Pseudo maximum likelihood estimation of structural models involving fixed-point problems

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1 Economics Letters 84 (2004) Pseudo maximum likelihood estimation of structural models involving fixed-point problems Victor Aguirregabiria* Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA Received 3 September 2003; received in revised form 24 February 2004; accepted 9 March 2004 Available online 20 May 2004 Abstract his paper deals with the estimation of structural econometric models where the probability distribution of endogenous variables is implicitly defined as an equilibrium of a fixed-point problem. It proposes a pseudo maximum likelihood (PML) procedure and studies its asymptotic properties. D 2004 Elsevier B.V. All rights reserved. Keywords: Fixed-points; Pseudo maximum likelihood estimation; Recursive methods JEL classification: C13; C63 1. Introduction his paper deals with the estimation of structural econometric models where the distribution of endogenous variables is implicitly defined as a solution of a fixed-point problem. his structure appears in Markov discrete decision processes (Rust, 1994), auction models (Guerre et al., 2000), empirical games of incomplete information (Seim, 2002), and discrete models with social interactions (Brock and Durlauf, 2001). his paper proposes a recursive pseudo maximum likelihood (PML) procedure for the estimation of this class of models. here are two main reasons why this method is of interest. First, it avoids the problem of indeterminacy associated with maximum likelihood estimation of models with multiple equilibria. And second, the procedure avoids the repeated solution of the fixed-point problem. In models where the dimension of the fixed-point is large, this can result in significant computational gains relative to maximum likelihood. * el.: ; fax: address: vagguire@bu.edu (V. Aguirregabiria) /$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi: /j.econlet

2 336 V. Aguirregabiria / Economics Letters 84 (2004) It is important to note that this paper considers a model where endogenous and exogenous variables are discrete. hough the method can be extended to the case of continuous variables, the proofs of consistency and asymptotic normality of the estimator are different, and additional conditions are needed to guarantee a parametric rate of convergence for the estimator of the structural parameters. 2. Econometric model Let yay be a vector of discrete random variables, where Y is a discrete and finite set, and let P 0 be the true probability distribution of y. 1 he structural model is a parametric family of probability distributions P(h), where hah is a finite vector of parameters and H is a compact set. he model does not provide a closed form analytical expression for P(h). Instead, this distribution is implicitly defined as a fixed-point of a mapping in probability space: PðhÞ ¼WðPðhÞ; hþ ð1þ where W(.jP,h) is a mapping from I H into I, where I is the space of probability distributions of y. In some models, and for some values of h, the mapping W(., h) can have more than one fixed-point. If that is the case, the model does not provide a unique probability distribution of y. Example (A model of market entry). Consider a retail industry with many independent local markets. N firms are the potential entrants in each local market. Let y it be the indicator of the event firm i operates in market t, define y t =(y 1t,y 2t,...,y Nt ), and let x t be the size of market t. Profits of firm i in market t are: 8 0 if y it ¼ 0 Y >< ¼ it h 0i þ h 1 x t h 2 log 1 þ X >: y jt e it if y it ¼ 1 jpi ð2þ where e it is private information of firm i and it is independently distributed over firms and over markets with distribution function F. he vector of structural parameters is h={h 01,...,h 0N, h 1, h 2 }. By the independence of e it across firms, the joint probability Pr(y t x t,h) can be described in terms of the set of individual entry probabilities P(x t,h) u {P i (x t,h): i = 1,2,...,N}. It is possible to show (see Aguirregabiria and Mira, 2003) that the equilibrium probabilities of entry are implicitly defined as the solution of the system: P i ðx i ; hþ ¼Fðh 0i þ h 1 x t h 2 H i ðp½x t ; hšþþ ð3þ 1 For notational simplicity we omit exogenous explanatory variables. However, all the results in this paper apply also to the case in which P 0 is a conditional probability distribution {P 0 (yjx):(y,x)ay X}.

3 V. Aguirregabiria / Economics Letters 84 (2004) where H i (P) is the expected value of log (1 + S j p i y j ) conditional on the information of firm i, and under the condition that the other firms behave according to their entry probabilities in P. hat is, H i ðpþ ¼ X y i! Y P y j j ½1 P jš 1 y j log jpi 1 þ X jpi y j! ð4þ and S y i represents the sum over all the possible actions of firms other than i. In this example, the fixed-point mapping is W(P, x t, h)={f(h 0i + h 1 x t h 2 H i (P)): i = 1,2,...,N}}. his example has two features that make PML estimation particularly useful. First, in general W(., x t, h) does not have a unique fixed-point. And second, when the number of firms is relatively large, the evaluation of W for different values of h and fixed P is much cheaper than the evaluation of W for different values of P and fixed h. his is because the main computational cost comes from the sum S y i, and this sum should be recalculated only when we change P but not when we change h. 3. Pseudo maximum likelihood estimators he problem is to estimate the vector of structural parameters h 0 given a random sample {y t : t =1,2,...,} from the population P 0. Let Pˆ 0 = Pˆ 0 (y):yay} be the nonparametric frequency estimator of P 0, i.e., Pˆ 0 (y)= 1 S t =1 I{y t = y}, where I{.} is the indicator function. For K z 1, the K-stage PML estimator is defined as: ĥ K X ¼ arg max lnwðy t j ˆP K 1 ; hþ ð5þ hah t¼1 where the sequence of probability distributions {Pˆ K :K z 1} are constructed recursively as: ˆP K ¼ Wð ˆP K 1 ; ĥk Þ: ð6þ he one-stage estimator of h 0 1 maximizes the pseudo likelihood S t =1 lnw(y t jpˆ,h). Given Pˆ and ĥ 0 0 we obtain a new estimate of P 0 by iterating in the fixed-point mapping, i.e., Pˆ 1 = W (Pˆ 0, ĥ 1 2 ). hen, ĥ maximizes the pseudo likelihood S t =1 ln W (y t Pˆ,h), and so on. 1 An alternative procedure consists in calculating one-stage PML estimator and then apply one Newton iteration for the maximization of the likelihood function. here are several reasons why PML iterations may be preferred. First, a Newton iteration requires the computation of the Jacobian matrix BW/BP V, and this can be computationally much more expensive than the successive iterations in the PML procedure. Second, in models with multiple equilibria the gradient of the likelihood function is not well defined, but the gradient of the pseudo likelihood is always well defined. And third, when the initial frequency estimator of P 0 is very imprecise, the Newton iteration estimator

4 338 V. Aguirregabiria / Economics Letters 84 (2004) can perform very poorly in finite samples. hough, one can apply successive Newton iterations in the likelihood function, this can be computationally very costly. Example. Let {y t,x t :t = 1,2,...,} be a sample of firms entry decisions and market sizes from independent local markets. For simplicity, suppose that our measure of market size is discrete. Let Pˆ (x) be the frequency estimator (S i0 t =1 y it I{x t = x}) / (S t =1 I{x t = x}), and let Pˆ 0(x) be the vector Pˆ i0 (x):i =1,2,...,N}. Given these frequency estimates, the one-stage estimator maximizes the pseudo likelihood function, X X N t¼1 i¼1 y it lnfðh 0i þ h 1 x t h 2 H i ð ˆP 0 ðx t ÞÞÞ þ ð1 y it Þln½1 Fðh 0i þ h 1 x t h 2 H i ð ˆP 0 ðx t ÞÞÞŠ: ð7þ When F is the cdf of a standard normal (logistic) random variable, this is just the likelihood of a Probit (Logit) model. Given this one-stage estimator we can get new estimates of firms entry probabilities as: Pˆ i1 (x t )=F(ĥ 1 0i + hˆ 1 1 x t hˆ2 1 H i (Pˆ 0(x t ))). Using these probabilities we can construct new values H i (Pˆ 1(x t )), obtain a two-stage estimator, and so on. Proposition 1 shows that the PMLEs are consistency and asymptotically normal under standard regularity conditions, and it provides a recursive expression for the sequence of asymptotic variance matrices. Proposition 2 presents a sufficient condition for the asymptotic efficiency of these estimators. Proposition 1. Let {y t :t = 1,2,...,} be a random sample of y, and let Pˆ 0 be the frequency estimator of P 0. Assume that: (a) W is twice continuously differentiable in P and h, and for any (y,p,h)a Y I H the probability W (y P,h) is strictly greater than zero; (b) {P(h):haH} is a correctly specified model, i.e., there is a value h 0 ah such that P 0 = W (P 0,h 0 ); (c) H is a compact set; and (d) (d) h 0 uniquely maximizes in H the function E(lnW(y P 0,h)). Under these conditions the PML estimators {ĥ K, Pˆ K :K z 1} are root- consistent and asymptotically normal with asymptotic variances: Varð ð ˆP K P0 ÞÞ ¼ A K RAV K ; Varð ð ĥ K h0 ÞÞ ¼ B K RBV K p ffiffiffi where R is the asymptotic variance of (Pˆ 0 P 0 ), and {A K :K z 1} and {B K :K z 1} are sequences of deterministic matrices which can be obtained recursively using the expressions: A K =(I W h M)W P A K 1 + W h M and B K = M(I W P A K 1 ), where: A 0 is the identity matrix; W h and W P are the Jacobian 0, h 0 )@hv 0,h 0 )/@PV, respectively; and M is the projection matrix (W h Vdiag{P 0 } 1 W h ) 1 W h Vdiag{P 0 } 1. Proof. By Lemma 24.1 and Property 24.2 in Gourieroux and Monfort (1995), and an induction argument, the proof of consistency is straightforward. I derive here the asymptotic distribution. First order conditions of optimality imply that the sequence of estimators {ĥ K, Pˆ K :K z 1} can be obtained using the recursive expressions S t =1 BlnW(y t Pˆ 1 K, ĥ K )/Bh = 0 and Pˆ K = W(Pˆ 1 K, ĥ K ). Since W is twice continuously differentiable, we can apply the stochastic mean value theorem to these conditions

5 between (Pˆ K 1, ĥ K ) and (P 0,h 0 ). By consistency of (Pˆ K 1, ĥ K ), the stochastic mean value theorem implies that: and ð ĥ K h0 Þ¼X 1 hh ½ X p hp ð ˆP K 1 P 0 Þþð1= ffiffiffi X Þ ð ˆP K P0 Þ¼W P ð ˆP K 1 P 0 ÞþW h ð ĥ K h0 Þþo p ð1þ t¼1 BlnW 0 t =BhŠþo pð1þ where C 0 t u W(y t P 0,h 0 ), X hh u E({B lnw 0 t /Bh}{BlnW 0 t /Bh}V), and X hp u E({BlnW 0 t /Bh}{BlnW p 0 t /BP}V). Solving (Eq. 8) into (Eq. 9) we get the following recursive expression for the sequence { ffiffiffi (Pˆ K P 0 ): K z 1}: ð ˆP K P0 Þ¼½W P W h X 1 V. Aguirregabiria / Economics Letters 84 (2004) hh X hpš þ W h X 1 hh ð1= p ffiffiffi X Þ ð ˆP K 1 t¼1 P 0 Þ BlnW 0 t =Bh þ 0 pð1þ: p aking into account that Pˆ 0 (y) is the frequency estimator (1/)S t =1 I{y t = y}, we can write (1 ffiffiffi p ffiffiffi ) p S t ffiffiffi =1 Bln W 0 t /Bh as S yay BlnW(y P 0,h 0 )/Bh (Pˆ 0 (y) P 0 (y)), or in matrix form as WV h diag{p 0 } 1 (Pˆ P 0 ). Also, the matrices X 0 hh u E({BlnW 0 t /Bh} {BlnW 0 t /Bh}V) and X hp u E({BlnW 0 t /Bh} {Bln W 0 t /BP}V) can be written as: X hh ¼ X yayfblnwðy j P 0 ; h 0 Þ=BhgfBlnWðy j P 0 ; h 0 Þ=BhVgP 0 ðyþ ¼WV h diagðp 0 Þ 1 W h ð11þ ð8þ ð9þ ð10þ X hp ¼ X yayfblnwðy j P 0 ; h 0 Þ=BhgfBlnWðy j P 0 ; h 0 Þ=BhVgP 0 ðyþ ¼WV h diagðp 0 Þ 1 W P : ð12þ herefore, we have: ð ˆP K p P0 Þ¼ðI W h MÞW P ð ˆP K 1 P 0 ÞþW h M ffiffiffi ð ˆP 0 P0 Þþo p ð1þ ð13þ p ffiffiffi where p ffiffiffim = X 1 hh W h Vdiag(P 0 ) 1. Solving this difference equation backwards we get (Pˆ P 0 )= K A K (Pˆ 0 P 0 )+o p (1), where Ap 0 = ffiffiffii and, for K >0, p ffiffiffia K =(I W h M) W P A K 1 + W h M. Solving this expression in Eq. (8), we get that K (ĥ h 0 )=B K (Pˆ 0 P 0 )+o p ffiffiffi p (1), where B K = M(I W P A K 1 ). Finally, p ffiffiffi by Mann Wald heorem, it is straightforward that (Pˆ K Pp 0 ffiffiffi )! d N(0,A K SAV K ) and K (ĥ h 0 )! d N(0,B K SB K V), where S is the asymptotic variance of (Pˆ 0 P 0 ). 5 Proposition 2. If the Jacobian matrix BW(P 0,h 0 )/BPV is zero, then all the estimators in the sequence {ĥ K :K z 1} are asymptotically equivalent to the maximum likelihood estimator (MLE).

6 340 V. Aguirregabiria / Economics Letters 84 (2004) Proof. First, notice that applying the implicit function theorem to P(h 0 )=W(P(h 0 ),h 0 ) we have that: BPðh 0 Þ=BhV ¼ðI BWðP 0 ; h 0 Þ=BPVÞ 1 BWðP 0 ; h 0 Þ=BhV: ð14þ herefore, if BW(P 0,h 0 )/BPV= 0, the score and pseudo-score are equal, i.e., BlnP(y t h 0 )/Bh = BlnW (y t P 0,h 0 )/Bh, and we can write the variance of the MLE as: V MLE ¼ X! 1 fblnpðy j h 0 Þ=BhgfBlnPðy j h 0 Þ=BhVg P 0 ðyþþ 1 ¼ðW h V diagðp 0 Þ 1 W h yay ¼ X 1 hh : Second, BW(P 0,h 0 )/BPV= 0 implies that A K = W h M and B K = M for any K z 1. herefore, Varð ð ĥ K ÞÞ ¼ MRMV ¼ X 1 hh W hv diagðp 0 Þ 1 R diagðp 0 Þ 1 W h X 1 hh ð15þ ð16þ R = diag(p 0 ) P 0 P 0 Vis the asymptotic variance of the frequency estimator p ffiffiffi of P 0, and it is simple to show that diag(p 0 ) 1 Rdiag(P 0 ) 1 = diag(p 0 ) 1. herefore, Var( K (ĥ h 0 )) = X 1 hh, which is the variance of the MLE.. he zero Jacobian condition holds in single-agent dynamic programming models with conditional independence of unobservables (see Aguirregabiria and Mira, 2002), but it does not hold in static or dynamic games of incomplete information. However, even when the one-stage estimator is asymptotically efficient, Montecarlo experiments show that iterating in the PML procedure can provide estimators with significantly better finite sample properties (see Aguirregabiria and Mira, 2003). 5 Acknowledgements his Paper is based upon work supported by the National Science Foundation under Grant No References Aguirregabiria, V., Mira, P., Swapping the nested fixed point algorithm: a class of estimators for Markov decision models. Econometrica 70, Aguirregabiria, V., Mira, P., Sequential estimation of dynamic discrete games, Manuscript, Department of Economics, Boston University. Brock, W., Durlauf, S., Discrete choice with social interactions. Review of Economics Studies 68, Gourieroux, C., Monfort, A., Statistics and Econometric Models. Cambridge Univ. Press, Cambridge, UK. Guerre, E., Perrigne, I., Vuong, Q., Optimal nonparametric estimation of first-price auctions. Econometrica 68, Rust, J., Estimation of dynamic structural models, problems and prospects: discrete decision processes. In: Sims, C. (Ed.), Advances in Econometrics, Sixth World Congress. Cambridge Univ. Press, Cambridge, UK. Seim, K Geographic Differentiation and Firms Entry Decisions: he Video Retail Industry. Manuscript, Stanford University.