Semi-nonparametric Estimation of First-Price Auction using Simulated Method of Moments∗ Herman J. Bierens†and Hosin Song‡ August 27, 2006

Abstract In this paper, we propose an SNP-SSMM(Semi-NonParametric Sieve Simulated Method of Moments) estimation based on the nonparametric identification results of the independently repeated identical first-price auctions in Bierens and Song (2006). Our proposed SNP-SSMM estimator is built on a well-known theorem of the characteristic function that two distributions are equivalent if and only if both characteristic functions are equal for all real values. Based on the construction of SNP density functions on the unit interval in Bierens (2006), we can generate simulated bids randomly and get the empirical characteristic function of simulated bids from each density estimate. Therefore, we can construct an integrated moment condition by using the difference between empirical characteristic functions of bids and simulated bids. Moreover, we propose a method to determine a truncation order of Legendre polynomials in SNP density based ∗

We are grateful to Quang Vuong, Isabelle Perrigne and Joris Pinkse for their valuable

comments. † Department of Economics, The Pennsylvania State University, University Park, PA 16802, [email protected] ‡ Department of Economics, The Pennsylvania State University, University Park, PA 16802, [email protected]

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on information criteria. We show that the resulting density estimator and distribution estimator are consistent. The proposed SNP-SSMM estimation shows a good performance in Monte-Carlo experiments. Keywords: Accept-reject method, empirical characteristic function, first-price auction, integrated moment, Legendre polynomials, semi-nonparametric estimation, sieve estimation, simulated method of moments, selection of polynomial truncation order, weak convergence

1. Introduction As Laffont and Vuong (1993) point out, the distribution of bids determines the structural elements of the model provided identification is achieved. In the first-price auction model with symmetric independent private value, the structural element of interest is the value distribution. Much research has been done on the identification and the estimation of the value distribution. Donald and Paarsch (1992) apply ML estimation to first-price auctions and Dutch auction. In spite of non-parametric identification of values distribution, they need parametric specification for values distribution to implement ML estimation.1 Because of general difficulty of ML estimation in the firstprice auction2 , Laffont and Vuong (1993) suggest SNLLS(Simulated NonLinear Least Squares) estimation and SMM(Simulated Method of Moment) estimation for a descending price auction model with symmetric independent private value.3 Their SNLLS requires to replace the expectation of winning 1

In the numerical example, they assume that the value distribution is a uniform distribution on the interval [0, v l ] where v l = exp(θ0 + θ1 Zl ), Zl is a covariate vector of lth auction. 2 The difficulty mainly comes from two reasons. The first reason is that equilibrium bid is highly non-linear in value and its distribution and thus the implementation of ML is computationally challenging. The second reason that the support of winning bid affected by the parameter of values distribution which violates the condition of consistency of ML estimator. 3 Descending price (Dutch) auction is strategically equivalent to the first-price auction.

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bid with simulated one. They also suggest that the expectation of higher moments of a winning bid can be used for SMM when the expectation of a winning bid only is not sufficient to identify all parameters. Both SNLLS and SMM needs some parametric specification for the value distribution. Laffont, Ossard and Vuong (1995) apply the SNLLS suggested by Laffont and Vuong (1993) to the egg plant auction which is a descending price auction. They specify the value distribution to be a log-normal distribution.4 Li (2005) considers the first-price auctions with entry and binding reserve price.5 He proposes an MSM(Method of Simulated Moments) to estimate the parameters of structural elements.6 Guerre, Perrigne and Vuong (2000) show a nonparametric identification and propose a nonparametric estimation using a kernel. They focus on the estimation of values distribution whose support is an interval in R+ .7 Bierens and Song (2006) show a nonparametric identification which requires minimal assumptions that values distribution has finite expectation and values distribution is absolutely continuous distriSo, the equilibrium bid function is the same in symmetric independent private value paradigm. 4 They determine the standard deviation of log value by using the retail price data of egg plant for the identification of parameters, that is, mean standard deviation of log value. Moreover, the conditional mean of log value is assumed to be the function of covariates regarding egg plant and its market. 5 The auction consists of two stage. At the first stage, the potential bidder decides whether he or she enters the auction with payment of entry cost. At the second stage, the bidder gets to know his or her value and then decides to bid following the equilibrium bid function which is the same function in usual first-price auction model. 6 The structural elements in the paper are the entry probability and values distribution. Both are specified to be parametric in the estimation. One interesting thing is that one conditional moment is a function of an upper bound of bid in each auction. The computation of the bid upper bound in estimation can be done by following a simulation in Laffont, Ossard and Vuong (1995). The other moment conditions are related to the number of active bidders. In this paper, the bidder is categorized as 3 types, that is, potential bidder, active bidder and actual bidder. The potential bidder indicates a bidder before deciding the participation. The active bidder is a potential bidder who decides to participate in the auction. The actual bidder is a bidder who submits a bid which is greater than the seller’s reserve price. 7 For general nonparametric identification of first-price auctions with symmetric IPV see Guerre, Perrigne and Vuong (1995) or Athey and Haile (2005).

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bution with connected support. They achieved the identification of values distribution even when values are unbounded. Based on the identification result of the repeated identical first-price auctions in Bierens and Song (2006), we propose an SNP-SSMM(Semi-NonParametric Sieve Simulated Method of Moments) estimation in this paper. Particularly, it is not necessary to set an upper bound of values to be finite unless there is additional certain prior information about the upper bound since Bierens and Song (2006) allow bidders’ values to be unbounded above. Our SNP-SSMM estimator is very reliable to recover the private values distribution from the bids.8 To avoid unnecessary parametric assumption on the values distribution, we propose a simulated method of moment estimation. The moment condition is the squared difference of the characteristic function of bids and 8

Some people might wonder what is different between the nonparametric approach using a kernel proposed in Guerre, Perrigne and Vuong (2000) and our approach. Our proposed estimator is more reliable than the nonparametric estimation using a kernel in the following sense. Guerre, Perrigne and Vuong (2000) draw a simple relationship between 1 Λ(b) bids and private values v = b + I−1 λ(b) where v is a private value, b is a corresponding bid, Λ is a distribution function of bids and λ is an associated density function. From observed bids, they recover the private values which are called pseudo-private values via a ˆ 1 Λ(b) kernel estimation v˜ = b + I−1 . Then, they estimate the distribution of private values ˆ λ(b) using those obtained pseudo private values v˜’s via a kernel estimation again. Hence, their ˆ is true for all proposed method depends heavily on the assumption that their estimate Λ(b) ˆ λ(b) b. But, it is likely that the estimate

ˆ Λ(b) ˆ λ(b)

is pretty sensitive particularly near the boundary. To solve the problem, they introduce a trimming of some of pseudo private values that seem to be extreme values. This trimming process basically comes from the assumption of ˆ the bounded support of private values. Without assuming that Λ(b) is true, our approach ˆ λ(b) exploits the best by v˜ = b +

ˆ Λ(b) . ˆ λ(b)

ˆ 1 Λ(b) , ˆ I−1 λ(b)

That is, we use the following process: if

then the estimate

to the original estimate

ˆ Λ(b) ˆ λ(b)

ˆ ˆ ˜ Λ( b) ˆ ˆ ˜ λ( b)

ˆ Λ(b) ˆ λ(b)

recovers pseudo values v˜

from equilibrium bids ˜b = β(˜ v ) should be equal

eventually. By trying every possible candidates for values

density function, we can find the best density (or distribution ) estimator which is believed to generate the sample bids. To sum up, our proposed estimator tries all possible density estimators while the non-parametric estimator in Guerre, Perrigne and Vuong (2000) use only one-time estimated density. In this sense, our proposed semi-nonparametric simulated method of moments estimation can give us more reliable information for the private values distribution than nonparametric estimation using kernels.

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simulated bids.9 As a sample moment condition, we propose using empirical characteristic functions of bids and simulated bids where the former are from the true values distribution and the latter come from the SNP(SemiNonParametric) density estimator. Each moment condition should be zero for all real value, we can construct an integrated moment condition which is similar to one in Bierens and Ploberger (1997). Our SNP-SSMM estimation is different from SMM in Laffont and Vuong (1993) and MSM in Li (2005) in that the two latter approaches require parametric specification for the value distribution while ours does not. Moreover, our approach exploits an infinite number of moment conditions while Laffont and Vuong (1993) and Li(2005) use the finite number of moment conditions. It gives our SNP-SSMM estimator the consistency. As a by-product, we do not have a boundary problem which occurs in the estimation using a kernel.10 It follows from the semi-nonparametric property of our estimation.11 The boundary problem in kernel methods basically comes from the small number of observations near the tail of the support. But, our method does not suffer from the boundary problem directly because the initial guessed distribution G(v) can play a role of parametric specification. Moreover, we are not concerned with the trimming of simulated values like in kernel density estimation in Guerre, Perrigne and Vuong (2000), since our proposed estimator can accommodate any sufficiently large support. Finally, we propose a procedure for the selection of truncation polynomial order which is robust to both an over-parameterized case and an underparameterized case asymptotically. In the sieve simulated method of moments estimation, the choice of truncation order of basis polynomials is not 9

This comes from the well-known theorem that if two characteristic functions are the

same, then their distributions are the same and vice versa. 10 Guerre, Perrigne and Vuong (2000) use a nonparametric kernel density estimation assuming that the support of the underlying distribution is bounded from above and below. So the density estimator is not consistent at the values near the boundary. But our proposed estimator is consistent at all values on the support. 11 Strictly saying, it comes from the property of parametric specification in the seminonparametric specification.

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well developed.12 In this sense, our proposed procedure may shed light on the related research. This paper is organized as follows. In section 2, we introduce our proposed SNP-SSMM estimation. First, we propose a simulated method of moment using an empirical characteristic function of sample bids and simulated bids in the first subsection. In the second subsection, we briefly address the construction of SNP density and SNP distribution following Bierens (2006). In the third subsection, we propose a method of random drawing of simulated values from an arbitrary SNP density estimate. In section 3, we show the consistency of SNP-SSMM estimator and construct a criterion function for the implementation of SNP-SSMM estimation. We also propose a procedure to the truncation order of polynomials. In section 4, we show the performance of our proposed SNP-SSMM estimation via Monte Carlo experiments. In concluding remarks, we suggest the direction for some future research. Throughout the paper, we denote a random variable in upper-case and P

a non-random variable in lower-case. Regarding some notations, Xn −→ X or p lim Xn = X indicates that Xn converges to X in probability. Similarly, d

Xn −→ X indicates that Xn converges to X in distribution. To clarify the space of random elements, sometimes we use the notation Xn =⇒ X to indicate that Xn converges weakly to X instead of using a terminology R 1 convergence in distribution. Norm || · ||p denotes || · ||p = ( | · |p ) p where 0 < p < ∞.

2. Semi-nonparametric Sieve Simulated Method of Moments (SNP-SSMM) Estimation of FirstPrice Auctions Throughout this paper, we confine our interests to first-price sealed bid auctions where values are independent, private and bidders are symmetric 12

Particulary, it is very hard to find the related research regarding simulated method of moments. Most of related literature focus on sieve maximum likelihood estimation.

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and risk-neutral. Suppose that we have independently repeated identical auctions. Under the assumption that the values distribution has finite expectation and values distribution is confined to the class of absolutely continuous distributions with connected support, the values distribution is identified form the bids.13 For details, see Bierens and Song (2006). The equilibrium bid is defined as Z v 1 β(v) = v − F (x)I−1 dx, v > max(p0 , v). F (v)I−1 max(p0 ,v)

(1)

Note that the equilibrium bid is different depending on whether the seller’s reserve price is binding or not.14 The reserve price is not binding when p0 < v. Then, the number of potential bidders I is observed. When the reserve price is binding, only actual bidders whose value is greater than p0 participates in the auction. In binding reserve price case, we assume that the number of potential bidders is known to be I. In this section, we propose the semi-nonparametric estimation using simulated method of moments. We introduce our simulated method of moments using empirical characteristic function of bids and simulated bids. Simulated bids come from a density estimator which is an element of a compact set of density functions on the unit interval. See Bierens (2006). For the random generating of simulated bids, we propose a accept-reject method which implements a random drawing of simulated values from each density estimator. Using these randomly drawn simulated bids from each density estimator, we can compute the corresponding simulated bids. With the use of these simulated bids from each density estimator, we can implement simulated method of moments estimation. Since the approximation of a density function and distribution function have a property of sieve estimation, we call our proposed estimation method a semi-nonparametric sieve simulated method of moments(SNP-SSMM) estimation. 13

When the seller’s reserve price is binding, it is identified for the value greater than the

reserve price. 14 For the derivation of the equilibrium bid in (1), see Krishna (2002) or Riley and Samuelson (1981).

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2.1.

Simulated Method of Moments using Empirical

Characteristic Function The observed bid is assumed to be an equilibrium bid Z vj 1 F (x)I−1 dx for all vj ≥ p0 bj = β(vj , F ) = vj − I−1 F (vj ) p0

(2)

We have independently repeated L auctions with I potential bidders in each auction. Suppose that F˜ is an estimator for the true distribution F and v˜ is the simulated value which is randomly drawn from F˜ .15 We can get the simulated bid ˜b from v˜ and F˜ , which is ˜b = β(˜ v , F˜ ) = v˜ −

1 ˜ F (˜ v )I−1

Z

v˜

F˜ (x)I−1 dx.

(3)

p0

ˆ We propose a complex-valued function Ψ(t|F, F˜ ) which is the difference of the empirical characteristic functions between bids and simulated bids: N N 1 X 1 X ˆ ˜ Ψ(t|F, F ) = exp(itbj ) − exp(it˜bj ) for all t ∈ R. N j=1 N j=1

(4)

where N = L × I and a sample bid bj is defined in (2) and a simulated bid ˜bj is defined in (3). Note that the bids (bj )N are from the true distribution j=1 N ˜ F (v) and simulated bids (bj ) are from an distribution estimator F˜ (v). j=1

There is a well-known theorem of the characteristic function.16 Theorem 1. The random vector X and Y in Rk is equal in distribution if and only if E(exp(itX)) = E(exp(itY )) for all t ∈ Rk . ˆ Ψ(t|F, F˜ ), we can construct the following SMM(Simulated Method of Moments) to estimate the true distribution F : Z ∞ 2 ˆ ˆ ˜ ˆ F = arg min Ω(F ) ≡ | Ψ(t|F, F˜ ) | w(t)dt F˜

−∞

where w(t) is a positive weight function. 15 16

(5)

The method of random generating simulated values is addressed in subsection 3.3. For proof, see p15 in van der Vaart (1998).

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The moment condition has a form of integrated conditional moment which makes use of the infinite number of moment conditions.17 As one application example, think of a case where the weight function is a standard normal density function. Then, the objective becomes a tractable real-valued function of bids and simulated bids: Z ∞ Z 2 ˆ F˜ ) = ˆ Ω( | Ψ(t|F, F˜ ) | w(t)dt =

∞

1 1 ˆ ˆ F˜ ) √ exp(− t2 )dt Ψ(t|F, F˜ )Ψ(t|F, 2 2π −∞ −∞ N N N N 1 XX 1 1 XX 1 2 2 = 2 exp(− (bj1 − bj2 ) ) − 2 exp(− (bj1 − ˜bj2 ) ) N j =1 j =1 2 N j =1 j =1 2 −

1 N2

1

2

1

N X

N X

N X N X

j1 =1 j2

1 1 2 exp(− (˜bj1 − bj2 ) ) + 2 2 N =1

2

j1 =1 j2

1 2 exp(− (˜bj1 − ˜bj2 ) ) 2 =1 (6)

where N = L × I and a sample bid bj is defined in (2) and a simulated bid ˜bj is defined in (3).18 Moreover, we can construct the integrated moment on the some compact set containing zero19 using a uniform probability measure as follows: Z κ 2 1 ˆ F˜ ) ≡ ˆ Fˆ = arg min Ω( |Ψ(t|F, F˜ )| dt 2κ −κ F˜

(7)

This type objective function in (7) will be used in this paper. Hereafter, we ˆ ˆ denote Ψ(t|F, F˜ ) by Ψ(t) for simple notation. In the following subsection, we address how to construct a semi-nonparametric density and distribution following Bierens (2006).

2.2. Semi-nonparametric density and distribution 17

Therefore, the resulting estimator is consistent. See Appendix for the derivation of equation (6). For the derivation, we need to use the fact that if a random variable X follows a standard normal distribution then its characteristic function ξ(t) ≡ E[exp(itX)] is exp(− 12 t2 ) for any t ∈ R. 19 It is a well known fact that two bounded random variables are the same if and only if the moment generating function of both random variables are the same on any compact 18

set containing zero on R. This holds automatically when the characteristic function is the same on the compact set containing zero on R. See p390 in Billingsley (1995).

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Absolutely continuous distribution function F (v) can be expressed as F (v) = H(G(v)) where G is an initial guess for F and H is a distribution on the unit interval. Therefore, we can estimate F by estimating H given an initial guess G. For the estimation of H, we need to approximate any density function on the unit interval by some basis. In this subsection, we show how to approximate any density function on the unit interval using the orthonormal Legendre polynomials. 2.2.1. Legendre Polynomials Legendre polynomials can be constructed recursively by Pn (z) =

(2n − 1)zPn−1 (z) − (n − 1)Pn−2 (z) n

where n ≥ 2 and P0 (z)=1, P1 (z) = z with z on [−1, 1]. Using the fact that these Pn (z)’s are orthogonal, we can construct an orthonormal Legendre polynomial ρn (u) on [0, 1] by defining ρn (u) =

√

2n + 1Pn (2u − 1).

The orthonormal Legendre polynomial ρ(u) satisfies the following property: ( Z 1 1 if j = k ρj (u)ρk (u)du = 0 otherwise. 0 Like orthogonal Legendre polynomial Pn (z), the orthonormal Legendre polynomial ρn (u) can also be constructed recursively by √ √ √ 2n − 1 2n + 1 (n − 1) 2n + 1 √ ρn (u) = (2u − 1)ρn−1 (u) − ρn−2 (u) n n 2n − 3 with ρ0 (u) = 1 and ρ1 (u) =

√

(8)

3(2u − 1).

The domain of the orthonormal Legendre polynomial ρn (u) is [0,1], hence we can approximate any absolutely continuous distribution H(G(v)) by orthonormal Legendre polynomials.

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2.2.2. Approximation of H(u) using Orthonormal Legendre Polynomials

Specify the true distribution F (v) = H(G(v)) =

Ru 0

h(t)dt where u =

G(v) on [0, 1]. Theorem 1 in Bierens (2006) states that the orthonormal Legendre polynomials on [0,1] form a complete orthonormal basis for the Hilbert space L2B (0, 1) where L2B (0, 1) is the set of square integrable Borel measurable real functions on [0, 1]. Hence, any square-integrable Borel measurable real funcP tion q(u) on [0,1] can be represented as q(u) = ∞ k=0 γk ρk (u) on [0,1] a.e. R1 where γk is a Fourier coefficient γk = 0 ρk (u)q(u)du and ρk (u) is an orthonormal Legendre polynomial. Therefore, every density function h(u) on [0,1] R1 can be written as h(u) = q(u)2 such that q(u) ∈ L2B (0, 1) and 0 q(u)2 du=1.20 To eliminate the restriction

P∞ k=0

γk2 = 1, reparameterize the γk ’s: 1 P∞

γ0 = q 1+ and for k = 1, 2, 3, . . . γk = q 1+

k=1 δk

δk P∞

k=1 δk

2

2

.

Therefore, a density function h(u) on [0, 1] can be represented as P 2 (1 + ∞ k=1 δk ρk (u)) h(u) = P∞ . 1 + k=1 δk 2

(9)

For a density function h(u) in (9) and its associated sequence {δk }∞ k=1 , there exists a truncated density hn (u) with δ n = (δ1 , . . . , δn ) : P 2 (1 + nk=1 δk ρk (u)) Pn hn (u) = hn (u|δn ) = 1 + k=1 δk 2 such that

Z

1

| hn (u) − h(u) | du = 0.

lim

n→∞ 20

(10)

0

From this integration, the restriction

P∞ k=0

11

γk2 = 1 comes out.

(11)

We will call this hn (u) in (10) a SNP density function following Gallant and Nychka (1987). The associated distribution function is Hn (u) which is called a SNP distribution function such that ¡1¢ Ru Pn 0 2 (1, δ )A (u) (1 + δ ρ (v)) dv n n+1 m m δn m=1 Pn Hn (u) = Hn (u|δn ) = 0 = , 0 2 1 + m=1 δm 1 + δn δn (12) where u ∈ [0, 1], δn = (δ1 , ..., δn )0 ∈ Rn , and  Ru   An+1 (u) =   

R0u 0 Ru 0

ρ0 (v)ρ0 (v)dv ρ1 (v)ρ0 (v)dv .. . ρn (v)ρ0 (v)dv

Ru R0u 0 Ru 0

ρ0 (v)ρ1 (v)dv · · · ρ1 (v)ρ1 (v)dv · · · .. .. . . ρn (v)ρ1 (v)dv · · ·

Let ρm (u) =

m X

Ru R0u 0 Ru 0

ρ0 (v)ρn (v)dv ρ1 (v)ρn (v)dv .. .

     

ρn (v)ρn (v)dv (13)

`m,k uk

(14)

√ `1,1 = 2 3

(15)

k=0

From (8), `0,0 = 1

√ `1,0 = − 3, For m ≥ 2,

√ √ m m−1 2 4m2 − 1 X 4m2 − 1 X k `m,k u = `m−1,k−1 u − `m−1,k uk m m k=0 k=1 k=0 √ m−2 X (m − 1) 2m + 1 √ `m−2,k uk . (16) − m 2m − 3 k=0

m X

k

Hence, we can get the recursive relation of the coefficients `m,k ’s: for 0 < k ≤ m, √ `m,k =

√ (m − 1) 2m + 1 4m2 − 1 √ (2`m−1,k−1 − `m−1,k ) − `m−2,k m m 2m − 3

12

(17)

where (15) is given and letting `m,k = 0 for (m, k) satisfying k > m or k < 0. Using this recursive relation of `m,k ’s, for 0 ≤ m ≤ n, 0 ≤ k ≤ n, we can construct a (n + 1) × (n + 1) lower triangular matrix Ln+1 with m-th row (`m,0 , . . . , `m,n ). Observe that Z

Z

u

u

ρk (v)ρm (v)dv = 0

0

k X

`k,i v

i=0

i

m X

j

`m,j v dv =

j=0

k X m X

Z

u

`k,i `m,j

v i+j dv

0

i=0 j=0

k X m X `k,i `m,j i+j+1 u = i + j + 1 i=0 j=0

= (`k,0 , . . . , `k,n )Πn+1 (u)(`m,0 , . . . , `m,n )0

(18)

ui+j+1 where Πn+1 (u) is the (n + 1) × (n + 1) matrix with (i, j)-th element . i+j+1 Ru Therefore, 0 ρk (v)ρm (v)dv is equal to the (k, m) element of the matrix, Ln+1 Πn+1 (u)L0n+1 , hence An+1 (u) = Ln+1 Πn+1 (u)L0n+1 . Thus, ¡ ¢ (1, δn 0 )Ln+1 Πn+1 (u)L0n+1 δ1n Hn (u) = Hn (u|δn ) = 1 + δn 0 δn

(19)

where u ∈ [0, 1], δn = (δ1 , . . . , δn ) ∈ Rn . Thus, we can denote SNP distribu(1,δn 0 )Ln+1 Πn+1 (G(v))L0n+1 (δ1 ) n tion estimator for F (v) by F˜n (v) = Hn (G(v)) = . 0 1+δn δn

We can get the compact set of F˜n (v) whose compactness comes from the compactness of the set Dn (0, 1) which is defined in the following way.21 Dn (0, 1) = {hn : hn (u) = |δk | ≤

√c 1+ k ln k

P δ ρ (u))2 (1+ n k=1 Pn k k2 , 1+ k=1 δk

for c > 0 for k = 1, 2, . . . , n}.

Note that the restriction of |δk | ≤ P∞ 2 k=1 δk < ∞.

√c 1+ k ln k

(20)

for k = 1, 2, . . . , n guarantees

2.3. Random Drawing of Simulated Values based on the SNP density 21

For sufficiently large c, Dn (0, 1) is compact since it is totally bounded by the condition |δk | ≤ 1+√ck ln k and every sequence δn has a convergent subsequence.

13

Through subsection 2.2-2.3, the estimation is reduced to the following problem: ˆ n (u) ≡ Hn (u|δc Fˆn (v) = H n) Z 2 1 c ˆ c such that δn = arg min |Ψ(t)| dt δ˜n 2c −c

(21)

ˆ where Ψ(t) is defined in (4) and bj comes from the true distribution F (v) = H(u) and ˜bj comes from an SNP distribution estimator F˜n = Hn (G(v)|δ˜n ). For SNP-SSMM estimation we need to generate simulated bids from an SNP distribution estimator F˜n = Hn (G(v)|δ˜n ). It can be done by 2 steps. First, we can draw simulated values randomly from an SNP distribution estimator F˜n (v) = Hn (G(v)) using an accept-reject method and then we can obtain the corresponding simulated bids by importance sampling. In this subsection, we propose an accept-reject method to draw simulated values randomly from an arbitrary SNP distribution estimator F˜n (v) = Hn (G(v)|δ˜n ) and address the importance sampling to obtain simulated bids. 3.3.1. Random Drawing of Simulated Values with Accept-Reject Method

The following proposition is an applied version of general accept-reject random drawing method. Proposition 1. Assume that ϕ(·) is a density function from which we want to draw random variables. Also assume that ψ(·) is a density function from which it is easy to draw a random variable X. The random variable from the proposed accept-reject method of (step 1) - (step 4) below then delivers the random variable which follows the distribution ϕ(·). (Step 1): Find a constant c such that ϕ(x) ≤ cψ(x) for all x. (Step 2): Draw an X0 from ψ(x). (Step 3): Draw a U from the uniform distribution on [0,1]. (Step 4): If U ≤

ϕ(X0 ) , cψ(X0 )

then set X = X0 , else draw a new pair (X0 , U )

and redo (step 1) - (step 4). 14

Proof. See Appendix.

Corollary 1. Suppose the density estimator f˜n (v) = hn (G(v))g(v) where hn (u) is in (10) and G(v) is an initial guess for F (v). Then, we can randomly draw simulated values from an estimator f˜n (v) by applying the accept-reject method in proposition 1. Proof of Corollary 1. Let ϕ(v) = h(G(v))g(v) where G(v) is a given distribution function with support (0, ∞)22 and density g(v), and h is a density function on [0,1] with its associated distribution H(u). Ra 0

ϕ(v)dv = =

Ra R0a 0

h(G(v))g(v)dv h(G(v))dG(v)

(22)

= H(G(a)) With an inverse function G−1 (u),23 we can draw v = G−1 (u) which comes from the distribution G(v) where u is drawn from a uniform distribution on [0,1]. Moreover, P 2 (1 + nk=1 δk ρk (u)) hn (u) = hn (u|δn ) = where δn = (δ1 , . . . , δn )0 Pn 2 1 + k=1 δk which is defined in (10). For each δn = (δ1 , . . . , δn )0 , we need to find a number c such that c ≥ sup hn (u|δn ) 0≤u≤1

which can be done by a grid search over [0,1]. Then, let ψ(v) = g(v), and follow the same procedure proposed in proposition 1 as follows

(Step 1) : Draw U1 from the uniform distribution on [0,1]. (Step 2) : Let V = G−1 (U1 ) 22

The support of G(v) should be at least as large as that of F (v). For example, if G(v) is an exponential distribution, i.e., G(v) = 1 − exp(−v) then, G−1 (u) = − ln(1 − u). 23

15

(Step 3) : Compute

hn (G(V ))g(V ) cg(V )

=

hn (G(V )) c

=

hn (U1 ) c

(Step 4) : Draw another U2 from the uniform distribution on [0,1], and accept V if U2 <

hn (U1 ) . c

Otherwise, reject V and redo (step 1)-(step 4). Q.E.D.

Since simulated values can be drawn from the density estimator for f (v), f˜n (v) = hn (G(v))g(v), by the method proposed in corollary 1, all that remains is to compute the corresponding simulated bid β(˜ v , F˜n ) which is, for all v˜ > 0, β(˜ v , F˜n ) = v˜ −

Z

1

v˜

I−1 F˜n (x) dx.

(23) 0 F˜n (˜ v) To obtain the simulated bid, we use the importance sampling which is adI−1

dressed in the following subsubsection. 3.3.2. Simulated Bids from Simulated Private Values via Importance Sampling.

Simulated values v˜ can be drawn from the SNP distribution estimator ˜ Fn (v) = Hn (G(v)) by the proposed accept-reject method in corollary 1 once we choose the initial guess G(v).24 Since the corresponding equilibrium bid in a simulated auction is (23), we need to deal with numerical integration. The integration term can be written as follows: #I−1 Z v Z v" ˜ 1 F (x) n I−1 F˜n (x) dx = dx. F˜n (v)I−1 0 F˜n (v) 0 Let y = xv , then Z " v

0

F˜n (x) F˜n (v)

#I−1

Z

1

dx = v 0

"

F˜n (vy) F˜n (v)

#I−1 dy.

(24)

Note y can be drawn from the uniform distribution on [0,1]. We will focus on the lth simulated auction using SNP distribution. 24

Note that the corresponding density estimator for f (v) is f˜n (v) = hn (G(v))g(v).

16

1. Draw v˜l,j , j = 1, ..., I, randomly from a SNP distribution estimator F˜n = Hn (G(v)) using the proposed accept/reject method. R 1 h F˜n (˜vy) iI−1 25 2. For each v˜l,j , we can get the integration term , v˜ 0 F˜ (˜v) dy, by n

an importance sampling: " #I−1 N0 1 X F˜n (ui v˜l,j ) v˜l,j N0 i=1 F˜n (˜ vl,j ) where ui is drawn from a uniform distribution on [0, 1], i = 1, . . . , N0 . 3. We can get the following simulated bid ˜bl,j : ˜bl,j

Note that

v˜ N10

" #I−1 N0 1 X F˜n (ui v˜l,j ) = v˜l,j − v˜l,j N0 i=1 F˜n (˜ vl,j )  " #I−1  N0 X ˜ Fn (ui v˜l,j ) 1  = v˜l,j 1 − N0 i=1 F˜n (˜ vl,j )

PN0 h F˜n (ui v˜) iI−1 i=1

F˜n (˜ v)

(25)

is an unbiased estimator for v˜

R 1 h F˜n (y˜v) iI−1 0

F˜n (˜ v)

dy.26

2.3.3. Binding Reserve Price Case In binding price case, we need to divide simulated bids into two types. One is the actual bid when the bidder’s value is greater than the seller’s reserve price p0 . The other bid is zero when the bidder’s value is lower than the seller’s reserve price. Since the number of potential bidders is known as I, we can estimate the true distribution consistently for the value greater than the reserve price p0 by redefining sample bids and simulated bids as follows. Suppose that the lth auction has Il∗ actual bids and I − Il∗ zero bids, say, bl = (bl,1 , ..., bl,Il∗ , 0l,Il∗ +1 , ..., 0I ). In the lth simulated auction, draw I simulated values randomly from an estimator F˜n . Let the vector of simulated values 25

Sometimes it is called bid shade PN0 h F˜n (ui v˜) iI−1 26 The variance of v˜ N10 i=1 is F˜ (˜ v) n

" v ˜ N0

17

R 1 h F˜n (˜vy) i2(I−1) 0

F˜n (˜ v)

· dy −

R 1 ³ F˜n (˜vy) ´I−1 0

F˜n (˜ v)

¸2 # dy

.

in the simulated lth auction be v ˜l = (˜ vl,1 , . . . , v˜l,I ). Then, the equilibrium simulated bid in the lth simulated auction is as follows Z v˜l,j 1 I−1 ˜bl,j = v˜l,j − F˜n (x) dx27 if v˜l,j ≥ p0 I−1 p0 F˜n (˜ vl,j ) (26) = 0 if v˜l,j < p0 . In computation of simulated bid in binding reserve price, the integration R v˜ I−1 part in simulated bid with simulated value v˜, ˜ 1 I−1 p0l,j F˜n (x) dx, can Fn (˜ vl,j ) PN0 h F˜n (ui v˜) iI−1 1 where ui is drawn from an uniform be computed by v˜ N0 i=1 F˜ (˜v) distribution on [0, 1].28

n

Therefore, we can redefine the vector of bids in the lth auction and the vector of simulated bids in the lth simulated auction as follows when the seller’s reserve price is binding. bl = (bl,1 , ..., bl,Il∗ , 0l,Il∗ +1 , ..., 0l,I ) where Il ∗ is the number of actual bidders in the lth auction. ∗ ˜l = (˜bl,1 , ..., ˜bl,I ) where ˜bl,i = max{β(˜ b vl,j , F˜n )I(˜ vl,j > p0 ), 0} and β(˜ vl,j , F˜n ) is defined in (23).

(27)

Throughout the paper, we mean both bid vector and simulated bid vector by these redefined bid and simulated bid vectors when a reserve price is binding.29

3. Asymptotic Property SNP-SSMM Estimator and Implementation 28

The computation for an actual bid is the same as the bid with non-binding reserve

price. 29 In non-binding reserve price case, the redefined bid vector is the same as the original bid vector since the seller’s reserve price is the lower bound of values.

18

In this section, we go over some asymptotic properties related to SNPSSMM estimator and propose a criterion function to implement the SNPSSMM estimation.

3.1. Consistency of fˆn (v) and Fˆn (v) with use of infinite dimensional Sieve ˜ n (G(v)) ≡ Hn (G(v)|δ˜n ), the subscript Regarding the notation F˜n (v) = H n denotes a truncation order of SNP density function in (10). The SNPˆ n (G(v)) = Hn (G(v)|δc SSMM distribution estimator Fˆn (v) = H n ) is such that Z κ Z κ 2 1 1 1 c ˆ δn = arg min |ΨN,n (t)| dt ≡ |WN,n (t)|2 dt (28) δn 2κ −κ N 2κ −κ where I

I

L

I

X1X 1X1X ˆ N,n (t) = 1 Ψ exp(itbl,j ) − exp(it˜bl,j ) L l=1 I j=1 L l=1 I j=1 N N 1 X 1 X exp(itbj ) − exp(it˜bj ), and N j=1 N j=1 √ ˆ N,n (t) WN,n (t) = N Ψ

=

(29) (30)

where bids bj and simulated bids ˜bj are redefined bids in (27), I is the number of potential bidders, L is the number of auctions, N = L × I and n is the dimension of δn in (10). Note that our SNP-SSMM density estimate fˆ(v) and distribution estimate Fˆ (v) can be called substitution estimate in the c c ˆ sense that fˆ(v) = hn (u|δc n )g(v) and F (v) = Hn (G(v)|δn ) where δn is defined (28) above.30 ˆ N,n (t) comes from N = L×I Since I is fixed, the asymptotic property of Ψ through L.31 You may consider the case where I goes to infinity but it is a very rare case.32 We do not consider it here. 30

Shen (1997) uses the terminology of substitution estimate. In this paper, we think of asymptotic property in terms of N . But, using L instead of N brings the same result. 32 If I goes to the infinity, bid equals the value itself. 31

19

Assumption A. The truncation order n = n(N ), which is associated with the SNP distribution Hn (u) in (12) and the SNP density hn (u) in (10), increases as N → ∞. Assumption A is a fundamental assumption in sieve estimation for the ˆ N (t) in (29) consistency. After suppressing the subscript n, we can write Ψ as follows. N N X 1 X ˆ N (t) = 1 (Zj,1 (t) − µ1 (t)) − (Zj,2 (t) − µ2 (t)) − (µ1 (t) − µ2 (t)) Ψ N j=1 N j=1

(31) where Zj,1 (t) = exp(itbj ), Zj,2 (t) = exp(it˜bj ) for j = 1, . . . , N, and µ1 (t) = E(exp(itb1 )), µ2 (t) = E(exp(it˜b1 )). Note that Zj,1 comes from the true distribution F (v) = H(u) and Zj,2 comes from an estimator F˜n (v) = Hn (u) ≡ Hn (u|δn ) and that both are independent. The following proposition shows that the estimator from (28) is consistent. ˆ N,n (t)| Proposition 2. Under assumption A, the moment condition function |Ψ P

−→ 0 as N → ∞ for each t ∈ R if and only if the distribution of simulated bids is the same as the true distribution function of bids. Proof. See Appendix. The consistency of the density estimator fˆn (v) can be verified by using a L1 semi-metric which is defined on the particular density function space which consists of square integrable Borel measurable functionsPon [0,1]. Suppose (1+

∞

δj ρj (G(v)))2

j=1 P that the true density function is f (v) = h(G(v))g(v) = g(v). 2 1+ ∞ j=1 δj ˆ n (G(v))g(v) ≡ Then, the SNP-SSMM density estimator for f (v) is fˆn (v) = h

c hn (G(v)|δc n )g(v) where δn is from (28). Corollary 2. Suppose F (v) = H(G(v)) and SNP-SSMM estimator distribˆ n (G(v)) ≡ Hn (G(v)|δc ution Fˆ (v) = H n ) from (28) where G(v) is an initial 20

guess for the true distribution. Under assumption A, the density estimator ˆ n (G(v))g(v) ≡ hn (G(v)|δc fˆn (v) = h n )g(v) is pointwise consistent for the true density f (v) = h(G(v))g(v). Proof. See Appendix.

Corollary 3. Suppose F (v) = H(G(v)) and SNP-SSMM distribution estiˆ n (G(v)) ≡ Hn (G(v)|δc mator Fˆ (v) = H n ) from (28) where G(v) is an initial guess for the true distribution. Under assumption A, the distribution esˆ n (G(v)) is uniformly consistent for the true distribution timator Fˆ (v) = H F (v) = H(G(v)). Proof. See Appendix. P Corollary 3 immediately implies that Fˆn (v) −→ F (v) for each v.

3.2. Construction of a Test Statistic TˆN,n To construct a test statistic, we consider the asymptotic property of √ ˆ N,n (t) = N Ψ ˆ N,n (t) in (29). W ˆ N,n (t) ≡ 3.2.1. Asymptotic Property of W √ ˆ N,n (t) = N Ψ ˆ N,n (t) by Define W

√

ˆ N,n (t) NΨ

N N 1 X 1 X ˆ WN,n (t) = √ exp(itbj ) − √ exp(it˜bj ) N j=1 N j=1

(32)

where bj is a sample bid and ˜bj is a simulated bid and N = L × I 33 . If the reserve price is binding, these bids are redefined bids in (27). Note that the bids and simulated bids are independent across auctions as well as within each auction whether an arbitrary distribution estimator ˆ N,n (t) as follows F˜ (v) = Hn (G(v)|δn ) is true or not. Rearrange W ˆ N,n (t) = W ˆ N,n,1 (t) − W ˆ N,n,2 (t) + W 33

√

N (E[exp(itb1 )] − E[exp(it˜b1 )])

Throughout, L is the number of auctions and I is the number of potential bidders.

21

N 1 X ˆ where WN,n,1 (t) = √ (exp(itbj ) − E[exp(itb1 )]), N j=1 N X ˆ N,n,2 (t) = √1 W (exp(it˜bj ) − E[exp(it˜b1 )]). N j=1

Note that the bids bj ’s come from the true distribution F (v) = H(u) and simulated bids ˜bj ’s come from the distribution estimator F˜n (v) = Hn (u) ≡ Hn (u|δn ) where u = G(v). Under the null hypothesis Hnull : H(u) = Hn (u), ˆ N,n (t) follows a complex-valued Gaussian process Z(t) an empirical process W which is completely characterized by its covariance Γ(t1 , t2 ): i h ˆ ˆ Γ(t1 , t2 ) = E WN,n (t1 )WN,n (t2 )

(33)

Now, suppose that a compact subset K in R is [−κ, κ]. Then, we can have a metric space of continuous functions on K = [−κ, κ], C(K), which has a metric ρ(x, y) = sup |x(t) − y(t)| for x, y ∈ C(K). t∈K

ˆ N,n (t) is an random element on C(K), we can consider the weak Since W ˆ N,n (t) on the metric space C(K). It follows from theorem convergence of W ˆ N,n (t) for the weak 7.1 in Billingsley (1999) that we need a tightness of W ˆ N,n =⇒ Z in the metric space C(K). convergence W ˆ N by suppressing the subscript n in W ˆ N,n . For simpler notation, we use W ˆ N has a sequence of the corresponding probability Lemma 1. A sequence of W measure PN . Then, the sequence {PN } is tight. Proof. See Appendix.

3.2.2. Test Statistic Therefore, we can propose a test statistic as follows ¯2 R κ ¯¯ 1 ˆ N,n (t)¯¯ dt W ¯ 2κ −κ TˆN,n = Rκ 1 ˆ t)dt Γ(t, 2κ −κ N N 1 X 1 X ˆ where WN,n (t) = √ exp(itbj ) − √ exp(it˜bj ). N j=1 N j=1

22

(34)

Note that the integrated variance is introduced in (34) for standardization of the value. E|b|k < ∞ for all k ∈ Z+ since the values distribution is ˆ N,n (t) =⇒ Z(t). Moreover, assumed to have a finite expectation.34 Then, W the statistic TˆN,n in (34) has the following property: Rκ 1 |Z(t)|2 dt under H0 TˆN,n =⇒ 2κ1 R−κ κ Γ(t, t)dt 2κ −κ

(35)

which comes from the continuous mapping theorem. Using the test statistic in (34), we can construct a new criterion function as follows: TˆN,n δc ≡ n = arg min δn N

¯2 R κ ¯¯ ¯ ˆ W (t)¯ dt −κ ¯ N,n . Rκ 1 ˆ t)dt Γ(t, 2κ −κ

1 1 N 2κ

(36)

Note that TˆN,n is a function of δn since the simulated bids come from the distribution estimator F˜n (v) = Hn (u|δ˜n ) in (19). Proposition 3. Under the null hypothesis that sample bids and simulated bids have the same distribution, the test statistic

TˆN,n N

goes to zero in proba-

bility while it goes to some positive number when the null hypothesis is not true. Proof. See Appendix. We need to estimate 34

1 2κ

¯2 R κ ¯¯ ˆ N,n (t)¯¯ dt and W ¯ −κ

1 2κ

Rκ −κ

E(v) < ∞ implies β(v) < ∞. See Bierens and Song (2006).

23

ˆ t)dt consistently. Γ(t,

1 2κ

¯2 R κ ¯¯ ¯ ˆ W (t) ¯ dt can be computed as follows: −κ ¯ N,n Z κ¯ ¯2 1 ¯ˆ ¯ ¯WN,n (t)¯ dt 2κ −κ ¸ N N · sin (κ × (bj1 − bj2 )) 1 XX = I(bj1 = bj2 ) + I(bj1 6= bj2 ) × N j =1 j =1 κ(bj1 − bj2 ) 1 1 ³ ´  ˜bj ) N X N sin κ × (b − X j 1 2 2  I(bj1 = ˜bj2 ) + I(bj1 6= ˜bj2 ) × − ˜ N j =1 j =1 κ(b − b ) j1 j2 1 2 ³ ´  ˜bj − ˜bj ) N X N sin κ × ( X 1 2 1 I(˜bj1 = ˜bj2 ) + I(˜bj1 6= ˜bj2 ) × . + ˜bj − ˜bj ) N j =1 j =1 κ( 1 2 1 1

(37)

ˆ t) can be See Appendix for the derivation of (37). The variance function Γ(t, estimated consistently as follows N N i 1 XXh ˆ cos(t(bj1 − bj2 )) + cos(t(˜bj1 − ˜bj2 )) Γ(t, t) = 2 − 2 (38) N j =1 j =1 1 2 Rκ 1 ˆ t)dt is Γ(t, The integrated variance 2κ −κ Z κ 1 ˆ t)dt Γ(t, 2κ −κ ¸ N N · 1 XX sin(κ(bj1 − bj2 )) = 2− 2 I (bj1 = bj2 ) + I(bj1 6= bj2 ) N j =1 j =1 κ(bj1 − bj2 ) 1 2 " # N N ´ ˜bj − ˜bj ) 1 X X ³˜ sin(κ( 1 2 I bj1 = ˜bj2 + I(˜bj1 6= ˜bj2 ) − 2 ) (39) ˜ ˜ N j =1 j =1 κ( b − b ) j j 1 2 1 2 R κ ˆ t)dt, see Appendix. ˆ t) and 1 Γ(t, For derivation of Γ(t, 2κ −κ We can infer the statistic TˆN,n based on the following proposition. Proposition 4. Under the null hypothesis, the statistic TˆN,n = converges in distribution to a random variable T where T = E(T ) = 1 where pj =

π P∞ j

j=1

πj

1 2κ

Rκ

2 ˆ −κ WN,n (t) dt Rκ 1 ˆ Γ(t,t)dt 2κ −κ

P∞

j=1

|

|

pj ε2j and

> 0, πj ’s are eigenvalues of the covariance func-

tion Γ(t, t) and εj ∼ i.i.d. N (0, 1). But, under the alternative hypothesis, TˆN,n N

P

−→ η(n) > 0.

Proof. See Appendix.

24

3.3.

Truncation Order Selection and Consistency of SNP-SSMM Estimators fˆnˆ (v), Fˆnˆ (v) The implementation of SNP-SSMM with infinite dimensional parameters is not possible. In this subsection, we propose a practical way to select a truncation order for SNP-SSMM estimation. In order to do it, first we need to relax the assumption A by replacing it with assumption A’. Assumption A’. The truncation order in SNP density n is a non-decreasing function of sample size N . We also need to assume that the true distribution F (v) = H(G(v)) can be approximated with a SNP distribution which has a finite truncation order of orthonormal Legendre polynomials. Assumption B. SNP density hn0 (u) approximates the true density h(u) with L1 semi-metric within ε0 > 0 in the sense that for some preassigned R1 error ε0 > 0, 0 |h(u) − hn0 (u)|du < ε0 where n0 is unknown. Since the preassigned error can be close to zero, we treat the SNP density hn0 and SNP distribution Hn0 as the true density and distribution respectively. In practice, we use SNP density estimator with arbitrary truncation order n and propose a method to select an appropriate truncation order. Moreover, we assume that if the truncation order is specified to be at least n0 , the approximation is good while the approximation is not good when the truncation order is below n0 . Based on the statistic TˆN,n , we can determine the polynomial order following the information criteria such as Schwarz (1978) and Hannan and Quinn (1979). Assumption C. There is a true finite truncation order of SNP density, n0 such that F (v) = Hn (G(v))+op (1) and f (v) = hn (G(v))g(v)+op (1) provided if n ≥ n0 .

25

Note that assumption C can be justified by setting very small approximation error ε0 in assumption B since ||h(u) − hn (u)||1 = ||h(u) − hn0 (u) + hn0 (u) − hn (u)||1 ≤ ||h(u) − hn0 (u)||1 + ||hn0 (u) − hn (u)||1 P

P

−→ ε0 provided ||hn0 (u) − hn (u)||1 −→ 0

(40)

which will be shown in this subsection soon. In practice, the approximation seems very good with a very small truncation order from the results in our Monte Carlo experiments. ˆN (n) by Define a criterion function C φ(N ) TˆN,n +n CˆN (n) = N N

(41)

where TˆN,n is defined in (34), N = L × I, n is the dimension of δn in SNP density and the penalty term satisfies two conditions:

φ(N ) N

→ 0 and

φ(N ) → ∞ as N → ∞. Lemma 2. The criterion function CˆN (n) in (41) gives a consistent order selection which is robust either to a misspecified case(n < n0 ) or overparameterized case(n > n0 ) under assumption A’ and assumption C. Proof. See Appendix. By increasing truncation order n gradually of the criterion function CˆN (n) in (41), we can find a true truncation order n0 with probability one. That is, n ˆ = max{n}

(42)

subject to min CˆN (n) < min CˆN (n − 1), n ≥ 2, and δn

min δ1

δn−1

TˆN,0 TˆN,1 < min . δ0 N N

Then, limN →∞ P [ˆ n = n0 ] = 1. In practice, we use φ(N ) = ln N/2 and φ(N ) = ln(ln N ) following Schwarz (1978) and Hannan-Quinn (1979).

26

Proposition 5. Suppose assumption A’ and assumption C hold. Denote ˆ nˆ (G(v))g(v) ≡ hnˆ (G(v)|δc SNP-SSMM density estimator by fˆnˆ (v) = h n ˆ )g(v) where n ˆ is from (42), G(v) is an initial guess for the true distribution and g(v) is an associated density. Then, the density estimator fˆnˆ (v) is pointwise consistent for the true density f (v) = hn0 (G(v))g(v) + op (1). Proof. See Appendix.

Proposition 6. Suppose assumption A’ and assumption C hold. Denote ˆ nˆ (G(v)) ≡ Hnˆ (G(v)|δc SNP-SSMM distribution estimator by Fˆnˆ (v) = H n ˆ) from (42) where G(v) is an initial guess for the true distribution. Then, the distribution estimator Fˆnˆ (v) is uniformly consistent for the true distribution F (v) = Hn0 (G(v)) + op (1). Proof. See Appendix.

3.3.1. Binding Reserve Price Case

Every consistency argument in a non-binding reserve price case is carried over to the binding reserve price case with simple modification. 35 The only ˆ nˆ (u) = Hnˆ (u|δc difference is the limiting distribution of H n ˆ ) is not unique since it is not identified on [0, G(p0 )].

4. Monte Carlo Experiments Note that we have to use a relationship sup |F (v) − Fˆnˆ (v)| = sup |H(u) − v≥p0 u∈[G(p0 ),1] ¯ ¯ R1 ¯ ˆ nˆ (u)¯¯ du when the seller’s reserve price is binding. But, Hnˆ (u)| ≤ G(p0 ) ¯h(u) − h ¯ ¯ R1 ¯ ˆ nˆ (u)¯¯ du < the essence does not change because of the relationship G(p0 ) ¯h(u) − h ¯ R 1 ¯¯ ˆ nˆ (u)¯¯ du. − h ¯h(u) 0 35

27

In this section, we apply our proposed SNP-SSMM estimation to Monte Carlo experiments. First, we generate 200 auctions independently and each auction consists of 5 bids whose private values come from a chi-square distribution with degree of freedom R. We treat these 1000 bids as sample bids. We have 3 experiments where R = 3, 4 and 5. The environment in each experiment can be described as follows. Environment of Experiment. i. There are 200 independent identical auctions.(L = 200) ii. Assume that there are 5 potential bidders in each auction.(I = 5). iii. Assume the seller’s reserve price is zero. iv. Set κ = 1 where κ is in (36). v. The true distribution is chi-square distribution with degree of freedom R. We have 3 separate experiments with R = 3, 4 and 5.

4.1. Procedures of SNP-SSMM Estimation. Under the experiment environment, we can have 1000 observed bids in each example. Given these bids, we can recover the true value distribution with SNP-SSMM estimation. We use the following procedure. (1) Choose G(v) among exponential distribution family. • See the shape of bid distribution in figure 2, figure 4 and figure 6, then choose the initial distribution G(v) whose support is presumed to be sufficiently larger than that of true values distribution.36 In this experiment, we will use G2 (v) = 1 − exp(− 12 v) and G3 (v) = 1 − exp(− 31 v). 36

Since the bid distribution has a finite support as long as the value distribution has a

finite expectation, it is desirable to choose an initial distribution which has a sufficiently large support.

28

From the comparison from both results, we can confirm the role of the initial distribution G(v).37 (2) Begin with the truncation order n = 1. (3) For given δ n , we can find the corresponding simulated bids via SNP density and SNP distribution by our proposed accept-reject method and importance sampling. (4) For given δ i = (δ1i , δ2i , . . . , δni ), we have the corresponding value of our criterion function CˆN (n). The optimal δc n can be found via simplex method.38 We use three types of CˆN (n) =

TˆN,n N

) + n φ(N : (a). No N

penalty case with φ(N ) = 0, (b). Exogenous moderate penalty with φ(N ) =

ln N , 2

(c). Exogenous soft penalty with φ(N ) = ln(ln N ).

(5) Increase the truncation order to n + 1 and repeat (2) - (4) up to some orders. 4.1.1. Some Discussion about the Initial Distribution G(v)

For SNP-SSMM estimation, we need to choose an appropriate initial guess G(v) for the true distribution. The support of the initial distribution should be larger than that of the distribution of interest. Otherwise, some values can not be generated via G(v) even though they can come from the true distribution F (v). Note that the appropriate initial guess can deliver a consistent SNP-SSMM density estimator with a smaller truncation order.39 Regarding 37

In this example, it is clearly undesirable to choose an exponential distribution with mean 1 for the case of R = 4 and R = 5 since its density vanishes very quickly beyond 5, but there are some bids greater than 5 in both examples. 38 We use the Nelder-Mead Algorithm. 39 By this reason, the choice G(v) may affect the convergence rate of fˆn (v).

4.2. Estimation Results depending on G(v). We have 3 cases where the bids come from a chi-square distribution of d.f. R with R = 3, 4, 5.

29

the selection of truncation order, both from using G2 (v) and G3 (v) suggests a small truncation order. See table 1 below. Table 1: Selected Truncation Order n ˆ depending on (R, G(v)) R=3 R=4 R=5 G(v) = G2 (v)

1

2

3

G(v) = G3 (v)

2

2

1

From the graph of density estimates, the both density estimates from G2 (v) and G3 (v) looks good when R = 3. See figure 8 and 11. But, when R = 4 or 5, density estimates from G3 (v) looks better than those from G2 (v). See figure 9 vs. figure 12 and figure 10 vs. figure 13 respectively. This implies that it is desirable to select an initial distribution G(v) whose support is sufficiently large. In this sense, the choice of G3 (v) is better than the choice of G2 (v).40

5. Concluding Remarks In this paper we propose a SNP-SSMM estimation to estimate the value distribution of the first-price auction based on the identification results in Bierens and Song (2006). Our SNP-SSMM estimation is a new nonparametric estimation approach in that most of non-parametric estimation approaches use a kernel. For general nonparametric approaches, see Athey and Haile (2005). Particularly, the integrated conditional moment using empirical characteristic function of bids and simulated bids delivers the consistent density estimator. Our proposed SNP-SSMM estimation can be extended to the case beyond the ideal situation. Particularly, we can think of a situation where there is some exogenous variation in the number of potential bidders even when identical auctions are repeated independently. Then, we can implement the 40

For comparison of exponential distribution and chi-square distribution, see figure 7.

30

ˆ F˜ ): SNP-SSMM estimation by adjusting the objective function as follows: Ω( Z ˆ ˜ ˆ F˜ )|2 w(ξ)dt Ω(F ) = |Ψ(ξ| (43) R2

where " I # Il L l X X X 1 ˆ F˜ ) = exp(it1 bl,j ) − exp(it1˜bj ) exp(it2 Il ) Ψ(ξ| TL l=1 j=1 j=1 where TL =

PL

l=1 Il ,

ξ = (t1 , t2 ) ∈ R2 , w(ξ) =

√

1 2 2π

(44)

exp(− 12 ξ 0 ξ), L is the

number of auctions and simulated auctions, Il is the number of potential bids in the lth auction and lth simulated auction. We can also define a criterion function on some compact set [−κ, κ]2 ⊂ R2 using uniform density w(ξ) = 1 (2κ)2

like (28).

We have to consider introducing heterogeneity into the auction model since it is rare to have independent replications of the identical auctions,. Based on the identification results of first-price auctions with auction-specific heterogeneity in Bierens and Song (2006), we can propose a SNP-SSMM estimation for such situation.

31

References Athey, S. and P. A. Haile (2002), “Identifications of Standard Auction Models”, Econometrica, 70(6), 2107-2140. Athey, S. and P.A. Haile (2005), “Nonparametric Approaches to Auctions ”, Handbook of Econometrics, Vol 6 (forthcoming) Bierens, H.J. (1990), “A Consistent Conditional Moment Test of Functional Form”, Econometrica, 58(6), 1443-1456. Bierens, H.J. (1994), Topics in Advanced Econometrics, Cambridge University Press. Bierens, H.J. and W. Ploberger (1997), “Asymptotic Theory of Integrated Conditional Moment Tests”, Econometrica, 65(5), 1129-1151. Bierens, H.J. (2004), Introduction to the Mathematical and Statistical Foundations of Econometrics, UK: Cambridge University Press. Bierens, H. J. (2006), ”Semi-Nonparametric Interval Censored Mixed Proportional Hazard Models: Identification and Consistency Results. Working paper(http://econ.la.psu.edu/˜hbierens/snpmphm.pdf). Bierens, H.J. and H. Song (2006), “Nonparametric Identification of FirstPrice Auction Models with Unbounded Values and Observed Auction-Specific Heterogeneity”, Working Paper, Pennsylvania State University Billingsley, P. (1999), Convergence of Probability Measures, John Wiley & Sons, Inc. Billingsley, P. (1995), Probability and Measure(3rd ed.), John Wiley & Sons, Inc. Chen, C. , “Large Sample Sieve Estimation of Semi-Nonparametric Models”, Handbook of Econometrics Vol 6, forthcoming. 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. Gallant, A. R. and R. W. Nychka. (1987) “Semi-Nonparametric Maximum Likelihood Estimation”, Econometrica, 55(2), 363-390. Guerre, E., I. Perrigne, and Q. Vuong. (2000) “Optimal Nonparametric Estimation of First-Price Auction”, Econometrica, 68, 525-574. Hannan, E. J. and B. G. Quinn. (1979) “The Determination of the Or32

der of an Autoregression”, Journal of the Royal Statistical Society. Series B(Methodological), 41(2), 190-195. Krishna, V., Auction Theory (2002), Academic Press. Laffont, J. J. and Q. Vuong (1993), ”Structural econometric analysis of descending auctions”, European Economic Review, 37, 329-341. Laffont, J. J., H. Ossard and Q. Vuong (1995), ”Econometrics of FirstPrice Auctions”, Econometrica, 63, 953-980. Li, T.(2005) “Econometrics of first-price auctions with entry and binding reservation prices”, Journal of Econometrics, 126, 173-200. Press,W.H., B.P.Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipe (1986), Cambridge University Press. Rao, B.L.S. Prakasa, Nonparametric Functional Estimation (1983), Academic Press. Riley, G.J., W.F.Samuelson. (1981) “Optimal Auctions”, American Economic Review, 71, 381-392. Roehrig, C.S. (1988) “Conditions for identification in Nonparametric and Parametric Models”, Econometrica, 56, 433-447. Rubinstein, R.Y., Simulation and The Monte Carlo Method (1981), John Wiley & Sons, Inc. Schwarz, Gideon (1978), “Estimating the Dimension of a Model”, Annals of Statistics, 6(2), 461-464. Shen, Xiaotong (1997), “On Methods of Sieves and Penalization ”, Annals of Statistics,25(6), 2555-2591. van der Vaart, A.W. and J. A. Wellner, Weak Convergence and Empirical Processes with Applications to Statistics (1996), Springer. van der Vaart, A.W., Asymptotic Statistics (1998), Cambridge University Press.

33

APPENDIX Derivation of Equation (6). Z ∞ ˆ ˜ ˆ | F˜ ) |2 w(t)dt Ω(F ) = | Ψ(t Z−∞ ∞ ˆ | F˜ )Ψ(t ˆ | F˜ ) √1 exp(− 1 t2 )dt = Ψ(t 2 2π −∞ Z ∞X N X N 1 1 1 = 2 exp(it(bj1 − bj2 )) √ exp(− t2 )dt N −∞ j =1 j =1 2 2π 1 2 Z ∞X N X N 1 1 1 exp(it(bj1 − ˜bj2 )) √ exp(− t2 )dt − 2 N −∞ j =1 j =1 2 2π 1 2 Z ∞X N X N 1 1 1 − 2 exp(it(˜bj1 − bj2 )) √ exp(− t2 )dt N −∞ j =1 j =1 2 2π 1 2 Z ∞X N X N 1 1 1 + 2 exp(it(˜bj1 − ˜bj2 )) √ exp(− t2 )dt N −∞ j =1 j =1 2 2π 1

1 = 2 N −

1 N2

2

N X N X j1 =1 j2 N X

N N 1 1 1 XX 2 2 exp(− (bj1 − bj2 ) ) − 2 exp(− (bj1 − ˜bj2 ) ) 2 N j =1 j =1 2 =1 1

N X

j1 =1 j2

1 1 2 exp(− (˜bj1 − bj2 ) ) + 2 2 N =1

2

N X N X j1 =1 j2

1 2 exp(− (˜bj1 − ˜bj2 ) ) 2 =1 Q.E.D.

34

Proof of Proposition 1. ¯ ¸ ϕ(X0 ) P[X0 ≤ a, U ≤ cψ(X ] ¯ ϕ(X0 ) 0) ¯ P[X ≤ a] = P X0 ≤ a ¯U ≤ = ϕ(X0 ) cψ(X0 ) P[U ≤ cψ(X ] 0) ·

h i i h ϕ(X0 ) ϕ(X0 ) E I(X0 ≤ a)I(U ≤ cψ(X ) E I(X ≤ a) 0 cψ(X0 ) 0) h i h i = = ϕ(X0 ) ϕ(X0 ) E I(U ≤ cψ(X ) E cψ(X0 ) 0) i h Ra ϕ(X0 ) E I(X0 ≤ a) ψ(X 0) −∞ h i = R∞ = ϕ(X0 ) E ψ(X0 ) −∞ Ra = R−∞ ∞ −∞

ϕ(x)dx ϕ(x)dx

Z

ϕ(x) ψ(x)dx ψ(x) ϕ(x) ψ(x)dx ψ(x)

a

=

ϕ(x)dx −∞

Q.E.D. Proof of Proposition 2. ¯ ¯ P ¯ˆ ¯ Denote ¯ΨN (t)¯ = |A1 (t) − A2 (t) − A3 (t)| where A1 (t) = N1 N j=1 (Zj,1 (t) − P N µ1 (t)), A2 (t) = N1 j=1 (Zj,2 (t) − µ2 (t)) and A3 (t) = µ1 (t) − µ2 (t). i. sufficiency(⇒) ¯ ¯ ¯ ¯2 ¯ˆ ¯ P ¯ˆ ¯ P Since ¯Ψ (t) −→ 0, Ψ (t) ¯ ¯ ¯ −→ 0 which is the equivalent to the folN N lowing statement that we need to show: A1 (t)2 + A2 (t)2 + A3 (t)2 − (A1 (t)A2 (t) + A1 (t)A2 (t)) P

+(A2 (t)A3 (t) + A2 (t)A3 (t)) − (A3 (t)A1 (t) + A3 (t)A1 (t)) −→ 0. P

(45) P

By weak law of large numbers(WLLN), A1 (t) −→ 0 and A2 (t) −→ 0. P

Therefore, (45) becomes the statement that A3 (t)2 −→ 0 which implies that P

|A3 (t)| −→ 0. Since µ1 (t) and µ2 (t) are non-random, |A3 (t)| = |µ1 (t) − P

µ2 (t)| −→ 0 implies that µ1 (t) = µ2 (t) for all t. Hence, it implies that the ˜ are equivalent. distribution of random variables bid B and simulated bid B ii. necessity(⇐) 35

For all ε > 0 and for all t ∈ R, h i ˆ N (t)| > ε = P [|A1 (t) − A2 (t) − A3 (t)| > ε] P |Ψ h h h εi εi εi ≤ P |A1 (t)| > + P | − A2 (t)| > + P | − A3 (t)| > 3 3 h 3 h h εi εi εi + P |A2 (t)| > + P |A3 (t)| > = P |A1 (t)| > 3 3 3 (46) where A1 (t) =

1 N

PN

j=1 (Zj,1 (t)

− µ1 (t)), A2 (t) =

1 N

PN

j=1 (Zj,2 (t)

− µ2 (t)) and

A3 (t) = µ1 (t) − µ2 (t). It suffices to show the last equation in (46) converges to zero. If the distribution of simulated bids and bids are the same, µ1 (t) = µ2 (t) for all t, and thus |A3 (t)| = 0 for all t. Moreover, WLLN implies that h h εi εi −→ 0 and P |A2 (t)| > −→ 0 as N → ∞. P |A1 (t)| > 3 3

(47)

Therefore, the last equation in (46) converges to zero as N → ∞. Hence, P ˆ N,n (t)| −→ |Ψ 0. Q.E.D. Proof of Corollary 2. Suppose that the space of SNP density functions on [0,1] consists of square integrable Borel measurable functions on [0,1] where L1 semi-metric is deP ˆj ρj (G(v)))2 (1+ n δ j=1 ˆ n (G(v))g(v) = hn (G(v)|δc Pn g(v) fined. Note that fˆn (v) = h n )g(v) = 2 and f (v) = h(G(v))g(v) =

1+

P 2 (1+ ∞ j=1 δj ρj (G(v))) P∞ g(v). 2 1+ j=1 δj

ˆ

j=1 δj

ˆ n (G(v))g(v)||1 ||f (v) − fˆn (v)||1 = ||h(G(v))g(v) − h ˆ n (u)g(G−1 (u))||1 = ||h(u)g(G−1 (u)) − h Z 1 ˆ n (u))g(G−1 (u))|du = |(h(u) − h 0

µZ

1

≤

ˆ n (u)|2 du |h(u) − h

¶ 21 µZ

1

−1

2

¶ 12

|g(G (u))| du 0

0

(48) where u = G(v). 36

¯ ¯ ¯¯ Pn ˆ (1+ δj ρj (u))2 ¯ ¯ ˆ Pn ˆ2 Note that ¯h(u) − hn (u)¯ = ¯¯ 1+j=1 − δ j=1 j

¯

P 2¯ (1+ ∞ j=1 δj ρj (u)) ¯ P∞ 2 ¯ 1+ j=1 δj

P

−→ 0 as N →

∞ by assumption A. This implies that the last equation in (48) converges to R1 zero in probability as N → ∞ since 0 |g(G−1 (u))|2 du is bounded. Hence, P fˆn (v) −→ f (v) pointwise as N → ∞.41 Q.E.D. Proof of Corollary 3. When the initial guess distribution G(v) is given, we can have a one-toone relationship H(u) = F (G−1 (u)), equivalently, F (v) = H(G(v)) = H(u). Note the SNP distribution H is the distribution function on the unit interval [0,1]. sup |F (v) − Fˆn (v)| = sup |H(u) − Hn (u)| v≥v

u∈[0,1]

Z

¯ ¯ ¯ ¯ ˆ ≤ ¯h(u) − hn (u)¯ du 0 ¯ Pn ˆ P∞ Z 1 ¯¯ 2¯ 2 (1 + δ ρ (u)) (1 + δ ρ (u)) ¯ ¯ j=1 j j j=1 j j P∞ = − ¯ ¯ du Pn ˆ2 2 ¯ 1 + j=1 δj 1 + j=1 δj 0 ¯ 1

(49) ¯ P ¯ (1+ n δˆj ρj (u))2 Pn ˆ2 − Note ¯¯ 1+j=1 j=1 δj

¯

P 2¯ (1+ ∞ j=1 δj ρj (u)) ¯ P∞ 2 ¯ 1+ j=1 δj

P

−→ 0 as N → ∞ by assumption B.

P P Hence, sup |H(u) − Hn (u)| −→ 0 as N → ∞. Therefore, Fˆn (v) −→ F (v)

u∈[0,1]

uniformly on [v, ∞).

Q.E.D.

Proof of Lemma 1. Following the Lemma 4 in Bierens (1990), we need to show that the following two conditions hold: (i) For each δ > 0 and an arbitrary t0 ∈ T , there exists an ε such that ˆ N (t0 ) > ε) ≤ δ sup P (W N

and 41

We use the simple inequality: If |A| ≤ |B| , then for all ε > 0, P[|A| > ε] ≤ P[|B| > ε]. In addition, if P[|B| > ε] → 0, then P[|A| > ε] → 0.

37

(ii) for each δ > 0 and ε > 0, there exists an ξ > 0 such that à ! ˆ N (t1 ) − W ˆ N (t2 )| ≥ ε ≤ δ. sup P sup |W N

|t1 −t2 |<ξ

ˆ N (t) converges to a normal distribCondition (i) comes from the fact that W ution for each t ∈ T . Condition (ii) comes from the Chebishev’s inequality: Ã ! ˆ N (t1 ) − W ˆ N (t2 )| Ã ! E sup |W ˆ N (t1 ) − W ˆ N (t2 )| ≥ ε sup |W

P

≤

|t1 −t2 |<ξ

|t1 −t2 |<ξ

|ε|

Note that ˆ N (t1 ) − W ˆ N (t2 )| sup |W |t1 −t2 |<ξ

=

N ¯ ¯ 1 X ¯ ¯ sup ¯(exp(it1 bj ) − exp(it1˜bj )) − (exp(it2 bj ) − exp(it2˜bj ))¯ N j=1 |t1 −t2 |<ξ

N ∞ ¯o X 1 n¯¯ 1 X ¯ sup = ¯(it1 )k (bkj − ˜bkj ) − (it2 )k (bkj − ˜bkj )¯ N j=1 |t1 −t2 |<ξ k=0 k! N ∞ ¯o X 1 X 1 n¯¯ k k ˜k k k ¯ = sup ¯(i) (bj − bj )(t1 − t2 )¯ , N j=1 |t1 −t2 |<ξ k=0 k!

Moreover note that ∞ ¯o X 1 n¯¯ k k ˜k k ¯ sup ¯(i) (bj − bj )(t1 − tk2 )¯ k! |t1 −t2 |<ξ k=0 ∞ X ¯o 1 n¯¯ k ¯¯ ¯¯ k ˜k ¯¯ ¯¯ k ≤ sup (i) ¯(bj − bj )¯ (t1 − tk2 )¯ |t1 −t2 |<ξ k=0 k! ) ( ∞ X ¯ ¯ k 1 ¯¯ k ¯¯ ¯¯ k ˜k ¯¯ (i) ¯(bj − bj )¯ sup ¯(t1 − tk2 )¯ ≤ k! |t1 −t2 |<ξ k=0

38

and that ¯ ¯ sup ¯(tk1 − tk2 )¯ =

|t1 −t2 |<ξ

≤

¯ ¯ sup ¯(t1 − t2 + t2 )k − tk2 ¯

|t1 −t2 |<ξ

k µ ¶ X k j=1

j

sup |(t1 − t2 )|j sup |t|k−j t∈T

|t1 −t2 |<ξ

µ

¶k

≤ ξ 1 + sup |t|

.

(50)

t∈T

Hence, "

# ˆ N (t1 ) − W ˆ N (t2 )| sup |W

E

|t1 −t2 |<ξ ∞ N 1 1 XX 1 ≤ |ε| N j=1 k=0 k! N ∞ 1 1 XX 1 ≤ |ε| N j=1 k=0 k!

Setting δ =

ξ 1 |ε| N

∞ N P P j=1 k=0

( 1 k!

( (

) ¯ ¯ k ¯ ¯¯ k ¯ ¯ ¯i ¯ E ¯(bj − ˜bkj )¯¯ sup ¯(tk1 − tk2 )¯ |t1 −t2 |<ξ

¶k ¯µ ¯ k ¯ ¯¯ k ¯i ¯ E ¯(bj − ˜bkj )¯¯ 1 + sup |t|

) ξ.

t∈T

¶k ¯ ¯µ ¯ k ˜k ¯ E ¯(bj − bj )¯ 1 + sup |t|

) completes the proof

t∈T

of (ii). Q.E.D. Proof of Proposition 3. Under the null hypothesis

ˆ N,n (t) W √ N

P

ˆ W

(t)

P

√ −→ 0 while N,n −→ η(t) > 0 unN ˆ der the alternative hypothesis. Moreover, Γ(t, t) = Op (1) and positive and R Rκ ˆ t)dt converges to κ Γ(t, t)dt which is also positive. Γ(t, Q.E.D. −κ −κ

Derivation of 1 2κ

Z

1 2κ

¯2 R κ ¯¯ ¯ ˆ WN,n (t)¯ dt. −κ ¯

à N !¯2 Z κ ¯¯ N ¯2 ¯ ¯ X X 1 ¯ ¯ 1 ¯ˆ ¯ ˜ exp(itΦ(bj )) − exp(itΦ(bj )) ¯ dt. ¯√ ¯WN,n (t)¯ dt = ¯ 2κ −κ ¯ N l=1 −κ j=1 κ

(51)

39

Note that ¯2 ¯ N N ¯ 1 X ¯ X 1 ¯ ¯ ˜ exp(itbj ) − √ exp(itbj )¯ ¯√ ¯ N ¯ N j=1 j=1 N N N N 1 XX 1 XX exp(it[bj1 ) − bj2 ]) − exp(it[bj1 − ˜bj2 ]) = N j =1 j =1 N j =1 j =1 1

−

1 N

2

1

2

N X N X

N X N X

j1 =1 j2

j1 =1 j2 =1

1 exp(it[˜bj1 ) − bj2 ]) + N =1

exp(it[˜bj1 − ˜bj2 ])

(52)

and note that exp(it[bj1 − bj2 ]) = cos(t[bj1 − bj2 ]) + i sin(t[bj1 − bj2 ]) exp(it[bj1 − ˜bj2 ]) = cos(t[bj1 − ˜bj2 ]) + i sin(t[bj1 − ˜bj2 ]) exp(it[˜bj1 − bj2 ]) = cos(t[˜bj1 − bj2 ]) + i sin(t[˜bj1 − bj2 ]) exp(it[˜bj − ˜bj ]) = cos(t[˜bj − ˜bj ]) + i sin(t[˜bj − ˜bj ]) 1

2

1

2

1

2

(53) Using the fact

Rκ −κ

sin(t)dt = 0 and cos(−x) = cos(x) with (51), (52) and

(53), we can get the following relation: Z κ N N ¯ ¯2 1 1 XX ¯ˆ ¯ cos(t[bj1 − bj2 ])dt ¯WN,n (t)¯ dt = 2κ −κ N j =1 j =1 −κ 1 2 Z κ Z κ N N N N 2 XX 1 XX 1 1 ˜ cos(t[bj1 − bj2 ])dt + cos(t[˜bj1 − ˜bj2 ])dt − 2κ −κ N j =1 j =1 2κ −κ N j =1 j =1 1 2 1 2 Z Z N N N N 1 2 XX κ 1 1 XX κ cos(t[bj1 − bj2 ])dt − cos(t[bj1 − ˜bj2 ])dt = κ N j =1 j =1 0 κ N j =1 j =1 0 1 2 1 2 Z N N κ 1 1 XX + cos(t[˜bj1 − ˜bj2 ])dt. (54) κ N j =1 j =1 0

1 2κ

Z

κ

1

2

Note that bj1 − bj2 , bj1 − ˜bj2 and ˜bj1 − ˜bj2 appear in the denominator after

40

integration in (54), and we need to consider when they are zero. Therefore, N N Z 1 1 XX κ cos(t[bj1 − bj2 ])dt κ N j =1 j =1 0 1 2 ¸ N N · sin (κ × (bj1 − bj2 )) 1 XX I(bj1 6= bj2 ) × = + I(bj1 = bj2 ) , N j =1 j =1 κ(bj1 − bj2 ) 1

1

N N Z 1 2 XX κ cos(t[bj1 − ˜bj2 ])dt κ N j =1 j =1 0 1 2 ³ ´   ˜bj ) N X N sin κ × (b − X j 1 2 2 I(bj1 6= ˜bj2 ) × = + I(bj1 = ˜bj2 ) , ˜ N j =1 j =1 κ(bj1 − bj2 ) 1

2

N N Z 1 1 XX κ cos(t[˜bj1 − ˜bj2 ])dt κ N j =1 j =1 0 1 2 ³ ´   ˜ ˜ N N sin κ × ( b − b ) j1 j2 1 XX ˜ I(bj1 6= ˜bj2 ) × = + I(˜bj1 = ˜bj2 ) . N j =1 j =1 κ(˜bj1 − ˜bj2 ) 1

1

Plugging these three terms above in (54) leads to (37).

Q.E.D.

ˆ t) and the Integrated Variance Derivation of the Variance Γ(t, R κ 1 ˆ t)dt. Γ(t, 2κ

−κ

ˆ N,n (t), Γ(t, t), is as follows. The variance of W à ! N 1 X Γ(t, t) = V ar √ (exp(itbj ) − exp(it˜bj )) N j=1 à ! à ! N N 1 X 1 X = V ar √ exp(itbj ) + V ar √ exp(itbj ) N j=1 N j=1 which is due to the independence of bj and ˜bj . Note that à ! N 1 X V ar √ exp(itbj ) = V ar (exp(itb1 )) N j=1 = E |exp(itb1 ) − E(exp(itb1 ))|2 41

(55)

(56)

V ar (exp(itb1 )) in (56) can be consistently estimated as follows ¯ ¯2 N ¯ N ¯ X X 1 1 ¯ ¯ Vd ar (exp(itb1 )) = exp(itb ) − exp(itb ) ¯ j k ¯ ¯ N j=1 ¯ N k=1

(57)

Moreover, ¯ ¯2 N ¯ ¯ X 1 ¯ ¯ exp(itbk )¯ ¯exp(itbj ) − ¯ ¯ N k=1 Ã !Ã ! N N 1 X 1 X = exp(itbj ) − exp(itbk ) exp(−itbj ) − exp(−itbk ) N k=1 N k=1 N N 1 1 X 1 1 X = 1 − exp(itbj ) exp(−itbk ) − exp(−itbj ) exp(itbk ) N N k=1 N N k=1 N N 1 X 1 X + exp(itbk ) exp(−itbk ) N k=1 N k=1

(58)

It follows from (57) and (58) that ¯ ¯2 N ¯ N ¯ X X 1 1 ¯ ¯ Vd ar (exp(itb1 )) = exp(itb ) exp(itb ) − ¯ k ¯ j ¯ N j=1 ¯ N k=1 N N 1 X 1 X exp(itbj1 ) exp(−itbj2 ) =1− N j =1 N j =1 1

=1−

1 N2

2

N X N X

exp(it(bj1 − bj2 ))

j1 =1 j2 =1

N N 1 XX cos(t(bj1 − bj2 )) =1− 2 N j =1 j =1 1

(59)

2

Note that the last equation in (59) follows from the fact that exp(it(bj1 − PN P bj2 )) = cos(it(bj1 −bj2 ))+i sin(t(bj1 −bj2 )) and N j2 =1 sin(t(bj1 −bj2 )) = j1 =1 0. It follows from (56) and (59) that à ! N N N X 1 1 XX d V ar √ exp(itbj ) = 1 − 2 cos(t(bj1 − bj2 )) N j =1 j =1 N j=1 1 2

42

(60)

Similarly, Ã Vd ar

! N N N 1 X 1 XX ˜ √ exp(itbj ) = 1 − 2 cos(t(˜bj1 − ˜bj2 )) N N j=1 j1 =1 j2 =1

(61)

We can estimate Γ(t, t) in (55) consistently: follows N N N N 1 XX 1 XX ˆ Γ(t, t) = 2 − 2 cos(t(bj1 − bj2 )) − 2 cos(t(˜bj1 − ˜bj2 )) N j =1 j =1 N j =1 j =1 1

=2−

1 N2

2

N X N X

1

2

h i ˜ ˜ cos(t(bj1 − bj2 )) + cos(t(bj1 − bj2 ))

(62)

j1 =1 j2 =1

Therefore, we can have the integrated variance function

1 2κ

Rκ −κ

ˆ t)dt as Γ(t,

follows. Z κ 1 ˆ t)dt Γ(t, 2κ −κ ! Z κà N N i 1 1 XXh = cos(t(bj1 − bj2 )) + cos(t(˜bj1 − ˜bj2 )) dt 2− 2 2κ −κ N j =1 j =1 1 2 ! Z κà N X N h i X 1 1 cos(t(bj1 − bj2 )) + cos(t(˜bj1 − ˜bj2 )) dt 2− 2 = 2× 2κ 0 N j =1 j =1 1 2 " µ ¶#κ N N 1 XX 1 sin(t(bj1 − bj2 )) 2t − 2 I(bj1 = bj2 )t + I(bj1 6= bj2 ) = κ N j =1 j =1 (bj1 − bj2 ) 1 2 " à !#κ 0 N N ˜ ˜ 1 1 XX ˜bj = ˜bj )t + I(˜bj 6= ˜bj ) sin(t(bj1 − bj2 )) I( − 1 2 1 2 κ N 2 j =1 j =1 (˜bj1 − ˜bj2 ) 1 2 0 " µ ¶# N N X X 1 1 sin(κ(bj1 − bj2 )) 2κ − 2 I(bj1 = bj2 )κ + I(bj1 6= bj2 ) = κ N j =1 j =1 (bj1 − bj2 ) 1 2 " !# à N N ˜bj − ˜bj )) 1 1 XX sin(κ( 1 2 − I(˜bj1 = ˜bj2 )κ + I(˜bj1 6= ˜bj2 ) ˜ ˜ κ N 2 j =1 j =1 (bj1 − bj2 ) 1 2 ¶ µ N N 1 XX sin(κ(bj1 − bj2 )) = 2− 2 I(bj1 = bj2 ) + I(bj1 6= bj2 ) N j =1 j =1 κ(bj1 − bj2 ) 1 2 ! à N N ˜bj − ˜bj )) 1 XX sin(κ( 1 2 (63) − 2 I(˜bj1 = ˜bj2 ) + I(˜bj1 6= ˜bj2 ) ˜ ˜ N j =1 j =1 κ( b − b ) j j 1 2 1 2 43

Q.E.D. Proof of Proposition 4. ¯ ¯2 P ¯ˆ ¯ ˆ t) −→ Note that Γ(t, E(¯W (t) ¯ ) under the null hypothesis as N inN,n d creases. Therefore, TˆN,n −→ T with E(T ) = 1 by the continuous mapping P 2 theorem. T = ∞ j=1 pj εj follows Bierens and Ploberger (1997). Turning to ¯ ¯2 ¯√ ³ P ´¯2 PN ¯ˆ ¯ ¯ ¯ N 1 1 ˜ the second statement, note that ¯W (t) = N exp(itb ) − exp(it b ) ¯ ¯ ¯. N,n j j j=1 j=1 N N ¯ P ¯ 2 PN ¯ 1 ˜ ¯ P Under the alternative hypothesis, ¯ N1 N j=1 exp(itbj ) − N j=1 exp(itbj )¯ −→ η(n) which is some positive real number which may depend on the truncation order n. Therefore,

TˆN,n N

P

−→ η(n) > 0. Q.E.D.

Proof of Lemma 2. (i). Suppose n > n0 . We need to show that lim P[CˆN (n) > CˆN (n0 )] = 1.

N →∞

It follows that # " h i ˆN,n0 ˆN,n T T φ(N ) φ(N ) +n > + n0 lim P CˆN (n) > CˆN (n0 ) = lim P N →∞ N →∞ N N N N · ¸ 1 ˆ φ(N ) = lim P (TN,n − TˆN,n0 ) > (n0 − n) N →∞ N N " # (TˆN,n − TˆN,n0 ) = lim P > n0 − n N →∞ φ(N ) =1 since

(TˆN,n −TˆN,n0 ) φ(N )

(64)

P −→ 0 which follows TˆN,n − TˆN,n0 = Op (1) and φ(N ) → ∞

as N → ∞ while n0 − n < 0. (ii). Suppose n < n0 . We need to show that P[CˆN (n0 ) < CˆN (n)] = 1.

44

(65)

It follows that h

"

i ˆ ˆ lim P CN (n0 ) < CN (n) =

φ(N ) TˆN,n φ(N ) TˆN,n0 lim P + n0 < +n N →∞ N N N N " # ˆ ˆ TN,n TN,n0 φ(N ) = lim P − > (n0 − n) N →∞ N N N

N →∞

= 1 since p lim

³ˆ

TN,n N

−

TˆN,n0 N

´

#

(66)

) > 0 and lim (n0 − n) φ(N = 0. N

Q.E.D.

N →∞

Proof of Proposition 5. Suppose that the space of SNP density functions on [0,1] consists of square integrable Borel measurable functions on [0,1] where L1 semi-metric is dePˆ ˆ 2 (1+ n j=1 δj ρj (G(v))) ˆ ˆ c Pn g(v) fined. Note that fnˆ (v) = hnˆ (G(v))g(v) = hnˆ (G(v)|δnˆ )g(v) = ˆ 2 and f (v) = hn0 (G(v))g(v)+op (1) ≡ hn0 (G(v)|δn0 )g(v)+op (1) = op (1).

1+ j=1 δˆj P 2 (1+ n j=1 δ0,j ρj (G(v))) P g(v)+ 2 1+ ∞ δ j=1 0,j

ˆ nˆ (G(v))g(v)||1 ||f (v) − fˆnˆ (v)||1 = ||hn0 (G(v))g(v) + op (1) − h ˆ nˆ (u)g(G−1 (u))||1 = ||hn0 (u)g(G−1 (u)) + op (1) − h Z 1 ˆ nˆ (u))g(G−1 (u))|du + op (1) = |(hn0 (u) − h 0

µZ

1

≤ 0

ˆ nˆ (u)|2 du |hn0 (u) − h

¶ 12 µZ

1

|g(G−1 (u))|2 du

¶ 21 + op (1)

0

(67) where u = G(v). ¯ ¯ ¯¯ Pn0 (1+ δ0,j ρj (u))2 ¯ ¯ ˆ Pn0 2 Note that ¯hn0 (u) − hnˆ (u)¯ = ¯¯ 1+j=1 − δ j=1 0,j

¯

Pˆ ˆ 2¯ (1+ n j=1 δj ρj (u)) ¯ P ˆ2 ¯ 1+ ∞ δ j=1 j

P

−→ 0 since

lim P [ˆ n = n0 ] = 1 as N → ∞. This implies that the last equation in (67) conR1 verges to zero in probability as N → ∞ since 0 |g(G−1 (u))|2 du is bounded. P Hence, fˆnˆ (v) −→ f (v) pointwise as N → ∞.42 Q.E.D. 42

We use the simple inequality: If |A| ≤ |B| , then for all ε > 0, P[|A| > ε] ≤ P[|B| > ε]. In addition, if P[|B| > ε] → 0, then P[|A| > ε] → 0.

45

Proof of Proposition 6.

ˆ nˆ (u)| sup |F (v) − Fˆnˆ (v)| = sup |Hn0 (u) + op (1) − H v≥v

u∈[0,1]

Z

¯ ¯ ¯ ¯ ˆ (1) − h (u) h (u) + o ¯ du ¯ n0 p n ˆ 0 ¯ P n0 Pnˆ ˆ Z 1 ¯¯ 2 2¯ (1 + δ ρ (u)) (1 + δ ρ (u)) 0,j j j j ¯ ¯ j=1 j=1 = P n0 + op (1) − ¯ ¯ du Pnˆ ˆ2 2 ¯ ¯ 1 + j=1 δ0,j 1 + j=1 δj 0 1

≤

¯ Pn ¯ (1+ 0 δ0,j ρj (u))2 Pn0 + op (1) − Note ¯¯ 1+j=1 2 j=1 δ0,j

¯

Pˆ ˆ 2¯ (1+ n j=1 δj ρj (u)) ¯ Pn ˆ 2 ¯ 1+ j=1 δˆj

P

−→ 0 as lim P [ˆ n = n0 ] = 1

P as N → ∞. Hence, sup |F (v) − Fˆnˆ (v)| −→ 0 as N → ∞. Therefore,

v≥v

Fˆnˆ (v) −→ F (v) uniformly on [v, ∞). P

Q.E.D.

46