Estimating the First-Price Auction Model with Entry:

A Parametric Approach∗

Shinichi Sakata; Pai Xu†

April 5, 2010

Preliminary. Please do not cite.

Abstract

This paper studies the structural estimation of first-price auctions with endogenous entry.

Most of the empirical auction literature to date is based on the theoretical work of Levin and

Smith (1994, LS hereafter) in which the potential bidders make their entry decisions before

knowing their own valuations. This paper, instead, assumes that a potential bidder already

knows his own valuation when deciding to pay an entry cost and submit a bid. Such an assump-

tion reflects the reality in certain important auctions.

We propose estimating the distribution of bidders’ private valuations in the entry model by

using the simulated method of moments. We implement the proposed estimator on a dataset of

highway procurement auctions in Oklahoma.

∗We are grateful to Han Hong, Quang Vuong, Harry Paarsch, Vadim Marmer, Brian Krauth and Unjy Song for

useful comments, as well as the seminar participants at the University of Hong Kong. We also thank Kun Li for his

excellent research assistance. The second author gratefully acknowledges the financial support of HKU under the

seed funding scheme for basic research.†Sakata: Department of Economics, University of British Columbia; ssakata@interchange.ubc.ca. Xu: School of

Economics and Finance, University of Hong Kong, paixu@hku.hk.

1 Introduction

A fact in the real-world auctions is that not all the eligible bidders actually submit a bid, even

though they have initially shown their interest on the auctioned object. This observation suggests

that it may be costly participating the auction. Moreover, it is often of a researcher’s interest to

understand the bidder entry behavior for the desirable economic policies. In particular, a policy

maker often cares whether any favorable treatment to some group of bidders may induce inefficiency

in the market.

Most recently, several empirical works have attempted to study the auction models with endoge-

nous entry in the highway procurement industry. Krasnokutskaya and Seim (2006)[10] consider

two types of bidders deciding to enter the auctions by introducing bid preference programs. Li and

Zheng (2009)[13] propose a semi-parametric Bayesian method to jointly estimate entry and bid-

ding. Bajari, Hong and Ryan (2009)[3] use a likelihood-based estimation approach in the presence

of multiple equilibria. All of these works estimate variants of an entry model based on Levin and

Smith (1994)[12] [hereafter, LS], in which all the potential bidders have no information on their

valuation of the auctioned object when deciding whether or not to enter. By incurring a cost,

they then can become informed on the valuations and submit a bid. The equilibrium of the model

predicts that all the potential bidders randomize their decisions on entry so that their expected

payoffs from entry become zero.

In many applications, however, the equilibrium conditions for LS model are hard to justify. For

example, consider the highway procurement auctions in US, which are the applications studied

in those recent empirical works. The Department of Transportation (DOT) in the state usually

announces the project under auction a few weeks before the letting date. The DOT also describes

the jobs and time frame involved in the project. Though actual participants in the auction obtain

more information later, the description provided at this stage still contains valuable information

for typical potential bidders, who are experienced construction companies. Therefore, it does not

seem reasonable to assume that potential bidders have no information on the projects upon their

decision on participation.

1

This paper considers an alternative entry model in auctions. Our model is based on a theoretical

work by Samuelson (1985)[19] [hereafter, S model], in which the bidders already know their valua-

tions when making the entry decisions. In equilibrium, the bidder does not submit a bid if and only

if his expected profit from participation is not large enough to cover the bidding cost. Participants

in auctions are allowed to have different (positive) expected profits form joining the auction in

accordance with their own valuations. Therefore, the zero expected profit from participation in the

LS model is discarded in our framework.

Besides the different equilibrium characterizations of the models, the entry costs have different

interpretations between the two entry models. The entry cost in the LS model is related to informa-

tion acquisition. In the practice of highway procurement auctions, such an information acquisition

process occurs when a firm purchases and studies the bidding plan. The costly entry of this kind

may possibly be the reason why not all qualified firms made the purchases of bidding plans.

Instead, the costly entry in our S model reflects the fact that bid preparation and submission are

costly, as the bidders already know their valuations. In the highway procurement auctions, such

entry costs entail all the time and efforts needed to compile the bidding plan, check for the errors,

travel to the bid meetings, and monetary costs a firm may have to devote to the bidding submission

process. These entry costs can help to explain why not all the bidders actually decided to submit

a bid, even though potentially interested. This paper is motivated to study the entry phenomenon

of this kind.

Most recently, Xu (2009) [20] adopts the same entry model to study Michigan Highway procure-

ment auctions. There we propose a non-parametric estimate for the entry cost, with which we

further suggest how to implement optimal auctions. Within the nonparametric framework, certain

types of counterfactual experiments are impossible, because we have no way to nonparametrically

recover the distribution function below the cut-off point. For this reason, we sometimes employ a

parametric model to investigate the consequences of policy changes in an auction. In this paper,

we develop a method to estimate the auction model with endogenous entry based on a parametri-

cally specified value distribution. With the estimated parametric auction model, we can conduct

various counterfactual experiments involving the entire value distribution. This, for example, lets

2

us investigate the effectiveness of asymmetric optimal mechanism (Celik and Yilankaya (2009)[5]).

A challenge in estimating an auction model is computation. We need to calculate multi-fold

integral each time we evaluate the optimal bidding function, for example. A remedy for this type of

problems is the use of a simulation-based method. In a seminal work by Laffont, Ossard and Vuong

(1995)[11][hereafter, LOV], a simulated nonlinear least squares (SNLLS) estimator is proposed

to estimate one of the simplest theoretical auction models, the first-price sealed bid independent

private-value auctions (without entry).

The endogenous entry in our model, however, makes our framework differ from LOV. The equilib-

rium bidding strategy takes a different form, which in turn induces the different estimation strategy.

In particular, we have a truncation parameter to estimate in our entry context, besides the dis-

tributional parameters in the model. We therefore propose a simulated-GMM estimator, instead

of extending LOV’s SNLLS. Moreover, the truncation point of the bid distribution in our model

depends on the parameters in estimation, because the entry decision depends on the private-value

distribution. It turns out that this feature of our problem complicates the large sample analysis

of the simulated GMM method, making the estimation objective function non-smooth in the pa-

rameters. We derive a set of sufficient conditions for the asymptotic properties of our estimator.

Therefore, our work compliments to the literature by showing the applicability of simulation-based

estimation framework to the context of endogenous entry. We apply the proposed estimation strat-

egy to Oklahoma Highway Procurement Auctions. Within the specified lognormal distribution

family, we are able to estimate the parameters that govern the bidders valuation distribution, as

well as the truncation parameter due to the entry cost considered in the model, in the Oklahoma

highway procurement auctions.

The rest of this paper is organized as follows. In the next section, we introduce the Samuelson

(1985)[19] entry model and discuss the equilibrium characterizations. In Section 3, we propose a

method to estimate the model elements and establish its large sample properties. In Section 4,

we apply the proposed estimation methodology to the Oklahoma Highway Procurement Auctions.

Moreover, we illustrate how to implement the asymmetric optimal auctions in section 6. The last

section concludes. The mathematical proofs are collected in the appendix.

3

2 A First-Price Auction Model with Entry Costs

2.1 Equilibrium Characterization

We consider a first-price sealed-bid auction of a single indivisible good. Within the symmetric

independent private-value (IPV) framework, each potential risk neutral participant i ∈ 1, 2, ..., N

knows her own value vi for the object, but only knows the distribution of the values to the other

potential bidders. It is assumed that the values to individuals are independently drawn from

the absolutely continuous distribution F (v) with support [v, v] ⊂ R+. Bidders submit their bids

simultaneously and the object goes to the highest bidder. The winner pays her bid to the seller,

provided that the bid is no less than the reserve price r.

Our analysis deviates from the standard IPV framework by allowing for the presence of a common

entry cost, κ, with which each bidder has to pay to join the auction. Given her private value,

the bidder decides whether or not to submit a bid (paying κ) and becomes an actual bidder.

All potential bidders make this decision simultaneously. Therefore, they make their participation

decisions without knowing how many competitors they are actually going to face.

We will focus on the unique symmetric Bayesian Nash equilibrium (Milgrom, 2004)[16], in which

each potential participant joins the auction if her value is no less than, vρ, the cut-off point (common

to all bidders), otherwise chooses not to participate. The cut-off point is such that the participant

with value vρ is indifferent in entering the auction or not: so, the marginal type, vρ, is the unique

zero of ζ : R→ R defined by

ζ(v) ≡ (v − r)F (v)N−1 − κ. (2.1)

To verify the unique existence of the zero of ζ, note that the zero cannot be less than r. Also, it is

easy to verify that the function ζ is strictly increasing on the interval [r,∞), and it maps [r,∞) to

[−κ,∞). It follows that (2.1) has a unique solution, which belongs to [r,∞).

The equilibrium bidding strategy B, which maps each bidder’s value of the object to his bidding

price, is strictly increasing and differentiable at each point in the support above vρ. Given B, a

bidder’s expected payoff function with value v from bidding b is

(v − b)F (maxB−1(b), vρ)N−1 − κ.

4

Because the expected payoff must be maximized when b is set equal to B(v), the derivative of the

expected payoff with respect to b must vanish at b = B(v). This fact yields that for each v in the

support of F exceeding vρ,

(v −B(v))(N − 1)F (v)N−2 f(v)

B′(v)− F (v)N−1 = 0.

Rearranging terms in this equality, we get

v (N − 1)F (v)N−2f(v) = B′(v)F (v)N−1 +B(v) (N − 1)F (v)N−2 f(v)

=d

dv(B(v)F (v)N−1).

By solving this differential equation with the condition that B(vρ) = r, we obtain that if v > vρ,

B(v) = F (v)−(N−1)

(∫ v

vρ

y (N − 1)F (y)N−2 F (dy) + r F (vρ)N−1

)(2.2)

= F (v)−(N−1)

(∫ v

−∞maxy, vρ (N − 1)F (y)N−2 F (dy) + (r − vρ)F (vρ)N−1

)= F (v)−(N−1)

(∫ v

−∞maxy, vρ (N − 1)F (y)N−2 F (dy)− κ

)(2.3)

= E[maxY(1), vρ |Y(1) < v]− κ/F (v)N−1 (2.4)

where the third equality follows by the fact that ζ(vρ) = 0; and Y(1) is the highest of N − 1 random

draws from F . The above equality can be further rewritten as

F (v)N−1 (E[maxY(1), vρ |Y(1) < v]−B(v)) = κ. (2.5)

(2.5) provides an intuitive interpretation of equilibrium bidding strategy in the S entry model. It

says that the bidder with value v chooses the bid level at which his best “expected” competitor’s

profit would be nonpositive, if the competitor outbids him. (2.5) is the counterpart of the well-

known Revenue Equivalence Theorem for the auctions in a standard setup in which there are no

endogenous entries associated with participation costs of any format.

Though a potential bidder with v < vρ does not bid, we artificially assign r to his bid value for

convenience in our econometric analysis, so that B(v) = r whenever v < vρ. By rewriting (2.3) and

5

using r as the bids of non-bidders, we obtain that

B(v) =1(v ≥ vρ)F (v)−(N−1)

(∫1(y ≤ v) maxy, vρ (N − 1)F (y)N−2 F (dy)− κ

)+ 1(v < vρ) r

=m(v)

≡∫ (

1(v ≥ vρ)F (v)N−1

(1(y ≤ v) maxy, vρ(N − 1)F (y)N−2 − κ

)+ 1(v < vρ) r

)F (dy). (2.6)

The jth moment of the bid is therefore equal to∫m(v)j F (dv). Such construction provides a basis

and convenience for our design of estimators in the entry context.

2.2 Identification

In general, the observables in our current model are the bids, the number of potential bidders

and the number of actual bidders, while the private values and their distributions are not observed.

The identification problem here amounts to a discussion on whether the observed variables can

uniquely determine both the latent distribution F and the participation cost κ.1

Following GPV (2000)[7], the inverse equilibrium bidding function can be written as

v = ε(b∗) ≡ b∗ +1− p+ pG∗(b∗)(N − 1)pg∗(b∗)

(2.7)

where b∗ is observed bids. G∗ and g∗ are their CDF and pdf, respectively. And the rate of

participation p = 1− F (vρ). (2.7) implies that if both p and G∗ (therefore, g∗) are identified, then

so is ε for v ≥ vρ.

Furthermore, as F (vρ) = 1−p and vρ is identified (from 2.7), then (2.1) implies that κ is identified.

It is clear from above argument that the identification hinges on the facts that p and vρ are both

identified. However, p is always identified, as we assume econometricians can observe both the

number of potential bidders and the number of actual bidders.1We provide a heuristic argument on the identification of the S model. More thorough discussion can be found in

Xu (2009)[20], Marmer, Shneyerov and Xu (2010)[15].

6

3 Estimation of the Auction Model with Entry Costs

3.1 An Econometric Model

This paper considers a parametric framework for the power of counterfactual analysis as outlined

in the introduction. We therefore assume that a parametric specification for the value distribution is

given. We also allow for heterogeneity of auctions through the dependence of the value distribution

and the entry cost on a k × 1 vector consisting of exogenous variables. In a formal format, the

following assumption summarizes the elements of an econometric model in the auction context.

Assumption 1 The parameter space Θ is a nonempty compact subset of Rp, and F : R×Rk×Θ→

R is a function measurable B ⊗ Bk ⊗ B(Θ)/B that satisfies that for each z ∈ Rk and θ ∈ Θ,

F (·, z, θ) : R → R is a cumulative distribution function with a pdf f with respect to the Lebesgue

measure, and for each (v, z) ∈ R × Rk, F (v, z, ·) : Θ → R and f(v, z, ·) : Θ → R are continuous.

Also, κ : Rk×Θ→ [0,∞) is a function measurable Bk⊗B(Θ)/B([0,∞)) such that for each z ∈ Rk,

κ(z, ·) : Θ→ R is continuous.

The value distribution of an auction with the exogenous variables z is modeled as F (·, z, θ), where

θ ranges over Θ.

In terms of data generating process, we assume that the number of potential bidders (N), all of

the bids (B), the exogenous variables (Z), and the reserved price (R) for each auction, where the

bids are given through the mapping m of the unobserved values (V ) in (2.6) suitably modified to

allow for the heterogeneity of auctions.

Assumption 2 (i) On a probability space (Ω,F, P ), (Nt, Rt, Zt, Vt) : Ω → N × R × Rk ×

R∞t∈N is an independently and identically distributed (i.i.d.) process such that there uniquely

exists θ0 ∈ Θ such that each element of v1 are independently and identically distributed with

F (·, Z1, θ0) conditionally given N1, R1, and Z1.

(ii) For each θ ∈ Θ, F (R1, Z1, θ) > 0 a.s.-P .

Remarks.

(i) In Assumption 2, the conditions imposed on the first observation (indicated by subscript 1)

applies to all observations, because (Nt, Rt, Zt, Vt)t∈N is i.i.d.

7

(ii) Despite that in each auction t, we only observe Nt bids, which are mapped from vt1, . . . , vtNt ,

the random vector vt consists of infinitely many elements in Assumption 2. This is merely

for technical convenience, and it imposes no restrictions on the applicability of our approach.

The special feature of endogenous entry in our auction model introduces a threshold (vρ) on the

valuation distribution F . We define it in the econometric model accordingly. vρ : R×Rk×N×Θ→ R

such that

vρ(r, z, n, θ) ≡ infv ∈ R : (v − r)F (v, z, θ)n−1 − κ(z, θ) ≥ 0

Because v 7→ (v − r)F (v, z, θ)n−1 − κ(z, θ) is nondecreasing and continuous, and its range con-

tains [0,∞), vρ(r, z, n, θ) is the minimum zero of v 7→ (v − r)F (v, z, θ)n−1 − κ(z, θ). Under As-

sumption 2(ii), vρ(R1, Z1, N1, θ) is a singleton a.s.-P for each θ ∈ Θ, so that for each θ ∈ Θ,

vρ(R1, Z1, N1, θ) is a unique zero of v 7→ (v −R1)F (v, Z1, θ)n−1 − κ(Z1, θ) a.s.-P .

For purpose of structural analysis, the observed bids are assumed to be generated following the

symmetric equilibrium strategy. Define m : R× R× Rk × N×Θ→ R by

m(v, r, z, n, θ) ≡∫ (

1(v ≥ vρ(r, z, n, θ)

)F (v, z, θ)n−1

(1(y ≤ v) maxy, vρ(r, z, n, θ)(n− 1)F (y, z, θ)n−2 − κ(z, θ)

)+ 1(v < vρ(r, z, n, θ)

)r

)F (dy, z, θ).

Then:

Assumption 3 For each (t, i) ∈ N2, Bti ≡ m(Vti, Rt, Zt, Nt, θ0), where Vti denotes the ith element

of Vt.

3.2 A simulated GMM estimator

A way to estimate the model is to use the generalized method-of-moments (GMM) estimator,

which minimizes

GT (θ) = JT (θ)′WJT (θ),

8

where W is the suitably chosen, possibly data-dependent weighting matrix (for the moment, we

ignore the issue of W and take it as an identity matrix), and ∀k ∈ N > 1,

JT (θ) =1T

T∑t=1

Nt∑i=1

Bti − l1(Zt, θ)h1(Zt)

· · ·

Bkti − lk(Zt, θ)hk(Zt)

1(Bti = r)− F (vρ(Zt; θ))hρ(Zt)

.

l1, ..., lk denote, respectively, the first, ... and k-th moments of bid in the t-th auction conditional on

the covariates. h1(Zt), · · · , hρ(Zt) are deterministic functions of Zt, which serve as instrumental

variables in the GMM framework. The last moment condition in JT is due to the participation

decision by the bidders in our entry context.

The moments of bids in the t-th auction can be written as

l1(Zt; θ) =∫ v

vm(vi;Zt, θ)f(vi;Zt, θ) dvi,

...

lk(Zt; θ) =∫ v

vmk(vi;Zt, θ)f(vi;Zt, θ) dvi, (3.8)

where

m(vi;Zt, θ) =∫ v

v

[1(vi ≥ vρ(Zt; θ))F (vi;Zt, θ)Nt−1

(1(y ≤ vi) maxy, vρ(Zt; θ)(Nt − 1)F (y;Zt, θ)Nt−2 − κ(Zt; θ)

)+ 1(vi < vρ(Zt; θ)) r

]f(y;Zt, θ) dy.

Nevertheless, the evaluation of l1(Zt; θ) and lk(Zt; θ) is costly. To keep the computation burden

low, we replace these with simulators X [1](Zt; θ) and X [k](Zt; θ) so that E[X [1](Zt; θ)] = l1(Zt; θ)

and E[X [k](Zt; θ)] = lk(Zt; θ).

First, we consider simulating l1(Zt; θ). For each t = 1, ..., T , S1 independent pairs of independent

random variables are drawn from U(0, 1) the uniform distribution over the interval (0, 1). Let u[1]t,j

denote the j-th pair drawn for t-th auction. u[1]t,j,1 and u

[1]t,j,2 are the first and second elements of

u[1]t,j , respectively, where j = 1, 2, ..., S1. Then, for each t, we approximate l1(Zt; θ) by

X[1]t (u[1]

t,1, ..., u[1]t,S1

, Zt, θ) =1S1

S1∑j=1

X[1]1t (u[1]

t,j , Zt, θ), (3.9)

9

where

X[1]1t (u[1]

t,j , Zt, θ) =1F−1(u[1]

t,j,1;Zt, θ) ≥ vρ(Zt; θ)

u[1]t,j,1

Nt−1

(1u[1]

t,j,2 ≤ u[1]t,j,1maxF−1(u[1]

t,j,2;Zt, θ), vρ(Zt; θ)

(Nt − 1)u[1]t,j,2

Nt−2 − κ(Zt; θ))

+ 1F−1(u[1]t,j,1;Zt, θ) < vρ(Zt; θ) r. (3.10)

The higher moments can be approximated in a similar fashion as what is done for l1. We

consider the k-th moment, lk(Zt; θ). For each t = 1, ..., T , we draw Sk independent (k + 1)-

tuples of independent random variables from U(0, 1). Let u[k]t,j denote the j-th tuple drawn for

the t-th auction, where j = 1, 2, ..., Sk. The q-th random variable in u[k]t,j is denoted u

[k]t,j,q, where

q = 1, ..., k + 1. Then, for each t, we approximate lk(Zt; θ) =∫ vv m

k(vi;Zt, θ)f(vi;Zt, θ)dvi by

X[k]t (u[k]

t,1, ..., u[k]t,Sk

, Zt, θ) =1Sk

Sk∑j=1

[X

[k]1t (u[k]

t,j , Zt, θ) · · ·X[k]kt (u[k]

t,j , Zt, θ)], (3.11)

where

X[k]1t (u[k]

t,j , Zt, θ) =1F−1(u[k]

t,j,1;Zt, θ) ≥ vρ(Zt; θ)

u[k]t,j,1

Nt−1

(1u[k]

t,j,2 ≤ u[k]t,j,1maxF−1(u[k]

t,j,2;Zt, θ), vρ(Zt; θ)

(Nt − 1)u[k]t,j,2

Nt−2 − κ(Zt; θ))

+ 1F−1(u[k]t,j,1;Zt, θ) < vρ(Zt; θ) r,

· · · · · ·

X[k]kt (u[k]

t,j , Zt, θ) =1F−1(u[k]

t,j,1;Zt, θ) ≥ vρ(Zt; θ)

u[k]t,j,1

Nt−1

(1u[k]

t,j,k+1 ≤ u[k]t,j,1maxF−1(u[k]

t,j,k+1;Zt, θ), vρ(Zt; θ)

(Nt − 1)u[k]t,j,k+1

Nt−2 − κ(Zt; θ))

+ 1F−1(u[k]t,j,1;Zt, θ) < vρ(Zt; θ) r

Then, our estimator θ is the simulated method-of-moments estimator that minimizes

GS,T (θ) = JS,T (θ)′WJS,T (θ), (3.12)

where

JS,T (θ) =1T

T∑t=1

Nt∑i=1

Bti − X [1]

t (u[1]t,1, ..., u

[1]t,S1

, Zt, θ)h1(Zt)

· · ·

Bkti − X

[k]t (u[k]

t,1, ..., u[k]t,Sk

, Zt, θ)hk(Zt)

1(Bti = r)− F (vρ(Zt; θ))hρ(Zt)

.

10

Different from LOV (1995)[11], the bias correction is not required in our case, though the sim-

ulated moments are used. To see this, we take the first moment of bids for example and notice

that

E[Bti − Xt(u1, ...uS , Zt, θ)h1(Zt)]

= E[Bti − l1(Zt; θ) + l1(Zt; θ)− Xt(u1, ...uS , Zt, θ)h1(Zt)]

= E[Bti − l1(Zt; θ)h1(Zt)] + E[l1(Zt; θ)− Xt(u1, ...uS , Zt, θ)h1(Zt)] (3.13)

As suggested by the second expectation in (3.13), there requires no bias correction by using simu-

lated moments in the objective function, as long as the simulation error is orthogonal to h1(Zt).

In general, h1 needs to be chosen as the gradient of l1 to attain the efficiency for GMM estimation.

(3.13) further implies that, regarding the choice (of estimation) of h1(Zt), the simulation method

may also be applied, provided that the simulation errors from simulating moments of bids and

from simulating h1 are uncorrelated. However, in this paper, we have to avoid the choice of such

an efficient estimator, as l1 is non-smooth in parameters, therefore may be non-differentiable.

3.3 Large-Sample Properties

To ease the exhibition, we first define the population counterpart of GS,T as G(θ) = J(θ)′WJ(θ),

where

J(θ) = E

Bti − l1(Zt, θ)h1(Zt)

B2ti − l2(Zt, θ)h2(Zt)

· · ·

Bkti − lk(Zt, θ)hk(Zt)

1(Bti = r)− F (vρ(Zt; θ))hρ(Zt)

.

To ensure the desired asymptotic properties of our simulated GMM estimator, we assume the

following regularity conditions:

Assumption 4 (i) W is positive semi-definite and WJ(θ) 6= 0 for θ 6= θ0.

(ii) v vρ(Z, θ) v, ∀Z, θ.

(iii) fz the marginal density of Z is bounded.

11

(iv) FzZ : F−1(., Z, θ)− vρ(Z, θ) = 0 = 0

(v) S1 ... Sk are fixed natural numbers no less than 2 and independent of t.

As G is continuous on the compact parameter set, assumption 4(i) requires there be only one

minimizer of G. This by and large is a typical identification condition for the GMM type estimators.

4(ii) states that the truncation introduced by the entry cost is bounded away from the supports

of the valuation distribution. This assumption is generally assumed in the auction setup with

endogenous entry, as otherwise the entry issue will become trivial (either full entry or no entry).

Assumption 4 (iii) and (iv) are not restrictive in the sense that the boundedness and continuity of

covariates are usually assumed in the auction theory. The last item in assumption 4 reinforces that

our asymptotic analysis is for a fixed number of simulations as T →∞.

The following theorem states the consistency property of θ.

Theorem 1 Given Assumptions 1 to 4, θ converges in probability to θ0 as T →∞.

We further impose the following assumption for the asymptotic distribution.

Assumption 5 (i) θ0 is an interior point of Θ.

(ii) ∀Z, ∂F (., Z, θ)/∂θ exsits and is bounded ∀θ ∈ Θ.

(iii) ∀Z, ∂f(., Z, θ)/∂θ exists and is bounded ∀θ ∈ Θ.

(iv) (Zt, θ) has a density that is bounded above uniformly over θ ∈ Θ.

Assumption 5 (ii) and (iii) impose conditions on f and F over the parameter set Θ. These

restrictions can be satisfied if f is chosen from the commonly used distributions. Especially, in our

empirical applications and all of other empirical auction works with parametric approaches, the

private values are usually specified to follow lognormal distribution. Then these assumptions hold.

The last item of assumption 5 asks the boundedness of joint density of covaiates and parameters.

This is not generally restrictive in applications.

12

We now present the following theorem, which states the asymptotic normality for θ.

Theorem 2 Given Assumptions 1 to 5,√T (θ−θ0)→ N(0,Σ), where Σ = (′W)−1′WΩW(′W)−1

and = E[∇θJ(Z, θ0)], and Ω = E[J(Z, θ0)J(Z, θ0)′].

To use the asymptotic normality of θ to conduct inference on θ0, it is necessary to estimate

the asymptotic covariance matrix Σ. This matrix depends on , Ω and W . Following the same

reasoning and spirit, we propose consistent estimators of and Ω. We postpone the discussion of

estimating W later for the efficiency of simulated GMM estimation. For now, we simply assume

we have a consistent estimator of W , W .

Our proposed estimator that relies on simulating first order derivatives and that is also consistent

for a fixed number of simulations as the number of auctions increases to infinity. We define Y [ι]t =

∂X[ι]t (θ)/∂θ and Fθ = ∂F (v;θ)

∂θ . Then,

(θ) = − 1T

T∑t=1

Nt∑i=1

Y

[1]t (θ)h1(Zt)

· · ·

Y[k]t (θ)hk(Zt)

Fθ(θ))hρ(Zt)

.

and

Ω(θ) =1T

T∑t=1

Jt(θ)Jt(θ)′

where

Jt(θ) =

Bti − X [1]

t (u[1]t,1, ..., u

[1]t,S1

, Zt, θ)h1(Zt)

· · ·

Bkti − X

[k]t (u[k]

t,1, ..., u[k]t,Sk

, Zt, θ)hk(Zt)

1(Bti = r)− F (vρ(Zt; θ))hρ(Zt)

.

Therefore, Σ = (′W )−1′W ΩW (′W )−1. The following theorem states that Σ is a consistent

estimator of Σ.

Theorem 3 Σ converges to Σ in probability as T →∞.

13

A practical advantage of the proposed Σ is that it does not require the number of simulations to

be large, as otherwise the computation burden may become incredibly heavy due to the complex

of bidding strategy. Moreover, Ω can typically be obtained from the last iteration of minimization

routines with which θ was obtained.

3.4 Comparison with LOV (1995)

LOV (1995) [11] considers a first-price auction model without endogenous entry. In there, the

value distribution is truncated by the presence of a binding reserve price. Conditional on partici-

pation, the Revenue Equivalence Theorem implies:

b = ε(v,N, F ) = E[max(v(2), r)|v(1) = v,N, F ] (3.14)

where v(1) and v(2) are the largest and second largest order statistic among v1, v2, ..., vN . Considering

only winning bids and extending to R, (3.14) gives

bw = E[max(v(2), r)|v(1) = v,N, F ]1[v ≥ vρ] + r · 1[v < vρ]

= E[max(v(2), r)|v(1) = v,N, F ]. (3.15)

Then, taking the expectation with respect to v, (3.15) gives

E[bw|N,F ] = E[max(v(2), r)|N,F ]

which can be written as an integral with respect to f :

E(bw|N,F ) =∫ v

vmaxu(2), rf(u1)...f(uN )du1...duN . (3.16)

LOV propose a simulated non-linear least squares estimator for the parameters of F based on the

moment condition (3.16). Though our work is related to LOV, there are subtle differences between

the two:

(i) Bidding strategies. (2.4) vs. (3.14).

There is no endogenous entry issue in LOV’s model. They assume that the bidders know

the number of competitors they are actually facing when submitting the bids. Then, r

just serves as the lower bound for the truncated value distribution. Therefore, the revenue

14

equivalence theorem takes the regular form as (3.14). However, in our model, the revenue

equivalence theorem implies a different form as (2.4), which reflects the free entry condition

in the equilibrium of the model.

(ii) Asymptotics. The objective function maximized by LOV’s estimator, which is based on

(3.16), is smooth, as long as F is smooth in parameters. In our model, on the other hand,

the truncation point corresponding to r in (3.16) is vρ, which is one of the parameters to

be estimated. Our estimation objective function is therefore non-smooth in the parameters.

This feature makes the asymptotic analysis of our estimator more involved.

(iii) Estimation strategy. The model primitive in LOV is the parameters that characterize the

distribution of valuations. In their specification of lognormal distribution, let µ and σ2 be the

mean and variance, respectively. LOV suggest a SNLLS estimator to estimate the parameters.

However, in essence, their LS estimator only matches the first moment of observed (winning)

bids with data. µ and σ are not jointly identified. Therefore, in practice, LOV use the market

data to fix σ first, then apply their SNLLS estimator to recover µ.

Besides the distributional parameters µ and σ, we also have a truncation parameter κ (or,

equivalently, vρ) to estimate in our entry context. We propose a simulated-GMM estimator

for estimating both the distributional and truncation parameters, instead of extending LOV’s

SNLLS. By doing so, we are able to recover both µ and σ jointly.

(iv) Simulation method. LOV proposed using the importance sampling to simulate the first

moment of bids, while we provide a general approach to simulate higher moments by extending

LOV’s method.

An alternative way to estimate the model is to extend LOV’s SNLLS. Focusing on the first

moment and suppressing the covariates, a SNLLS estimator θ, for any fixed number of simulations

S, minimizes

Q1S,T (θ) =

1T

T∑t=1

Nt∑i=1

[(Bti − Xt(θ)

)2 − 1S(S − 1)

S∑s=1

(Xt(θ)− Xt(θ)

)2].

We next compare our simulated GMM method with simulated SNLLS by LOV up to front. Such

comparison allows us to see more clearly the advantage of using our approach in practice.

15

(i) Bias correction. The second term in bracket is the bias correction term, by incorporating

which enables Q1S,T converges in probability to Q1

T . To see this point, we notice that

E[Bti − Xt(θ)2]

= E[Bti − l1t(θ) + l1t(θ)− Xt(θ)2]

= E[Bti − l1t(θ)2] + E[l1t(θ)− Xt(θ)2] (3.17)

E[l1t(θ) − Xt(θ)2] in (3.17) corresponds to the bias correction term in Q1S,T . The cross

product term disappears due to the fact that B is always uncorrelated with the simulation

errors.

The bias correction is needed as SNLLS takes the expectation of squares. Therefore, the bias

can always be formulated as the variance of simulation errors. Instead, the objective function

of Simulated GMM takes the form of squares of the moments. Then no bias is occurred in

the objective function as long as the moments are unbiasedly simulated.

(ii) Computation. If we consider higher moments of bids, the simulation and bias correction

required in NLLS approach are demanding. The computation in NLLS is more involved than

GMM.

(iii) the objective function approximated in LOV is unbiased but bears a huge variance when F

puts a large probability mass in the area where G puts a low probability mass.

4 Empirical Application

Our dataset consists of 277 auctions for surface paving and grading contracts let by Oklahoma

Department of Transportation (ODoT) during the period of January 2002 to December 2005.2 The

available data items include all bids, the engineer’s estimate, the time length of the contract (in

days), the number of items in the proposal and the length of the road. The ODoT implements a

policy under with all bids over 7% of the engineer’s estimate are typically rejected, so there is a2Our choice of surface paving and grading contracts is motivated by the fact that Hong and Shum (2002)[8],

in their study of New Jersey highway procurement auctions, find little support for common values for this type of

contracts. This is important because in this paper, we assume the independent private value framework.

16

binding reserve price. In reality, we do observe bids above the reserve price (although extremely

few winning bids were above the reserve price). We treat these bids as non-serious.

Importantly, we observe the list of eligible bidders (planholders) for each auction. In the vo-

cabulary of this paper, these eligible bidders are the potential bidders. the list of planholders is

published on the ODoT website prior to bidding. A firm becomes a planholder through the follow-

ing process. All projects to be auctioned are advertised by the ODoT 4 to 10 weeks prior to the

letting date. These advertisements include the engineer’s estimate, a brief summary of the project,

location of the work and the type of the work involved.

Interested firms can then submit a request for plans and bidding proposals, the documents that

contain the specifics of the project (in particular, the items schedule). An important feature of

the qualification process is that only eligible firms are allowed the access to these documents. A

firm is deemed eligible if it satisfies certain qualification requirements. The goal of the qualification

process is to ensure that the winning firm will have sufficient expertise and capacity to undertake

the project. While the expertise part is typically determined at the pre-qualification stage, the

capacity part is project-specific. An important requirement is that the prospective bidder is not

qualified for the aggregate amount of work that exceeds 2.5 times its current working capital. Given

that the bidders know the sizes of all projects to be let but not the project specifics, it is plausible

that the decision of a firm to request the plan for a particular project is primarily determined by

the project size as well as the sizes of other projects for which it is pre-qualified, in relation to the

available capacity of the firm. The capacity may be determined by a number of factors, such as for

example the amount of resources committed to other projects, including but not limited to those

previously contracted with ODoT.

Before turning to our parametric specification, we investigate the importance of various observ-

able covariates on bid levels and the decisions to submit a bid with the help of usual OLS and logit

regression. The description of the variables used in the regressions and their summary statistics

are reported in Table 1. The results of the OLS bidding regressions and the entry logit regression

are presented in Tables 2 and 3.

17

In the OLS regression, the dependent variable is logarithm of bid, where bid is in millions of

dollars. The size of the project (EngEst) has a strong and positive effect on the bids. Clearly, it

is the most important variable in the OLS regression, so the impact of the other variables is much

smaller. In the order of importance, the next variable is the number of business days needed for

the completion of project (Ndays). Though small, its impact is statistically significant to explain

the variation in bids. We also mention that the project size has a negative (but not statistically

significant) effect on the probability of submitting a bid.

The complexity of the project is captured by the number of items in the construction plan

(Nitems). Table 1 shows that there is a substantial variation in Nitems, with a mean of 49 and

standard deviation of 58 items. One may expect that the cost of preparing a bid is an increasing

function of Nitems. The conjectured effect of Nitems is therefore to reduce the probability of

submitting a bid. The estimate of the Nitems coefficient in the logit regression confirms this

conjecture. Increasing Nitems by 1% at mean level, reduces the odds of bidding by 55%.

A discussion of the variable NPotential, the number of potential bidder, is in demand. Its effect

is statistically significant in both regressions. Having more potential bidders reduces the odds of

submitting a bid, and also results in lower bids. Whether such observed effects are consistent

with the auction model considered in the paper is yet beyond the scope of the paper.3 Instead,

this paper focuses on the participation and bidding behaviors for auctions with same number of

potential bidders. This effectively treats NPotential as an exogenous parameter which simply

defines the bidding environment as in the common knowledge of bidders. By doing so, we avoid the

strong assumption of the exogenous variation in NPotential. Table 4 shows the empirical frequencies

and sample sizes (observed bids) for each N. As a practical matter of implementing our estimator,

it is desirable that our estimation results and interpretation on counterfactual experiments are

comparable across the auctions with different number of potential bidders. Therefore, we choose

to focus our data analysis on the auctions with N = 3, 4, 5, 6.3Interested readers may refer to Li and Zheng (2009) [13] and Marmer, Shneyerov and Xu (2010) [15] for the

comparative studies on the equilibrium bidding with respect to the number of potential bidder.

18

We restrict our empirical analysis in Log-Normal distribution family, that is, F ∼ LogN(µ, σ).

Based on the previous preliminary data analysis, we assume the following functional form specifi-

cations:

µ = θ1 + θ2 · Log(EngEst) + θ3 · Log(Ndays)

σ = θ4 + θ5 · Log(EngEst) + θ6 · Log(Ndays)

κ = θ7 + θ8 · Log(EngEst) + θ9 · Log(Nitems)

The estimation results are presented in Table 5. The EngEst is significant in determining all the

parameters. It has positive impacts on (µ, σ). In other words, the larger the project, the higher

value of moments for bids. However, the larger project reduces the entry cost κ, which may suggest

an economy of scale in the bidding preparation process. Unlike the results in preliminary analysis,

we found that Ndays has no statistically significant impact on the bids. The effect of Nitems on

the bids is interesting. Only for auctions with 6 potential bidders, we found a significant impact of

Nitems. We suspect that for auctions with few potential participants, say 2 or 3, they are large and

complicated in nature already. Therefore, the impact of adding more bidding items does not change

the entry behavior of bidders. Instead, for auctions with certain smaller scale, which could attract

a few bidders in the first place, getting more bidding items into the plan significantly increases the

bidding costs.

5 Potential counterfactual (policy) analysis

There is a small literature that examines the cost of government intervention in the procurement

markets. In particular, these works have been focusing on the small business programs, in which

governments favor the firms with certain characteristics in the procurement process. Looking at

federal dredging contracts, Denes (1997) [6] estimates the effects on the winning bids when some

contracts are set aside for small businesses. Marion (2007) [14] studies the cost effect of the

bid preference program for small businesses in California highway procurement auctions. Both

of these works find that ”assessments of intervention to the programs for disadvantaged firms

(minorities or women) in contracting deserve special attention.” Discrimination by contractors, or

the vulnerability of these programs to fraud through the formation of nonminority front companies,

could weaken or reverse the estimated effect on the procurement costs. For example, Blanchflower

19

and Wainwright (2005) [4] suggest the existence of front companies as a possible reason, when they

find little empirical evidence supporting the impact of the preference programs on minority self-

employment. These messages suggest a great difficulty in studying the preference programs through

a reduced-form approach, when directly analyzing the data observations under the program.

Structural assessment of the bid preferential program has been studied by Krasnokutskaya and

Seim (2006) [10]. They directly investigate the effect of the preference program in the highway

procurement auctions in California. A critical problem they have encountered in their study is

multiplicity of the equilibrium. When bidders are treated asymmetrically, the participation and

bidding equilibrium in the auction model is not unique. Through simulation experiments, they

statistically find one equilibrium which might generate the data.

This paper provides an alternative approach to look at the bid preference program. In a sub-

sample of auctions, the involved bidding firms in Oklahoma can be regarded as symmetric bidders.

Then the unique entry and bidding equilibrium allows one to use the proposed estimation method-

ology to recover bidders’ valuation distribution and entry costs. At the end, one can use these

estimated model elements to conduct policy experiments. In particular, it will make it possible

to predict how the bid preference program may affect the bidding outcome, i.e., the market price

(winning bids). In the policy experiments, the preferential treatment is reflected by the differ-

ent truncation parameters for bidders. By doing so, one no longer face the problem of multiple

equilibriums, which is a fundamental and difficult issue to handle in structural analysis.

The counterfactual exercise in this case amounts to focusing on how to implement asymmetric

auctions towards the optimal outcome. The possible improvement of asymmetric equilibrium on

the auction outcomes has been documented by Celik and Yilankaya (2009) [5] in economic theory.

The researcher can compute the actual improvement a state could have made if the state had been

effectively imposing entry barriers for different groups of bidders.

20

6 Conclusion and extensions

A seminal work by LOV provides a new parametric estimation strategy for analyzing auction data.

They propose a simulated NLLS estimator to approximate the bidders private-value distributions.

Their work greatly broaden the class of distributions that empirical researchers can handle when

analyzing auction data sets.

This paper can be viewed as an extension to LOV’s work and complement to the methodological

literature on the auction data analysis. We propose a simulated-GMM estimation method to a first-

price auction model with endogenous entry. The particular entry pattern in such models makes

LOV’s objective function no longer twice continuously differentiable. The loss of smoothness on the

statistical objective demands another investigation on the asymptotic properties for simulated-based

estimators. We derive a set of sufficient conditions for the consistency and asymptotic normality

to remain valid. We also apply our estimation method to Oklahoma highway procurement auction

data to estimate the model parameters.

The proposed estimation method involves simulating the moments of the observed bids. It is a

vital role that the number of random draws S plays in the simulation. Specifically, S appears in

the variance-covariance matrix. Therefore, it is not a trivial question how the choice of S is going

to affect the estimation accuracy in the finite sample. We leave this for the future research.

Our estimation framework is quite general and the empirical findings are by and large intuitive.

But there is also an important limitation that the future research should address. In auction

datasets, one typically finds that the variation in bids is only partially explained by their variation

within auctions. The between-auction variation is typically present. It is also observed in our

dataset. We deal with this issue by pooling auctions with same number of potential bidders together

and controlling the observed auction heterogeneity. Though our appraoch reasonable and intuitive,

the observed auction heterogeneity may also present and be important in explaining the bids

variation, within IPV paradigm. Recently, krasnokutskaya (2010) [9] has developed a structural

estimation method that can be applied even in the presence of unobserved auction heterogeneity.

21

Another extension would be to allow bidder asymmetries. The obvious difficulty here would be

the necessity to deal with multiple equilibria. Bajari, Hong and Ryan (2009) [3] obtain a number

of identification results in this direction and estimate a parametric model with multiple equilibria

for highway procurement auctions. Finally, incorporating dynamic feature as in Pesendorfer and

Jofre-Bonet (2003) [18] is also left for future research.

22

References

[1] Andrews D.W.K. Consistency in Nonlinear Econometric Models: a Generic Uniform Law of

Large Numbers. Econometrica, 55:1465–1471, 1987.

[2] Andrews D.W.K. Empirical Process Methods in Econometrics. Handbook of Econometrics,

IV, 1994.

[3] Bajari, P. and H. Hong and S. Ryan. Identification and Estimation of a Discrete Game of

Complete Information. Econometrica, 2009. Forthcoming.

[4] Blanchflower D. and J. Wainwright. An Analysis of the Impact of Affirmative Action Programs

on Self-Employment in Construction Industry. NBER Working Paper, 2005.

[5] Celik G. and O. Yilankaya. Optimal Auctions with Simultaneous and Costly Participation.

The B.E. Journal of Theoretical Economics, 9(1):Article 24, 2009.

[6] Denes T. Do Small Business Set-Asides Increase the Cost of Government Contracting? Public

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tions. Econometrica, 68:525–574, 2000.

[8] Hong, H. and M. Shum. Increasing Competition and the Winner’s Curse: Evidence from

Procurement. Review of Economic Studies, 69(4):871–898, 2002.

[9] Krasnokutskaya, E. Identification and Estimation in Highway Procurement Auctions under

Unobserved Auction Heterogeneity. Review of Economic Studies, 2010. Forthcoming.

[10] Krasnokutskaya, E. and K. Seim. Bid Preference Programs and Participation in Highway

Procurement Auctions. 2006. Working paper, University of Pennsylvania.

[11] Laffont J., H. Ossard and O. Vuong. Econometrics of First-Price Auctions. Econometrica,

63(4):953–980, 1995.

[12] Levin, D. and J. Smith. Equilibrium in Auctions with Entry. The American Economic Review,

84(3):585–599, 1994.

23

[13] Li, T. and X. Zheng. Entry and Competition Effects in First-Price Auctions: Theory and

Evidence from Procurement Auctions. Review of Economic Studies, 2009. Forthcoming.

[14] Marion J. Are Bid Preferences Benign? The Effect of Small Business Subsidies in Highway

Procurement Auctions. Journal of Public Economics, 91(7-8):1591–1624, 2007.

[15] Marmer, V., and S. Shneyerov and P. Xu. What Model for Entry in First-Price Auctions? A

Nonparametric Approach. 2010. Working Paper, University of British Columbia.

[16] Milgrom, P. Putting Auction Theory to Work. Cambridge University Press, 2004.

[17] Newey M.K. and D. McFadden. Large Sample Estimation and Hypothesis Testing. Handbook

of Econometrics, IV, 1994.

[18] Pesendorfer, M. and M. Jofre-Bonet. Estimation of a Dynamic Auction Game. Econometrica,

71(5):1443–1489, 2003.

[19] Samuelson, W. Competitive Bidding with Entry Costs. Economics Letters, 17(1):53–57, 1985.

[20] Xu, P. Nonparametric Estimation of the Entry Cost in First-Price Procurement Auctions.

2009. Working Paper, University of Hong Kong.

24

A Proofs.

Proof of Theorem 1. The proof amounts to checking all the conditions of the The-

orem 2.1 in Newey and McFadden (1994) [p.2121] [17]. To ease the exhibition, we suppress the

subscriptions at the places where the confusions may not be caused.

Both identifiability condition and the compactness of parameter space Θ hold under the assump-

tions. We now verify that l1, ...,lk are continuous with respect to θ. To this end, it suffices to show

l1(Z, θ) is continuous in θ. We note that l1 = L1 + L2 + L3 + L4, where

L1 =∫ v

vρ(Z,θ)

∫ v

vρ(Z,θ)

y(N − 1)F (y;Z, θ)N−2

F (v;Z, θ)N−1f(y;Z, θ)f(v;Z, θ) dy dv

L2 =∫ v

vρ(Z,θ)

∫ vρ(Z,θ)

v

vρ(Z, θ)(N − 1)F (y;Z, θ)N−2

F (v;Z, θ)N−1f(y;Z, θ)f(v;Z, θ) dy dv

L3 =∫ vρ(Z,θ)

vr f(v;Z, θ) dv = r · F (vρ(Z, θ);Z, θ)

L4 =∫ v

vρ(Z,θ)

−κ(Z, θ)F (v;Z, θ)N−1

f(v;Z, θ) dv

Given the continuity of F , f and κ in the assumption 1, one can easily see that L1, ..., L4 are

continuous in θ. So are all the l’s.

Then we are left to show the uniform convergence of GS,T (θ), that is, supθ∈Θ |GS,T (θ)−G(θ)| → 0

in probability. To this end, we invoke the generic uniform law of large numbers of Andrews (1987)

[1]. We first note that,

|X [1]| ≤ | v(N − 1)F (vρ;Z, θ)N−1

|+ r <∞

where the second inequality follows from the assumption that the cutoff vρ is strictly bounded away

from lower bound v. Therefore, for ı ∈ 1, 2, ..., k, X [ı] is bounded. Together with the boundedness

of fz in assumption, the A6(b) in Andrews (1987) [1] is satisfied, i.e.,∫

supθ∈Θ |JS,T (Z, θ)|fz(Z)dZ <

∞. Moreover, the assumption FzZ : F−1(., Z, θ)− vρ(Z, θ) = 0 = 0 implies that the Assumption

A6(a) in Andews (1987) [1] holds, that is, JS,T (Z, θ)fz(Z) is continuous in θ almost everywhere

with respect to fz. This completes the proof.

25

Proof of Theorem 2. For the asymptotic normality results, we use the Theorem 7.2 in

Newey and McFadden (1994) [p.2186] [17]. The proof amounts to checking the following conditions

(i) J(θ) is differentiable at θ0 with derivative such that ′W is nonsingular.

(ii) for any δT → 0,

sup‖θ−θ0‖≤δT

√T‖ JS,T (θ)− JS,T (θ0)− J(θ) ‖

1 +√T ‖ θ − θ0 ‖

→ 0.

(iii)√TJS,T (θ0)→ N(0,Ω).

To derive , we first investigate ∂l1(θ)∂θ . To this end, we note that the derivative of l1, can be

decomposed into the following,

∂(L1 + L2)∂θ

=∫ v

vρ

∫ v

vmaxy, vρ

∂φ(θ)∂θ

dy dv +∫ v

vρ

∫ vρ

vφ(θ) dy dv

∂vρ∂θ− vρ · f(vρ)

∂vρ∂θ

∂L3

∂θ= r · f(vρ)

∂vρ∂θ

+ r · ∂F (vρ, θ)∂θ

∂L4

∂θ=

∫ v

vρ

κ(θ)F (v; θ)N−1

f(v; θ), dv =κ(θ)N − 2

[1− F (vρ; θ)2−N]

where φ(θ) = (N−1)F (y,θ)N−2

F (v;θ)N−1 f(y; θ) f(v, θ). Therefore,

∂l1(θ)∂θ

=∫ v

vρ

∫ v

v1(v ≥ y) maxy, vρ

∂φ(θ)∂θ

dy dv +1− F (vρ)2−N

N − 2∂κ(θ)∂θ

+F (vρ, θ)− F (vρ, θ)N−1

N − 2∂vρ∂θ

+ vρ ·∂F (vρ, θ)

∂θ

The differentiability of F and f in assumptions ensures that l1(θ) is differentiable at θ0. Extending

the argument to higher order moments of bids, lι∀ι ∈ 1, 2, ..., k, we note that lι can be written as

lι =∫ v

vρ

∫ v

vρ

y(N − 1)F (y, θ)N−2

F (v; θ)N−1f(y, θ) dy +

∫ vρ

v

vρ(N − 1)F (y, θ)N−2

F (v; θ)N−1f(y, θ) dy

− κ(θ)F (v; θ)N−1

ιf(v; θ) dy +∫ vρ

vrι f(v; θ) dv

Again, it is not hard to see that the assumed differentiability allows the condition (i) to hold in our

context.

Verify the non-singularity here.

26

Next, we show that the stochastic equicontinuity condition (ii) holds. To this end, we invoke the

Andrews (1994) [2] empirical process methods by checking the JS,T is type IV class function, which

satisfies Ossiander’s entropy condition and, in turn, implies condition (ii) to hold. For the desired

result, we want ∀Z, and ∀θ ∈ Θ,

(E supθ1:‖θ1−θ0‖<δ

| JS,T (θ1)− JS,T (θ0) |2)1/2 ≤ Cδψ (A.18)

where ∀δ > 0 in a neighborhood of 0, for some finite positive constants C and ψ. We note that

for all ι ∈ 1, 2, ..., k, the non-smoothness of X [ι], and therefore JS,T , comes from the indicator

function of the form 1F−1(u, Z, θ)− vρ(Z, θ) ≷ 0. It can be shown that condition (A.18) holds,

if

supt

supθ∈Θ‖ ∂[F−1(u, Z, θ)− vρ(Z, θ)]/∂θ ‖≤ C1 <∞

for some constant C1. In turn, the assumed boundedness of f and F ensures the condition (ii) to

hold. Condition (iii) follows directly from a central limit theorem in the i.i.d case. This completes

the proof.

Proof of Theorem 3. It suffices to verify that Ω→ Ω and → . The first convergence

is an immediate consequence of unbiased estimation of covariance. Therefore, we are left to show

the second convergence for the desired results. To this end, first, we note that the moments of bids,

lι∀ι ∈ 1, 2, ..., k, can be written as

lι =∫ v

vρ

∫ v

vρ

y(N − 1)F (y, θ)N−2

F (v; θ)N−1f(y, θ) dy +

∫ vρ

v

vρ(N − 1)F (y, θ)N−2

F (v; θ)N−1f(y, θ) dy

− κ(θ)F (v; θ)N−1

ιf(v; θ) dy +∫ vρ

vrι f(v; θ) dv

Taking the first order derivative with respect to θ yields

∂lι

∂θ=∫ v

vρ

ι · φι−1(v, θ)∂φ(v, θ)∂θ

f(v) dv +∫ v

vρ

φι(v, θ)∂f(v, θ)∂θ

dv + rι∂F (vρ; θ)

∂θ

where

φ(v, θ) =∫ v

vρ

y(N − 1)F (y, θ)N−2

F (v; θ)N−1f(y) dy +

∫ vρ

v

vρ(N − 1)F (y, θ)N−2

F (v; θ)N−1f(y) dy − κ(θ)

F (v; θ)N−1

27

and

∂φ(v, θ)∂θ

=∫ v

v1y < vmaxy, vρ

∂

∂θ

[(N − 1)F (y, θ)N−2

F (v, θ)N−1f(y, θ)

]dy

+F (vρ)(N − 1)F (v)N−1

∂vρ∂θ− 1F (v)N−1

∂κ(θ)∂θ

+κ(θ)

(N − 1)F (v)N∂F (v; θ)∂θ

It is not hard to see that the sample counterparts converge to their population, as the way we

decompose the formula. Therefore, the second convergence holds. This completes the proof.

28

c

Table 1: Description and Summary Statistics of Variables

Variable Description mean Std. Dev Min Max

EngEst The engineers' estimate 2.45E+06 3.47E+06 66500 2.48E+07

Bid Submitted Bid 2.74E+06 3.61E+06 69635 2.65E+07

Nitems Number of pay items in the project ad 49.0361 57.6513 1 363

Ndays Number of business days to complete the proje 146.3682 127.3332 10 681

Length Length of the road in miles 5.2125 4.9361 0 36.63

Distance Distance in miles from the headquarters of 344.2369 382.4686 0 1.70E+03the bidding firm to the project site

Backlog Total amount of unfinised work on a given day 0.2188 0.2971 0 1and normalized by the bidder specific maximum

Npotential Number of potential bidders 6.2635 3.5772 1 26

Nactual Number of actual bidders 3.0217 1.3485 1 7

Table 2: OLS Estimates of Log(Bid) on Explanatory Variables

Estimate Sd. Error t-stat

log(EngEst) 0.98079 0.00750 130.71log(Nitem) 0.00287 0.00612 0.47Distance 0.00000 0.00001 -0.42Backlog -0.11489 0.01231 -0.93Ndays 0.00015 0.00006 2.51Length 0.00143 0.00092 1.56Npotential -0.00477 0.00131 -3.65Constant 0.26758 0.08624 3.10

Adj R-squared = 0.9931

Table 3: Logit Eestimates of Entry Decision on Explanatory Variables

Estimate Sd. Error z-stat

log(EngEst) -0.04014 0.10843 -0.37log(Nitems) -0.26903 0.09632 -2.79Distance -0.00010 0.00013 -0.78Backlog 0.13713 0.17088 0.8Ndays 0.00052 0.00082 0.63Length -0.00830 0.01392 -0.6Npotential -0.09145 0.01832 -4.99Constant 2.18045 1.24751 1.75

Log likelihood = -1125.961

significant at 5% (marked in bold)

Table 4: Sample Summaries by Number of Potential Bidders

N Number of Auctions Number of Bids Rate of Participation

2 12 20 0.833 46 99 0.724 39 106 0.685 31 92 0.596 35 125 0.607 19 59 0.448 22 67 0.389 18 65 0.40

10 9 39 0.4311 13 44 0.3112 7 31 0.3713 7 28 0.3114 3 17 0.4015 3 15 0.3316 1 6 0.3817 1 6 0.3518 1 4 0.2226 1 5 0.19

Table 5: Simulated GMM Estimation Results

Variable N=3 N=4 N=5 N=6mu sigma kappa mu sigma kappa mu sigma kappa mu sigma kappa

Log(EngEst) 0.97 0.43 -1.67 0.98 0.32 -2.68 0.99 0.31 -2.76 0.98 0.29 -1.59(std. error) 0.00 0.11 0.13 0.01 0.10 0.12 0.01 0.09 0.11 0.00 0.19 0.13

Log(Ndays) 0.02 0.02 0.02 0.03 0.02 0.01 0.01 0.03(std. error) 0.02 0.14 0.02 0.14 0.03 0.14 0.01 0.14

Log(Nitems) 0.15 0.12 0.12 0.17(std. error) 0.13 0.10 0.11 0.13

Constant 0.18 -4.42 7.41 0.19 -4.43 8.01 0.15 -4.50 7.90 0.18 -4.29 7.41(std. error) 0.11 1.24 1.58 0.14 1.13 1.67 0.12 1.22 1.34 0.09 1.24 1.02