optimal nonparametric estimation of first-price auctions 30pt by

Optimal Nonparametric Estimation ofFirst-Price Auctions

by Emmanuel Guerre, Isabelle Perrigne and Quang Vuong

Presenter: Andreea ENACHE1

1CREST-LEI and Paris School of Economics

25 April 2012

OutlineObjectives of the paper

Economic framework

Identification issue

Estimation

Advantages of GPV’s procedure

1/20 GUERRE / PERRIGNE/ VUONG

Objectives of the paper

• Does a theoretical auction model place any restrictions on observabledata to be tested?

• Does a structural approach require a priori parametric information aboutthe structural elements to identify the model?

• Propose an estimation procedure that does not rely upon parametric as-sumptions and that is computationally feasible.


Economic framework

Hypothesis of the first-price auction model

• A selling auction of a single and indivisible object.

• All bids are submitted simultaneously.

• The bidder with the highest bid wins and pays its bid.

• Bids are taken into account only if they are at least as high as a reserva-tion price p0.

• Each bidder has a private valuation vi for the auctioned object.

• Each bidder doesn’t know other bidders’ private values, but knows thatall private values including his own have been independently drawn froma common distribution F(·) (IPV environment).


Economic framework

Hypothesis of the first-price auction model

• F(·) is absolutely continuous with density f (·) and support [v, v].

• F(·), I (the number of potential bidders) and the reservation price p0 arecommon knowledge with p0 ∈ [v, v]

• Each bidder is assumed to be risk neutral.

• The equilibrium bid bi corresponding to the symmetric Bayesian NashEquilibrium is given by:

bi = s(vi,F, I, p0) ≡ vi −1(

F(vi))I−1

vi∫p0

(F(u)

)I−1 du (1)


Economic framework

Resolution of the model

• The objective function of the bidder

maxbi

(vi − bi) Pr(bi wins) = (vi − bi)FI−1(

s−1(bi))

leads to the following FOC:

1 = (vi − s(vi))(I − 1)f (vi)

F(vi)

1s′(vi)

(2)

with boundary condition s(p0) = p0.

• (2) is a first order differential equation in s(·) whose solution givesus the equilibrium bid in (1).



"A rose by any other name maynot be a rose!"

(Gujarati, Porter, 2009, Essentials of Econometrics)



What is identification?

Identification...

• Allows to verify whether the underlying structure (distribution, pa-rameters...) can be recovered from the observed random vari-ables =⇒ it is an existence problem.

• Precedes estimation and it is invariant to the estimation proce-dure.

• Is based on the population version of the stochastic system andnot on a particular sample.



Some definitions...

Definition 1The parameters a1 and a2 in F0 are observationally equivalent (a1 ∼ a2) iff G1 = G2,where Gi = Fai ◦ s−1

ai (i = 1, 2).

Definition 2The parameter a ∈ F0 is globally identified iff ∀a∗ ∈ F0, a∗ ∼ a⇒ a∗ = a.The model (s, a) is globally identified (for a given functional s) if all a’s in F0 are globallyidentified.

Definition 3

The parameter a ∈ F0 is locally identified iff there exists a neighborhood V(a) in F0such that ∀a∗ ∈ V(a), a∗ ∼ a⇒ a∗ = a.



Nonparametric identification of first-price sealed bid auction

Assume that p0 = v⇒ number of potential bidders (I)=number of actual bidders. Hence:

• I and bi are observed by the econometrician.

• F is the unknown structural element which needs to be identified.

Technical issue: s(·) depends also on the unknown parameter F(·), as we can seefrom (1).

Solution provided by GPV: If G is the distribution function of the bids and g the proba-bility density function of the bids, then:

G(b) = Prob(bi ≤ b) = Prob(s−1(bi) ≤ s−1(b)) = Prob(vi ≤ s−1(b)) = F(

s−1(b))= F(v)




Then:

g(b) =f (s−1(b))

s′(v)=

f (v)s′(v)

Therefore:G(b)g(b)

=F(v)s′(v)

f (v)

Using (2) we rewrite:

vi = s(vi) +1

I − 1F(vi)s′(vi)

f (vi)(3)




If we replace the expression ofG(b)g(b)

in (3) we get the central result of the paper:

vi = ξ(bi,G, I) ≡ bi +1

I − 1G(bi)

g(bi)(4)

Theorem

Let I ≥ 2. Let the joint distribution of bids G(·) belong to the set P I with support [b, b]I .There exists a distribution of bidders’ private values F(·) ∈ P such that G(·) is thedistribution of the equilibrium bids in a first price sealed bid auction with independentprivate values and a nonbinding reservation price if and only if:C1: G(b1, b2, ..., bI )=

∏Ii=1 G(bi).

C2: The function ξ(·,G, I) defined in (4) is strictly increasing on [b, b] and its in-verse is differentiable on [v, v] ≡ [ξ(b,G, I), ξ(b,G, I)]Moreover, when F(·) exists, it is unique with support [v, v] and satisfiesF(v) = G

(ξ−1(v,G, I)

)for all [v, v]. In addition, ξ(·,G, I) is the quasi inverse of

the equilibrium strategy in the sense that ξ(b,G, I) = s−1(b,F, I, for all b ∈ [b, b] .




Proof :• bi = s(vi,F, I) and vi are iid⇒ bi are also iid and thus C1 must hold.

• s(·,F, I) is the strictly increasing differentiable and BNE correspondingto F(·) on [v, v].

G(b) = F(

s−1(b,F, I))

for every b ∈ [b, b] ≡ [s(v,F, I), s(v,F, I)].

s(·,F, I) solves (2), (3) follows from (2)⇒ ξ(s(v,F, I),G, I) = v⇒ξ(b,G, I) = s−1(b,F, I) ∀b ∈ [b, b].




The knowledge of the joint distribution of the private values, F, allows one to:

• Simulate outcomes under alternative market mechanisms;

• Assess efficiency and the division of surplus;

• Determine the optimal reserve price.

• Evaluate the "market power" of the bidders v− b.

• Analyze how this margin decrease as the number of bidder increases.

• Testing between CV and PV.


Estimation

Nonparametric estimation

Two step estimation:

• Construction of a sample of pseudo private values using (3).

• Obtain the density of bidders’ private values using the pseudosample constructed previously.


Estimation


First stageWe begin by the nonparametric estimation of G and g:

G̃(b) =1IL

L∑l=1

I∑p=1

1[Bpl≤b] (5)

g̃(b) =1

ILhg

L∑l=1

I∑p=1

Kg

(b− Bpl

hg

)(6)

where L is the number of homogeneous auctions with he same numberof bidders, I.This will give us the following pseudo-values:

V̂pl = Bpl +1

I − 1G̃(Bpl)

g̃(Bpl)≡ ξ̂(Bpl, I) (7)


Estimation


Intermediate step (sample trimming)

Redefine V̂pl =∞ for each V̂pl = ξ̂(Bpl, I) such that:

Bpl ∈ [Bmin,Bmin + hg] ∪ [Bmax − hg,Bmax]

Final step

Estimate the density of the private value distribution according to:

f̂ (v) =1

ILhf

L∑l=1

I∑p=1

Kf

(v− V̂pl

hf

)(8)


Estimation

Some drawbacks of the sample trimming procedure

• Sample trimming involves a non random loss of data.

• The boundary effect problem is compounded at the second stageof the estimation. Within a width of ∆(hg, hf ) from the boundaries,the kernel density estimator will be downward biased.

• The usual bandwidth selection rule (MISE) are designed for onestage estimator and not for two step estimator.

• Consistency of the GPV estimator is achieved only on closed sub-sets of the interior of private values support.



Advantages of GPV’s estimation procedure

• A nonparametric procedure robust to misspecifications of the underlying

distribution.

• Each step of the structural estimation procedure consists of nonparamet-

ric techniques.

• Derivation of the best rate of uniform convergence of nonparametric es-

timates of the density of latent variables from the unobserved bids.

• GPV’s estimation procedure avoids numerical difficulties.



Conclusions

• GPV show that the distribution of bidders’ private values withinIPV is identified from observables, without any parametric as-sumptions.

• They obtain a global identification result.

• Their methodology can be applied to other types of auctions (seeGPV, 1995 on Dutch auctions).

• Their methodology avoids the determination of the equilibriumstrategy.

• GPV’s identification results provide necessary and sufficient con-ditions for the existence of a latent distribution that "rationalizes"the distribution of bids.



La fin...


optimal nonparametric estimation of first-price auctions 30pt by

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