Identification Problem in First-Price Auctions: Ideal Situation and Observed Auction-Specific Heterogeneity Herman J. Bierens and Hosin Song April 27, 2006
1. Introduction For the structural estimation of the economic model, we need to identify the structure of interests. In this paper, we consider the nonparametric identification problems in the first-price auctions1 where values are independent and private and bidders are ex-ante identical.2 In particular, we go after the nonparametric identification under mild assumptions. Generally, two situations will be investigated in the paper. One is the ideal situation where identical auctions are repeated independently among the identical number of potential bidders. The other situation is more realistic environment where auction-specific characteristics are observed and the number of potential bidders are allowed to change. In this case, the identification problem breaks into two parts. The first step is the identification of the conditional distribution of values given covariates. The second step is to identify the semi-nonparametric structures which are a covariate function and semi-nonparametric(SNP) distribution. In particular, we find that a covariate function is nonparametrically identified up to a constant in general. Hence, for the nonparametric identification of semi-nonparametric structures, we come 1 2
Throughout the paper, the first-price auction means the first-price sealed bid auction. Sometimes we just call this environment IPV paradigm.
1
up with two restrictions. One is that covariate functions are assumed to have a certain same expectation. The other restriction is that a quantile restriction on SNP distribution. Particularly, we can use a binding reserve price as a quantile restriction. The contribution of this paper can be summarize as follows. First, we prove the identification of the values distribution in a new and convincing way. Second, we extend the nonparametric identification to the semi-nonparametric structures of the conditional distribution of values when the auction-specific characteristics are observed. In particular, we incorporate the auction-specific covariates into the semi-nonparametric specification which is a new approach as fa as we know. The remaining of this paper is organized as follows. In section 2, we analyze the identification of values distribution in the ideal situation. We consider two cases respectively: a binding reserve price case vs a non-binding reserve price case respectively. In section 3, we study the identification problem when the auctionspecific characteristics are observed. Like section 2, a binding reserve price case and a non-binding reserve price case are reviewed. In concluding remarks, we suggest some ideas for the test of a covariate function. Throughout the paper, we denote a random variable or a distribution function in upper-case and denote a realization of random variable or a density function in lower-case throughout the paper. We will put the proofs of all lemmas, corollaries and propositions in the appendix except for the case where the proof itself has contribution.
2. First-Price Auctions in Ideal Situation Suppose there are I ex-ante identical bidders and there is an indivisible object to sell. Assume that bidders are risk-neutral. Bidders’ values are assumed to be independent and private. Moreover, bidders’ values follow the distribution F (V ). There is a seller’s reserve price p0 . Then, the equilibrium bid of a bidder with value v is as follows: for all v > p0 β(v) = v −
Z
1 I−1
F (v)
2
v p0
F (x)I−1 dx.
(1)
For details, see Riley and Samuelson (1981) or Krishna (2002). Note that the equilibrium bid depends on the value and the value distribution. Since the value V is a random variable, the equilibrium β(V ) is also a random variable. Suppose it is known that there are I potential bidders and one seller who announces the reserve price p0 . Also, suppose that the lower bound of the support of private values distribution F is v. Then, there are two cases regarding the seller’s reserve price. If the seller sets the reserve price p0 below v, every bidder’s value drawn from F is greater than p0 . Hence, every potential bidder will enter the auction.3 This type of reserve price is called a non-binding reserve price. On the other hand, the seller can set the reserve price p0 above v. If the seller does so, some bidders whose drawn values are above p0 will enter the auction and make their bids while some bidders whose values are below p0 will not enter the auction.4 This type of reserve price is called a binding reserve price. . In this section, we assume that we can observe many first-price auctions which are identical and independent. We call it the ideal situation. We study the identification problem of the first-price auction in the ideal situation.
2.1. Identification Problem Assuming that every observed bid is an equilibrium bid in (1), we want to recover the values distribution from the observed bids. For this purpose, we need to check whether the values distribution can be determined uniquely from the observed bids. This is the identification problem in this section. We will consider the identification problem in first-price auctions with non-binding reserve prices and with binding reserve prices respectively. 2.1.1. Identification: Non-binding Reserve Price Case5
3
This means that the bidder submits his bid which is above p0 . Hence, the bidder does not submit his bid. 5 A non-binding reserve price means that the seller’s reserve price is not effective. 4
3
In a first-price auction with a non-binding reserve price, the sequence of actions is as follows: (a) There are I ex-ante identical potential buyers and one seller with an object. (b) The auctioneer announces the seller’s reserve price p0 which is below the possible lowest value v. (c) Each buyer draws the value from the distribution F (V ). Every buyer participates in the auction since the reserve price is lower than their values. (d) Every bidder submits his own sealed bid following the equilibrium bidding function (1). (e) The highest bidder is awarded the object. (f) All submitted bids can be observed by an econometrician. Definition 1. Denote X ∼ Y if the random variables X and Y follow the same distribution. Definition 2. The support of any distribution G(X) is defined by {x : g(x) > 0, x ∈ R} where g is a density function associated with G. In an auction with a non-binding reserve price, we can assume that the seller sets p0 = v without loss of generality. Consider two equilibrium bidding functions β1 (V1 ) and β2 (V2 ) such that β1 (V1 ) ≡ β(V1 ; F1 ) and β2 (V2 ) ≡ β(V2 ; F2 ) where Z V1 1 β1 (V1 ) = V1 − F1 (x)I−1 dx I−1 F1 (V1 ) p0 and β2 (V2 ) = V2 −
Z
1 I−1
V2
F2 (x)I−1 dx.
F2 (V2 ) p0 Following Roehrig(1988), we can define the notion of observationally equivalent as follows. Definition 3. Suppose there are two different distributions F1 (V1 ) and F2 (V2 ) such that V1 follows F1 and V2 follows F2 . Then F1 and F2 are observationally RV equivalent if β1 (V1 ) ∼ β2 (V2 ) where β1 (V1 ) = V1 − F (V1)I−1 p01 F1 (x)I−1 dx and 1 1 RV β2 (V2 ) = V2 − F (V1)I−1 p02 F2 (x)I−1 dx. 2
2
4
Definition 4. Let F be the family of private values distributions with common support. The private values distribution function is identified from the observed bids if the following statement is true: for any (F1 , F2 ) ∈ F × F , β1 (V1 ) ∼ β2 (V2 ) implies that F1 (V ) = F2 (V ) a.s. Note that the support can be unbounded such as [v, ∞). Lemma 1. If a random variable X follows a distribution F (X) which is continuous and strictly monotone, then U = F (X) follows a uniform distribution on [0,1]. Proof. F is continuous and strictly monotonic, hence F −1 exists. Then, P[U ≤ u0 ] = P[F (X) ≤ u0 ] = P[X ≤ F −1 (u0 )]
(2)
= F (F −1 (u0 )) = u0 .
Q.E.D. Assumption 1. (1-i) Values distribution F (V ) and bids distribution Λ(B) are absolutely continuous with respect to Lebesgue measure with its density f and λ respectively. (1-ii) The support of F (V ) is a connected set in R+ . (1-iii) The number of potential bidders I is a constant.6 (1-iv) E(β(V )) < ∞ where β(·) is the equilibrium bidding strategy in equation (1). Note that definition 2 and assumption (1-ii) makes the distribution F strictly monotone and continuous on the support. Assumption (1-iv) is necessary particularly when the distribution F has the unbounded support such as [v, ∞). 6 7
I is known to the econometrician because the reserve price is not binding. We just want to avoid the unbounded expected payment by a bidder.
5
7
Proposition 1. Consider a first-price sealed bid auction where bidders’ values are independent and private and bidders are risk-neutral. Suppose that the reserve price is not binding, that is, p0 ≤ v. Then the distribution of values F (V ) is identified for all v ≥ p0 under Assumption 1. Proof.
Without loss of generality, we can assume that the lower bound of the
support v is zero and the reserve price p0 is zero. Denote the equilibrium bid by B which is also a random variable because it is a function of random variable V . Suppose that F1 and F2 are observationally equivalent. That is, β1 (V1 ) ∼ β2 (V2 ) where V1 follows the distribution F1 and V2 follows the distribution F2 : Z V1 Z V2 1 1 I−1 V1 − F1 (x) dx ∼ V2 − F2 (x)I−1 dx. I−1 I−1 F1 (V1 ) F2 (V2 ) 0 0 Since V1 = F1−1 (F1 (V1 )) = F1−1 (U1 ) and V2 = F2−1 (F2 (V2 )) = F2−1 (U2 ) from Lemma 1, we can write β1 (V1 ) and β2 (V2 ) as follows: β1 (V1 ) =
β2 (V2 ) =
F1−1 (U1 )
F2−1 (U2 )
−
−
Z
1 U1 I−1
F1−1 (U1 )
U2 I−1
(3)
F2 (x)I−1 dx where U2 = F2 (V2 )
(4)
0
Z
1
F1 (x)I−1 dx where U1 = F1 (V1 )
F2−1 (U2 ) 0
Denote β1 (V1 ) and β2 (V2 ) by ϕ1 (U1 ) and ϕ2 (U2 ) respectively, then ϕ1 (U1 ) =
F1−1 (U1 )
−
1 U1
I−1
and ϕ2 (U2 ) =
F2−1 (U2 )
−
Z
1 U2 I−1
F1−1 (U1 )
F1 (x)I−1 dx
0
Z
F2−1 (U2 )
F2 (x)I−1 dx.
0
Then, by the hypothesis, B ∼ ϕ1 (U1 ) ∼ ϕ2 (U2 )
(5)
where U1 and U2 follow a uniform distribution on [0,1]. By the definition of bids distribution Λ and the relation in equation (5), P[B ≤ b0 ] = Λ(b0 ) = P[ϕ1 (U1 ) ≤ b0 ] = P[ϕ2 (U2 ) ≤ b0 ]. 6
(6)
Since ϕ1 (U1 ) is invertible8 , it follows from Lemma 1 that −1 P[ϕ1 (U1 ) ≤ b0 ] = P[U1 ≤ ϕ−1 1 (b0 )] = ϕ1 (b0 ).
Similarly, P[ϕ2 (U2 ) ≤ b0 ] = ϕ−1 2 (b0 ). Therefore, we can get the following result: for all b0 > 09 , −1 Λ(b0 ) = ϕ−1 1 (b0 ) = ϕ2 (b0 ).
Hence, ϕ1 (u) = ϕ2 (u) a.e. on (0, 1). Now we need to show the following statement: ϕ1 (u) = ϕ2 (u) a.e. on (0,1) implies F1 (v) = F2 (v) for almost all v > 0. Without the loss of generality, suppose the support of any distribution in F is [0, ∞). Since ϕ1 (u) = ϕ2 (u) a.e. on (0,1), we can write for almost all u ∈ (0, 1), F1−1 (u) −
1
uI−1
R F1−1 (u) 0
F1 (x)I−1 dx = F2−1 (u) −
1
uI−1
R F2−1 (u) 0
F2 (x)I−1 dx.
(7)
Multiplying both sides by uI−1 yields Z u
I−1
F1−1 (u)−
F1−1 (u) 0
Z I−1
F1 (x)
dx = u
I−1
F2−1 (u)−
F2−1 (u)
F2 (x)I−1 dx a.e. on (0, 1)
0
(8)
Differentiating both sides with respect to u yields (I − 1)uI−2 F1−1 (u) + uI−1 − (F1 (F1−1 (u)))I−1
dF1−1 (u) du
= (I − 1)uI−2 F2−1 (u) + uI−1 − (F2 (F2−1 (u)))I−1
dF2−1 (u) du
dF1−1 (u) du
dF2−1 (u) du
From this, we get F1−1 (u) = F2−1 (u) a.e. on (0, 1). 8 9
Note that ϕ0i (u) > 0 for all u ∈ (0, 1),i = 1, 2. Note that the boundary condition is β(v) = v and we assume v = 0
7
(9)
Hence, F1 (v) = F2 (v) a.e. on [0, ∞).
(10) Q.E.D.
2.2.2. Identification: Binding Reserve Price Case
Now we turn to the identification problem when the reserve price p0 is above the lower bound of the value v. The sequence of actions is similar to the binding reserve case except for the following: (a) P[vi < p0 ] > 0: Some bidders’ values are above p0 while some bidders’ values are below p0 . The former bidders submit their bids following equilibrium bidding function β(v) = v −
1 F (v)I−1
Z
v
F (x)I−1 dx for all v ≥ p0
(11)
p0
while the latter bidders do not submit the bid.10 For the latter case, we can assume that they submit zero bids. The bidder in the former case is called an actual bidder and the bid by an actual bidder is called an actual bid. (b) After the auction, the econometrician can observe I ∗ actual bids greater than p0 and I − I ∗ zero bids. Note that the actual bidder’s value and bid are greater than the seller’s reserve price. Assumption 2. (2-i) Values distribution F (V ) and bids distribution Λ(B) are absolutely continuous with respect to Lebesgue measure with its density f and λ respectively. (2-ii) The support of values distribution F is a connected set in R+ . (2-iii) The number of potential bidders I is a known constant and it is given. 10
We can say that the equilibrium bid in the latter case is any bid lower than the reserve price
p0 .
8
(2-iv) E(β(V, F )) < ∞ where β is the equilibrium bidding strategy in (11).11 (2-v) The seller’s reserve price p0 is given. (2-vi) F (p0 ) is given. If we have many independent and identical auctions, F (p0 ) can be nonparametrically determined. Assumption (2-vi) can justified by the following lemma. Lemma 2. The participation ratio 1 − F (p0 ) ≡ 1 − α is determined nonparametrically in the ideal situation. Proof. Since I ∗ follows a binomial distribution with (I, 1 − α), 1 − α is the expecL ∗ P ∗ il ˆ = 1 − L1 where L tation of II . Hence, the nonparametric estimator for α is α I is the number of auctions in the ideal situation.
l=1
Q.E.D.
The following lemma 3 states that once a potential bidder becomes an actual bidder, then the actual bidder’s value follows F ∗ (V ) ≡ F (V |V ≥ p0 ) where actual bidder’s value V is greater than p0 . Lemma 3. Suppose the actual bidder’s value V follows F ∗ (V ). Then, F ∗ (v) = Proof.
F (v)−F (p0 ) 1−F (p0 )
for all v ≥ p0 .
Let the actual bidder’s value be V which is greater than p0 , then for all
v ≥ p0 , F ∗ (v) = P(V ≤ v|V ≥ p0 ) P(p0 ≤ V ≤ v) P(V ≥ p0 ) F (v) − F (p0 ) = 1 − F (p0 ) =
Q.E.D. Lemma 4. An actual bidder’s private value distribution F ∗ (V ) follows a uniform distribution on [0,1]. 11
It is necessary if the support of private values distribution F is unbounded.
9
Proof. Let U ≡ F ∗ (V ). Suppose F (p0 ) = α. Then, for all u0 ∈ [0, 1], P[U ≤ u0 ] = P[F ∗ (V ) ≤ u0 ] )−F (p0 ) = P[ F (V ≤ u0 ] 1−F (p0 )
= P[V ≤ F −1 ((1 − α)u0 + α)] ∗
= F (F
−1
(12)
((1 − α)u0 + α))
= u0 . Q.E.D. From lemma 3 and lemma 4, we have the relationship: for all v ≥ p0 , F (v) = (1 − F (p0 ))F ∗ (v) + F (p0 ) = (1 − F (p0 ))u + F (p0 )
(13)
where u is a realization of U on [0,1]. The following proposition says that the private values distribution can be identified for all values above the reserve price p0 when the reserve price is binding. Proposition 2. Suppose that the reserve price is binding, that is, p0 > v. Then, the distribution of the value F (V ) is identified for all v ≥ p0 if assumption 2 holds. Proof.
Note that bids are submitted by actual bidders whose values are greater
than the reserve price p0 . The equilibrium bid of an actual bidder is β(V, F ) = RV V − F (V1)I−1 p0 F (x)I−1 dx for V = v ≥ p0 . Suppose there are observationally equivalent distributions F1 (V1 ) and F2 (V2 ) such that β1 (V1 ) ∼ β2 (V2 ) where β1 (V1 ) ≡ β(V1 , F1 ) = V1 − β2 (V2 ) ≡ β(V2 , F2 ) = V2 −
Z
1 I−1
F1 (V1 ) 1
I−1
F2 (V2 )
Z
V1 p0 V2
F1 (x)I−1 dx, F2 (x)I−1 dx.
(14)
p0
That is, B ∼ β1 (V1 ) ∼ β2 (V2 ) where B is an observed equilibrium bid which is greater than p0 . Without loss of generality, suppose the support of private value is (0, ∞). From lemma 3 and lemma 4, we can get the following relation: for all v ≥ p0 , F (v) = (1 − α)F ∗ (v) + α = (1 − α)u + α 10
(15)
where u = F ∗ (v). Using the equation (15), we can rewrite β1 (V1 ) and β2 (V2 ) in (14) as follows: Z V1 1 β1 (V1 ) = V1 − F1 (x)I−1 dx I−1 F1 (V1 ) p0 Z F1 −1 (F1 (V1 )) 1 −1 F1 (x)I−1 dx = F1 (F1 (V1 )) − F1 (V1 )I−1 F1 −1 (F1 (p0 )) Z F1 −1 ((1−α)U1 +α) 1 −1 = F1 ((1 − α)U1 + α) − F1 (x)I−1 dx ((1 − α)U1 + α)I−1 p0 and
(16)
β2 (V2 ) = F2 ((1 − α)U2 + α) −
Z
1
−1
I−1
((1 − α)U2 + α)
F2 −1 ((1−α)U2 +α)
F2 (x)I−1 dx
p0
Redefining βj (Vj ) by ϕj (Uj ) with j = 1, 2, then ϕ1 (U1 ) = F1 ((1 − α)U1 + α) −
Z
1
−1
I−1
((1 − α)U1 + α)
F1 −1 ((1−α)U1 +α)
F1 (x)I−1 dx
p0
(17) ϕ2 (U2 ) = F2 −1 ((1 − α)U2 + α) −
Z
1 I−1
((1 − α)U2 + α)
F2
−1 ((1−α)U
2 +α)
F2 (x)I−1 dx
p0
(18) From the definition of Λ(B) and the hypothesis of B ∼ ϕ1 (U1 ) ∼ ϕ2 (U2 ), it follows that for all b0 ≥ p0 12 , P[B ≤ b0 ] = Λ(b0 ) = P(ϕ1 (U1 ) ≤ b0 ) = P(ϕ2 (U2 ) ≤ b0 )
(19)
From the property of U1 and the invertibility of ϕ1 (U1 ) on [0,1], it follows that P(ϕ1 (U1 ) ≤ b0 ) = P(U1 ≤ ϕ1 −1 (b0 )) = ϕ1 −1 (b0 ).
(20)
P(ϕ2 (U2 ) ≤ b0 ) = ϕ2 −1 (b0 ).
(21)
Similarly, Therefore, the equation (19),(20) and (21) lead to the following result: for all actual bid b0 ≥ p0 , P(B ≤ b0 ) = Λ(b0 ) = ϕ1 −1 (b0 ) = ϕ2 −1 (b0 ) 12
Note that the boundary condition β(p0 ) = p0 .
11
(22)
Hence, ϕ1 −1 (b0 ) = ϕ2 −1 (b0 ) for all b0 ≥ p0 .
(23)
Therefore, we can get the following relation: ϕ1 (u) = ϕ2 (u) a.e. on [0, 1].
(24)
It remains to show that: ϕ1 (u) = ϕ2 (u) a.e. on [0, 1] implies F1 (v) = F2 (v) a.e. on [p0 , ∞). From the equation (17), (18) and (24), it follows that for all u ∈ (0, 1), Z F1 −1 ((1−α)u+α) 1 −1 F1 ((1 − α)u + α) − F1 (x)I−1 dx I−1 ((1 − α)u + α) p0 Z F2 −1 ((1−α)u+α) 1 =F2 −1 ((1 − α)u + α) − F2 (x)I−1 dx ((1 − α)u + α)I−1 p0
(25)
Multiply both sides of equation (25) by ((1 − α)u + α)I−1 , then Z F1 −1 ((1−α)u+α) −1 I−1 ((1 − α)u + α) F1 ((1 − α)u + α) − F1 (x)I−1 dx Z I−1
=((1 − α)u + α)
−1
p0 F2 −1 ((1−α)u+α)
F2 ((1 − α)u + α) −
F2 (x)I−1 dx
(26)
p0
Differentiating both sides of equation (26) with respect to u gives us (I − 1)((1 − α)u + α)I−2 (1 − α)F1 −1 ((1 − α)u + α) I−1 dF1
−1
I−1 dF2
−1
((1 − α)u + α) du dF1 −1 ((1 − α)u + α) − (F1 (F1 −1 ((1 − α)u + α)))I−1 du −1 I−2 = (I − 1)((1 − α)u + α) (1 − α)F2 ((1 − α)u + α) + ((1 − α)u + α)
((1 − α)u + α) du dF2 −1 ((1 − α)u + α) − (F2 (F2 −1 ((1 − α)u + α)))I−1 du It follows that + ((1 − α)u + α)
F1 −1 ((1 − α)u + α) = F2 −1 ((1 − α)u + α) for almost all u ∈ [0, 1].13 13
Note that (1 − α)u + α ∈ (α, 1).
12
(27)
(28)
Hence, from equation (15) and (28), we obtain: F1 (v) = F2 (v) a.e. on [p0 , ∞).
(29) Q.E.D.
3. First-Price Auctions with Observed AuctionSpecific Heterogeneity So far we have studied the ideal situation where the identical auctions are repeated independently. Now we study the auctions where auction-specific characteristics are observed by bidders. We consider a nonparametric identification by incorporating the covariates into the semi-nonparametric distribution which is equivalent to the true conditional distribution. In the next subsection, we show that the auction-specific covariates can be incorporated into the conditional distribution via semi-nonparametric approach. The environment is the same as the ideal situation except that bidders can observe the auction-specific characteristics. Therefore, the structure of interest is the conditional values distribution given covariates.
3.1. Semi-nonparametric Specification of Conditional Distribution Suppose a lower bound of bidders’ values is v = 0. Let X be the vector of auctionspecific characteristics for each auctioned item and ε be bidders’ idiosyncratic term. Suppose that bidders’ private values are the function of auction-specific characteristics and an idiosyncratic term which is a latent variable. A random variable ε comes from some unknown distribution. Let Y = ln(V ). Assume that Y = γ(X) + ε It follows from (30) that V = exp(γ(X) + ε). 13
(30)
Note that exp(γ(X)) is a contribution of auction-specific characteristics on bidder’s private value V and exp(ε) is a contribution of idiosyncratic term on the value V respectively. If we want to denote variables either by bidders or by auctions, we can write as Ylj = γ(Xl ) + εlj
(31)
Subscript j denotes a jth bidder and subscript l denotes a lth auction. For the simple notation, we will suppress the subscripts unless they are necessary. Assumption 3. A covariate random vector X is independent of a random variable ε.14 Then, the conditional distribution given auction-specific characteristics X is F (v|X) = P [V ≤ v|X] = P [exp(Y ) ≤ v|X] = P [exp(ε) ≤ v exp(−γ(X))|X] = Υ (v exp(−γ(X)))
(32)
where Υ is a distribution function of exp(ε) whose support is (0, ∞). Note that the independence of X and ε is used in the last equation in (32). We can have a semi-nonparametric specification of F (v|X) if we can specify Υ(v exp(−γ(X))) as H(G∗ (v exp(−γ(X)))) where a distribution G∗ (v exp(−γ(X))) is an initial guess for Υ(v exp(−γ(X))) on [0, 1]. Since this semi-nonparametric specification can be done for arbitrary initial guess G∗ asymptotically, the seminonparametric structures is H(U ) and γ(X). We call H SNP(seminonparametric) distribution and call γ a covariate function. For the identification proof here, we choose G∗ (v exp(−γ(X))) ≡ 1−exp[−v exp(−γ(X))]. Hence, we can have a semi-nonparametric specification of the conditional distribution of values F (V |X) as follows: F (v|X) ≡ Υ (v exp(−γ(X))) = H (1 − exp[−v exp(−γ(X))]) 14
(33)
We may assume that the function of covariates γ(X) is independent of the idiosyncratic term
ε.
14
The semi-nonparametric specification in (33) is possible since the support of U ≡ 1 − exp(−V exp(−γ(X))) is (0,1) and u ≡ 1 − exp(−v exp(−γ(X))) is strictly monotone in v on (0, ∞) when X is given and the condition 0 < exp(−γ(X)) < ∞ holds. Note that we denote a random variable in upper-case and a non-random variable in lower-case throughout the paper. The only one exception is an idiosyncratic term ε which is a random variable.
3.2. Identification Problem The identification problem here breaks into two parts. The first part is to show that the conditional distribution F (V |X) is identified from observed bids when the auction-specific heterogeneity X is observed. The second part is to show whether the corresponding semi-nonparametric specification structures H(·) and γ(·) are unique. After the conditional distribution F (V |X) is identified from the observed bids in the first step, we can take the true conditional distribution as given. Then, the identification problem can be thought in terms of semi-nonparametric structure (γ(·), H(·)). In particular, we will consider the nonparametric identification of a covariate function γ(X). 3.2.1. Identification: Non-binding Reserve Price Case
Suppose that a seller’s reserve price p0 is not binding.15 So every potential bidder will participate in the auction. An observed bid is assumed to an equilibrium bid: B ≡ β(V |X) = V −
Z
1 I−1
F (V |X)
V
F (y|X)I−1 dy
(34)
0
First, we introduce the following well known lemma which is trivial but essential for the identification proof. 15
For this, suppose p0 = 0 here. It is is due to the assumption v = 0. Generally, a non-binding
reserve price means P[p0 ≤ v] = 1 if v > 0.
15
Lemma 5. The conditional distribution F (V |X) follows a uniform distribution on [0, 1] if it is continuous and strictly monotone on the support (0, ∞). Proof. Let U ≡ F (V |X) when the realization of random variable X is observed. Then, for any u ∈ [0, 1], P[U ≤ u] = P[F (V |X) ≤ u] = P[V ≤ F −1 (u|X)] = F (F −1 (u|X)|X) =u Q.E.D. For the first part identification, we need the following assumption. Assumption 4. (4-i) Values distribution given covariates F (V |X) and bids distribution given covariates Λ(B|X) are absolutely continuous with respect to Lebesgue measure with its density f (V |X) and λ(B|X) respectively. (4-ii) The support of F (V ) is connected set in R+ . (4-iii) The number of potential bidders I is given and known.16 (4-iv) E(β(V |X)) < ∞ where β(V |X) is the equilibrium bidding strategy in equation (34). Proposition 3. Consider a first-price sealed bid auction where bidders are risk neutral and their values are symmetric, independent and private and auctionspecific characteristics X are observed. If assumption 4 holds, then the true conditional distribution of private values, F (V |X), is identified for all v > 0. Proof.
Given observed covariates X, suppose that F1 (V1 |X) and F2 (V2 |X) are
observationally equivalent. That is, β1 (V1 |X) ∼ β2 (V2 |X) where V1 follows the distribution F1 (V1 |X) and V2 follows the distribution F2 (V2 |X): Z V1 Z V2 1 1 I−1 V1 − F1 (y|X) dy ∼ V2 − F2 (y|X)I−1 dy I−1 I−1 F1 (V1 |X) F2 (V2 |X) 0 0 16
I is known to each bidder and the econometrician.
16
where V1 comes from F1 (V1 |X) and V2 comes from F2 (V2 |X). Since V1 = F1−1 (F1 (V1 |X)|X) = F1−1 (U1 ) and V2 = F2−1 (F2 (V2 |X)|X) = F2−1 (U2 ) from Lemma 5, we can write β1 (V1 |X) and β2 (V2 |X) as follows: β1 (V1 |X) =
F1−1 (F1 (V1 |X)|X)
= F1−1 (U1 ) −
Z
1 U1
−
I−1
Z
1 I−1
F1 (V1 |X)
F1−1 (U1 )
F1−1 (F1 (V1 |X)|X)
F1 (y|X)I−1 dy
0
F1 (y|X)I−1 dy where U1 = F1 (V1 |X)
0
and β2 (V2 |X) =
F2−1 (F2 (V2 |X)|X)
= F2−1 (U2 ) −
−
Z
1 U2 I−1
Z
1 F2 (V2 |X)I−1
F2−1 (U2 )
F2−1 (F2 (V2 |X)|X)
F2 (y|X)I−1 dy
0
F2 (y|X)I−1 dy where U2 = F2 (V1 |X)
0
Denote β1 (V1 |X) and β2 (V2 |X) by ϕ1 (U1 ) and ϕ2 (U2 ) respectively, then ϕ1 (U1 ) =
F1−1 (U1 )
−
1 U1
I−1
and ϕ2 (U2 ) =
F2−1 (U2 )
−
1 U2
Z
I−1
F1−1 (U1 )
F1 (y|X)I−1 dy
0
Z
F2−1 (U2 )
F2 (y|X)I−1 dy.
0
Then, by the hypothesis, given X, B ∼ ϕ1 (U1 ) ∼ ϕ2 (U2 )
(35)
where U1 and U2 follow a uniform distribution on [0,1]. By the definition of bids distribution Λ(B|X) and the relation (35), for all b0 > 0 P[B ≤ b0 |X] = Λ(b0 |X) = P[ϕ1 (U1 ) ≤ b0 |X] = P[ϕ2 (U2 ) ≤ b0 |X]. Since ϕ1 (U1 ) is invertible17 , it follows that −1 P[ϕ1 (U1 ) ≤ b0 ] = P[U1 ≤ ϕ−1 1 (b0 )] = ϕ1 (b0 ).
Similarly, P[ϕ2 (U2 ) ≤ b0 ] = ϕ−1 2 (b0 ). Therefore, for all b0 > 0, −1 Λ(b0 |X) = ϕ−1 1 (b0 |X) = ϕ2 (b0 |X). 17
Note that ϕ0i (u) > 0 for all u ∈ (0, 1),i = 1, 2.
17
(36)
Therefore, ϕ1 (u) = ϕ2 (u) a.e. on (0, 1) since Λ(B|X) follows a uniform distribution. Now we need to show that: ϕ1 (U ) = ϕ2 (U ) a.s. implies F1 (V |X) = F2 (V |X) a.s.. Suppose the support of bidders’ values V is (0, ∞). Since ϕ1 (u) = ϕ2 (u) a.e. on (0,1), we can write: for almost all u ∈ (0, 1), F1−1 (u)
−
Z
1 uI−1
F1−1 (u)
I−1
F1 (y|X)
dy =
0
F2−1 (u)
−
Z
1 uI−1
F2−1 (u)
F2 (y|X)I−1 dy.
0
(37) This can be re-expressed as follows: Z Z F1−1 (u) I−1 I−1 −1 I−1 −1 F1 (y|X) dy = u F2 (u) − u F1 (u) − 0
F2−1 (u)
F2 (y|X)I−1 dy. (38)
0
Differentiating both sides with respect to u gives dF1−1 (u) dF −1 (u) − (F1 (F1−1 (u)))I−1 1 du du −1 dF (u) dF −1 (u) =(I − 1)uI−2 F2−1 (u) + uI−1 2 − (F2 (F2−1 (u)))I−1 2 du du (I − 1)uI−2 F1−1 (u) + uI−1
Since F (F −1 (u)) = u, it follows that F1−1 (u) = F2−1 (u) a.e. on (0, 1). Hence, F1 (v|X) = F2 (v|X) a.e. on (0, ∞).
(39) Q.E.D.
Now, we move on to the second part identification issue in a non-binding reserve price case. Assumption 5. (5-i) V ar[X] is positive definite where X is a k-dimensional covariate vector. (5-ii) A Borel measurable function of a covariate vector γ: Rk → R has a positive variance. 18
(5-iii) The support of value V is (0, ∞).18 (5-iv) The distribution H in the semi-nonparametric specification (33) is absolutely continuous with its density h.19 Assumption (5-i) is necessary to identify both H(·) and covariate function γ(·). The positive definite variance of a covariate vector X implies that X does not have a constant. Note that we do not impose any strong restriction on covariates X in that X may be either discrete or continuous or both. Assumption (5-ii) implies that γ(X) has a variation through a variation of X. Assumption (5-iii) can be justified when we do not have any particular information about the lower bound and upper bound of private values respectively. When the true support of private values is an interval [v, v] on R, we can still have a semi-nonparametric specification by some transformation of values.20 Assumption (5-iv) says that we can choose a semi-nonparametric specification following (33). The following proposition says about the second part identification in a nonbinding reserve case. Proposition 4. Suppose assumption 5 holds. Then, the semi-nonparametric specification of the conditional distribution is not identified nonparametrically since the function of a covariate function γ(X) is nonparametrically identified up to a constant. Proof. Suppose that there are two different pairs (H1 , γ1 ) and (H2 , γ2 ) such that for all v > 0 and for given X, F (v|X) = H1 [1 − exp(−v exp(−γ1 (X)))]
(40)
= H2 [1 − exp(−v exp(−γ2 (X)))]
(41)
where (H1 , γ1 ) 6= (H2 , γ2 ). 18
The support of V is defined as {v ∈ R+ : f (v|x) > 0}. Distribution function H(·) is absolutely continuous with respect to Lebesgue measure. 20 Practically, assuming the support to be (0, ∞) can make the estimation of a density with bounded support easier than trying to estimate the lower bound and upper bound of the support. 19
19
It follows from the relation (40) that H1−1 [F (v|X)] = 1−exp(−v exp(−γ1 (X))). So, we get the following equation: for all v > 0, µ ¶ ¡ ¢ 1 v = exp(γ1 (X)) ln = − exp(γ1 (X)) ln 1 − H1−1 [F (v|X)] −1 1 − H1 [F (v|X)] (42) Similarly, it follows from (41) that for all v > 0, ¶ µ ¢ ¡ 1 −1 [F (v|X)] = − exp(γ (X)) ln 1 − H v = exp(γ2 (X)) ln 2 2 1 − H2−1 [F (v|X)] (43) Dividing (42) by (43), then we get the following equation: for all u ∈ (0, 1), ¡ ¢ ln 1 − H1−1 (u) ¢ (44) 1 = exp(γ1 (X) − γ2 (X)) ¡ ln 1 − H2−1 (u) where u = F (v|X) for all v > 0 and X is given. Note that the random variable U ≡ F (V |X) follows a uniform distribution on [0, 1] so equation (44) holds for arbitrary realization u of a random variable U . Seeing RHS on (44), we know that the second term depends on u while the first term does not depend on u. Since equation (44) holds for arbitrary realization u, it must be true that:
Let
ln(1−H1−1 (U )) ln(1−H2−1 (U ))
¡ ¢ ln 1 − H1−1 (U ) ¡ ¢ is constant a.s. ln 1 − H2−1 (U )
(45)
= C where C is some positive constant. Using this fact and taking
the log of both sides in (44) gives us the following equation: for all x and for all u ∈ (0, 1), "
¡ ¢# ln 1 − H1−1 (u) 0 = (γ1 (x) − γ2 (x)) + ln ln(1 − H2−1 (u)) = (γ1 (x) − γ2 (x)) + ln(C)
(46)
where C is some positive constant. It follows that γ2 (X) − γ1 (X) = ln(C) a.s.. 20
(47)
Hence, γ2 (X) = γ1 (X) + ln(C) a.s.
implies that γ(X) is nonparametrically
identified up to a constant. Accordingly, H(·) is not identified since H1−1 (u) = 1 − C[1 − H2−1 (u)] holds a.e. on [0,1] for some positive C 6= 1.
Q.E.D.
For the nonparametric identification, we need to restrict the family of a covariate function γ(X). Assumption 6. The family of a covariate function, G, is restricted to G = {γ|E(γ(X)) = c∗ } where c∗ is an arbitrary constant. Proposition 5. Suppose assumption 5 and assumption 6 hold. Then, a covariate function γ(X) is nonparametrically identified. Hence, a pair of the seminonparametric specification structures (H(·), γ(·)) is nonparametrically identified in the sense that H1 [1 − exp(−v exp(−γ1 (x))] = H2 [1 − exp(−v exp(−γ2 (x)))] a.e. in v > 0 and x implies that H1 (·) = H2 (·)a.s. and γ1 (·) = γ2 (·) a.s.. Proof.
The equation (46) from the previous proof reads, in terms of random
variables X and U , "
¡ ¢# ln 1 − H1−1 (U ) 0 = γ1 (X) − γ2 (X) + ln ln(1 − H2−1 (U )) = γ1 (X) − γ2 (X) + ln(C)
(48)
where C is some positive constant. Taking expectation of both sides, then assumption 6 leads to C = 1. Hence, γ1 (X) = γ2 (X) a.s. and H1 (·) = H(·) a.e. on [0,1].
Q.E.D.
The family satisfying assumption 7 is a particular case which is nested by the family satisfying assumption 6. Assumption 7. The function of a covariate vector γ(X) is linear function, i.e., γ(X) = θ0 X. With assumption 7 and F (V |X) = H[1 − exp(−V exp(−θ0 X)], the structure of a semi-nonparametric specification becomes a pair of (θ, H(·)). 21
Corollary 1. Suppose assumption 5 and assumption 7 hold. Then, a pair of the semi-nonparametric specification structures (θ, H(·)) is unique in the sense that H1 [1 − exp(−v exp(−θ10 x)] = H2 [1 − exp(−v exp(−θ20 x))] a.e. in v ≥ 0 and x implies that H1 = H2 and θ1 = θ2 . Proof.
Along the same line of the previous proof with γ(X) = θ0 X, we get the
equation (46) in terms of random variables X and U , " ¡ ¢# −1 ln 1 − H (U ) 1 0 = (θ1 − θ2 )0 X + ln ln(1 − H2−1 (U )) = (θ1 − θ2 )0 X + ln(C)
(49)
where C is some positive constant. Assumption (5-i) leads to θ1 = θ2 which implies C = 1. Hence, H1 (·) = H(·) a.e. on [0,1].
Q.E.D.
3.2.2. Identification: Binding Reserve Price Case
Suppose that a seller’s reserve price is binding in the sense that the reserve price p0 is above the lower bound of values, i.e., P[V < p0 |X] > 0. Then, bidders who are participating in the auction have values which are greater than the reserve price p0 . These participants are called actual bidders. Note that any information about values below the reserve price p0 is not available when the reserve price is binding. Without loss of generality, we can assume that v = 0. It follows from the semi-nonparametric specification in (33) that F (V |X) = H(1 − exp(−V exp(−γ(X))))
(50)
The actual bidders’ values follow the distribution F (V |V ≥ p0 , X). Denote F (V |V ≥ p0 , X) by F ∗ (V |p0 ). Then, the conditional distribution of actual bidders’
22
value given auction-specific characteristics X is: for all v ≥ p0 , F ∗ (v|X) ≡ F (v|V ≥ p0 , X) = P [V ≤ v|V ≥ p0 , X] =
F (v|X) − F (p0 |X) 1 − F (p0 |X)
(51)
The following lemma shows that the actual bidders’ values follow a uniform distribution. Lemma 6. For all actual bidders’ value V = v ≥ p0 , F ∗ (V |X) follows a uniform distribution on [0, 1]. Proof. Let U = F ∗ (V |X) where V = v ≥ p0 . Then, for any u ∈ [0, 1], P[U ≤ u] = P[F ∗ (V |X) ≤ u] · ¸ F (V |X) − F (p0 |X) =P ≤u 1 − F (p0 |X) = P[F (V |X) ≤ (1 − F (p0 |X))u + F (p0 |X)] = P[V ≤ F −1 ((1 − F (p0 |X))u + F (p0 |X))] ¡ ¢ = F ∗ F −1 [(1 − F (p0 |X))u + F (p0 |X)|X]
(52)
=u Q.E.D. Assume the number of potential bidders I is given and known to a researcher as well as bidders. Since a researcher has access to the submitted bids which are greater than reserve price p0 , a researcher also knows how many potential bidders did not participate in the auction. Without loss of generality, we can assume that the bid from the non-participant is zero. Therefore, a researcher knows the number of zero bids and positive bids by actual bidders whose values are greater than the reserve price p0 . So, if we experiment sufficiently many replications of the same auction in the sense that each replication has the same characteristics and the same number of potential bidders, we can determine unique F (p0 |X) nonparametrically since the participation ratio, 1−F (p0 |X), is the conditional expectation of the ratio 23
of the number of positive bids to the number of potential bidders given covariates. Therefore, F (p0 |X) is identified nonparametrically and then, we can assume that F (p0 |X) is given. Denote F (p0 |X) ≡ α(X), then (51) becomes the following: for all v ≥ p0 , F ∗ (v|X) =
F (v|X) − α(X) 1 − α(X)
(53)
Like a non-binding reserve price case, the identification problem consists of two parts. The first part is to show that the conditional distribution of actual bidders’ values is uniquely determined from observed bids when the reserve price is binding and auction-specific characteristics are observed. The second part of identification problem is whether the semi-nonparametric specification structure (γ(·), H(·)) is unique. When the reserve price is binding, i.e., p0 > 0, the equilibrium bid function given covariates β(v|X) is as follows β(v|X) = v −
Z
1 I−1
F (v|X)
v
F (y|X)I−1 dy for v ≥ p0
(54)
p0
Assumption 8. (8-i) Values distribution F (V |X) and bids distribution Λ(B|x) are absolutely continuous with respect to Lebesgue measure with its density f (·|X) and λ(·|X) respectively. (8-ii) The number of potential bidders I is given and known.21 (8-iii) E(β(V, F (·|X)) < ∞ where β(V, F (·|X)) is the equilibrium bidding strategy in (54).22 (8-iv) The seller’s reserve price p0 is given. (8-v) F (p0 |X) is given.23 21
I is known to each bidder and to an econometrician. It is necessary if the support of private values distribution F is unbounded. 23 Of course, p0 itself may depend on x.
22
24
The following proposition shows that the conditional distribution of actual bidders’ value identified from observed bids given auction-specific covariates. Proposition 6. Suppose the reserve price is binding in a first-price sealed bid auction where bidders are risk-neutral and bidders’ values are symmetric, independent and private. Moreover, auction-specific characteristics are observed. If assumption 8 holds, then the conditional distribution given auction-specific characteristics F (V |X) is identified from the observed bids for almost all v ≥ p0 . proof.
Suppose the reserve price is binding. Note that the equilibrium bidding
function β(V |X) follows β(v|X) = v −
Z
1 I−1
F (v|X)
v
F (y|X)I−1 dy for all v ≥ p0 .
p0
Note that for all V = v ≥ p0 , F (V |X) = (1 − F (p0 |X))F ∗ (V |X) + F (p0 |X) = (1 − F (p0 |X))U + F (p0 |X) by lemma 6 and equation (53) where V is an actual bidder’s private value. Rewriting β(V |X) as follows: Z V 1 F (y|X)I−1 dy β(V |X) = V − I−1 F (V |X) p0 Z F −1 (F (V |X)|X) 1 −1 = F (F (V |X)|X) − F (y|X)I−1 dy I−1 F (V |X) p0 R F −1 ((1−α(X))U +α(X)) F (y|X)I−1 dy p (55) = F −1 ((1 − α(X))U + α(X)) − 0 [(1 − α(X))U + α(X))]I−1 where α(X) ≡ F (p0 |X). Since β(V |X) depends on V and F (V |X), we can denote it by β(V, F (V |X)). Suppose that β(V1 , F1 (V1 |X)) and β(V2 , F2 (V2 |X)) are observationally equivalent. Denote β(V1 , F1 (V1 |X)) and β(V2 , F2 (V2 |X)) by ϕ1 (U1 ) and ϕ2 (U2 ) respectively. R F1 −1 ((1−α(X))U1 +α(X)) F1 (y|X)I−1 dy p0 −1 ϕ1 (U1 ) = F1 ((1 − α(X))U1 + α(X)) − [(1 − α(X))U1 + α(X))]I−1 and R F2 −1 ((1−α(X))U2 +α(X)) −1
ϕ2 (U2 ) = F2 ((1 − α(X))U2 + α(X)) − 25
p0
F2 (y|X)I−1 dy
[(1 − α(X))U2 + α(X))]I−1
From the definition of actual bids distribution Λ(B|X) and the relation β(V1 , F1 (V1 |X)) ∼ β(V2 , F2 (V2 |X)), for all b0 ≥ p0 , given X Λ(b0 |X) = P(B ≤ b0 |X) = P(ϕ1 (U1 ) ≤ b0 ) = P(ϕ2 (U2 ) ≤ b0 ).
(56)
Since ϕ(U ) is invertible for given X, (56) implies ϕ1 −1 (b0 ) = ϕ−1 2 (b0 ) for all b0 . Hence ϕ1 (U ) = ϕ2 (U ) a.s. for given X. Now, it remains to show that ϕ1 (u) = ϕ2 (u) a.e. on [0,1] implies that F1 (v|x) = F2 (v|x) for all v ≥ p0 . It follows from ϕ1 (u) = ϕ2 (u) a.e. on [0,1] that R F1 −1 ((1−α(s))u+α(x)) p0
F1 −1 ((1 − α(x))u + α(x)) −
F1 (y|x)I−1 dy
[(1 − α(x))u + α(x)]I−1
R F2 −1 ((1−α(x))u+α(x)) p0
−1
= F2 ((1 − α(x))u + α(x)) −
F2 (y|x)I−1 dy
[(1 − α(x))u + α(x)]I−1
(57)
where α(x) = F (p0 |x). Multiplying both sides by [(1 − α(x))u + α(x)]I−1 gives Z I−1
[(1 − α(x))u + α(x)]
−1
F1 ((1 − α(x))u + α(x)) − Z
I−1
=[(1 − α(x))u + α(x)]
F1 −1 ((1−α(x))u+α(x))
−1
F1 (y|x)I−1 dy
p0 F2 −1 ((1−α(x))u+α(x))
F2 ((1 − α(x))u + α(x)) −
F2 (y|x)I−1 dy
p0
(58) Differentiating both sides in (58) with respect to u delivers the following equation: for almost all u ∈ [0, 1], (I − 1)[(1 − α(x))u + α(x)]I−2 F1 −1 ((1 − α(x))u + α(x)) dF1 −1 ((1 − α(x))u + α(x)) du −1 −1 I−1 dF1 ((1 − α(x))u + α(x)) −F1 (F1 ((1 − α(x))u + α(x))) du −1 I−2 = (I − 1)[(1 − α(x))u + α(x)] F2 ((1 − α(x))u + α(x)) +[(1 − α(x))u + α(x)]I−1
−1
((1 − α(x))u + α(x)) du −1 dF ((1 − α(x))u + α(x)) −F2 (F2 −1 ((1 − α(x))u + α(x)))I−1 2 du I−1 dF2
+[(1 − α(x))u + α(x)]
26
(59)
It follows from (59) that for almost all u ∈ [0, 1], F1 −1 ((1 − α(x))u + α(x)) = F2 −1 ((1 − α(x))u + α(x)) Hence F1 (v|x) = F2 (v|x) for almost all v ≥ p0 and for all x.
(60) Q.E.D.
The following proposition says about the identification of semi-nonparametric structures (γ(·), H(·). For the identification of semi-nonparametric specification, we need the following assumption. Assumption 9. (9-i) V ar[X] is positive definite where X is a k-dimensional covariate vector. (9-ii) A Borel measurable function of a covariate vector γ: Rk → R has a positive variance. (9-iii) The support of private value V is (0, ∞) and the seller’s reserve price p0 is above v. (9-iv) H is absolutely continuous with its density h. Proposition 7. If assumption (8-v) and assumption 9 hold, then the semi-nonparametric structures (γ(X), H(U )) are not identified nonparametrically since a covariate function γ(X) is nonparametrically identified up to a constant for almost all v ≥ p0 . Proof. Suppose that there are two different pairs (H1 , γ1 ) and (H2 , γ2 ) such that for all v ≥ p0 and for X = x, F (v|x) = H1 [1 − exp(−v exp(−γ1 (x)))]
(61)
= H2 [1 − exp(−v exp(−γ2 (x)))]
(62)
where (H1 , γ1 ) 6= (H2 , γ2 ). Note that F (V |X) is already given by proposition 6. It follows from the relation (61) that H1−1 [F (v|x)] = 1 − exp(−v exp(−γ1 (x)). So, we get the following equation: for almost all v ≥ p0 , µ ¶ 1 v = exp(γ1 (x)) ln 1 − H1−1 [F (v|x)] ¡ ¢ = − exp(γ1 (x)) ln 1 − H1−1 [F (v|x)] 27
(63)
Similarly, it follows from (62) that for almost all v ≥ p0 , µ ¶ 1 v = exp(γ2 (x)) ln 1 − H2−1 [F (v|x)] ¡ ¢ = − exp(γ2 (x)) ln 1 − H2−1 [F (v|x)]
(64)
Dividing (63) by (64), then we get the following equation: for all u ∈ (0, 1), ¡ ¢ ln 1 − H1−1 [α(x) + (1 − α(x))u] ¢ 1 = exp(γ1 (x) − γ2 (x)) ¡ (65) ln 1 − H2−1 [α(x) + (1 − α(x))u] where α(x) ≡ F (p0 |x) and u = F ∗ (v|x) for almost all v ≥ p0 . Note that the random variable U ≡ F (V |X) follows a uniform distribution on [0, 1] so equation (65) holds for arbitrary realization u. Seeing RHS in (65), we know that the second term depends on u while the first term does not depend on u. Since equation (65) holds for arbitrary realization u, it must be true that: ¡ ¢ ln 1 − H1−1 [α(x) + (1 − α(x))u] ¡ ¢ is constant for any u ∈ (0, 1). (66) ln 1 − H2−1 [α(x) + (1 − α(x))u] Let
ln(1−H1−1 [α(x)+(1−α(x))u]) ln(1−H2−1 [α(x)+(1−α(x))u])
= C where C is some positive constant and u is arbi-
trary realization. Take the log of both sides in (65), then we can have the following equation: for almost all x and for almost all u ∈ (0, 1), " ¡ ¢# ln 1 − H1−1 [α(x) + (1 − α(x))u] ¡ ¢ 0 = (γ1 (x) − γ2 (x)) + ln ln 1 − H2−1 [α(x) + (1 − α(x))u] = (γ1 (x) − γ2 (x)) + ln(C)
(67)
where C is some positive constant. Equation (67) can be rewritten in terms of random variables as follows (γ2 (X) − γ1 (X)) = ln(C) a.s.
(68)
Equation (68) implies that γ(x) is nonparametrically identified up to a constant. Therefore, a pair (γ(·), H(·)) is not nonparametrically identified in general. Q.E.D. After restricting the family of a covariate function, we can have a nonparametric identification. 28
Proposition 8. Suppose assumption 6, assumption (8-v) and assumption 9 hold. Then, a covariate function γ(X) is nonparametrically identified. Hence, the pair of the semi-nonparametric specification structures (H(·), γ(·)) is nonparametrically identified in the sense that H1 [1 − exp(−v exp(−γ1 (x))] = H2 [1 − exp(−v exp(−γ2 (x)))] a.e. in v > p0 and x implies that γ1 (X) = γ2 (X) a.s. and H1 (u) = H2 (u) a.e. on the partial interval of [0, 1]. Proof. From the previous proof, (68) reads (γ2 (X) − γ1 (X)) = ln(C) a.s. where C =
ln(1−H1−1 [α(X)+(1−α(X))U ]) ln(1−H2−1 [α(X)+(1−α(X))U ])
. Then, assumption 6 leads to ln(C) = 0
and γ1 (X) = γ2 (X) a.s.. Accordingly, H1−1 (·) = H2−1 (·) a.e. on [α(x), 1] and thus, H1 (·) = H2 (·) a.e. on [1 − exp(−p0 exp(−γ(x)), 1].
Q.E.D.
Corollary 2. Suppose assumption 7, assumption (8-v) and assumption 9 hold. Then, a pair of the semi-nonparametric specification structures (θ, H(·)) is unique in the sense that H1 [1 − exp(−v exp(−θ10 x))] = H2 [1 − exp(−v exp(−θ20 x))] a.e. in v ≥ p0 and x implies that θ1 = θ2 and H1 (u) = H2 (u) a.e. on the partial interval on [0, 1]. Proof. From the previous proof, (68) reads (γ2 (X) − γ1 (X)) = ln(C) a.s. where C =
ln(1−H1−1 [α(X)+(1−α(X))U ]) ln(1−H2−1 [α(X)+(1−α(X))U ])
.
Then, assumption (9-i) leads to ln(C) = 0 and θ1 = θ2 . Therefore, H1−1 (·) = H2−1 (·) a.e. on [α(x), 1]. Therefore H1 (·) = H2 (·) a.e. on [1−exp (−p0 exp(−θ0 x)) , 1] where θ = θ1 = θ2 .
Q.E.D.
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3.2.3. Some Discussion about Identification Problem In the second part identification, we can consider the identification problem in terms of location and scale of a covariate function γ(·). Since log V = γ(X) + ² and γ(X) and ε are independent, we can think of the identification problem in terms of either V or exp(²).24 Since we already have a given F (v|X) by the first identification step25 , we need to go over the covariate function γ(X) in terms of location and scale. Once the covariate function γ(X) is identified, the SNP distribution on the unit interval, H(U ), is automatically identified.26 3.2.3.1. Location and Scale of γ(X) First, we see the location problem of γ(X). Suppose there is a constant c0 such ˜ − exp(−v(˜ that γ˜ (X) = γ(X) + c0 and H(1 − exp(−vγ(X))) = H(1 γ (X))). Since F (v|X) is given from the first identification part, we can have the following way in the similar way used in the proof of proposition . µ ¶ ln(1 − H −1 (F (v|X))) γ˜ (X) − γ(X) = ln for almost all v ˜ −1 (F (v|X))) ln(1 − H
(69)
Note the LHS of (69) is c0 which is a constant. Therefore, there can exist different ˜ and H such that the RHS of (69) holds for some nonzero constant c0 . Therefore, H we have a location problem for the nonparametric identification of (γ, H). So we need some restriction like assumption 6 to avoid a location problem. ˜ = c1 γ(X). Similarly, we have the followFor the scale problem, suppose γ(X) ing relation. Ã γ(X) − γ˜ (X) = ln
˜ −1 (F (v|X))) ln(1 − H ln(1 − H −1 (F (v|X)))
! for all v
(70)
for all v.
(71)
It follows à (1 − c1 )γ(X) = ln
˜ −1 (F (v|X))) ln(1 − H ln(1 − H −1 (F (v|X)))
24
!
Or we can think it in terms of log V or ². Note that F (v|X) ≡ Υ(v exp(−γ(X))) where Υ is a distribution of exp(²). 26 Note F (v|X) ≡ Υ(v exp(−γ(X))) = H(1 − exp(−v exp(−γ(X)))).
25
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Note that RHS in (71) should be constant in spite of its variation of v since LHS does not depend on v. Therefore, LHS should be constant too. But, c1 must be 1 due to the assumption that the variance of γ(X) is positive. Therefore, we do not have a scale problem of γ(X) because of a positive variance of a covariate function γ(X). For the identification of semi-nonparametric structure (γ, H), it suffices to the location problem. Instead of assumption 6, we can consider a quantile restriction. In a binding reserve price case, we can use the condition F (p0 |X) = α(X) as a quantile restriction. But, we need an arbitrary quantile condition in a non-binding reserve price case. 3.2.3.2. Identification via Quantile Restriction When the reserve price is binding, we can use the assumption F (p0 |X) ≡ α(X) is given.27 Hence, we can have a restriction α(X) = H(1 − exp(−p0 exp(−γ(X)))). We can use this restriction instead of assumption 6 used in proposition 8. Definition 5. The family Hα(X) = {H|H(1 − exp(−p0 exp(−γ(X)))) = α(X)}. Then, we can use assumption instead of assumption 5 in proposition 6. Proposition 9. Suppose assumption (8-v) and assumption 9 hold and the reserve price is binding. Then, the semi-nonparametric structures H(·) and γ(·) are nonparametrically identified in the sense that H1 [1 − exp(−v exp(−γ1 (x))] = H2 [1 − exp(−v exp(−γ2 (x)))] a.e. in v > p0 and x implies that γ1 (X) = γ2 (X) a.s. and H1 (u) = H2 (u) a.e. on the partial interval of [0, 1]. Proof. Suppose that there are two different pairs (H1 , γ1 ) and (H2 , γ2 ) such that for all v ≥ p0 and for all x,
27
F (v|x) = H1 [1 − exp(−v exp(−γ1 (x)))]
(72)
= H2 [1 − exp(−v exp(−γ2 (x)))]
(73)
Note that F (p0 |X) is determined nonparametrically.
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where (H1 , γ1 ) 6= (H2 , γ2 ) and H1 , H2 ∈ Hα(X) = {H|H(1−exp(−p0 exp(−γ(X)))) = α(X)} since α(X) = F (p0 |X) is already given. Like the previous proof of proposition 8, we can have the following equation ¡ ¢ ln 1 − H1−1 [α(X) + (1 − α(X))u] ¢. (γ2 (X) − γ1 (X)) = ln(C) a.s. where C = ¡ ln 1 − H2−1 [α(X) + (1 − α(X))u] (74) Note that C = 1 since H1 , H2 ∈ Hα(X) and RHS of the equation above holds when u = 0. Therefore, γ1 (X) = γ2 (X) which makes H1 (u) = H2 (u) a.e. on [α(X), 1].
Q.E.D.
Moreover, we can generalize proposition 9 in the following proposition. The following assumption 10 is more general than assumption (5-i) or assumption (9-i) since it allows a constant constant in covariates vector X. Assumption 10. V ar[X] is positive semidefinite where X is a k-dimensional covariate vector. Proposition 10. Define the family of SNP distribution H by Hz = {H|H(1 − exp(−v exp(−γ(X)))) = z} where z = F (v|X) for some v. Then, the seminonparametric structures (γ, H) are nonparametrically identified if the following conditions hold: (i) Assumption (5-ii)-(5-iv) and assumption 10 hold when the reserve price is not binding. (ii) Assumption (8-v) and assumption (9-ii)-(9-iv) hold when the reserve price is binding. Proof. (i) Suppose that there are two different pairs (H1 , γ1 ) and (H2 , γ2 ) such that for all v > 0 and for all x, F (v|x) = H1 [1 − exp(−v exp(−γ1 (x)))] = H2 [1 − exp(−v exp(−γ2 (x)))]
32
where (H1 , γ1 ) 6= (H2 , γ2 ) and HF (v0 |X) = {H|H(1 − exp(−v0 exp(−γ(X)))) = F (v0 |X)} for some v0 > 0. By similar way used in the proof of proposition 9, we can have the following equation: ¡ ¢ ln 1 − H1−1 [F (v|X)] ¢. (γ2 (X) − γ1 (X)) = ln(C) a.s. where C = ¡ ln 1 − H2−1 [F (v|X)]
(75)
At v = v0 , we have H1 (1−exp(−v0 exp(−γ1 (X)))) = H2 (1−exp(−v0 exp(−γ2 (X)))). Therefore, C = 1 and γ1 (X) = γ2 (X) which makes H1 (u) = H2 (u) a.e. on [0, 1]. Q.E.D. (ii) Setting v0 to be a reserve price p0 . Then, the proof is identical to that of proposition 10.
Q.E.D.
Note that the proposition 10 can be also verified in terms of location and scale of a covariate function γ(X).
4. Concluding Remarks We study the identification problem in the ideal situation and auction-specific characteristics observed situation. We can think of other situation such as bidderspecific characteristics observed situation or some other environment than IPV paradigm.28 In spite of many other identification problems, we would like to suggest some ideas which are believed to be very relevant to the identification results of auction-specific heterogeneity case. First, we estimate the covariate function γ(X) semi-nonparametrically using a neural net or wavelet. Second, we can test the functional form of a covariate function γ(X) if we want to specify γ(X) parametrically. Specifically, if we specify γ(X) = θ0 X initially, we get the estimator Fˆ1 (V |X) and θˆ1 . We can have the conditional expectation of log V from a distribution Fˆ1 (V |X) and compare the estimate of the conditional expectation of E(log V |X) and γˆ (X) = θˆ10 X. If the 28
See Athey, S. and P. A. Haile (2002) for the general identification.
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functional form is correct, the difference E(log V |X) − θˆ10 X would be zero. So, we can have a test statistic.
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References [1] Athey, S. and P. A. Haile (2002), “Identifications of Standard Auction Models”, Econometrica, 70(6), 2107-2140. [2] Athey, S. and P.A. Haile (2005), “Nonparametric Approaches to Auctions ”, Handbook of Econometrics, Vol 6 (forthcoming) [3] Campo,S. , Guerre, E., Perrigne, I. and Q. Vuong (2002),“Semiparametric Estimation of First-Price Auctions wiht Risk Averse Bidders”, Mimeo. [4] Chen, C. , “Large Sample Sieve Estiamtion of Semi-Nonparametric Models”, Handbook of Econometrics Vol 6, forthcoming. [5] Donald, G.S and J.H. Paarsch. (1996) “Identification, Estimation, and Testing in Parametric Empirical Models of Auctions within the Independent Private Values Paradigm”, Econometric Theory, 12, 517-567. [6] Guerre, E., I. Perrigne, and Q. Vuong. (2000) “Optimal Nonparametric Estimation of First-Price Auction”, Econometrica, 68, 525-574. [7] Krishna, V., Auction Theory (2002), Academic Press. [8] Li,T. , I. Perrigne and Q. Vuong (2000), “Conditionally independent private information in OCS wildcat auctions”, Journal of Econometrics, 98, 129-161. [9] Riley, G.J., W.F.Samuelson. (1981) “Optimal Auctions”, American Economic Review, 71, 381-392. [10] Roehrig, C.S. (1988) “Conditions for identification in Nonparametric and Parametric Models”, Econometrica, 56, 433-447.
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