identification and estimation in a third-price auction model · 2014. 2. 21. · identification...
TRANSCRIPT
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Identification and Estimation in a Third-Price Auction
Model ∗
Andreea ENACHE†and Jean-Pierre FLORENS‡
Abstract
We show global identification of the distribution of private values in the sealed-bid third priceauction model using a fully nonparametric approach. We consider an i.i.d. environment for theprivate values with risk-seeking bidders. In a first place, we consider the case where the risk-seeking parameter is known.We show that the speed of convergence in process of our nonparametric estimator produces at theparametric rate
√n.
Although the model is globally identified, we present also the functional local approach whichis useful when the model is not globally tractable. This is the case if one searches to identify alsothe risk-seeking parameter. Using the variation in the number of participants we are able to achievelocal identification of the unknown risk-seeking parameter. Not only this methodology is morerigorous, but it can be applied for different types of games which involve complicated nonlinearinverse problems.
We extend also to the case where we observe only the winning bids and we generalize themodel as to account for the presence of exogenous variables.
Our results are important from a policy perspective, as some authors recommend the use of thethird-price auction format in some Internet auctions.Moreover, the paper provides a methodological toolbox for analyzing local and global identificationfor other types of games in incomplete formation.
Keywords: structural nonparametric estimation, nonlinear inverse problems, global and local
identification, functional convergence of estimator
JEL Classification: C73,C40
∗For useful comments and suggestions, we wish to thank Xavier D’Haultefoeuille, Philippe Février and DavidMartimort. Any remaining mistakes are ours.
†Paris School of Economics and CREST-LEI, 48 Boulevard Jourdan, 75014 Paris, phone: +33(0)1 43 13 6393, email: andreea.enache@parisschoolofeconomics.
‡Toulouse School of Economics, email: [email protected].
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1 Introduction
Past and recent years have witnessed a growing interest in the structural analysis of auction data.
Its success is motivated by a coherent body of theory and also by its usefulness in providing
insights into many practical policy issues. The structural approach allows the econometrician
to estimate the data generating process, i.e. the joint distribution of valuations and signals
for bidders. Thus this approach is extremely useful for the normative aspect of designing an
optimal selling mechanism. It is well known that the design of optimal auctions has been
criticized for relying upon the distributions of the bidders’ valuations which are unknown to
the auction organizer. Therefore the structural econometrics can compensate for this drawback
by estimating the law of probability governing these valuations. In a next step, the estimated
distribution can be used to determine the revenue-maximizing selling mechanism.
The first seminal works in the field of structural econometrics are owned to [23], [6], [7]
who worked in a parametrical environment using the method of maximum likelihood (MLE).
In order to address some criticism of the MLE approach proposed by [6], [16] proposed the
method of simulated nonlinear least-squares based on the Revenue Equivalence Principle.
But when speaking about estimation in a structural model, a strong related concept is the
notion of identification. In structural econometrics, the models contain the economic informa-
tion underlying the process studied. Therefore, before estimating the model, one has to be sure
that there is a unique correspondence between the body of data and the elements to be esti-
mated. Otherwise said, identification trivially means that there is only one structure that can
explain the process under observation.
In the field of econometrics for auctions, a milestone concerning the identification was the
paper by [14], where the authors proved that the distribution of the unobserved valuations in
the first-price sealed-bid auctions, within the independent private values paradigm, is identified
from observables without imposing any parametric assumptions. It should be noticed that
nonparametric identification is valid whether all bids are observed or only the winning bid.
Another contribution to the identification issue was brought by the article of [11] where
the authors studied the general conditions of identification for empirical games in incomplete
information. They presented the results for the identification of the parameter that characterizes
the distribution of private values. An extension of this work has been conducted by [12] who
introduced additional parameters to be identified (e.g. parameter of risk aversion) and studied
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some theoretical situations as: partial observations of the exogenous, randomized strategies and
non closed form solutions for first-order conditions.
More complicated formats of auction have been treated by [10] who obtained important
results for the case of multi-unit auctions with common values. An important policy question
that can be addressed through this model is how government should auction treasury securities
in order to maximize its revenues. Usually, most of the literature on this topic focuses on the
private auction environment, whence the relevance of the paper.
Our paper uses a structural approach to study the identification and the estimation in the
case of a third-price sealed-bid auction model. The third-price auction is considered to be a
theoretical type of auction, as for the moment there is no known instance (up to our knowledge)
of such mechanism in practice. The rule of this game is such that the highest bidder wins the
object (efficient mechanism), but he pays a price equal to the third highest bid.
[15] is the seminal paper that formalizes for the first time the body of theory underly-
ing the third-price auction under the assumption of uniformly distributed private values and
risk-neutrality or constant absolute risk aversion. Thus, the authors show that in a third-price
auction: bids exceed private values, the marginal effect on bids of an increase in private values
is greater than one and increasing the number of bidders reduces the bids. It is easy to see why
one should afford to bid more than one’s valuation, as the winner will end up paying the third
highest bid. Still, this doesn’t preclude the situation for the winning bidder to incur a loss. The
probability of having a negative payoff increases as the number of participants increases, since
also the probability that the third-highest bid to be close to the winner’s valuation increases.
These peculiar properties of the model permit, at least, a better understanding of how the other
usual types of auctions (first price and second price auctions) work.
Besides this, the paper by [15] is a valuable source of data as it also contains a section where
the authors describe a laboratory experiment using the third-price auction format.
In an earlier paper, [19] extend the results obtained by [15] for any distribution function of
private values and any attitude towards the risk (including risk-seeking). The authors show that
in the presence of risk-seeking bidders, the revenue of the seller is higher than in the second
price auction (the latter type of auction outperforms the first price auction in case of risk-
seeking). Thus a third-price auction is appropriate for situations where bidders are risk-seeking
as the seller could better exploit the love for risk of the agents.
Moreover, [19, 20] provide recommendations for the use of such design in the case of Internet
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auctions for relatively cheap items.
For more arguments in this sense, see mainly [18] and also [21] where the authors consider that
buyers of cheap items on the Internet are fun-seeking, somehow like players in a casino.
Thus, the risk-seeking attitude of the Internet bidders can be supported by the presence of
numerous forms of Internet auction fraud: the non-delivery of the good, the delivery of a good
with different characteristics than previously advertised by the seller (misrepresentation), fee
stacking, triangulation, shill bidding...
From another point of view, the extensive use in practice of English auction format for the
Internet auctions is not consistent with the result of [17] showing that when bidders are risk
averse, a first-price auction yields higher revenue than a second-price auction.
Thus, the uncertainty of the bidding environment, the lack of seller’s commitment (coming from
the seller’s anonymity on the Consumer2Consumer auction web-sites) make authors believe
that Internet bidders may be risk-seeking.
[13] realize a comparison of the auction formats in function of the revenue they generate to
the charity organizer. When analyzing the winner-pay auctions (where bidders get utility from
the revenue they produce) the authors show that the third-price auction dominates in terms
of revenue the second-price auction and the first-price auction. We mention this reference, but
nevertheless, the equilibrium strategy for a charity auction under the third-price rule is different
from the strategy for standard auctions under the same format (but where the bidder do not value
the utility he generates).
Others works on third price auctions include [26] and [29]. The latter article claims that
in the presence of corruption the bidders might behave accordingly to a third-price auction,
although the auction was designed as a second-price auction.
Our paper proceeds as follows: in the next part we present the model of a third-price auction,
in the third section we proceed to the global identification approach, assuming the parameter of
risk-seekin as being given. Section 4 provides the corresponding estimation methodology along
with a Monte Carlo experiment. Section 5 provides the asymptotic statistics for our estimator.
Section 6 to 9 illustrate some extensions, namely the local functional identification approach,
the estimation of the risk-seeking parameter, the case where only the winning bids are observed
and the case with exogenous variables. Section 10 concludes.
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2 Model
A third-price auction is a game with at least 3 players (N≥ 3) and where each player, j = 1, ...N,
has a private information (his type) denoted by ν j. This private information is unknown to
the other players and do not influence the valuation for the good that the other players have
(situation which describes a game with incomplete information in a private value environment).
The equilibrium concept adequate for such situations is the Bayesian Nash Equilibrium.
We assume that the private information is independently and identically distributed, drawn
from a common cumulative distribution function ν j ∼ F , which is common knowledge for the
players, but unobservable for the statistician.
At equilibrium each player plays an action, x j, which is influenced by his private value ν j
and by the joint distribution function, F . The latter aspect constitutes the strategic component
of the game.
Otherwise said: x j = σ j(ν j,F), where function σ j is the solution of a Bayesian Nash Equi-
librium (known by the statistician). We consider symmetric bidders and whence σ j = σ .
Following [19], we consider bidders with constant absolute risk-seeking (CARS) attitude1
and therefore the equilibrium bid in the case of a third price auction with risk-seeking bidders
is:
x = ν +1λ
ln{
1+λM
F(ν)f (ν)
}(1)
where N ≥ 3, f is the probability density function (hereafter p.d.f.) of F , λ ≤ 0 is the CARS
parameter and, in order to alleviate notations, we put N−2 = M.
We make the following assumptions for the cumulative distribution function, F (some of
this assumptions can be found in the theoretical model of [19]):
Assumption 1. F ∈ C 3[0,A], i.e. F is three times continuously differentiable on [0,1].
Assumption 2. f (ν) = F ′(ν)> 0 for every ν ∈ (0,1].
Assumption 3.Ff
is an increasing function and limν→0
F(ν)f (ν)
= 0.
Assumption 4. As λ ≤ 0, we have the additional assumption: 1+ λM
F(ν)f (ν)
> 0.
1For buyers with constant absolute risk-seeking, the utility function has the following shape u(x) =a(
1− e−λx)
, with λ ≤ 0 and a≤ 0.
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We consider the risk-seeking parameter as being known (otherwise, the model will be
clearly nonparametrically non-identified).
Relation (1) can be written as:
x = σ(ν ,F,λ ) (2)
or under an implicit form2:
G◦σ(ν)−F(ν) = 0 (3)
or
G(x) = F ◦σ−1(x) (4)
where G denotes the distribution of bids x j.
3 Identification
Before the estimation of any theoretical game model, we must ensure that the model is well
identified. Identification consists in being able to uniquely recover the distribution of the latent
variables from the observables.
We assume that we observe all the bids of several games. Then the actual data is x jl that is
the bid of player j in game l. The number of players is assumed to be identical for each game,
but the players may be different for different games. In this section, the distribution F is iden-
tical for different games and we actually observe iid sample x jl of the c.d.f. G. For simplicity
we denote by i the couple ( j, l) and i = 1, ...,n (n = NL, where l = 1, ...,L).
Then the data identify the c.d.f. G. We recall that we assume that the strategy function σ is
known and that λ is also given. We relax this assumption later on.
Equation (4) defines a nonlinear functional equation where the unknown element is F .
We may consider global identification, i.e. the uniqueness of the solution, but we may also
analyze this equation locally. Obviously, global identification implies local identification. Nev-
ertheless, the local analysis is necessary for the asymptotic property of the estimator. Moreover,
it is well known that the ill-posedness of a nonlinear inverse problem should be analyzed locally
(see [9]).2This implicit form can be easily obtained from:
G(x) = Prob(x j ≤ x) = Prob(σ−1(x j)≤ σ−1(x)) = Prob(ν j ≤ σ−1(x)) = F(
σ−1(x))
where σ−1 defines the inverse of the strategy function, σ .
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The following proposition illustrates the first important result of the paper:
Proposition 1. In a third-price auction model with risk-seeking bidders, the joint cumulative
distribution function of the private values, F(ν), is globally nonparametrically identified with-
out any restrictions and the quantile function corresponding to it, F−1(t), is described by the
following relation:
F−1(t) =1λ
ln1
tM
t∫0
MsM−1eλG−1(s) ds. (5)
where t ∈ [0,1], F−1(t) is the quantile function of the private values and G−1(t) the quantile
function of the bids.
Proof: We rewrite equation (4) as:
G−1(t) = F−1(t)+1λ
ln
1+ λMF(
F−1(t))
f(F−1(t)
) (6)
Equation (6) can be written as the following differential equation:
β (t)+t
Mβ ′(t) = eλG
−1(t) (7)
where β (t) = eλF−1(t).
Hence, we managed to reduce our complex equation to a simple first order differential equation.
To solve this equation note that:
(tMβ (t))′
MtM−1= β (t)+
tM
β ′(t) (8)
and
tMβ (t) =t∫
0
MsM−1eλG−1(s) ds+C (9)
Since β (0) = 1 implies that C = 0, the constant is identified. Moreover, using l’Hôpital’s rule
we have that that:
limx→0+
1tM
t∫0
MsM−1eλG−1(s) ds = 1 (10)
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Thus the solution of (6) is given by:
F−1(t) =1λ
ln1
tM
t∫0
MsM−1eλG−1(s) ds. (11)
The previous equation highlights a link between the quantile function of private values and the
quantile of bids. Therefore we obtain an explicit form for the solution of our structural equation
and we can state the global identification for the sealed bid third-price auction model (in the
spirit of [14]). �
4 Estimation
The next step is to use the explicit form for the quantile function and to show how we can
estimate it.
Our approach is to use the empirical quantile function as a natural estimator for G−1(t) and
then to plug it in (11):
F̂−1(t) =1λ
ln1
tM
t∫0
MsM−1eλ Ĝ−1(s) ds (12)
where Ĝ−1(t) is the sample quantile function for the observables (data on bids) and is given by
Ĝ−1(t) = inf{
x : Ĝ(x)≥ t}
.
Before analyzing the asymptotic properties of our estimator, we will briefly discuss some
practical computation issues.
We will also conduct a Monte Carlo experiment as to see the performance of our estimator in a
finite sample.
The evaluation of our estimator on a discrete grid of k elements, t1 = 0, ..., tk = 1 and using
Riemann sums to approximate our integral leads to the following shape for our estimator:
F̂−1 (tk) =1λ
ln(
1tk
)M tkm
m
∑j=1
M(
jtkm
)M−1e
λ Ĝ−1
jtkm
(13)
where n is the number of observations available in the sample, t = i/n and m is the number of
elements of partitions used to approximate the integral. For simplicity, m doesn’t depend on tk
in this expression.
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For the Monte Carlo experiment, we used a beta distribution of parameters 2 and 1 from
which we drew i.i.d. The coefficient of risk-seeking was fixed equal to 1. We conducted two
configurations: one with 50 observations and another one with 100 observations. Under each
design we replicated the experiment 1000 times. Below we give the results illustrated on the
two different sample sizes:
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
11000 Monte Carlo simulations on 50 observations
Private values
Pro
ba
Mean of the estimatorTheoretical distribution
Figure 1: The true cumulative distribution function and the mean of the estimator for 1000simulations with a sample of 50 observations.
As we can see, the estimator performs well in small samples and we highlight the fact that,
usually in the literature for auction, simulations are conducted with larger samples compared
with those chosen by us. From a computation point of view, the execution time of 1000 repli-
cations with n = 50 was of ≈ 9350(s), which is quite fast.
5 Asymptotic statistics
Before deriving the asymptotics for our estimator using sample quantile process, we need to
check that the distribution of bids verifies certain properties.
From (6) and under the Assumption 1, we have that the cumulative distribution function of
the bids, G, satisfies the following properties:
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0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
11000 Monte Carlo simulations on 100 observations
Private values
Pro
ba
Mean of the estimatorTheoretical
Figure 2: The true cumulative distribution function and the mean of the estimator for 1000simulations with a sample of 100 observations.
Property 1. G has a compact support [0,xmax] with
xmax = 1+1λ
ln{
1+λM
1f (1)
}< ∞. (14)
Property 2. G ∈ C 2 on [0,xmax] and g > 0 on (0,xmax].
Thus, G verifies the necessary conditions in order to apply the following Property:
Property 3. (Lemma 21.4, [27]) G has compact support [0,xmax] and is continuously differ-
entiable on its support with strictly positive derivative g. Then, ∀ a > 0,√
n(
Ĝ−1−G−1)
converges in distribution in L∞(a,1) on [a,xmax] to a processBB
g(G−1(t)
) , where BB is the usualbrownian bridge on [0,1].
The covariance of this process is:
Σ(s, t) =s(1− t)
g(G−1(s)
)g(G−1(t)
) (15)where 0 < s≤ t ≤ 1.
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Property 4. Furthermore, G satisfies the conditions of Lemma 1.4.1 from [4] on the interval
[a,xmax], a > 0, as G(x)(1−G(x))|g′(x)|g2(x)
is continuous on the compact [a,xmax] and it is hence
bounded by a γ > 0.
Property 5. This property refers to an extension of Dvoretzky-Kiefer-Wolfowitz inequality (see
[8]) to the sample quantile process. Given that G satisfies the Property 4, we have that (see
Theorem 1.4.3 from [4]):
√n sup
t∈[a,b]|Ĝ−1(t)−G−1(t)| ∼ O(1),∀ 0 < a < b < xmax (16)
Under the conditions imposed by Property 3 we have the following proposition, which
represents the second important result of the paper:
Proposition 2. The empirical process of the quantile function, of the derivative of the quantile
function and of the density of private values converge in distribution to a gaussian process at
the parametric speed of convergence,√
n:
√n(
F̂−1(t)−F−1(t)) N(0,ϒ) (17)
√n(
F̂−1′(t)−F−1′(t)) N(0,Φ) (18)
√n(
f̂ (ν)− f (ν)) N(0,Γ) (19)
where ϒ, Φ, and Γ denote the asymptotic variance which will be computed in the following
lines.
Proof: We recall that: F−1(t) =1λ
ln1
tM
t∫0
MsM−1eλG−1(s) ds. Thus, we may consider that the
quantile function of the private values is a functional of the quantile of the bids: F−1 = B(G−1).
Hence, we differentiate the functional B with respect to G−1 in the direction of G̃−1 = Ĝ−1−
G−1:
B′G−1(
G̃−1)=
∂∂α
1λ
ln1
tM
t∫0
MsM−1eλ(
G−1+αG̃−1)(s) ds
∣∣∣∣∣∣α=0
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and we obtain:
√n(
F̂−1(t)−F−1(t))≈
t∫0
sM−1eλG−1(s)√n
(Ĝ−1−G−1
)(s)ds
t∫0
sM−1eλG−1(s) ds
(20)
The residual term of this first order Taylor expansion is negligible using Proposition 5.
Using Von Mises Calculus (see [24] and [27]) we conclude that:
√n(
F̂−1(t)−F−1(t)) N(0,ϒ) (21)
where the asymptotic variance of the process is given by:
ϒ(s, t) =1
b(t)b(s)
1∫0
1∫0
1[u≤t]1[v≤s]a(u)a(v)Σ(u,v)dudv (22)
We denoted a(s) = sM−1eλG−1(s), b(t) =
t∫0
sM−1eλG−1(s) ds, δ (t) =
√n(
Ĝ−1(t)−G−1(t))
and
Σ(u,v) has been defined in (15).
More details about this computation can be found in the Appendix A.1.
But the object of our interest is the weak convergence of the density of the private valu-
ations. Before computing it, we will need to perform an intermediary step which consists in
studying the asymptotic properties for the derivative of the quantile function for the private
values.
We have that: F−1′(t) =−M
λ t+
1λ
tM−1eλG−1(t)
t∫0
sM−1eλG−1(s) ds
.
This means that F−1′(t) is a functional of G−1(t) and thus we can differentiate it accord-
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ingly :
√n(
F̂−1′(t)−F−1′(t))≈
tM−1eλG−1(t)√n
(Ĝ−1(t)−G−1(t)
)t∫
0
sM−1eλG−1(s) ds
−
−
tM−1eλG−1(t)
t∫0
sM−1eλG−1(s)√n
(Ĝ−1(s)−G−1(s)
)ds
t∫0
sM−1eλG−1(s) ds
2(23)
We have the following result of convergence in distribution:
√n(
F̂−1′(t)−F−1′(t)) N(0,Φ) (24)
where the asymptotic variance of the process is computed as follows:
Φ(s, t) = Cov
a(s)δ (s)
b(s)−
a(s)s∫
0
a(v)δ (v)dv
(b(s))2,a(t)δ (t)
b(t)−
a(t)t∫
0
a(u)δ (u)du
(b(t))2
(25)
We denoted a(s) = sM−1eλG−1(s), b(t) =
t∫0
sM−1eλG−1(s) ds and δ (t) =
√n(
Ĝ−1(t)−G−1(t))
.
Finally we get that:
Φ(s, t) =a(s)a(t)b(s)b(t)
Σ(s, t)−2 a(s)a(t)b(s)(b(t))2
E
δ (s) t∫0
a(u)δ (u)du
+
a(s)a(t)(b(s))2(b(t))2
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)Σ(v,u)dvdu
(26)
A detailed computation is provided in A.2.
As we will see later on, we will need also covariance between the previous two empirical
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processes and we obtain (see A.3) :
∆(s, t) =Cov[√
n(
F̂−1′(s)−F−1′(s)),√
n(
F̂−1(t)−F−1(t))]
=a(s)
b(s)(b(t)E
δ (s) t∫0
a(u)δ (u)du
− a(s)(b(s))2b(t)
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)Σ(v,u)dvdu
(27)
The final step consists in describing the weak convergence for the density function of the
private values, f .
From F−1′(t) =
1f(F−1(t)
) we have that f (ν) = 1F−1′
(F(ν)
) = 1F−1′
[(F−1(t)
)−1] .From the last relation we can see it clearly that the density is a functional of the quan-
tile function and the derivative of the quantile function f = D(
F−1′,F−1
)and thus the total
derivative can be computed as d f = dDF−1′(
F̃−1′)+dDF−1
(F̃−1
).
Using Von Mises Calculus once more, we obtain (see appendix A.3):
√n(
f̂ (ν)− f (ν))≈−√
n(
F̂−1′−F−1′)(F)+F−1
′′(F)×
√n(
F̂−1−F−1)(F)
F−1′(F)[F−1′(F)
]2 (28)We will have the following result of weak convergence:
√n(
f̂ (ν)− f (ν)) N(0,Γ) (29)
where
Γ(s, t) =1[
F−1′(F(s))]2 Φ(s, t) 1[F−1′(F(t))]2 + F
−1′′(F(s))[F−1′(F(s))
]3 ϒ(s, t) F−1′′(F(t))[F−1′(F(t))]3−2 1[
F−1′(F(s))]2 ∆(s, t) F−1′′(F(t))[F−1′(F(t))]3
(30)
and ∆(s, t) is the covariance computed in (27). �
The√
n convergence of f might seem paradoxical. However it is not in contradiction with
usual statistical results. It is well known that the minimax speed of convergence of the density
of data is not√
n, but it is slower and it depends on the regularity of the problem. In our case,
the density f is not the density of the actual data, but it is a structural parameter. The paradox
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comes from the fact we cannot do better if we observe the latent variable ν . The√
n speed of
convergence is coming from the fact that our estimation is obtained as integrals of densities of
observed data.
Moreover, as all our estimators converge as stochastic processus to gaussian processus, non-
parametric bootstrap procedure is justified.
6 Local approach
Our initial problem defined by (4) is a nonlinear inverse problem and Proposition 1 basically
proves that this problem is well-posed because in terms of quantile function a unique solution
exists and is a continuous transformation of G−1.
In the estimation part we have implicitly proved that this inverse problem is also locally well-
posed.
Even if the main result has been derived, we present now a local analysis of our problem. This
presentation doesn’t contain any new important result, but it is interesting for the following
reasons.
Firstly the framework is more precise: the spaces are carefully defined and the concept of
derivation are more rigorous. For example, we show that the density of the latent variable
should be strictly positive at the boundary which takes values in [0,∞[
Secondly, in most of nonlinear inverse problem, the global analysis is difficult and only local
identification may be considered (see [3]). Thus the functional methodology presented here
can be applied for the identification of other types of games where the model is not globally
tractable.
Our approach is in style of [3] and [12].
Thirdly, extension of our model to unknown risk-seeking parameter or to exogenous variables
need a local analysis.
6.1 Description of the parameter space
We construct a geometrical setting based on a hilbertian space in which we will study the local
identification. The Hilbert space will allow us to perform the analysis of our statistical model
in a infinite dimension space, as a Hilbert space is the generalization of the Euclidean space to
an infinite number of dimensions.
15
-
Let us assume the following classical hilbertian space:
H ={
a : [0,A]→R : a ∈ C 1and is square integrable w.r.t. the Lebesgue measure}
(31)
The A value may be a finite positive number. The lower bound is conventionally chosen equal
to 0. Thus, since we are interested in functions a that are square integrable to one, we will work
with a subset of the unit sphere as in [5]. So, we denote:
F =
a ∈H : a(ν)> 0,A∫
0
a2(ν)dν = 1
(32)The unit sphere of a hilbertian space is a curved Hilbert manifold as the tangent space at any
point on the unit sphere is locally isomorphic to the surface of the sphere, but it does not
correspond to it. We consider that the support of the c.d.f. with density f (ν) = a2(ν) is [0,A].
We needed to consider a squared form for f as a hilbertian structure is generated by squared
integrable functions (and not by integrable functions). Hence, we have defined the space of
densities equipped with the Hellinger distance 3.
Actually, more structure is required on F : the existence of a Bayesian Nash Equilibrium
requires thatFf
is an increasing function and therefore we denote:
F0 =
a ∈F :
ν∫0
a2(t)dt
a2(ν)is increasing
(33)
where F0 ⊂F and F0 bears some additional restrictions when compared with F .
Note that F is a manifold in H and for any a ∈F there exists a tangent space to F in a
described by the following property:
Ta =
b ∈H : 〈a,b〉=A∫
0
a(t)b(t)dt = 0
(34)3The Hellinger distance between f1 and f2 equals
∫ (√f1(ν)−
√f2(ν)
)2dν .
16
-
Geometrically, the above expression says that a radius of our unit sphere is orthogonal to a tan-
gent vector to the sphere. The tangent space will provide us with a locally linear approximation
of the manifold and will allow us to conduct our analysis in the tangent space to the manifold.
The tangent affine space will be a+Ta.
The observations x are i.i.d. with a c.d.f. denoted by G which is obviously identifiable. The
strategy function is now considered as a function of a and denoted by σa(ν)=σ
ν , ν∫0
a2(t)dt
and it is an injective application for all F ∈F . Hence, the game strategy is the following:
ν → σa(ν) = ν +1λ
ln
1+
λM
ν∫0
a2(t)dt
a2(ν)
(35)
which is a strictly increasing one to one function.
As the c.d.f. G is related to the c.d.f. F we will have the following relation:
G◦σa(ν)−ν∫
0
a2(t)dt = 0 (36)
The identification issue reduces to the study of the uniqueness of the solution of this relation
considered as a function of a. Following [12], we consider a local identification approach in a
nonparametric specification4.
6.2 The local nonparametric identification procedure
Let us first consider the purely nonparametric model with N fixed and known and we assume
that the parameter λ is also given. Our objective is to locally identify a, i.e. to show that for
any a there exists a neighborhood of a such as the solution of (36) is unique in this domain.
This condition is usually found using the implicit function theorem.
4As we know nonparametric statistics permits the estimation of a statistical functional, otherwise said, theestimation of a functional T (F).
17
-
After we compose the functions in the relation (36), we obtain:
G
ν +
1λ
ln
1+
λM
ν∫0
a2(t)dt
a2(ν)
−
ν∫0
a2(t)dt = 0 (37)
The previous equation is in fact another way of writing G◦σ −F = 0.
First step Following the procedure described by [12], we start by computing the Gâteaux
derivatives 5 of the left hand side (hereafter LHS) of (11). Hence we replace a by a+αb and
we take the derivatives w.r.t. α in α = 0.
Therefore, our operator of interest will be:
Ta(b)(ν) = σ ′a(g◦σa)(ν)−2ν∫
0
a(t)b(t)dt (38)
where g is the p.d.f. associated to G and σ ′a(b,ν) =∂
∂ασa+αb(ν)
∣∣∣∣α=0
Knowing that:
f (ν) = (G◦σa)′ =∂
∂νσa(ν)× (g◦σa)(ν) (39)
we can rewrite our operator as:
Ta(b)(ν) =f (ν)
∂∂ν
σa(ν)σ ′a(b,ν)−2
ν∫0
a(t)b(t)dt (40)
After some computations (see the Appendix B.1), we obtain the following shape for our oper-
ator (remember that f (ν) = a2(ν) and F(ν) =ν∫
0
a2(t)dt):
5 From a mathematical point of view, the Gâteaux derivative is the generalization of the directional derivativein a infinite dimensional space. Statistically, the Gâteaux derivative gives us the rate of change in a statisticalfunctional following a small amount of contamination by another distribution function, H.
18
-
Ta(b)(ν) = Sa(ν)ν∫
0
a(t)b(t)dt +Ra(ν)a(ν)b(ν) (41)
where
Sa(ν) =−2M f 2(ν)+λF(ν) f (ν)−2a(ν)a′(ν)F(ν)
M f 2(ν)+λF(ν) f (ν)−2a(ν)a′(ν)F(ν)+ f 2(ν)(42)
Ra(ν) =−2F(ν) f (ν)
M f 2(ν)+λF(ν) f (ν)−2a(ν)a′(ν)F(ν)+ f 2(ν)(43)
Second step The next point of our analysis is to verify if Ta(b) is the Fréchet derivative6 of
the LHS of (36). First, note that Ta(b) is a bounded continuous linear operator defined on Ta
(the tangent space of F in a). Moreover Ta(b)(ν) is also a continuous operator as a function
of a in C 0[0,A]. Then Ta(b) is actually the Fréchet derivative of the LHS of (36) (see [22]).
Final step The following step in the identification analysis is to check if Ta(b) is one to one
on Ta as a function of b. As Ta is a linear operator, the problem boils down to check if the
equation Ta(b) = 0 has or not a solution different from 0 in Ta. From (41) we can say that the
equation Ta(b) = 0 is equivalent to:
(M f 2(ν)+λF(ν) f (ν)−2a(ν)a′(ν)F(ν)
) ν∫0
a(t)b(t)dt +F(ν) f (ν)a(ν)b(ν) = 0 (44)
The unknowns of the above equation are:
H(ν) =ν∫
0
a(t)b(t)dt
h(ν) = a(ν)b(ν) (45)
The solution in H for this first order linear differential equation is (for more details concerning
6We cannot use Hadamard differentiation because we use chain rule, nor Gâteaux differentiation as we applyimplicit function theorem.
19
-
the computations see the appendix B.2):
H = KF−M(ν) f (ν)e−νλ (46)
Remember now, that our interest is to have the solution in b. Therefore:
b(ν) =1
a(ν)H ′(ν) (47)
Next we have to see if b ∈ T sa , where T sa denotes the tangent space to the unit sphere (we can
see the sphere as a manifold which embeds our sub-manifold F . Note that:
Ta ⊂
b ∈H : 〈a,b〉=A∫
0
a(ν)b(ν)dν = 0
= T sa (48)
Then we have to check the conditionA∫
0
a(ν)b(ν)dν = 0 or equivalently:
A∫0
a(ν)(
1a(ν)
H ′(ν))
dν = 0 (49)
which leads to:
H(A)−H(0) = 0 (50)
From (45) we have that H(0) = 0. Thus b ∈ T sa if and only if H(A) = 0 or equivalently if
f (A) = 0. Then if f (A) 6= 0 the intersection of the null space of Ta(b) and of T sa reduces to
{0}. This implies that the intersection of N (Ta(b)) with Ta is {0}. If the solution of the
differential equation is not included in the tangent space to F , it will also be the case for F0,
as the latter is a subspace of the former.
Proposition 3. Let us assume that f is bounded from below by a strictly positive number. Then
the third price auction model is nonparametrically locally identified in a neighborhood of a =
f12 .
Proof. This result follows from the implicit function theorem (see Appendix B.3 for a statement
20
-
of the classical implicit function theorem). The parameter a is the solution of the nonlinear
equation G ◦σa(ν)−ν∫
0
a2(t)dt = 0. In a neighborhood of a, the Fréchet derivative is Ta(b).
This linear operator is one to one in Ta (the tangent space of the parameter space in a) and its
inverse is continuous. Then the implicit function theorem applies.
6.3 Estimation
Usually, the fact that we cannot observe the private values of the individuals leads us to a situ-
ation in which the speed of convergence of our estimator changes. In order to characterize how
this speed of convergence is affected by the existence of unobservables, we will use a concept
called "degree of ill-posedness".
As the linear operator associated with our game described by σa and F is the inverse operator
T−1a we need to check its properties. Thus we will study the degree of ill-posedness of T−1
a .
In order to find the inverse operator and to look at its properties, we have to solve the equation
Ta(b) = u. This equation is equivalent to:
(M f 2(ν)+λF(ν) f (ν)−2a(ν)a′(ν)F(ν)
)H(ν)+F(ν) f (ν)h(ν) = v (51)
where v = −(
M f 2(ν)+λF(ν) f (ν)−2a(ν)a′(ν)F(ν)+ f 2(ν)) u
2. Using elementary prop-
erties of linear first order differential equations, we get the following family of solutions (see
Appendix B.4 for more details) :
Hc(ν) = β (ν)
ν∫
0
α(t)u(t)dt +C
and bc(ν) = 1a(ν)H ′c(ν) (52)
where β (ν)= e−P(ν)=F−M(ν)e−λν f (ν), C∈R and α(ν)=−12
M+λF(ν)f (ν)
−2a(ν)a′(ν) F(ν)f 2(ν)
+1
F1−M(ν)e−λν.
As we only look at the solutions in T sa , the constant C is uniquely defined under the assumption
f (A) 6= 0. Indeed,A∫
0
bc(ν)ac(ν)dν = 0 implies that:
A∫0
β (ν) ν∫0
α(t)u(t)dt +β (ν)C
′ dν = 0 (53)
21
-
Then,
β (ν)ν∫
0
α(t)u(t)dt
∣∣∣∣∣∣A
0
+β (ν)C
∣∣∣∣∣A
0
= 0
So,
β (A)
A∫0
α(ν)u(ν)dν +C
= 0
Under f (A) 6= 0, we have that β (A) 6= 0. Hence we obtain that C = −A∫
0
α(ν)u(ν)dν and
therefore, the unique solution in T sa (and implicitly in Ta) is given by:
H(ν) = β (ν)
ν∫
0
α(t)u(t)dt−A∫
0
α(ν)u(ν)dν
(54)Hence, the form of our inverse operator is the following:
T−1a = b(ν) =1
a(ν)β ′(ν)
ν∫
0
α(t)u(t)dt−A∫
0
α(ν)u(ν)dν
+ 1a(ν)β (ν)α(ν)u(ν) (55)Hence T−1a is a continuous operator and our model (σ ,a) has a degree of ill-posedness equal
to 0.
As our model is well identified, we know that a is the unique local solution of the equation (36)
which can be written in the following implicit form: Ψ(G,a) = 0. In the following lines we
will solve locally this equation and afterwards apply the Delta Theorem. Therefore:∂Ψ∂G
dG+
∂Ψ∂a
da = 0.
Next:
da =−[
∂Ψ∂a
]−1 ∂Ψ∂G
dG (56)
where da = â− a, dG = Ĝ−G and we recall that[
∂Ψ∂a
]−1is exactly the operator we have
computed in the previous lines, i.e., T−1a . So we want to find out what is the impact of our
22
-
equation of interest over the quantity da.
As our structural equation is of form G ◦σa−F = 0 we obtain dG = Ĝ(σa)−G(σa) and this
is a known quantity.
Using (56) and (55) we obtain that:
â−a =− 1a(ν)
β ′(ν)
ν∫
0
α(t)(
Ĝ(σa)−G(σa))(t)dt−
A∫0
α(ν)(
Ĝ(σa)−G(σa))(ν)dν
− 1a(ν)
β (ν)α(ν)(
Ĝ(σa)−G(σa))(ν) (57)
As√
n(Ĝ−G)⇒ BB, we can write the following result of uniform convergence:
√n(â−a)⇒ T−1a (BB◦σa) (58)
As we can see above, we obtained a parametric speed of convergence for our nonparametric
estimator, as we were expecting given a model with a degree of ill-posedness equal to 0.
7 Estimation of the risk-seeking parameter
Here we are back on a support ν ∈ [0,1]. If λ is unknown, the precedent scheme of observations
will not allow us anymore to identify λ . Indeed, the only parameter that can be identified is
the function G−1, as G−1 depends on F−1 and λ in a way that do not allow us to identify
nonparametrically F and λ .
A possible configuration of data which allows us to estimate λ is to observe auctions with
different number of participants. To simplify the problem, we assume that we observe two
values of N (N1 and N2), corresponding to the same cumulative distribution function of private
values, F . We will have thus two values of the quantile function of the bids, G−11 and G−12 , and
λ will satisfy the following relation:
1tM1
t∫0
M1sM1−1eλG−11 (s) ds− 1
tM2
t∫0
M2sM2−1eλG−12 (s) ds = 0 ∀t ∈ [0,1] (59)
23
-
where M1 = N1−2 and M2 = N2−2.
Using the change of variable G−1(s) = x, (59) can be written as:
1tM1
G−11 (t)∫0
M1G1(x)M1−1eλxg1(x)dx−1
tM2
G−12 (t)∫0
M2G2(x)M2−1eλxg2(x)dx= 0 ∀t ∈ [0,1] (60)
We write (60) using expectations corresponding to the distribution functions G1 and G2:
M1tM1E1
(1[G1(x)≤t]G1(x)
M1−1eλx)=
M2tM2E2
(1[G2(x)≤t]G2(x)
M2−1eλx)∀t ∈ [0,1] (61)
The parameter λ is identified if this equation (for given G−11 , G−12 , M1 and M2) has only one
solution.
The nonlinearity of this equation makes difficult the characterization of the uniqueness
condition. The solution of this equation is locally unique if the derivative of the LHS term of
(59) is not 0 for the true value.
This condition can be written as:
M1tM1E1
(1[G1(x)≤t]G1(x)
M1−1xeλx)− M2
tM2E2
(1[G2(x)≤t]G2(x)
M2−1xeλx)6= 0 (62)
for at least one value of t.
A sufficient condition of local identification can be obtained for t = 1:
M1E1(
G1(x)M1−1xeλx)−M2E2
(G2(x)M2−1xeλx
)6= 0 (63)
The estimation can be done by replacing G1 and G2 with their estimators. In fact, the equation
(61) defines an infinity of equations for ∀t, but we can restrain ourselves to the case t = 1.
If data will consist of pairs (xi,M), i = 1, ...n, with M ∈ (M1,M2), the λ parameter will be
the solution of:
M1∑ni=11[Mi=M1]
n
∑j=1
1[M j=M1]
(1n
n
∑i=1
1[xi≤x j]1[Mi=M1]
)M1−1eλx j
− M2∑ni=11[Mi=M2]
n
∑j=1
1[M j=M2]
(1n
n
∑i=1
1[xi≤x j]1[Mi=M2]
)M2−1eλx j = 0 (64)
24
-
We denote (64) with rn(λ )= 0 and we have that√
n(
λ̂ −λ)
behaves like−(
lim∂ rn∂λ
)−1√nrn(λ ).
8 Winning bids
Let us suppose now that we observe L auctions, but for each of this auction we observe only
the winning bid, that is the bid corresponding to the third price proposed. In this case, the dis-
tribution of ν corresponding to the observed price is the distribution of the third order statistics
of an i.i.d. sample of L values. Thus the number of observation is n = L×1. The distribution
of a third order statistic is given by:
FN,3(ν) =N
∑k=3
CkNFk(ν)(1−F(ν))N−k (65)
and its density:
fN,3(ν) =N!
2(N−3)!F2(ν)(1−F(ν))N−3 f (ν) (66)
Hence, the method is the following: using the distribution of prices paid by the winner, G,
we estimate FN,3(ν) using (5). This estimator follows the gaussian process as shown previ-
ously. Afterwards, we compute F using (65). This transformation has been studied by [1], who
showed that the relation between F and FN,3 is bijective (see also [2]). Otherwise said, if we
have an estimation of FN,3, we can compute F as the unique solution of the equation:
FN,3(ν) =F(ν)∫0
N!2(N−3)!
t2(1− t)N−3 dt (67)
Many methods of resolution of this problem have been proposed in the literature (see [1]). The
speed of estimation of F and its asymptotic properties are deduced from this equation. Briefly,
we can see this problem as a nonlinear inverse problem T (F) = FN,3 whose properties are
characterized by the Fréchet differential of T :
T ′F(F̃) =N!
2(N−3)!F2(1−F)N−3F̃ (68)
This differential shows that the problem is well-posed and, using a Taylor expansion, we obtain:
√n(
F̂−F)=
2(N−3)!N!F2(1−F)N−3
√n(
F̂N,3−FN,3)+o(1) (69)
25
-
We have thus a convergence in√
n of F̂ − F and the fact that√
n(
F̂−F)
converges to a
centered gaussian process, whose variance can be inferred from the above expression. We can
conduct a similar analysis for f using a linear approximation of:
fN,3 =N!
2(N−3)!F2(1−F)N−3 f (70)
We have that:
√n(
f̂ − f)=
[N!
(N−3)!F (1−F)N−3 f + N!
2(N−4)!F2(1−F)N−4 f+
+N!
2(N−3)!F2(1−F)N−3
]√n(
f̂N,3− fN,3)+o(1)
(71)
We have thus the convergence of√
n(
f̂ − f)
to a gaussian process.
9 Exogenous variables
Let us suppose that we observe L games with N players and where all bids are observed. Thus
we will have n = L×N observations. Moreover, let us suppose that for each game the distri-
bution of types is identical for all players, but it varies between the games conditionally on the
observed exogenous variables. The data allows us therefore to identify the conditional distri-
bution G(x|Z = z j) and the conditional quantiles, G−1(t|z). Whence, we identify F(ν |z) using
the generalization of (5):
F−1(t|z) = 1λ
ln1
tM
t∫0
MsM−1eλG−1(s|z) ds (72)
We generalize the estimation procedure presented in section 4. The conditional quantile func-
tion Ĝ−1(t|z) may be estimated by several techniques and F−1(t|z) is then estimated by plug-in
method if we replace in (72) G−1 by its estimator. We do not detail the asymptotic properties
of this estimator. Briefly speaking, for almost sure value of z,√
nhqn(
Ĝ−1(t|z)−G−1(t|z))
converges to a gaussian process, where hn is a bandwidth of the estimation of G−1(t|z) (if ker-
nel smoothing is used or hn is any regularization parameter) and q is the dimension of z (see
[25] and [28]). Then the results of Proposition 2 applies using a comparable argument where√
n is replaced by√
nhqn and where the covariance operators of the gaussian processes become
26
-
functions of z. Thus for a fixed z and a fixed M, G−1(t|z) still converges in process but at a
nonparametric speed. If N→∞, Ĝ−1(t|z) converges in process to G−1(t|z) at a speed of√
N. If
M varies between the games we could estimate also λ using the arguments of this section and
of section 7. In case λ is not a constant and depends on the exogenous variables of the game,
we would need to have exclusion conditions to be able to identify λ (zl) and F(ν |zl).We leave
this last case for further research.
10 Conclusion
This paper solved both for identification and estimation in a sealed-bid third price auction
model using a fully nonparametric approach. We considered an i.i.d. private values environ-
ment with risk-seeking bidders and we treated the case where the parameter of risk-seeking
is known. Thus, using a global approach, we obtained nonparametric identification by link-
ing the quantiles of bids and valuations. Moreover, the speed of convergence of our estimator
computed using the functional delta method is a parametric one.
We provide also a necessary condition for local identification in a infinite dimensional func-
tional framework. Thus, as long as the value of the density function computed in the upper limit
of the support of the private values is not null, f (A) 6= 0, the model under consideration is lo-
cally identified. The inverse problem generated by the model is well-posed in Hadamard’s
sense, so we find a speed of convergence equal to√
n. Thus, the results for the asymptotic
convergence are the same as in the case of the global approach.
The case of this auction is very interesting also from an econometric point of view. In the
case of a first price auction, the speed of convergence for an estimator F̂ is less than the speed
of convergence for Ĝ (in fact it equals the speed of convergence for ĝ, the two speeds are equal
in the case of a second price auction, whereas in the case of a third-price auction both F̂ and f̂
converge to the speed of Ĝ.
Up to our knowledge, the identification issues concerning this types of auction has not been
studied before. Even if third-price auction is an "exotic" type of game, solving for its identifi-
cation helps our understanding of more subtle models.
Given the gambling aspect of the third-price auction, [19] recommend this type of auction to
be used for situations where bidders are risk-seeking. Thus, the seller could exploit better the
love for risk of agents and, as the authors show in their paper, seller’s revenue is higher than
27
-
in the second price auction (the latter type of auction outperforms the first price auction in
case of risk-seeking). The authors argue that the bidders in an Internet auction can be char-
acterized as being risk-loving, and therefore they recommend the use of a third-price auction
instead of an English auction design (as it is the case in reality for most of the Internet auctions).
Even though there is no real data on this kind of auction (as far as we know), [15] conducted
an experiment on third price auction using different numbers of participants and hence we can
use our procedure to estimate not only the c.d.f. of private values, but also the parameter that
characterizes the attitude towards the risk.
Thus this article contributes to the literature of structural analysis of auction data especially
by providing identification results for an interesting format of auction, by discussing the esti-
mation issues and different extensions.
28
-
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Appendix
A Global approach: weak convergence
A.1 Part I
We have that:
√n(
F̂−1(t)−F−1(t))≈
t∫0
sM−1eλG−1(s)√n
(Ĝ−1(s)−G−1(s)
)ds
t∫0
sM−1eλG−1(s) ds
(A.1)
and if we denote a(s)= sM−1eλG−1(s), b(t)=
t∫0
sM−1eλG−1(s) ds and δ (t)=
√n(
Ĝ−1(t)−G−1(t))
we may compute the covariance operator as follows:
ϒ(s, t) = Cov
1b(s)
s∫0
a(v)δ (v)dv
, 1b(t)
t∫0
a(u)δ (u)du
=1
b(s)b(t)E
1∫0
1[v≤s]a(v)δ (v)dv×1∫
0
1[u≤t]a(u)δ (u)du
=
1b(s)b(t)
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)E[δ (v)δ (u)
]dvdu
=1
b(s)b(t)
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)Σ(v,u)dvdu
(A.2)
A.2 Part II
The expression for the derivative of the quantile function for the private values is given by:
F−1′(t) =−M
λ t+
1λ
tM−1eλG−1(t)
t∫0
sM−1eλG−1(s) ds
(A.3)
32
-
Its derivative with respect to G−1 in the direction of G̃−1 = Ĝ−1−G−1 is computed as follows:
C′G−1(
G̃−1)=
∂∂α
−M
λ t+
1λ
tM−1eλ(
G−1+αG̃−1)(t)
t∫0
sM−1eλ(
G−1+αG̃−1)
ds
∣∣∣∣∣∣∣∣∣∣∣∣∣α=0
where we used the fact that F−1′
is a functional of G−1: F−1′= C(G−1). Von Mises calculus
gives us:
√n(
F̂−1′(t)−F−1′(t))≈
tM−1eλG−1(t)√n
(Ĝ−1(t)−G−1(t)
)t∫
0
sM−1eλG−1(s) ds
−
−
tM−1eλG−1(t)
t∫0
sM−1eλG−1(s)√n
(Ĝ−1(s)−G−1(s)
)ds
t∫0
sM−1eλG−1(s) ds
2(A.4)
To alleviate notations in the computation of the covariance operator we denote a(s)= sM−1eλG−1(s),
b(t) =t∫
0
sM−1eλG−1(s) ds and δ (t) =
√n(
Ĝ−1(t)−G−1(t))
.
In the following lines, we provide the explicit computation for the covariance of the empir-
33
-
ical process√
n(
F̂−1′(t)−F−1′(t))
:
Φ(s, t) = Cov
a(s)δ (s)
b(s)−
a(s)s∫
0
a(v)δ (v)dv
(b(s))2,a(t)δ (t)
b(t)−
a(t)t∫
0
a(u)δ (u)du
(b(t))2
= Cov[
a(s)δ (s)b(s)
,a(t)δ (t)
b(t)
]+2Cov
a(s)δ (s)
b(s),−
a(t)t∫
0
a(u)δ (u)du
(b(t))2
+Cov
a(s)
s∫0
a(v)δ (v)dv
(b(s))2,
a(t)t∫
0
a(u)δ (u)du
(b(t))2
=
a(s)a(t)b(s)b(t)
Σ(s, t)−2 a(s)a(t)b(s)(b(t))2
E
δ (s) t∫0
a(u)δ (u)du
+
a(s)a(t)(b(s))2(b(t))2
E
1∫0
1[v≤s]a(v)δ (v)dv×1∫
0
1[u≤t]a(u)δ (u)du
(A.5)
The last term of the sum can be written as:
a(s)a(t)(b(s))2(b(t))2
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)E[δ (v)δ (u)
]dvdu
=a(s)a(t)
(b(s))2(b(t))2
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)Σ(v,u)dvdu (A.6)
34
-
Thus, we obtain:
Φ(s, t) =a(s)a(t)b(s)b(t)
Σ(s, t)−2 a(s)a(t)b(s)(b(t))2
E
δ (s) t∫0
a(u)δ (u)du
+
a(s)a(t)(b(s))2(b(t))2
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)Σ(v,u)dvdu
(A.7)
A.3 Part III
∆(s, t) =Cov[√
n(
F̂−1′(s)−F−1′(s)),√
n(
F̂−1(t)−F−1(t))]
= Cov
a(s)δ (s)
b(s)−
a(s)s∫
0
a(v)δ (v)dv
(b(s))2,
t∫0
a(u)δ (u)du
b(t)
=
a(s)b(s)(b(t)
E
δ (s) t∫0
a(u)δ (u)du
− a(s)(b(s))2b(t)
1∫0
1∫0
1[v≤s]1[u≤t]a(v)a(u)Σ(v,u)dvdu
(A.8)
A.4 Part IV
Let a = F−1′and b = F−1, then we have that f =
1a(b−1) . The partial derivatives will be:
D′a(ã) =∂
∂α
[1
(a+α ã)(b−1)]∣∣∣∣∣
α=0
=−ã(
b−1)
[a(b−1)]2 (A.9)
D′b(b̃) =∂
∂α
[1
a◦ (b+α b̃)−1
]∣∣∣∣∣α=0
=a′(
b−1)× b̃(b
−1)b′(b−1)[
a(b−1)]2 (A.10)
The result in (A.10) was obtained using the fact that b ◦ b−1(ν) = ν and taking the functional
derivative leads to (we assume that b is differentiable, with density b′ such that b′(
b−1)> 0):
(b−1)′(b̃) =−
b̃(
b−1)
b′(b−1) (A.11)
35
-
B Local approach
B.1 Computation of different elements of the operator
After some tedious computation we obtain the components of our operator of interest. In the
cases where we took the derivative w.r.t. a function, Gâteaux derivatives have been used.
σ ′a(b,ν) =∂
∂ασa+αb(ν)
∣∣∣∣α=0
=1M
1
1+λM
ν∫0
a2(t)dt
a2(ν)
2
a2(ν)ν∫
0
a(t)b(t)dt−a(ν)b(ν)ν∫
0
a2(t)dt
a4(ν)
∂∂ν
σa(ν) = 1+1M
1
1+λM
ν∫0
a2(t)dt
a2(ν)
a4(ν)−2a(ν)a′(ν)ν∫
0
a2(t)dt
a4(ν)
B.2 Solution of the equation Ta(b) = 0
Our equation of interest (22) can be rewritten as:
(M f 2(ν)+λF(ν) f (ν)−2a(ν)a′(ν)F(ν)
)H +F(ν) f (ν)h = 0
Hence:
36
-
hH
=−M f (ν)F(ν)
−λ + 2a(ν)a′(ν)
f (ν)
lnH =−M lnF(ν)−λν + ln f (ν)+C
H = e−M lnF(ν)−λν+ln f (ν)+C
So, our solution is:
H = KF−M(ν) f (ν)e−νλ
B.3 Mathematical recalls of different results used in the paper
The implicit function theorem of Hildebrandt and Graves(1927)
Suppose that:
i) the mapping H : U(x0,y0) ⊆ X ×Y → Z is defined on an open neighborhood U(x0,y0), and
H(x0,y0) = 0, where X ,Y, and Z are real or complex Banach spaces.
ii) Hy exists as a partial Fréchet-derivative on U(x0,y0) and Hy(x0,y0 : Y → Z is bijective,
which is equivalent to say that the inverse operator Hy(x0,y0)−1 : Z→Y , exists as a continuous
linear operator.
iii) H and Hy are continuous at (x0,y0).
Then the following result of existence and uniqueness is true: there exist positive numbers
r0 and r such that for every x ∈ X satisfying‖x− x0‖ ≤ r0 and there is exactly one y(x) ∈ Y for
which∥∥y(x)− y0∥∥≤ r and H(x,y(x)) = 0.
37
-
B.4 Solution of the equation Ta(b) = u
We solve the equation Ta(b) = u for b by finding an inverse operator T−1 such that b = T−1(u).
Therefore the solution to the inhomogeneous equation gives us the inverse operator. We provide
a brief recall of the solution of a first order linear differential equation. Assume the following
first order linear differential equation:
y′+ p(t)y = f (t)
Solution of such kind of equation is given by:
y(t) = e−P(t){∫
eP(t) f (t)dt +C}
where P(t) is an antiderivative of p(t), P(t) =∫
p(t)dt, which leads to an integrating factor
eP(t).
Coming back to our equation of interest (51), we can rewrite it as:
h(ν)+
(M
f (ν)F(ν)
+λ − 2a(ν)a′(ν)
f (ν)
)H =
v(ν)F(ν) f (ν)
If we denote p(ν)=
(M
f (ν)F(ν)
+λ − 2a(ν)a′(ν)
f (ν)
), we have the following antiderivative P(ν)=
M lnF(ν)+λν − ln f (ν). Therefore e−P(ν) = F−M(ν)e−λν f (ν) and the general solution of
our equation will be:
Hc(ν) = β (ν)
ν∫
0
α(t)u(t)dt +C
where β (ν)= e−P(ν)=F−M(ν)e−λν f (ν),C∈R and α(ν)=−12
M+λF(ν)f (ν)
−2a(ν)a′(ν) F(ν)f 2(ν)
+1
F1−M(ν)e−λν.
38
IntroductionModelIdentificationEstimationAsymptotic statisticsLocal approachDescription of the parameter spaceThe local nonparametric identification procedureEstimation
Estimation of the risk-seeking parameterWinning bidsExogenous variablesConclusionReferencesGlobal approach: weak convergencePart IPart IIPart IIIPart IV
Local approachComputation of different elements of the operatorSolution of the equation Ta(b)=0Mathematical recalls of different results used in the paperSolution of the equation Ta(b)=u