Semiparametric Estimation of the Optimal Reserve Price in First-Price Auctions Tong Li Indiana University Isabelle Perrigne University of Southern California Quang Vuong University of Southern California and INRA December 1999
* The authors are grateful to K. Hendricks and R. Porter for providing the data analyzed in this paper and to S. Campo for research assistance. Financial support from the National Science Foundation under Grant SBR-9631212 is gratefully acknowledged. Correspondence to: Isabelle Perrigne, USC, Department of Economics, Los Angeles, CA 90089-0253. Email:
[email protected].
Abstract Until now the optimal reserve price in the independent private value paradigm has been expressed as a function of the latent distribution of private signals which is by nature unobserved to the analyst. This feature has limited considerably the empirical implementation of this result. In this paper, we consider rst-price auctions within the aliated private values paradigm, which includes the independent private value case. Relying upon the identi cation of auction models, we show that the seller's expected pro t can be written as a functional of the observed bids distribution. This can be used to estimate the optimal reserve price from observed bids only. Speci cally, we propose a semiparametric extremum estimator for estimating consistently the optimal reserve price from observed bids. As an illustration, we consider the OCS wildcat auctions, and obtain an estimate of the optimal reserve price. Our empirical ndings show that the optimal reserve price would generate signi cantly higher revenues and pro ts for the federal government.
JEL Classi cation: C14, D44, L70 Keywords: First-Price Auctions, Optimal Reserve Price, Aliated Private Value, Independent Private Value, Semiparametric Extremum Estimation, OCS Wildcat Auctions.
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Semiparametric Estimation of the Optimal Reserve Price in First-Price Auctions T. Li, I. Perrigne and Q. Vuong
1 Introduction Given the monopoly power of the seller in an auction, the economic literature has focused on how the seller can extract the largest revenue/pro t. The problem of optimal auctions is to nd the auction mechanism that provides the greatest revenue/pro t for the seller. Such a result is of considerable practical value as many commodities are sold through auctions. Because any auction mechanism such as the rst-price, second-price, English, and Dutch auctions, generates the same revenue for the seller in the independent private value (IPV) paradigm with risk neutral bidders, as shown by Vickrey (1961), the problem of optimal mechanism design reduces to determining the optimal reserve price. The latter is expressed as a function of the distribution of private values and its corresponding density function. See La ont and Maskin (1980), Harris and Raviv (1981), Myerson (1981) and Riley and Samuelson (1981). Despite its economic importance, this result has been of little practical usefulness in empirical studies. As argued by McAfee and Vincent (1992) and Hendricks and Paarsch (1995), a major diculty for implementing the optimal reserve price is the use of unobservables such as the latent distribution of private signals. Hence, very few empirical studies have attempted to assess the optimal reserve prices using eld auction data.1 Moreover, 1
McAfee and Vincent (1992) give an approximation of the optimal reserve price for OCS auctions in
1
to our knowledge, the optimal reserve price has been obtained for the IPV paradigm only. This can be restrictive given its independence assumption as there are auctions where the independent private value paradigm does not apply. In this paper, we consider the aliated private values (APV) paradigm, which includes the IPV paradigm as a special case. As shown by La ont and Vuong (1996), the APV model is the most general auction model that can be identi ed from observed bids only without additional restrictions on the latent distributions. In particular, we characterize the optimal reserve price in rst-price sealed-bid auctions with symmetric and risk neutral bidders.2 Using results related to the identi cation of the APV model from observed bids established by Li, Perrigne and Vuong (1999), we show that the seller's expected pro t can be written as a functional of the observed bids distribution. This can be used to derive the optimal reserve price from observed bids only as the maximizer of this function. We then propose a semiparametric extremum estimator for estimating consistently the optimal reserve price from observed bids. Speci cally, the estimated optimal reserve price is obtained as a maximizer of an estimated pro t function in which some unknown distributions and densities have been nonparametrically estimated in a rst step. Our procedure is computationally simple as it does not require the computation of the equilibrium strategy. Moreover, it does not require either a parametric speci cation of the distribution of unobserved private values, or explicit estimation of the latter. We show that our estimator is consistent. We illustrate our results with the US gas lease auctions o the coast of Louisiana and Texas, which have been largely studied in the literature.3 We focus on wildcat auctions the common value framework. Paarsch (1997) proposes an estimate of the optimal reserve price from English auction data relying on parametric assumptions on the distribution of private values. 2 The reserve price is optimal in the sense that it generates the highest expected pro t for the seller in a rst-price auction. Thus we do not consider the optimality of the auction mechanism as for instance a second-price auction would generate higher pro t for the seller. See Milgrom and Weber (1982). Regarding risk averse bidders, the optimal auction mechanism is much more complex and involves some transfer payments. See Maskin and Riley (1984). 3 See Porter (1995) for a survey. More recently, Li, Perrigne and Vuong (1999) have analyzed these
2
between 1954 and 1969 as bidders in these auctions can be considered as symmetric and the crude oil market was relatively stable during that period. It is widely recognized that the reserve price in these auctions is too low. See, e.g. McAfee and Vincent (1992). Our empirical ndings indicate that the optimal reserve price (i) would allow the federal government to extract a larger proportion of oil companies' willingnesses-to-pay, and (ii) would generate a higher pro t and higher revenues for the federal government despite more tracts remaining unsold. The paper is organized as follows. A second section is devoted to the derivation of the optimal reserve price in rst-price auctions using observed bids. A third section develops our semiparametric extremum estimator. A fourth section illustrates our method. A fth section concludes. An appendix collects the proofs of our propositions and corollaries.
2 Optimal Reserve Price in First-Price Auctions 2.1 The APV Case We rst consider a general paradigm for auction models, which was introduced by Wilson (1977) and then developed by Milgrom and Weber (1982) under the name of the aliated value (AV) paradigm. The AV paradigm includes the well known common value (CV) and independent private value (IPV) paradigms as polar cases. Throughout the paper we consider rst-price sealed-bid auctions. A single and indivisible object is auctioned to n bidders who are assumed to be risk neutral. All sealed bids are collected simultaneously. Provided his bid is at least as high as a possible reserve price, the highest bidder wins the auction and pays his bid. The utility of each buyer i = 1; . . . ; n for the object is Ui = U (vi; c), where U () is increasing in both arguments, vi denotes the ith-player private signal or information, and c represents a common component a ecting the utility of all n bidders. The vector (v1; . . . ; vn; c) is a auctions within the APV paradigm.
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realization of a random vector whose (n +1)-dimensional cumulative distribution function is F () with support [v; v]n [c; c]. This distribution is assumed to be aliated as de ned by Milgrom and Weber (1982). Roughly, aliation means that large values for some of the components make the other components more likely to be large than small. Each bidder knows the value of his signal vi, the utility function U () and the distribution F () from which v1; . . . ; vn; c are jointly drawn but does not know c as well as other bidders' private signals. We restrict our study to the case where F () is symmetric in its rst n arguments. In this case the model is said to be symmetric. In terms of identi cation, La ont and Vuong (1996) have shown that the AV model is not identi ed from observed bids without additional restrictions on the pair (U; F ) or additional information beyond observed bids. On the other hand, they have shown that the most general identi ed model is the APV model where U (v; c) = v in the sense that, given a number of bidders, observed bids can be explained as well using an APV than an AV model, when the analyst observes only bids. Hereafter, we concentrate on the symmetric APV model.4 At the Bayesian Nash equilibrium, bidder i chooses his bid bi to maximize E [(vi , bi)1I(Bi bi)jvi], where Bi = s(yi), yi = maxj6=i vj , and s() is the strictly increasing equilibrium strategy. Assume rst that the reserve price is nonbinding. The equilibrium strategy satis es the rst-order di erential equation
s0(vi) = [vi , s(vi)]fy1jv1 (vijvi)=Fy1jv1 (vijvi);
(1)
for all vi 2 [v; v] subject to the boundary condition s(v) = v, where Fy1jv1 (j) denotes the conditional distribution of y1 given v1, fy1 jv1 (j) denotes the corresponding density, and the index \1" refers to any bidder among the n bidders because all bidders are ex ante As far as we know, despite its generality, the APV model has been seldom studied in the literature except in experimental studies by Kagel, Harstad and Levin (1987) and in empirical studies by Li, Perrigne and Vuong (1999, 2000). 4
4
identical. From Milgrom and Weber (1982), the explicit solution of (1) is Zv bi = s(vi) = vi , v L( jvi)d ; where L( jv ) = exp[, R v f (uju)=F (uju)du]. i
i
i
y1 jv1
(2)
y1 jv1
We next introduce a binding reserve price p0 > v. The introduction of a reserve price acts as a screening device for participating to the auction. The screening level is a function of p0 such that those biddders with private signals strictly below this level will not bid. From Milgrom and Weber (1982), it can be easily shown that the screening level in the APV model is exactly the reserve price p0. The equilibrium strategy becomes Zv bi = s(vi; p0) = vi , L( jvi)d ; for vi p0: (3) i
p0
In particular, s(vi) given in (2) is identical to s(vi; v). The choice of the reserve price constitutes an important instrument for the seller to take advantage of his monopoly power to increase his pro ts from the auction. As far as we know, the characterization of the optimal reserve price in a rst-price sealed-bid auction within the APV paradigm has not been proposed in the literature. This is the purpose of the next proposition.
Proposition 1: In a rst-price sealed-bid auction with n 2 bidders within the APV
paradigm, the optimal reserve price p0 that maximizes the expected pro t for the seller satis es R v L(pju)F (uju)f (u)du p0 = v0 + p0 F0 (py1jjvp1)f (pv1) ; (4) y1 jv1 0 0 v1 0 if v < p0 < v , where v0 denotes the private value of the seller for the auctioned object, fv1 () is the marginal density of vi and Fy1 jv1 (j) and L(j) are de ned as above.5
This calls for some comments. First, note that whatever his private value v0, the seller does not lose in terms of expected pro t by restricting himself to choosing a reserve price Equation (4) may have multiple roots. If this is the case, it is necessary to evaluate the expected pro t at each root to determine the global maximum as is usually advised in the theoretical literature. 5
5
belonging to the interval [v; v]. The intuition is quite simple as any reserve price larger than v or smaller than v will generate the same expected pro t as when p0 = v and p0 = v, respectively (see (A.3) in the appendix). Moreover, if the seller's private value v0 belongs to [v; v), then the optimal reserve price p0 must belong to (v; v) as required in Proposition 1. Indeed, the seller will always increase his pro t by setting a reserve price slightly larger than v as the derivative of his expected pro t is strictly positive in the neighborhood of this point. A similar argument applies for the upper bound v. It follows that v0 < p0 < v for any v0 2 [v; v).6 As is well known, a special case of the APV paradigm is the IPV paradigm rst considered by Vickrey (1961), where private values are drawn independently from a common distribution F (). All the other assumptions made previously for the APV case remain valid. In the IPV paradigm, the optimal reserve price has been widely studied, see La ont and Maskin (1980), and Riley and Samuelson (1981) among others. The optimal reserve price p0 in a rst-price auction solves (5) p0 = v0 + 1 ,f (Fp()p0) 0 if v < p0 < v. It can be easily checked that (5) follows from (4) by noting that L( jv) and Fy1jv1 (vjv) reduce to (F ( )=F (v))n,1 and F n,1(v), respectively, in the IPV case. A remarkable property of p0 is that it does not depend on the number of bidders as well as the auction mechanism used because of the Revenue Equivalence Theorem. As expected, the optimal reserve price in the APV paradigm crucially depends on the latent distribution of private signals through Fy1jv1 (j), fy1jv1 (j) and fv1 ().7 Unfortunately, these functions are unknown to the analyst, which limits the application of (4) or (5) on eld data. This is a typical problem in the optimal auction literature whatever the paradigm under consideration. Such a diculty has been mentioned by McAfee and In particular, this implies that a nonbinding optimal reserve price, i.e. p0 v, implies v0 < v. Levin and Smith (1996) studied the optimal reserve price in the APV model without giving its expression. An interesting question is the e ect of the number n of bidders on the optimal reserve price. Their results suggest that the optimal reserve price decreases with n in second-price auctions. 6
7
6
Vincent (1992), Hendricks and Paarsch (1995) and Paarsch (1997) among others. The structural econometric approach developed by Paarsch (1992), La ont, Ossard and Vuong (1995) and Guerre, Perrigne and Vuong (2000) o ers econometric methods for estimating the latent distribution in the IPV model. More recently, Li, Perrigne and Vuong (1999) extended the structural approach to the APV model by developing a two-step procedure for estimating the latent distribution nonparametrically. Though this procedure can be used to determine the optimal reserve price, in this paper we propose a more direct and simpler method with superior statistical properties.
2.2 A Reformulation from Observed Bids Assume that we observe bids from a rst-price sealed-bid auction with a nonbinding reserve price.8 As postulated in the structural approach, such bids are the equilibrium bids of the corresponding game and hence are given by bi = s(vi), where s() is de ned by (2). Thus because private values are random, bids are naturally random with a ndimensional joint distribution G(). Moreover, because s() is strictly increasing, we have GB1jb1 (X jx) = Fy1jv1 (s,1(X )js,1(x)), where B1 = maxj6=1 bj = s(y1) and GB1jb1 (j) denotes the conditional distribution of B1 given b1. It follows that the corresponding density is given by gB1jb1 (X jx) = fy1 jv1 (s,1(X )js,1(x))=s0(s,1(X )). Thus the di erential equation (1) can be written as (b) = b + Gg B1jb1((bbjjbb)) : (6) B1 jb1 As noted by Li, Perrigne and Vuong (1999), the function () is the inverse bidding strategy s,1() in a rst-price sealed-bid auction with a nonbinding reserve price. The next proposition characterizes the optimal reserve price in terms of the observed If observed bids are coming from a rst-price auction with a reserve price, the function () de ned below becomes more involved. See Guerre, Perrigne and Vuong (2000) for the IPV case. Nonetheless, a similar argument can be applied to obtain a characterization of the optimal reserve price from the observed bids' distribution. 8
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bids distribution. Let GB1 ;b1 (; ) be the joint distribution of (B1; b1) with density gB1;b1 (; ) and support [b; b]2. Let GB1b1 (; ) = @GB1;b1 (; )=@b1 = GB1jb1 (j)gb1 ().
Proposition 2: The optimal reserve price p0 in a rst-price sealed-bid auction with n 2 bidders within the APV paradigm can be written as
p0 = (x0);
(7)
for b x0 b, where () is de ned as (6) and x0 maximizes with respect to x 2 [b; b] the expected pro t
Zb (x) = v0GB1;b1 (x; x) + n [t + ((x) , x)(xjt)] GB1 b1 (t; t)dt; (8) x h = E v01I(B1 x)1I(b1 x) i +n[b1 + ((x) , x)(xjb1)]1I(B1 b1)1I(b1 x) ; (9) R where (rjt) = exp , rt gB1 ;b1 (u; u)=GB1b1 (u; u)du , E denotes the expectation with respect to (B1 ; b1) and 1I() denotes the indicator of the event in parentheses. Moreover, if v < p0 < v, then x0 is a solution in x 2 (b; b) of the following rst-order condition Zb 0 = ((x) , v0)GB1 b1 (x; x) , 0(x) (xjw)GB1b1 (w; w)dw; (10) x = E [((x) , v0)GB1 b1 (x; x) , 1I(B1 b1)1I(b1 x)0(x)(xjb1)] : (11) Note that if v0 2 [v; v), then x0 2 (b; b) = (s(v); s(v)) from (7) because v < p0 < v as noted before. In contrast to Proposition 1, which requires the knowledge of the underlying distributions Fy1 jv1 (j) and fv1 () of private signals, the main advantage of Proposition 2 is that it expresses the optimal reserve price in terms of the bids distribution, which can be estimated directly from observed bids. This is the purpose of Section 3. In practice, any one of (8){(11) can be used to determine x0 and hence the optimal reserve price p0 by (7). 8
Proposition 2 applies a fortiori to the IPV case. In this case, the inverse bidding strategy () in (6) reduces to (b) = b + n ,1 1 Gg((bb)) ; (12) where G() and g() are the c.d.f. and the density of an individual bid bi as bids are independently and identically distributed, respectively. As noted by Guerre, Perrigne and Vuong (2000), the function () is the inverse bidding strategy s,1() in a rst-price sealed-bid auction with a nonbinding reserve price. We then have the following corollary. Corollary 1: The optimal reserve price p0 in the IPV model can be written as in (7) where () is now de ned by (12) and x0 maximizes with respect to x 2 [b; b] the expected pro t Zb n n (x) = v0G (x) + b , xG (x) , x Gn (t)dt + n((x) , x)Gn,1(x)(1 , G(x)); (13) h = E v01I(B1 x)1I(b1 x) !n,1 i G ( x ) + n[b1 + ((x) , x) G(b ) ]1I(B1 b1)1I(b1 x) ; (14) 1 where E denotes the expectation with respect to (B1; b1 ). Moreover, if v < p0 < v , x0 solves the following rst-order condition in x 2 (b; b) 0 = ((x) , v0)g(x) , (1 , G(x))0 (x); = E[((x) , v0)g(x) , 0(x)1I(b1 x)]:
(15) (16)
The same remarks for Proposition 2 apply to Corollary 1. In addition, because of the Revenue Equivalence Theorem, p0 applies to any auction mechanism.
3 Econometric Implementation In agreement with Section 2.2, we consider L rst-price auctions with a nonbinding reserve price. We assume that private values from di erent auctions are mutually independent to 9
prevent dynamic considerations, and that observed bids obey the equilibrium strategy (2) within each auction, as postulated in the structural approach. To simplify the presentation, we also assume that auctions are homogeneous in the sense that there is no need to control for heterogeneity of the auctioned objects through some vector of characteristics.9 Let ` index the `-th auction, and n` be the number of bidders in the `-th auction with n` 2. The observed bids are fbi`; i = 1; . . . ; n`; ` = 1; . . . ; Lg. The purpose of this section is to show how the optimal reserve price p0 in rst-price sealed-bid auctions within the APV and IPV paradigms can be estimated consistently from such observations without making parametric assumptions on the latent distribution of private values. Our methods rely upon Proposition 2 and Corollary 1. Speci cally, as p0 = (x0), a natural estimator of p0 is p^0 = ^(^x0), where ^() and x^0 are estimators of () and x0, respectively. Regarding (), which is the inverse of the equilibrium strategy s(), we follow Li, Perrigne and Vuong (1999) for the APV model, and Guerre, Perrigne and Vuong (2000) for the IPV model. This is presented in the rst subsection. Regarding x0, Proposition 2 and Corollary 1 indicate that, if the pro t function () or the rst-order condition can be estimated suciently well, then x0 could be estimated by x^0, where x^0 either maximizes the estimated pro t function or solves the estimated rst-order condition. In this paper we shall focus on the maximization of the estimated pro t function. This is the purpose of the second subsection. As the expected pro t () involves unknown functions, namely, GB1b1 (; ) and gB1 ;b1 (; ) for the APV case, or G() and g() for the IPV case, this leads naturally to considering semiparametric M estimation, in which these distributions and densities are estimated nonparametrically in a rst step. In each subsection, we distinguish the APV case from the IPV case.10 Allowing for heterogeneity of the auctioned objects through a vector z 2 IRd of characteristics is straightforward. In this case, our methods estimate the optimal price for an arbitrary value z0 . For instance, if z = (z1 ; . . . ; zd ) includes only continuous variables and z` denotes the value of z for the `-th Q object, it suces to insert the term dk=1 K ((zk0 , zk` )=hG) in (18) and (19). Similar adjustements apply to (22) and (23) for the IPV case. 10Considering the rst-order conditions (10) and (11) for the APV case or (15) and (16) for the IPV 9
10
3.1 Nonparametric Estimation of ( )
Our rst step consists in estimating the inverse equilibrium strategy () from observed bids. Note that this function actually depends on the number of bidders. To emphasize such a dependence, hereafter we denote it by (jn) when n 2 is the number of bidders. Hereafter, following Li, Perrigne and Vuong (1999) and Guerre Perrigne and Vuong (2000), we use kernel estimators though any other nonparametric estimators can be employed. As () involves di erent unknown functions in the APV case and IPV case (see (6) and (12)), we treat these two cases successively.
3.1.1 The APV Case To estimate (jn), we consider only the bids from auctions with n bidders, namely fbi`; i = 1; . . . ; n; ` = 1; . . . ; Ln g, where Ln is the number of such auctions. Noting that GB1jb1 (; )=gB1jb1 (; ) = GB1b1 (; )=gB1;b1 (; ), it follows from (6) that the inverse equilibrium strategy (jn) can be estimated by ^
^(bjn) = b + Gg^B1b1((b;b;bb)) ;
(17)
B1 ;b1
for any b 2 (b; b), where
! L 1X n X 1 b , b i` G^ B1 b1 (B; b) = L h 1I(Bi` B )K hG ; n G `=1 n i=1 ! ! L 1X n X 1 B , B b , b i` i` g^B1;b1 (B; b) = L h2 n K h K h ; n
n
n g `=1
i=1
g
g
(18) (19)
for any value (B; b) where hG and hg are some smoothing parameters called bandwidths, and K () is a weight function called kernel satisfying assumption A2 given in the appendix. case would lead to semiparametric GMM estimation as considered by Newey (1994) and Olley and Pakes (1995), in which the unknown distributions and densities are estimated nonparametrically in a rst step. Note that such an estimation method would require estimation of the derivative 0 (). This method is left for future research.
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Note that the symmetry of bidders has been used through averaging over i in (18) and (19). As shown by Li, Perrigne and Vuong (1999, Proposition A1), if assumption A1 in the appendix holds, then GB1 b1 (; ) and gB1;b1 (; ) have both R + n , 2 continuous bounded partial derivatives on any compact subsets of (b; b)2, where R is the di erentiability order of the joint density f (v1; . . . ; vn) of private values. Let
hG = cG (log Ln =Ln )1=(2R+2n,3);
hg = cg (log Ln =Ln )1=(2R+2n,2);
(20)
where cG and cg are some constants. From Stone (1982), these bandwidths are optimal in the sense of delivering the fastest rate of uniform convergence for estimating GB1 b1 (; ), gB1;b1 (; ), and hence the inverse equilibrium strategy (). See Lemma A.1 in the appendix.
3.1.2 The IPV Case As for the APV case, we estimate (jn) using bids from auctions with n 2 bidders. From (12), the inverse bidding strategy, which depends on n, can be estimated by ^ ^(bjn) = b + n ,1 1 Gg^((bb)) ; (21) for any b 2 (b; b), where L X n X G^ (b) = 1 1I(b b); (22) n
nLn `=1 i=1 il ! L X n X 1 b , b il g^(b) = nL h K h ; (23) n `=1 i=1 for any value of b where h is a bandwidth and K () is a kernel satisfying assumption A2. n
As shown by Guerre, Perrigne and Vuong (2000, Proposition 1), if assumption A1 holds, then g() has R +1 continuous bounded derivatives on any compact subset of (b; b), where R is the di erentiabily order of the density f (v) of private values. Let
h = c(log Ln=Ln )1=(2R+3); 12
(24)
where c is a constant. As for the APV case, this bandwidth delivers the fastest rate of uniform convergence for estimating g(). See Lemma A.2.
3.2 Semiparametric M-Estimation of x0 As shown in Proposition 2 and Corollary 1, x0 maximizes the expected pro t function (). It is thus natural to estimate x0 by the value that maximizes an estimated pro t function. This motivates our estimator as an extremum or M-estimator preceded by a nonparametric rst step. Note that x0 generally depends on the number n of bidders even in the IPV paradigm, hence the notation x0n used hereafter.
3.2.1 The APV Case Proposition 2 provides two equivalent expressions (8) and (9) for the expected pro t function (). Though either expression can be used for developing a semiparametric estimator, (9) is preferrable as it is written in expectation form, which leads to a computationally less demanding estimator. Speci cally, noting that (x) , x = GB1b1 (x; x)=gB1;b1 (x; x), (9) can be written as (x) = E [ (B1; b1; x; gB1;b1 =GB1 b1 )];
(25)
where () is given by the term in brackets of (9). This leads to considering the estimated expected pro t function by averaging over bids the function ^() obtained by replacing GB1b1 and gB1 ;b1 by their nonparametric estimates (18)-(19), namely, L n ^(x) = 1 X 1 X ^(Bi`; bi` ; x; g^B1;b1 =G^ B1b1 )]; Ln `=1 n i=1 L 1X n X 1 = L v01I(Bi` x)1I(bi` x) + nbi`1I(Bi` bi`)1I(x bi`) n `=1 n i=1 ! ^B1 b1 (x; x) ^ G +n g^ (x; x) (xjbi`)1I(Bi` bi`)1I(x bi` bmax , ) ; (26) n
n
B1 ;b1
13
R where ^ (xjt) = exp , xt g^B1 ;b1 (u; u)=G^ B1b1 (u; u)du , max(hg ; hG), and bmax is the maximum of the bids bi`, i = 1; . . . ; n, ` = 1; . . . ; Ln . As in Li, Perrigne and Vuong (1999), the trimming associated with the requirement bi` bmax , arises from the fact that GB1 b1 (u; u), gB1 ;b1 (u; u) and hence (xjt) are not well estimated uniformly if u and hence t are too close to the upper boundary b. See Lemma A.1. De ne x^0n as maximizing the estimated expected pro t ^ () over [bmin + ; bmax , ], where bmin is the minimum of the bids bi`, i = 1; . . . ; n, ` = 1; . . . ; Ln. More rigorously, x^0n is de ned as any value in [bmin + ; bmax , ] that satis es ^ (^x0n)
sup
x2[bmin+;bmax,]
^ x) + o(Ln ); (
(27)
where o(Ln ) converges to zero as Ln ! 1. Because ^ () is piecewise continuous in x, the term o(Ln ) actually insures the existence of x^0n. As x0n maximizes the expected pro t (), while x^0n essentially maximizes the estimated expected pro t ^ (), it is expected that the latter is a consistent estimator of the former. This is formally established in the next proposition, which relies on Lemma A.3. This lemma extends Amemiya (1985) Theorem 4.1.1, which is fundamental for establishing the consistency of extremum estimators. Our lemma di ers in two respects: It allows for (i) a discontinuous statistical objective function, and (ii) a maximization and uniform consistency over an expanding subset.
Proposition 3: Suppose that p0 is the unique maximizer of the seller's expected pro t over [v; v] and that v < p0 < v. If assumptions A1 and A2 hold with R n, then x^0n is a strongly consistent estimator of x0n as Ln ! 1. As noted earlier, there is no loss in requiring that p0n 2 [v; v]. The uniqueness assumption is necessary for identi cation, while v < p0 < v is satis ed if v v0 < v. Combining Proposition 3 and the uniform consistent estimation of () in section 3.1.1
delivers a consistent estimator of the optimal reserve price p0n . The additional subscript n emphasizes that p0n generally depends on n in the APV paradigm. Speci cally, under 14
the assumptions of Proposition 2, p^0n = ^(^x0njn) is a strongly consistent estimator of p0n as Ln ! 1.
3.2.2 The IPV Case Corollary 1 provides two equations (13) and (14) for the expected pro t function () in the IPV model. As for the APV case, we prefer to use (14) because of its expectation form. Noting that (x) , x = G(x)=((n , 1)g(x)), (14) can be written as (x) = E[ (B1; b1; x; g=G)];
(28)
where () is given by the term between brackets of (14). Replacing G() and g() by their kernel estimators (22) and (23), we can average over bids the function (). This gives L n ^ x) = 1 X X ^(Bi`; bil; x; g^=G^ ); ( nL n
= L1
n `=1 i=1 L 1X n X
n i=1 v01I(Bi` x)1I(bi` x) + nbi`1I(Bi` bi`)1I(x bi`) ! ^ n (x) n G 1I(B b )1I(x bi` bmax , h) ; (29) + (n , 1)^g (x)G^ n,1(bi` ) i` i` where bmax is the maximum of the bids bi` ; i = 1; . . . ; n; ` = 1; . . . ; Ln . As in Guerre, Perrigne and Vuong (2000), the trimming associated with the restriction bi` bmax , h arises from the fact that the density g() is not well estimated for values too close to the upper boundary b. See Lemma A.2. De ne x^0n as maximizing the estimated expected pro t ^ () over [bmin + h; bmax , h], where bmin is the minimum of all the bids bi`. Using similar notations, x^0n is de ned as any value in [bmin + h; bmax , h] that satis es n
n `=1
^ (^x0n)
sup
x2[bmin+h;bmax ,h]
^ (x) + o(Ln );
where o(Ln ) converges to zero as Ln ! 1. The next corollary establishes that x^0n is a consistent estimator of x0n. 15
Corollary 2: Suppose that p0 is the unique maximizer of the seller's expected pro t over [v; v] and that v < p0 < p. If assumptions A1 and A2 hold with R 1, then x^0n is a strongly consistent estimator of x0n as Ln ! 1. Comments similar to the APV case apply here. In particular, p^0n ^(^x0njn) is a
strongly consistent estimator of p0n . Note that we have let p0n depend on n. This is because we have allowed the latent distribution F () to depend on n, as is the case when n is endogenous. This contrasts with the statement that (5) is independent of n, which implicitly assumes that the latent distribution F () is independent of n. In particular, assesssing di erences in the estimated optimal reserve price p^0n for every n can indicate possible endogeneity of the number of bidders. On the other hand, if one is willing to accept that F () is independent of n, then one can average the various estimates p^0n to obtain a consistent estimator of the optimal reserve price p0.
4 An Aplication to OCS Wildcat Auctions It has been widely recognized by economists that the reserve price used by the federal government in OCS auctions was too low. See McAfee and Vincent (1992). Hendricks, Porter and Spady (1989) and Hendricks, Porter and Wilson (1994) also agree on the low level of the reserve price for drainage auctions. As argued by McAfee and Vincent (1992), however, a roadblock to applying auction theory and especially to determining the optimal reserve price is the need for unobservables. As our results circumvent such a diculty, OCS auctions provide an interesting illustration for assessing the optimal reserve price.
4.1 Data The U.S. federal government began auctioning its mineral rights on oil and gas on o shore lands o Texas and Louisiana coasts in 1954. The auctioned objects are usually tracts of about 5000 acres. The bidders are oil companies. The auction is organized as a rstprice sealed-bid auction with a reserve price of $15 per acre held constant across auctions 16
during 1954-1979. As we focus our study on symmetric auctions, we restrict our analysis to wildcat auctions which consist of tracts whose geology is not well-known. As argued by McAfee and Vincent (1992), bidders' information tend to be more symmetric in wildcat auctions than in drainage auctions which have been extensively studied by Hendricks and Porter. See Porter (1995) for a survey. Moreover, we exclude from our study all auctions held after 1970 as market conditions after 1970 were unstable. The data set provides the number of bidders for each auction and the corresponding bids in 1972 dollars. The summary statistics of bids per acre are given in the following table for two and three bidders. Table 1: Summary Statistics on Bids per Acre # bidders # auctions mean STD min. max. n=2 217 145.78 255.72 19.70 2220.28 n=3 108 175.99 250.54 18.98 2132.56 Table 1 indicates that there is a great variability in bids as measured by the standard deviation and the range of bids. As a matter of fact, the empirical distribution of bids is very skewed and a large proportion of bids falls between $50 and $200 per acre. Hereafter, we adopt the APV paradigm. The APV paradigm allows for aliation among private values through some common component while retaining some private aspects arising from idiosyncratic rms' characteristics. See Li, Perrigne and Vuong (1999) for a discussion. As a rst approximation, we consider that the current reserve price is nonbinding and hence does not act as an e ective screening device to participating to the auction. Indeed, the reserve price of $15 per acre is much lower than the average bid. Moreover, there are very few bids clustered around the actual reserve price of $15. Speci cally, for n = 2, there is less than 0.25% of bids in the interval [15; 20] and less than 1% in the interval [15; 30]. For n = 3 these percentages are similar. Hence, the actual reserve price can be viewed as nonbinding. Second, the federal government rejected the highest bid in some auctions despite it being larger than the announced reserve price. Hence, the actual reserve price can be 17
viewed as random. For wildcat auctions between 1954 and 1969, we observe a 1.8% rejection rate for auctions with two and three bidders, and no rejection for auctions with more than three bidders. As we consider wildcat auctions with more than one bidder, we can reasonably consider that the random reserve price does not have much e ect on bidding strategies.11 Third, we have to consider the problem of possible heterogeneity across tracts as it a ects the value of the optimal reserve price. To assess possible heterogeneity across tracts, we follow Porter (1995) and regress the log of bids on a complete set of tract speci c dummies. Using all the data including auctions with one bidder, we obtain a weak rejection of tract homogeneity based on a F -test.12 Moreover, a similar regression controlling for the number of bidders shows that the F -value drops signi cantly. To assess further tract heterogeneity given a number of bidders, we conduct separate regressions for each size of bidders. We nd that tracts dummies explain an even lower percentage of the variability of the log of bids. Hence we can reasonably neglect heterogeneity among tracts when controlling for the number of bidders. This will be automatically satis ed as our econometric analysis will be conducted successively for 2 and 3 bidders.13
4.2 Practical Issues To implement our kernel estimators (18) and (19), we need to choose a kernel function. The triweight kernel has a compact support and two continuous derivatives following assumption A2. It is de ned as 2 )3 1I(juj 1): K (u) = 35 (1 , u 32
See Hendricks, Porter and Spady (1989) and Hendricks, Porter and Wilson (1994) for an analysis of random reserve prices in drainage auctions. 12As the F -test requires independence among bids, this statistic tends to over-reject the null assumption of homogeneity because bids are not independent in the presence of aliation. 13Although our analysis can be performed for all sizes of bidders, we focus on auctions with 2 or 3 bidders because of the large number of observations for these two cases, 217 and 108 auctions, respectively. 11
18
We assume that R = n.14 Equation (20) thus gives the following bandwidths
hG = cGL,1=5; hg = cg L,1=6; for n = 2 hG = cGL,1=9; hg = cg L,1=10; for n = 3 where cG and cg are some constants. These two constants are obtained by the so-called rule of thumb. Speci cally, cG = cg = 2:978 1:06^b, where ^b is the empirical standard deviation of bids, and the factor 2.978 is the correction for the triweight kernel. See Hardle (1991). As is the case for many auction data, the bids density for each number of bidders is highly skewed. To reduce such skewness e ects, we transform the data using a logarihmic function. Let d log(1 + b), D = log(1 + B ) and X = log(1 + x).15 Using this transformation, (26) becomes L X n X v01I(Di` X )1I(di` X ) + n(exp(di` ),1)1I(Di` di` )1I(X di` ) ^ (X ) = L1 n1 n `=1 i=1
^ X; X ) ^ +n exp(di` ) Gg^Dd((X; X ) (exp(X ),1j!exp(di` ),1) D;d 1I(Di` di` )1I(X di` dmax , ) ;
exp(d),1 g^ (U; U )=(exp(U )G ^Dd (U; U ))dU , = where ^ (exp(X ),1; exp(d),1) = exp(, Rexp( X ),1 D;d max(hG; hg ) with hG and hg obtained from the empirical standard deviation of the log(1+bids), and dmax is de ned as the maximum of the log(1+bids). The solution of the maximization of ^ () is denoted by X^0. To compute the estimated optimal reserve price, we have two options. A rst option would be to compute p^0 = ^(^x0) with x^0 = exp(X^0) , 1. This possibility requires the
The triweight kernel is of order 2. Though assumption A2-(iii) would require a kernel of order 4 when n = 3, the choice of the kernel does not have much e ect in practice. Moreover, we have preferred using a kernel of order 2 to obtain nonnegative estimates from (18) and (19). 15As in Li, Perrigne and Vuong (1999), we use a log(1 + ) transformation to ensure the compactness and the positiveness of the support of the transformed bids. 14
19
estimation of the function (). A second possibility is to note that v (d), where () is given by ! G Dd (u; u) (u) = exp(u) 1 + g (u; u) , 1 : D;d See Li, Perrigne and Vuong (1999). Thus p^0 = ^(X^0 ).
4.3 Empirical Results Any determination of the optimal reserve price requires the knowledge of the value of the auctioned object v0 for the seller. This valuation must be nonnegative. On the other hand, because the seller may receive negative pro ts if the reserve price is set below his value v0, the seller is always better o by setting a reserve price above v0. Hence we can conclude that the federal government valuation must be between 0 and $15 per acre. Below we propose an estimate of the optimal reserve price for both values 0 and 15. The results are given in the following table. Table 2: Optimal Reserve Price Seller's Valuation p0 ; n = 2 p0 ; n = 3 v0 = 0 $297 $407 $315 $414 v0 = 15 McAfee and Vincent (1992) found that the reserve price should be approximately $600 in 1992 dollars, i.e. about 40-fold the current level. To be comparable, as the results in Table 2 are in 1972 dollars, they need to be divided by 0.339. For two bidders we nd that the optimal reserve price should be between $897 and $929. For three bidders, which is about the average number of bidders across all tracts, these bounds are larger, namely $1,201 and $1,221, respectively. Hence our estimates are larger than those found by McAfee and Vincent (1992). The choice of the reserve price is an important instrument for the federal government to take advantage of its monopoly power in the auction as an unique seller. A higher 20
reserve price induces rms to bid more aggressively but increases the probability of not selling the auctioned tract. The optimal reserve price balances these two opposite e ects and maximizes the seller's expected gain. Thus, in terms of economic policy conclusions, simulating the rst-price sealed-bid auctions with the optimal reserve price is of great interest for assessing the potential gain for the federal government. This is possible by combining our current results with results of previous studies. Speci cally, Li, Perrigne and Vuong (1999) have recovered the private values of most bidders by using the equality v = (b) given in (6) for 174 auctions with 2 bidders and 77 auctions with 3 bidders.16 We simulate the bids of the oil companies following (3) when facing the optimal reserve prices given above. As expected, rms' bids take larger values whenever rms' private values are above the reserve price, but some tracts remain unsold because of the higher reserve price. The following table gives the results in terms of percentage of informational rents, noted IR, left to the winning oil companies for sold tracts. The informational rent is computed as the ratio of the winning bidder's private value minus his bid divided by his private value. Table 3: Average Informational Rents for the Optimal Reserve Price # bidders v0 # sold tracts IR n=2 0 87 36.02% n = 2 15 86 33.68% n=3 0 36 39.07% n = 3 15 35 39.80% We observe that about half of the tracts remain unsold and that informational rents left to the rms decrease signi cantly from 62.2% and 57% found in Li, Perrigne and Vuong (1999). Hence, implementing the optimal reserve price would allow the federal government to be much more successful in capturing the willingnesses-to-pay of the oil companies. Because nonparametric methods are used, some observations had to be trimmed out because of boundary e ects. 16
21
Another issue, which is of great interest, is to compare revenues and pro ts between the two mechanisms. Table 4 gives the revenue from an average sold tract (in terms of acreage) while Table 5 gives the pro t from an average sold tract for each mechanism. Again, we provide revenues and pro ts for the optimal auction whether the federal government has a private value per acre equal to 0 or 15. The pro t is computed as the winning bid minus the value of the tract for the federal government. Table 4: Revenues in $ for an Average Sold Tract # bidders v0 Actual Mechanism Optimal Mechanism n=2 0 514,184.92 1,632,402.01 15 514,184.92 1,704,023.61 n=3 0 871,403.03 2,703,639.46 15 871,403.03 2,726,880.94 Table 5: Pro ts in $ for an Average Sold Tract # bidders v0 Actual Mechanism Optimal Mechanism n=2 0 514,184.92 1,632,402.01 15 444,756.21 1,633,224.66 n=3 0 871,403.03 2,703,639.46 15 800,935.56 2,653,254.51 It is important to note that the revenue and pro t for an average sold tract both signi cantly increase. On the other hand, as noted earlier, half of the tracts become unsold under the optimal reserve price. However, if we multiply the gures in Tables 4 and 5 by the number of sold tracts, the overall pro t and revenue for the federal government is much higher than the actual ones. For instance, if we consider auctions with 3 bidders and we assume that the value of a tract for the federal government is $15 per acre, the optimal reserve price allows to generate a pro t around 50% larger than the pro t out of the actual mechanism despite the fact that more tracts remain unsold. 22
5 Conclusion A well known diculty in applying theoretical results on the optimal reserve price to eld data is that they make use of latent distributions which are unknown to the analyst. In this paper we circumvent such a diculty by showing that the seller's expected pro t can be expressed as a functional of the observed bids distribution. Hence, the seller's expected pro t function can be identi ed and estimated from observed bids. It follows that a natural estimator of the optimal reserve price is obtained by maximizing the estimated pro t function. This leads us to propose a semiparametric extremum estimator of the optimal reserve price, which is shown to be strongly consistent. Possible future lines of research are as follows. First, our estimation procedure does not impose any parametric assumptions on the distribution of private values and hence on the observed bids distribution. A possible extension of our method is to specify a parametric family for the distribution of observed bids. Second, as we mentioned, an alternative semiparametric procedure would be to use the rst-order conditions characterizing the optimal reserve price so as to develop a semiparametric GMM estimator. Third, as our result relies on the identi cation of the model from which bids are generated, it would be interesting to consider bids arising from a rst-price auction with a binding reserve price. Guerre, Perrigne and Vuong (2000) have solved such an identi cation issue within the IPV paradigm. When considering the APV paradigm, identi cation may require additional restrictions/information. Fourth, as is the case for many auction situations such as procurement auctions, our application involves a public seller. Because of its public nature, it may be possible that the seller has other objectives than pro t maximization. Although of great interest, this issue requires more theoretical research.
23
Appendix 1. Proofs of Theoretical Results Proof of Proposition 1: Using a similar argument as in Riley and Samuelson (1981),
we maximize the expected pro t for the seller. For any bidder, say bidder 1, his expected payment if he wins the auction given v1 p0 is
p(v1) = s(v1; p0) Pr(v2 v1; . . . ; vn v1jv1) = s(v1; p0)Pr(y1 v1jv1) = s(v1; p0)Fy1jv1 (v1jv1);
(A.1)
where s(v1; p0) is given by (3). Therefore, using (3) the expected revenue for the seller from bidder 1 is Zv R1(p0) = p p(v1)f (v1)dv1 Z 0v Z v1 = v1 , L( jv1)d Fy1jv1 (v1jv1)fv1 (v1)dv1: (A.2) p0
p0
The total expected pro t for the seller is the sum of the seller's private value for the object if the latter remains unsold and the expected revenue for the sellers if it is sold to any of the n bidders. Because of the symmetry among bidders, we obtain = v0F (p0; . . . ; p0) + nR1(p0):
(A.3)
Di erentiating with respect to p0 , the expected pro t of the seller is maximized for some p0 satisfying the following rst-order condition Zv nFy1 jv1 (p0jp0)fv1 (p0)(v0 , p0) + n p L(p0jv1)Fy1 jv1 (v1jv1)fv1 (v1)dv1 = 0; 0
since @F (x; . . . ; x)=@v1 = Fy1jv1 (xjx)fv1 (x). Equation (4) follows.
Proof of Proposition 2: The proof is in four parts corresponding to the derivations of (9), (8), (10) and (11), respectively.
24
Part (i). Combining (A.1), (A.2) and (A.3), and using y1 = maxj6=1 vj , the expected pro t for the seller (p0) given n is equal to (p0) = v0Pr[v1 p0 ; . . . ; vn p0] Zv +n p s(v1; p0)Pr[v2 v1; . . . ; vn v1jv1]fv1 (v1)dv1 0 = v0E[1I(y1 p0 )1I(v1 p0 )] ! Zv Zv +n 1I(v1 p0)s(v1; p0) 1I(y1 v1)fy1 jv1 (y1 jv1)dy1 fv1 (v1)dv1 v v = E [v01I(y1 p0)1I(v1 p0) + ns(v1; p0)1I(y1 v1)1I(v1 p0)] : (A.4) Now, for p0 and v1 such that v p0 v1 v, we have from (2) and (3) Z p0 s(v1; p0) = s(v1) + v L( jv1)d Z p0 = s(v1) + L(p0 jv1) v L( jp0)d = s(v1) + L(p0 jv1)(p0 , s(p0));
(A.5)
where the second equality follows from L( jv1) = L( jp0 )L(p0jv1), and the third equality follows from (2) evaluated at vi = p0. Because p0 = (x) and v1 = (b1), L(p0 jv1) is also equal to L((x)j(b1 )). The latter is denoted (xjb1) and can be expressed as a function of the observed bids distribution. Indeed, because Fy1v1 (t; t) = GB1 b1 (s(t); s(t)) and fy1 ;v1 (t; t) = gB1;b1 (s(t); s(t))s0(t), using a change of variable u = s(t), we obtain Z b1 gB ;b (u; u) ! (xjb1) L((x)j(b1 )) = exp , G 1 1 (u; u) du : (A.6) x B1 b1 Combining (A.4), (A.5) and (A.6), and using B1 = s(y1), b1 = s(v1), p0 = (x), x = s(p0) with s() increasing, (A.4) can be written as in (9), as desired. Part (ii). Equation (9) follows from (8) by noting that the expectation in (8) corresponds to integration with respect to the joint density gB1 ;b1 (; ) of (B1; b1). Part (iii). Note that dGB1 ;b1 (x; x)=dx = nGB1 b1 (x; x). Thus, using (8) the derivative of 25
(x) with respect to x is, after some algebra,
d(x) = nv G (x; x) + n(0 (x) , 1) Z b (xjt)G (t; t)dt 0 B1 b1 B1b1 dx x Zb ,n(x)GB1 b1 (x; x) + n((x) , x) GgB1;b1 ((x;x;xx)) x (xjt)GB1b1 (t; t)dt: B1 ;b1
Noting that (x) = x + (GB1;b1 (x; x)=gB1;b1 (x; x)), the above expression simpli es to d(x) = ,n((x) , v )G (x; x) + n0 (x) Z b (xjt)G (t; t)dt: 0 B1 b1 B1b1 dx x
Equating this to zero gives the rst-order condition (10). Part (iv). Using indicators and the joint density gB1;b1 (; ), the second term of (10) can be written as Z bZ b 0 (x) b b 1I(b x)1I(B b)(x; b)gB1;b1 (B; b)dBdb: We note that the rst term of (10) is a constant as it does not depend on (B; b). Thus, (10) can be written as in (11).
Proof of Corollary 1: We proceed in four parts as in the proof of Proposition 2. To
derive (14), (13), (15) and (16), we consider their counterparts within the APV model and rewrite them within the IPV model. Part (i). Because gB1;b1 (b; b) = (n , 1)Gn,2 (b)g2(b) and GB1b1 (b; b) = Gn,1 (b)g(b), (A.6) reduces to !n,1 Z b1 g(u) ! G ( x ) (xjb1) = exp ,(n , 1) G(u) du = G(b ) : (A.7) x 1 Using this in (9) gives (14). Part (ii). Note that GB1;b1 (b; b) and GB1b1 (b; b) become Gn (b) and Gn,1 (b)g(b), respectively in the IPV case. Using then (A.7) in (8) and integrating by parts give (13). Part (iii). We can either di erentiate (13) with respect to x or apply (10) to the IPV case. Doing the latter, and replacing (xjb1) and GB1 b1 (b; b) by (G(x)=G(b))n,1 and Gn,1 (b)g(b), respectively, we obtain (15) after simpli cation. 26
R Part (iv). As 1 , G(x) = bb 1I(b1 x)g(b)db, (16) follows from (15).
2. Proofs of Statistical Results We make the following assumptions. Throughout we assume that n 2.
Assumption A1: The joint density f (v1; . . . ; vn) has R 1 continuous partial derivatives on [v; v]n. Moreover, f (v1; . . . ; vn ) c > 0 on [v; v]n. Assumption A2: (i) The kernel K () is symmetric with support [,1; +1] with continuous bounded rst and second derivatives, R (ii) K (u)du = 1, (iii) K () is of order R + n , 2 and R + 1 for the APV model and the IPV model, respectively, i.e. moments of order strictly smaller than the given order vanish.
Note that assumption A1 implies that the marginal density f () fv1 () has R continuous derivatives on [v; v] and that f () c > 0 on [v; v]. Assumption A1 also implies that v < 1 so that b < 1. The next lemmas state the uniform consistency of the nonparametric estimators (18), (19), (22) and (23) of GB1b1 (; ), gB1;b1 (; ), G() and g(), respectively, on arbirary xed compact inner subsets of their supports as well as on expanding subsets. These lemmas are proven in Li, Perrigne and Vuong (1999, Lemma A4) and Guerre, Perrigne and Vuong (2000, Lemma B2) for the APV and IPV models, respectively. Let C be an arbitrary but xed inner compact subset of [b; b], and CL be the expanding subset [b + ; b , ], where = maxfhG; hg g with (hG; hg ) de ned by (20) for the APV case and = h with h de ned by (24) for the IPV case.
Lemma A.1: Under assumptions A1-A2 with R 1, we have (i) supC 2 jG^ B b , GBbj = O(hRG+n,2 ), supC2 jg^B;b , gB;b j = O(hRg +n,2 ), (ii) supC 2 jG^ B b , GBbj = O(hRG ), supC 2 jg^B;b , gB;b j = O(hRg ). Lemma A.2: Under Assumptions A1-A2 with for R 1, we have L
L
27
(i) supC jg^(b) , g( b)j = O(hR+1 ), (ii) supC jg^(b) , g(b)j = O(hR ). L
L
Next, we need another lemma similar to Amemiya (1985) Theorem 4.1.1, which is fundamental for establishing the consistency of extremum estimators. Our lemma di ers, however, in two respects: It allows for (i) a discontinuous statistical objective function, and (ii) a maximization and uniform consistency over an expanding subset.
Lemma A.3: Let Q() and QL() be nonstochastic and stochastic functions de ned on
the compact sets and L , where L forms an increasing sequence of possibly stochastic subsets of converging almost surely to o so that o [1 L=1 L almost surely. For every L = 1; 2; . . ., let ^L be a value in satisfying
QL(^L) sup QL(^) + o(L): 2
L
If (i) Q() is continuous on with a unique maximizer on at 0 2 o and (ii) sup2 jQL() , Q()j ! 0 almost surely as L ! 1, then ^ ! 0 almost surely. L
Proof of Lemma A.3: Adapting it to our case, the proof is largely inspired by the proof
of Theorem 4.1.1 in Amemiya (1985). Let N be an arbitrary (open) neighborhood of 0, and N its complement in . Let
= Q(0) , max Q(): 2N \
(A.8)
Thus > 0 because 0 is the unique maximizer of Q() and Q() is continuous on the compact N \ . Now, from (ii) we have almost surely for all L suciently large sup jQL() , Q()j < =3:
2
L
But, we have almost surely 0 2 L for all L suciently large as 0 2 o = [1L=1L. Hence, applying the above inequality at = 0 and = ^ 2 L, we obtain almost surely 28
for all L suciently large
Q(^) > QL(^) , =3; QL(0) > Q(0) , =3:
(A.9) (A.10)
On the other hand, we have sup2 QL() QL(0) since, as noted above, 0 2 L for all L suciently large. Hence, using the de nition of ^ we have QL(^) QL(0) + o(L). We can pick Ln suciently large so that o(L) > ,=3. Hence, almost surely and for all L suciently large, we have QL(^) > QL(0) , =3. Using (A.10), it follows that QL(^) > Q(0) , 2=3. Using (A.9), it follows that Q(^) > Q(0) , . Hence, using (A.8), it follows that ^ 2 N \ almost surely. Indeed, for all 2 N \ , we have Q() max2N \ Q() = Q(0) , . L
Proof of Proposition 3: We show that the assumptions of Lemma A.3 are satis ed with L = Ln , = [b; b], L = [bmin + ; bmax , ], Q() = () and QL() = ^ (). Note that bmin # b, bmax " b and # 0 almost surely as Ln ! 1. Hence L forms an increasing
sequence of compact subsets of converging to o = (b; b) almost surely. Regarding (i), () is clearly continuous on in view of (8) and assumption A1. Moreover, because p0n is the unique maximizer of the seller's expected pro t on [v; v], and because x0n = s(p0n) from (7), then () admits a unique maximizer on = [b; b] at 0 = x0n. Since v < p0n < v by assumption and s() is strictly increasing, then b < x0n < b, i.e. x0n 2 o. Regarding (ii), assumption A1 and the abstract Glivenko-Cantelli Theorem in van der Vaart (1998, p.270) imply L 1X n 1 X v 1I(B x)1I(b x) ! E [v 1I(B x)1I(b x)]; (A.11) n
i` 0 Ln `=1 n i=1 0 i` L 1X n 1 X Ln `=1 n i=1 nbi` 1I(Bi` bi`)1I(x bi`) ! E [nb1I(B b)1I(x b)]; (A.12) uniformly in x 2 almost surely as Ln ! 1.17 Turning to the third term in (26), we n
17
Following Example 19.6 in van der Vaart (1998), it can be readily shown that the bracketing numbers
29
rst recall from the proof of Proposition A2 in Li, Perrigne and Vuong (1999) that ^ G G B1 b1 (x; x) B1 b1 (x; x) sup g^ (x; x) , g (x; x) O(hRg ,(n,2) ): (A.13) B1 ;b1 x2[b+;b,] B1 ;b1 Because L [b + ; b , ] almost surely as L ! 1, it follows that ^ G G B1 b1 (x; x) B1 b1 (x; x) sup , g (x; x) ! 0 x2 g^B1 ;b1 (x; x) B1 ;b1
(A.14)
L
almost surely as L ! 1, provided R n , 1 as assumed. Moreover, From Li, Perrigne and Vuong (1999, Lemma A1) we also have GB1 b1 (x; x) = a(x , b) + o(x , b); (A.15) gB1;b1 (x; x) where a = 1=(n , 1) so that GB1b1 (x; x)=gB1;b1 (x; x) is continuous in x 2 [b; b]. Thus, dropping the subscript i because of symmetry, it remains to study the almost sure uniform convergence on [bmin + ; bmax , ] of L 1 X ^ L (xjb`)1I(B` b`)1I(x b` bmax , ) n
n `=1
L X 1 = L [^ (xjb`) , (xjb`)]1I(B` b` )1I(x b` bmax , ) n
n `=1 L 1 X
+L
n
,L
n
n `=1 L 1 X n `=1
(xjb`)1I(B` b`)1I(x b` b) (xjb`)1I(B` b`)1I(bmax , b` b):
(A.16)
We now study the three terms on the right-hand side, denoted A1(x), A2(x) and A3(x). Consider A1(x). we have sup jA1(x)j
x2[b+;b,]
sup
b+rtb ,
j^ (rjt) , (rjt)j;
of families of functions of the form v01I(B x)1I(b x) or b1I(B b)1I(x b), where x 2 , is nite.
30
where
! Z t gB ;b (u; u) 1 1 ^ (rjt) , (rjt) = exp , G [1 , (u)]du , (rjt); r B1 b1 (u; u) ^ B1b1 GB1 b1 ! G 1 (u) = G G^ G g^ , g ; 1 1 + 1 1 , 1 1 B1 ;b1 B1 ;b1 g 1 1 g^ 1 1 g 1 1 where we have dropped the argument (u; u) to simplify. From (A.15) and Lemma A.5 in Li, Perrigne and Vuong (1999) we have inf u2[b+;b,] GB1b1 (u; u)=gB1;b1 (u; u) c, where c > 0. Thus, if R n as assumed, it follows from (A.13) that the rst term is uniformly strictly positive and dominated by 1=[c + O(R,(n,2) )] for suciently large Ln , almost surely. Because the second term is an O(R,(n,2)) by (A.13), it follows that (u) = O(R,n+1 ) uniformly in u 2 [b + ; b , ]. Hence, for arbitrarily small, we have j(u)j uniformly in u 2 [b + ; b , ], for Ln suciently large, almost surely. Thus (rjt)1+ , (rjt) ^ (rjt) , (rjt) (rjt)1, , (rjt); B
b
B ;b
B
B
b
;b
B
b
B ;b
uniformly in (r; t) such that b + r t b , , for Ln suciently large, almost surely. But the functions x1+ , x and x1, , x attains their minimum and maximum on [0; 1] when x = (1 + ),1= and x = (1 , )1=, respectively. Since (rjt) 2 [0; 1], we obtain 1 1= 1 1 1= 1 ^ 1+ 1 + , 1 (rjt) , (rjt) 1 , 1, ,1 : Because the lower bound and upper bound converge to zero as ! 0, and because can be chosen arbitrarily small, it follows that supb+rtb, j^ (rjt) , (rjt)j ! 0 as Ln ! 1, almost surely. Hence supx2[b+;b,] jA1(x)j ! 0 as Ln ! 1, almost surely. Next, consider A2(x). From the abstract Glivenko-Cantelli Theorem in van der Vaart (1998, p.270) we obtain h i A2(x) ! E (xjb)1I(B b)1I(x b b) uniformly in x 2 = [b; b], as Ln ! 1, almost surely.18 Note that the right-hand side is continuous on . Combining Examples 19.6 and 19.7 in van der Vaart (1998), it can be shown that the bracketing numbers of families of functions of the form (xjb)1I(B b)1I(x b b), where x 2 , is nite. 18
31
Lastly, consider A3(x). Because 0 (rjt) 1, we have L X 0 A3(x) 1 1I(bmax , b` b): L n
n `=1
for all x 2 . Because bmax , ! b almost surely, it follows from the standard GlivenkoCantelli Theorem that the right-hand side converges almost surely to zero as Ln ! 1. Hence supx2 jA3(x)j ! 0 as Ln ! 1, almost surely. Collecting results, it follows from (A.14) and (A.15) that L X n G^ B1 b1 (x; x) 1 X ^ (xjbi`)1I(Bi` bi`)1I(x bi` bmax , ) ; g^B1;b1 (x; x) Ln `=1 i=1 h i ! n GgB1 b1((x;x;xx)) E (xjb)1I(B b)1I(x b b) (A.17) B1 ;b1 uniformly in x 2 [b + ; b , ], and hence in x 2 L = [bmin + ; bmax , ], as L ! 1, almost surely. Thus, combining (A.11), (A.12) and (A.17) into (26) and using (9) show that ^ (x) ! (x), uniformly in x 2 L = [bmin + ; bmax , ], as L ! 1, almost surely. This establishes (ii) of Lemma A.3 and hence Proposition 3. n
Proof of Corollary 2: The proof is similar to that of Proposition 3 and consists in
verifying the conditions of Lemma A.3. The rst part of that proof still applies, while (A.11) and (A.12) continue to hold. It remains to consider the third term of (29). From Proposition 3-(i) in Guerre, Perrigne and Vuong (2000) we have ^ G ( x ) G ( x ) sup g^(x) , g(x) O(hR); (A.18) x2 which plays the same role as (A.13). Moreover, (A.16) still holds, where ^ (xjb) = [G^ (x)=G^ (b)]n,1 and (xjb) = [G(x)=G(b)]n,1. h i As before, the terms A2(x) and A3(x) converge to E (xjb)1I(B b)1I(x b b) and zero, respectively, uniformly on as Ln ! 1, almost surely. Regarding A1(x), it suces to consider ^ n,1 n,1 G ( x ) G ( x ) ^ j(xjb) , (xjb)j = ^ n,1 , G(b)n,1 G(b) L
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0 n,2 n,3 ^ ^ n,21 ^ G ( x ) G ( x ) G ( x ) G ( x ) G ( x ) ^ , G(b) @ G(b) + G(b) ^ + . . . + G^(x) A G(b) G(b) G(b) ^ (n , 1) G^(x) , GG((xb)) ; (A.19) G(b) where we have used G^ (x)=G^ (b) 1 and G(x)=G(b) 1 because x b. Now ^ G(x) , G(x) G^ (x) , G(x) + G^ (x) G^ (b) , G(b) G^ (b) G(b) G(b) G^ (b) G(b) G^ (x) , G(x) : (A.20) inf 2 G(b) sup x2 b2[b+h;b] But, inf b2[b+h;b] G(b) O(h) > 0 by Lemma A.5 in Li, Perrigne and Vuong (1999), while supx2 G^ (x) , G(x) O(log log Ln =Ln )1=2 as Ln ! 1 almost surely by the law of the iterated logarithm (see van der Vaart (1998, p.268)). Hence the right-hand side of (A.20) is dominated by an O[(log log Ln=Ln )1=2=h], which is an o(1) using (24). Thus jG^ (x)=G^ (b) , G(x)=G(b)j ! 0 uniformly in x and b such that b + h x b b, as Ln ! 1 almost surely. Therefore, by (A.19) supb+hxbb j^ (xjb) , (xjb)j ! 0 as Ln ! 1 almost surely. This implies that supx2[b+h;b] jA1(x)j ! 0 as Ln ! 1 almost surely. Combining this with the limits of A2(x), A3(x), (A.18), (A.12) and (A.11) show that ^ (x) ! (x), uniformly in x 2 L, as L ! 1, almost surely, as desired.
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References Amemiya, T. (1985): Advanced Econometrics, Harvard University Press: Cambridge. Guerre, E., I. Perrigne, and Q. Vuong (2000): \Optimal Nonparametric Estimation of First-Price Auctions," Econometrica, forthcoming.
Hardle, W. (1991): Smoothing Techniques with Implementation in S, Springer-Verlag, New York.
Harris, M. and A. Raviv (1981): \Allocation Mechanisms and the Design of Auctions," Econometrica, 49(6), 1477-1499.
Hendricks, K. and H.J. Paarsch (1995): \A Survey of Recent Empirical Work Concerning Auctions," Canadian Journal of Economics, 28, 403-426.
Hendricks, K., R.H. Porter and R.H. Spady (1989): \Random Reservation Prices and Bidding Behavior in OCS Drainage Auctions," Journal of Law and Economics, 32, 83-106.
Hendricks, K., R.H. Porter and C.A. Wilson (1994): \Auctions for Oil and Gas
Leases with an Informed Bidder and a Random Reservation Price," Econometrica, 62, 1415-1444.
Kagel, J.H., R.M. Harstad and D. Levin (1987): \Information Impact and Allocation Rules in Auctions with Aliated Private Values: A Laboratory Study," Econometrica, 55(6), 1275-1304.
La ont, J.J. and E. Maskin (1980): \Optimal Reservation Price in the Vickrey Auction," Economics Letters, 6, 309-313.
La ont, J.J., H. Ossard and Q. Vuong (1995): \Econometrics of First-Price Auctions," Econometrica, 63, 953-980.
La ont, J.J. and Q. Vuong (1996): \Structural Analysis of Auction Data," American Economic Review, Papers and Proceedings, 86(2), 414-420.
34
Levin, D. and J.L. Smith (1996): \Optimal Reservation Prices in Auctions," The Economic Journal, 106, 1271-1283.
Li, T., I. Perrigne and Q. Vuong (1999): \Structural Estimation of the Aliated
Private Value Model with an Application to OCS Wildcat Auctions," mimeo, University of Southern California.
Li, T., I. Perrigne and Q. Vuong (2000): \Conditionally Independent Private Information in OCS Wildcat Auctions," Journal of Econometrics, forthcoming.
Maskin, E. and J. Riley (1984): \Optimal Auctions with Risk Averse Buyers," Econometrica, 52, 1473-1518.
McAfee, R.P. and D. Vincent (1992): \Updating the Reserve Price in Common-Value Auctions," American Economic Review, Papers and Proceedings, 82(2), 512-518.
Milgrom, P.R. and R.J. Weber (1982): \A Theory of Auctions and Competitive Bidding," Econometrica, 50(5), 1089-1122.
Myerson, R.B. (1981): \Optimal Auction Design," Mathematics of Operations Research, 6(1), 58-73.
Newey, W. (1994): \The Asymptotic Variance of Semiparametric Estimators," Econometrica, 62, 1349-1382.
Olley, S. and A. Pakes (1995): \A Limit Theorem for a Smooth Class of Semiparametric Estimators," Journal of Econometrics, 65, 295-332.
Paarsch, H.J. (1992): \Deciding between the Common and Private Value Paradigms in Empirical Models of Auctions," Journal of Econometrics, 51, 191-215.
Paarsch, H.J. (1997): \Deriving an Estimate of the Optimal Reserve Price: An Application to British Columbian Timber Sales," Journal of Econometrics, 78, 333-357.
Porter, R.H. (1995): \The Role of Information in U.S. O shore Oil and Gas Lease Auctions," Econometrica, 63(1), 1-27.
35
Riley, J. and W. Samuelson (1981): \Optimal Auctions," American Economic Review, 71, 381-392.
Stone, C.J. (1982): \Optimal Rates of Convergence for Nonparametric Regressions," Annals of Statistics, 10, 1040-1053.
van der Vaart, A.W. (1998): Asymptotic Statistics, Cambridge University Press: Cambridge.
Vickrey, W. (1961): \Counterspeculation, Auctions, and Sealed Tenders," Journal of Finance, 16, 8-37.
Wilson, R. (1977): \A Bidding Model of Perfect Competition," Review of Economic Studies," 44(3), 511-518.
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