Revisiting the Nested Fixed-Point Algorithm in BLP Random Coeffi cients Demand Estimation

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1 Revisiting the Nested Fixed-Point Algorithm in BLP Random Coeffi cients Demand Estimation Jinhyuk Lee Kyoungwon Seo September 9, 016 Abstract This paper examines the numerical properties of the nested fixed-point algorithm (NFP in the estimation of Berry, Levinsohn, and Pakes s (1995 random coeffi cient logit demand model. Dubé, Fox, and Su (01 find the bound on the errors of the NFP estimates computed by contraction mappings (NFP/CTR has the order of the square root of the inner loop tolerance. Under our assumptions, we theoretically derive an upper bound on the numerical bias in the NFP/CTR, which has the same order of the inner loop tolerance. We also discuss that, compared with NFP/CTR, NFP using Newton s method has a smaller bound on the estimate error. Keywords: random coeffi cients logit demand; numerical methods; nested fixed-point algorithm; Newton s method JEL Classification: C1, L1 1 Introduction This paper examines the numerical properties of the nested fixed-point approach (NFP in the estimation of the random coeffi cient logit demand model by Berry, Levinsohn, and Pakes (1995; BLP hereafter. BLP propose a nested contraction mapping algorithm, which we refer to as NFP/CTR. We find that the upper bound on the numerical error in the estimate of NFP/CTR has the same order of the inner loop tolerance while Dubé, Fox, and Su (01; DFS hereafter find that the upper bound has the order of the square root of the inner loop tolerance. We have a different Corresponding author: Kyoungwon Seo, Business School, Seoul National University, Seoul, South Korea, seo840@snu.ac.kr. Jinhyuk Lee is at Department of Economics, Korea University, Seoul, South Korea, jinhyuklee@korea.ac.kr. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government NRF-014S1A5A (Seo and Korea University Grant K (Lee. We thank Kyoo il Kim, Daniel Ackerberg and in particular Jeremy Fox for very helpful discussions. 1

2 set of technical assumptions from DFS, which leads to a different error bound from that of DFS. We believe that all our assumptions as well as theirs are satisfied in most applications of the BLP model. We also discuss another version of NFP, which we refer to as NFP/NT. It uses Newton s method to solve the inner fixed-point problem. 1 We briefly discuss that the estimate error of NFP/NT can be smaller than NFP/CTR. For the computational speed of NFP/NT, see Iskhakov et al. (016. Model and Estimation.1 Random Coeffi cients Logit Demand Model We assume a set of independent markets, t = 1,..., T, where each market has the same set of products, j = 1,..., J. The utility of consumer i from consuming product j in market t is U ijt = X jt β i + ξ jt + ε ijt, where X jt is a vector of observed product characteristics, β i is a vector of consumer i s preference for observed product characteristics, ξ jt is a product characteristic or a demand shock that is unobserved by the econometrician, and ε ijt is an idiosyncratic shock. We also denote the option not to purchase as j = 0. The utility of consumer i from not purchasing in market t is U i0t = ε i0t. Following the standard random coeffi cient logit demand model, we further assume that the vector of random coeffi cients β i is drawn independently from the distribution F (β i ; θ, and ε ijt follows the independent Type I extreme value distribution. Under the assumptions, the market share function of product j in market t is exp ( X jt β i + ξ jt s j (X t, ξ t ; θ = 1 + J j =1 exp ( df (β i ; θ, (1 X j tβ i + ξ j t where X t (X 1t,..., X Jt and ξ t (ξ 1t,..., ξ Jt. In practice, we approximate the integral using simulation; that is, we generate n s draws of β i and then evaluate (1 by calculating 1 n s exp ( X jt β i + ξ jt n s 1 + J j =1 exp (. ( X j tβ i + ξ j t i=1 For simplicity, we write s j (ξ t ; θ instead of s j (X t, ξ t ; θ. We define s (ξ t ; θ (s 1 (ξ t ; θ,..., s J (ξ t ; θ as the predicted market share functions in market t, and S t (S 1t,..., S Jt, where S jt is the observed market share of product j in market t. 1 When we refer to both NFP/CTR and NFT/NT, we will simply write NFP.

3 . Estimation Procedures Prices are likely to be correlated with the unobserved product characteristics ξ jt. To address the endogeneity, BLP propose the generalized method of moments (GMM estimation with the moment conditions E [ ] ξ jt z jt = 0, where z jt is a vector of instrumental variables. The moment conditions are often implemented as E [ ξ jt h (z jt ] = 0, where h ( is a vector-valued function. To form the moment, BLP invert the market share equations S t = s (ξ t ; θ for a given θ, and obtain the solution, denoted as ξ t (θ (ξ 1t (θ,..., ξ Jt (θ. The sample moment is g T (ξ (θ = 1 T T g (ξ t (θ = 1 T t=1 T J ξ jt (θ h (z jt, t=1 j=1 where ξ (θ ( ξ 1 (θ,..., ξ T (θ and should be close to 0 when T is large. We can write the BLP GMM problem as follows: min Q (ξ (θ = min g T (ξ (θ W g T (ξ (θ, θ Θ θ Θ where W is a weight matrix. The BLP estimator θ is the minimizer of the BLP GMM problem in the finite sample. BLP propose the following NFP approach: invert the market share equations for a given θ to obtain ξ t (θ, t = 1,..., T in the inner loop, and search for θ that minimizes the GMM objective function Q (ξ (θ in the outer loop. To obtain ξ t (θ in the inner loop, BLP suggest the contraction mapping iterations: such that ξ h+1 ( ξ h CT R = ξ h 1 CT R + ln S t ln s ξ h 1 CT R ; θ, h = 1,, CT R ξh CT R L (θ ξ h CT R ξ h 1, ξ h CT R, h = 1,, (3 with a Lipschitz constant L (θ [0, 1. They show that iterative applications of the contraction CT R mapping converge to ξ t (θ. This is what we refer to as NFP/CTR. As an alternative, we consider NFP/NT, which starts with contraction mapping iterations and switches to Newton s method to ensure global convergence. Newton s method in NFP/NT begins with an initial guess ξ 0 NT obtained by the contraction mapping, and the subsequent iterate is computed using the iteration rule: ( ] 1 ( ] ξ h NT = ξ h 1 NT [ + ξ s ξ h 1 NT [S ; θ t s ξ h 1 NT ; θ, h = 1,, (4 3

4 ( where ξ s ξ h 1 NT ; θ is the matrix of first partial derivatives of s (ξ; θ with respect to ξ at the iterate ξ h 1 NT. The Newton iterate converges if s (ξ; θ is continuously differentiable with respect to ξ, ξ s (ξ; θ is invertible, and the initial guess is suffi ciently close to a solution ξ t (θ. 3 Bounds on Errors in the Estimate In this section, we derive upper bounds on the errors in the BLP estimate when it is computed with NFP/CTR and NFP/NT. We follow the same notations as in DFS. For notational simplicity, we omit the market index t. When we solve for ξ (θ in the inner loop, the exact value of the solution is not available. Instead, we impose a stopping rule of the inner loop, which requires the change of two successive iterates to be less than a given inner loop tolerance, ɛ in : ξ h ξ h 1 ɛin. (5 This rule is commonly used to check convergence in the inner loop. Following DFS, let ξ (θ, ɛ in be the first iterate ξ h satisfying the stopping rule (5. When the contraction mapping iterations are applied, we write ξ CT R (θ, ɛ in. Similarly, we write ξ NT (θ, ɛ in for Newton s method. We are ready to study an effect of the inner loop and outer loop tolerances on the estimate errors for NFP. An optimization routine stops and reports θ if the norm of the gradient of the objective function is less than the outer loop tolerance, ɛ out. Computing the gradient of the objective function, θ Q (ξ (θ, is not free of numerical error from the inner loop. To deal with this explicitly, define, for any ξ and θ (even when ξ ξ (θ, [ ( ] s (ξ; θ 1 s (ξ; θ Q (ξ Γ (ξ, θ. ξ θ ξ We can show that θ Q (ξ (θ = Γ (ξ (θ, θ using the implicit function theorem. However, the error-free value of ξ (θ (= ξ (θ, 0 is infeasible in most applications, and so is θ Q (ξ (θ. Thus, in practice, the approximated value, Γ (ξ (θ, ɛ in, θ, is used instead of the true value θ Q (ξ (θ and is understood as an analytical gradient of the objective function. Then, the optimizer reports θ if Let θ (ɛ in, ɛ out be a parameter θ satisfying (6. Γ (ξ (θ, ɛ in, θ ɛ out. (6 As noted previously, θ CT R (ɛ in, ɛ out and θnt (ɛ in, ɛ out denote the computed estimates from the contraction mapping iterations and Newton s method, respectively. Recall that θ is the true value of the BLP estimate in the finite sample, and θ = θ (0, 0. 4

5 In what follows, we often suppress (ɛ in, ɛ out in θ (ɛ in, ɛ out, and just write θ. Theorem 1. Suppose that (a θ is an interior solution, (b θθ Q (ξ (θ θ=θ is nonsingular, and (c Γ(ξ,θ ξ (ξ,θ=(ξ, θ is bounded. Then, ( θ θ Γ O (ξ ( ξ + O. (7 Proof: For simple notations, let θ = θ (ɛ in, ɛ out and write Γ(ξ,θ ξ (ξ, θ instead of Γ(ξ,θ ξ (ξ,θ=(ξ, θ, and similarly for other derivatives. Let θ = (θ 1,..., θ K and, for k = 1,..., K, Γ k (ξ, θ be the k-th component of Γ (ξ, θ. For any k = 1,..., K, compute Γ k (ξ = Γ k (ξ ( Γ k ξ, θ ( + Γ k ξ, θ Γ k (ξ (θ, θ = Γ k (ξ ( Γ k ξ, θ dq (ξ (θ dq (ξ (θ + dθ θ= θ k dθ θ=θ k [ ] Γk (ξ, θ [ ] ( ξ = ξ (ξ,θ=(ξ, θ ξ + O [ d ] Q (ξ (θ + dθdθ θ=θ [ θ ] ( θ + O θ θ. k The first and second equalities follow because Γ k (ξ (θ, θ = dq(ξ(θ dθ k θ=θ = 0. Stacking for all j = 1,..., J and rearranging the terms, we obtain [ θθ Q (ξ (θ θ=θ ] [ θ ] ( θ + O θ θ 1 K ( = Γ ξ [ Γ (ξ, θ ξ ( ξ +O, ɛin ξ 1 K, (ξ,θ=(ξ, θ ] [ ξ, ɛin ] ξ where 1 K is the K-dimensional column vector of 1 s. Since θθ Q (ξ (θ θ=θ is nonsingular, ( θ + O θ θ 1 K ( = [ θθ Q (ξ (θ θ=θ ] 1 Γ ξ [ ] [ θθ Q (ξ (θ θ=θ ] 1 Γ (ξ, θ [ ] ξ (ξ,θ=(ξ, θ ξ ( ξ +O, ɛin ξ 1 K. 5

6 Then, since Γ(ξ,θ ξ (ξ,θ=(ξ, θ is bounded, θ θ, ɛin, θ [ θθ Q (ξ (θ θ=θ ] 1 Γ (ξ + [ θθ Q (ξ (θ θ=θ ] 1 Γ (ξ, θ ξ ( ξ +O O ( Γ ( ξ, ɛin, θ (ξ,θ=(ξ, θ ξ ( ξ + O., ɛin ξ ( Here, O θ θ ( ξ and O, ɛin ξ are ignored because there are lower order terms. Subsequently, we investigate the effect of the inner loop errors on the estimates for NFP/CTR and NFP/NT. 3.1 Numerical Errors in NFP/CTR DFS begin their analysis of the error propagation of NFP/CTR by observing the following bound on the error from (5: ξ h CT R ξ (θ L (θ ξ h CT R ξ h 1 CT R 1 L (θ L (θ 1 L (θ ɛ in, (8 where L (θ is a Lipschitz constant defined in (3. That is, ξ CT R (θ, ɛ in ξ (θ DFS set ɛ out = 0 when defining θ (ɛ in θ CT R (ɛ in, 0. following corollary. L(θ 1 L(θ ɛ in. We use the same definition in the Corollary. Suppose that the assumptions of Theorem 1 hold. Then, θ (ɛ in θ ( θ O L (ɛin ɛ in. (9 1 L ( θ (ɛin Proof: Let ɛ out = 0. Recall that we write θ instead of θ (ɛ in = θ CT R (ɛ in, 0. Because θ satisfies (6, ( Γ ξ = 0. 6

7 This and Theorem 1 imply ( θ θ ξ O O L ɛ in 1 L where the latter inequality follows from (8. We compare our estimate error in (9 with that of DFS. Theorem 3 in DFS states that, under mild conditions, O ( θ (ɛin θ Q (ξ CT R ( θ (ɛin, ɛ in Q (ξ (θ ( θ, 0 + O L (ɛin ɛ in. (10 1 L ( θ (ɛin The first term in the RHS of (10 is the bias due to the objective function values, and the second is the numerical bias due to the demand shocks, ξ. In contrast, (9 has only the numerical bias due to the demand shocks. Our error bound does not have the bias due to the objective function values because we apply the Taylor expansion on θ Q (ξ (θ which vanishes at θ because of the first order condition. But DFS do so on Q (ξ (θ which does not necessarily vanish. It is worth noting that Corollary, unlike (10, relies on the assumption that the outer loop is solved exactly and (10 may be applied even when an optimization routine does not report convergence. A more important difference lies in an order of magnitude of the error bounds. The error bound in (9 has the same order as the inner loop tolerance when the contraction mapping iterations are applied. This is different from that in (10, which has an order of the square-root of the inner loop tolerance. We discuss briefly a main difference between DFS s idea and ours. DFS argue that Q (ξ (θ, ɛ in is not differentiable with respect to ɛ in, and conjecture that ξ (θ, ɛ in and Q (ξ (θ, ɛ in are not differentiable with respect to θ for ɛ in > 0, even though the denoted differentiability can induce a sharper bound, as they discuss (see also Theorem in Ackerberg, Geweke, and Hahn (009. Taking this nondifferentiability into account and assuming differentiability of Q (ξ (θ with respect to θ instead, they approximate the objective function Q (ξ (θ with the Taylor expansion. In one part of their proof (Theorem 3 in DFS, they derive ( Q ξ ( θ (ɛin, 0 Q (ξ (θ, 0 = [ θ Q (ξ (θ ] ( θ (ɛin θ + O ( θ (ɛin θ. Kim and Park (010 also assume the differentiability and find that the error bound in the estimate has the same order as the inner loop tolerance in large samples. We (and DFS study small sample errors. 7

8 Then, because θ Q (ξ (θ = 0, the term involving ( θ (ɛin θ vanishes, and thus ( O θ (ɛin θ ( = Q ξ ( θ (ɛin, 0 Q (ξ (θ, 0 which is shown to be not greater than the RHS of (10. Thus their error bound has an order of the square-root of the inner loop tolerance. We agree with DFS on the aforementioned nondifferentiability. However, we make other assumptions. We assume not only that Q (ξ (θ is differentiable with respect to θ as DFS do, but also that its gradient, θ Q (ξ (θ, is differentiable with respect to θ. Under our assumption, we can apply the Taylor approximation to the gradient, and obtain ( θ Q ξ ( θ (ɛin θ Q (ξ (θ = [ θθ Q (ξ (θ ] ( θ (ɛin θ + O ( θ (ɛin θ. The second-order condition of the GMM minimization problem is guaranteed by nonsingularity of θθ Q (ξ (θ, which is often assumed. Then, the term involving ( θ (ɛin θ does not vanish and, unlike in DFS, our error bound can be expressed in terms of ( θ (ɛin θ. Since ( θ (ɛin θ dominates θ (ɛ in θ, the latter may be ignored and then ( θ (ɛin θ is in the same order ( of θ Q ξ ( θ (ɛin θ Q (ξ (θ which is in the same order of the inner loop tolerance. The proofs of Theorem 1 and Corollary contain more detail. It is worth noting that the theorems in DFS are not incorrect. We just have different sets of assumptions and theorems. We believe that all our assumptions as well as their assumptions are satisfied in most applications. 3. Numerical Errors in NFP/NT We discuss Newton s method briefly. When Newton s method is applied, the rate of convergence is known to be quadratic. In our setup, ξ NT (θ, ɛ in ξ (θ O ( ɛ in. To prove this formally, see Yamamoto (1986, Lemma.4 and Corollary for example. Then, we obtain the following result by applying Theorem 1 with ɛ out = 0: θ NT (ɛ in θ ( O ɛ in. Thus, we have a smaller upper bound of the estimate error for NFP/NT than NFP/CTR. In practice, ɛ out > 0 and if ɛ out is in the same order of ɛ in, ɛ out will be dominant in the numerical 8

9 error bound of θ NT (ɛ in. In NFP/CTR, both ɛ in and ɛ out are dominant when they are in the same order. The quadratic convergence rate of Newton s method implies that in the region near the solution, one more iteration may reduce the inner loop error significantly, for example, from 10 6 to This property makes NFP/NT solve the inner loop more quickly and tightly than NFP/CTR given an inner loop tolerance. For example, under the inner loop tolerance of 10 6, NFP/CTR tends to achieve the numerical error below but close to 10 6 in the inner loop while NFP/NT achieves the error much lower than 10 6 because of its quadratic convergence. Thus, NFP/NT easily minimizes the error propagation into the outer loop objective function, and thus an optimization routine is more likely to converge to a local optimum. Finally, we add that the original version of Newton s method converges locally only and one way to guarantee global convergence is to start with contraction mapping iterations and then switch to Newton s method. See Rust (1987 and Iskhakov et al. ( Concluding Remarks In this paper, we study the numerical performance of NFP/CTR and NFP/NT in BLP s random coeffi cients logit demand model. We show the theoretical result that under our assumptions, the upper bound on the numerical error in the estimate of NFP/CTR is different from that of DFS. Also NFP/NT can achieve a very tight inner loop error easily, and thus converge more often in the outer loop. References [1] Ackerberg, D., J. Geweke, and J. Hahn (009: Comments on Convergence Properties of the Likelihood of Computed Dynamic Models, Econometrica, 77(6, [] Berry, S., J. Levinsohn, and A. Pakes (1995: Automobile Prices in Market Equilibrium, Econometrica, 63(4, [3] Dubé, J. P., J. T. Fox, and C. L. Su (01: Improving the Numerical Performance of BLP Static and Dynamic Discrete Choice Random Coeffi cients Demand Estimation, Econometrica, 80(5, [4] Iskhakov, F., J. Lee, J. Rust, B. Schjerning, and K. Seo (016: Comment on Constrained Optimization Approaches to Estimation of Structural Models, Econometrica, 84(1,

10 [5] Kim, K. and M. Park (010: Approximation error in the nested fixed point algorithm for BLP model estimation, Working paper. [6] Rust, J. (1987: Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher, Econometrica, 55(5, [7] Yamamoto, T. (1986: A Method for Finding Sharp Error Bounds for Newton s Method Under the Kantorovich Assumptions, Numerische Mathematik, 49,