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Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Semiparametric Estimation of the Optimal Reserve
Price in First-Price Auctions
Tong Li, Isabelle Perrigne & Quang Vuong
To cite this article: Tong Li, Isabelle Perrigne & Quang Vuong (2003) Semiparametric
Estimation of the Optimal Reserve Price in First-Price Auctions, Journal of Business & Economic
Statistics, 21:1, 53-64, DOI: 10.1198/073500102288618757
To link to this article: http://dx.doi.org/10.1198/073500102288618757
Published online: 01 Jan 2012.
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Semiparametric Estimation of the Optimal
Reserve Price in First-Price Auctions
Tong Li
Department of Economics, Indiana University, Bloomington, IN
(toli@indiana.edu)
47405-5301
Isabelle Perrigne and Quang Vuong
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Department of Economics, University of Southern California, Los Angeles, CA
(perrigne@usc.edu, qvuong@usc.edu)
90089-0253
The optimal reserve price in the independent private value paradigm is generally expressed as a functional of the latent distribution of private signals, which is by nature unobserved. This feature has limited the implementation of the optimal reserve price in practice. In this article, we consider rst-price
auctions within the general afliated private values paradigm. We show that the seller’s expected prot
can be written as a functional of the observed bid distribution. We propose a semiparametric extremum
estimator for estimating consistently the optimal reserve price from observed bids. As an illustration,
we consider the Outer Continental Shelf (OCS) wildcat auctions.
KEY WORDS:
Afliated private value; OCS wildcat auction; Semiparametric extremum estimator.
Given the monopoly power of the seller in an auction, the
economic literature has focused on how the seller can extract
the largest revenue or prot. The problem of optimal auctions
is to nd the auction mechanism that provides the greatest
revenue or prot for the seller. This is of considerable practical
value because many commodities are sold through auctions.
Because any auction mechanism, such as rst-price, secondprice, 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 functional of the distribution of private values and its corresponding density function. See the work of Laffont 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 difculty in implementing the optimal reserve
price is the use of unobservables such as the latent distribution of private signals. Hence, very few empirical studies were
attempted to assess the optimal reserve prices using eld auction data. In particular, McAfee and Vincent (1992) gave an
approximation of the optimal reserve price for Outer Continental Shelf (OCS) auctions in the common value framework.
Paarsch (1997) proposed an estimate of the optimal reserve
price from English auction data relying on parametric assumptions on the distribution of private values. Moreover, to our
knowledge, the optimal reserve price has been obtained for
the IPV paradigm only. This can be restrictive given its independence assumption because there are auctions in which the
IPV paradigm does not apply.
In this article, we consider the afliated private values
(APV) paradigm, which includes the IPV paradigm as a special case, and we propose a direct method for estimating the
optimal reserve price. In particular, we characterize the optimal reserve price in rst-price sealed-bid auctions with sym-
metric and risk neutral bidders. The reserve price is optimal
in the sense that it generates the highest expected prot for
the seller in a rst-price auction. Thus we do not consider the
optimality of the auction mechanism because, for instance, a
second-price auction would generate a higher prot for the
seller; see Milgrom and Weber (1982). With risk averse bidders, the optimal auction mechanism is much more complex
and involves some transfer payments; see Maskin and Riley
(1984). Next, we show that the seller’s expected prot can
be written as a functional of the observed bid distribution.
This can be used to derive the optimal reserve price from
observed bids as the maximizer of this function. We then
propose a semiparametric extremum estimator for estimating
consistently the optimal reserve price from observed bids.
Specically, the estimated optimal reserve price is obtained as
a maximizer of an estimated prot function in which some
unknown distributions and densities have been nonparametrically estimated in a rst step. Our procedure is computationally simple because it does not require computation of the
equilibrium strategy. It does not require a parametric specication of the distribution of unobserved private values or
nonparametric estimation of the latter, as in our other work
(Li, Perrigne, and Vuong, 2002). We show that our estimator
is consistent. In addition to being more direct and computationally simpler, our estimator can converge at a faster rate
than the one based on the estimation of the underlying private
values distribution.
We illustrate our methodology with the U.S. gas lease auctions off the coast of Louisiana and Texas, which have been
studied in the literature. See Porter (1995) for a survey. We
focus on wildcat auctions between 1954 and 1969 because
bidders in these auctions can be considered as symmetric and
© 2003 American Statistical Association
Journal of Business & Economic Statistics
January 2003, Vol. 21, No. 1
DOI 10.1198/073500102288618757
53
54
Journal of Business & Economic Statistics, January 2003
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the crude oil market was relatively stable during that period.
It is widely recognized that the reserve price in these auctions
was too low; see, for example, the work of McAfee and
Vincent (1992). Our empirical ndings indicate that the optimal reserve price would (i) allow the federal government to
extract a larger proportion of oil companies’ willingness to
pay and (ii) generate a higher prot and higher revenues for
the federal government despite more tracts remaining unsold.
The article is organized as follows. Section 1 is devoted to
the derivation of the optimal reserve price in rst-price auctions using observed bids. Section 2 develops our semiparametric extremum estimator. Section 3 illustrates our method.
Section 4 concludes. The Appendix collects the proofs of our
propositions as well as their applications to the IPV case.
1.
1.1
OPTIMAL RESERVE PRICE IN
FIRST-PRICE AUCTIONS
The Private Value Model
Throughout the article we consider rst-price sealed-bid
auctions within the private value paradigm. Formally, a single and indivisible object is auctioned to n bidders who are
assumed to be risk neutral. All sealed bids are collected simultaneously. Provided that bid is at least as high as a reserve
price, the highest bidder wins the auction and pays the bid.
The utility of each buyer i D 11 : : : 1 n for the object is vi . The
vector 4v1 1 : : : 1 vn 5 is a realization of a random vector whose
n-dimensional cumulative distribution function is F 4¢5 with
support 6v1 v7n . This distribution is assumed to be afliated
as dened by Milgrom and Weber (1982). Roughly, afliation
means that large values for some of the components make the
other components more likely to be large than small. Each bidder knows his or her private value vi and the distribution F 4¢5
from which v1 1 : : : 1 vn are jointly drawn but does not know
other bidders’ private values. We restrict our study to the case
in which F 4¢5 is exchangeable in its n arguments. This model
is known as the symmetric afliated private value model. As
far as we know, despite its generality, the APV model has
seldom been studied in the literature, except in experimental studies by Kagel, Harstad, and Levin (1987), in empirical
studies by Li, Perrigne, and Vuong (2000), and in simulation
studies by Li et al. (2002).
At the Bayesian Nash equilibrium, bidder i chooses bid
bi to maximize E64vi ƒ bi 5 4Bi µ bi 5—vi 7, where Bi D s4yi 5,
yi D maxj6Di vj , and s4¢5 is the strictly increasing equilibrium
strategy. Assume rst that the reserve price is nonbinding. The
equilibrium strategy satises the rst-order differential equation
s 0 4vi 5 D 6vi ƒ s4vi 57fy1 —v1 4vi —vi 5=Fy1 —v1 4vi —vi 5
(1)
for all vi 2 6v1 v7 subject to the boundary condition s4v5 D
v, where Fy1 —v1 4¢—¢5 denotes the conditional distribution of y1
given v1 , fy1 —v1 4¢—¢5 denotes the corresponding density, and the
index 1 refers to any bidder among the n bidders because
all bidders are ex ante identical. From Milgrom and Weber
(1982), the explicit solution of (1) is
Z vi
L4—vi 5d1
bi D s4vi 5 D vi ƒ
(2)
v
where L4—vi 5 D exp6ƒ
R vi
fy1 —v1 4u—u5=Fy1 —v1 4u—u5du7.
We next introduce a binding reserve price p0 > v. The introduction of a reserve price acts as a screening device for participating in the auction. The screening level is a function of
p0 such that those bidders 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
Z vi
bi D s4vi 1 p0 5 D vi ƒ
L4—vi 5d for vi ¶ p0 0
(3)
p0
In particular, s4vi 5 given in (2) is identical to s4vi 1 v5.
1.2
The Optimal Reserve Price
The choice of the reserve price constitutes an important instrument with which the seller can take advantage of
monopoly power and increase prots 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 derived 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 p0ü that maximizes the expected prot for the seller satises
Rv
L4p0ü —u5Fy1 —v1 4u—u5fv1 4u5du
pü
ü
p0 D v0 C 0
(4)
Fy1 —v1 4p0ü —p0ü 5fv1 4p0ü 5
if v < p0ü < v, where v0 denotes the private value of the seller
for the auctioned object, fv1 4¢5 is the marginal density of vi ,
and Fy1 —v1 4¢—¢5 and L4¢—¢5 are as dened earlier. [Equation (4)
may have multiple roots. If this is the case, it is necessary
to evaluate the expected prot at each root to determine the
global maximum, as is usually advised in the theoretical literature.]
The condition v < pü0 < v arises from the characterization
of a maximum through its rst-order condition (4). It is not a
primitive condition and thus calls for some comments. First,
note that whatever a seller’s private value v0 , the seller does
not lose expected prot by only choosing a reserve price
belonging to the closed interval 6v1 v7. The intuition is quite
simple as any reserve price larger than v or smaller than v will
generate the same expected prot as when p0 D v and p0 D v,
respectively. The latter arises from the boundary condition
s4v1 p0 5 D v for any p0 µ v so that the equilibrium strategy is
identical to (2) in this case. Second, if the seller’s private value
v0 belongs to 6v1 v5, then the optimal reserve price p0ü must
belong to 4v1 v5 as required in Proposition 1: the seller will
always increase the prot by setting a reserve price slightly
larger than v because the derivative of the expected prot is
strictly positive in the neighborhood of this point. It follows
that v0 < p0ü < v for any v0 2 6v1 v5. Third, if v0 ¶ v, then pü0 D v
because the seller does not gain by setting a higher reserve
price. Fourth, if v0 < v, then p0ü D v or p0ü 2 4v1 v5 depending
on the underlying distribution of private values. In particular,
this implies that a nonbinding optimal reserve price, that is,
p0ü µ v, implies v0 < v.
Li, Perrigne, and Vuong: Semiparametric Estimation
55
As is well known, a special case of the APV paradigm is
the IPV paradigm rst considered by Vickrey (1961), wherein
private values are drawn independently from a common distribution F 4¢5. All other assumptions made previously for the
APV case remain valid. In the IPV paradigm, the optimal
reserve price has been widely studied (Laffont and Maskin
1980; Riley and Samuelson 1981). The optimal reserve price
p0ü in a rst-price auction solves
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p0ü D v0 C
1 ƒ F 4p0ü 5
f 4p0ü 5
(5)
if v < p0ü < v. It can be easily checked that (5) follows
from (4) by noting that L4—v5 and Fy1 —v1 4v—v5 reduce to
4F 45=F 4v55nƒ1 and F nƒ1 4v5, respectively, in the IPV case.
A remarkable property of p0ü 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 Fy1 —v1 4¢—¢5, fy1 —v1 4¢—¢5 and fv1 4¢5. [Levin and Smith
(1996) studied the optimal reserve price in the APV model
without giving its expression. An interesting question is the
effect 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.] 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 difculty was mentioned by McAfee and
Vincent (1992), Hendricks and Paarsch (1995), and Paarsch
(1997) among others. The structural econometric approach
developed by Paarsch (1992), Laffont, Ossard, and Vuong
(1995), and Guerre, Perrigne, and Vuong (2000) offers econometric methods for estimating the latent distribution in the IPV
model. More recently, Li et al. (2002) extended the structural
approach to the APV model by developing a two-step procedure for estimating the latent distribution nonparametrically.
See Perrigne and Vuong (1999) for a survey of recent developments. Although this procedure can be used to determine
the optimal reserve price, in this article we propose a more
direct and simpler method with superior statistical properties.
1.3
A Reformulation from Observed Bids
Assume that we observe bids from a rst-price sealed-bid
auction with a nonbinding reserve price. If observed bids are
coming from a rst-price auction with a reserve price, the function 4¢5 dened in the following becomes more involved; see
the IPV case of Guerre et al. (2000). Nonetheless, a similar
argument can be applied to obtain a characterization of the optimal reserve price from the observed bid distribution. As postulated in the structural approach, such bids are the equilibrium
bids of the corresponding game and hence are given by bi D
s4vi 5, where s4¢5 is dened by (2). Because private values are
random, bids are naturally random with an n-dimensional joint
distribution G4¢5. Moreover, because s4¢5 is strictly increasing, we have GB1 —b1 4X —x5 D Fy1 —v1 4s ƒ1 4X5—s ƒ1 4x55, where B1 D
maxj6D 1 bj D s4y1 5 and GB1 —b1 4¢—¢5 denotes the conditional distribution of B1 given b1 . It follows that the corresponding density
is given by gB1 —b1 4X —x5 D fy1 —v1 4s ƒ1 4X5 —s ƒ1 4x55=s 0 4s ƒ1 4X55.
Thus the differential equation (1) can be written as
v D bC
GB1 —b1 4b —b5
² 4b50
gB1 —b1 4b —b5
(6)
As noted by Li et al. (2002), the function 4¢5 is the inverse
bidding strategy s ƒ1 4¢5 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 bid distribution. Let GB1 1b1 4¢1 ¢5
be the joint distribution of 4B1 1 b1 5 with density gB1 1b1 4¢1 ¢5
and support 6b1 b72 . Let GB 1 b1 4¢1 ¢5 D ¡GB1 1b1 4¢1 ¢5=¡b1 D
GB 1 —b1 4¢—¢5gb1 4¢5.
Proposition 2. The optimal reserve price pü0 in a rstprice sealed-bid auction with n ¶ 2 bidders within the APV
paradigm can be written as p0ü D 4x0 5 for b µ x0 µ b, where
4¢5 is dened as (6) and x0 maximizes with respect to
x 2 6b1 b7 the expected prot
£
ç4x5 D E v0 4B1 µ x5 4b1 µ x5
C n6b1 C 44x5 ƒ x5å4x—b1 57
¤
4B1 µ b1 5 4b1 ¶ x5 1
(7)
Rt
where å4r —t5 D exp4ƒ r gB1 1b1 4u1 u5=GB1 b1 4u1 u5du5, E6¢7
denotes the expectation with respect to 4B1 1 b1 5, and 4¢5
denotes the indicator of the event in parentheses. [The third
term in (7) arises from adopting a game theoretic approach,
which takes into account the dependence of bidders’ behavior
on the reserve price. In contrast, a decision theoretic approach
would consider the rst two terms only in (7) and would give
p0ü D v0 .]
Note that if v0 2 6v1 v5, then x0 2 4b1 b5 D 4s4v51 s4v55
because x0 D s4p0ü 5 and v < pü0 < v as noted before. In contrast
to Proposition 1, which requires the knowledge of the underlying distributions Fy1 —v1 4¢—¢5 and fv1 4¢5 of private signals, the
main advantage of Proposition 2 is that it expresses the optimal reserve price in terms of the bid distribution, which can
be estimated directly from observed bids. This is the basis of
our direct procedure for estimating the optimal reserve price,
as presented in the next section.
2.
ECONOMETRIC IMPLEMENTATION
In agreement with Section 1.3, we consider L rst-price
auctions with a nonbinding reserve price and we assume that
bidders’ payoffs for each auction do not depend on bidders’
past actions to prevent dynamic considerations. As postulated
in the structural approach, observed bids are assumed to obey
the equilibrium strategy (2) within each auction. To simplify
the presentation, we also assume that auctions are homogeneous so that there is no need to control for heterogeneity of
the auctioned objects through some vector of characteristics.
Allowing for heterogeneity of the auctioned objects through
a vector z 2 òd of characteristics is straightforward. In this
case, our methods estimate the optimal price for an arbitrary
value z0 . For instance, if z D 4z1 1 : : : 1 zd 5 includes only continuous variables and z` denotes the
Q value of z for the `th
object, it sufces to insert the term dkD1 K44zk0 ƒ zk` 5=hz 5 in
(8) and (9).
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
56
Journal of Business & Economic Statistics, January 2003
Let ` index the `th auction, and let n` be the number of
bidders in the `th auction with n` ¶ 2. The observed bids
are 8bi` 1 i D 11 : : : 1 n` 1 ` D 11 : : : 1 L9. This section proposes a
direct method for estimating the optimal reserve price p0ü in
rst-price sealed-bid auctions within the APV paradigm from
such observations without making parametric assumptions.
Our method relies on Proposition 2. Specically, as p0ü D
O xO 5, where 4¢5
O
4x0 5, a natural estimator of p0ü is pO 0ü D 4
0
O
and x0 are estimators of 4¢5 and x0 , respectively. Regarding 4¢5, which is the inverse of the equilibrium strategy
s4¢5, we follow Li et al. (2002). Regarding x0 , Proposition 2
indicates that, if the prot function ç4¢5 can be estimated
sufciently well, then x0 could be estimated by xO 0 , where
xO 0 maximizes the estimated prot function. This is the purpose of the second subsection. [Alternatively, we could use
the rst-order condition corresponding to the maximization
of (7) with respect to x. This would lead to semiparametric Generalized Method of Moments (GMM) estimation of
x0 (Newey 1994; Newey and MacFadden 1994; 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 4¢5, which can be estimated only at a slower rate
than 4¢5. This method is left for future research. Specically, when x0 2 4b1 b5, which occurs if v < pü0 < v, then x0
can be characterized by its rst-order condition 0 D E644x5 ƒ
v0 5GB1 b1 4x1 x5 ƒ 4B1 µ b1 5 4b1 ¶ x5 0 4x5å4x—b1 57. This
rst-order condition is obtained after some algebra by differentiating (7) with respect to x and using dGB1 1b1 4x1 x5=dx D
nGB1 b1 4x1 x5, GB1 —b1 4x—x5gb1 4x5 D GB 1 b1 4x1 x5, 4x5 ƒ x D
GB 1 b1 4x1 x5=gB1 1b1 4x1 x5, and (A.6).] Because the expected
prot ç4¢5 involves unknown functions, namely, GB1 b1 4¢1 ¢5
and gB1 1b1 4¢1 ¢5, this leads naturally to considering semiparametric M-estimation, in which these distributions and densities
are estimated nonparametrically in a rst step. Thus our semiparametric M-estimator corresponds more to the terminology
used by Andrews (1994) and Powell (1994) than to that used
by Newey and MacFadden (1994).
2.1
Nonparametric Estimation of 4¢5
Our rst step is to estimate the inverse equilibrium strategy
4¢5 from observed bids. Note that this function depends on
the number n of bidders. To emphasize such a dependence,
we denote it by 4¢—n5. Following Li et al. (2002) and Guerre
et al. (2000), we use kernel estimators, although any other
nonparametric estimators can be employed.
To estimate 4¢—n5, we consider only bids from auctions
with n bidders, namely, 8bi` 1 i D 11 : : : 1 n1 ` D 11 0 0 0 1 Ln 9,
where Ln is the number of such auctions. Noting that
GB 1 —b1 4¢1 ¢5=gB1 —b1 4¢1 ¢5 D GB1 b1 4¢1 ¢5=gB1 1b1 4¢1 ¢5, it follows from
(6) that the inverse equilibrium strategy 4¢—n5 can be
O —n5 D b C G
bB b 4b1 b5=gO B 1b 4b1 b5 for any
estimated by 4b
1 1
1
1
b 2 4b1 b5, where
´
³
Ln
n
b ƒ bi`
1X
1 X
b
1 (8)
4Bi` µ B5K
GB1 b1 4B1 b5 D
Ln hG `D1 n iD1
hG
´ ³
´
³
Ln
n
b ƒ bi`
B ƒ Bi`
1 X
1X
gO B1 1b1 4B1 b5 D
K
K
(9)
Ln h2g `D1 n iD1
hg
hg
for any value 4B1 b5 with hG and hg some smoothing parameters called bandwidths, and K4¢5 a weight function called
kernel satisfying Assumption A2 in the Appendix. Note that
the symmetry of bidders was used by averaging over i in (8)
and (9).
As shown by Li et al. (2002, prop. A1), if Assumption A1
in the Appendix holds, then GB1 b1 4¢1 ¢5 and gB1 1b1 4¢1 ¢5 have
both R C n ƒ 2 continuous bounded partial derivatives on any
compact subsets of 4b1 b52 , where R is the differentiability
order of the joint density f 4v1 1 : : : 1 vn 5 of private values. Let
hG D cG 4log L=L51=42RC2nƒ35 1
hg D cg 4log L=L51=42R C2nƒ25 1
(10)
where cG and cg are some constants. From Stone (1982), these
bandwidths are optimal in the sense of delivering the fastest
rates of uniform convergence for estimating GB1 b1 4¢1 ¢5,
gB1 1b1 4¢1 ¢5, and hence the inverse equilibrium strategy 4¢5. See
Lemma A.1 in the Appendix.
2.2
Semiparametric M-Estimation of x0
As shown in Proposition 2, x0 maximizes the expected prot
function ç4¢5. It is thus natural to estimate x0 by the value
that maximizes an estimated prot 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.
Proposition 2 provides an expression (7) for the expected
prot function ç4¢5. Specically, noting that 4x5 ƒ x D
GB 1 b1 4x1 x5=gB1 1b1 4x1 x5, (7) can be written as ç4x5 D
E6–4B1 1 b1 1 x1 gB 1 1b1 =GB1 b1 57, where –4¢5 is given by the term
in brackets in (7). This leads to considering the estimated
expected prot function by averaging over bids the function
–4¢5, where GB1 b1 and gB1 1b1 are replaced by their nonparametric estimates (8) and (9), namely,
L
n
1 Xn 1 X
O
b
b
O
D
–4B
ç4x5
i` 1 bi` 1 x1 gB1 1b1 =GB1 b1 5
Ln `D1 n iD1
L
n ³
1 Xn 1 X
D
v 4Bi` µ x5 4bi` µ x5
Ln `D1 n iD1 0
C nbi` 4Bi` µ bi` 5 4x µ bi` 5
bB b 4x1 x5
G
1
1
b
å4x —bi` 5 4Bi` µ bi` 5
gO B 1 1b1 4x1 x5
´
4x µ bi` µ bmax ƒ „5 1
(11)
Cn
Rt
bB b 4u1 u5du5, „ ²
å4x—t5 D exp4ƒ x gO B 1 1b1 4u1 u5=G
where b
1
1
max4hg 1 hG 5, and bmax is the maximum of the bids bi` ,
i D 11 0 0 0 1 n, ` D 11 0 0 0 1 Ln . The trimming associated with
the requirement bi` µ bmax ƒ „ arises from the fact that
GB 1 b1 4u1 u5, gB1 1b1 4u1 u5 and hence å4x—t5 are not well estimated uniformly if u and hence t are too close to the upper
boundary b. See Lemma A.1.
b
Dene xO 0n as maximizing the estimated expected prot ç4¢5
over 6bmin C „1 bmax ƒ „7, where bmin is the minimum of the
Li, Perrigne, and Vuong: Semiparametric Estimation
57
bids bi` , i D 11 0 0 0 1 n, ` D 11 0 0 0 1 Ln . More rigorously, xO 0n is
dened as any value in 6bmin C „1 bmax ƒ „7 that satises
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b xO 0n 5 ¶
ç4
sup
x26bmin C„1bmax ƒ„7
b
C o4151
ç4x5
(12)
b
where o415 converges to zero as Ln ! ˆ. Because ç4¢5
is
piecewise continuous in x, the term o415 ensures the existence
O
of xO 0n . [Because the statistical objective function 4¢5
is not
continuous, our estimator cannot be characterized by its rstorder conditions.]
Because x0n maximizes the expected prot ç4¢5, whereas
b
xO 0n essentially maximizes the estimated expected prot ç4¢5,
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.2. This lemma extends
Theorem 4.1.1 of Amemiya (1985), which is fundamental for
establishing the consistency of extremum estimators. Besides
allowing for a discontinuous statistical objective function,
our lemma allows for maximization and uniform consistency
over an expanding subset. [Andrews (1995) also considered
expanding subsets to trim out observations for which the estimated density is too close to zero. In contrast, our expanding subsets arise from trimming close to the boundaries of
the compact support of the bid distribution. Alternatively, we
could consider maximization of (12) over a xed compact subset 6a1 a7 included in 4b1 b5. The interval 4b1 b5 is, however,
unknown, although it can be estimated by 4bmin 1 bmax 5. More
important, x0n is not guaranteed to belong to the chosen xed
compact subset. Requiring x0n 2 6a1 a7 imposes some restrictions on the seller’s private value v0 and the underlying distribution F 4¢5.]
Proposition 3. Suppose that p0ü is the unique maximizer
of the seller’s expected prot over 6v1 v7 and that v < pü0 < v.
If Assumptions A1 and A2 hold with R ¶ n ¶ 2, then xO 0n is a
strongly consistent estimator of x0n as Ln ! ˆ.
ü
2
As noted earlier, there is no loss in requiring that p0n
6v1 v7. The uniqueness assumption is necessary for identication, whereas v < p0ü < v is satised if v µ v0 < v.
Combining Proposition 3 and the uniform consistent estimation of 4¢—n5 in Section 2.1 delivers a consistent estimator of the optimal reserve price pü0n . The additional subü
script n emphasizes that p0n
generally depends on n in the
APV paradigm. Specically, under the assumptions of Propoü D O O —
4x0n n5 is a strongly consistent estimator
sition 2, pO 0n
ü
ü
of p0n as Ln ! ˆ. The rate of convergence of pO 0n
is at
O
4RCnƒ25=42RC2nƒ25
—
best that of 4¢ n5, namely, Ln
from
Li
et
al.
p
(2002, prop. 2(i)). In particular, if x0n is Ln -consistent as in
Andrews (1994), our estimator pO ü0n would be consistent at the
rate L4RCnƒ25=42RC2nƒ25
.
n
As mentioned, an alternative method for estimating the
optimal reserve price is rst to estimate the APV model,
namely, the underlying joint density f 4¢1 : : : 1 ¢5, and then
to use (4), where fv1 4¢5 and fy1 —v1 4¢—¢5 are replaced by
their corresponding nonparametric estimates. From Li et al.
(2002, prop. 3), the marginal density estimator fOv1 4¢5
converges at the rate Ln4RCnƒ25=42RC2nC25 while the conditional density estimator fOy1 —v1 4¢—¢5 converges at the rate
2
Ln4RCnƒ25 =64RCnC1542RC2nƒ257 , which is strictly slower. The latter
consistency rate is established by following the proof of
Proposition 3 of Li et al. (2002) with a bandwidth rate of
the form c4log Ln =Ln 54RCnƒ25=64RCnC1542RC2nƒ257 when estimating the bivariate density fyv 4¢1 ¢5, which has R C n ƒ 2 continuous derivatives from Lemma A1(iii) in that article. Hence,
ü
the convergence rate of this alternative estimator of p0n
is at
2
best Ln4RCnƒ25 =64RCnC1542RC2nƒ257 , which is strictly slower than
.
the previous rate L4RCnƒ25=42RC2nC25
n
3.
AN ILLUSTRATION OF OCS
WILDCAT AUCTIONS
It was widely recognized by economists that the reserve
price used by the federal government in OCS auctions was too
low (Hendricks, Porter, and Spady 1989; McAfee and Vincent
1992; Hendricks, Porter, and Wilson 1994). As argued by
McAfee and Vincent (1992), however, a roadblock to applying auction theory and especially to determining the optimal
reserve price is the knowledge of unobservables. This section
proposes an estimate of the optimal reserve price in OCS wildcat auctions using our semiparametric estimator within the
private value paradigm. [The previous literature on OCS data
used the so-called mineral rights model or (pure) common
value (CV) model. In the APV model, afliation among private values can arise from an unobserved common component.
See Li et al. (2000) for the formalization of such a model.
The basic difference between an APV model and a common
value model is the specication of a bidder’s utility, which
is the bidder’s own private value when considering a private
value model and the unknown common component when considering a pure common value model. Consequently, the APV
model does not incorporate a winner’s curse, which is present
in the CV model. Although the APV model does not exclude
the existence of a possible common component through the
afliation of bidders’ private values, it assumes that each company’s utility is idiosyncratic and related to rms’ opportunity
costs and efciency. On the other hand, the CV model assumes
that each company’s utility is identical whatever the rm’s
opportunity costs and efciencies, which could be restrictive.
It is likely that the real situation is between these two polar
cases in that the utility of each bidder is a function of both
common and private components.]
3.1
Data
The U.S. government began auctioning its mineral rights
to offshore oil and gas off the Texas and Louisiana coasts in
1954. The auctioned tracts are usually of about 5,000 acres.
The bidders are oil companies. The auction is organized as a
rst-price sealed-bid auction with a reserve price of $15 per
acre held constant across auctions from 1954 through 1979.
As we focus 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 tends to be more symmetric in wildcat auctions than in drainage auctions. The latter were extensively studied by Hendricks and Porter; see Porter (1995) for
a survey. Moreover, we exclude from our study all auctions
held after 1970 because market conditions were unstable. The
dataset provides the number of bidders for each auction and
58
Journal of Business & Economic Statistics, January 2003
Table 1. Summary Statistics on Bids per Acre
No. bidders
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
n D2
No. auctions
Mean
STD
Min.
Max.
217
145.78
255.72
19.70
2220.28
the corresponding bids in 1972 dollars. As an illustration, we
consider auctions with two bidders only. We could consider
auctions with a higher number of bidders, but the number of
corresponding auctions Ln tends to reduce dramatically. The
summary statistics of bids per acre are given in Table 1 for
two bidders.
Table 1 indicates that there is a great variability in bids as
measured by the standard deviation and the range of bids; a
large proportion of bids falls between $50 and $200 per acre.
A test for independence among bids within auctions using the
Blum, Kiefer, and Rosenblatt (1961) nonparametric test gives
a test statistic equal to 6.00 corresponding to a p-value equal
to .14%. This constitutes a clear rejection of independence and
hence of the IPV model. This leads us to consider the APV
model.
As a rst approximation, we assume that the current reserve
price is nonbinding and hence does not act as an effective
screening device to participating to the auction because the
reserve price of $15 per acre is much lower than the average
bid. Moreover, very few bids are clustered around the actual
reserve price of $15. Specically, less than .25% of bids are
in the interval 6151 207 and less than 1% are in the interval
6151 307. Hence, the actual reserve price can be viewed as nonbinding. Second, the federal government sometimes rejected
the highest bid in some auctions although that bid was larger
than the announced reserve price. Hence, the actual reserve
price can be viewed as random. For wildcat auctions between
1954 and 1969, we observe a 1.8% rejection rate for the 217
auctions analyzed here. We can reasonably consider that the
random reserve price does not have much effect on bidding
strategies.
Third, we assume that there is no dynamic consideration
in these auctions. This can be justied by the fact that the
different sales were spread widely over the 1954–1969 period,
reducing dynamic linkages across sales. Within each sale, we
follow the earlier literature, which neglects oil companies’
capacity constraints given their large sizes; see Porter (1995).
Fourth, we have to address the possible heterogeneity across
tracts because this affects the value of the optimal reserve
price. To assess possible heterogeneity across tracts, we followed Porter (1995) and regress the log of bids on a complete set of tract-specic dummies. Using the full dataset of
1,147 auctions, we obtain a weak rejection of tract homogeneity based on an F -test. Because the F -test requires independence among bids, this statistic tends to overreject the null
assumption of homogeneity because bids are not independent
in the presence of afliation. Moreover, a similar regression
controlling for the number of bidders shows that the F -value
drops signicantly. To assess further tract heterogeneity given
a number of bidders, we conduct separate regressions for each
size of bidders. We nd that tract 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 is automatically
satised because our econometric analysis is conducted separately for each number of bidders.
3.2
Practical Issues
We assume that the underlying joint density f 4v1 1 v2 5 is
twice continuously differentiable, that is, R D 2. To implement our kernel estimators (8) and (9), we need to choose
a kernel function. The triweight kernel has a compact support and two continuous derivatives. It is dened as K4u5 D
435=32541 ƒ u2 53 4—u— µ 15 and is of order 2 in agreement
with Assumption A2. For n D 2, Equation (10) gives the bandwidths hG D cG Lƒ1=5 and hg D cg Lƒ 1=6 , where cG and cg are
some constants. These two constants are obtained by the socalled rule of thumb. Specically, cG D cg D 20978 1006‘O b ,
where ‘O b is the empirical standard deviation of bids, and the
factor 2.978 is the correction for the triweight kernel; see
Hardle (1991).
For inference purposes, it is useful to provide condence
O xO 5, where xO maximizes the estimated
intervals for pO 0ü D 4
0
0
expected prot (11). Because asymptotic distributions of estimators in structural models are frequently difcult to establish,
bootstrap methods are commonly used as an alternative for
constructing approximate condence intervals. See Horowitz
(in press) for a survey on bootstrap methods. Our bootstrap
procedure is as follows. We resample the original data for
auctions with two bidders to obtain 500 datasets, each of size
equal to 434 bids. For each resampled dataset, we apply our
semiparametric estimator and thus obtain 500 different values
for pO 0ü . To construct a 90% condence interval, we use the
Efron (1979) method relying on percentiles. This method does
not impose any symmetry restriction.
3.3
Empirical Results
Any determination of the optimal reserve price requires the
knowledge of the value of the auctioned object v0 for the
seller. On the other hand, this valuation must be nonnegative.
Moreover, because the seller may receive negative prots if
the reserve price is set below the seller’s value v0 , the seller is
always better off setting a reserve price above v0 . Hence we
can infer that the federal government valuation is between $0
and $15 per acre. In the following, we provide an estimate of
the optimal reserve price for both values $0 and $15 as well
as for bootstrap condence intervals. The results are given in
Table 2.
The bootstrap condence intervals are relatively tight. It
is interesting to compare our results with the approximation
found by McAfee and Vincent (1992) when considering a
pure common value model with endogenous entry and stochastic participation of bidders. McAfee and Vincent found that
Table 2. Optimal Reserve Price
Seller’s valuation
v0 D 0
v 0 D 15
pü0
Con’dence interval
$272.83
$290.07
[$2180711 $322033]
[$2240801 $336046]
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
Li, Perrigne, and Vuong: Semiparametric Estimation
the reserve price should be approximately $600 in 1992 dollars, that is, about 40-fold the current level. To be comparable, the results in Table 2, which are in 1972 dollars, need
to be divided by .339. For two bidders we nd that the optimal reserve price should be $805 with a condence interval
6$6451 $9517 when v0 D 0. This discrepancy can be explained
by the absence of correction for the winner’s curse in a
private value framework: when considering a pure common
value model, bidders have to adjust their bids downward relative to their private estimates because of the winner’s curse.
This can explain why we obtain somewhat greater optimal
reserve prices. Both approaches agree, however, that the actual
reserve price used by the federal government is far from being
optimal.
The choice of the reserve price is an important instrument with which the federal government can take advantage
of its monopoly power in the auction as a unique seller. A
higher reserve price induces rms to bid more aggressively
but increases the probability of the auctioned tract not being
sold. The optimal reserve price balances these two opposite
effects and maximizes the seller’s expected gain. Thus, in
terms of economic policy conclusions, simulating the rstprice sealed-bid auctions with the optimal reserve price is of
great interest for assessing the potential gain for the federal
government. Using the estimator proposed by Li et al. (2002),
we can recover the bidders’ private values by using the equality v D 4b5 given in (6) for 174 auctions as well as the
underlying distribution of private values. Because nonparametric methods are used, some auctions had to be trimmed out
because of boundary effects.
For the 174 auctions for which we are able to recover
bidders’ private values, we simulate the oil companies’ bids
following (3) when facing our estimated optimal reserve
prices. As expected, rms bid more aggressively whenever
their private values are above the reserve price, but some tracts
remain unsold because of the higher reserve price. Table 3
gives the percentage of informational rents, noted IR, left to
the winning oil companies for sold tracts. The informational
rent in percentage is computed as the ratio of the winning
bidder’s private value minus the bid divided by the bidder’s
private value. We perform the simulation 500 times using
resampled data to obtain condence intervals.
We observe that about half of the tracts remain unsold.
Moreover, the informational rents decrease signicantly from
the average informational rent under the actual reserve price,
which is 62%. The latter gure is computed from the observed
winning bids and the estimated private values for the 174 auctions. Hence, implementing the optimal reserve price would
allow the federal government to be much more successful in
capturing the willingness to pay of the oil companies.
Another issue of great interest is to compare revenues
and prots between the two mechanisms. Table 4 gives the
59
Table 4. Revenues in Dollars for an Average Sold Tract
v0
Actual
mechanism
Optimal
mechanism
Con’dence
interval
0
15
$5141184092
$5141184092
$112671528001
$114411549060
[$1108514980101 $115681295060]
[$1110913380701 $116011344060]
revenues from an average sold tract (in terms of acreage) and
Table 5 gives the prots from an average sold tract for each
mechanism. Again, we provide revenues and prots with the
optimal reserve price when the federal government has a private value per acre equal to $0 or $15. Revenues are computed
as the average winning bid times the average acreage of a
tract, whereas prots are computed as the average winning bid
minus the value of the tract for the federal government times
the average acreage of a tract. We also provide 90% bootstrap
condence intervals.
It is important to note that revenues and prots for an average sold tract both signicantly increase as expected. On the
other hand, as noted, half of the tracts become unsold under
the optimal reserve price. Nevertheless, if we multiply the
gures in Tables 4 and 5 by the number of sold tracts, the
overall prot and revenue for the federal government is much
higher than those from the actual mechanism. For instance, if
we assume that the value of a tract for the federal government
is $15 per acre, the optimal reserve price would allow prots
of around 50% larger than actual prots although more tracts
remain unsold.
4.
CONCLUSION
A well-known difculty in implementing optimal reserve
prices on eld data is that the latent distributions are unknown
to the analyst. In this article we circumvent such a difculty
by showing that the seller’s expected prot can be expressed
as a functional of the observed bids distribution. Hence, the
seller’s expected prot function can be identied and estimated from observed bids. It follows that a natural estimator
of the optimal reserve price can be obtained by maximizing
the estimated prot function. This leads us to propose a semiparametric extremum estimator of the optimal reserve price,
which is shown to be strongly consistent. In addition to being
more direct and computationally simpler, this estimator can
converge at a faster rate than can one relying on the estimation of the underlying distribution of private values.
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 bid distribution. A possible extension of our
method is to specify a parametric family for the distribution of
observed bids. Second, as mentioned, an alternative semiparametric procedure would be to use the rst-order conditions
Table 5. Pro’ts in Dollars for an Average Sold Tract
Table 3. Average Informational Rents for the Optimal Reserve Price
v0
0
15
No. sold tracts Con’dence interval
90
88
[851 107]
[771 103]
IR
Con’dence interval
v0
Actual
mechanism
Optimal
mechanism
Con’dence
interval
43.87%
39.31%
[35033%1 45031%]
[37024%1 45082%]
0
15
$5141184092
$4441756021
$112671528001
$113721968080
[$1108514980101 $115681295060]
[$1104017570901 $115321763090]
60
Journal of Business & Economic Statistics, January 2003
characterizing the optimal reserve price to develop a semiparametric GMM estimator. Third, because our result relies on
the identication of the model from which bids are generated,
it would be interesting to consider bids arising from a rstprice auction with a binding reserve price. Guerre et al. (2000)
solved such an issue within the IPV paradigm, and Hendricks,
Pinkse, and Porter (2000) considered the CV model. Fourth,
as is the case for many auction situations, such as procurement
auctions, our illustration involves a public seller. Because of
its public nature, it may be possible that the seller has objectives other than prot maximization. Although of great interest, this issue requires more theoretical research.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
ACKNOWLEDGMENTS
The authors are grateful to K. Hendricks and R. Porter
for providing the data analyzed in this article as well as to
M. Pesendorfer, the editor, and two referees for helpful and
constructive comments. This article was presented at the Far
Eastern Meeting of the Econometric Society in Singapore,
June 1999, and at the Winter Meeting of the Econometric
Society in Boston, January 2000. Financial support from the
National Science Foundation under grant SES-0001663 to the
rst author and grant SBR-9631212 to the third author is
gratefully acknowledged.
condition
nFy1 —v1 4p0ü —p0ü 5fv1 4p0ü 54v0 ƒ p0ü 5
Zv
C n L4p0ü —v1 5Fy1 —v1 4v1 —v1 5fv1 4v1 5 dv1 D 01
p ü0
because dF 4x1 : : : 1 x5=dx D nFy1 —v1 4x —x5fv1 4x5. Equation (4)
follows.
Proof of Proposition 2. Combining (3), (A.1), (A.2), and
(A.3), and using y1 D maxj6D1 vj , the expected prot for the
seller ç4p0 5 is equal to
ç4p0 5 D v0 E6 4y1 µ p0 5 4v1 µ p0 57
Zv
Cn
4v1 ¶ p0 5s4v1 1 p0 5
v
³Z
v
v
´
4y1 µ v1 5fy1 —v1 4y1 —v1 5 dy1 fv1 4v1 5 dv1
D E6v0 4y1 µ p0 5 4v1 µ p0 5
C ns4v1 1 p0 5 4y1 µ v1 5 4v1 ¶ p0 570
(A.4)
Now, for p0 and v1 such that v µ p0 µ v1 µ v, we have from
(2) and (3)
Z p0
L4—v1 5 d
s4v1 1 p0 5 D s4v1 5 C
v
D s4v1 5 C L4p0 —v1 5
APPENDIX
A.1 Proofs of Theoretical Results
Proof of Proposition 1. Using an argument similar to that
by Riley and Samuelson (1981), we maximize the expected
prot for the seller. For any bidder, say, Bidder 1, the expected
payment if he or she wins the auction given v1 ¶ p0 is
p4v1 5 D s4v1 1 p0 5 Pr 4v2 µ v1 1 : : : 1 vn µ v1 —v1 5
D s4v1 1 p0 5Fy1 —v1 4v1 —v1 51
(A.1)
where s4v1 1 p0 5 is given by (3). Therefore, using (3), the
expected revenue for the seller from Bidder 1 is
R1 4p0 5 D
D
Z
p0
p4v1 5f 4v1 5 dv1
p0
v1 ƒ
Z
v1
p0
L4—v1 5 d
¶
Fy1 —v1 4v1 —v1 5fv1 4v1 5 dv1 0
(A.2)
The total expected prot ç 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
ç4p0 5 D v0 F 4p0 1 : : : 1 p0 5 C nR1 4p0 50
p0
v
L4—p0 5 d
D s4v1 5 C L4p0 —v1 54p0 ƒ s4p0 551
(A.3)
Differentiating ç4p0 5 with respect to p0 , the expected prot of
the seller is maximized for some p0ü satisfying the rst-order
(A.5)
where the second equality follows from L4—v1 5 D L4—p0 5
L4p0 —v1 5 and the third equality follows from (2) evaluated at
vi D p0 .
Because p0 D 4x5 and v1 D 4b1 5, L4p0 —v1 5 is also
equal to L44x5—4b1 55. The latter is denoted å4x—b1 5 and
can be expressed as a function of the observed bids
distribution. Specically, let Fy1 v1 4y1 v5 D Fy1 —v1 4y —v5fv1 4v5.
Because Fy1 v1 4t1 t5 D GB1 b1 4s4t51 s4t55 and fy1 1v1 4t1 t5 D
gB1 1b1 4s4t51 s4t55s 0 4t5, using a change of variable u D s4t5, we
obtain
å4x —b1 5 ² L44x5—4b1 55
´
³ Zb
1 g
B1 1b1 4u1 u5
D exp ƒ
du 0
x GB b 4u1 u5
1
1
v
Z vµ
Z
(A.6)
Combining (A.4), (A.5), and (A.6), and using B1 D s4y1 5, b1 D
s4v1 5, p0 D 4x5, x D s4p0 5 with s4¢5 increasing, (A.4) can be
written as in (7), as desired.
A.2 Proofs of Statistical Results
We make the following assumptions. Throughout we assume
that n ¶ 2.
Assumption A1. For each n ¶ 2, the private values vectors 4V1` 1 : : : 1 Vn` 51 ` D 11 21 : : : 1 are independently and identically distributed (iid) with joint density f 4v1 1 : : : 1 vn 5. The
latter has R ¶ 1 continuous partial derivatives on 6v1 v7n with
f 4v1 1 : : : 1 vn 5 ¶ c > 0 on 6v1 v7n .
Li, Perrigne, and Vuong: Semiparametric Estimation
61
Assumption A2.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
1. The kernel K4¢5 is symmetric with support 6ƒ11 C17 with
continuous
bounded rst and second derivatives.
R
2. K4u5du D 1.
3. K4¢5 is of order R C n ƒ 2 (R C 1 for the IPV case), that is,
moments of order strictly smaller than the given order vanish.
Note that Assumption A1 implies that the bids vectors
4B1` 1 : : : 1 Bn` 51 ` D 11 21 : : : 1 are iid. Note also that the
marginal density f 4¢5 ² fv1 4¢5 has R continuous derivatives on
6v1 v7 and that f 4¢5 ¶ c > 0 on 6v1 v7. Moreover, Assumption
A1 implies that v < ˆ so that b < ˆ.
The next lemma states the uniform consistency of the nonparametric es
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Semiparametric Estimation of the Optimal Reserve
Price in First-Price Auctions
Tong Li, Isabelle Perrigne & Quang Vuong
To cite this article: Tong Li, Isabelle Perrigne & Quang Vuong (2003) Semiparametric
Estimation of the Optimal Reserve Price in First-Price Auctions, Journal of Business & Economic
Statistics, 21:1, 53-64, DOI: 10.1198/073500102288618757
To link to this article: http://dx.doi.org/10.1198/073500102288618757
Published online: 01 Jan 2012.
Submit your article to this journal
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Citing articles: 1 View citing articles
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Download by: [Universitas Maritim Raja Ali Haji]
Date: 13 January 2016, At: 00:38
Semiparametric Estimation of the Optimal
Reserve Price in First-Price Auctions
Tong Li
Department of Economics, Indiana University, Bloomington, IN
(toli@indiana.edu)
47405-5301
Isabelle Perrigne and Quang Vuong
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
Department of Economics, University of Southern California, Los Angeles, CA
(perrigne@usc.edu, qvuong@usc.edu)
90089-0253
The optimal reserve price in the independent private value paradigm is generally expressed as a functional of the latent distribution of private signals, which is by nature unobserved. This feature has limited the implementation of the optimal reserve price in practice. In this article, we consider rst-price
auctions within the general afliated private values paradigm. We show that the seller’s expected prot
can be written as a functional of the observed bid distribution. We propose a semiparametric extremum
estimator for estimating consistently the optimal reserve price from observed bids. As an illustration,
we consider the Outer Continental Shelf (OCS) wildcat auctions.
KEY WORDS:
Afliated private value; OCS wildcat auction; Semiparametric extremum estimator.
Given the monopoly power of the seller in an auction, the
economic literature has focused on how the seller can extract
the largest revenue or prot. The problem of optimal auctions
is to nd the auction mechanism that provides the greatest
revenue or prot for the seller. This is of considerable practical
value because many commodities are sold through auctions.
Because any auction mechanism, such as rst-price, secondprice, 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 functional of the distribution of private values and its corresponding density function. See the work of Laffont 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 difculty in implementing the optimal reserve
price is the use of unobservables such as the latent distribution of private signals. Hence, very few empirical studies were
attempted to assess the optimal reserve prices using eld auction data. In particular, McAfee and Vincent (1992) gave an
approximation of the optimal reserve price for Outer Continental Shelf (OCS) auctions in the common value framework.
Paarsch (1997) proposed an estimate of the optimal reserve
price from English auction data relying on parametric assumptions on the distribution of private values. Moreover, to our
knowledge, the optimal reserve price has been obtained for
the IPV paradigm only. This can be restrictive given its independence assumption because there are auctions in which the
IPV paradigm does not apply.
In this article, we consider the afliated private values
(APV) paradigm, which includes the IPV paradigm as a special case, and we propose a direct method for estimating the
optimal reserve price. In particular, we characterize the optimal reserve price in rst-price sealed-bid auctions with sym-
metric and risk neutral bidders. The reserve price is optimal
in the sense that it generates the highest expected prot for
the seller in a rst-price auction. Thus we do not consider the
optimality of the auction mechanism because, for instance, a
second-price auction would generate a higher prot for the
seller; see Milgrom and Weber (1982). With risk averse bidders, the optimal auction mechanism is much more complex
and involves some transfer payments; see Maskin and Riley
(1984). Next, we show that the seller’s expected prot can
be written as a functional of the observed bid distribution.
This can be used to derive the optimal reserve price from
observed bids as the maximizer of this function. We then
propose a semiparametric extremum estimator for estimating
consistently the optimal reserve price from observed bids.
Specically, the estimated optimal reserve price is obtained as
a maximizer of an estimated prot function in which some
unknown distributions and densities have been nonparametrically estimated in a rst step. Our procedure is computationally simple because it does not require computation of the
equilibrium strategy. It does not require a parametric specication of the distribution of unobserved private values or
nonparametric estimation of the latter, as in our other work
(Li, Perrigne, and Vuong, 2002). We show that our estimator
is consistent. In addition to being more direct and computationally simpler, our estimator can converge at a faster rate
than the one based on the estimation of the underlying private
values distribution.
We illustrate our methodology with the U.S. gas lease auctions off the coast of Louisiana and Texas, which have been
studied in the literature. See Porter (1995) for a survey. We
focus on wildcat auctions between 1954 and 1969 because
bidders in these auctions can be considered as symmetric and
© 2003 American Statistical Association
Journal of Business & Economic Statistics
January 2003, Vol. 21, No. 1
DOI 10.1198/073500102288618757
53
54
Journal of Business & Economic Statistics, January 2003
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the crude oil market was relatively stable during that period.
It is widely recognized that the reserve price in these auctions
was too low; see, for example, the work of McAfee and
Vincent (1992). Our empirical ndings indicate that the optimal reserve price would (i) allow the federal government to
extract a larger proportion of oil companies’ willingness to
pay and (ii) generate a higher prot and higher revenues for
the federal government despite more tracts remaining unsold.
The article is organized as follows. Section 1 is devoted to
the derivation of the optimal reserve price in rst-price auctions using observed bids. Section 2 develops our semiparametric extremum estimator. Section 3 illustrates our method.
Section 4 concludes. The Appendix collects the proofs of our
propositions as well as their applications to the IPV case.
1.
1.1
OPTIMAL RESERVE PRICE IN
FIRST-PRICE AUCTIONS
The Private Value Model
Throughout the article we consider rst-price sealed-bid
auctions within the private value paradigm. Formally, a single and indivisible object is auctioned to n bidders who are
assumed to be risk neutral. All sealed bids are collected simultaneously. Provided that bid is at least as high as a reserve
price, the highest bidder wins the auction and pays the bid.
The utility of each buyer i D 11 : : : 1 n for the object is vi . The
vector 4v1 1 : : : 1 vn 5 is a realization of a random vector whose
n-dimensional cumulative distribution function is F 4¢5 with
support 6v1 v7n . This distribution is assumed to be afliated
as dened by Milgrom and Weber (1982). Roughly, afliation
means that large values for some of the components make the
other components more likely to be large than small. Each bidder knows his or her private value vi and the distribution F 4¢5
from which v1 1 : : : 1 vn are jointly drawn but does not know
other bidders’ private values. We restrict our study to the case
in which F 4¢5 is exchangeable in its n arguments. This model
is known as the symmetric afliated private value model. As
far as we know, despite its generality, the APV model has
seldom been studied in the literature, except in experimental studies by Kagel, Harstad, and Levin (1987), in empirical
studies by Li, Perrigne, and Vuong (2000), and in simulation
studies by Li et al. (2002).
At the Bayesian Nash equilibrium, bidder i chooses bid
bi to maximize E64vi ƒ bi 5 4Bi µ bi 5—vi 7, where Bi D s4yi 5,
yi D maxj6Di vj , and s4¢5 is the strictly increasing equilibrium
strategy. Assume rst that the reserve price is nonbinding. The
equilibrium strategy satises the rst-order differential equation
s 0 4vi 5 D 6vi ƒ s4vi 57fy1 —v1 4vi —vi 5=Fy1 —v1 4vi —vi 5
(1)
for all vi 2 6v1 v7 subject to the boundary condition s4v5 D
v, where Fy1 —v1 4¢—¢5 denotes the conditional distribution of y1
given v1 , fy1 —v1 4¢—¢5 denotes the corresponding density, and the
index 1 refers to any bidder among the n bidders because
all bidders are ex ante identical. From Milgrom and Weber
(1982), the explicit solution of (1) is
Z vi
L4—vi 5d1
bi D s4vi 5 D vi ƒ
(2)
v
where L4—vi 5 D exp6ƒ
R vi
fy1 —v1 4u—u5=Fy1 —v1 4u—u5du7.
We next introduce a binding reserve price p0 > v. The introduction of a reserve price acts as a screening device for participating in the auction. The screening level is a function of
p0 such that those bidders 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
Z vi
bi D s4vi 1 p0 5 D vi ƒ
L4—vi 5d for vi ¶ p0 0
(3)
p0
In particular, s4vi 5 given in (2) is identical to s4vi 1 v5.
1.2
The Optimal Reserve Price
The choice of the reserve price constitutes an important instrument with which the seller can take advantage of
monopoly power and increase prots 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 derived 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 p0ü that maximizes the expected prot for the seller satises
Rv
L4p0ü —u5Fy1 —v1 4u—u5fv1 4u5du
pü
ü
p0 D v0 C 0
(4)
Fy1 —v1 4p0ü —p0ü 5fv1 4p0ü 5
if v < p0ü < v, where v0 denotes the private value of the seller
for the auctioned object, fv1 4¢5 is the marginal density of vi ,
and Fy1 —v1 4¢—¢5 and L4¢—¢5 are as dened earlier. [Equation (4)
may have multiple roots. If this is the case, it is necessary
to evaluate the expected prot at each root to determine the
global maximum, as is usually advised in the theoretical literature.]
The condition v < pü0 < v arises from the characterization
of a maximum through its rst-order condition (4). It is not a
primitive condition and thus calls for some comments. First,
note that whatever a seller’s private value v0 , the seller does
not lose expected prot by only choosing a reserve price
belonging to the closed interval 6v1 v7. The intuition is quite
simple as any reserve price larger than v or smaller than v will
generate the same expected prot as when p0 D v and p0 D v,
respectively. The latter arises from the boundary condition
s4v1 p0 5 D v for any p0 µ v so that the equilibrium strategy is
identical to (2) in this case. Second, if the seller’s private value
v0 belongs to 6v1 v5, then the optimal reserve price p0ü must
belong to 4v1 v5 as required in Proposition 1: the seller will
always increase the prot by setting a reserve price slightly
larger than v because the derivative of the expected prot is
strictly positive in the neighborhood of this point. It follows
that v0 < p0ü < v for any v0 2 6v1 v5. Third, if v0 ¶ v, then pü0 D v
because the seller does not gain by setting a higher reserve
price. Fourth, if v0 < v, then p0ü D v or p0ü 2 4v1 v5 depending
on the underlying distribution of private values. In particular,
this implies that a nonbinding optimal reserve price, that is,
p0ü µ v, implies v0 < v.
Li, Perrigne, and Vuong: Semiparametric Estimation
55
As is well known, a special case of the APV paradigm is
the IPV paradigm rst considered by Vickrey (1961), wherein
private values are drawn independently from a common distribution F 4¢5. All other assumptions made previously for the
APV case remain valid. In the IPV paradigm, the optimal
reserve price has been widely studied (Laffont and Maskin
1980; Riley and Samuelson 1981). The optimal reserve price
p0ü in a rst-price auction solves
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p0ü D v0 C
1 ƒ F 4p0ü 5
f 4p0ü 5
(5)
if v < p0ü < v. It can be easily checked that (5) follows
from (4) by noting that L4—v5 and Fy1 —v1 4v—v5 reduce to
4F 45=F 4v55nƒ1 and F nƒ1 4v5, respectively, in the IPV case.
A remarkable property of p0ü 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 Fy1 —v1 4¢—¢5, fy1 —v1 4¢—¢5 and fv1 4¢5. [Levin and Smith
(1996) studied the optimal reserve price in the APV model
without giving its expression. An interesting question is the
effect 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.] 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 difculty was mentioned by McAfee and
Vincent (1992), Hendricks and Paarsch (1995), and Paarsch
(1997) among others. The structural econometric approach
developed by Paarsch (1992), Laffont, Ossard, and Vuong
(1995), and Guerre, Perrigne, and Vuong (2000) offers econometric methods for estimating the latent distribution in the IPV
model. More recently, Li et al. (2002) extended the structural
approach to the APV model by developing a two-step procedure for estimating the latent distribution nonparametrically.
See Perrigne and Vuong (1999) for a survey of recent developments. Although this procedure can be used to determine
the optimal reserve price, in this article we propose a more
direct and simpler method with superior statistical properties.
1.3
A Reformulation from Observed Bids
Assume that we observe bids from a rst-price sealed-bid
auction with a nonbinding reserve price. If observed bids are
coming from a rst-price auction with a reserve price, the function 4¢5 dened in the following becomes more involved; see
the IPV case of Guerre et al. (2000). Nonetheless, a similar
argument can be applied to obtain a characterization of the optimal reserve price from the observed bid distribution. As postulated in the structural approach, such bids are the equilibrium
bids of the corresponding game and hence are given by bi D
s4vi 5, where s4¢5 is dened by (2). Because private values are
random, bids are naturally random with an n-dimensional joint
distribution G4¢5. Moreover, because s4¢5 is strictly increasing, we have GB1 —b1 4X —x5 D Fy1 —v1 4s ƒ1 4X5—s ƒ1 4x55, where B1 D
maxj6D 1 bj D s4y1 5 and GB1 —b1 4¢—¢5 denotes the conditional distribution of B1 given b1 . It follows that the corresponding density
is given by gB1 —b1 4X —x5 D fy1 —v1 4s ƒ1 4X5 —s ƒ1 4x55=s 0 4s ƒ1 4X55.
Thus the differential equation (1) can be written as
v D bC
GB1 —b1 4b —b5
² 4b50
gB1 —b1 4b —b5
(6)
As noted by Li et al. (2002), the function 4¢5 is the inverse
bidding strategy s ƒ1 4¢5 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 bid distribution. Let GB1 1b1 4¢1 ¢5
be the joint distribution of 4B1 1 b1 5 with density gB1 1b1 4¢1 ¢5
and support 6b1 b72 . Let GB 1 b1 4¢1 ¢5 D ¡GB1 1b1 4¢1 ¢5=¡b1 D
GB 1 —b1 4¢—¢5gb1 4¢5.
Proposition 2. The optimal reserve price pü0 in a rstprice sealed-bid auction with n ¶ 2 bidders within the APV
paradigm can be written as p0ü D 4x0 5 for b µ x0 µ b, where
4¢5 is dened as (6) and x0 maximizes with respect to
x 2 6b1 b7 the expected prot
£
ç4x5 D E v0 4B1 µ x5 4b1 µ x5
C n6b1 C 44x5 ƒ x5å4x—b1 57
¤
4B1 µ b1 5 4b1 ¶ x5 1
(7)
Rt
where å4r —t5 D exp4ƒ r gB1 1b1 4u1 u5=GB1 b1 4u1 u5du5, E6¢7
denotes the expectation with respect to 4B1 1 b1 5, and 4¢5
denotes the indicator of the event in parentheses. [The third
term in (7) arises from adopting a game theoretic approach,
which takes into account the dependence of bidders’ behavior
on the reserve price. In contrast, a decision theoretic approach
would consider the rst two terms only in (7) and would give
p0ü D v0 .]
Note that if v0 2 6v1 v5, then x0 2 4b1 b5 D 4s4v51 s4v55
because x0 D s4p0ü 5 and v < pü0 < v as noted before. In contrast
to Proposition 1, which requires the knowledge of the underlying distributions Fy1 —v1 4¢—¢5 and fv1 4¢5 of private signals, the
main advantage of Proposition 2 is that it expresses the optimal reserve price in terms of the bid distribution, which can
be estimated directly from observed bids. This is the basis of
our direct procedure for estimating the optimal reserve price,
as presented in the next section.
2.
ECONOMETRIC IMPLEMENTATION
In agreement with Section 1.3, we consider L rst-price
auctions with a nonbinding reserve price and we assume that
bidders’ payoffs for each auction do not depend on bidders’
past actions to prevent dynamic considerations. As postulated
in the structural approach, observed bids are assumed to obey
the equilibrium strategy (2) within each auction. To simplify
the presentation, we also assume that auctions are homogeneous so that there is no need to control for heterogeneity of
the auctioned objects through some vector of characteristics.
Allowing for heterogeneity of the auctioned objects through
a vector z 2 òd of characteristics is straightforward. In this
case, our methods estimate the optimal price for an arbitrary
value z0 . For instance, if z D 4z1 1 : : : 1 zd 5 includes only continuous variables and z` denotes the
Q value of z for the `th
object, it sufces to insert the term dkD1 K44zk0 ƒ zk` 5=hz 5 in
(8) and (9).
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56
Journal of Business & Economic Statistics, January 2003
Let ` index the `th auction, and let n` be the number of
bidders in the `th auction with n` ¶ 2. The observed bids
are 8bi` 1 i D 11 : : : 1 n` 1 ` D 11 : : : 1 L9. This section proposes a
direct method for estimating the optimal reserve price p0ü in
rst-price sealed-bid auctions within the APV paradigm from
such observations without making parametric assumptions.
Our method relies on Proposition 2. Specically, as p0ü D
O xO 5, where 4¢5
O
4x0 5, a natural estimator of p0ü is pO 0ü D 4
0
O
and x0 are estimators of 4¢5 and x0 , respectively. Regarding 4¢5, which is the inverse of the equilibrium strategy
s4¢5, we follow Li et al. (2002). Regarding x0 , Proposition 2
indicates that, if the prot function ç4¢5 can be estimated
sufciently well, then x0 could be estimated by xO 0 , where
xO 0 maximizes the estimated prot function. This is the purpose of the second subsection. [Alternatively, we could use
the rst-order condition corresponding to the maximization
of (7) with respect to x. This would lead to semiparametric Generalized Method of Moments (GMM) estimation of
x0 (Newey 1994; Newey and MacFadden 1994; 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 4¢5, which can be estimated only at a slower rate
than 4¢5. This method is left for future research. Specically, when x0 2 4b1 b5, which occurs if v < pü0 < v, then x0
can be characterized by its rst-order condition 0 D E644x5 ƒ
v0 5GB1 b1 4x1 x5 ƒ 4B1 µ b1 5 4b1 ¶ x5 0 4x5å4x—b1 57. This
rst-order condition is obtained after some algebra by differentiating (7) with respect to x and using dGB1 1b1 4x1 x5=dx D
nGB1 b1 4x1 x5, GB1 —b1 4x—x5gb1 4x5 D GB 1 b1 4x1 x5, 4x5 ƒ x D
GB 1 b1 4x1 x5=gB1 1b1 4x1 x5, and (A.6).] Because the expected
prot ç4¢5 involves unknown functions, namely, GB1 b1 4¢1 ¢5
and gB1 1b1 4¢1 ¢5, this leads naturally to considering semiparametric M-estimation, in which these distributions and densities
are estimated nonparametrically in a rst step. Thus our semiparametric M-estimator corresponds more to the terminology
used by Andrews (1994) and Powell (1994) than to that used
by Newey and MacFadden (1994).
2.1
Nonparametric Estimation of 4¢5
Our rst step is to estimate the inverse equilibrium strategy
4¢5 from observed bids. Note that this function depends on
the number n of bidders. To emphasize such a dependence,
we denote it by 4¢—n5. Following Li et al. (2002) and Guerre
et al. (2000), we use kernel estimators, although any other
nonparametric estimators can be employed.
To estimate 4¢—n5, we consider only bids from auctions
with n bidders, namely, 8bi` 1 i D 11 : : : 1 n1 ` D 11 0 0 0 1 Ln 9,
where Ln is the number of such auctions. Noting that
GB 1 —b1 4¢1 ¢5=gB1 —b1 4¢1 ¢5 D GB1 b1 4¢1 ¢5=gB1 1b1 4¢1 ¢5, it follows from
(6) that the inverse equilibrium strategy 4¢—n5 can be
O —n5 D b C G
bB b 4b1 b5=gO B 1b 4b1 b5 for any
estimated by 4b
1 1
1
1
b 2 4b1 b5, where
´
³
Ln
n
b ƒ bi`
1X
1 X
b
1 (8)
4Bi` µ B5K
GB1 b1 4B1 b5 D
Ln hG `D1 n iD1
hG
´ ³
´
³
Ln
n
b ƒ bi`
B ƒ Bi`
1 X
1X
gO B1 1b1 4B1 b5 D
K
K
(9)
Ln h2g `D1 n iD1
hg
hg
for any value 4B1 b5 with hG and hg some smoothing parameters called bandwidths, and K4¢5 a weight function called
kernel satisfying Assumption A2 in the Appendix. Note that
the symmetry of bidders was used by averaging over i in (8)
and (9).
As shown by Li et al. (2002, prop. A1), if Assumption A1
in the Appendix holds, then GB1 b1 4¢1 ¢5 and gB1 1b1 4¢1 ¢5 have
both R C n ƒ 2 continuous bounded partial derivatives on any
compact subsets of 4b1 b52 , where R is the differentiability
order of the joint density f 4v1 1 : : : 1 vn 5 of private values. Let
hG D cG 4log L=L51=42RC2nƒ35 1
hg D cg 4log L=L51=42R C2nƒ25 1
(10)
where cG and cg are some constants. From Stone (1982), these
bandwidths are optimal in the sense of delivering the fastest
rates of uniform convergence for estimating GB1 b1 4¢1 ¢5,
gB1 1b1 4¢1 ¢5, and hence the inverse equilibrium strategy 4¢5. See
Lemma A.1 in the Appendix.
2.2
Semiparametric M-Estimation of x0
As shown in Proposition 2, x0 maximizes the expected prot
function ç4¢5. It is thus natural to estimate x0 by the value
that maximizes an estimated prot 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.
Proposition 2 provides an expression (7) for the expected
prot function ç4¢5. Specically, noting that 4x5 ƒ x D
GB 1 b1 4x1 x5=gB1 1b1 4x1 x5, (7) can be written as ç4x5 D
E6–4B1 1 b1 1 x1 gB 1 1b1 =GB1 b1 57, where –4¢5 is given by the term
in brackets in (7). This leads to considering the estimated
expected prot function by averaging over bids the function
–4¢5, where GB1 b1 and gB1 1b1 are replaced by their nonparametric estimates (8) and (9), namely,
L
n
1 Xn 1 X
O
b
b
O
D
–4B
ç4x5
i` 1 bi` 1 x1 gB1 1b1 =GB1 b1 5
Ln `D1 n iD1
L
n ³
1 Xn 1 X
D
v 4Bi` µ x5 4bi` µ x5
Ln `D1 n iD1 0
C nbi` 4Bi` µ bi` 5 4x µ bi` 5
bB b 4x1 x5
G
1
1
b
å4x —bi` 5 4Bi` µ bi` 5
gO B 1 1b1 4x1 x5
´
4x µ bi` µ bmax ƒ „5 1
(11)
Cn
Rt
bB b 4u1 u5du5, „ ²
å4x—t5 D exp4ƒ x gO B 1 1b1 4u1 u5=G
where b
1
1
max4hg 1 hG 5, and bmax is the maximum of the bids bi` ,
i D 11 0 0 0 1 n, ` D 11 0 0 0 1 Ln . The trimming associated with
the requirement bi` µ bmax ƒ „ arises from the fact that
GB 1 b1 4u1 u5, gB1 1b1 4u1 u5 and hence å4x—t5 are not well estimated uniformly if u and hence t are too close to the upper
boundary b. See Lemma A.1.
b
Dene xO 0n as maximizing the estimated expected prot ç4¢5
over 6bmin C „1 bmax ƒ „7, where bmin is the minimum of the
Li, Perrigne, and Vuong: Semiparametric Estimation
57
bids bi` , i D 11 0 0 0 1 n, ` D 11 0 0 0 1 Ln . More rigorously, xO 0n is
dened as any value in 6bmin C „1 bmax ƒ „7 that satises
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b xO 0n 5 ¶
ç4
sup
x26bmin C„1bmax ƒ„7
b
C o4151
ç4x5
(12)
b
where o415 converges to zero as Ln ! ˆ. Because ç4¢5
is
piecewise continuous in x, the term o415 ensures the existence
O
of xO 0n . [Because the statistical objective function 4¢5
is not
continuous, our estimator cannot be characterized by its rstorder conditions.]
Because x0n maximizes the expected prot ç4¢5, whereas
b
xO 0n essentially maximizes the estimated expected prot ç4¢5,
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.2. This lemma extends
Theorem 4.1.1 of Amemiya (1985), which is fundamental for
establishing the consistency of extremum estimators. Besides
allowing for a discontinuous statistical objective function,
our lemma allows for maximization and uniform consistency
over an expanding subset. [Andrews (1995) also considered
expanding subsets to trim out observations for which the estimated density is too close to zero. In contrast, our expanding subsets arise from trimming close to the boundaries of
the compact support of the bid distribution. Alternatively, we
could consider maximization of (12) over a xed compact subset 6a1 a7 included in 4b1 b5. The interval 4b1 b5 is, however,
unknown, although it can be estimated by 4bmin 1 bmax 5. More
important, x0n is not guaranteed to belong to the chosen xed
compact subset. Requiring x0n 2 6a1 a7 imposes some restrictions on the seller’s private value v0 and the underlying distribution F 4¢5.]
Proposition 3. Suppose that p0ü is the unique maximizer
of the seller’s expected prot over 6v1 v7 and that v < pü0 < v.
If Assumptions A1 and A2 hold with R ¶ n ¶ 2, then xO 0n is a
strongly consistent estimator of x0n as Ln ! ˆ.
ü
2
As noted earlier, there is no loss in requiring that p0n
6v1 v7. The uniqueness assumption is necessary for identication, whereas v < p0ü < v is satised if v µ v0 < v.
Combining Proposition 3 and the uniform consistent estimation of 4¢—n5 in Section 2.1 delivers a consistent estimator of the optimal reserve price pü0n . The additional subü
script n emphasizes that p0n
generally depends on n in the
APV paradigm. Specically, under the assumptions of Propoü D O O —
4x0n n5 is a strongly consistent estimator
sition 2, pO 0n
ü
ü
of p0n as Ln ! ˆ. The rate of convergence of pO 0n
is at
O
4RCnƒ25=42RC2nƒ25
—
best that of 4¢ n5, namely, Ln
from
Li
et
al.
p
(2002, prop. 2(i)). In particular, if x0n is Ln -consistent as in
Andrews (1994), our estimator pO ü0n would be consistent at the
rate L4RCnƒ25=42RC2nƒ25
.
n
As mentioned, an alternative method for estimating the
optimal reserve price is rst to estimate the APV model,
namely, the underlying joint density f 4¢1 : : : 1 ¢5, and then
to use (4), where fv1 4¢5 and fy1 —v1 4¢—¢5 are replaced by
their corresponding nonparametric estimates. From Li et al.
(2002, prop. 3), the marginal density estimator fOv1 4¢5
converges at the rate Ln4RCnƒ25=42RC2nC25 while the conditional density estimator fOy1 —v1 4¢—¢5 converges at the rate
2
Ln4RCnƒ25 =64RCnC1542RC2nƒ257 , which is strictly slower. The latter
consistency rate is established by following the proof of
Proposition 3 of Li et al. (2002) with a bandwidth rate of
the form c4log Ln =Ln 54RCnƒ25=64RCnC1542RC2nƒ257 when estimating the bivariate density fyv 4¢1 ¢5, which has R C n ƒ 2 continuous derivatives from Lemma A1(iii) in that article. Hence,
ü
the convergence rate of this alternative estimator of p0n
is at
2
best Ln4RCnƒ25 =64RCnC1542RC2nƒ257 , which is strictly slower than
.
the previous rate L4RCnƒ25=42RC2nC25
n
3.
AN ILLUSTRATION OF OCS
WILDCAT AUCTIONS
It was widely recognized by economists that the reserve
price used by the federal government in OCS auctions was too
low (Hendricks, Porter, and Spady 1989; McAfee and Vincent
1992; Hendricks, Porter, and Wilson 1994). As argued by
McAfee and Vincent (1992), however, a roadblock to applying auction theory and especially to determining the optimal
reserve price is the knowledge of unobservables. This section
proposes an estimate of the optimal reserve price in OCS wildcat auctions using our semiparametric estimator within the
private value paradigm. [The previous literature on OCS data
used the so-called mineral rights model or (pure) common
value (CV) model. In the APV model, afliation among private values can arise from an unobserved common component.
See Li et al. (2000) for the formalization of such a model.
The basic difference between an APV model and a common
value model is the specication of a bidder’s utility, which
is the bidder’s own private value when considering a private
value model and the unknown common component when considering a pure common value model. Consequently, the APV
model does not incorporate a winner’s curse, which is present
in the CV model. Although the APV model does not exclude
the existence of a possible common component through the
afliation of bidders’ private values, it assumes that each company’s utility is idiosyncratic and related to rms’ opportunity
costs and efciency. On the other hand, the CV model assumes
that each company’s utility is identical whatever the rm’s
opportunity costs and efciencies, which could be restrictive.
It is likely that the real situation is between these two polar
cases in that the utility of each bidder is a function of both
common and private components.]
3.1
Data
The U.S. government began auctioning its mineral rights
to offshore oil and gas off the Texas and Louisiana coasts in
1954. The auctioned tracts are usually of about 5,000 acres.
The bidders are oil companies. The auction is organized as a
rst-price sealed-bid auction with a reserve price of $15 per
acre held constant across auctions from 1954 through 1979.
As we focus 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 tends to be more symmetric in wildcat auctions than in drainage auctions. The latter were extensively studied by Hendricks and Porter; see Porter (1995) for
a survey. Moreover, we exclude from our study all auctions
held after 1970 because market conditions were unstable. The
dataset provides the number of bidders for each auction and
58
Journal of Business & Economic Statistics, January 2003
Table 1. Summary Statistics on Bids per Acre
No. bidders
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
n D2
No. auctions
Mean
STD
Min.
Max.
217
145.78
255.72
19.70
2220.28
the corresponding bids in 1972 dollars. As an illustration, we
consider auctions with two bidders only. We could consider
auctions with a higher number of bidders, but the number of
corresponding auctions Ln tends to reduce dramatically. The
summary statistics of bids per acre are given in Table 1 for
two bidders.
Table 1 indicates that there is a great variability in bids as
measured by the standard deviation and the range of bids; a
large proportion of bids falls between $50 and $200 per acre.
A test for independence among bids within auctions using the
Blum, Kiefer, and Rosenblatt (1961) nonparametric test gives
a test statistic equal to 6.00 corresponding to a p-value equal
to .14%. This constitutes a clear rejection of independence and
hence of the IPV model. This leads us to consider the APV
model.
As a rst approximation, we assume that the current reserve
price is nonbinding and hence does not act as an effective
screening device to participating to the auction because the
reserve price of $15 per acre is much lower than the average
bid. Moreover, very few bids are clustered around the actual
reserve price of $15. Specically, less than .25% of bids are
in the interval 6151 207 and less than 1% are in the interval
6151 307. Hence, the actual reserve price can be viewed as nonbinding. Second, the federal government sometimes rejected
the highest bid in some auctions although that bid was larger
than the announced reserve price. Hence, the actual reserve
price can be viewed as random. For wildcat auctions between
1954 and 1969, we observe a 1.8% rejection rate for the 217
auctions analyzed here. We can reasonably consider that the
random reserve price does not have much effect on bidding
strategies.
Third, we assume that there is no dynamic consideration
in these auctions. This can be justied by the fact that the
different sales were spread widely over the 1954–1969 period,
reducing dynamic linkages across sales. Within each sale, we
follow the earlier literature, which neglects oil companies’
capacity constraints given their large sizes; see Porter (1995).
Fourth, we have to address the possible heterogeneity across
tracts because this affects the value of the optimal reserve
price. To assess possible heterogeneity across tracts, we followed Porter (1995) and regress the log of bids on a complete set of tract-specic dummies. Using the full dataset of
1,147 auctions, we obtain a weak rejection of tract homogeneity based on an F -test. Because the F -test requires independence among bids, this statistic tends to overreject the null
assumption of homogeneity because bids are not independent
in the presence of afliation. Moreover, a similar regression
controlling for the number of bidders shows that the F -value
drops signicantly. To assess further tract heterogeneity given
a number of bidders, we conduct separate regressions for each
size of bidders. We nd that tract 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 is automatically
satised because our econometric analysis is conducted separately for each number of bidders.
3.2
Practical Issues
We assume that the underlying joint density f 4v1 1 v2 5 is
twice continuously differentiable, that is, R D 2. To implement our kernel estimators (8) and (9), we need to choose
a kernel function. The triweight kernel has a compact support and two continuous derivatives. It is dened as K4u5 D
435=32541 ƒ u2 53 4—u— µ 15 and is of order 2 in agreement
with Assumption A2. For n D 2, Equation (10) gives the bandwidths hG D cG Lƒ1=5 and hg D cg Lƒ 1=6 , where cG and cg are
some constants. These two constants are obtained by the socalled rule of thumb. Specically, cG D cg D 20978 1006‘O b ,
where ‘O b is the empirical standard deviation of bids, and the
factor 2.978 is the correction for the triweight kernel; see
Hardle (1991).
For inference purposes, it is useful to provide condence
O xO 5, where xO maximizes the estimated
intervals for pO 0ü D 4
0
0
expected prot (11). Because asymptotic distributions of estimators in structural models are frequently difcult to establish,
bootstrap methods are commonly used as an alternative for
constructing approximate condence intervals. See Horowitz
(in press) for a survey on bootstrap methods. Our bootstrap
procedure is as follows. We resample the original data for
auctions with two bidders to obtain 500 datasets, each of size
equal to 434 bids. For each resampled dataset, we apply our
semiparametric estimator and thus obtain 500 different values
for pO 0ü . To construct a 90% condence interval, we use the
Efron (1979) method relying on percentiles. This method does
not impose any symmetry restriction.
3.3
Empirical Results
Any determination of the optimal reserve price requires the
knowledge of the value of the auctioned object v0 for the
seller. On the other hand, this valuation must be nonnegative.
Moreover, because the seller may receive negative prots if
the reserve price is set below the seller’s value v0 , the seller is
always better off setting a reserve price above v0 . Hence we
can infer that the federal government valuation is between $0
and $15 per acre. In the following, we provide an estimate of
the optimal reserve price for both values $0 and $15 as well
as for bootstrap condence intervals. The results are given in
Table 2.
The bootstrap condence intervals are relatively tight. It
is interesting to compare our results with the approximation
found by McAfee and Vincent (1992) when considering a
pure common value model with endogenous entry and stochastic participation of bidders. McAfee and Vincent found that
Table 2. Optimal Reserve Price
Seller’s valuation
v0 D 0
v 0 D 15
pü0
Con’dence interval
$272.83
$290.07
[$2180711 $322033]
[$2240801 $336046]
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
Li, Perrigne, and Vuong: Semiparametric Estimation
the reserve price should be approximately $600 in 1992 dollars, that is, about 40-fold the current level. To be comparable, the results in Table 2, which are in 1972 dollars, need
to be divided by .339. For two bidders we nd that the optimal reserve price should be $805 with a condence interval
6$6451 $9517 when v0 D 0. This discrepancy can be explained
by the absence of correction for the winner’s curse in a
private value framework: when considering a pure common
value model, bidders have to adjust their bids downward relative to their private estimates because of the winner’s curse.
This can explain why we obtain somewhat greater optimal
reserve prices. Both approaches agree, however, that the actual
reserve price used by the federal government is far from being
optimal.
The choice of the reserve price is an important instrument with which the federal government can take advantage
of its monopoly power in the auction as a unique seller. A
higher reserve price induces rms to bid more aggressively
but increases the probability of the auctioned tract not being
sold. The optimal reserve price balances these two opposite
effects and maximizes the seller’s expected gain. Thus, in
terms of economic policy conclusions, simulating the rstprice sealed-bid auctions with the optimal reserve price is of
great interest for assessing the potential gain for the federal
government. Using the estimator proposed by Li et al. (2002),
we can recover the bidders’ private values by using the equality v D 4b5 given in (6) for 174 auctions as well as the
underlying distribution of private values. Because nonparametric methods are used, some auctions had to be trimmed out
because of boundary effects.
For the 174 auctions for which we are able to recover
bidders’ private values, we simulate the oil companies’ bids
following (3) when facing our estimated optimal reserve
prices. As expected, rms bid more aggressively whenever
their private values are above the reserve price, but some tracts
remain unsold because of the higher reserve price. Table 3
gives the percentage of informational rents, noted IR, left to
the winning oil companies for sold tracts. The informational
rent in percentage is computed as the ratio of the winning
bidder’s private value minus the bid divided by the bidder’s
private value. We perform the simulation 500 times using
resampled data to obtain condence intervals.
We observe that about half of the tracts remain unsold.
Moreover, the informational rents decrease signicantly from
the average informational rent under the actual reserve price,
which is 62%. The latter gure is computed from the observed
winning bids and the estimated private values for the 174 auctions. Hence, implementing the optimal reserve price would
allow the federal government to be much more successful in
capturing the willingness to pay of the oil companies.
Another issue of great interest is to compare revenues
and prots between the two mechanisms. Table 4 gives the
59
Table 4. Revenues in Dollars for an Average Sold Tract
v0
Actual
mechanism
Optimal
mechanism
Con’dence
interval
0
15
$5141184092
$5141184092
$112671528001
$114411549060
[$1108514980101 $115681295060]
[$1110913380701 $116011344060]
revenues from an average sold tract (in terms of acreage) and
Table 5 gives the prots from an average sold tract for each
mechanism. Again, we provide revenues and prots with the
optimal reserve price when the federal government has a private value per acre equal to $0 or $15. Revenues are computed
as the average winning bid times the average acreage of a
tract, whereas prots are computed as the average winning bid
minus the value of the tract for the federal government times
the average acreage of a tract. We also provide 90% bootstrap
condence intervals.
It is important to note that revenues and prots for an average sold tract both signicantly increase as expected. On the
other hand, as noted, half of the tracts become unsold under
the optimal reserve price. Nevertheless, if we multiply the
gures in Tables 4 and 5 by the number of sold tracts, the
overall prot and revenue for the federal government is much
higher than those from the actual mechanism. For instance, if
we assume that the value of a tract for the federal government
is $15 per acre, the optimal reserve price would allow prots
of around 50% larger than actual prots although more tracts
remain unsold.
4.
CONCLUSION
A well-known difculty in implementing optimal reserve
prices on eld data is that the latent distributions are unknown
to the analyst. In this article we circumvent such a difculty
by showing that the seller’s expected prot can be expressed
as a functional of the observed bids distribution. Hence, the
seller’s expected prot function can be identied and estimated from observed bids. It follows that a natural estimator
of the optimal reserve price can be obtained by maximizing
the estimated prot function. This leads us to propose a semiparametric extremum estimator of the optimal reserve price,
which is shown to be strongly consistent. In addition to being
more direct and computationally simpler, this estimator can
converge at a faster rate than can one relying on the estimation of the underlying distribution of private values.
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 bid distribution. A possible extension of our
method is to specify a parametric family for the distribution of
observed bids. Second, as mentioned, an alternative semiparametric procedure would be to use the rst-order conditions
Table 5. Pro’ts in Dollars for an Average Sold Tract
Table 3. Average Informational Rents for the Optimal Reserve Price
v0
0
15
No. sold tracts Con’dence interval
90
88
[851 107]
[771 103]
IR
Con’dence interval
v0
Actual
mechanism
Optimal
mechanism
Con’dence
interval
43.87%
39.31%
[35033%1 45031%]
[37024%1 45082%]
0
15
$5141184092
$4441756021
$112671528001
$113721968080
[$1108514980101 $115681295060]
[$1104017570901 $115321763090]
60
Journal of Business & Economic Statistics, January 2003
characterizing the optimal reserve price to develop a semiparametric GMM estimator. Third, because our result relies on
the identication of the model from which bids are generated,
it would be interesting to consider bids arising from a rstprice auction with a binding reserve price. Guerre et al. (2000)
solved such an issue within the IPV paradigm, and Hendricks,
Pinkse, and Porter (2000) considered the CV model. Fourth,
as is the case for many auction situations, such as procurement
auctions, our illustration involves a public seller. Because of
its public nature, it may be possible that the seller has objectives other than prot maximization. Although of great interest, this issue requires more theoretical research.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
ACKNOWLEDGMENTS
The authors are grateful to K. Hendricks and R. Porter
for providing the data analyzed in this article as well as to
M. Pesendorfer, the editor, and two referees for helpful and
constructive comments. This article was presented at the Far
Eastern Meeting of the Econometric Society in Singapore,
June 1999, and at the Winter Meeting of the Econometric
Society in Boston, January 2000. Financial support from the
National Science Foundation under grant SES-0001663 to the
rst author and grant SBR-9631212 to the third author is
gratefully acknowledged.
condition
nFy1 —v1 4p0ü —p0ü 5fv1 4p0ü 54v0 ƒ p0ü 5
Zv
C n L4p0ü —v1 5Fy1 —v1 4v1 —v1 5fv1 4v1 5 dv1 D 01
p ü0
because dF 4x1 : : : 1 x5=dx D nFy1 —v1 4x —x5fv1 4x5. Equation (4)
follows.
Proof of Proposition 2. Combining (3), (A.1), (A.2), and
(A.3), and using y1 D maxj6D1 vj , the expected prot for the
seller ç4p0 5 is equal to
ç4p0 5 D v0 E6 4y1 µ p0 5 4v1 µ p0 57
Zv
Cn
4v1 ¶ p0 5s4v1 1 p0 5
v
³Z
v
v
´
4y1 µ v1 5fy1 —v1 4y1 —v1 5 dy1 fv1 4v1 5 dv1
D E6v0 4y1 µ p0 5 4v1 µ p0 5
C ns4v1 1 p0 5 4y1 µ v1 5 4v1 ¶ p0 570
(A.4)
Now, for p0 and v1 such that v µ p0 µ v1 µ v, we have from
(2) and (3)
Z p0
L4—v1 5 d
s4v1 1 p0 5 D s4v1 5 C
v
D s4v1 5 C L4p0 —v1 5
APPENDIX
A.1 Proofs of Theoretical Results
Proof of Proposition 1. Using an argument similar to that
by Riley and Samuelson (1981), we maximize the expected
prot for the seller. For any bidder, say, Bidder 1, the expected
payment if he or she wins the auction given v1 ¶ p0 is
p4v1 5 D s4v1 1 p0 5 Pr 4v2 µ v1 1 : : : 1 vn µ v1 —v1 5
D s4v1 1 p0 5Fy1 —v1 4v1 —v1 51
(A.1)
where s4v1 1 p0 5 is given by (3). Therefore, using (3), the
expected revenue for the seller from Bidder 1 is
R1 4p0 5 D
D
Z
p0
p4v1 5f 4v1 5 dv1
p0
v1 ƒ
Z
v1
p0
L4—v1 5 d
¶
Fy1 —v1 4v1 —v1 5fv1 4v1 5 dv1 0
(A.2)
The total expected prot ç 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
ç4p0 5 D v0 F 4p0 1 : : : 1 p0 5 C nR1 4p0 50
p0
v
L4—p0 5 d
D s4v1 5 C L4p0 —v1 54p0 ƒ s4p0 551
(A.3)
Differentiating ç4p0 5 with respect to p0 , the expected prot of
the seller is maximized for some p0ü satisfying the rst-order
(A.5)
where the second equality follows from L4—v1 5 D L4—p0 5
L4p0 —v1 5 and the third equality follows from (2) evaluated at
vi D p0 .
Because p0 D 4x5 and v1 D 4b1 5, L4p0 —v1 5 is also
equal to L44x5—4b1 55. The latter is denoted å4x—b1 5 and
can be expressed as a function of the observed bids
distribution. Specically, let Fy1 v1 4y1 v5 D Fy1 —v1 4y —v5fv1 4v5.
Because Fy1 v1 4t1 t5 D GB1 b1 4s4t51 s4t55 and fy1 1v1 4t1 t5 D
gB1 1b1 4s4t51 s4t55s 0 4t5, using a change of variable u D s4t5, we
obtain
å4x —b1 5 ² L44x5—4b1 55
´
³ Zb
1 g
B1 1b1 4u1 u5
D exp ƒ
du 0
x GB b 4u1 u5
1
1
v
Z vµ
Z
(A.6)
Combining (A.4), (A.5), and (A.6), and using B1 D s4y1 5, b1 D
s4v1 5, p0 D 4x5, x D s4p0 5 with s4¢5 increasing, (A.4) can be
written as in (7), as desired.
A.2 Proofs of Statistical Results
We make the following assumptions. Throughout we assume
that n ¶ 2.
Assumption A1. For each n ¶ 2, the private values vectors 4V1` 1 : : : 1 Vn` 51 ` D 11 21 : : : 1 are independently and identically distributed (iid) with joint density f 4v1 1 : : : 1 vn 5. The
latter has R ¶ 1 continuous partial derivatives on 6v1 v7n with
f 4v1 1 : : : 1 vn 5 ¶ c > 0 on 6v1 v7n .
Li, Perrigne, and Vuong: Semiparametric Estimation
61
Assumption A2.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
1. The kernel K4¢5 is symmetric with support 6ƒ11 C17 with
continuous
bounded rst and second derivatives.
R
2. K4u5du D 1.
3. K4¢5 is of order R C n ƒ 2 (R C 1 for the IPV case), that is,
moments of order strictly smaller than the given order vanish.
Note that Assumption A1 implies that the bids vectors
4B1` 1 : : : 1 Bn` 51 ` D 11 21 : : : 1 are iid. Note also that the
marginal density f 4¢5 ² fv1 4¢5 has R continuous derivatives on
6v1 v7 and that f 4¢5 ¶ c > 0 on 6v1 v7. Moreover, Assumption
A1 implies that v < ˆ so that b < ˆ.
The next lemma states the uniform consistency of the nonparametric es