Identification in Ascending Auctions, with an Application to Digital Rights Management

NBER WORKING PAPER SERIES

IDENTIFICATION IN ASCENDING AUCTIONS, WITH AN APPLICATION TO

DIGITAL RIGHTS MANAGEMENT

Joachim Freyberger

Bradley J. Larsen

Working Paper 23569

http://www.nber.org/papers/w23569

NATIONAL BUREAU OF ECONOMIC RESEARCH

1050 Massachusetts Avenue

Cambridge, MA 02138

July 2017

We thank Dominic Coey, Dan Quint, Yoshi Rai, and Caio Waisman for helpful comments. This

project was supported by NSF Grant SES-1530632. The views expressed herein are those of the

authors and do not necessarily reflect the views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been

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Identification in Ascending Auctions, with an Application to Digital Rights Management

Joachim Freyberger and Bradley J. Larsen

NBER Working Paper No. 23569

July 2017

JEL No. C1,C57,D44,L0,L96,O3

ABSTRACT

This study provides new identification and estimation results for ascending (traditional English or

online) auctions with unobserved auction-level heterogeneity and an unknown number of bidders.

When the seller's reserve price and two order statistics of bids are observed, we derive conditions

under which the distributions of buyer valuations, unobserved heterogeneity, and number of

participants are point identified. We also derive conditions for point identification in cases where

reserve prices are binding (in which case bids may be unobserved in some auctions) and present

general conditions for partial identification. We propose a nonparametric maximum likelihood

approach for estimation and inference. We apply our approach to the online market for used

iPhones and analyze the effects of recent regulatory changes banning consumers from

circumventing digital rights management technologies used to lock phones to service providers.

We find that buyer valuations for unlocked phones dropped after the unlocking ban took effect.

Joachim Freyberger

Department of Economics

University of Wisconsin-Madison

1180 Observatory Drive

Madison, WI 53706

[email protected]

Bradley J. Larsen

Department of Economics

Stanford University

579 Serra Mall

Stanford, CA 94305

and NBER

[email protected]

2 FREYBERGER AND LARSEN

1. Introduction

This paper presents an approach to jointly solve two identiﬁcation challenges to

empirical auctions work in ascending auctions: unobserved heterogeneity at the auc-

tion level and an unknown number of bidders. Unlike sealed-bid auctions, ascending

auctions—both traditional English auctions as well as online auctions—proceed se-

quentially, and some potential bidders planning to place a bid are not observed doing

so. Hence the number of bidders (N)—a key element for identiﬁcation arguments

in empirical auctions methods—is often unobserved to the researcher. Previously

proposed solutions to this problem of unknown N rely on the assumption of indepen-

dent, private values (IPV), and consequently little empirical work exists in English or

online auctions with unknown N outside of the IPV framework. The IPV framework

does not allow for bidder’s values to be correlated through unobserved heterogeneity

in the auctioned items, but such unobserved heterogeneity is common in practice.

Previous research has suggested solutions to this challenge of unobserved heterogene-

ity, but these methods require that the researcher observe N and, furthermore, while

useful in ﬁrst price auctions (where often all bids are observed by the researcher)

these methods do not immediately apply to English or online auctions given the in-

completeness of bid data in ascending auctions (where the researcher rarely observes

all bids—i.e., the thresholds at which each player would drop out of the bidding).

In this paper, we provide a uniﬁed framework for nonparametric identiﬁcation and

estimation when both problems exist. In particular, we derive conditions for point

identiﬁcation of the distributions of bidder valuations, unobserved heterogeneity, and

the number of bidders, as well as partial identiﬁcation results when these conditions

are not met.

We build on the identiﬁcation arguments of Song (2004), who suggested an ap-

proach to handling settings where the number of bidders is unknown and the re-

searcher observes multiple order statistics of bids in English or online auctions. The

For example, in online auctions, listings often contain pictures and detailed descriptions about

characteristics of the items sold that both the seller and potential buyers can observe, but such

information is diﬃcult for the econometrician to quantify. Therefore, items in diﬀerent auctions can

diﬀer dramatically in ways that are observable to the seller and bidders but not to the econometrician.

IDENTIFICATION IN ASCENDING AUCTIONS 3

Song (2004) approach relies on the assumption that bidders have independent private

values, in which case the density of a higher order statistic conditional on a lower order

statistic will not depend on N. We demonstrate that the same argument holds even

if bidders valuations are only independent conditional on auction-level heterogeneity

and such heterogeneity is unobserved by the econometrician. We also demonstrate

that the distribution of the number of bidders is identiﬁed.

To nonparametrically identify the distribution of unobserved heterogeneity, we use

a similar approach to Li and Vuong (1998), Li, Perrigne, and Vuong (2000), and Kras-

nokutskaya (2011), relying as they do on the deconvolution result of Kotlarski (1967).

These approaches require that the researcher observe two bids that are independent

conditional on auction-level unobserved heterogeneity. This approach has been ap-

plied to ﬁrst price auctions in a number of papers (see, for example, Decarolis 2017,

Krasnokutskaya and Seim (2011), and others). Various studies (Athey and Haile 2002;

Athey, Levin, and Seira 2011; Aradillas-L´opez, Gandhi, and Quint 2013), however,

have highlighted that the deconvolution approach to unobserved heterogeneity can-

not be applied to English or online auctions using bids alone because bids represent

order statistics and not all order statistics are observed, leading to correlation in the

observed bids even when individual valuations represent independent draws from the

same underlying distribution.

Our approach circumvents this issue of correlated order statistics by relying on

an alternative measure of unobserved heterogeneity available to the researcher in

many settings. Speciﬁcally, we rely on sellers’ reserve prices reported to the auction

platform. We demonstrate that when reserve prices are secret or non-binding, the

distributions of unobserved heterogeneity and buyer valuations are nonparametrically

point identiﬁed. When reserve prices are public, this can introduce correlation be-

tween reserve prices and observed bids, as bids are only recorded if they exceed the

public reserve price. We demonstrate how these binding reserve prices aﬀect the like-

lihood of observed bids and we derive support conditions under which we still obtain

point identiﬁcation. When these conditions are not met, our results yield partial iden-

tiﬁcation. The data requirements for all of our identiﬁcation arguments, in which the

4 FREYBERGER AND LARSEN

researcher is concerned with both unobserved heterogeneity and an unknown number

of bidders, are the following: 1) the econometrician observes the seller’s reserve price;

2) if reserve prices are secret, the econometrian observes at least two order statis-

tics of bids; and 3) if reserve prices are public, the econometrian observes two order

statistics of bids if these exceed the reserve price.

For estimation we propose a nonparametric maximum likelihood approach to jointly

estimate the distibutions of unobserved heterogeneity and buyer valuations. Our es-

timator diﬀers from previous studies relying on the result of Kotlarski (1967), in that

these previous studies have generally applied an explicit deconvolution approach, di-

rectly estimating a joint characteristic function and then applying inverse Fourier

transforms to recover underlying distributions. We take an alternative approach

based on nonparametric sieve maximum likelihood estimation (see Gallant and Ny-

chka 1987), which allows us to estimate all parameters in one step. Our framework

also nests the possibility of using ﬂexible parametric or semiparametric models for

the unknown distributions.

We apply the approach to study the impact of recent legislation regarding con-

sumers circumventing digital rights management. Digital rights management refers

to technological locks restricting how consumers use software or hardware. These

digital locks are used in computer software, e-books, music, ﬁlm, cell phones, and

in many other products. The US Digital Millenium Copyright Act (DMCA) bans

circumvention of these digital locks or production of technologies intended to aid con-

sumers in circumventing digital locks. However, tools and tips for how to cirumvent

digital locks are readily available on the Internet, and punishment mechanisms for

violators of these laws are not necessarily salient to consumers. Therefore, it is un-

clear whether the DMCA or related legislation has any eﬀect in practice on market

primitives, such as consumers’ willingness to purchase—or sellers’ willingness to sell—

potentially illegally tampered products. Using data from auctions of used iPhones, we

analyze the impact of a recent regulatory change banning smartphone unlocking on

bidder valuations for unlocked phones. While our application is primarily included as

an illustration of our methodology, it provides insights into this previously unstudied

IDENTIFICATION IN ASCENDING AUCTIONS 5

question. In particular, we ﬁnd that buyer valuations for unlocked smartphones were

lower after the ban was put into place than before. The estimated diﬀerence in the

means of the distributions of buyer valuations for unlocked smartphones in the pre

and post periods corresponds to a decrease in the dollar valuation for the phones of

about 27%. This diﬀerence suggests that the regulatory change may indeed have had

real eﬀects on consumers’ willingness to engage in potentially shady behavior.

Related Literature

This paper contributes to several strands of the empirical auctions literature. First,

it extends the approach of Song (2004), which Hortacsu and Nielsen (2010) argued

has long been “the standard to beat in the empirical online auctions literature” due to

its distinct ability to handle an unknown number of bidders. Other papers building

on the Song (2004) approach include Kim and Lee (2014), who developed a test

of the independent private values assumption in the Song (2004) environment, and

Sailer (2006), who embedded the Song (2004) approach in a repeated auction setting.

Other approaches to handling an unknown number of bidders include An, Hu, and

Shum (2010), who considered ﬁrst price auctions and demonstrated identiﬁcation

when the econometrician observes an instrument for the number of potential bidders

(not all of whom necessaily place bids).

Canals-Cerd´a and Pearcy (2013), Platt

(2015) and Hickman, Hubbard, and Paarsch (2016) provided approaches to estimating

the distributions of valuations in online auctions with independent private values,

obtaining identiﬁcation by exploiting the arrival order of bidders. Our nonparametric

identiﬁcation argument is the ﬁrst of which we are aware for ascending auctions with

unobserved heterogeneity and an unknown number of bidders.

Hu, McAdams, and Shum (2013) extended the results of An, Hu, and Shum (2010) to apply to

settings with non-separable unobserved auction-level heterogeneity (where the number of potential

bidders in An, Hu, and Shum (2010) can be considered a form of unobserved heterogeneity in their

model) when three bids are observable in ﬁrst price auctions. Additional work studying unobserved

heterogeneity in ﬁrst price auctions includes Armstrong (2013), and Balat (2015, 2016). As explained

above, existing deconvolution approaches (Li and Vuong 1998; Li, Perrigne, and Vuong 2000; Kras-

nokutskaya 2011) have thus far been applied primarily in ﬁrst price auctions (with a known number

of bidders) where, unlike ascending auctions, independent bids are available.

Canals-Cerd´a and Pearcy (2013) provide a parametric identiﬁcation result that incorporates unob-

served heterogeneity.

6 FREYBERGER AND LARSEN

Second, our paper illustrates the use of variation in potentially binding reserve

prices to obtain partial identiﬁcation. Athey and Haile (2007) discussed several al-

ternative uses of reserve price information. Recent work by Decarolis (2017) applied

the Krasnokutskaya (2011) approach using the reserve price and transaction price

from ﬁrst price auctions as two measurements of unobserved heterogeneity, focusing

on a sample in which reserve prices were nearly always non-binding in order to avoid

the issue of correlation between bids and reserve prices that we address in this pa-

per. Roberts (2013) presented a control function approach to handling unobserved

heterogeneity in settings in which the reserve price is monotonic in the unobserved

heterogeneity, which does not apply in the settings we study given that reserve prices

are chosen by sellers who may have a privately known valuation for the good, and

hence reserve prices may not be monotonic in the realization of unobserved auction-

level heterogeneity.

Third, our approach provides new positive identiﬁcation results for ascending auc-

tions. As demonstrated in Athey and Haile (2002), even when the number of bidders

is known, the joint distribution of bidder valuations is not identiﬁed in ascending

auctions under arbitrary correlation because the willingness to pay of the highest-

valuation bidder is rarely observed (it is never observed in English auctions, but may

be observed in some online auction data) and often bids of lower-value bidders are

also not observed. Instead of considering arbitrary correlation, we focus on a par-

ticular form of correlation among valuations through additively (or multiplicatively)

separable auction-level heterogeneity unobserved to the econometrician. Several re-

cent papers have demonstrated that certain objects of interest, such as bounds on

optimal reserve prices, or buyer and seller surplus, are identiﬁed in ascending auc-

tion settings with correlated private values under the assumption that the number

of bidders is known (Aradillas-L´opez, Gandhi, and Quint 2013; Coey, Larsen, and

Sweeney 2016; Coey, Larsen, Sweeney, and Waisman 2017). Unlike these studies, our

approach yields estimates of the underlying valuation distributions, which are useful

for studying revenue and welfare under counterfactual auction formats. Quint (2015)

provides an alternative argument, related to these papers, that also yields valuation

IDENTIFICATION IN ASCENDING AUCTIONS 7

identiﬁcation when the number of bidders is unobserved to the econometrian and

when observations are available from auctions under at least two diﬀerent known

probability distributions for the number of bidders.

Our application contributes to a small literature on digital rights management

(e.g. Liu, Safavi-Naini, and Sheppard 2003; Von Lohmann 2004; Walker 2003; Stall-

man 1997) and the literature on piracy and copyright enforcement more broadly

(Harbaugh and Khemka 2010). It remains an open question in this literature how

eﬀective regulation is at altering consumers’ willingness to engage in circumvention.

The speciﬁc application of cellphone unlocking relates to a variety of previous stud-

ies that have examined the role digital locks play in raising switching costs of con-

sumers. This work has focused on non-US markets, including, among others, Tallberg,

H¨amm¨ainen, T¨oyli, Kamppari, and Kivi (2007) (studying Finland), Maicas, Polo, and

Sese (2009) (studying Spain), Nakamura (2010) (studying Japan), and Park and Koo

(2016) (studying South Korea). Baker (2007) describes several costs consumers face

when unlocking a phone, including time and monetary costs and potential invalidation

of the handset’s warranty. Farrell and Klemperer (2007) describe general theoretical

arguments for how lock-in practices such as handset locking can create ineﬃcien-

cies and increase ﬁrm proﬁts, in particular in settings with network eﬀects such as

telecommunications markets. Finally, in focusing our application on smartphones,

we contribute to a nacent literature on this industry more broadly. Sinkinson (2014)

and Zhu, Liu, and Chintagunta (2015) examine exclusive contracting deals between

Apple and AT&T. Fan and Yang (2016) provides a broad study of the welfare eﬀects

of product proliferation and competition in the smartphone industry.

2. Identification

2.1. Introduction of Model. We analyze static, single-unit ascending auctions

where bidders have symmetric private values. For each bidder i, we specify the value

of bidder i to take the following form:

= X + U

8 FREYBERGER AND LARSEN

In many settings, as in our empirical application, the researcher may prefer to model

valuations in a multiplicative form, V

= e

and work with logs; all our results

hold under multiplicative separability as well, but we state the additively separable

version here for ease of exposition. The random variable X is independent of U

for

all i and represents a common component through which bidders’ valuations (and the

seller’s reserve price, described below) are correlated. This X is observed by bidders

and the seller but is unobserved to the econometrician. Let U

∼ F

with density f

X ∼ F

with density f

, and V

∼ F

with density f

. We assume bidders follow

the weakly dominant strategy of bidding their valuations, and hence we will refer to

bids and valuations interchangeably.

Let B

be the bid of bidder i.

Let N be a random variable with realizations n representing the number of bidders

in an auction. We assume N is independent of X and U. Let U

refer to the j

highest

U in an auction with at least j bidders but unconditional on the actual realization

of the number of bidders N . We refer to U

as the j

unconditional order statistic,

to distinguish it from the traditional use of the term order statistic, which refers to

an order statistic conditional on a realization or N = n.

Thus, U

is the maximum

unconditional order statistic, U

is the second unconditional order statistic, etc., and

similarly for order statistics of other random variables (e.g. B and V ).

We allow for the seller’s reserve price to be either secret or public. To be precise, in

this paper a reserve price is termed to be public if the auction is such that only bids

exceeding the reserve price are recorded, and a reserve price is termed to be secret if

As the primary focus in this study is unobserved heterogeneity and the unknown number of bidders,

we do not focus on bidders potentially bidding below their values as in Haile and Tamer (2003).

We use this notation and terminology, rather than the traditional notation U

n:n

, U

n−1:n

, etc.,

because in our case the order statistics come from samples of varying sizes. That is, U

is the j

highest U among N bidders, unconditional on the realization of the random variable N , and is thus

a draw from the distribution

(u) ≡

Pr(N = n|N ≥ j)F

n−j+1:n

(u)

where F

n−j+1:n

is the distribution of the j

highest bid conditional on N = n, which, given that

draws of U are i.i.d., is given by the following (see David and Nagaraja 2003):

n−j+1:n

(u) ≡





k=n−j+1





(u)

(1 − F

(u))

n−k





IDENTIFICATION IN ASCENDING AUCTIONS 9

the auction is such that bids need not exceed the reserve price in order to be recorded.

We specify the seller’s reserve price as

R = X + W.

We assume that W is independent of (X, U, N). Let W ∼ F

with density f

and

R ∼ F

with density f

. We do not directly model the seller’s valuation or choice

of reserve price (nor assume that these reserve prices are optimal), but rather simply

assume reserve prices take the above form, as in Decarolis (2017). Roberts (2013)

takes a diﬀerent approach, assuming away the seller-speciﬁc term W and assuming

instead that R is an unknown monotonic function of X. Under standard auction

rules, optimality of the reserve price combined with an additively (or multiplicatively)

separable valuation for the seller would be suﬃcient conditions for the reserve price

to take the form we assume.

We assume the econometrician observes realizations of R. In the secret reserve

price case, we assume the econometrician also observes at least two unconditional

order statistics of bids, B

and B

, with j < k. In the public reserve price case, the

econometrian observes B

and B

if these exceed R. In other words, in the secret

reserve price case, we assume that at least k bidders wish to place a bid and indeed

do place a bid, whereas, in the public reserve price case, at least k bidders wish to

place a bid but may be prevented from doing so if the reserve price exceeds their

valuation. The econometrician does not observe realizations of X, U, W , or N. We

demonstrate identiﬁcation of the distributions of each of these random variables. We

If the auction rules are such that the highest bidder wins the good if and only if B

≥ R, paying

R when B

≥ R > B

and paying B

otherwise, then the optimal reserve price for a seller of value

X + S, where S is independent of X and U

for all i, would satisfy

R = X + S +

1 − F

(R)

= X + S +

1 − F

(R − X)

Letting W be the random variable such that W = S +

1−F

(W )

will yield the form R = X + W ,

as above, with W independent of X and U . A common alternative rule for ascending auctions (e.g.

Larsen 2014) with secret reserve prices is that the highest bidder wins the good if and only if B

> R

and pays B

. In this case a seller with value X + S would optimally choose a reserve price of X + S,

yielding again the form R = X + W above, with W = S.

10 FREYBERGER AND LARSEN

describe three main identiﬁcation results: First, we obtain point identiﬁcation when

reserve prices are secret. Second, we obtain point identiﬁcation when reserve prices

are public and a support condition is satisﬁed. Third, we obtain partial identiﬁcation

when reserve prices are public and the support condition is not satisﬁed.

Throughout, we will let subscripts j and k, with j < k, be any ﬁxed, positive

integers. We denote the lower bound of the support of any random variable Y by Y

and the upper bound by Y . Notice that V

= V

= V and V

= V

= V because

and V

are unconditional order statistics from the same distribution as V , and

similarly for the supports of U, U

, and U

. Let φ

(t) denote the characteristic

function of a random variable Y .

We summarize the key assumptions from our discussion thus far in Assumption 1.

Assumption 1. (i) For an auction with n bidders, R = X + W and V

= X + U

for

i = 1, . . . n, where X, W , U

, . . . , U

are mutually independent, (ii) N is independent

of X, W , and U

for all i = 1, . . . , N, (iii) B

= V

and B

= V

We also assume the following:

Assumption 2. E[|B

| + |B

| + |R|] < ∞ and E[X] = 0.

Assumption 3. (i) φ

and φ

have only isolated real zeros. (ii) The real zeros of

and φ

are disjoint.

Notice that the means of X, W , U, are not identiﬁed without a location normal-

ization. To see why let

X = X − c,

W = W + c,

= U

+ c. Then V

X +

and R =

X +

W . To identify the distributions we therefore impose in Assumption 2

that E[X] = 0, but normalizing the mean of W instead yields analogous results. The

other moment condition in Assumption 2 is a mild regularity condition and Assump-

tion 3 imposes technical conditions on characteristic functions, which are satisﬁed by

standard distributions. These type of conditions are common in models with multiple

measurements; see, for example, Li and Vuong (1998).

IDENTIFICATION IN ASCENDING AUCTIONS 11

2.2. Identiﬁcation with Secret Reserve Prices. In the case of secret reserve

prices, when the econometrician observes B

, B

, and R, we obtain the following

result:

Theorem 1. Suppose that Assumptions 1-3 hold. Then F

, F

, and F

are identi-

ﬁed from the joint distribution of bids B

and B

and secret reserve prices R. If the

number of points of support of N is ﬁnite and F

is continuous, then the distribution

of N is identiﬁed as well.

The theorem implicitly assumes that we always observe B

and B

, which implies

that N ≥ k. We could allow for the possibility that realizations of N may be less

than k, in which case we have to assume that, whenever N ≥ k, we always observe B

and B

. We then can simply do our analysis conditional on N ≥ k and still identify

, F

, and F

because X, W , and U are independent of N. The distribution of the

number of bidders is then identiﬁed conditional on N ≥ k (that is P (N = n | N ≥ k)

for all n ≥ k) under the assumptions of Theorem 1.

The formal proof, which is in the appendix, proceeds in three steps. First we

use one observed bid and the reserve price to identify the distributions of X and

W , which follows from an extension of Kotlarski’s Lemma; see Kotlarski (1967) and

Evdokimov and White (2012). While the formal arguments are more involved, it is

easy to see that the ﬁrst two moments of X and W are identiﬁed from E[X] = 0,

E[W ] = E[R], var(X) = cov(W, U

), and var(W ) = var(R) − var(X). Second, we

show that knowledge of the characteristic function of X implies identiﬁcation of the

joint distribution of U

and U

. Finally, arguments related to those in Song (2004)

then yield identiﬁcation of the distribution of valuations and the number of bidders.

Also notice that since X, W , and U are independent, identiﬁcation of the marginal

distributions is equivalent to identiﬁcation of the joint distribution. The arguments in

the proof of the theorem can also be used to demonstrate that it is generally possible

to identify a parametric distribution of N, even if the support is inﬁnite, for example,

if N followed a Poisson distribution.

12 FREYBERGER AND LARSEN

2.3. Identiﬁcation with Public Reserve Prices. In the case of public reserve

prices, bids will only be observed if they lie above R. Deﬁne D

= 1(R > B

≥ B

= 1(B

≥ R > B

), and D

= 1(B

≥ B

≥ R). We assume that the observed

data is a random sample from the distribution of (R, D

, D

, B

·(D

), B

·D

)

with D

+ D

= 1. Notice that we therefore assume that N ≥ k and that if

is not observed, then B

< R. Point identiﬁcation is still achieved in this case as

long as the support of B is greater than that of R in the strong set order, as we state

in the following result.

Theorem 2. Suppose Assumptions 1, 2, and 3(i) hold and that R ≤ B < ∞. Then

, F

, and P (U ≤ u

| U ≥ u

) for all u

, u

≥ W are identiﬁed from the joint

distribution of (R, D

, D

, B

·(D

+ D

), B

·D

). Moreover, if in addition B ≥

R > −∞, then (i) F

is identiﬁed and (ii) the distribution of N is identiﬁed if N has

ﬁnite support and F

is continuous.

The intuition for the identiﬁcation result is as follows. By the additivity assumption

and independence, conditioning on B

= B is equivalent to conditioning on X = X

and U

= U. Since W is independent of X and U, it follows that

P (R ≤ r | B

= B) = P (R ≤ r | X = X, U

= U)

= P (W ≤ r − X).

Hence, the distribution of W is identiﬁed up to a location shift, which is ﬁxed by the

assumption that E[R] = E[W ]. Similarly, the joint distribution of U

and U

is iden-

tiﬁed by considering P (B

≥ b

, B

≥ b

| R = R). Finally, using similar arguments

as in the proof of Theorem 1 we can then show identiﬁcation of the distributions of

X, U, and N. If B < R, however, then only P (U ≤ u

| U ≥ u

) for all u

, u

≥ W

is identiﬁed, but we cannot point identify P (U ≤ u).

3. Estimation and Inference

In this section we discuss estimation of the unknown densities f

, f

, and

in both the secret and the public reserve price cases using a nonparametric or

IDENTIFICATION IN ASCENDING AUCTIONS 13

semiparametric maximum likelihood approach. Our approach does not require esti-

mating P (N = n). We also describe an inference procedure for certain features of

the auctions, which is robust to a lack of point identiﬁcation. Finally, we describe

our speciﬁc recommendation of a nonparametric or semiparametric estimator.

3.1. Estimation and Inference in the Secret Reserve Case. In the secret re-

serve price case, the likelihood of the joint distribution of B

, B

and R can be

obtained by ﬁrst writing

P (B

≤ b

, B

≤ b

, R ≤ r)

P (B

≤ b

, B

≤ b

, R ≤ r | X = x)f

(x)dx

P (U

≤ b

− x, U

≤ b

− x, W ≤ r − x | X = x)f

(x)dx

P (U

≤ b

− x, U

≤ b

− x)P (W ≤ r − x)f

(x)dx.

The ﬁrst step uses the law of iterated expectations and the remaining steps the

independence assumptions. It follows that

, b

, r) =

− x, b

− x)f

(r − x)f

(x)dx.

− x | b

− x)f

(r − x)f

(x)dx.

Notice that f

− x | b

− x) is a function of f

only. For example, when j = 2

and k = 3,

− x, | b

− x) =

2(1 − F

− x))f

− x)

(1 − F

− x))

Denote the data by Z

= (B

, B

, R

), where t = 1, . . . , T denotes an auction.

Let θ

= (f

, f

) ∈ Θ, where Θ denotes the parameter space. Deﬁne the

contribution of an individual auction t to the log-likelihood as

(θ

, Z

) = ln



− x | B

− x)f

(x)dx



14 FREYBERGER AND LARSEN

where the s subscript on l

(·) denotes the secret reserve price case. Thus, given

a random sample of T auctions {B

, B

, R

}

t=1

, we can estimate θ

by maximum

likelihood. Searching over the entire parameter space of densities satisfying certain

regularity conditions would prove infeasible. We instead maximize the likelihood over

a ﬁnite-dimensional approximation. In particular, let Θ

be a ﬁnite-dimensional sieve

space of Θ, which depends on the sample size T and has the property that θ

can

be approximated arbitrary well by some element in Θ

when T is large enough (see

Section 3.3 for a speciﬁc recommendation, as well as Chen (2007) for an overview on

sieve estimation). We can now estimate θ

= arg max

θ∈Θ

t=1

(θ, Z

Consistency of the estimator follows from standard arguments and regularity con-

ditions, such as Theorem 3.1 in Chen (2007). Furthermore, we can conduct inference

about certain functionals of the densities by inverting a likelihood ratio test. Let

g(θ

) : Θ → R be some functional of interest, such as moments of the distributions,

the optimal reserve price, or bidders’ surplus. Suppose we are interested in testing

: g(θ

) = m for some m ∈ R. Deﬁne L

(θ) =

t=1

(θ, Z

). The likelihood ratio

test is based on

= 2

sup

θ∈Θ

(θ) − sup

θ∈Θ

:g(θ)=m

(θ)

Shen and Shi (2005) provide conditions under which

→ χ

if H

is true. Hence,

we reject H

: g(θ

) = m if

> c

1−α

, where c

1−α

is the 1 − α quantile of the χ

distribution. Finally, let

g(θ

)

= {m ∈ R : do not reject H

: g(θ

) = m at level α}.

Then, by construction, CI

g(θ

)

is a 1 − α conﬁdence set for g(θ

An alternative is to specify the model semiparametrically, for example, by assuming

parametric distributions for X, W, U but allowing for a nonparametric distribution

of U

, thus retaining the ﬂexibility to accommodate an unknown distribution of the

IDENTIFICATION IN ASCENDING AUCTIONS 15

number of bidders. In such a model, inference about the ﬁnite-dimensional parameters

can be carried out by using the outer product form of the estimated covariance matrix

(see Ackerberg, Chen, and Hahn 2012).

3.2. Estimation and Inference in the Public Reserve Case. If the reserve price

is public we can still derive the log-likelihood function. To do so, deﬁne

(r) =

∂

∂r

P (R ≤ r, D

= 1),

(r, b

) =

∂

∂b

∂r

P (B

≤ b

, R ≤ r, D

= 1),

and

(r, b

, b

) =

∂

∂b

∂r

P (B

≤ b

, B

≤ b

, R ≤ r, D

= 1).

We show in Appendix B that

(r) =

∞

−∞

r−x

−∞

(r − x | u

)du

(r − x)f

(x)dx,

(r, b

) =

∞

−∞

r−x

−∞

− x | u

)du

(r − x)f

(x)dx,

and

(r, b

, b

) =

∞

−∞

− x | b

− x)f

(r − x)f

(x)dx.

Notice that since f

and F

depend on f

only, all three expressions only

depend on the four densities f

, f

, and f

. Now deﬁne

(θ

) =

t:D

ln(p

)) +

t:D

ln(p

, B

)) +

t:D

ln(p

, B

)),

where p in L

(·) denotes the public reserve price case. Under the condition of Theorem

2, where θ

is point identiﬁed, and the regularity conditions of Theorem 3.1 in Chen

(2007), we can consistently estimate θ

= arg max

θ∈Θ

(θ). (1)

16 FREYBERGER AND LARSEN

Moreover, we can obtain conﬁdence intervals for certain functionals of θ

by inverting

a likelihood ratio test, analogous to the secret reserve price setting, or do inference in

a semiparametric setting.

As shown in Theorem 2 and the related discussion, in general θ

might not be

point identiﬁed if the reserve price is public. In this case,

will not be a consistent

estimator of θ

, but we can still conduct inference using a likelihood ratio test. Similar

to before, let g(θ

) : Θ → R be some functional and suppose we are interested in

testing H

: g(θ

) = m for some m ∈ R. The likelihood ratio test is based on

= 2

sup

θ∈Θ

(θ) − sup

θ∈Θ

:g(θ)=m

(θ)

If θ

is not point identiﬁed,

does not converge to a χ

in distribution. Nevertheless,

we can obtain the critical values by using a weighted bootstrap as shown by Chen,

Tamer, and Torgovitsky (2011). While this procedure is very ﬂexible and allows for

partial identiﬁcation under weak assumptions, a downside is that it is computationally

very demanding.

3.3. Computational and Practical Details. In our application in Section 4 below,

we choose as our sieve space the space of normalized, orthogonal Hermite polynomials,

as in Gallant and Nychka (1987). This allows us to ﬂexibly approximate the density

function for any random variable Y ∈ {W, U

, U, X} by

(y) ≈

k=0



y − µ



√

2π

−



y−µ



where K is a smoothing parameter and θ

, µ

, and σ

are estimated. H

are

Hermite polynomials deﬁned by H

(x) = 1, H

(x) = x, and, for k > 2, H

(x) =

√

[xH

k−1

(x) −

√

k − 1H

k−2

(x)].

Plugging in these approximating polynomials, we maximize the above likelihood

expressions subject to the constraints

i=1

(θ

)

= 1 for each random variable

Y ∈ {W, U

, U, X}, which ensures each approximated function is indeed a density

IDENTIFICATION IN ASCENDING AUCTIONS 17

function, and also subject to the constraint E[X] = 0. The location and scale pa-

rameters {µ

, σ

}

Y ∈{W,U

,U,X}

are not required for consistent estimation but improve

the performance of the estimator. We estimate them in a parametric initial step,

maximizing the likelihood while ﬁxing the vectors θ

to have the ﬁrst element θ

= 1

and the remaining elements θ

= 0 for all k > 0. That is, each density f

is approxi-

mated by a N(µ

, σ

). We then plug in the estimated values of {ˆµ

, ˆσ

}

Y ∈{W,U

,U,X}

into the likelihood expression and maximize over {θ

}

Y ∈{W,U

,U,X}

. We perform the

integration in the likelihood by Gauss-Hermite quadrature (see Judd 1998).

In the application below, we also present results from estimating the model semi-

parametrically. In the semiparametric estimation, we specify the distributions of X,

U, and W to be normally distributed and we approximate the density of U

using Her-

mite polynomials, as above, retaining ﬂexibility to allow for an unknown distribution

for the number of bidders.

4. Application to Used Smartphone Auctions

4.1. Background on Digital Rights Management and Smartphone Unlock-

ing. Digital rights management (DRM) refers to technological locks placed on soft-

ware or hardware to restrict its use or modiﬁcation. The use of these locks has been

highly controversial. Proponents of DRM argue that these restrictions are necessary

to prevent copyright infringement of digital intellectual property (Liu, Safavi-Naini,

and Sheppard 2003). Opponents argue that DRM takes a step beyond traditional

they have legally purchased, suggesting that these laws instead serve primarily to

restrict competition between producers (Von Lohmann 2004; Walker 2003; Stallman

1997). A number of products are controlled through DRM, including, among many

others, computer software, with digital locks enforcing limited installs or requiring

activation keys; e-books, music, or ﬁlm, with limits on sharing or on device compati-

bility; and cellular handsets, with digital locks between the subscriber identiﬁcation

module (SIM) and the phone’s software, restricting the handset to only function on

a particular provider’s cellular service network.

18 FREYBERGER AND LARSEN

In the United States, the key law regarding DRM is the Digital Millennium Copy-

right Act (DMCA) of 1998. This law was implemented in response to the 1996

its members (including the United States) to adopt measures to prohibit tamper-

ing with digital locks. In the early years of the DMCA, cellular handset unlocking

was granted an explicit exemption, and consumers could legally unlock their out-of-

contract phones through a variety of do-it-yourself or third-party services (Van Camp

2013). In late 2012, the copyright oﬃce of the Library of Congress failed to renew

this exemption, arguing that phone unlocking tampers with copyrighted ﬁrmware

and hence is arguably in violation of the law (Federal Register 2012; Couts 2012).

This change made phone unlocking illegal as of January 26, 2013, imposing a ﬁne of

$500,000 and a sentence of ﬁve years in prison for unauthorized unlocking (Wyatt

2013). Contemporary conversations among consumers online suggest that consumers

were nervous as to how this massive ﬁne and prison sentence would be enforced and

to whom it would apply (Velazco 2013; Khanna 2013). In response to backlash from

consumer advocates (Wyatt 2013), a bill was eventually signed into law in 2014 to

re-allow consumer unlocking of phones.

Although laws such as the DMCA have arisen to prohibit the production or dis-

tribution of technology intended to circumvent these digital locks, these laws and

2010) and violations are diﬃcult to police. Given this enforcement challenge, it

remains an open question whether these laws are eﬀective in altering individuals’

(buyers and sellers) willingness to engage in DRM circumvention. We contribute to

this question by examining buyers’ willingness to pay and sellers’ pricing for DRM-

tampered goods—unlocked smartphones—before and after the January 2013 ban on

unlocking. While the purpose of this exercise is primarily to illustrate our method

rather than provide a deﬁnitive answer to the eﬀects of this regulatory change, the

analysis provides several insights that we discuss below.

IDENTIFICATION IN ASCENDING AUCTIONS 19

4.2. Data on Used Smartphone Auctions. We use a new dataset of eBay auctions

for used iPhones. The sample consists of used iPhone 4 and 4S models with 8, 16, or

32 gigabytes (GB) of memory and of black or white color. These models were the most

frequently auctioned iPhone models during our sample period, September 22, 2012

to May 21, 2013. We choose this sample period as it begins after the introduction

of the iPhone 5 (September 21, 2012) and includes the date of the regulatory change

banning phone unlocking, January 26, 2013. The models we study were released

between June 24, 2010 and October 14, 2011, prior to the start date of our sample,

and thus a large number of used (unlocked and locked) iPhone 4 and 4S handsets had

accumulated and were being sold in our sample period.

As with many real-world ascending-like auctions, the number of bidders is not

observed in our setting: potential bidders who arrive at the auction after the standing

bid has passed their values will not be observed placing a bid. These auctions also

exemplify a setting in which it is important to account for unobserved heterogeneity,

as used smartphones can diﬀer dramatically in ways that are observable to the seller

and bidders but not to the econometrician, such as through a cracked screen, scratched

surface, missing USB adapter, or faulty battery, or in positive ways, such as a lack

of wear and tear.

Pictures and detailed descriptions posted by the seller contain

information about these characteristics that both the seller and potential buyers can

observe, but such information is diﬃcult for the econometrician to quantify.

We focus our application on auctions in which the seller used a secret reserve price.

For each auction, the data contains the second and third unconditional order statistics

of bids, the seller’s secret reserve price, the shipping fee, and an indicator for whether

The precise release dates were June 24, 2010 for the black iPhone 4 16 GB and 32 GB; April 28,

2011 for the white iPhone 4 16 GB and 32 GB; and October 14, 2011 for either color of the iPhone

4 8GB and either color of the iPhone 4S 16 or 32 GB. The iPhone 4S 8 GB was not released until

after our sample, on September 20, 2013.

Our focus on used phones is also due to the fact that new unlocked phones are more likely to have

been unlocked (legally) by the original vendor, and thus the alteration of the DMCA is less likely to

have impacted new phones.

When observable to the econometrican, text descriptions could be exploited using natural language

parsing algorithms or images could be analyzed with image processing algorithms, and this could

aid in accounting for item-level heterogeneity. In such cases our approach would remain useful to

account for remaining unobserved heterogeneity.

20 FREYBERGER AND LARSEN

the phone was locked to a particular carrier (Verizon or AT&T) or unlocked. We

drop all auctions in which the bids, the reserve price, or the shipping fee lies outside

of their respective 0.01 and 0.99 quantiles. We also drop auctions in which the bids

or the reserve price lie above the contemporary price of a brand new iPhone 5 ($649)

sold at the Apple Store (see Wyatt 2013). This leaves a sample of 12,890 auctioned

iPhones.

4.3. Estimation Results. Using this data, we estimate, for diﬀerent carriers and

for unlocked phones, the distributions of unobserved heterogeneity, bidder valuations,

and reserve prices, where the latter two distributions are net of the unobserved het-

erogeneity. Therefore, the objects of interest are F

, F

, and F

. By Theorem 1,

each of these objects is point identiﬁed. We note that our method treats bidders

in these auctions as though they are short lived, exiting the auction market after

one attempt.

We also treat bidders as having private values; as with many auc-

tion settings, allowing for interdepencies in valuations would be preferable but would

be beyond the state of the methodological literature. However, we believe that the

private values assumption is a reasonable approximation to reality here in that all

buyers have access to the same information on the website about the product. Also,

lemons-like interdependencies—arising from the seller withholding information from

the buyer about the quality of the good—are less likely to be a concern in our data

than in many other auction settings due to buyer protection plans and sanctions

against deceptive sellers, which eBay has incorporated in recent years.

To implement our method, we adopt the log (i.e. multiplicatively separable) speci-

ﬁcation of the model described at the beginning of Section 2. We account for shipping

fees simply by adding them to the observed bids and reserve prices. We then control

for observable heterogeneity in the smartphones using the standard homogenization

step of Haile, Hong, and Shum (2003) by regressing log reserve prices on observable

characteristics, consisting of fully saturated model of indicators for all combinations

In practice, this appears to be a reasonable approximation, as 71% of bidders in our data bid in

at most one auction for a given phone speciﬁcation.

IDENTIFICATION IN ASCENDING AUCTIONS 21

of iPhone model (4 or 4S), memory size (8, 16, or 32 GB), and color (black or white).

We then compute homogenenized bids and reserve prices by subtracting the regres-

sion’s predicted value from the log bids and log reserve prices.

With these homogenized bids and reserve prices, we apply the steps described in

Section 3 to estimate the model’s primitives, estimating F

, F

, and F

separately

for each carrier code (AT&T, Verizon, and unlocked) and separately for each time

period, with the pre period (prior to the unlocking ban) being September 22, 2012

through January 25, 2013 and the post period (after the unlocking ban took eﬀect)

being January 26, 2013 through May 21, 2013.

We begin by estimating the semiparametric model. We specify the marginal den-

sities f

, f

, and f

, to all be Normal, but we specify f

to be a fourth-degree

Hermite polynomial. This ﬂexibility in the k

unconditional order statistic density

permits the model to accommodate the unknown distribution of the number of

bidders. The results of the semiparametric model are displayed in Table 1. Panel A

displays the estimated parameters for the pre period and panel B displays the results

for the post period.

We ﬁnd that, in the pre period, the means of buyer valuations and seller reserve

prices are higher for unlocked phones than for AT&T or Verizon phones (although, in

the case of buyer valuations for unlocked vs. AT&T phones, this diﬀerence is small

in magnitude). This ordering of locked vs. unlocked phones is intuitive, as unlocked

phones should obtain a premium given that they can be used on any carrier. The

fact that consumers value Verizon phones less than AT&T phones is likely simply an

artifact of Apple phone contracts for Verizon being a relatively new phenomenon dur-

ing our sample period when compared to such contracts for AT&T. We ﬁnd that the

standard deviations of seller reserve prices and unobserved heterogeneity are similar

across carriers.

We perform this homogenization step using using log reserve prices as the left hand side variable,

but the log of bids could have be used instead of (or in addition to) the log of reserve prices.

Recall that the mean of the unobserved heterogeneity distribution is mechanically the same for

each carrier, but the standard deviation is allowed to diﬀer.

22 FREYBERGER AND LARSEN

Comparing across the two time periods, we see that several model parameters are

quite similar before and after the unlocking ban for all carriers and unlocked phones,

including the standard deviation of seller reserve prices and the standard deviation

of unobserved heterogeneity. The distributions of buyer valuations and seller reserve

prices for AT&T phones also stayed roughly constant before and after the ban. For

Verizon phones, the mean of buyer valuations decreased from −0.287 to −0.340 log

points and the mean of seller reserve prices decreased from −0.072 to −0.145 log

points. For unlocked phones, the mean of seller reserve prices changed only slightly,

but the mean of buyer valuations had a much larger shift, dropping from 0.036 to

−0.155 log points after the ban went into aﬀect. The diﬀerence of 0.119 is roughly

equal to a decrease of 0.5 standard deviations of the estimated distribution of log-

valuations, or a decrease in the dollar valuation for the phone of about 11%. Moreover,

the standard deviation increased by roughly 20%.

To examine whether any of the above implications are driven by parametric restric-

tions, we now turn to the results of the nonparametric model. For this exercise, we

specify f

, f

, and f

each as fourth-degree Hermite polynomials. For the density

we use a third-degree Hermite polynomial.

Figure 1 includes all of the non-

parametric estimates for AT&T, Verizon, and unlocked phones in the pre period, and

Figure 2 contains the analogous estimates in the post period.

Figures 1 and 2 suggest, as did the results of the semiparametric model, that the

distribution of unobserved heterogeneity is similar for AT&T, Verizon, and unlocked

phones, both before and after the unlocking ban. We also ﬁnd again that distributions

of seller reserve prices have a clear stochastic ordering in both the pre and post periods,

with unlocked phones being highest, then AT&T phones, and then Verizon phones.

The ordering for buyer valuations in the pre period is roughly the same, although

This latter choice is driven by the fact that X, the unobserved heterogeneity, is the dimension

over which we integrate in computing the likelihood, and, even with many Gauss-Hermite nodes,

there can be few nodes close to zero. With an even-degree polynomial, this can yield an estimated

density with little mass exactly at zero and a mode on either side of zero. With a third-degree

polynomial approximation, the integration and estimation result in an estimated density that is

single-peaked. The qualitative results and the pre and post comparison are similar when all densities

are approximated with fourth-degree polynomials. Figures 7 and 8 in the Appendix displays these

results.

IDENTIFICATION IN ASCENDING AUCTIONS 23

buyer valuations for AT&T and unlocked phones are quite similar to one another.

In the post period, however, we see that the ordering for buyer valuations changes,

with unlocked phones being valued much less than AT&T phones, although still more

than Verizon phones. This shift is suggestive that buyers may have been less willing

to pay for technologically circumvented handsets after the unlocking ban took eﬀect.

Figures 3–5 display these same estimates separately by carrier code for the pre

vs. post time period. In each of these three ﬁgures, the distribution of unobserved

heterogeneity is similar pre and post the unlocking ban. In Figure 3 the distributions

of buyer valuations for AT&T phones is slightly lower in the post period, which is

likely due in part to the passage of time, depreciating buyers’ willingness to pay for a

ﬁxed model. The distribution of seller reserve prices for AT&T phones is similar pre

and post. Figure 4 displays a similar decrease in buyer valuations for Verizon phones,

and also a decrease in seller valuations, in the post period relative to the pre period.

In Figure 5, however, there is a dramatic shift in buyer valuations for unlocked phones.

The estimated diﬀerence in means of the distributions in the pre and post periods is

0.31, corresponding to about 0.9 standard deviations of the estimated distribution of

log-valuations, or a decrease in the dollar valuation for the phone of about 27%. The

estimated mean decreases in the dollar valuations for AT&T and Verizon phones are

only 5.1% and 4.6%, respectively. This is again suggestive evidence that buyers may

have incorporated into their valuations a distaste for violations of digital rights laws.

Thus far we have not attempted to identify seller valuations directly, but rather

worked only with sellers’ reported reserve prices. We now consider the seller val-

uations that would be implied by the data if the reserve prices are interpreted as

having been set optimally by the seller (given the distribution of buyer valuations).

As explained in Section 2, if the seller has a valuation that is additively separable

in the auction-level heterogeneity, the optimal reserve price (net of auction-level het-

erogeneity) would satisfy W = S +

1−F

(W )

, and thus the seller’s net valuation is

given by S = MR

(W ) ≡ W −

1−F

(W )

, where the notation MR

(·) denotes the

24 FREYBERGER AND LARSEN

buyer’s marginal revenue as in Bulow and Roberts (1989).

The distribution of seller

valuations is then given by F

(s) = F

(MR

−1

(s)).

We compute the implied seller distributions for AT&T, Verizon, and unlocked

phones in the pre and post periods using the estimated parameters from the semi-

parametric model from Table 1.

We show these distributions in Figure 6. We ﬁnd

that implied seller valuations for AT&T phones, the top panel, increased slightly and

implied seller valuations for Verizon phones, the middle panel, decreased slightly from

the pre period to the post period. Implied seller valuations for unlocked phones ap-

pear nearly identical before and after the unlocking ban. These ﬁndings are consistent

with our ﬁndings above that the primary change in market valuations for these used

smartphones occurred on the buyers’ side and only for unlocked phones.

5. Conclusion

This paper introduces a new approach to identiﬁcation in auctions with unobserved

auction-level heterogeneity and an unknown number of bidders. The methodology re-

lies on deconvolution ideas for handling unobserved heterogeneity—which have been

applied extensively to ﬁrst price auctions but not to ascending auctions due the

complicating factor of correlation between order statistics. The approach also re-

lies on order statistics comparisons which have previously been limited to settings

that do not allow for unobserved heterogeneity. We bring these ideas together in a

uniﬁed framework, exploiting information contained in reserve prices—either secret

or public—chosen by the seller. We provide point identiﬁcation as well as partial

identiﬁcation results for these settings and propose a nonparametric sieve maximum

likelihood approach or semiparametric approach for estimation and inference.

This same property holds in logs for the case where valuations are multiplicatively separable in

the unobserved heterogeneity term.

We use the semiparametric rather than nonparametric model to simplify the inversion from ob-

served reserve prices to seller valuations. Speciﬁcally, the normality assumption for F

in the semi-

parametric model ensures that F

is regular in the sense of Myerson (1981), i.e. ψ(w) ≡ w −

1−F

(w)

will be strictly increasing in w and hence will correspond to marginal revenue. If ψ(w) is not strictly

increasing (i.e., the distribution is irregular) then marginal revenue, M R

(w), would instead be

computed as the derivative of the convex hull of

ψ(y)dy (the ironed marginal revenue function).

IDENTIFICATION IN ASCENDING AUCTIONS 25

We apply this framework to analyze changes in bidders’ willingness to pay and

sellers’ reserve prices before and after 2013 regulatory changes banning the removal

of digital locks on cellphones. We ﬁnd that, for phones locked to AT&T or Verizon

phones, as well as for unlocked phones, the distributions of unobserved heterogeneity

and of seller reserve prices were quite similar before and after the unlocking ban. We

also ﬁnd that the distribution of buyer valuations was similar before and after the ban

for AT&T and Verizon phones. However, for unlocked phones—those which, after the

law took eﬀect, were potentially in violation of digital rights legislation—there was a

clear drop in buyer valuations, suggestive that digital rights management laws, while

diﬃcult to enforce in practice, may have real eﬀects on consumers’ willingness to pay.

We believe our methodology has applications to ascending auctions more broadly,

whether they be traditional English auctions, such as for timber, cattle, or used cars;

or online auctions for e-commerce or display advertising. In many of these settings,

unobserved heterogeneity in the auctioned items is a major concern, and the number

of bidders is often unknown to the econometrician, as many such auctions do not

require all bidders to register in any fashion prior to the auction and hence their

would-be bids are unobserved.

26 FREYBERGER AND LARSEN

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IDENTIFICATION IN ASCENDING AUCTIONS 31

Table 1. Estimated Parameters From Semiparametric Model

AT&T Verizon Unlocked

A. Pre Unlocking Ban

Mean of buyer valuations (U) 0.032 -0.287 0.036

Std of buyer valuations (U) 0.145 0.220 0.187

Mean of seller reserve (W ) 0.024 -0.072 0.183

Std of seller reserve (W ) 0.240 0.221 0.220

Std of unobs heterogeneity (X) 0.100 0.105 0.102

B. Post Unlocking Ban

Mean of buyer valuations (U) 0.010 -0.340 -0.155

Std of buyer valuations (U) 0.166 0.204 0.254

Mean of seller reserve (W ) 0.027 -0.145 0.167

Std of seller reserve (W ) 0.235 0.232 0.254

Std of unobs heterogeneity (X) 0.104 0.106 0.107

Notes: Table displays estimates from semiparametric model, where densities f

, f

, and

are each Normally distributed and f

is a fourth-degree Hermite polynomial. Units

are log points, after homogenization (i.e. subtracting oﬀ observable auction-level

heterogeneity).

32 FREYBERGER AND LARSEN

Figure 1. Pre Unlocking Ban: Distribution Functions and Densities

for AT&T, Verizon, and Unlocked Phones

-2 -1.5 -1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1.5 -1 -0.5 0 0.5 1

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

Notes: Each plot shows nonparametric estimates for AT&T, Verizon, and unlocked phones in the

pre period. Panels on the left show distribution functions and on the right, densities, for the

noncommon component of buyer valuations F

(top row) and reserve prices F

(middle row), and

for the unobserved heterogeneity F

(bottom row). Units on the horizontal axes are log points,

after homogenization (i.e. subtracting oﬀ observable auction-level heterogeneity).

IDENTIFICATION IN ASCENDING AUCTIONS 33

Figure 2. Post Unlocking Ban: Distribution Functions and Densities

for AT&T, Verizon, and Unlocked Phones

-2 -1.5 -1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1.5 -1 -0.5 0 0.5 1

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

Notes: Each plot shows nonparametric estimates for AT&T, Verizon, and unlocked phones in the

post period. Panels on the left show distribution functions and on the right, densities, for the

noncommon component of buyer valuations F

(top row) and reserve prices F

(middle row), and

for the unobserved heterogeneity F

(bottom row). Units on the horizontal axes are log points,

after homogenization (i.e. subtracting oﬀ observable auction-level heterogeneity).

34 FREYBERGER AND LARSEN

Figure 3. Estimates for AT&T-locked iPhones Before and After Un-

locking Ban

-2 -1.5 -1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1.5 -1 -0.5 0 0.5 1

Pre

Post

-2 -1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1 0 1

Pre

Post

-1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-1 0 1

Pre

Post

Notes: AT&T phones only. Each plot shows the nonparametric estimate before and after the

regulatory change. Panels on the left show distribution functions and on the right, densities, for

the noncommon component of buyer valuations F

(top row) and reserve prices F

(middle row),

and for the unobserved heterogeneity F

(bottom row). Units on the horizontal axes are log

points, after homogenization (i.e. subtracting oﬀ observable auction-level heterogeneity).

IDENTIFICATION IN ASCENDING AUCTIONS 35

Figure 4. Estimates for Verizon-locked iPhones Before and After Un-

locking Ban

-2 -1.5 -1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1.5 -1 -0.5 0 0.5 1

Pre

Post

-2 -1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1 0 1

Pre

Post

-1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-1 0 1

Pre

Post

Notes: Verizon phones only. Each plot shows the nonparametric estimate before and after the

regulatory change. Panels on the left show distribution functions and on the right, densities, for

the noncommon component of buyer valuations F

(top row) and reserve prices F

(middle row),

and for the unobserved heterogeneity F

(bottom row). Units on the horizontal axes are log

points, after homogenization (i.e. subtracting oﬀ observable auction-level heterogeneity).

36 FREYBERGER AND LARSEN

Figure 5. Estimates for Unlocked iPhones Before and After Unlocking Ban

-2 -1.5 -1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1.5 -1 -0.5 0 0.5 1

Pre

Post

-2 -1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1 0 1

Pre

Post

-1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-1 0 1

Pre

Post

Notes: unlocked phones only. Each plot shows the nonparametric estimate before and after the

regulatory change. Panels on the left show distribution functions and on the right, densities, for

the noncommon component of buyer valuations F

(top row) and reserve prices F

(middle row),

and for the unobserved heterogeneity F

(bottom row). Units on the horizontal axes are log

points, after homogenization (i.e. subtracting oﬀ observable auction-level heterogeneity).

IDENTIFICATION IN ASCENDING AUCTIONS 37

Figure 6. Estimates of Implied Seller Valuations Before and After

Unlocking Ban, Semiparametric Model

-2 -1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1 0 1

0.2

0.4

0.6

0.8

Pre

Post

-2 -1 0 1

0.2

0.4

0.6

0.8

Pre

Post

Notes: CDF of seller valuations implied by assuming that seller reserve prices are set optimally.

Each plot shows the semiparametric estimate before and after the regulatory change. Upper panel

shows AT&T phones, middle panel shows Verizon phones, and lower panel shows unlocked phones.

Units on the horizontal axes are log points, after homogenization (i.e. subtracting oﬀ observable

auction-level heterogeneity).

38 FREYBERGER AND LARSEN

Appendix A. Proofs

Proof of Theorem 1. First write

= X + U

R = X + W.

By Lemma 1 of Evdokimov and White (2012), φ

is identiﬁed. Moreover, notice that

by independence of X and (U

, U

, W ) we get

, t

, t) = φ

+ t

+ t)φ

, t

, t)

for all (t

, t

, t) ∈ R

. Therefore, for all (t

, t

, t) ∈ R

such that φ

+ t

+ t) 6= 0,

, t

, t) =

, t

, t)

+ t

+ t)

It follows that φ

, t

, t) is identiﬁed for all (t

, t

, t) ∈ R

such that φ

+t) 6= 0. Since the zeros of φ

are isolated and since φ

, t

, t) is continuous,

, t

, t) is identiﬁed for all (t

, t

, t) ∈ R

. Identiﬁcation of the characteristic

function is equivalent to identiﬁcation of the density f

Since we know the joint distribution of U

and U

, we can use arguments as in Song

(2004) to identify the distribution of valuations. Speciﬁcally, Song (2004) demon-

strated that the density of U

conditional on a realization of U

does not depend on

the realization of N:

) =

, u

)

(k − 1)!(F

) − F

))

k−j−1

(1 − F

))

j−1

)

(k − j −1)!(j −1)!(1 − F

))

k−1

Letting u

converge to the lower bound of the support of U identiﬁes

(k − 1)!F

)

k−j−1

(1 − F

))

j−1

)

(k − j −1)!(j −1)!

IDENTIFICATION IN ASCENDING AUCTIONS 39

which is the density of the (j −k)’th order statistic from a sample of size (j −1) with

distribution function F

. Hence, from Theorem 1 in Athey and Haile (2002), F

identiﬁed.

Finally, note that if N ∈ {n, . . . , ¯n} with n ≥ k and ¯n < ∞, then

(u) =

¯n

n=n

Pr(N = n)

i=n−k+1





(u)

(1 − F

(u))

n−i

and therefore for all z ∈ [0, 1]

−1

(z)) =

¯n

n=n

Pr(N = n)

i=n−k+1





(1 − z)

n−i

The terms



i=n−k+1





(1 − z)

n−i



are polynomials of order n. Hence, the term

in F

−1

(z)) belonging to the highest order polynomial, z

¯n

, is

¯n

Pr(N = ¯n)

¯n

i=¯n−k+1



¯n



(u)

¯n

(−1)

¯n−i

Therefore, if

¯n

i=¯n−k+1



¯n



(−1)

¯n−i

6= 0, then Pr(N = ¯n) is identiﬁed from the ¯n’th

derivative of F

−1

(z)) with respect to z. Indeed,

¯n

i=¯n−k+1



¯n



(−1)

¯n−i

k−1

j=1



¯n

¯n − j



(−1)

k−1

j=1



¯n



(−1)

= (−1)

k−1



¯n − 1

k − 1



6= 0.

Given identiﬁcation of Pr(N = ¯n), we know

−1

(z)) − Pr(N = ¯n)

¯n

i=¯n−k+1



¯n



(1 − z)

¯n−i

40 FREYBERGER AND LARSEN

which equals

¯n−1

n=n

Pr(N = n)

i=n−k+1





(1 − z)

n−i

Analogous arguments as above now show identiﬁcation of Pr(N = ¯n −1). Repeating

these steps yield identiﬁcation of P (N = n) for all n ∈ {n

, . . . , ¯n}.

Proof of Theorem 2. Since B ≥ R, B is identiﬁed from the largest possible bid.

Notice that if B

= B, then U

= U and X = X. We can now observe for all

r ∈ [R, R] = [X + W , X + W ]

P (R ≤ r | B

= B) = P (R ≤ r | X = X, U

= U)

= P (W ≤ r − X).

It follows that the distribution of

W = W + X is identiﬁed. But since E[

W ] =

E[W ] + X = E[R] + X and since E[R] is identiﬁed, it follows that X and hence the

distribution of W is identiﬁed. By independence of X and W ,

(t) = φ

(t)φ

(t)

and for all t with φ

(t) 6= 0,

(t) =

(t)

Since φ

(t) and φ

(t) are identiﬁed and since φ

(t) has only isolated zeros, φ

(t) is

identiﬁed. Thus, F

is identiﬁed.

Moreover,

P (B

≥ b

, B

≥ b

| R = R) = P (B

≥ b

, B

≥ b

| X + W = X + W )

= P (B

≥ b

, B

≥ b

| X = X, W = W )

= P (U

≥ b

− X, U

≥ b

− X | X = X, W = W )

= P (U

≥ b

− X, U

≥ b

− X)

IDENTIFICATION IN ASCENDING AUCTIONS 41

Given X = X, the support of the observed B

is [X +W , X +U]. Therefore, if B ≥ R

(and thus U ≥ W), we can identify

P (U

≥ u

, U

≥ u

) = P (B

≥ u

+ X, B

≥ u

+ X | R = R).

for all u

, u

∈ [U, U]. In this case, just as in the proof of Theorem 1, identiﬁcation

of F

yields identiﬁcation of F

and P (N = n).

If instead B < R, we can only identify P (U

≥ u

, U

≥ u

) for all u

≥ u

≥ W >

U. Analogous arguments identify P (U

≥ u

) and P (U

≥ u

) for all u

, u

≥ W >

U. Now since

| u

) = 1 −

(z, u

)

dz,

| u

) for all u

≥ u

≥ W is identiﬁed as well. Therefore, by the arguments

from Song (2004), knowing F

| u

) implies knowledge of

) − F

))/(1 − F

))

for all u

≥ u

≥ W . In other words, we can identify P (U ≤ u

| U ≥ u

) for all

, u

≥ W .

Appendix B. Likelihood Derivation

Here we derive the expression for p

, p

, and p

given in Section 3.2. For the ﬁrst

part write

P (R ≤ r, D

= 1) = P (R ≤ r, B

≤ B

< R)

= P (R ≤ r, B

< R)

∞

−∞

P (W ≤ r − x, U

≤ W )f

(x)dx

∞

−∞

∞

−∞

P (W ≤ r − x, u ≤ W | U

= u

(x)du

∞

−∞

r−x

−∞

P (u

≤ W ≤ r − x)f

(x)du

∞

−∞

r−x

−∞

(P (W ≤ r − x) − P (W ≤ u

)) f

(x)dudx

42 FREYBERGER AND LARSEN

Taking the derivative with respect to r (using Leibniz’s rule) yields

(r) =

∞

−∞

r−x

−∞

(r − x)f

(x)du

∞

−∞

(r − x)f

(x)dx.

If the number of bidders is unknown, we need expressions in terms of the distributions

of U

| U

and U

and we use

) =

−∞

| u

)du

Speciﬁcally, when B

is the second highest and B

is the third highest bid, then

, | u

) =

2(1 − F

))f

)

(1 − F

))

Thus,

, | u

) =

2(1 − F

(z))f

(z)

(1 − F

))

(2F

) − F

)

) − (2F

) − F

)

(1 − F

))

1(u

≥ u

Similarly,

P (B

≤ b

, R ≤ r, D

= 1)

= P (B

≤ b

, R ≤ r, B

≥ R > B

)

∞

−∞

P (U

≤ b

− x, W ≤ r − x, U

≥ W > U

(x)dx

∞

−∞

∞

−∞

P (U

≤ b

− x, w ≤ r − x, U

≥ w > U

| W = w)f

(w)f

(x)dudx

∞

−∞

min{b

−x,r−x}

−∞

P (w ≤ U

≤ b

− x, U

< w)f

(w)f

(x)dwdx

∞

−∞

min{b

−x,r−x}

−∞



P (U

≤ b

− x, U

< w) − P (U

≤ w, U

< w)



(w)f

(x)dwdx.

IDENTIFICATION IN ASCENDING AUCTIONS 43

If b

< r, then P (B

≤ b

, R ≤ r, D

= 1) does not depend on r and p

(r, b

) = 0. If

≥ r, then

(b, r) =

∂

∂r∂b

P (B

≤ b

, R ≤ r, D

= 1)

∞

−∞

∂

∂b

P (U

≤ b

− x, U

< r − x)f

(r − x)f

(x)dx

∞

−∞

(r − x | b

− x)f

(r − x)f

(x)dx.

Alternatively in terms of the distributions of U

| U

and U

write

∂

∂b

P (U

≤ b

− x, U

< r − x) =

∂

∂b

−x

−∞

r−x

−∞

, u

)du

r−x

−∞

− x, u

)du

r−x

−∞

− x | u

)du

Then

(r, b

) =

∞

−∞

r−x

−∞

− x | u

)du

(r − x)f

(x)dx.

Finally, using the same arguments as before

(r, b

, b

) =

∞

−∞

− x, b

− x)f

(r − x)f

(x)dx

∞

−∞

− x | b

− x)f

(r − x)f

(x)dx.

44 FREYBERGER AND LARSEN

Figure 7. Pre Unlocking Ban: Distribution Functions and Densities

for AT&T, Verizon, and Unlocked Phones, with Fourth Degree Hermite

Polynomial for Distribution of X

-2 -1.5 -1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1.5 -1 -0.5 0 0.5 1

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

Notes: Each plot shows nonparametric estimates for AT&T, Verizon, and unlocked phones in the

pre period. Panels on the left show distribution functions and on the right, densities, for the

noncommon component of buyer valuations F

(top row) and reserve prices F

(middle row), and

for the unobserved heterogeneity F

(bottom row). Units on the horizontal axes are log points,

after homogenization (i.e. subtracting oﬀ observable auction-level heterogeneity). Estimates in this

ﬁgure use a fourth degree polynomial for X rather than a third-degree polynomial as in Figure 1.

IDENTIFICATION IN ASCENDING AUCTIONS 45

Figure 8. Post Unlocking Ban: Distribution Functions and Densities

for AT&T, Verizon, and Unlocked Phones, with Fourth Degree Hermite

Polynomial for Distribution of X

-2 -1.5 -1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1.5 -1 -0.5 0 0.5 1

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-2 -1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

0.2

0.4

0.6

0.8

ATT Phones

Verizon Phones

Unlocked Phones

-1 0 1

ATT Phones

Verizon Phones

Unlocked Phones

Notes: Each plot shows nonparametric estimates for AT&T, Verizon, and unlocked phones in the

post period. Panels on the left show distribution functions and on the right, densities, for the

noncommon component of buyer valuations F

(top row) and reserve prices F

(middle row), and

for the unobserved heterogeneity F

(bottom row). Units on the horizontal axes are log points,

after homogenization (i.e. subtracting oﬀ observable auction-level heterogeneity). Estimates in this

ﬁgure use a fourth degree polynomial for X rather than a third-degree polynomial as in Figure 2.