Estimation AMPL/KNITRO Output KNITRO 6.0.0: alg=1 opttol=1.0e-6 feastol=1.0e-6 Problem Characteristics Objective goal: Minimize Number of variables: 2338 bounded below: 0 bounded above: 0 bounded below and above: 0 fixed: 0 free: 2338 Number of constraints: 2300 linear equalities: 44 nonlinear equalities: 2256 linear inequalities: 0 nonlinear inequalities: 0 range: 0 Number of nonzeros in Jacobian: Number of nonzeros in Hessian: 58609

58 Random-Coefficients Demand Estimation AMPL/KNITRO Output Iter Objective FeasError OptError Step CGits e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02 0 EXIT: Locally optimal solution found.

59 Random-Coefficients Demand Estimation AMPL/KNITRO Output Final Statistics Final objective value = e+00 Final feasibility error (abs / rel) = 1.20e-08 / 1.20e-08 Final optimality error (abs / rel) = 6.36e-08 / 3.21e-09 # of iterations = 7 # of CG iterations = 9 # of function evaluations = 8 # of gradient evaluations = 8 # of Hessian evaluations = 7 Total program time (secs) = ( CPU time) Time spent in evaluations (secs) = =================================================== KNITRO 6.0.0: Locally optimal solution. objective ; feasibility error 1.2e-08 7 iterations; 8 function evaluations

60 Random-Coefficients Demand Estimation Monte Carlo in DFS11: Simulated Data Setup x 1,j,t x 2,j,t x 3,j,t N ξ j,t N (0, 1) p j,t = 0.5 ξ j,t + e j,t , k=1 x k,jt,t z j,t,d N ( 1 4 p j,t, 1 ), D = 6 instruments F β (β; θ): 5 independent normal distributions (K = 3 attributes, price and the intercept) β i = { β 0 i, β1 i, β2 i, β3 i, βp i } : E [βi ] = {0.1, 1.5, 1.5, 0.5, 3} and Var [β i ] = {0.5, 0.5, 0.5, 0.5, 0.2}

61 Random-Coefficients Demand Estimation Implementation Details MATLAB, highly vectorized code, available at http: //faculty.chicagobooth.edu/jean-pierre.dube/vita/mpec%20code.htm Optimization software KNITRO Professional quality optimization program Can be called directly from R2008a version of MATLAB We call from TOMLAB We provide sparsity pattern for c(x) and 2 L(x) for MPEC We code exact first-order and second-order derivatives Important for performance of smooth optimizers With both 1st and 2nd derivatives, NFP is 3 to 10 times faster than using only 1st order derivatives Same component functions for derivatives Helpful for standard errors

62 Random-Coefficients Demand Estimation Loose v.s. Tight Tolerances for NFXP NFXP NFXP NFXP Truth Loose Loose Tight Inner Both Fraction Convergence Frac.< 1% > Global Min Mean Own Price Elasticity Std. Dev. Own Price Elasticity Lowest Objective Elasticity for Lowest Obj starting values for one dataset NFXP loose inner loop:ɛ in = 10 4, ɛ out = 10 6 NFXP loose both: ɛ in = 10 4, ɛ out = 10 2 NFXP tight: ɛ in = 10 14, ɛ out = 10 6

63 Random-Coefficients Demand Estimation Lessons Learned Loose inner loop causes numerical error in gradient Failure to diagonose convergence of outer loop Leads to false estimates Making outer loop tolerance loose allows convergence But to false solution

64 Random-Coefficients Demand Estimation Speeds, # Convergences and Finite-Sample Performance T = 50, J = 25, nn = 1000, 20 replications, 5 starting points/replication Intercept Lipsch. Alg. CPU Elasticities Out. E [ ] βi 0 Const (min) Bias RMSE Value Share NFP MPEC NFP MPEC NFP MPEC NFP MPEC NFP MPEC NFP MPEC NFP MPEC

65 Random-Coefficients Demand Estimation # of Function/Gradient/Hessian Evals and # Contraction Mapping Iterations Intercept Alg. Func Grad/Hess Contraction E [ ] βi 0 Eval Eval Iter -2 NFP ,400 MPEC NFP ,100 MPEC NFP ,200 MPEC NFP ,000 MPEC NFP ,000 MPEC NFP ,000 MPEC NFP ,000 MPEC

66 Random-Coefficients Demand Estimation Lessons Learned For low Lipschitz constant, NFXP and MPEC about the same speed For high Lipschitz constant, NFXP becomes very slow 1 hour per run for Intercept = 4 Reminder: you need to use more starting points if you want to find a good solution MPEC speed relatively invariant to Lipschitz constant No contraction mapping in MPEC

67 Random-Coefficients Demand Estimation Speed for Varying # of Markets, Products, Draws T J nn Lipsch. Alg Runs CPU Outside Const. (hr) Share NFP 80% MPEC 100% NFP 100% MPEC 100% NFP 80% MPEC 100% NFP 80% MPEC 100% NFP 100% MPEC 100% NFP 100% MPEC 100% NFP 100% MPEC 100% 2.3

68 Random-Coefficients Demand Estimation # of Function/Gradient/Hessian Evals and # Contraction Mapping Iterations T J nn Alg # Iter. Func. Grad Contrac. Eval. Eval. Mapping NFP ,278 MPEC NFP ,000 MPEC NFP ,980 MPEC NFP ,578 MPEC NFP ,000 MPEC NFP ,114 MPEC NFP ,741 MPEC

69 Random-Coefficients Demand Estimation Summary Constrained optimization formulation for the random-coefficients demand estimation model is min (θ,ξ,g,) subject to g T W g s(δ; θ 2 ) = S Z T ξ = g The constrained optimization approach (with good solvers) is reliable and has speed advantage It allows researchers to access best optimization solvers

70 Part IV General Formulations

71 Standard Problem and Current Approach Individual solves an optimization problem Econometrician observes states and decisions Want to estimate structural parameters and equilibrium solutions that are consistent with structural parameters Current standard approach Structural parameters: θ Behavior (decision rule, strategy, price): σ Equilibrium (optimality or competitive or Nash) imposes G (θ, σ) = 0 Likelihood function for data X and parameters θ max L (θ; X) θ where equilibrium can be presented by σ = Σ(θ)

72 NFXP Applied to Single-Agent DP Model Rust (1987) G (θ, σ) = 0 represents the Bellman equations Σ(θ) is the expected value function and is single-valued as a function of θ Outline of NFXP Given θ, compute σ = Σ(θ) by solving G (θ, σ) = 0 For each θ, define L(θ; X) : likelihood given σ = Σ(θ) Solve max L(θ; X) θ

73 NFXP Applied to Random-Coefficeints Demand Estimation BLP (1995) G (θ, σ) = 0 represents the demand system of share equations Σ(θ) is the unobserved demand shock and is single-valued as a function of θ Outline of NFXP Given θ, compute σ = Σ(θ) by solving G (θ, σ) = 0 For each θ, define Q(θ; X) : GMM given σ = Σ(θ) Solve min θ Q(θ; X)

74 NFXP Applied to Games with Multiple Equilibria G (θ, σ) = 0 characterizes Nash equilibrium for a given θ Σ(θ) is set of Nash equilibria given θ and can be multi-valued Outline of NFXP Given θ, compute all σ Σ(θ) For each θ, define Solve L(θ; X) = max likelihood over all σ Σ(θ) max L(θ; X) θ If Σ(θ) is multi-valued, then L can be nondifferentiable and/or discontinuous

75 NFXP Applied to Games with Multiple Equilibria

76 NFXP and Related Methods to Games NFXP requires finding all σ that solve G (θ, σ) = 0, compute the likelihood at each such σ, and report the max as the likelihood value L(θ) Finding all equilibria for arbitrary games is an essentially intractable problem - see Judd and Schmedders (2006) One fundamental issue: G-S or G-J type methods are often used to solve for an equilibrium. This implicitly imposes an undesired equilibrium selection rule: converge only to equilibria that are stable under best reply Two-step estimator : Computationally light, very popular, biased in small samples NPL Ag-M(2007): Iterating over the two-step estimator in an attempt to improve the small-sample bias

77 Constrained Optimization Applied to ML Estimation Suppose the game has parameters θ. Let σ denote the equilibrium strategy given θ; equilibrium conditions impose G (θ, σ) = 0 For a data set, X,Denote the augmented likelihood by L (θ, σ; X) L (θ, σ; X) decomposes L(θ; X) so as to highlight the seperate dependence of likelihood on θ and σ In fact, L(θ; X) = L (θ, Σ(θ); X) Therefore, maximum likelihood estimation is max (θ,σ) L (θ, σ; X) subject to G (θ, σ) = 0

78 Equivalent Reformulation for Games (or Models) with Multiple Equilibria? ML-NFXP: where for each θ, max L(θ; X), θ L(θ; X) = max likelihood over all σ Σ(θ) ML-Constrained Optimization: max (θ,σ) L (θ, σ; X) subject to G (θ, σ) = 0 Are these two formulations equivalent? Proof? Are the first-order conditions of these two formulations equivalent?

79 Advantanges of Constrained Optimization Both L and G are smooth functions Do not require that equilibria are stable under best-reply iteration Do not need to solve for all equilibria σ for every θ Use multi-start to attempt to find the global solution Using a constrained optimization approach allows one to take advantage of the best available methods and software

80 So... What is NFXP? NFXP is equivalent to nonlinear elimination of variables Consider max (x,y) f(x, y) subject to g(x, y) = 0 Define Y (x) implicitly by g(x, Y (x)) = 0 Solve the unconstrained problem max f(x, Y (x)) x Used only when memory demands are too large Often creates very difficult unconstrained optimization problems

81 Part V Estimation of Games

82 Estimation of Discrete Games of Incomplete Information of Games of Incomplete Information An active research topic in Applied Econometrics/Empirical Industrial Organization Statis entry/exit games of incomplete information: Seim (2006), Su (2012) Dynamic games of incomplete information: Aguirregabiria and Mira (2007), Bajari, Benkard, Levin (2007), Pakes, Ostrovsky, and Berry (2007), Pesendorfer and Schmidt-Dengler (2008), Pesendorfer and Schmidt-Dengler (2010), Kasahara and Shimotsu (2012), Egesdal, Lai and Su (2007) : Two main econometric issues appear in the estimation of these models the existence of multiple equilibria need to find all of them computational burden in the solution of the game repeated solving for equilibria for every guessed of structural parameters

83 Estimation of Discrete Games of Incomplete Information Example: Static Game of Incomplete Information - due to John Rust Two firms: a and b Actions: each firm has two possible actions: { 1 if firm a choose to enter the market d a = 0 if firm a choose not to enter the market { 1 if firm b choose to enter the market d b = 0 if firm b choose not to enter the market

84 Estimation of Discrete Games of Incomplete Information Example: Static Game of Incomplete Information Utility: Ex-post payoff to firms [α + d b (β α)] x a + ε a1, if d a = 1, u a (d a, d b, x a, ɛ a ) = 0 + ε a0, if d a = 0; u b (d a, d b, x b, ɛ b ) = (α, β): structural parameters to be estimated [α + d a (β α)] x b + ε b1, if d b = 1, (x a, x b ): firms observed types; common knowledge 0 + ε b0, if d b = 0; ε a = (ε a0, ε a1 ), ε b = (ε b0, ε b1 ): firms unobserved types, private information (ε a, ε b ) are observed only by each firm, but not by their opponent firm nor by the econometrician

85 Estimation of Discrete Games of Incomplete Information Example: Static Game of Incomplete Information Assume the error terms (ε a, ε b ) have a standardized type III extreme value distribution A Bayesian Nash equilibrium (p a, p b ) satisfies p a = = p b = exp[p b βx a + (1 p b )αx a ] 1 + exp[p b βx a + (1 p b )αx a ] exp[ x a α + p b x a (α β)] Ψ a (p b, x a ; α, β) exp[ x b α + p a x b (α β)] Ψ b (p a, x b ; α, β).

86 Estimation of Discrete Games of Incomplete Information Static Game Example with One Market: Solving for Equilibria The true values of the structural parameters are (α, β) = (5, 11) There is only 1 market with observed types (x a, x b ) = (0.52, 0.22) p a = p b = exp{0.52( 5) + p b 0.52(16)} exp{0.22( 5) + p a 0.22(16)}

87 Estimation of Discrete Games of Incomplete Information Static Game Example: Three Bayesian Nash Equilibria Eq1: (p a, p b ) = ( , ) stable under BR Eq2: (p a, p b ) = ( , ) unstable under BR Eq3: (p a, p b ) = ( , ) stable under BR

88 Estimation of Discrete Games of Incomplete Information Static Game Example: Data Generation and Identification Data Generating Process (DGP): the data are generated by a single equilibrium The two players use the same equilibrium to play 1000 times Data: X = {(d i a, d i b )1000 i=1, (x a, x b ) = (0.52, 0.22)} Given data X, we want to recover structural parameters α and β

89 Estimation of Discrete Games of Incomplete Information Static Game Example: Maximum Likelihood Estimation Maximize the likelihood function max (α,β) logl (p a (α, β), p b (α, β); X) 1000 ( = d i a log(p a (α, β)) + (1 d i a) log(1 p a (α, β)) ) i= ( + d i b log(p b (α, β)) + (1 d i b ) log(1 p b(α, β)) ) i=1 (p a (α, β), p b (α, β)) are the solutions of the Bayesian-Nash Equilibrium equations 1 p a = 1 + exp{ 0.52(α) + p b 0.52(α) β} = Ψ a(p b, x a, α, β) p b = exp{ 0.22(α) + p a 0.22(α) β} = Ψ b(p a, x b, α, β)

90 Estimation of Discrete Games of Incomplete Information Static Game Example: MLE via NFXP Outer loop: Choose (α, β) to maximize the likelihood function logl (p a (α, β), p b (α, β); X) Inner loop: For a given (α, β), solve the BNE equations for ALL equilibria: (p k a(α, β), p k b (α, β)), k = 1,..., K Choose the equilibrium that gives the highest likelihood value: k = argmax logl ( p k a(α, β), p k b (α, β); X ) {k=1,...,k} (p a (α, β), p b (α, β)) = (p k a (α, β), p k b (α, β))

91 Estimation of Discrete Games of Incomplete Information NFXP s Likelihood as a Function of (α, β) Eq Log Likelihood " !

92 Estimation of Discrete Games of Incomplete Information NFXP s Likelihood as a Function of (α, β) Eq Likelihood " ! 4

93 Estimation of Discrete Games of Incomplete Information NFXP s Likelihood as a Function of (α, β) Eq Log Likelihood ! 4 6 "

94 Estimation of Discrete Games of Incomplete Information Constrained Optimization Formulation for Maximum Likelihood Estimation max (α,β,p a,p b ) logl (p a, p b ; X) 1000 ( = d i a log(p a ) + (1 d i a) log(1 p a ) ) i= ( + d i b log(p b ) + (1 d i b ) log(1 p b) ) i=1 1 subject to p a = 1 + exp{0.52(α) + p b 0.52(α β)} 1 p b = 1 + exp{ 0.22(α) + p a 0.22(α β )} 0 p a, p b 1 Log-likelihood function is a smooth function of (p a, p b ).

95 Estimation of Discrete Games of Incomplete Information Monte Carlo Results with Eq Truth for alpha is 5 Truth for beta is Truth for p1 in Equilibrium #2 is Truth for p2 in Equilibrium #2 is

96 Estimation of Discrete Games of Incomplete Information Monte Carlo Results with Eq Truth for alpha is 5 Truth for beta is Truth for p1 in Equilibrium #1 is Truth for p2 in Equilibrium #1 is

97 Estimation of Discrete Games of Incomplete Information Monte Carlo Results with Eq Truth for alpha is 5 Truth for beta is Truth for p1 in Equilibrium #3 is Truth for p2 in Equilibrium #3 is

98 Estimation of Discrete Games of Incomplete Information Estimation with Multiple Markets There 256 different markets, i.e., 256 pairs of observed types (x m a, x m b ), m = 1,..., 256 The grid on x a has 16 points equally distributed between the interval [0.12, 0.87], and similarly for x b Use the same true parameter values: (α 0, β 0 ) = ( 5, 11) For each market with (x m a, x m b ), solve BNE conditions for (pm a, p m b ). There are multiple equilibria in most of 256 markets For each market, we (randomly) choose an equilibrium to generate 250 data points for that market The equilibrium used to generate data can be different in different markets

99 Estimation of Discrete Games of Incomplete Information # of Equilibria with Different (x m a, x m b ) 5 4 # of Equilibria x b x a

100 Estimation of Discrete Games of Incomplete Information Estimation with Multiple Markets Constrained optimization formulation for MLE max L ({p m (α,β,{p m a,pm b }) a, p m b }, X) subject to p m a = Ψ a (p m b, xm a, α, β) p m b = Ψ b (p m a, x m b, α, β) 0 p m a, p m b 1, m = 1,..., 256.

101 Estimation of Discrete Games of Incomplete Information Static Game Example: Monte Carlo Results with Multiple Markets #! ()*+,-./)-012,0-34-!! #$! $!!"#!!"$!!!!%"&!!%"&!%"'! #! ()*+,-./) ## #$! $ #$"5 #$"' #$"& ## ##"# ##"6 ##"7

102 Estimation of Discrete Games of Incomplete Information 2-Step Methods Recall the constrained optimization formulation for FIML is max L (p a, p b, X) ({α,β,p a,p b }) subject to p a = Ψ a (p b, x a, α, β) p b = Ψ b (p a, x b, α, β) 0 p a, p b 1 Denote the solution as (α, β, p a, p b ) Suppose we know (p a, p b ), how do we recover (α, β )?

103 Estimation of Discrete Games of Incomplete Information 2-Step Methods: Recovering (α, β ) Idea 1: Solve the BNE equations for (α, β ): Idea 2: Choose (α, β) to p a = Ψ a (p b, x a, α, β) p b = Ψ b (p a, x b, α, β) max L (Ψ a(p b, x a, α, β), Ψ b (p a, x b, α, β), X) (α,β)

104 Estimation of Discrete Games of Incomplete Information 2-Step Methods Idea 1 Step 1: Estimate ˆp = ( ˆp a, ˆp b ) from the data Step 2: Solve ˆp a = Ψ a ( ˆp b, x b, α, β) ˆp b = Ψ b ( ˆp b, x b, α, β) Idea 2 Step 1: Estimate ˆp = ( ˆp a, ˆp b ) from the data Step 2: max (α,β) L (Ψ a( ˆp b, x a, α, β), Ψ b ( ˆp a, x b, α, β), X)

105 Estimation of Discrete Games of Incomplete Information 2-Step Methods: Potential Issues to be Addressed How do we estimate ˆp = ( ˆp a, ˆp b )? Different methods give different ˆp One method is the frequency estimator: ˆp a = 1 N ˆp b = 1 N N i=1 N i=1 I {d i a =1} I {d i b =1} If ( ˆp a, ˆp b ) (p a, p b ), then (ˆα, ˆβ) (α, β ) For a given ( ˆp a, ˆp b ), there might not be a solution to the BNE equations ˆp a = Ψ a ( ˆp b, x a, α, β) ˆp b = Ψ b ( ˆp a, x b, α, β)

106 Estimation of Discrete Games of Incomplete Information 2-Step Methods: Pseudo Maximum Likelihood In 2-tep methods Step 1: Estimate ˆp = ( ˆp a, ˆp b ) Step 2: Solve Or equivalently Step 1: Estimate ˆp = ( ˆp a, ˆp b ) Step 2: Solve max L (p a, p b, X) ({α,β,p a,p b }) subject to p a = Ψ a ( ˆp b, x a, α, β) p b = Ψ b ( ˆp a, x b, α, β) 0 p a, p b 1 max ({α,β}) L (Ψ a ( ˆp b, x a, α, β), Ψ b ( ˆp a, x b, α, β), X)

107 Estimation of Discrete Games of Incomplete Information 2-Step Methods: Least Square Estimators Pesendofer and Schmidt-Dengler (2008) Step 1: Estimate ˆp = ( ˆp a, ˆp b ) from the data Step 2: min (α,β) { ( ˆpa Ψ a ( ˆp b, x a, α, β)) 2 + ( ˆp b Ψ b ( ˆp b, x b, α, β)) 2} For dynamic games, Markov perfect equilibrium conditions are characterized by p = Ψ(p, θ) Step 1: Estimate ˆp from the data Step 2: min θ [ˆp Ψ(ˆp, θ)] W [ˆp Ψ(ˆp, θ)]

108 Estimation of Discrete Games of Incomplete Information Static Game Example: ML v.s. 2-Step PML 50 Truth for alpha is Truth for beta is Pakes, Ostrovsky, and Berry (2007): pseudo likelihood function is not a suitable criterion function in a 2-step estimator and can lead to large bias.

109 Estimation of Discrete Games of Incomplete Information Nested Pseudo Likelihood (NPL): Aguirregabiria and Mira (2007) NPL iterates on the 2-step methods 1. Estimate ˆp 0 = (ˆp 0 a, ˆp 0 b ), set k = 0 2. REPEAT 2.1 Solve ( (α k+1, β k+1 ) = arg max L Ψ a(ˆp k b, x a, α, β), Ψ b (ˆp k a, x b, α, β), X (α,β) 2.2 One best-reply iteration on ˆp k ˆp k+1 ˆp k+1 a = Ψ a(ˆp k b, x a, α k+1, β k+1 ) b = Ψ b (ˆp k a, x b, α k+1, β k+1 ) 2.3 Let k := k + 1; UNTIL convergence in (α k, β k ) and (ˆp k a, ˆp k b ) )

110 Estimation of Discrete Games of Incomplete Information Static Game Example: ML, 2-Step PML and NPL 200 Truth for alpha is Truth for beta is Some equilibria in the data are NOT best-reply (Lyapunov) stable Pesendofer (2010): Best-reply stable is not a reasonable equilibrium selection rule in games of incomplete information

111 Estimation of Discrete Games of Incomplete Information Design of Data Generating Process in Monte Carlo Experiments Monte Carlo 1: Randomly selected equilibrium in each market In each market, we randomly choose an equilibrium to generate data Monte Carlo 2: Best-response stable equilibrium with lowest probabilities of entering for firm a In each market, we choose the equilibrium that results in the lower probability of entering for firm a to generate data. These equilibria are stable under Best-Reply iteration. Monte Carlo 3: Best-response stable equilibrium with lowest probabilities of entering for firm a In each market, we randomly choose an equilibrium that is stable under Best-Reply iteration to generate data. In each experiment, we vary the number of repeated observations T. For each experiment and each T, we generate 100 datasets and estimate the model using each of the 100 datasets

112 Estimation of Discrete Games of Incomplete Information Monte Carlo 1: Random Equilibrium in Each Market In each market, we randomly choose an equilibrium to generate data

113 Estimation of Discrete Games of Incomplete Information Monte Carlo 1: T = 5 and 10 for Each Market T Estimator Estimates RMSE CPU # of Avg. NPL α β (sec.) Data Sets Iter. Truth ML (Cons. Opt.) (0.179) (0.585) 5 2-Step PML (0.208) (0.512) 5 2-Step LS (0.203) (0.654) 5 NPL N/A N/A N/A (freq. prob.) (N/A) (N/A) 10 ML (Cons. Opt.) (0.126) (0.326) 10 2-Step PML (0.165) (0.403) 10 2-Step LS (0.164) (0.531) 10 NPL N/A N/A N/A (freq. prob.) (N/A) (N/A)