selective entry in first-price auctionsselective entry in first-price auctions ... ∗we thank ken...
TRANSCRIPT
Selective Entry in First-Price Auctions
Artyom Shneyerov and Pai Xu∗
University of British Columbia
October 2006;Revised November 29, 2006
Abstract
We develop a model of selective entry in first-price auctions. The model nests theenvironment of Levin and Smith (1994) but also allows for selective entry. We derivetestable restrictions of the model and develop a nonparametric test of selective entry.We also derive sharp nonparametric bounds on the CDF of bidders’ valuations. Weimplement the test on a dataset of highway procurement auctions in Oklahoma andfind some evidence of selective entry.
1 Introduction
A robust feature of many real-world auctions is that not all potential bidders choose tosubmit a bid. A plausible explanation is that entry is costly. In this paper we argue thatcostly entry creates incentives for firms to enter only if they have sufficiently favorableestimates of their values, which may lead to selectivity effects. If the model is misspecified,these selectivity effects may result in biased estimates of key structural parameters, suchas the distribution of bidders’ valuations and the entry cost.
Auctions with costly entry have received considerable attention in the theoretical lit-erature. The seminal paper for most of the empirical work is Levin and Smith (1994). Intheir model, potential bidders are initially uninformed about their true valuations of thegood, but may become informed and submit a bid at a cost k > 0. Levin and Smith (1994)characterize a unique symmetric equilibrium in mixed strategies. The entrants randomizetheir entry decisions and earn zero expected profit.
Samuelson (1985) considers a model whose information environment is polar to that ofLevin and Smith (1994): bidders make their entry decisions after they have learned theirvaluations. The entry cost k is interpreted solely as the cost of preparing a bid. Biddersenter provided that their valuations exceed a cutoff sN that is increasing in the numberof potential bidders N , implying that the set of entrants is a selected sample, biased moretowards avid bidders as N gets bigger. We are not aware of any papers applying this
∗We thank Ken Hendricks, Vadim Marmer, Mike Peters and Larry Samuelson for useful comments, aswell as participants of the CEA meeting in Montreal (2006) and the Summer Theory Workshop at UBC(2006). We thank SSHRC for financial support made available through grants 12R27261 and 12R27788.
1
model to data, a possible reason being that in Samuelson (1985) there is no room for costlyinformation acquisition.
The empirical literature on auctions with entry is quite recent. Hendricks, Pinkse, andPorter (2003), HPP hereafter, estimate a model of bidding for off-shore oil. The model ofentry that they outline occupies a middle ground between the polar cases of Levin and Smith(1994) and Samuelson (1985). As Levin and Smith (1994), costly information acquisitionis part of the model. Potential bidders are the firms that have commissioned a costly area-wide survey. As in Samuelson (1985), selective entry is also part of the model. The actualbidder are those for whom the information contained in the area-wide survey is sufficientlyfavorable to undertake a tract-specific survey. The focus of Hendricks, Pinkse, and Porter(2003) is however not on selective entry but on testing the equilibrium model of bidding;the entry model is in the background.
Our model has private values and brings the entry process outlined in HPP to theforeground. It inherits the realistic feature of selective entry, but dispenses with the commonvalue component.1 The private values assumption simplifies the model considerably, andenables us to make progress in a number of directions. As in HPP, there are N potentialbidders each of whom receives some private information (signal) Si about their values Vibefore they decide to submit a bid. A bidder may learn Vi and submit a bid only afterincurring a cost k > 0. We allow Si to be informative about Vi, as captured by the ”goodnews” property of Milgrom (1981). We show that in equilibrium, entry occurs if the signalSi is above a cutoff sN . If in addition k is sufficiently large so that entry occurs withprobability strictly between 0 and 1, we say that equilibrium has selective entry.
With selective entry, we show that the model is not non-parametrically identified.2
Moreover, if the model is misspecified as random entry, as in the Levin and Smith case, theestimated distribution of valuations will be biased downwards relative to the true marginaldistribution. This is of course the kind of selectivity bias that one should expect: the biddersthat have chosen to enter on average have higher values than those who have not entered.Following the recent literature on partial identification (Manski (2003)), we construct sharpinformative bounds on the marginal distribution of bidders’ valuations.3 If variation in Nis exogenous, these bounds can be further refined.
We next show that the estimate of the entry cost is also upward biased, and thatthis bias can potentially be large. The rough intuition here is that the entry cost underselective entry is equal to the expected profit of the marginal entrant, i.e. the bidder who isindifferent between entering or not. This marginal entrant has expected profit smaller thanthe average entrant. When misspecifying the model to have random entry, one substitutesthe expected profit of the average entrant for the expected profit of the marginal entrant,thus overestimating the entry cost.
Given these important consequences, the question arises whether selective entry is
1The private values assumption is realistic in many applications, for example in procurement. See forexample Bajari and Ye (2003), Pesendorfer and Jofre-Bonet (2003), Krasnokutskaya (2003), Li and Zheng(2005), Krasnokutskaya and Seim (2006), Bajari, Hong, and Ryan (2004). One important exception is Hongand Shum (2002), who find some evidence of common values for certain types of projects.
2 By identification we mean a standard notion of nonparametric identification. See Athey and Haile(2002), Athey and Haile (2005).
3The approach of putting bounding the distribution also appears in Haile and Tamer (2003) who consideran incomplete econometric model of bidding in English auctions.
2
testable. Even broader, regardless of the selectivity of entry, can the model itself be tested?One easily testable prediction is that our model implies that the entry probabilities pNdecline in N provided that variation in N is exogenous. (We implement this test using ourdata.)
We build on the insight in Haile, Hong, and Shum (2003) and propose to use exogenousvariation in N to test for selective entry. We nest the Levin and Smith’s environment byallowing the signal Si to be uninformative about Vi. Our test relies on monotonicity of thecutoff sN : larger values of N imply lower expected profits and therefore a more favorablesignal is needed in order for a bidder to enter the auction. We look at the distribution ofvaluations conditional on entry. Following the approach of Guerre, Perrigne, and Vuong(2000) (GPV hereafter) we show that this distribution can be non-parametrically identifiedfrom data if N and all bids in each auction are observed. We show if Si and Vi areindependent, then the set of entrants is a random sample and consequently the distributionof entrants’ valuations is the same for all N . If, on the other hand, Si is informativeabout Vi, then the set of entrants is a selected sample biased more towards higher valuesas N increases. This implies that the distributions of bidders’ valuations conditional on Nare ordered in the sense of first-order stochastic dominance. This enables us to develop anonparametric test of selective entry in the spirit of Haile, Hong, and Shum (2003).
Although our approach shares with Haile, Hong, and Shum (2003) the basic idea thatexogenous variation in the number of bidders can be used for testing the information en-vironment of the game, it is different. Haile, Hong, and Shum (2003) consider a differentmodel in which bidders’ valuations may have a common component. They propose a testfor a common values based on the variation in the number of actual bidders, while we testfor selective entry using the variation in the number of potential bidders. Our approach isalso different in its implementation: we use the estimation framework recently proposed inMarmer and Shneyerov (2006) that allows for arbitrary form of dependence on covariates.45
In a Monte-Carlo study of the bounds and the test we find that the bias in the estimationof the entry cost can be substantial if the model is misspecified as random entry. If variationin N is exogenous, the identified bounds can be tight in large samples, but less informativein smaller samples. But even in moderate samples, the approach to the inference on F (v)based on the bounds appears to be superior to the approach that ignores the selectivitybias. We find that the test of selective entry performs reasonably well both with respect tosize and power when several quantiles are considered simultaneously.
A recent empirical literature has focussed on Levin and Smith’s model or its variants.Bajari and Hortacsu (2003) study bidding behavior in eBay auctions, assuming commonvalues. A Bayesian estimation method is implemented on a dataset of mint and proof setsof US coins. They estimate the entry cost, simulate the expected seller revenues underdifferent reserve prices and measure the extent of the winner’s curse. Athey, Levin, andSeira (2004) estimate a model of bidding in timber auctions with costly entry. An importantpoint of departure from the Levin and Smith’s model is the focus on asymmetric equilibrium
4Haile, Hong, and Shum (2003) show how to incorporate covariates in a number of parametric specifica-tions. It is not known how this approach could be extended to a model with entry like ours.
5On the other hand, Haile, Hong, and Shum (2003) consider tests of the Kolmogorov-Smirnov varietythat may be more powerful than the tests proposed here. We have not succeeded in developing such testsfor our model.
3
in which mills enter with probability one, and make positive profit, while loggers randomizebetween entering or not, and make zero profit.6 The model is estimated on a sample ofsealed-bid auctions, and its predictions are compared to a control sample of ascending-bidauction. They find that the estimated model can explain (counterfactually) several featuresof bidding behavior in ascending-bid auctions.
Two recent empirical papers that have addressed entry in auctions are Li and Zheng(2005) and Krasnokutskaya and Seim (2006).7 Li and Zheng (2005) study entry and biddingfor lawn mowing contracts using a variant of the Levin and Smith model. It is the firstpaper in the literature that utilizes the number of planholder as a measure of potentialcompetition. This is an important advance relative to earlier studies that had to relyon coarser measures. Li and Zheng (2005) propose and implement a Bayesian estimationmethod. Krasnokutskaya and Seim study bid preference programs and bidder participationusing California data. Their paper also uses the Levin and Smith model, but as in Athey,Levin, and Seira (2004), the focus is on asymmetric equilibria.
In our empirical application, we test for selective entry using a dataset of auctionsconducted by the Oklahoma Department of Transportation. In addition to all winningand losing bids and certain project characteristics, we also observe the number of eligiblebidders, a variable that can serve as a reasonable proxy for the number of potential bidders.This feature makes our nonparametric tests empirically feasible. The test of selective entryhas a p-value of 0.082, providing some empirical support for selective entry in OklahomaHighway Procurement auctions.
2 The model and testable implications
The model has an entry stage and a bidding stage. The game begins with the entry stage. Npotential risk-neutral bidders obtain preliminary estimates (signals) Si of their true valuesVi; it is assumed that this information is available to them for free. Upon observing Si, abidder may spend k dollars to learn its true value Vi and enter the auction; for simplicity, kis assumed to be the same for everyone. Only the bidders that have learned Vi are eligibleto submit a bid in the auction; these bidders will be called active bidders, or sometimesactual bidders.
We assume that the pairs (Vi, Si) are identically and independently distributed acrossbidders and are drawn from distribution F (v, s) with density f (v, s), positive on [v, v] ×[0, 1]. For convenience, we assume that the marginal distribution of the signals is uni-form on [0, 1]. Since the informational content of signals is preserved under a monotonetransformation, this is without loss of generality.
The entry stage is followed by the bidding stage, which is a standard first-price auction.Active bidders draw their values Vi, and then simultaneously and independently submitsealed bids. Active bidders do not know the number of active bidders, only the number ofpotential bidders N . The good is awarded to the highest bidder who pays its bid.
6Their theoretical model assumes that the entry cost is private information of potential bidders, but inthe estimated model a fixed entry cost is assumed. They argue that in timber auctions, the heterogeneityin the entry cost is expected to be small.
7In addition, Li (2005) develops a general parametric approach to the estimation of auctions with (ran-domized) entry.
4
We consider the following two alternative assumptions about the joint density f (v, s).
Assumption 1 (Uninformative Signals) Si is not informative about Vi: (Vi, Si) areindependent: ∀ (v, s) ∈ [v, v]× [0, 1], f (v, s) = fV (v), where fV (·) is the marginal densityof Vi.
Assumption 2 (Informative Signals) Si is informative about Vi: ∀ (v, s) ∈ [v, v]×[0, 1],s < s0 ⇐⇒ F (v|s) > F (v|s0).
The equilibrium under Assumption 1 is completely analyzed in Levin and Smith (1994).Their equilibrium characterization can be summarized as follows.8 In a symmetric equi-librium, each firm enters with probability pN ∈ [0, 1], and makes the profit in the biddinggame that can be shown to be equal
ΠLSN (p) ≡Z v
vF (v)
¡1− pF (v)
¢N−1dv.
where F (v) ≡ 1 − F (v). If the cost of entry k ≤ ΠLSN (1), then all N potential biddersenter with probability pN = 1. If, on the other hand, the cost of entry is moderate so thatΠLSN (1) < k < ΠLSN (0), then entry occurs with probability pN ∈ (0, 1). Finally, there is alsoan uninteresting case of a very large entry cost, k ≥ ΠLSN (0), so that none enters: pN = 0.
Assumption 2 is equivalent to requiring that distributions F (v|s) are ordered in the senseof strict first-order stochastic dominance. This means that Si is ”good news” about Vi, inthe sense of Milgrom (1981). In this environment, a symmetric equilibrium is characterizedby a signal cutoff sN such that only the bidders for whom the news are sufficiently good,Si ≥ sN , enter. Since the bidders who have entered know their true values Vi, the biddinggame is essentially a first-price auction with a random number of bidders. The distributionof the number of actual bidders is binomial B (N, pN) with entry probability
pN = 1− sN
(recall that the signals are normalized to have a uniform [0, 1] distribution).In the auction, the equilibrium winning probability against a potential bidder with value
v is 1−pN F ∗ (v|N), where F ∗ (·|N) is the distribution of the rival bidder’s value conditionalon entry,
F ∗ (v|N) = F (v|Si ≥ sN)
=1
1− sN
Z 1
sN
F (v|s) ds,
and we use ”-” to denote upper cdfs, i.e. F ∗ (v|N) = 1−F ∗ (v|N). Since entry and biddingdecisions are independent, it follows by the envelope theorem that the expected profit of
a bidder with value v is equal toR v0
¡1− pN F∗ (v|N)
¢N−1dv. The expected profit at the
entry stage is, after some simple algebra,
ΠN (sN ) ≡Z v
vF (v|sN )
¡1− (1− sN ) F
∗ (v|N)¢N−1 dv.8See for example the discussion in Milgrom (2004).
5
where we use the notation F (v|s) = 1− F (v|Si ≥ s).By standard envelope theorem arguments (e.g., Milgrom (2004)), the bidding strategy
of actual bidders is given by
BN (v) = v − 1¡1− (1− sN) F ∗ (v|N)
¢N−1 Z v
v
¡1− (1− sN) F
∗ (v|N)¢N−1 dv. (1)
As in the Levin and Smith model, the form of equilibrium depends on the magnitudeof the entry cost k. If k is very small, k ≤ ΠN (0), then all N potential bidders enter; ifk is moderate, ΠN (0) < k < ΠN (1), then each potential bidder enters with probabilitypN = 1− sN ∈ (0, 1), where sN solves the equationZ v
vF (v|sN )
¡1− (1− sN) F
∗ (v|N)¢N−1 dv = k, (2)
and if k is very large, k ≥ ΠN (1) =R vv F (v|1) dv, then none enters, pN = 0. Our maintained
assumption will be that k is not that large, i.e. k < ΠN (1).Now suppose that initially, for N = 2, the entry cost is small so that p2 = 1 (i.e.,
sN = 0.) As N increases there will be a change of the entry regime. This change will occurwhen N crosses the threshold
N∗ ≡ max {N : pN = 1} .
For N ≥ N∗, sN begins to increase so that the probability of entry falls from 1; the cutoffsN solves entry equation (2).
It is not too hard to check that the same cutoff interpretation continues to hold even ifthe signals are uninformative, i.e. under Assumption 1. This way the Levin and Smith modelis nested within ours: we purify their mixed-strategy equilibrium (their entry probabilitiespN ) with a signal cutoff sN = 1−pN . The next Proposition gives a summary of these facts.
Proposition 1 For a given number of potential bidders N , a symmetric equilibrium ischaracterized by a cutoff sN such that a bidder with signal Si > sN enters and doesn’t enterotherwise. The equilibrium may either have the feature that firms enter with probabilitypN = 1 (i.e. sN = 0), or the feature that firms enter with probability pN ∈ (0, 1) (i.e.,sN ∈ (0, 1)). In both types of equilibria, the bidding strategy is given by equation (1). Thereexists a threshold N∗ such that pN = 1 for N ≤ N∗ and pN ∈ (0, 1) for N > N∗. ForN ≥ N∗, the cutoff sN is increasing in N and pN is decreasing in N .
3 Testable implications of selective entry
Assume that the number of potential bidders N is bounded, from below by N and fromabove by N > N . We next define what we mean by selective entry.
Definition 2 The equilibrium has selective entry if the signals are informative, i.e. As-sumption 2 holds, and the entry cost is moderate so that N∗ < N . Otherwise, we say thatthe equilibrium has non-selective entry.
6
In other words, for entry to be selective it is not sufficient that the signals are infor-mative; the entry cost must be small enough so that selective entry actually occurs inequilibrium for some N < N . The basis of our test is the observation from the previoussection that sN is increasing in N for N ≥ N∗. If the entry is selective, it must then be thecase that the entering bidders have, on average, higher valuations that the non-entrants,and progressively more so as N increased beyond the threshold N∗. If, on the other hand,entry is non-selective, then the entrants have the same distribution of valuations for allN ≤ N .
We assume that the econometrician can observe all the bids and therefore also thenumber of active bidders n. An important additional information that is assumed to bealso available is the number of potential bidders N . In other words, we assume that the datagenerating process identifies pN and G∗ (·|N) where pN = E [n|N ] /N is the probability ofentry and G∗ (b|N) is the distribution of entrants’ bids conditional on N .
Bidders’ valuations are not directly observable, but can be recovered from the first-orderconditions following the approach of GPV. Consider first-order equilibrium conditions ofthe bidding game. A bidder with value v who submits a bid b has a probability of winningover a given rival equal to 1 − pN G
∗ (b|N), i.e. it is complementary to the probabilitythat the rival submits a bid times the probability that this bid is above b. Since thereare N − 1 identical rivals, it follows by independence that the probability of winning is¡1− pNG
∗ (b|N)¢N−1, and the expected profit isΠ (b, v) = (b− v)
¡1− pNG
∗ (b|N)¢N−1 .Writing out the first-order condition, i.e. taking the derivative of eΠ (b, v) with respect to band setting it equal to 0, gives the formula for the inverse bidding strategy
ξN (b) = b+1− pNG
∗ (b|N)(N − 1) pNg∗ (b|N) . (3)
The reader will notice that ξN is identified from the observables, and that ξN is an increasingfunction so that there is an inverse ξ−1N .
Because ξN is identified and is increasing, the distribution of entrants’ valuationsF ∗ (v|N) is also identified:
F ∗ (v|N) = G∗¡ξ−1N (b) |N¢ .
Another important implication of the above is that the joint distribution F (v, s) is notidentified. This is because even if the signals are informative about the values, we caninterpret the distribution F ∗ (v|N) as the marginal distribution of values in the world withuninformative signals, so that the joint distribution of bids would be F (v, s) = sF ∗ (v|N),different from the true distribution.
Moreover, if the econometrician wrongly assumed random entry in the environment withselective entry, he would bias the estimate of the valuations downwards, F ∗ (v|N) < F (v).In fact, the only information about the true value of F (v) is that it is within the bound:
F (v) ∈ [F ∗ (v|N) , F ∗ (v|N) pN + 1− pN ] . (4)
7
To see why, follow Manski (2003) and write the law of total probability
F (v) = F ∗ (v|N) Pr [Si ≥ sN |N ] + F (v|Si < sN) Pr [Si < sN |N ]= F ∗ (v|N) pN + F (v|Si < sN) (1− pN) .
Under selective entry, F ∗ (v|N) < F (v|Si < sN), implying the lower bound in (4). Also, thequantity F (v|Si < sN ) can be anything between 0 and F (v|Si < sN) = 1, so in particular0 is a conceivable possibility. This implies the upper bound in (4). This argument alsoimplies that the bounds F ∗ (v|N) and F ∗ (v|N) pN + 1− pN are sharp.
A similar systematic bias would also be present in the estimate of the entry cost; itwould be biased upwards. To see why, let’s once again abstract from sampling issues andassume that the econometrician can directly observe pN and G∗ (·|N). In the world withuninformative signals in which bidders enter with probability between 0 and 1, he woulduse F ∗ (v|N) in place of F (v|sN) to estimate the entry cost using formula (2),
kN =
Z v
vF ∗ (v|N) ¡1− (1− sN ) F
∗ (v|N)¢N−1 dv. (5)
But since the true equilibrium has selective entry,
F ∗ (v|N) =1
1− sN
Z 1
sN
F (v|s) ds
>1
1− sN
Z 1
sN
F (v|sN) ds= F (v|sN) ,
and the true value of k is smaller than the estimated value k:
k =
Z v
vF (v|sN)
¡1− (1− sN) F
∗ (v|N)¢N−1 dv<
Z v
vF ∗ (v|N) ¡1− (1− sN) F
∗ (v|N)¢N−1 dv= kN .
As the next Proposition shows, the bias can potentially be very severe.
Proposition 3 Assume selective entry and a fixed N . An ε > 0 exists such that anyk ∈ (0, ε] is compatible with the identified quantities pN and G∗ (·|N).
Proof. The proof is constructive. Pick any distribution F (v|s) that satisfies Assump-tion 2. Let
F 0ε (v|s) =½0, s ≥ s∗ε (v)1, s < s∗ε (v)
(6)
where s∗ε (v) is to be determined, pick a small ε > 0 and consider the distribution
Fε (v|s) = (1− ε)F 0ε (v|s) + εF (v|s) .
8
(see Figure 1). Note that Fε (v|s) satisfies our ”good news” Assumption 2. For this distri-bution, we have
1
1− sN
Z 1
sN
Fε (v|s) ds = (1− ε)s∗ε (v)− sN1− sN
+ εF ∗ (v) , (7)
where
F ∗ (v|N) =Z 1
sN
F (v|s) ds.
Solving equation (7) for s∗ε (v) yields
s∗ε (v) = sN +1− sN1− ε
³F ∗ (v|N)− εF ∗ (v|N)
´= 1− pN +
pN1− ε
³F ∗ (v|N)− εF ∗ (v|N)
´Note that, if ε > 0 is sufficiently small, d
dv
hF ∗ (v|N)− εF ∗ (v|N)
i≥ 0, while for all ε,
s∗ε (0) = sN(= 1 − pN) and s∗ε (1) = 1. It follows that for all sufficiently small ε > 0,s∗ε (v) ∈ [0, 1], so the distribution in (6) is well-defined. By our construction, Fε (v|s)depends on the identified quantities pN and F ∗ (v|N). If we pick the entry cost to be
kε =
Z v
vFε (v|sN)
µZ sN
0Fε (v|s) ds
¶N−1dv
then in equilibrium, the distribution of values conditional on entry will be F ∗ (v|N), asrequired. Now note that Fε (v|sN ) = εF (v|sN )→ 0 as ε→ 0, so the entry cost kε → 0.¥
In the Proof of the Proposition above, we need to relax the assumption that the dis-tribution F (v|s) has a density: the distributions Fε (v|s) have atoms. A more elaborateargument would show that Fε (v|s) can be ”smoothed” to a distribution with a density,but we omit the tedious argument that would be necessary to show this. Even then, theform of Fε (v|s) would still be quite special and may not be empirically plausible. Butthe Proposition above does show that, in the ”worst-case” scenario, the potential bias inestimating the entry cost can be large if the model is misspecified as random entry. In theMonte-Carlo section, we show that the bias can be large also in an empirically plausiblespecification.
We show next that exogenous variation in the number of potential bidders can narrowthe bounds for F (v) and can be used to test for selective entry.
Assumption 3 (Exogenous variation in N) The tuples (Vi, Si) have the same distri-bution for all N ∈ N ≡ ©N, ...N
ª.
Under this assumption, we can refine the range of possible values for F (v) by takingthe intersection of the bounds across N :
F (v) ∈\N∈N
[F ∗ (v|N) , F ∗ (v|N) pN + 1− pN ] (8)
9
(this again follows Manski (2003)). Note that the lower bound on F (v|N) is equal to
F l (v) = maxN∈N
F ∗ (v|N) , (9)
while the upper bound is
Fu (v) = minN∈N
(F ∗ (v|N) pN + 1− pN) . (10)
Similarly, we can refine the upper bound for k: k ≤ minN∈N kN .We now show how the variation in N can be used to test for selective entry. Consider
the null hypothesis of non-selective entry. Under null, F ∗ (v|N) is the same as the marginaldistribution F (v), so it follows immediately that
F ∗ (v|N) = ...F ∗ (v|N∗) = F ∗ (v|N∗ + 1) ... = F ∗¡v|N¢ .
Under the alternative hypothesis of selective entry, the distribution F ∗ (v|N) = F (v|Si ≥ sN )is decreasing as a function of sN , while sN is increasing in N for N > N∗ by Proposition 1.It follows that under the alternative, the distributions F ∗ (v|N) are strictly ordered in thesense of first-order stochastic dominance for N ≥ N∗:
F ∗ (v|N) = ... = F ∗ (v|N∗) > F ∗ (v|N∗ + 1) > ... > F ∗¡v|N¢ , ∀v ∈ [v, v] . (11)
These orderings provide a basis for our test of selective entry.Our econometric implementation of such a test requires that we re-formulate the stochas-
tic dominance relation in terms of quantiles of the distributions F (v). Define
Q∗ (τ |N) ≡ F ∗−1 (τ |N)
to be the τ -th quantile of the distribution of entrants’ valuations. From equation (3) forthe inverse bidding strategy, it is easy to see that the (unobservable) value distributionquantiles are related to the (observable) bids distribution quantiles
q∗ (τ |N) ≡ G∗−1 (b|N) ,
via the formula
Q∗ (τ |N) = q∗ (τ |N) + 1− pN (1− τ)
(N − 1)pNg∗(q∗ (τ |N) |N) . (12)
Under the null of non-selective entry,
H0 : Q∗ (τ |N) = ... = Q∗
¡τ |N¢ , ∀τ ∈ [0, 1], (13)
while under the alternative of selective entry,
H1 : Q∗ (τ |N) = ... = Q∗ (τ |N∗) < Q∗ (τ |N∗ + 1) < ... < Q∗
¡τ |N¢ , ∀τ ∈ [0, 1], (14)
for some N∗ ∈©N + 1, ...N
ª(note that quantiles must be ordered in the reverse order
relative to the cdfs in (11)). For each τ ∈ [0, 1], the inequalities under the alternative H1
10
form an open set. Since our test will be based on a maximum of a certain log-likelihoodstatistic, it is necessary to take a closure of these sets, which results in the alternative H1,
H1 : Q∗ (τ |N) ≤ ... ≤ Q∗
¡τ |N¢ , ∀τ ∈ [0, 1]. (15)
We now show how our approach can be extended to auctions with a reserve price. (Thisextension is necessary in view of our empirical application.) Suppose therefore that thereis a binding reserve price r > 0. Its presence creates an additional level of selectivity: thedecision not to submit a bid may be either because a bidder has not entered, or becausea bidder has entered but drawn a valuation Vi < r. Our approach extends neverthelesswith virtually no changes. The probability pN of submitting a bid is now interpreted asthe probability of participation, p∗N = Pr [Si ≥ sN , Vi ≥ r]. From the viewpoint of bidderi, his winning against rival j will occur either if j has not submitted a bid, or submitted abid below b. The probability of this event is 1 − p∗N + p∗NG (b|N), exactly as before. Thislogic leads to the identification of the distribution of valuations conditional on submittinga bid, i.e. to the identification of the distribution
F ∗∗ (v|N) = Pr (Vi ≤ v|Si ≥ sN , Vi ≥ r) .
Our bounds on F (v) transfer without any changes, and the testable implications alsotransfer if we are willing to strengthen our good news assumption (Assumption 2) byrequiring that Pr (Vi ≤ v|Si = s, Vi ≥ r) also satisfy the first-order stochastic dominancecondition.
4 Econometric implementation of the test
We assume that a sample of L auctions is available, and index each auction by l = 1, .., L. Ineach auction l, we observe the set of (anonymous) potential bidders indexed by i = 1, ..., Nl,where Nl is the number of potential bidders. Each auction is characterized by a covariatesvector xl ∈ X . Corresponding to each potential bidder there is a latent pair of signalsand valuations (sil, vil). The model provides a structural link between the ”potential” bidsbil of all potential bidders and their valuations vil: bil = B (vil|Nl, xl). The model alsoprovides a link between the entry decisions yil ∈ {0, 1} and signals: yil = 1 if and only ifsil ≥ s (Nl, xl), where s (Nl, xl) is the cutoff in auction l. Of course, the ”bids” of thosewho have not entered remain latent; only the entrants’ bids bil are observable:
bil =
⎧⎨⎩ bil, if yil = 1
not observable, if yil = 0.
We make the following assumption about the data generating process.
Assumption 4 (a) {(Nl, xl) : l = 1, . . . , l} are i.i.d.(b) The marginal PDF of xl, ϕ, is strictly positive, continuous and bounded away from zero
on its compact support X ⊂ Rd, and admits at least R = 2 continuous and boundedpartial derivatives on Interior (X ).
11
(c) The distribution of Nl conditional on xl, π (N |x) , has support N =©N, ..., N
ªfor all
x ∈ X , N ≥ 2.(d) (vil, sil) and Nl are independent conditional on xl.
(e) {(vil, sil) : i = 1, . . .Nl; l = 1, . . . , L} are i.i.d. conditional on (Nl, xl)
(f) For all x ∈ X , the density of valuations f (·|x) is strictly positive and bounded awayfrom zero on its support, a compact interval [v (x) , v (x)] ⊂ R+, and admits up to Rcontinuous and bounded derivatives its interior.
(g) F (·|·)and π (N |·) admit at least R = 2 continuous bounded partial derivatives onInterior(X ) for all v ∈ [v (x) , v (x)] and all N ∈ N respectively.
(h) The density of entrants’ bids conditional on (N,x), g∗ (b|N,x), and the entry probabilityconditional on (N,x), p (N,x), admit at least R = 2 continuous partial derivativesbounded away from 0 on an open subset X † ∈ Interior(X ) for all b in the interior ofthe support
£b (N,x) , b (N,x)
¤and all N ∈ N respectively.
Assumption (d) asserts that the number of potential bidders N is now exogenous con-ditional on X = x. The hypotheses become: under the null of non-selective entry,
Q∗ (τ |N,x) = .. = Q∗¡τ |N , x
¢, ∀τ ∈ [0, 1], x ∈ X
while under the alternative of selective entry, an N∗ exists such that
Q∗ (τ |N,x) ≤ .. ≤ Q∗¡τ |N , x
¢, ∀τ ∈ [0, 1], x ∈ X
Denote as p (N,x) the probability of entry conditional on x. Denote as q∗ (τ |N,x) theconditional quantile of bids:
q∗ (τ |N,x) = G∗−1 (τ |N,x)
≡ inf {b : G∗ (b|N,x) ≥ τ} .In order to implement our test, we need a consistent and asymptotically normal estimatorsof value quantiles Q∗ (τ |N,x). With covariates, (12) takes the form
Q∗ (τ |N,x) = q∗ (τ |N,x) +1− p (N,x) (1− τ)
(N − 1)p (N,x) g∗(q∗ (τ |N,x) |x) . (16)
Our approach to the estimation of Q∗ (τ |N,x) follows Marmer and Shneyerov (2006).9 Wefirst estimate g∗ (b|N,x), q∗ (τ |N,x) and p (N,x), and then insert them into equation (16).We use a kernel estimation method. We make the following assumption.
Assumption 5 The kernel K has up to R continuous and bounded derivatives on R,compactly supported on [−1, 1] and is of order R: R K (x) dx = 1,
RxjK (x) dx = 0 for
j = 1, ..., R− 1.9Marmer and Shneyerov (2006) consider a setting that corresponds to our ”uninformative signal” case,
i.e. the Levin and Smith model. Our model shares the sam crucial i.i.d. assumption on the valuations vi ,implying that all their asymptotic results in Section 5 are also valid in our more general setting.
12
Denote the number of actual bidders in auction l as nl =PNl
i=1 yil. The estimators ofg∗ and G∗ proposed in Marmer and Shneyerov (2006) are:
g∗ (b|N,x) =
PLl=1
PNli=1 yil1 {Nl = N}K
³bil−bh
´Qdk=1K
¡xk−xkl
h
¢p (N,x) π (N |x) ϕL (x)hd+1NL
, (17)
G∗ (b|N,x) =
PLl=1
PNli=1 yil1 {Nl = N} 1 (bil ≤ b)
Qdk=1K
¡xk−xkl
h
¢p (N,x) π (N |x) ϕL (x)hdNL
, (18)
where ϕL (x) and p (N,x), are estimated as standard nonparametric regressions,
ϕL (x) =1
hdL
LXl=1
Qdk=1K
µxk − xkl
h
¶,
p (N,x) =1
ϕL (x) π (N |x)hdLLXl=1
nl1 {Nl = N}Qdk=1K
µxk − xkl
h
¶,
and
π (N |x) = 1
ϕL (x)hdL
LXl=1
1 {Nl = N}Qdk=1K
µxk − xkl
h
¶To understand the intuition for these estimators, consider the function
g0 (b|N,x) =1
hd+1L
LXl=1
NlXi=1
yil1 {Nl = N}Kµbil − b
h
¶Qdk=1K
µxk − xkl
h
¶that appears on the right-hand side of the estimator g∗ (b|N,x). Following Marmer andShneyerov (2006) (see also Newey (1994)), g0 (b|N,x) is a consistent estimator of the func-tion
N · g∗ (b|N,x)E [yi1 {Nl = N} |x]ϕ (x) .Since
E [yi1 {Nl = N} |x] = Pr [yi = 1, Nl = N |x]= p (N,x)π (N |x) ,
this function reduces to
N · g∗ (b|N,x) p (N,x)π (N |x)ϕ (x) .
It follows that
g∗ (b|N,x) =g0 (b|N,x)
Np (x)π (N |x)ϕ (x) ,
which corresponds exactly to our estimator (17). The intuition for the estimator G∗ (b|N,x)is similar.
13
The estimator of bid quantiles q∗ (τ |N,x) is given by
q∗ (τ |N,x) = infnb : G (b|N,x) ≥ τ
o.
The quantiles Q∗ (τ |N,x) (16) can be consistently estimated by plugging in the estima-tors g∗, q∗ and p:
Q∗ (τ |N,x) = q∗ (τ) +1− p (N,x) (1− τ)
(N − 1)p (N,x) g∗(q∗ (τ |N,x) |x) .
In order to estimate the bounds F l (v|x) and F u (v|x), we need an estimator for F ∗ (v|N,x).Following the approach of Marmer and Shneyerov (2006), a consistent estimator of F ∗ (v|N,x)can be obtained in the following manner. Pick τ0 sufficiently far from 0 and 1, for example,τ0 = 1/2. We define a monotone version of the estimator Q
∗ as follows:
Q† (τ |N,x) =
(supt∈[τ0,τ ] Q
∗ (t|N,x) , τ0 ≤ τ < 1,
inft∈[τ,τ0] Q∗ (t|N,x) , 0 ≤ τ < τ0.
The estimator of the conditional CDF of the valuations conditional on entry is then givenby
F ∗ (v|N,x) = supτ∈(0,1)
nτ : Q† (τ |N,x) ≤ v
o.
The bounds F l (v|x) and F u (v|x) can then be consistently estimated by plugging in theestimator F ∗ (v|N,x) in their definitions:
F l (v|x) = maxN∈N
F ∗ (v|N,x) ,
Fu (v|x) = minN∈N
np (N,x) F ∗ (v|N,x) + 1− p (N,x)
o.
Theorem 4 in Marmer and Shneyerov (2006) implies that the estimators π, ϕ, g, q∗, Q∗, Q†
and F ∗ (v|N,x) as well as F l and F u are consistent with an appropriately chosen bandwidthsequence hL. (Theorem 4 in Marmer and Shneyerov (2006) also gives convergence rates ofthese estimators.)
We now turn to the implementation of the test. For this, we need asymptotic normal-ity of the quantile estimator Q∗ (τ |N,x). Applying delta-method to (16) one obtains thefollowing Taylor expansion (see Marmer and Shneyerov (2006), proof of Theorem 4):
Q∗ (τ |N,x)−Q∗ (τ |N,x) = − 1− p (N,x) (1− τ)
(N − 1)p (N,x) g∗2(q∗ (τ |N,x) |N,x)
× [g∗ (q∗ (τ |N,x))− g∗ (q∗ (τ |N,x))] + op
µ³Lhd+1
´− 12
¶.
(only the slowest-convergent term, the one that corresponds to the density g, has to beincluded). Following the same arguments as the proof of Lemma 2 in Marmer and Shneyerov(2006), the estimator g∗ (b|N,x) is asymptotically normal with variance
Vg (N, b, x) =g∗ (b|N,x)
Np (x,N)π (N |x)ϕ (x)µZ
K (x)2 dx
¶d+1
.
14
This leads to asymptotic normality of the estimator Q∗ (τ |N,x). Specifically, the followingresult follows from Theorem 4 in Marmer and Shneyerov (2006).
Proposition 4 Suppose that τ ∈ (0, 1) and x ∈ X †. Assume that the bandwidth h satisfiesas L→∞: Lhd+1 →∞ and
√Lhd+1h2 → 0. Then
√Lhd+1
³Q∗ (τ |N,x)−Q∗ (τ |N,x)
´→d N (0, VQ (N, τ, x)). (19)
where the asymptotic variance VQ (N, τ, x) is given by
VQ (N, τ, x) =
µ1− p (N,x) (1− τ)
(N − 1)p (N,x) g∗2(q∗ (τ |N,x) |N,x)
¶2Vg (N, q∗ (τ |N,x) , x) . (20)
Moreover, the estimators Q∗ (τ |N,x) are asymptotically independent for any distinct N,N 0 ∈©N, ...N
ª, any distinct τ , τ 0 ∈ Υ and any distinct x, x0 ∈ X †. The asymptotic variance
VQ (N, τ, x) can be consistently estimated by the plug-in method.
Our test of selective entry is based on a finite set of quantiles Υ ⊂ (0, 1), and a finiteset of covariates X ∗ ⊂ X †.The test is of the likelihood-ratio form. Consider the objective
Ξ (y) = Lhd+1XN∈N
Xτ∈Υ
Xx∈X∗
1
VQ (N, τ, x)
³Q∗ (τ |N,x)− yN,τ,x
´2,
for y = (yN,τ,x)N∈N ,τ∈Υ,x∈X∗ . Because the quantile estimators Q∗ (τ |N,x) are (asymptoti-
cally) normal, the maximum of log-likelihood under null is given by the statistic
T0 = −minyΞ (y) subject to. yN,τ,x = ... = yN,τ ,x (∀τ ∈ Υ,∀x ∈ X ∗)
and under alternative, by the statistic
T1 = −minyΞ (y) subject to yN,τ,x ≤ ... ≤ yN,τ ,x (∀τ ∈ Υ,∀x ∈ X ∗)
We test for selective entry using the log-likelihood ratio statistic
T = T1 − T0.
The asymptotic distribution of the test statistic T under null is not available in closed form,but can be computed using Monte-Carlo simulations.
5 Monte-Carlo Experiment
We specify the joint distribution of signals Si and valuations Vi to be bivariate normal,with the common mean μ = 1.8, common standard deviation σ = 0.25, and a correlationcoefficient ρ. To impose a bounded support, we truncate the distribution at the quantiles10−4 and 0.999. The entry cost is specified to be k = 0.17. The number of potential biddersis N = 2, ..., 6, distributed with equal probabilities π (N) = 1/5.
15
To begin, we explore the biases in estimation of a misspecified model (random entry).The true data generating process has ρ = 0.5. The misspecified model has ρ = 0. Considerfirst the case of a very large dataset in which the bounds can be estimated (almost) pre-cisely. In other words, we simply use equation (5) from Section 3 to compute kN from themisspecified model. The results are presented in Figure 2. The estimated value kN is thelargest, kN = 0.258, when N = 2 and declines with N , to the value of 0.204 when N = 6.The bias is quite substantial; in percentage terms, it is around 52% of the true value of kwhen N = 2, declining to 32% when N = 6. Of course, in practice the econometrician whouses the misspecified model would need to aggregate his estimates kN across N. Assumingthat he uses empirical frequencies πN for this purpose, he would estimate the entry cost as
k =PN
N=N πN kN = 0.222, again a substantial bias in the order of 30% of the true valuek = 0.17.
We now turn to the bias in the estimation of the distribution of bidders’ valuationsF (v). Once again, consider first the case of a very large sample so that the sampling errorcan be assumed away. In other words, we use the formulas (9) for the lower bound F l (v)and (10) for the upper bound F u (v). The bounds are presented graphically in Figure 3.Observe how tightly they surround the true curve F (v). Observe also how substantial thebias would be if the econometrician misspecified the model to have random entry.
We implement nonparametric kernel estimators with the twilight kernel functionK (u) =
(35/32)¡1− u2
¢31 {|u| ≤ 1}. The choice of the bandwidth sequences follows that in the
Monte-Carlo experiment reported in GPV: h = 1.06Vb (N) (π (N)NL)−15 , where Vb (N) is
the estimated standard deviation of bids in the subset of the sample with N potential bid-ders. Figure 4 presents the results of a Monte-Carlo experiment using a sample of L = 250auctions. It is apparent that this sample size is too small to learn much from the bounds.The estimated bound do not surround the true curve for most values of v, and deviatesubstantially. Also shown is the 95% band, which does surround the true curve but is verywide. Situation improves with larger samples (Figures 5 and 6). In a sample of L = 1000auctions, the estimated bound is tight and surrounds the true curve for values that are nottoo large. The benefits of using the bounds approach relative to the naive estimator of themisspecified model are obvious. The naive estimator would be the average of F ∗ (v|N) usingthe empirical frequencies πN : F (v) =
PNN=N πN F
∗ (v|N). In the graph, we abstract fromsampling variation and only show the exact the misspecified curve. When the sample sizeis increased to L = 5000 auctions, the main features remain; the only discernible differenceis that the 95% confidence band is more narrow.
Observe that this estimator is much closer to the true curve than the estimator from themisspecified random-entry model, Figure 4 also shows confidence regions for the bounds(given by the dotted curves), computed by the bootstrap based on 1000 resamples. Theconfidence regions are rather wide for those values of v that occur with a high probability.Figure 5 shows that the estimated bounds are correctly ordered, F u (v) > F l (v), and arequite informative, when the sample size is increased to 5000 auctions.
Table 2 shows the performance of our test. The experiment is performed with differentvalues of ρ: 0.1, 0.3, 0.5 and 0.7, and two different quantile sets, Υ1 = {0.5} and Υ2 ={0.3, 0.5, 0.7}. We construct tests with nominal sizes 5% and 10%, computing the criticalvalues under null in the following manner. We generate a M = 10, 000 samples of L = 500
16
auctions, and for each compute the value of statistic T (m) = T(m)1 − T
(m)0 under null.
Specifically, for each m = 1, ...,M we compute the variances V(m)Q (N, τ) using formula
(20), and then compute T(m)0 and T
(m)1 as
T(m)0 = −min
y
XN∈N
Xτ∈Υ
Lhd+1
V(m)Q (N, τ)
³n(m) (τ ,N)− yN,τ
´2
subject to yN,τ = ... = yN,τ (∀τ ∈ Υ),
and
T(m)1 = −min
y
XN∈N
Xτ∈Υ
Lhd+1
V(m)Q (N, τ)
³n(m) (τ ,N)− yN,τ
´2
subject to yN,τ ≤ ... ≤ yN,τ (∀τ ∈ Υ),
where y = (yN,τ )N∈N ,τ∈Υ and n(m) (τ ,N) are independent draws from normal distributions
with mean 0 and variance V(m)Q (N, τ). (The assumption that mean is 0 is without loss of
generality because any constant would cancel when the minimizations are performed.) Thesampling distribution of the statistic T (m) allows us to find the critical values cα of the testunder null at the level of significance α.
There are several findings. First, with a single quantile (the median), the size of the testis distorted. Size properties improve drastically once we include more quantiles, i.e. withthe set of quantiles Υ2. The power of the test increases when it is based on more quantiles,and/or when ρ is increased. For example, we reject 99% of the time when ρ is 0.7 and thetest is based on Υ2. When ρ is small (0.1), the test rejects 37-46% of the time if Υ1 is usedand 55-64% if Υ2 is used. Overall, the results with several quantiles are encouraging.
6 Empirical application
Our dataset consists of 547 auctions for surface paving and grading contracts let by Okla-homa Department of Transportation (ODOT) for the period of January, 2002 to December,2005.10 The available data items include all bids, the engineer’s estimate, the time lengthof the contract (in days), the number of items in the proposal and the length of the road.The ODOT implements a policy under which all bids over 7% of the engineer’s estimate aretypically rejected, so there is a binding reserve price. In reality, we do observe bids abovethe reserve price (although extremely few winning bids were above the reserve price). Wetreat these bids as non-serious and eliminate them from the sample.
10Our choice of surface paving and grading contracts is motivated by the fact that Hong and Shum (2002),in their study of highway procurement auctions in New Jersey, find little support for common values for thistype of contracts. This is important because in this paper, we assume independent private values (costs).
17
Importantly, we observe the list of eligible bidders (planholders) for each auction. Thelist of planholders is published on the ODOT website prior to bidding. A sample listof planholders is exhibited in the Appendix. A firm becomes a planholder through thefollowing process. All projects to be auctioned are advertised by the ODOT 4 to 10 weeksprior to the letting date. These advertisements include a brief summary of the project,including the general location of the work and the type of the work involved. A sampleadvertisement page is exhibited in the Appendix; one can see that the information in theadvertisements is quite imprecise. It only contains the location of the project, the numberof calendar days and sometimes the quality of asphalt to be used, but lacks the mostimportant information: the engineer’s estimate and the detailed schedule of items.
Interested companies can then submit a request for plans and bidding proposals, thedocuments that contain the specifics of the project (in particular, the engineer’s estimateand items schedule). An important feature of the qualification process is that only eligiblefirms are allowed access to these documents. A firm is deemed eligible if it satisfies certainqualification requirements. The goal of the qualification process is to ensure that a potentialbidder has sufficient expertise and capacity to undertake the project. While the expertisepart is typically determined at the pre-qualification stage, the capacity part is typicallyproject-specific. An important requirement is that the prospective bidder is not qualifiedfor more than 212 times its current working capital. Moreover, the prospective biddersare not permitted to bid on individual projects that in total exceed this working capitalrequirement.
The number of planholders is the variable that will be our proxy for the number ofpotential bidders N . Our view is that exogenous variation in the capacity relative to theproject size is what explains the variation inN conditional on the project size, this variationoccurs for reasons largely independent of the specifics of a given project, as captured inthe signal Si. Such a signal is obtained by the planholder from the initial inspection ofthe bidding documents. For this reason, we assume that the number of planholders N isstatistically independent from Si, conditionally on the observables.
Another variable deserving special mention is Nitems, the number of pay items in theproject advertisement. (The pay items are the various inputs needed for the constructionprocess.) On average, the projects have about 70 pay items, although even twice thisnumber is not uncommon- the standard deviation is also about 70 items. The reasonwhy we included this variable is because the number of pay items can affect the cost ofinformation acquisition, and therefore the entry cost k. One would expect that projects thathave more pay items would be more difficult to evaluate, because a potential bidder wouldneed to search for more prices. The theoretical prediction therefore is that the probabilityof entry is decreasing in Nitems.
The results of the entry logit regression and OLS bidding regression are presented inTable 3. A number of explanatory variables are included; their description is given in Table2.11 In the logit regression, EngEst is not significant, only N and Nitems are significant.They have signs that are consistent with the predictions of our model. The effect of addingone more potential bidder is to reduce the odds of submitting a bid by about 4%. Increasing
11The covariates are basically the same as in other papers on procurement auctions (e.g. Bajari and Ye(2003); Pesendorfer and Jofre-Bonet (2003); Krasnokutskaya (2003); Krasnokutskaya and Seim (2006); Liand Zheng (2005)).
18
Nitems by one standard deviation (adding about 70 pay items) reduces the odds by about1%. Although statistically significant, the effect of Nitems is empirically quite small.
In the OLS regression, the dependent variable is log(bid), where bid is the dollar biddivided by the engineer’s estimate. The most important explanatory variable is the en-gineer’s estimate. Using it alone produced R2 of about 0.79, so the impact of the othervariables is much smaller. In the order of importance, the next variable is the number ofpotential bidders N ; if it is included in the regression, R2 increases to about 0.94.
Theory also predicts that increasing the cost of entry should, ceteris paribus, result ina lower entry probability and therefore less aggressive bidding. This is confirmed by theresults of the OLS regression. Increasing Nitems by one standard deviation increases thebids by about 3%, a statistically significant but numerically small effect. A similar signifi-cant but small effect of Nitems is present in the logit regression. Because of this smallness,we do not condition on Nitems, or any other exogenous variables, in our implementationof the tests.
Included in both regressions are the dummy variables for top 20 firms. We define themas firms that appear on the planholders list most frequently. Observe that, although notall firms enter at the same rate and bid similarly, the empirical evidence of asymmetries isstrong only for out-of-state firms (the firms with headquarters outside the state of Okla-homa) that enter less frequently and also bid less, and for the following three firms: BroceConstruction, Glover Construction and Becco Contractors. In our effort to make potentialbidders symmetric, we eliminate the auctions in which either out-of-state firms or thesethree firms were on the planholders list. We also exclude auctions that have more thanN = 11 potential bidders (there are very few auctions for each N > 11). The working sam-ple was thereby reduced significantly, to 268 auctions, and all the results discussed belowthis point were obtained using this smaller sample.
Before we implement our test of selective entry, it is natural to ask if there is anyempirical evidence to support our model, regardless of the selectivity effects. A crude testof our theory can be obtained by considering the probabilities p∗N of bid submission. Inour model, the probability of submitting a bid is non-increasing in N . The average rate ofbid submission among the planholders is about 62%. The plot of the frequency of biddingas a function of N is exhibited in Figure 2. A strongly declining pattern is evident. Theprobability of submitting a bid is the highest (83%) when N = 2, and is reduced to about30% when N = 11. This pattern is consistent with the basic prediction of our model: ifthe entry cost is moderate, then the probability of bid submission must decline in N .
We now turn to the test of selective entry. In order to implement our test, we normalizethe bids, dividing them by the engineer’s estimate.12 Given the above discussion, we do notcondition on covariates. The estimated quantiles for τ ∈ Υ = {0.3, 0.5, 0.7} are shown inTable 4. As we described in Section 4, the null is Q∗2 (τ) = ... = Q∗11 (τ) ∀τ ∈ Υ, while thealternative is Q∗2 (τ) ≥ ... ≥ Q∗11 (τ) , ∀τ ∈ Υ. Note that the ordering is reversed to that in(15) because in low-bid auctions, larger values of N shift the distribution of costs upwards,and the quantile function downwards. The test has a p-value of value of 0.082, providingsome support for selective entry.
12This approach is pursued, for example, in Krasnokutskaya (2003).
19
7 Concluding remarks
In this paper, we have developed a model of selective entry in first-price auctions. We haveshown that ignoring selective entry can result in biased estimates of the entry cost . Onthe other hand, if the econometrician is primarily interested in the distribution of bidders’valuations, then this distribution can be bounded in an informative way. The bound isshown to be tight if the sample is large and exogenous variation in the number of potentialbidders is available. A non-parametric test of selective entry is proposed and implementedusing a dataset of highway procurement auctions. Some evidence of selective entry is found.
This research could be extended in a number of directions. One extension would beto allow bidder asymmetries, as in Krasnokutskaya and Seim (2006). Also, unobservedheterogeneity may be important (Krasnokutskaya (2003)). Yet another extension would beto allow for affiliated private values, as in Li, Perrigne, and Vuong (2002). Also, our modelis static, as are most other models in the literature. An important exception is Pesendorferand Jofre-Bonet (2003). Incorporating selective entry into a dynamic model would clearlybe important. We leave these extensions for future work.
20
References
Athey, S., and P. Haile (2002): “Identification of Standard Auction Models,” Econo-metrica, 70(6), 2107—2140.
Athey, S., and P. A. Haile (2005): “Nonparametric Approaches to Auctions,” Handbookof Econometrics, 6.
Athey, S., J. Levin, and E. Seira (2004): “Comparing Sealed Bid and Open Auctions:Theory and Evidence from Timber Auctions,” Department of Economics, Stanford Uni-versity.
Bajari, P., H. Hong, and S. Ryan (2004): “Identification and Estimation of DiscreteGames of Complete Information,” NBER Working paper.
Bajari, P., and A. Hortacsu (2003): “Winners Curse, Reserve Prices and EndogenousEntry: Empirical Insights from eBay Auctions,” RAND Journal of Economics, 34(2),329—355.
Bajari, P., and L. Ye (2003): “Deciding Between Competition and Collusion,” Reviewof Economics and Statistics, 85(4), 971—989.
Guerre, E., I. Perrigne, and Q. Vuong (2000): “Optimal Nonparametric Estimationof First-Price Auctions,”,” Econometrica, 68, 525—74.
Haile, P., H. Hong, and M. Shum (2003): “Nonparametric Tests for Common Valuesat First-Price Sealed-Bid Auctions,” NBER Working paper.
Haile, P. A., and E. Tamer (2003): “Inference with an Incomplete Model of EnglishAuctions,” Journal of Political Economy, 111(1), 1—51.
Hendricks, K., J. Pinkse, and R. Porter (2003): “Empirical Implications of Equilib-rium Bidding in First-Price, Symmetric, Common Value Auctions,” Review of EconomicStudies, 70(1), 115—145.
Hong, H., and M. Shum (2002): “Increasing Competition and the Winner’s Curse: Evi-dence from Procurement,” Review of Economic Studies, 69(4), 871—898.
Krasnokutskaya, E. (2003): “Identification and Estimation in Highway ProcurementAuctions under Unobserved Auction Heterogeneity,” Working Paper, University of Penn-sylvania.
Krasnokutskaya, E., and K. Seim (2006): “Bid Preference Programs and Participationin Highway Procurement Auctions,” Working paper, University of Pennsylvania.
Levin, D., and J. Smith (1994): “Equilibrium in Auctions with Entry,” The AmericanEconomic Review, 84(3), 585—599.
Li, T. (2005): “Econometrics of first-price auctions with entry and binding reservationprices,” Joural of Econometrics, 126(1), 173—200.
21
Li, T., I. Perrigne, and Q. Vuong (2002): “Structural Estimation of the AffiliatedPrivate Value Auction Model,” The RAND Journal of Economics, 33(2), 171—193.
Li, T., and X. Zheng (2005): “Procurement Auctions with Entry and an UncertainNumber of Actual Bidders: Theory, Structural Inference, and an Application,” WorkingPaper, Indiana University.
Manski, C. (2003): Partial Identification of Probability Distributions. Springer.
Marmer, V., and A. Shneyerov (2006): “Quantile-Based Nonparametric Inference forFirst-Price Auctions,” Working paper, UBC.
Milgrom, P. (1981): “Good News and Bad News: Representation Theorems and Appli-cations,” The Bell Journal of Economics, 12(2), 380—391.
Milgrom, P. (2004): Putting Auction Theory to Work. Cambridge University Press.
Newey, W. K. (1994): “Kernel Estimation of Partial Means and A General VarianceEstimator,” Econometric Theory, 10, 233—253.
Pesendorfer, M., and M. Jofre-Bonet (2003): “Estimation of a Dynamic AuctionGame,” Econometrica, 71(5), 1443—1489.
Samuelson, W. (1985): “Competitive Bidding with Entry Costs,” Economics Letters,17(1), 2.
22
Rejection probability under alternative
critical value Type I error 0.1 0.3 0.5 0.7
1 quantile: 0.5 (nominal size 5%) 4.91435 0.164 0.368 0.56 0.836 0.828(nominal size 10%) 3.54128 0.264 0.456 0.624 0.868 0.88
3 quantiles: 0.3, 0.5, 0.7(nominal size 5%) 10.2158 0.074 0.548 0.878 0.97 0.99(nominal size 10%) 8.32702 0.12 0.642 0.91 0.982 0.992
ρ
Table 1: Monte-Carlo Experiment: Testing Results
Variable Description Mean Std. Dev Min Max
EngEst The engineer's estimate for the project, in 3.647 4.488 0.066 24.800mil. Dollars
Bid Bid divided by the engineer's estimate 1.067 0.173 0.385 2.106
Nitems Number of pay items in the project ad 71.736 70.704 1.000 363.000
Ndays Number of business days to complete the 195.995 142.370 10.000 681.000project
Length Length of the road (in miles) 4.699 4.794 0.000 36.63
Distance Distance in miles from the headquarters of 344.237 382.469 0.000 1702.016the firm of the bidding firm to the project site
Backlog The total amount of unfinished work on a 0.219 0.297 0.000 1.000given day and normalized by the bidder-specific maximum, the value is between 0and 1
Npotential number of planholders 8.299 4.244 2.000 26.000
Nactual number of actual bidders 3.451 1.428 0.000 7.000
out-of-state dummy =1 if the firm has headquarters 0.136 0.343 0 1outside the state of Oklahoma
Table 2: Description of Variables
N 0.3 quantile s.e. 0.5 quantile s.e. 0.7 quantile s.e.
2 0.699 0.036 0.807 0.042 0.849 0.0193 0.818 0.066 0.921 0.030 1.030 0.0244 0.830 0.099 0.892 0.036 1.017 0.0325 0.726 0.060 0.819 0.039 0.992 0.0316 0.745 0.096 0.811 0.038 0.841 0.0347 0.762 0.037 0.822 0.031 0.985 0.0248 0.799 0.075 0.819 0.042 0.992 0.0349 0.693 0.081 0.810 0.048 0.910 0.04110 0.689 0.138 0.780 0.058 0.881 0.04611 0.828 0.146 0.880 0.071 1.003 0.059
Test statistic T: 23.873
Table 4: Estimated quantiles
Logit OLS
Variable Coefficient s.e. Coefficient s.e.
Intercept 3.565 1.148 0.497 0.123Log(EngEst) -0.010 0.034 0.970 0.010Npotential -0.043 0.019 -0.008 0.002Length 0.009 0.012 0.001 0.001Ndays 0.000 0.001 0.000 0.000Nitems -2.700E-04 0.000 4.580E-04 0.000
Distance 0.000 0.000 0.000 0.000Backlog 0.137 0.170 0.019 0.019
Out-of-state -0.359 0.166 -0.034 0.020
Fringe firm 0.250 0.175 0.027 0.034
FirmAPAC-OKLAHOMA, INC. 0.303 0.497 0.013 0.025THE CUMMINS CONST. CO., INC. 0.266 0.385 0.005 0.023HASKELL LEMON CONST. CO. -0.206 0.351 0.028 0.023BROCE CONSTRUCTION CO., INC. -1.803 0.313 0.052 0.029WESTERN PLAINS CONSTRUCTION COMPANY -1.338 0.756 -0.011 0.031BELLCO MATERIALS, INC. -0.672 0.411 -0.086 0.277OVERLAND CORPORATION -0.726 0.399 0.057 0.029GLOVER CONST. CO., INC. -0.919 0.395 0.061 0.003T & G CONSTRUCTION, INC. -0.974 0.992 -0.036 0.028TIGER INDUSTRIAL TRANS. SYS., INC. -0.061 0.479 -0.004 0.029HORIZON CONST. CO., INC. -0.181 0.473 0.009 0.033CORNELL CONST. CO., INC. -0.481 0.450 0.007 0.030SEWELL BROTHERS, INC -0.075 0.472 -0.018 0.034BECCO CONTRACTORS, INC. -1.818 0.876 -0.068 0.033EVANS & ASSOC. CONST. CO., INC. -0.228 0.582 -0.044 0.036SHERWOOD CONST. CO., INC. -0.561 0.416 0.006 0.035VANTAGE PAVING, INC. -1.619 0.962 0.030 0.047ALLEN CONTRACTING, INC. 0.245 0.493 0.014 0.035DUIT CONSTRUCTION CO., INC. 1.275 0.707 0.027 0.037MUSKOGEE BRIDGE CO., INC. -1.122 0.809 0.006 0.037
Observations 4485 1860Log-Likelihood -1543.750R2 0.983
Table 3: Logit and OLS regressions
Notes: Significant coefficients (at 5% level) are marked in bold. For the OLS regression, the bids were normalized by the engineer's estimate.
Ns
s
vv v
0
1
( )vs*
( ) ( )svFsvF |~| εε =
( ) ( )svFsvF |~1| εεε +−=
Figure 1: Construction for the proof of Proposition 3
Figure 2: Bias in entry cost estimation
0.15
0.17
0.19
0.21
0.23
0.25
0.27
2 3 4 5 6
N
estimated k from random entry model true k from DGP
Figure 3: Exact bounds on F(v) in the Monte-Carlo experiment
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c1.0
51.1
11.1
71.2
31.2
91.3
51.4
11.4
71.5
31.5
91.6
51.7
11.7
71.8
31.8
91.9
52.0
12.0
72.1
32.1
92.2
52.3
12.3
72.4
32.4
92.5
5
Values
Prob
abili
ty upper exactF(v)lower exactmisspecified
Figure 4: Estimated bounds on F(v). Monte-Carlo experiment, L=250 auctions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.51 1.598 1.686 1.774 1.862 1.95 2.038 2.126 2.214 2.302 2.39 2.478
Values
Prob
abili
ty estimate upperestimate lowerF(v)upper 97.95%lower 2.5%misspecified
Figure 5: Estimated bounds on F(v). Monte-Carlo experiment, L=1000 auctions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.40 1.49 1.58 1.66 1.75 1.84 1.93 2.02 2.10 2.19 2.28 2.37 2.46
Values
Prob
abili
ty
estimate upperestimate lowerF(v)upper 97.95%lower 2.5%misspecified
Figure 6: Estimated bounds on F(v). Monte-Carlo experiment, L=5000 auctions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.40 1.49 1.58 1.66 1.75 1.84 1.93 2.02 2.10 2.19 2.28 2.37 2.46
Values
Prob
abili
ty
estimate upperestimate lowerF(v)upper 97.95%lower 2.5%misspecified
Figure 7: Histogram of N
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
N
Num
ber o
f auc
tions
Figure 8: Probability of bidding
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2 3 4 5 6 7 8 9 10 11
N
Prob
abili
ty