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Parametric and Nonparametric Quantile
Regression Methods for First-Price Auctions:
A Signal Approach
N. Gimenes†
† University
of São Paulo
E. Guerre]
] Queen
Mary, University of London
Work in Progress
Overview
2/58
Plan of the talk
Notations
Quantile regression and additive/interactive quantile models
Quantile and auction
Identi…cation of linear (sieve) quantile speci…cation
Augmented (Sieve) Quantile Regression: dimension reduction and
boundary free estimation
A small simulation experiment
Extension to interdependent values
Overview
3/58
Sealed bids …rst-price auction
I
Auctioned good, with characteristics known to the bidders
and econometrician
I
Bidder forms a bid which is not observed by his opponents
I
Bids are sealed and collected
I
Bids are opened
I
Winner = largest bid
I
Paid price = bid of the winner = largest bid
Overview
4/58
Notations
` = auction, ` = 1, . . . , L
x` = auction good covariate
I` = number of bidders
i = bidders, i = 1, . . . , I`
Overview
5/58
Notations (cont’d): private value case
Private value Vi ` : iid given (x` , I` )
I
I
I
Common knowledge cdf F (v jx` , I` ), continuous pdf
f (v jx` , I` ) > 0 over its compact support
Conditional quantile V (αjx` , I` ), quantile level α 2 [0, 1]
Private value rank Ai ` = F (Vi ` jx` , I` ): prob. that an
opponent has a pv smaller than Vi `
Important property of Ai ` : [0, 1]-uniform and independent of
(x` , I` )
Overview
6/58
Notations (cont’d): observed bids
Observed bids Bi ` : iid bids given (x` , I` )
I
I
I
Cdf G (b jx` , I` ), pdf g (b jx` , I` )
Conditional quantile B (αjx` , I` ), α 2 [0, 1]
Bid rank Ui ` = G (Bi ` jx` , I` )
Overview
7/58
Why quantiles?
Fondamental Simulation Theorem: The private value rank Ai `
is independent of (x` , I` ) with a [0, 1]-uniform distribution, and
satis…es,
Vi ` = V (Ai ` jx` , I` )
I
Allow to simulate Vi ` in full generality
I
Since, in most case,
(Vi ` , Bi ` ) = (V (Ai ` jx` , I` ) , B (Ai ` jx` , I` ))
counterfactuals as the expected revenue E [Vi `
given mechanism
I
Bi ` jx` ] for a
Since V (αjx` , I` ) = F 1 (αjx` , I` ), quantile can be estimated
nonparametrically as fast as a c.d.f. and with faster rates than
a p.d.f
Overview
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Private value rank and Milgrom-Weber model
I
In Milgrom & Weber (1982),
e 0` , A
e 1` , . . . , A
e I ` , x` ,
Wi ` = Wi A
`
I
e 1` , . . . , A
e I ` bidders signals
A
e i ` but not A
e j ` , and
where the ith bidder knows A
e 1` , . . . , A
e I ` independent of x`
A
In the private value case
e i ` , x` ,
Vi ` = Vi A
e i ` independent r.v
A
) The private value rank Ai ` can be viewed as a standardized
signal
I
Issue: the general model is not identi…ed (La¤ont & Vuong,
1996)
) Extension of the paper: a new additive speci…cation for the
general model
Overview
9/58
An additive speci…cation
Wi ` = Wi A1 ` , . . . , AI` ` ; x`
I`
=
∑ πij Vj
Aj ` ; x ` ,
πii = 1
j =0
I
Vi (Ai ` , x` ) = Vi (Ai ` , x` , zi ` ), zi ` individual characteristic
(“capacity”)
I
Vi (Ai ` , x` ): intrinsic private value of the good for the ith
bidder
I
Interactions ) the …nal value must aggregate the intrinsic
private values (“prestige”, trading after auction, etc...).
Di¤ers from Somaini (2014), Wi = Wi A1 ` , . . . , AI` ` ; x` , zi `
Overview
10/58
Dimension reduction issues
Many covariate available in auction datasets:
I
Athey, Levin & Seira (2011) or Li & Zheng (2009, 2012): 5 to
15 covariates for 1,000 observations
I
Haile & Tamer (2003), Aradillas-López, Ghandi & Quint
(2013): 5-6 covariates for few thousands observations
Overview
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Not many dimension reduction methods for …rst-price auction
I
Paarsch & Hong (2006): implement p.d.f. estimation as in G.,
Perrigne & Vuong (2000) using an index assuming
Vi ` = g (x`0 β) + ε i ` . A quantile approach as in Chaudhuri,
Doksum & Samarov (1997) would be less restrictive
I
Haile, Hong & Shum (2003), Rezende (2008): Vi ` = x`0 β + vi `
implies Bi ` = x`0 β + bi ` where the p.d.f of vi ` can be estimated
from the ones of bi ` as in G., Perrigne & Vuong (2000)
Overview
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Additive quantile speci…cation
Various lower dimensional models have been proposed to restrict
the general quantile speci…cation
Vi ` = V (Ai ` jx` , I` )
I
Quantile regression (Koenker & Bassett, 1978);
Vi ` = xi0` β 1 (Ai ` jI` ) + β 0 (Ai ` jI` ) = Xi0` β (Ai ` jI` )
Nests Haile, Hong & Shum (2003), Rezende (2008)
(β 1 (Ai ` jI` ) = β 1 ) and allows for interactions between signal
and covariates
I
Additive speci…cation (Horowitz & Lee, 2005): for
x` = [x1 ` , . . . , xd ` ],
Vi ` = V1 (Ai ` jx1 ` , I` ) +
+ Vd (Ai ` jx1d , I` )
Overview
13/58
I
Additive interactive speci…cation (Andrews & Whang (1990)
for regression)
D
Vi ` =
∑ ∑
k =1 j 1 < <j k
Vj1 ,...,jD Ai ` jxj1 ` , . . . , xjk ` , I`
) A wide class of models ranging from parametric to
nonparametric
Overview
14/58
The general linear quantile speci…cation of the paper
All previous speci…cations can be nested in the linear sieve
speci…cation with D interactions (0 D dim x)
∞
Vi ` =
∑ Pk (x` ) γk (Ai ` jI` ) ,
Pk (x` ) = Pk xj1 (k )` , . . . , xjD (k )` ,
k =0
and where γk (αjI ) = hV (αjx, I ) , Pk (x )ix for orthonormal sieve
) An in…nite dimensional version of Koenker & Bassett (1978)
quantile regression
Overview
15/58
Other econometric issues
Econometric issues with G., Perrigne & Vuong (2000) two step
kernel density estimation method
I
Boundary bias for the upper and lower tails distribution
(Hickman & Hubbard, 2014)
I
Lack of clearcut bandwidth choice (Henderson, List, Millmet,
Parmeter & Price, 2012)
The proposed new quantile methodology is helpful regarding these
issues
Literature review
16/58
Quantile and auction in the econometric literature
I
Haile, Hong & Shum (2003): Quantile, dimension reduction
using a regression model. See also Rezende (2008)
I
Marmer & Shneyerov (2012): avoids estimation of private
values
I
G. & Sabbah (2012), Fan, Li & Pesendorfer (2013,WP): LP
quantile estimation
I
Menzel & Morganti (2013): order statistic (sample quantile)
approach for second-price auction
I
Gimenes (2013, WP): QR for ascending auction
Rest of the talk
17/58
Rest of the talk
I
Quantile identi…cation
I
A key property: Stability of linear quantile speci…cation
I
Augmented (Sieve) Quantile regression
I
Interdependent value extension
Quantile identi…cation for …rst-price auction under IPV
Quantile identi…cation: a preliminary lemma
Lemma Suppose that the values Wi are such,
Wi = Wi (A0 , A1 , . . . , AI , x, I ) , i = 1, . . . , I ,
where (A0 , A1 , . . . , AI ) is independent of (x, I ), each Ai are
[0, 1] uniform, and that each bidder plays a strictly increasing
strategy,
Bi = si (Ai jx, I ) , si ( jx, I ) " for all (x, I ) .
I
) No equilibrium assumption. Increasing strategy assumption
strong enough to identify Ai and si ( j , ) in a constructive
way
I
Bayesian Nash Equilibrium generates increasing strategies
(Reny & Zhamir, 2004)
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Quantile identi…cation for …rst-price auction under IPV
Lemma cont’d: Signal identi…cation
(i) The signal Ai , i
with,
1, can be recovered from the observed bids
Ai = Gi (Bi jx, I ) ,
where Gi ( jx, I ) is the conditional c.d.f of Bi ;
I
) the joint distribution of (A1 , . . . , AI ) is identi…ed
I
The signal Ai can be estimated (known identity or
Gi (Bi jx, I ) = G (Bi jx, I ))
19/58
Quantile identi…cation for …rst-price auction under IPV
Lemma cont’d: strategy identi…cation
Ai = Gi (Bi jx, I ) ) Bi = Bi (Ai jx, I )
(ii) the strategy si ( jx, I ) is identi…ed by the conditional bid
quantile function,
si (Ajx, I ) = Bi (Ajx, I ) , for any A 2 [0, 1] ;
I
Contrasts with strategies depending upon the private value for
symmetric IPV.
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Quantile identi…cation for …rst-price auction under IPV
21/58
Lemma cont’d: Probability of Winning
(iii) under symmetric IPV, that is if Ai independent,
Vi = V (Ai , x, I ) and Bi = B (Ai jx, I ) for all i = 2, . . . , I , the
probability that a bid B (Ajx, I ) wins is AI 1 .
I
Under asymmetry or interdependent value, the probability
that a bid B1 (Ajz, I ) is also identi…ed since it is
P B1 (Ajx, I ) > max Bi (Ai jx, I ) jA1 , x, I
i =2,...,I
which depends upon the identi…ed Bi ( jx, I ) and the
identi…ed joint distribution of (A1 , . . . , AI )0 . But no close
form expression in general
Quantile identi…cation for …rst-price auction under IPV
22/58
Quantile under symmetric IPV and Bayesian Nash
Equilibrium
) Under symmetric IPV and BNE, B ( jx, I ) is the optimal
strategy
This identi…es V (αjx, I ) in a simple linear way under risk
neutrality
The risk neutral expected utily of a bid B (ajx, I ) given …rst bidder
signal A1 = A is
( V1
B (ajx, I )) aI
1
= (V (Ajx, I )
B (ajx, I )) aI
1
Quantile identi…cation for …rst-price auction under IPV
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Since the optimal bid is B (Ajx, I ),
(V (Ajx, I )
B (ajx, I )) aI
1
(V (Ajx, I )
B (Ajx, I )) AI
1
for all a 2 [0, 1] .
Hence, for all A 2 (0, 1),
∂ h
(V (Ajx, I )
∂a
,
B (ajx, I )) aI
B (1 ) (Ajx, I ) AI
1
+ (I
1
i
=0
a =A
1) (V (Ajx, I )
B (Ajx, I )) AI
2
=0
Quantile identi…cation for …rst-price auction under IPV
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Rearranging gives the di¤erential equation,
V (Ajx, I ) = B (Ajx, I ) +
A
B (1 ) (Ajx, I )
,
I 1
B (0jx, I ) = V (0jx, I )
which is the quantile version of the identi…cation method in G.,
Perrigne & Vuong (2000)
Vi ` = Bi ` +
1
I`
G ( Bi ` j x ` , I ` )
1 g ( Bi ` j x ` , I ` )
Quantile identi…cation for …rst-price auction under IPV
Suggests to estimate V (αjx, I ) using
b (1 )
b (αjx, I ) = B
b (αjx, I ) + αB (αjx, I )
V
I 1
as in G. & Sabbah (2012) or Fan et al. (2013).
However,
I
Not so many good estimators of B (1 ) (αjx, I ) in the literature
I
It may be fruitful to solve the linear di¤erential equation
before estimating
25/58
Quantile identi…cation for …rst-price auction under IPV
26/58
A key lemma: (i) stability of linear speci…cation
(i) The conditional quantile function of optimal bids is given by the
linear operator,
B (αjx, I ) =
I 1
αI 1
Z α
0
tI
2
V (t jx, I ) dt.
linear speci…cation for V ( jx, I )
)
linear speci…cation for B ( jx, I )
as noted in Haile et al. (2003) and Rezende (2008) for the
particular case of regression.
Quantile identi…cation for …rst-price auction under IPV
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Example: if, for some γk (αjI ) = hV (αj , I ) , Pk ( )i
∞
V (αjx, I ) =
then
∑ Pk (x ) γk (αjI ) ,
k =0
∞
B (αjx, I ) =
with
β k ( α jI ) =
∑ Pk ( x ) β k ( α j I )
k =0
I 1
αI 1
Z α
0
tI
2
γk (t jI ) dt.
Quantile identi…cation for …rst-price auction under IPV
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A key lemma (ii): identi…cation
(ii) The conditional private values quantile function can be
recovered from the bid one,
V (αjx, I ) = B (αjx, I ) +
α
I
1
B (1 ) (αjx, I ) .
Quantile identi…cation for …rst-price auction under IPV
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Example (Cont’d): since
∞
B (αjx, I ) =
∑ Pk ( x ) β k ( α j I )
k =0
Z α
tI
with γk (αjI ) = β k (αjI ) +
I
I 1
with β k (αjI ) = I 1
α
2
0
γk (t jI ) dt,
then
∞
V (αjx, I ) =
∑ Pk (x ) γk (αjI )
k =0
α
(1 )
1
β k ( α jI ) .
Augmented quantile regression
30/58
Estimation methology
1. Postulate a quantile regression speci…cation for the private
values or set X = (P1 (x ) , . . . , PkL (x )),
) V (αjx, I ) = X 0 γ (αjI ) + biasV (no bias for QR)
2. By the stability property
B (αjx, I ) = X 0 β (αjI ) + biasB (no bias for QR)
3. Given an estimation of βb (αjI ) and βb(1 ) (αjI ), set
α βb(1 ) (αjI )
b (αjI ) = βb (αjI ) +
γ
,
I 1
b (αjx, I ) = X 0 γ
b ( α jI )
V
) Needs new techniques to …nd good estimation of β(1 ) (αjI ), an
issue mostly ignored in the literature.
Augmented quantile regression
31/58
Standard quantile regression
Check function ρα (t ) = t (α
I (t
0))
β (αjI ) = arg min E I (I` = I ) ρα Bi `
β
L
X`0 β
1
) βb (αjI ) = arg min ∑ I (I` = I ) ρα Bi `
β L
`=1
I
Does not give an estimator of β(1 ) (αjI )
I
Di¢ cult to de…ne for α = 0 or α = 1
X`0 β
Augmented quantile regression
Augmented quantile regression
I
Allow small variation of α in the check function ρα (t )
I
Expand β (α + ht ) to estimate β(1 ) (αjI ) by local polynomial
smoothing
32/58
Augmented quantile regression
33/58
For a = α + ht, h > 0 bandwidth, and β ( jI ) s + 2 di¤erentiable,
X 0 β (a jI )
(
= X0
β ( α jI ) + (a
+ O (a
= X (a
α ) β (1 ) ( α j I ) +
( a α ) s +1 (s +1 )
+
β
( α jI )
(s + 1) !
α )s +2
α ) 0 β ( α jI ) + O (a
2
6
α )s +2 , X (t ) = 4
1
..
.
t s +1
(s +1 ) !
h
i
0
0
0 0
where β (αjI ) = β (αjI ) , β(1 ) (αjI ) , . . . , β(s +1 ) (αjI ) .
3
7
5
X
)
Augmented quantile regression
34/58
Objective function of the augmented quantile regression
b ( β; α, I ) is
K ( ) kernel, h bandwidth ) objective function R
1
LIh
I`
L
∑ I (I` = I ) ∑
i =1 0
`=1
=
1
LI
Z 1
L
ρ a Bi `
I`
∑ I (I` = I ) ∑
`=1
i =1
Z
1 α
h
α
h
X` (a
ρα+ht Bi `
α)0 β K
a
α
h
da
X` (ht )0 β K (t ) dt.
Augmented quantile regression
35/58
The augmented quantile regression estimator is
b (αjI ) = arg min R
b ( β; α, I ) ,
β
β
2
6
b ( α jI ) = 6
β
6
4
b (αjx, I ) = X 0 βb(0 ) (αjI ) +
V
α
I
1
βb(0 ) (αjI )
βb(1 ) (αjI )
..
.
s
+
1
(
)
b
β
( α jI )
βb(1 ) (αjI )
3
7
7
7
5
Augmented quantile regression
Boundary behavior
b (0jI )
Smoothing gives a convex AQR function for α = 0, 1 ) β
b (1jI ) are well de…ned
and β
36/58
Theoretical results
37/58
Assumptions (QR case)
1. X in a compact set, ∞ < X 0 γ (0jI ) < X 0 γ (1jI ) < ∞,
supα X 0 γ(1 ) (αjI ) < ∞, inf α X 0 γ(1 ) (αjI ) > 0
) boundary bias for kernel estimation
2. α 2 [0, 1] 7! γ (αjI ) (s + 1)th continuously di¤erentiable )
β (αjI ) (s + 2) cont. di¤. except at α = 0.
Theoretical results
38/58
Theoretical results for quantile regression models
Theorem Suppose the private value quantile regression
speci…cation is correct. Then if h ! 0 with log3 L/ Lh2 = O (1)
!
1/2
log
L
b (αjx, I ) V (αjx, I ) = OP
V
sup
+ h s +1 .
LIh
(α,x )2[0,1 ] X
It also holds that
sup
(α,x )2[0,1 ] X
b (αjx, I )
B
B (αjx, I ) = OP
1
LI
1/2
+ h s +2
!
.
Theoretical results
39/58
Uniform consistency rate for private values
sup
(α,x )2[0,1 ] X
b (αjx, I )
V
V (αjx, I ) = OP
log L
LIh
1/2
+ h s +1
!
I
Rate given by βb(1 ) (αjI ). No boundary bias at α = 0 or 1.
I
L
Optimal rate = log
= minimax optimal rate of G.,
LI
Perrigne & Vuong (2000) with no covariate and for all s > 0.
Achieved when
1
log L 2(s +1)+1
h
.
LI
I
CLT + MSE decomposition allowing for plug in bandwidth
choice
s +1
2 (s +1 )+1
Private value estimation
40/58
Private value estimation
b i ` = arg min Bi `
A
α2[0,1 ]
bi ` = V
b A
b i ` j x` , I` .
V
b ( α j x` , I` ) ,
B
Lemma It holds that,
max
max
`=1,...,L i =1,...,I`
) OP
bi `
V
log L
L
Vi ` = OP
log L
LIh
1/2
+h
s +1
!
.
s +1
2 (s +1 )+1
for optimal bandwidth choice
Holds for all private values due to the absence of boundary bias
Sieve extension
41/58
Sieve interactive speci…cation
With D interactions and localized sieve as wavelets of cardinal B
splines and under suitable bandwdith (K = h D ) and smoothness
assumptions,
sup
(α,x )2[0,1 ] X
b (αjx, I )
V
V (αjx, I ) = OP
under conditions which imposes s >
h = (L/ ln L) 1/(2s +D +3 )
IMSE, MSE expansions and CLT
3
2
(D
log L
LIhD +1
1/2
+ h s +1
1) for the optimal
!
A small simulation experiment
42/58
Simulation example
L = 50 and I = 2
Second-order LP (s + 1 = 2), Epachnikov kernel, data-driven b
h
computed from a regression model with truncated exponential error
10,000 replications
V (αjx ) = γ0 (α) + x1 + γ2 (α) x2 ,
α
γ0 (α) = 0.1 log 1
,
e
γ2 (α) = 1 exp ( α) .
The covariates x1 and x2 are two independent uniform variables. x2
inactive for small α
A small simulation experiment
43/58
Quantiles
x1 = x2 2 f0.2, 0.5.0.8g
A small simulation experiment
44/58
Slope coe¢ cients
γ0 (α) =
0.1 log 1 eα , γ1 (α) = 1 and
γ2 (α) = 1 exp ( α)
Extension to additive interdependent value
Extension to additive interdependent value
I
I bidders with known identity from now on
I
zi : characteristic of ith bidder observed by all (“capacity”
variable as distance to the project, labor force, cash ‡ow,...)
z = (1, z1 , . . . , zI )0 full-rank
45/58
Extension to additive interdependent value
46/58
The general additive speci…cation
I
Wi (A; x, z ) = W0 (x, z ) + V0 (A0 ; x ) +
∑ πij Vj (Aj ; x, zj ) ,
j =1
πii = 1, πij
0
Vj ( ; x, zj ) ", Vj (0; x, zj ) = 0, and
Vj (Aj ; x, zj ) = vj (Aj ; zj ) +
Z zj
∂Vj (Aj ; x, t )
0
) Vj (Aj ; x, zj ) 6= V1j (Aj ; x ) + V2j (Aj ; x, zj )
) force an interaction between Aj and zj
∂zj
dt
Extension to additive interdependent value
47/58
A simple interdependent value speci…cation
I
Wi = γ0 (A0 ) +
∑ πij zj γj (Aj )
j =1
I
Aj : jth bidder private signal with a U[0,1 ] distribution
I
zj γj (Aj ): jth bidder “private” component of the ith bidder
value Wi , i = 1, . . . , I
Weighted by πij in Wi
I
γ0 (A0 ): common component of the values Wi , i = 1, . . . , I
A0 : U[0,1 ] distribution
Not identi…ed without a completness assumption
Parameter of interest: slope coe¢ cients γ1 ( ) , . . . , γI ( )
Extension to additive interdependent value
48/58
Assumption
1. The signals A0 , A1 , . . . , AI are a¢ liated with a conditional
c.d.f which is bounded away from 0 over [0, 1]I +1 . The
signals are independent of z
2. The slope co¢ cients γj ( ) are strictly increasing with
γj (0) = 0 and
πii = 1, πij
0
3. Each bidder plays a best-response strictly increasing and
di¤erentiable strategy si (Ai ; z ) (Reny and Zamir, 2004)
) si (Ai ; z ) = Bi (Ai ; z )
Extension to additive interdependent value
49/58
Expected pro…t and best response condition
I
Expected pro…t of a bid Bi (ajz ) given Ai = α
E (Wi
Bi (ajz )) I Bi (ajz )
= W i (ajα, z )
jAi = α, z
max Bj
j 6 =i
Bi (ajz ) ωi (ajα, z )
where
ωi (ajα, z ) = E I Bi (ajz )
max Bj
j 6 =i
jAi = α, z
= P (Bi (ajz ) winsjAi = α, z )
W i (ajα, z ) = E Wi I Bi (ajz )
I
Identi…cation issue: W i (ajα, z ) 6= Wi
max Bj
j 6 =i
jAi = α, z
Extension to additive interdependent value
50/58
W i (ajα, z ) = E γ0 (A0 ) I Bi (ajz )
jAi = α, z
max Bj
j 6 =i
I
+ ∑ πij zj E γj (Aj ) I Bi (ajz )
j =1
max Bj
j 6 =i
jAi = α, z
I
) W i (ajα, z ) = γi 0 (ajα, z ) + ∑ πij zj γij (ajα, z )
j =1
with
γij (ajα, z ) = E γj (Aj ) I Bi (ajz )
γii (ajα, z ) = γi (α) P Bi (ajz )
= γi (α) ωi (ajα, z )
max Bj
j 6 =i
jAi = α, z
max Bj jAi = α, z
j 6 =i
Extension to additive interdependent value
51/58
Best response condition
α = arg max W i (ajα, z )
Bi (ajz ) ωi (ajα, z )
α
FOC )
,
∂W i
(αjα, z )
∂a
∂ωi
(1 )
= Bi ( α j z )
(αjα, z ) + Bi (αjz ) ωi (αjα, z )
∂a
(1 )
Wi (α; z ) = Bi (αjz ) + Ωi (α; z ) Bi
where
Ωi (α; z ) =
Wi (α; z ) =
(αjz ) with I.C. Bi (0jz ) = 0
1
∂ωi
∂a
(αjα, z )
1
∂ωi
∂a
ωi (αjα, z )
∂W i
(αjα, z )
(αjα, z ) ∂a
Extension to additive interdependent value
52/58
Comparison with IPV
I
Symmetric IPV case:
PV Quantile (αjI ) = B (α) +
I
α
I
1
B (1 ) ( α )
Interdependent value case:
(1 )
Wi (α; z ) = Bi (αjz ) + Ωi (α; z ) Bi
where Ωi (α; z ) =
1
∂ωi
∂a
is identi…ed as
ωi (ajα, z ) = P Bi (ajz )
) Wi (α; z ) =
1
∂ωi
∂a
(αjα, z )
( α jz )
ωi (αjα, z )
max Bj jAi = α, z
j 6 =i
.
∂W i
(αjα, z ) is identi…ed
(αjα, z ) ∂a
but again Wi (α; z ) 6= Wi
Extension to additive interdependent value
53/58
Stability property for additive interdependent values (1)
I
W i (ajα, z ) = γi 0 (ajα, z ) +
∑ πij zj γij (ajα, z )
j =1
with γii (ajα, z ) = γi (α) ωi (ajα, z ) ,
Wi (α; z ) =
1
∂ωi
∂a
∂W i
(αjα, z )
(αjα, z ) ∂a
I
)Wi (α; z ) = γi 0 (α; z ) + ∑ πij zj γij (α; z ) with
j =1
γii (α; z ) = γi (α) (invariance of γi (α) )
∂γij
1
γij (α; z ) = ∂ω
(αjα, z )
i
(αjα, z ) ∂a
∂a
Extension to additive interdependent value
54/58
Stability property (2) and identi…cation
I
Wi (α; z ) = γi 0 (α; z ) +
∑ πij zj γij (α; z )
with γii (α; z ) = γi (α)
j =1
and solving the di¤erential equation in Bi (αjxz )
(1 )
Wi (α; z ) = Bi (αjz ) + Ωi (α; z ) Bi
( α jz ) ,
Bi ( 0 j z ) = 0
gives the “random coe¢ cient” quantile regression model
I
Bi (αjz ) = β i 0 (α; z ) +
∑ zj βij (α; z )
with
j =1
(1 )
πij γij (α; z ) = β ij (α; z ) + Ωi (α; z ) β ij (α; z ) (with πi 0 = 1)
(1 )
γi (α) = β ii (α; z ) + Ωi (α; z ) β ii (α; z )
Extension to additive interdependent value
55/58
I
) γi (α) is identi…ed for all i
I
) πij γij (α; z ) is identi…ed for all j
1
But γij (α; z ) is also identi…ed for all j
γij (α; z ) =
1, since
1
∂ωi
∂a
∂γij
(αjα, z ) where
(αjα, z ) ∂a
γij (ajα, z ) = E γj (Aj ) I Bi (ajz )
ωi (αjα, z ) = P Bi (ajz )
I
0
) πij is identi…ed
max Bk
k 6 =i
max Bk jAi = α, z
k 6 =i
jAi = α, z
Extension to additive interdependent value
56/58
Estimation method
I
Localised Augmented Quantile: For each i, estimate
(1 )
β ij (α; z ) and β ij (α; z ) from the “random coe¢ cient”
quantile regressions
I
Bi (αjz ) = β i 0 (α; z ) +
∑ zj βij (α; z )
j =1
using a kernel weighted AQR with weights K
z` z
h
Extension to additive interdependent value
I
57/58
Estimate
Ωi (α; z ) =
1
∂ωi
∂a
(αjα, z )
ωi (αjα, z ) where
ωi (αjα, z ) = P Bi (ajz )
I
max Bj jAi = α, z
j 6 =i
bi (αjz ) and A
bi `
from B
Structural parameters: compute
b i (α; z ) βb(1 ) (α; z )
bii (α; z ) = βbii (α; z ) + Ω
γ
ii
and average to improve convergence rate
bi (α) =
γ
1 L
∑ γbii (α; z` )
L `=
1
bij from γ
bi (α) and γ
bij (α; z` )
π
Conclusion
58/58
Final remarks
I
Flexible quantile regression speci…cations including
nonparametric components which can be estimated with fast
rate
I
A class of additive interactive speci…cation ranging from
quantile regression to the general nonparametric quantile
model. Can be tested from the data
I
Address the curse of dimensionality
I
No boundary bias. Allows estimation of all private values
I
Bandwidth choice
I
Additive interdependent value speci…cation
I
Statistical extension: dimension reduction for conditional p.d.f