CCP Estimation. Robert A. Miller. March Dynamic Discrete Choice. Miller (Dynamic Discrete Choice) cemmap 6 March / 27

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1 CCP Estimation Robert A. Miller Dynamic Discrete Choice March 2018 Miller Dynamic Discrete Choice) cemmap 6 March / 27

2 Criteria for Evaluating Estimators General principles to apply when assessing an estimator Three criteria for evaluating different estimators are: 1 Large sample properties: Is the estimator consistent? If so, what is the rate of convergence? What is its asymptotic distribution? 2 Finite sample properties: At what sample size do the finite sample properties accurately reflect the asymptotic distribution? For a given sample size, what is the standard deviation and mean squared error of the estimator? 3 Implementation: Is the estimator defined by an algorithm? A weaker definition is to satisify a set of conditions. Are numerical approximations involved? Does the estimator require tuning parameters or instruments? Miller Dynamic Discrete Choice) cemmap 6 March / 27

3 Maximum Likelihood Estimation Data and outside knowledge Suppose the data comes from a long panel either stationary or complete panel histories for finite lived agents). Also assume we know: 1 the discount factor β 2 the distribution of disturbances G t ɛ x ) 3 u 1t x) or more generally one of the payoffs for each state and time). 4 u 1t x) = 0 for notational convenience) Since the panel is long, p t x) and hence ψ jt x) are identified. There are, of course, alternative assumptions that deliver identification, and the methods described below are generic. Miller Dynamic Discrete Choice) cemmap 6 March / 27

4 Maximum Likelihood Estimation The likelihood To simplify the notation, consider a sample of N independently drawn observations on the whole history t {1,..., T } of individuals n {1,..., N}, with data on their state variables decisions denoted by x nt, and decisions denoted by d njt. The joint probability distribution of the decisions and outcomes is: ) N n=1 T t=1 Taking logs yields: N n=1 T t=1 J j=1 J X j=1 x =1 d njt {log p jt x nt )] + d njt I { x n,t+1 = x } p jt x)f jt x x) X x =1 I {x n,t+1 = x} log f jt x x nt )] } Miller Dynamic Discrete Choice) cemmap 6 March / 27

5 Maximum Likelihood Estimation The reduced form Note the choice probabilities are additively separable from the transition probabilities in the formula for the joint distribution of decisions and outcomes. Hence the estimation of the joint likelihood splits into one piece dealing with the choice probabilities conditional on the state, and another dealing with the transition conditional on the choice and state. Maximizing each additive piece separately with respect to f j x x) and p t x nt ) we obtain the unrestricted ML estimators: f jt x x ) = N n=1 I {x nt = x, d njt = 1, x n,t+1 = x } N n=1 I {x nt = x, d njt = 1} and: p jt x) = N n=1 I {x nt = x, d njt = 1} N n=1 I {x nt = x} Miller Dynamic Discrete Choice) cemmap 6 March / 27

6 Maximum Likelihood Estimation Estimating an intermediate probability distribution Following the notation of Lecture 3, let κ jtτ x t+τ+1 x t ) denote the probability of reaching x t+τ+1 at t + τ + 1 from x t by following action j at t and then always choosing the first action: { fjt x κ jtτ x t+τ+1 x t ) t+1 x t ) τ = 0 X x =1 f 1,t+τ x t+τ+1 x)κ jt,τ 1 x x t ) τ = 1,... Thus we can recursively estimate κ jtτ x t+τ+1 x t ) with: κ jtτ x t+τ+1 x t ) { f jt x t+1 x t ) τ = 0 X f x =1 1,t+τ x t+τ+1 x) κ jt,τ 1 x x t ) τ = t + 1,... Similarly we estimate ψ jt x t ) with ψ jt x t ) using the p jt x) estimates of the CCPs. Miller Dynamic Discrete Choice) cemmap 6 March / 27

7 Maximum Likelihood Estimation Unrestricted estimates of the primitives From previous lectures: u jt x t ) = ψ 1t x t ) ψ jt x t ) T t X + β τ t ψ 1,t+τ x) κ t1,τ 1 x x t ) κ tj,τ 1 x x t )] τ=1 x =1 Substituting κ τ 1 x x t, j) for κ τ 1 x x t, j) and ψ jt x t ) with ψ jt x t ) then yields: û jt x t ) ψ 1t x t ) ψ jt x t ) T t X + β τ t ψ 1,t+τ x) κ t1,τ 1 x x t ) κ tj,τ 1 x x t )] τ=1 x =1 The stationary case is similar and has the matrix representation we discussed in previous lectures). Miller Dynamic Discrete Choice) cemmap 6 March / 27

8 Large Sample or Asymptotic Properties Asymptotic effi ciency of the unrestricted ML estimator By the Law of Large Numbers f jt x x ) converges to f jt x x ) and p jt x) converges to p jt x), both almost surely. By the Central Limit Theorem both estimators converge at N and and have asymptotic normal distributions. Both f jt x x ) and p jt x) are ML estimators for f jt x x ) and p jt x) and obtain the Cramer-Rao lower bound asymptotically. Since and u jt x) is exactly identified, it follows by the invariance principle that û jt x) is consistent and asymptotically effi cient for u jt x t ), also attaining its Cramer Rao lower bound. The same properties apply to the stationary model. Note that greater effi ciency can only be obtained by making functional form assumptions about u jt x t ) and f jt x x ). False restrictions, such as adopting convenient functional forms for the payoffs, typically create misspecifications. Miller Dynamic Discrete Choice) cemmap 6 March / 27

9 Maximum Likelihood Estimation Restricted ML estimates of the primitives In practice applications further restrict the parameter space. For example assume θ θ 1), θ 2)) Θ is a closed convex subspace of Euclidean space, and: u jt x) u j x, θ 1) ) f jt x x nt ) f jt x x nt, θ 2) ) We now define the model by T, β, θ, g). Assume the DGP comes from T, β, θ 0, g) where θ 0 Θ interior ). The ML estimator, denoted by θ ML, maximizes: N n=1 T t=1 J j=1 d njt {ln p jt x nt, θ)] + X x =1 ] } I {x n,t+1 = x} ln f jt x x nt, θ 2) ) over θ Θ where p t x, θ) are the CCPs for T, β, θ, g). Miller Dynamic Discrete Choice) cemmap 6 March / 27

10 Maximum Likelihood Estimation A common variation on the ML estimator A common variation on the ML estimator is: 1 estimate f jt x x nt, θ 2) ) from the state transitions. 2 obtain a limited information ML estimator θ 2) LIML. 3 estimate θ 1) by searching over p t x, θ 1), θ 2) LIML ). More precisely we define: θ 2) LIML arg max θ 2 θ 1) N n=1 ML arg max θ 1 T t=1 N n=1 J j=1 T t=1 X x =1 J j=1 ] I {x n,t+1 = x} d njt log f jt x x nt, θ 2) ) ]} d njt {log p jt x nt, θ 1), θ 2) LIML ) Note that: when θ 2) 0, that is f jt x x nt ), is known, θ 1) ML = θ 1) ML ; otherwise θ 1) ML is less effi cient but computationally simpler than θ1) ML ; nevertheless both estimators solve for the optimal rule many times. Miller Dynamic Discrete Choice) cemmap 6 March / 27

11 Quasi Maximum Likelihood Estimation The steps Hotz and Miller, 1993) The essential difference between this estimator and ML is this estimator substitutes an estimator of the continuation value into the likelihood rather than computing it from the optimal policy function: 1 Estimate the reduced form p and f or θ 2) LIML ) as above; 2 Apply the Representation Theorem to obtain expressions for v jt x t ) v kt x t ); 3 Substitute the reduced form estimates into these differences to obtain v jt x, θ 1)) v kt x, θ 1)) for any given θ 1) ; 4 Replace v jt x t ) with v jt x, θ 1)) in the random utility model RUM) to obtain an estimate p jt x, θ 1)) for any given θ 1) ; 5 Maximize the quasi-likelihood with respect to θ 1). In effect we estimate a static RUM where differences in current utilities u j x, θ 1)) u k x, θ 1)) are augmented by a dynamic correction factor estimated in the first stage off the reduced form. Miller Dynamic Discrete Choice) cemmap 6 March / 27

12 Quasi Maximum Likelihood Estimation Notes on QML Estimation In the second step, appealing to the Representation theorem, and the slides above v jt x, θ 1)) v kt x, θ 1)) = u j x, θ 1)) u k x, θ 1)) T t X β τ κkt,τ 1 x x ψ 1,t+τ x) t ) κ τ=1 x =1 jt,τ 1 x x t ) In the last two steps we define: p jt x, θ 1)) ɛ t J I k=1 { ɛkt ɛ jt v jt x, θ 1)) v kt x, θ 1)) } ] dg t ɛ t x t ) and: θ 1) QML arg max θ 1 N n=1 T t=1 J j=1 ]} d njt {log p jt x nt, θ 1), θ 2) LIML ) Miller Dynamic Discrete Choice) cemmap 6 March / 27

13 Quasi Maximum Likelihood Estimation Exploiting finite dependence in QML Estimation Supposing there is ρ-period dependence, we could form: ṽ jt x, θ 1)) ṽ it x, θ 1)) = u j x, θ 1)) u i x, θ 1)) + ρ J,X ) τ=1 k,x t+τ ) β τ u i x t+τ, θ 1)) ] + ψ k τ x t+τ ) ωiktτ x t, x t+τ ) κ it,τ 1 x t+τ x t ) ω jktτ x t, x t+τ ) κ jt,τ 1 x t+τ x t ) Then p jt x, θ 1 ) is formed in an analogous manner to p jt x, θ 1 ) The last maximization step is defined in a similar way. Note this estimator does not have exactly the same interpretation as θ 1) QML, because the dynamic selection correction involves the current utility parameters. Miller Dynamic Discrete Choice) cemmap 6 March / 27 ]

14 Quasi Maximum Likelihood Estimation Adjusting the asymptotic covariance for pre-estimation Hotz and Miller, 1993) Form P θ 1), p, f ), a mapping from Θ 1) P F to P with: κ jtτ x t+τ+1 x t ) { fjt x t+1 x t ) τ = 0 X x =1 f 1,t+τ x t+τ+1 x)κ jt,τ 1 x x t ) τ = t + 1,... v jt x, θ 1)) v kt x, θ 1)) = u j x, θ 1)) u k x, θ 1)) T t X β τ κkt,τ 1 x x ψ 1,t+τ x) t ) κ τ=1 x =1 jt,τ 1 x x t ) ] p jt x, θ 1)) ɛ t { J ɛkt I ɛ jt v k=1 jt x, θ 1)) v kt x, θ 1)) } g t ɛ t x t ) dɛ t Miller Dynamic Discrete Choice) cemmap 6 March / 27

15 Quasi Maximum Likelihood Estimation Adjusting the asymptotic covariance for pre-estimation Let: ) { )]} π 1n θ 1), p, f = W N z n d n P θ 1), p, f where W N is a weighting matrix and z n are instruments. Define the CCP estimator for θ 1) CCP by solving: ) N n=1 π 1n θ 1), p, f = 0 Miller Dynamic Discrete Choice) cemmap 6 March / 27

16 Large Sample or Asymptotic Properties The asymptotic covariance matrix Newey, 1984) Write the cell estimators as the solution to: ] ) I 0 = N n=1 π 2n p, f = N d n d n p) ) n=1 f n f where: In d is a J 1) XT dimensional row vector indicator function matching the state variables of n to the relevant CCP components) in p; In f is a J 1) X 2 T dimensional row vector indicator function matching state variables and decisions) of n to f components; f n is the outcome from the n making a choice given her state variables. For k {1, 2} and k {1, 2} define: Ω kk E π kn π k ] n Γ 11 E π1n θ 1) ] I f n Γ 12 E π1n p, π ] 1n f Then the asymptotic covariance matrix for θ 1) CCP, denoted by Σ 1, is: Σ 1 = Γ11 1 Ω11 + Γ 12 Ω 22 Ω 21 Ω 12 ) Γ 12] Γ 1 11 Miller Dynamic Discrete Choice) cemmap 6 March / 27

17 Minimum Distance Estimators Minimizing the difference between unrestricted and restricted current payoffs Another approach is to match up the parametrization of u jt x t ), denoted by u jt x t, θ 1) ), to its representation as closely as possible: 1 Form the vector function where Ψ p, f ) by stacking: Ψ jt x t, p, f ) ψ 1t x t ) ψ jt x t ) T t + τ=1 X x =1 2 Estimate the reduced form p and f. 3 Minimize the quadratic form to obtain: θ 1) MD = arg min ux, θ 1) ) Ψ θ 1) Θ 1) = arg min θ 1) Θ 1) β τ ψ 1,t+τ x) p, f )] W κkt,τ 1 x x t ) κ jt,τ 1 x x t ) ] )] ux, θ 1) ) Ψ p, f ux, θ 1) ) W ) ] ux, θ 1) ) 2Ψ p, f W ux, θ 1) ) where W, is a square J 1) TX weighting matrix. Miller Dynamic Discrete Choice) cemmap 6 March / 27

18 Minimum Distance Estimators Notes on minimizing the difference between unrestricted and restricted payoffs From the Representation theorem u jt x t, θ 1) 0 ) = Ψ jt x t, p, f 0 ) if p are the CCPs for T, β, θ 0, g). Furthermore u jt x) is exactly identified from Ψ jt x, p, f 0 ) without imposing any additional restrictions. Therefore parameterizing u with θ 1) 0 imposes overidentifying restrictions so θ 1) MD is consistent if the restrictions are true. Note θ 1) MD has a closed form if ux; θ1) 0 ) is linear in θ1) 0. Miller Dynamic Discrete Choice) cemmap 6 March / 27

19 Minimum Distance Estimators A minimum distance estimator that exploits finite dependence Altug and Miller, 1998) We can adapt this estimator by exploiting finite dependence. Suppose utility is stable over time with u jt x, θ 1)) = u j x, θ 1)), and that ω iktτ x t, x t+τ ) achieves ρ dependence for all K choices at each x at t: ) 1 Form the K T ρ) X utility vector Λ θ 1), p, f from ) Λ ijt x t, θ 1), p, f = u j x t, θ 1)) u i x t, θ 1)) ψ jt x t ) + ψ it x t ) ρ J,X ) u k x t+τ, θ 1)) ] + ψ k,t+τ x t+τ ) β τ ] ωiktτ x t, x t+τ ) κ τ=1 k,x τ ) it,τ 1 x t+τ x t ) ω jktτ x t, x t+τ ) κ jt,τ 1 x t+τ x t ) 2 Given K T ρ) X weighting matrix W, minimize: ) ) Λ θ 1), f, p W Λ θ 1), f, p 1) ) Miller Dynamic Discrete Choice) cemmap 6 March / 27

20 Simulated Moments Estimators A simulated moments estimator Hotz, Miller, Sanders and Smith, 1994) We could form a Methods of Simulated Moments MSM) estimator from: 1 Simulate a lifetime path from x ntn onwards ] for each j, using f and p. 2 d Obtain estimates of Ê ɛ o jt jt = 1, x t. 3 Stitch together a simulated lifetime utility outcome from ) the j th choice at t n onwards for n, denoted v nj v jtn x ntn ; θ 1), f, p. ) 4 Form the J 1 dimensional vector h n x ntn ; θ 1), f, p from: ) h nj x ntn ; θ 1), f, p ) ) v jtn x ntn, θ 1), f, p v Jtn x ntn, θ 1), f, p + ψ jt x ntn ) ψ Jt x ntn ) 5 Given a weighting matrix W S and an instrument vector z n minimize: N 1 N )] )] n=1 z nh n x ntn ; θ 1) N, f, p WS n=1 z nh n x ntn ; θ 1), f, p Miller Dynamic Discrete Choice) cemmap 6 March / 27

21 Simulated Moments Estimators Notes on this MSM estimator In the first step, given the state simulate a choice using p, and simulate the next state ) using f. In this way generate x ns and d ns d n1s,..., d njs for all s {t n + 1,..., T }. Generating this path does not exploit] knowledge of G, only the CCPs. d In the second step Ê ɛ o jt jt = 1, x t pjt 1 x t ) ɛ t In Step 4 v jt x ntn, θ 1), f, p u jt x ntn, θ 1) ) + J } I { ψ jt x t ) ψ kt x t ) ɛ jt ɛ kt ɛ jt g ɛ t ) dɛ t k=1 ) T J s=t+1 j=1 is stitched together as: β t 1 1 { d } { njs = 1 u js x ns, θ 1) ) +Ê ɛ js xns, d njs = 1 The solution has a closed form if u jt x, θ 1) ) is linear in θ 1). Miller Dynamic Discrete Choice) cemmap 6 March / 27 ] }

22 Simulated Moments Estimators Another MSM estimator Indeed ɛ t could be simulated as well: 1 Draw a realization ɛ from G ɛ) for each s {t n,..., T } and n. 2 Set: J } d njs = I { ψ js x ns ) ψ ks x ns ) ɛ njs ɛ nks k =1 and stitch together: u jt x ntn, θ 1) ) + T s=t+1 J j=1 } } β t 1 1 { d njs = 1 {u js x ns, θ 1) ) + ɛ js 3 Minimize an analogous quadratic form to obtain θ 1). Bajari, Benkard and Levin 2007) estimate an approximate reduced form of the policy function without exploiting the CCPs pages , 2007), but acknowledge: "Our method requires that one be able to consistently estimate each firm s policy function, so this may limit our ability to estimate certain models page 1345, 2007)." Miller Dynamic Discrete Choice) cemmap 6 March / 27

23 Large Sample or Asymptotic Properties Adjusting the asymptotic covariance for simulation as well Pakes and Pollard, 1989) Simulation adds an additional, independent source of variation to the sample moments and hence the estimated asymptotic standard errors. Following the definition given in Lecture 7 suppose θ 1) minimizes: )] N 1 N )] n=1 z nh n x ntn, θ 1) N, f, p WS n=1 z nh n x ntn, θ 1), f, p Then the additional component to the covariance matrix for θ 1) is: Σ S 1 S 1 Υ W S Υ ) 1 Υ W S E z n h n h nz n ] WS Υ Υ W S Υ ) 1 where S is the number of simulations per observation): ) Υ = E z n h n x ntn, θ 1), 0 f 0, p 0 θ 1) Note that Σ S 1 0 as S. Miller Dynamic Discrete Choice) cemmap 6 March / 27

24 Asymptotic Effi ciency The Newton-Raphson Algorithm Recall that for any extremum estimator for a problem satisfying standard regularity conditions that: θ i+1) = θ i) 2 Q N θ i)) / ] 1 θ θ Q N θ i)) / ] θ where N indicates the sample, θ is the parameter value, and Q N θ) is the criterion function associated with the extremum estimator This algorithm converges to the maximand if the criterion function is strictly concave, and/or if θ i) is close enough to the maximum. The algorithm is based on the quadratic approximation: Q N θ) Q N θ i)) + Q N θ i)) / ] θ θ θ i)) + 1 θ θ i)) 2 Q N θ i)) / ] θ θ θ θ i)) 2 Miller Dynamic Discrete Choice) cemmap 6 March / 27

25 Asymptotic Effi ciency Iterating one step Amemiya, 1985, pp ) Suppose θ 1) is a N consistent estimator for the interior) true value θ 0 Θ, the parameter space. Then it is well known that θ 1) has the same asymptotic properties as θ ), the limit of the sequence, namely: N θ 2) θ 0 ) a.d. p lim 1 2 Q N θ 1)) N θ θ 1 p lim 1 Q N θ 1)) N θ Specializing Q N θ) = log L N θ), the log likelihood, θ 2) is asymptotically effi cient, and ) N θ 2) θ 0 is asymptotically normal with mean zero and covariance: { 1 2 ]} 1 log L N θ 0 ) lim E N θ θ Miller Dynamic Discrete Choice) cemmap 6 March / 27

26 Asymptotic Effi ciency Achieving asymptotic effi ciency from a CCP estimator Aguirregaberia and Miro, 2002) This general principle can be applied to dynamic discrete choice models: 1 Estimate the unrestricted) CCPs. 2 Use a CCP estimator to obtain θ 1), the parameters characterizing the primitives. 3 Solve for the CCPs as a mapping of θ 1) and the state variables. 4 Take one Newton-Raphson step to obtain θ 2). Note that this procedure asmptotically guarantees the global optimum is selected. Starting out with a trial guess of a θ 0) and then updating, the traditional way of implementing ML, reaches a local not global) optimum at best "Overshooting" cannot be ruled out). Miller Dynamic Discrete Choice) cemmap 6 March / 27

27 Summary Factors to consider when selecting an esimator There is a trade off between effi ciency and computational ease: 1 Asymptotic effi ciency ML and the 2 step CCP/Newton) estimators are the most effi cient. 2 Small sample properties Simulation induces additional variation that may increase the number of observations required to approximate the asymptotic distribution. 3 Computational ease MD with linear utility in the parameters has a closed form. ML is the most burdensome and typically requires numerical approximations. Finally there is an open question about how well these estimators perform when the model is misspecified. Miller Dynamic Discrete Choice) cemmap 6 March / 27