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Using all bids in parametric estimation of first-price auctions
Economics Letters 55 (1997) 321–325
Using all bids in parametric estimation of first-price auctions a b, Tong Li , Quang Vuong * a
University of Southern California, Los Angeles, CA, USA University of Southern California, INRA and IDEI, Los Angeles, CA, USA
b
Received 13 November 1996; received in revised form 17 December 1996; accepted 13 February 1997
Abstract Recently Laffont et al. (1995) have proposed an SNLLS estimator using winning bids only. Here, we derive a relationship between the means of observed bids and private values to propose an SNLLS estimator using all bids in first-price auctions. 1997 Elsevier Science S.A. Keywords: First-price auctions; Simulated nonlinear least-squares JEL classification: D44
1. Introduction Structural analysis of auction data has been limited until now despite the considerable progress of auction theory since the Vickrey (1961) seminal paper. This gap between theoretical and empirical work can be attributed to the computational difficulties arising from the high numerical complexity associated with the nonlinearity of the Bayesian Nash equilibrium strategy of the game. Recently, some progress has been made in overcoming these difficulties. See Donald and Paarsch (1993); Laffont et al. (1995). The key idea in the latter paper is to invoke the Revenue Equivalence Theorem to express the expectation of the winning bid as the expectation of the second highest private valuation so as to simulate the former in a nonlinear least-squares estimation method. While only winning bids are observed in descending auctions, in other auctions such as first-price sealed-bid auctions, one may observe all bids. Though the Laffont et al. (1995) method still applies as one can use only winning bids, an interesting question is whether all bids can be used in estimation.
2. The IPV first-price sealed-bid model A single and indivisible object is auctioned. All bids are collected simultaneously. The object is sold to the highest bidder who pays his bid to the seller, provided the bid is not less than a reservation *Corresponding author. Tel.: (1-213) 740-7432; fax: (1-213) 740-8543; e-mail: [email protected] 0165-1765 / 97 / $17.00 1997 Elsevier Science S.A. All rights reserved. PII S0165-1765( 97 )00089-X
T. Li, Q. Vuong / Economics Letters 55 (1997) 321 – 325
322
price p 0 . In the IPV paradigm, each bidder i (i 5 1,...,n) is assumed to have a private value vi for the auctioned object. When forming his bid, each bidder knows his private value but does not know others’ private values. On the other hand, each bidder knows that all private values including his own are drawn independently from a probability distribution F( ? ), which is absolutely continuous with respect to the Lebesgue measure, with density f( ? ) and support [v,v ] ¯ ] , [0, 1 `]. Therefore all bidders are identical a priori and the game is symmetric. All bidders are assumed to be risk neutral, while [n, 0 0 p , F(?)] is common knowledge with p [ [v,v ] ¯ ]. As usual, we restrict ourselves to strictly increasing differentiable symmetric Bayesian Nash equilibrium strategies. Let s(?) denote the equilibrium strategy. Then it is well known that the equilibrium strategy satisfies the first-order differential equation s9(vi ) 5 (vi 2 b i )(n 2 1)f(vi ) /F(vi )
(1)
for all vi [[ p ,v¯ ] subject to the boundary condition s( p )5p . From Riley and Samuelson (1981) among others, the solution of Eq. (1) is 0
1 b i 5 s(vi ) 5 vi 2 ]]] (F(vi ))n 21
0
E
vi
p0
(F(x))n 21 dx if vi $ p 0 ,
0
(2)
otherwise b i 5s(vi ) can be any value b 0 (say) strictly smaller than p 0 . The above theoretical model leads to a closely related structural econometric model. Namely, because vi is distributed as F(?), then the equilibrium strategy Eq. (2) induces a distribution G(?) for each of the n bids b i . When the reservation price p 0 is binding, i.e. p 0 .v, ] then G(?) has a mass point 0 ¯ ¯ ). Then parametric estimation proceeds by at b and is absolutely continuous on [ p0 ,b¯ ], where b5s(v specifying a parametric family for F(?). This is complicated by the high nonlinearity of Eq. (2) in the latent distribution F(?). To resolve such a difficulty, Guerre et al. (1995) have proposed an indirect nonparametric estimation procedure by noting that Eq. (1) can be rewritten as 0
F( p ) 1 G * (b) 1 1 ]] if b $ p 0 , v 5 b 1 ]] ]] 1 ]] ]]]] n 2 1 g * (b) n 2 1 1 2 F( p 0 ) g * (b)
(3)
where G * (b)5[G(b)2G( p0 )] / [12G( p0 )] and g * (?) is its density. In this paper, we rely on Eq. (3) to propose a parametric method for estimating F(?) using all bids without computing the equilibrium strategy Eq. (2).
3. A simulated nonlinear least squares estimator In an econometric investigation, heterogeneity across auctions typically arises as one considers more than one auction. Formally, let Fl (?) denote the distribution of private values for the lth auction, l51,...,L, where L is the number of auctions, and assume that Fl (?)5F(?uzl , u ), where u is an unknown parameter vector in Q , R k and zl is a vector of variables affecting bidders’ valuations. Let
T. Li, Q. Vuong / Economics Letters 55 (1997) 321 – 325
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n *l denote the number of bidders who actually bid in the lth auction, i.e. whose private values vil $p 0l . Hereafter, we assume that the number of (potential) bidders n is constant and known.1 Then for each l, Eq. (3) becomes F( p l0 uzl ,u ) 1 G * (b il uxl ,u ) 1 1 ]]]] if b il $ p l0 , vil 5 b il 1 ]] ]]]] 1 ]] ]]]]] n 2 1 g * (b il uxl ,u ) n 2 1 1 2 F( p 0l uzl ,u ) g * (b il uxl ,u )
(4)
where x 9l ;( p l0 , z l9 ) and G * (?uxl , u ) is the distribution of b il given b il $p l0 and xl . Now integrating both sides of Eq. (4) with respect to the density g * (b il uxl ,u ) on its support [ p l0 , b¯ (xl , u )] gives 1 E[vil uvil $ p ,xl ,u ] 5 E[b il ub il $ p ,xl ,u ] 1 ]] n21 0 l
0 l
E
¯ ,u ) b(x l
p 0l
G * (buxl ,u )db
F( p 0l uzl ,u ) 1 ¯ l ,u ) 2 p l0 ]. 1 ]] ]]]]] [b(x n 2 1 1 2 F( p 0l uzl ,u ) Provided n.2 (assumed hereafter) and b¯ (xl , u ),`, integrating by parts and rearranging give ¯ l ,u ) b(x pl F( p l uzlu ) n21 1 0 0 E[b il ub il $ p l ,xl ,u ] 5 ]]E[vil uvil $ p l ,xl ,u ] 1 ]] ]]]]] 2 ]] ]]]]] . 0 n22 n 2 2 1 2 F( p l uzl ,u ) n 2 2 1 2 F( p 0l uzl ,u ) 0
0
(5) Moreover, setting b 0 50, it follows from E[b il uxl , u ]5(12F( p 0l uzl ,u )) E[b il ub il $p 0l ,xl ,u ] that 0 ¯ l ,u ) pl b(x n21 0 0 0 E[b il uxl ,u ] 5 ]](1 2 F( p l uzl ,u ))E[vil uvil $ p l ,xl ,u ] 1 ]]F( p l uzl ,u ) 2 ]]. n22 n22 n22
(6)
Let m l* (u ) and ml (u ) denote the right-hand side of Eqs. (5) and (6), respectively. Eq. (5) or Eq. (6) can be used to estimate the structural parameters u by nonlinear least squares from the ‘truncated’ sample hb il ; i51,...,n *l ; l51,...,Lj or the full sample hbil ; i51,...,n; l51,...,Lj when the functional ¯ l ,u )5s(v¯ l ,F(?uxl ,u )) forms of m *l (?) and ml (?) are known. Typically, however, the upper boundary b(x is difficult to determine since it involves the equilibrium strategy Eq. (2). To simplify, we assume hereafter that all other terms in Eqs. (5) and (6) can be obtained explicitly, and we focus on ¯ l ,u ) only.2 simulating b(x max 0 max ¯ l ,u )5E[max(U (n To this end, we note from Laffont et al. (1995) that b(x 21)l , p l )], where U (n 21)l is 3 the largest order statistic in n21 independent draws from F(?uzl ,u ). For instance, using importance sampling as in Laffont et al. (1995), we can draw S independent samples, each of size n21, denoted Because maxl 51,...,L n l* 5 n with probability 1 as L →`, the assumption that n is known is not restrictive. When reservation prices are not binding, i.e. when p 0l 5vl for all l, the assumption that n is constant is unnecessary and our methods apply directly with a varying number nl 5 n l* of potential / actual bidders. 2 If not, then the methods presented next can easily be adapted since (1 2 F( p 0l uzl ,u )E[vil uvil $ p 0l ,xl ,u ] 5 ¯ ¯ v evll max[v, p 0l ] f(vuzl ,u )dv and F( p 0l uzlu ) 5 evvll I(v # p l0 )f(vuzl ,u )dv, which can be used to simulate the other terms of Eqs. (5) 0 ] ] and (6). If F( p l uzl ,u ) needs to be simulated, then only Eq. (6) can be used. 3 ¯ l ,u ),(n21)E(max[v il , p l0 ]uzl ,u ). Thus b(x ¯ l ,u ),` as soon as E(max(vil , p l0 uxl ,u ),`. Note that b(x 1
T. Li, Q. Vuong / Economics Letters 55 (1997) 321 – 325
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u s1l ,...,u s(n 21 )l where u sil is independently drawn from a known density gl (?) for s51,...,S. Then we can 1 ¯ estimate ]] b(xl ,u ) by X¯ l (u ) 5 ]1S o Ss 51 Xsl (u ), where n22 s fl (u 1l ...fl (u s(n 21 )l ) 1 s max 0 Xsl (u ) 5 ]]max[u (n 21 )l , p l ]]]]]]] , s s n22 gl (u 1l )...gl (u (n 21 ) ) smax s s and u (n21 ) is the highest value in u 1l ,...,u (n 21 )l . We now apply the SNLLS estimator in Laffont et al. (1995) to Eq. (5) or Eq. (6). For instance, using Eq. (6) leads to the objective function
OO
O
S 1 L 1 n 1 Q S,L (u ) 5 ] ] [(b il 2 r(xl ,u ) 1 X¯ l (u ))2 2 ]]] (Xsl (u ) 2 X¯ l (u ))2 ], L l 51 n i51 S(S 2 1) s51
(7)
where r(xl ,u ) is the sum of the first two terms of Eq. (6). The SNLLS estimator uˆ is obtained by minimizing Q S , L (u ) with respect to u. The objective function Q S , L (u ) uses all bids, in contrast to that in Laffont et al. (1995), which uses only winning bids and simulate their means through the second-order highest statistic in n independent draws from gl (?). Despite this difference, the asymptotic properties of our SNLLS estimator can be obtained similarly. Assuming that u is identified in Eq. (6), we obtain 4 Proposition 1: For any fixed S, uˆ is a strongly consistent estimator of u0 as L →`. As in Laffont et al. (1995), let m 0l ;ml (u0 ) / u. Let s 20 (xl ) denote the conditional variance of b il given xl . Let Y¯l (u ); ]1S o Ss51 Ys l (u ) where Ysl (u );Xs l (u ) / u. Let vl , Vl and Cl denote the variances of Xsl (u0 ), Ysl (u0 ), and their covariance conditional upon xl . / 2 Î] ˆ Proposition 2: For any fixed S, o 21 L(u 2 u0 ) converges in distribution to a normal N(0,Ik ) as S,L 21 21 L →`, where oS , L 5 A L BS , L A L with A L 5 ]L1 o Ll 5 1 E(m 0l m 09 l ),
OF
S
D
G
1 L 1 1 1 1 BS,L 5 ] E ]s 20 (xl )m 0l m 09 1 ] ]s 20 (xl )Vl 1 vl m 0l m 09 1 ]]](vl Vl 1 Cl C 9l ) , l l L l51 n S n S(S 2 1) and E[?] is the expectation with respect to x 9l 5 ( p l ,z 9l ). 21 Consistent estimation of the asymptotic variance covariance matrix o S,L 5 A 21 can be L BS,L A L obtained as in Laffont et al. (1995) (Proposition 3). 0
4. Conclusion Following Laffont et al. (1995), this paper proposes some SNLLS estimators for the distribution of private values in first-price auctions. In contrast to that paper, which uses only winning bids, our methods use all bids. On the other hand, as for their estimator, our estimators do not require the These results are obtained by noting that (i) r(xl ,u )2Xsl (u ) corresponds to Xsl (u ) in Laffont et al. (1995), where r(xl ,u) is fixed conditional upon xl , and (ii) that the SNLLS estimator uˆ can also be obtained by minimizing Eq. (7), where the average over n, is attached to bi l inside the squared term. This explains the use of s 20 (xl ) /n instead of s 20 (xl ) in Proposition 2 below. 4
T. Li, Q. Vuong / Economics Letters 55 (1997) 321 – 325
325
computation of the equilibrium strategy, are strongly consistent and are asymptotically normal. Though our estimators use more information, they cannot be ranked unambiguously with that of Laffont et al. (1995), as all use only first conditional moments and hence are asymptotically inefficient estimators. It is, however, possible to combine the estimators so as to propose a more efficient estimator. For instance, one can minimize the sum or more generally a weighted sum of the objective functions here and in Laffont et al. (1995). Alternatively, one can apply a simulated Generalized Method of Moments following McFadden (1988).
Acknowledgements Financial support was provided by the National Science Foundation under Grant SBR-9631212. The first author thanks INRA and IDEI for financial support and hospitality during completion of this paper.
References Donald, S., Paarsch, H., 1993. Piecewise maximum likelihood estimation in empirical models of auctions. International Economic Review 34, 121–148. Guerre, E., Perrigne I., Vuong, Q., 1995. Nonparametric estimation of first-price auctions, Working Papers No. 9503–9504 (University of Southern California). Laffont, J.J., Ossard, H., Vuong, Q., 1995. Econometrics of first-price auctions. Econometrica 63, 953–980. McFadden, D., 1988. A method of simulated moments for estimation of discrete response models. Econometrica 57, 995–1026. Riley, J., Samuelson, W., 1981. Optimal auctions. American Economic Review 71, 381–392. Vickrey, W., 1961. Counterspeculation, auctions, and sealed tenders. Journal of Finance 16, 8–37.
Using all bids in parametric estimation of first-price auctions a b, Tong Li , Quang Vuong * a
University of Southern California, Los Angeles, CA, USA University of Southern California, INRA and IDEI, Los Angeles, CA, USA
b
Received 13 November 1996; received in revised form 17 December 1996; accepted 13 February 1997
Abstract Recently Laffont et al. (1995) have proposed an SNLLS estimator using winning bids only. Here, we derive a relationship between the means of observed bids and private values to propose an SNLLS estimator using all bids in first-price auctions. 1997 Elsevier Science S.A. Keywords: First-price auctions; Simulated nonlinear least-squares JEL classification: D44
1. Introduction Structural analysis of auction data has been limited until now despite the considerable progress of auction theory since the Vickrey (1961) seminal paper. This gap between theoretical and empirical work can be attributed to the computational difficulties arising from the high numerical complexity associated with the nonlinearity of the Bayesian Nash equilibrium strategy of the game. Recently, some progress has been made in overcoming these difficulties. See Donald and Paarsch (1993); Laffont et al. (1995). The key idea in the latter paper is to invoke the Revenue Equivalence Theorem to express the expectation of the winning bid as the expectation of the second highest private valuation so as to simulate the former in a nonlinear least-squares estimation method. While only winning bids are observed in descending auctions, in other auctions such as first-price sealed-bid auctions, one may observe all bids. Though the Laffont et al. (1995) method still applies as one can use only winning bids, an interesting question is whether all bids can be used in estimation.
2. The IPV first-price sealed-bid model A single and indivisible object is auctioned. All bids are collected simultaneously. The object is sold to the highest bidder who pays his bid to the seller, provided the bid is not less than a reservation *Corresponding author. Tel.: (1-213) 740-7432; fax: (1-213) 740-8543; e-mail: [email protected] 0165-1765 / 97 / $17.00 1997 Elsevier Science S.A. All rights reserved. PII S0165-1765( 97 )00089-X
T. Li, Q. Vuong / Economics Letters 55 (1997) 321 – 325
322
price p 0 . In the IPV paradigm, each bidder i (i 5 1,...,n) is assumed to have a private value vi for the auctioned object. When forming his bid, each bidder knows his private value but does not know others’ private values. On the other hand, each bidder knows that all private values including his own are drawn independently from a probability distribution F( ? ), which is absolutely continuous with respect to the Lebesgue measure, with density f( ? ) and support [v,v ] ¯ ] , [0, 1 `]. Therefore all bidders are identical a priori and the game is symmetric. All bidders are assumed to be risk neutral, while [n, 0 0 p , F(?)] is common knowledge with p [ [v,v ] ¯ ]. As usual, we restrict ourselves to strictly increasing differentiable symmetric Bayesian Nash equilibrium strategies. Let s(?) denote the equilibrium strategy. Then it is well known that the equilibrium strategy satisfies the first-order differential equation s9(vi ) 5 (vi 2 b i )(n 2 1)f(vi ) /F(vi )
(1)
for all vi [[ p ,v¯ ] subject to the boundary condition s( p )5p . From Riley and Samuelson (1981) among others, the solution of Eq. (1) is 0
1 b i 5 s(vi ) 5 vi 2 ]]] (F(vi ))n 21
0
E
vi
p0
(F(x))n 21 dx if vi $ p 0 ,
0
(2)
otherwise b i 5s(vi ) can be any value b 0 (say) strictly smaller than p 0 . The above theoretical model leads to a closely related structural econometric model. Namely, because vi is distributed as F(?), then the equilibrium strategy Eq. (2) induces a distribution G(?) for each of the n bids b i . When the reservation price p 0 is binding, i.e. p 0 .v, ] then G(?) has a mass point 0 ¯ ¯ ). Then parametric estimation proceeds by at b and is absolutely continuous on [ p0 ,b¯ ], where b5s(v specifying a parametric family for F(?). This is complicated by the high nonlinearity of Eq. (2) in the latent distribution F(?). To resolve such a difficulty, Guerre et al. (1995) have proposed an indirect nonparametric estimation procedure by noting that Eq. (1) can be rewritten as 0
F( p ) 1 G * (b) 1 1 ]] if b $ p 0 , v 5 b 1 ]] ]] 1 ]] ]]]] n 2 1 g * (b) n 2 1 1 2 F( p 0 ) g * (b)
(3)
where G * (b)5[G(b)2G( p0 )] / [12G( p0 )] and g * (?) is its density. In this paper, we rely on Eq. (3) to propose a parametric method for estimating F(?) using all bids without computing the equilibrium strategy Eq. (2).
3. A simulated nonlinear least squares estimator In an econometric investigation, heterogeneity across auctions typically arises as one considers more than one auction. Formally, let Fl (?) denote the distribution of private values for the lth auction, l51,...,L, where L is the number of auctions, and assume that Fl (?)5F(?uzl , u ), where u is an unknown parameter vector in Q , R k and zl is a vector of variables affecting bidders’ valuations. Let
T. Li, Q. Vuong / Economics Letters 55 (1997) 321 – 325
323
n *l denote the number of bidders who actually bid in the lth auction, i.e. whose private values vil $p 0l . Hereafter, we assume that the number of (potential) bidders n is constant and known.1 Then for each l, Eq. (3) becomes F( p l0 uzl ,u ) 1 G * (b il uxl ,u ) 1 1 ]]]] if b il $ p l0 , vil 5 b il 1 ]] ]]]] 1 ]] ]]]]] n 2 1 g * (b il uxl ,u ) n 2 1 1 2 F( p 0l uzl ,u ) g * (b il uxl ,u )
(4)
where x 9l ;( p l0 , z l9 ) and G * (?uxl , u ) is the distribution of b il given b il $p l0 and xl . Now integrating both sides of Eq. (4) with respect to the density g * (b il uxl ,u ) on its support [ p l0 , b¯ (xl , u )] gives 1 E[vil uvil $ p ,xl ,u ] 5 E[b il ub il $ p ,xl ,u ] 1 ]] n21 0 l
0 l
E
¯ ,u ) b(x l
p 0l
G * (buxl ,u )db
F( p 0l uzl ,u ) 1 ¯ l ,u ) 2 p l0 ]. 1 ]] ]]]]] [b(x n 2 1 1 2 F( p 0l uzl ,u ) Provided n.2 (assumed hereafter) and b¯ (xl , u ),`, integrating by parts and rearranging give ¯ l ,u ) b(x pl F( p l uzlu ) n21 1 0 0 E[b il ub il $ p l ,xl ,u ] 5 ]]E[vil uvil $ p l ,xl ,u ] 1 ]] ]]]]] 2 ]] ]]]]] . 0 n22 n 2 2 1 2 F( p l uzl ,u ) n 2 2 1 2 F( p 0l uzl ,u ) 0
0
(5) Moreover, setting b 0 50, it follows from E[b il uxl , u ]5(12F( p 0l uzl ,u )) E[b il ub il $p 0l ,xl ,u ] that 0 ¯ l ,u ) pl b(x n21 0 0 0 E[b il uxl ,u ] 5 ]](1 2 F( p l uzl ,u ))E[vil uvil $ p l ,xl ,u ] 1 ]]F( p l uzl ,u ) 2 ]]. n22 n22 n22
(6)
Let m l* (u ) and ml (u ) denote the right-hand side of Eqs. (5) and (6), respectively. Eq. (5) or Eq. (6) can be used to estimate the structural parameters u by nonlinear least squares from the ‘truncated’ sample hb il ; i51,...,n *l ; l51,...,Lj or the full sample hbil ; i51,...,n; l51,...,Lj when the functional ¯ l ,u )5s(v¯ l ,F(?uxl ,u )) forms of m *l (?) and ml (?) are known. Typically, however, the upper boundary b(x is difficult to determine since it involves the equilibrium strategy Eq. (2). To simplify, we assume hereafter that all other terms in Eqs. (5) and (6) can be obtained explicitly, and we focus on ¯ l ,u ) only.2 simulating b(x max 0 max ¯ l ,u )5E[max(U (n To this end, we note from Laffont et al. (1995) that b(x 21)l , p l )], where U (n 21)l is 3 the largest order statistic in n21 independent draws from F(?uzl ,u ). For instance, using importance sampling as in Laffont et al. (1995), we can draw S independent samples, each of size n21, denoted Because maxl 51,...,L n l* 5 n with probability 1 as L →`, the assumption that n is known is not restrictive. When reservation prices are not binding, i.e. when p 0l 5vl for all l, the assumption that n is constant is unnecessary and our methods apply directly with a varying number nl 5 n l* of potential / actual bidders. 2 If not, then the methods presented next can easily be adapted since (1 2 F( p 0l uzl ,u )E[vil uvil $ p 0l ,xl ,u ] 5 ¯ ¯ v evll max[v, p 0l ] f(vuzl ,u )dv and F( p 0l uzlu ) 5 evvll I(v # p l0 )f(vuzl ,u )dv, which can be used to simulate the other terms of Eqs. (5) 0 ] ] and (6). If F( p l uzl ,u ) needs to be simulated, then only Eq. (6) can be used. 3 ¯ l ,u ),(n21)E(max[v il , p l0 ]uzl ,u ). Thus b(x ¯ l ,u ),` as soon as E(max(vil , p l0 uxl ,u ),`. Note that b(x 1
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u s1l ,...,u s(n 21 )l where u sil is independently drawn from a known density gl (?) for s51,...,S. Then we can 1 ¯ estimate ]] b(xl ,u ) by X¯ l (u ) 5 ]1S o Ss 51 Xsl (u ), where n22 s fl (u 1l ...fl (u s(n 21 )l ) 1 s max 0 Xsl (u ) 5 ]]max[u (n 21 )l , p l ]]]]]]] , s s n22 gl (u 1l )...gl (u (n 21 ) ) smax s s and u (n21 ) is the highest value in u 1l ,...,u (n 21 )l . We now apply the SNLLS estimator in Laffont et al. (1995) to Eq. (5) or Eq. (6). For instance, using Eq. (6) leads to the objective function
OO
O
S 1 L 1 n 1 Q S,L (u ) 5 ] ] [(b il 2 r(xl ,u ) 1 X¯ l (u ))2 2 ]]] (Xsl (u ) 2 X¯ l (u ))2 ], L l 51 n i51 S(S 2 1) s51
(7)
where r(xl ,u ) is the sum of the first two terms of Eq. (6). The SNLLS estimator uˆ is obtained by minimizing Q S , L (u ) with respect to u. The objective function Q S , L (u ) uses all bids, in contrast to that in Laffont et al. (1995), which uses only winning bids and simulate their means through the second-order highest statistic in n independent draws from gl (?). Despite this difference, the asymptotic properties of our SNLLS estimator can be obtained similarly. Assuming that u is identified in Eq. (6), we obtain 4 Proposition 1: For any fixed S, uˆ is a strongly consistent estimator of u0 as L →`. As in Laffont et al. (1995), let m 0l ;ml (u0 ) / u. Let s 20 (xl ) denote the conditional variance of b il given xl . Let Y¯l (u ); ]1S o Ss51 Ys l (u ) where Ysl (u );Xs l (u ) / u. Let vl , Vl and Cl denote the variances of Xsl (u0 ), Ysl (u0 ), and their covariance conditional upon xl . / 2 Î] ˆ Proposition 2: For any fixed S, o 21 L(u 2 u0 ) converges in distribution to a normal N(0,Ik ) as S,L 21 21 L →`, where oS , L 5 A L BS , L A L with A L 5 ]L1 o Ll 5 1 E(m 0l m 09 l ),
OF
S
D
G
1 L 1 1 1 1 BS,L 5 ] E ]s 20 (xl )m 0l m 09 1 ] ]s 20 (xl )Vl 1 vl m 0l m 09 1 ]]](vl Vl 1 Cl C 9l ) , l l L l51 n S n S(S 2 1) and E[?] is the expectation with respect to x 9l 5 ( p l ,z 9l ). 21 Consistent estimation of the asymptotic variance covariance matrix o S,L 5 A 21 can be L BS,L A L obtained as in Laffont et al. (1995) (Proposition 3). 0
4. Conclusion Following Laffont et al. (1995), this paper proposes some SNLLS estimators for the distribution of private values in first-price auctions. In contrast to that paper, which uses only winning bids, our methods use all bids. On the other hand, as for their estimator, our estimators do not require the These results are obtained by noting that (i) r(xl ,u )2Xsl (u ) corresponds to Xsl (u ) in Laffont et al. (1995), where r(xl ,u) is fixed conditional upon xl , and (ii) that the SNLLS estimator uˆ can also be obtained by minimizing Eq. (7), where the average over n, is attached to bi l inside the squared term. This explains the use of s 20 (xl ) /n instead of s 20 (xl ) in Proposition 2 below. 4
T. Li, Q. Vuong / Economics Letters 55 (1997) 321 – 325
325
computation of the equilibrium strategy, are strongly consistent and are asymptotically normal. Though our estimators use more information, they cannot be ranked unambiguously with that of Laffont et al. (1995), as all use only first conditional moments and hence are asymptotically inefficient estimators. It is, however, possible to combine the estimators so as to propose a more efficient estimator. For instance, one can minimize the sum or more generally a weighted sum of the objective functions here and in Laffont et al. (1995). Alternatively, one can apply a simulated Generalized Method of Moments following McFadden (1988).
Acknowledgements Financial support was provided by the National Science Foundation under Grant SBR-9631212. The first author thanks INRA and IDEI for financial support and hospitality during completion of this paper.
References Donald, S., Paarsch, H., 1993. Piecewise maximum likelihood estimation in empirical models of auctions. International Economic Review 34, 121–148. Guerre, E., Perrigne I., Vuong, Q., 1995. Nonparametric estimation of first-price auctions, Working Papers No. 9503–9504 (University of Southern California). Laffont, J.J., Ossard, H., Vuong, Q., 1995. Econometrics of first-price auctions. Econometrica 63, 953–980. McFadden, D., 1988. A method of simulated moments for estimation of discrete response models. Econometrica 57, 995–1026. Riley, J., Samuelson, W., 1981. Optimal auctions. American Economic Review 71, 381–392. Vickrey, W., 1961. Counterspeculation, auctions, and sealed tenders. Journal of Finance 16, 8–37.