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Federal Reserve Bank of Chicago
Characterizations in a random record
model with a non-identically
distributed initial record
Gadi Barlevy and H. N. Nagaraja
WP 2005-05
Characterizations in a random record model with a
non-identically distributed initial record
Gadi Barlevy∗
H. N. Nagaraja†
September 15, 2005
Abstract
We consider a sequence of random length M of independent absolutely continuous
observations Xi , 1 ≤ i ≤ M, where M is geometric, X1 has cdf G, and Xi , i ≥ 2, have
cdf F . Let N be the number of upper records and Rn , n ≥ 1, be the nth record value.
We show that N is free of F if and only if G(x) = G0 (F (x)) for some cdf G0 and that if
E(|X2 |) is finite so is E(|Rn |) for n ≥ 2 whenever N ≥ n or N = n. We prove that the
distribution of N along with appropriately chosen subsequences of E(Rn ) characterize
F and G, and along with subsequences of E(Rn − Rn−1 ) characterize F and G up to a
common location shift. We discuss some applications to the identification of the wage
offer distribution in job search models.
Key Words: Moment sequences; Number of records; Record spacings; Geometric distribution; Müntz-Szász theorem; Titchmarsh convolution theorem;
Job search models.
AMS 2000 Subject Classification: Primary 62E10; Secondary: 60G70,
62G32.
Abbreviated Title: Characterizations in a record model
∗ gbarlevy@frbchi.org; Economic Research Department, Federal Reserve Bank of Chicago, 230 South La
Salle Street, Chicago, IL 60604-1413, USA
† hnn@stat.ohio-state.edu; Department of Statistics, The Ohio State University, 1958 Neil Ave., Columbus
OH 43210-1247, USA.
1
1. Introduction
Let {Xi , i ≥ 1} be a sequence of independent random variables. Suppose that M is
a positive integer-valued random variable independent of the Xi , and assume that only
©
ª
{Xi , 1 ≤ i ≤ M } are observed. Define L (1) = 1 and L (n) = min k : Xk > XL(n−1) for
n > 1, and Rn = XL(n) for n ≥ 1. Then R1 is the initial record (sometimes called the trivial
record), and Rn for n ≥ 2 represent the upper record values from the sequence {Xi , i ≥ 1}.
The total number of records we observe is given by N = max {j : L (j) ≤ M } and is itself a
random variable.
When the Xi are identically distributed, this model is called the random record model
(see, e.g., Arnold et al., 1998, p. 224). When we further assume that M has a geometric
distribution, i.e. Pr (M = m) = q m−1 p for m ≥ 1, where 0 < p < 1 and q = 1 − p,
we have a geometric random record (GRR) model. Nagaraja and Barlevy (2003) derived
several characterization results for the GRR model using record moments. In this paper,
we consider a variation of the GRR model in which the initial observation, X1 , has a
potentially different distribution from remaining observations {Xi , i ≥ 2}. We refer to this
as a GRR model with a non-identically distributed initial record or a modified GRR model.
Our purpose in this paper is to determine whether there exist analogous characterization
results for this alternative formulation and to discuss some applications of this variation
concerning identification of job search models.
Formally, let X1 be distributed with continuous cumulative distribution function (cdf) G,
and {Xi , i ≥ 2} be independent and identically distributed (i.i.d.) with continuous cdf F .
Define a mapping Γ from the set of continuous distribution functions into itself so that
G = Γ (F ). This notation allows us to view the model as being parameterized by a single
cdf F . We impose the following assumptions on Γ:
Assumption 1: the probability measure implied by G = Γ (F ) is absolutely continuous
with respect to the probability measure implied by F .
¡
¢
Assumption 2: the composite function G F −1 (u) is absolutely continuous in u ∈ (0, 1).
The first assumption implies that the support of G must always form a subset of the support
2
of F . The second assumption implies that without loss of generality we can assume X1 has
a well-defined density function, since we can always normalize F (x) = x. We denote this
d
density function g (x; F (·)) =
G (x; F (·)).
dx
Remark 1: The identity mapping Γ : F → F satisfies Assumptions 1 and 2. Our model
thus includes the GRR model as a special case.
Remark 2: Our formulation is itself a special case of the Pfeifer (1982) model, in which
the distribution of the underlying observations changes after each record is set. Here, the
distribution changes only after the first record, and the distribution of the first record
G = Γ (F ) is required to satisfy Assumptions 1 and 2. Although Pfeifer assumes M = ∞,
Bunge and Nagaraja (1991) subsequently generalized Pfiefer’s model to allow the number
of observations M to be random.
Remark 3: Our assumptions do not require Γ to be one-to-one, as illustrated by Example
2 below. However, Assumption 1 implies Γ cannot assign a single G to all cdf’s F . Thus, G
cannot be free of F .
Here are some examples of functions G (x) that satisfy Assumptions 1-2. The motivation
for these examples will become clear in Section 6, when we discuss how the model can be
applied to estimate job search models.
1. G (x; F (·)) =
F (x)
for some constant κ
1 + κ (1 − F (x))
2. G (x; F (·)) = F (x) /z if F (x) ≤ z and 1 if F (x) > z for some constant z ∈ (0, 1)
Rx
¢
R F (x) ¡ −1
H (w) dF (w)
H F (u) du
−∞
0
R
= R1
where H (·) is a cdf.
3. G (x; F (·)) = ∞
H (w) dF (w)
H (F −1 (u)) du
−∞
0
In Example 1, G(x) =
P∞
i=1
pq i−1 F (x)i , where p = (1 + κ)−1 , or X1 has the same distribu-
tion as the maximum of a random (geometric) number of i.i.d. random variables distributed
like X2 . In Example 2, G arises from F by the truncation of its upper tail, and in Example
3, G has the form of a weighted distribution. Note that in the first two examples, G = Γ (F )
assumes the form G0 (F (x)) for some function G0 , i.e. the cdf G evaluated at x depends
on F (x) but not on the value of F at any point other than x. This is not true for the last
example.
3
We show in Section 2 of the paper that Γ (F ) = G0 (F (x)) if and only if the distribution of
the number of observed records N is independent of F . In Section 3, we focus on mappings
Γ where Γ (F ) = G0 (F (x)), and show that under an additional assumption on Γ, the
distribution F is characterized by subsequences of the following moments:
a. E (Rn | N ≥ n)
b. E (Rn | N = n)
c. E (Rn − Rn−1 | N ≥ n)
d. E (Rn − Rn−1 | N = n) .
In demonstrating this result, we appeal not only to the Müntz-Szász theorem, which is often
invoked in moment-based characterization theorems (see Kamps, 1998), but also to a convolution theorem due to Titchmarsh (1926). In Section 4, we consider arbitrary mappings
Γ that satisfy Assumptions 1 and 2, and provide characterizations of F and G using subsequences of these moments together with the distribution Pr (N = n). All of these results
are premised on a fixed distribution for M . In Section 5, we derive conditions that jointly
characterize F and G as well as the distribution for M . Section 6 then discusses how our
results can be used to non-parametrically identify the wage offer distribution in job search
models when wage data can only be measured with noise.
2. Characterization results for Γ and N
We begin with results that characterize the mapping Γ. Our first result shows that the
number of observed records N is identical for all continuous cdf’s F if and only if Γ is such
that G = Γ (F ) evaluated at x can be expressed as a function of F (x).
Proposition 1: In the modified GRR model, the number of observed records N is independent of F if and only if G (x; F (·)) = G0 (F (x)) for some absolutely continuous cdf G0
with support [0, 1].
Proof : Building on Bunge and Nagaraja (1991) and Nagaraja and Barlevy (2003), we can
4
express the likelihood of exactly n observed records with values r1 through rn as
n
h (r1 , ..., rn ∩ N = n) =
(1 − q) g (r1 ; F (·)) Y qf (ri )
.
1 − qF (r1 )
1 − qF (ri )
i=2
Next, we integrate out r2 through rn to get
h (r1 ∩ N = n) =
1−q
1
(n − 1)! 1 − qF (r1 )
µ µ
¶¶n−1
1 − qF (r1 )
g (r1 ; F (·)) .
ln
1−q
Hence, Pr (N = n) can be expressed as
· µ
¶¸n−1
Z ∞
1−q
1
1 − qF (r1 )
g (r1 ; F (·)) dr1 .
ln
Pr (N = n) =
(n − 1)! −∞ 1 − qF (r1 )
1−q
Suppose G (x; F (·)) = G0 (F (x)) where G0 (·) is an absolutely continuous function. We
want to show that Pr (N = n) is independent of F (·). Since G0 is absolutely continuous, it
d
has a related density function g0 (x) =
G0 (x). This implies
dx
¯
¯
d
g (r1 ; F (·)) =
= g0 (F (r1 )) f (r1 ) .
G0 (F (x))¯¯
dx
x=r1
Substituting this in and using the change of variables u = F (r1 ), we find
· µ
¶¸n−1
Z 1
1 − qu
1
1−q
g0 (u) du
ln
Pr (N = n) =
(n − 1)! 0 1 − qu
1−q
which is indeed independent of F (·).
Next, suppose Pr (N = n) is independent of F (·). We want to show this implies G (x; F (·)) =
G0 (F (x)) where G0 (·) is an absolutely continuous cdf. Given Assumption 1, we can rewrite
Pr (N = n) using the change of variables u = F (r1 ) so that
¢
· µ
¶¸n−1 ¡ −1
Z 1
g F (u)
1 − qu
1−q
1
ln
du.
Pr (N = n) =
(n − 1)! 0 1 − qu
1−q
f (F −1 (u))
(1)
Since {Pr (N = n) , n ≥ 1} does not depend on the distribution of F (·), then for any two
distributions F1 (·) and F2 (·), we have for n = 1, 2, 3, ...
¢
· µ
¶¸n−1 ¡ −1
Z 1
g F1 (u) ; F1 (·)
1 − qu
1−q
¢ du
¡ −1
ln
1−q
f1 F1 (u)
0 1 − qu
¢
· µ
¶¸n−1 ¡ −1
Z 1
g F2 (u) ; F2 (·)
1 − qu
1−q
¢ du.
¡ −1
=
ln
1−q
f2 F2 (u)
0 1 − qu
Let F1 (·) = F (·) be any continuous distribution function, and let F2 (·) to be the uniform
distribution, i.e. F2−1 (u) = u and f2 (u) = 1 for all u ∈ (0, 1). Let us further define
5
h (u) = g (u; u), i.e. the density function g (·) evaluated when F (·) is uniform. By the
Müntz-Szász theorem, it follows that for almost all u ∈ (0, 1),
¢
¡
g F −1 (u) ; F (·)
= h (u) .
f (F −1 (u))
Rx
Since G (x; F1 (·)) = −∞ g (y; F1 (·)) dy, we have
G (x; F (·)) =
=
=
Z
Z
x
g (y; F (·)) dy
−∞
x
h (F (y)) f (y) dy
−∞
Z F (x)
h (z) dz
0
≡ G0 (F (x)) .
Since G (·) is a cdf, it follows that G0 (·) is non-decreasing, G0 (0) = 0, and G0 (1) = 1.
Hence, G0 is a cdf with support [0, 1]. Absolute continuity of G0 is immediate. ¥
The probability mass function (pmf) of N in (1) implies N − 1 has a mixed Poisson distri-
bution. An interesting property of this pmf, stated below, will be useful for us later.
Lemma 1: In the modified GRR model, the sequence {Pr(N = nj ), j ≥ 1} such that
P∞ −1
j=1 nj = ∞ uniquely determines the pmf of N .
Proof : Consider two modified GRR models with cdfs F1 , G1 and F2 , G2 . If Pr(N = nj )
remains the same for these two models, from (1) we have
¢
¢#
¡
· µ
¶¸nj −1 " ¡ −1
Z 1
g2 F2−1 (u)
g1 F1 (u)
1 − qu
1−q
¢ − ¡ −1
¢ du = 0.
¡
ln
1−q
f1 F1−1 (u)
f2 F2 (u)
0 1 − qu
Set t = ln
µ
1 − qu
1−q
¶
and rewrite the equation above as
Z
− ln(1−q)
h1 (t)tnj −1 dt = 0.
0
It follows from the Müntz-Szász theorem that h1 (t) = 0 almost everywhere or
¡
¡
¢
¢
g1 F1−1 (u)
g2 F2−1 (u)
¢ = ¡ −1
¢
¡
f1 F1−1 (u)
f2 F2 (u)
(2)
for almost all u ∈ (0, 1). Again, upon appealing to (1) we conclude that Pr(N = n), and
thus Pr(N ≥ n), remain the same for all n ≥ 1 under the two models. ¥
6
The mapping Γ can also be characterized by the dependence structure of the record indicators derived from {Xi , i ≥ 1}. Define the m-th record indicator Im as a random variable that
takes on a value of 1 if Xm is a record, i.e. if Xm > max {X1 , ..., Xm−1 }, and 0 otherwise.
Previous work by Nevzorov (1986) has shown that if Xi are independent random variables
and Xi has cdf
α(i)
Fi (x) = {F (x)}
,
(3)
for some common F , then {Im , m ≥ 2} are independent and Im is Bernoulli with success
P
probability α(m)/{ m
i=1 α(i)}. Conversely, Nevzorov shows that if the supports of Xi are
not disjoint and {I2 , ..., In−1 } and In are independent for any Fn , then (3) holds for m =
1, . . . , n − 1. (see also Arnold et al (1998), p. 219). In our model, (3) implies that
G(x) = Γ (F (x)) = {F (x)}α
(4)
for some α > 0. Our next proposition establishes that Γ satisfies (4) if and only if the record
indicator I2 is independent of the record indicators {I3 , ..., Im } for all m, conditional on at
least m observations.
Proposition 2: In a random record model with a non-identically distributed initial record,
Pr(I3 = 1, . . . , Im = 1 | M ≥ m) = Pr(I3 = 1, . . . , Im = 1 | I2 = 1, M ≥ m)
for all m = mj (≥ 3), j ≥ 1 such that
P
j
(5)
m−1
j = ∞, if and only if (4) holds.
Proof : The given condition (5) can be expressed as
Pr(I2 = 1, . . . , Im = 1) = Pr(I3 = 1, . . . , Im = 1) Pr(I2 = 1).
Without loss of generality we take F (x) = x. Upon conditioning on X1 , the LHS can be
simplified to
1
(m − 1)!
Z
0
1
(1 − x)m−1 dG(x) or
1
(m − 2)!
Z
0
1
(1 − x)m−2 G(x)dx.
Further Pr(I3 = 1, . . . , Im = 1) = LHS + Pr(I2 = 0, I3 = 1, . . . , Im = 1). The second term,
upon conditioning on X1 , can be written as
Z 1
1
(1 − x)m−2 xg(x)dx.
(m − 2)! 0
R1
Let c = Pr(I2 = 1)(= 0 G(x)dx). Then (5) reduces to the condition
Z 1
(1 − x)m−2 [(1 − c)G(x) − cxg(x)]dx = 0, m = mj , j ≥ 1.
0
7
(6)
By the Müntz-Szász theorem, it follows that for almost all x ∈ (0, 1),
(1 − c) 1
g(x)
=
.
G(x)
c x
This differential equation clearly shows that (4) holds with α = (1 − c)/c.
Conversely, if (3) holds, with F (x) = x, G(x) = xα , then set c = (1 + α)−1 . It follows that
(1 − c)G(x) = cxg(x) for all x ∈ (0, 1). Thus (6) holds for all m ≥ 1, or (5) holds. ¥
Corollary: If (5) holds, then (4) holds, and in turn the record indicators {Im , m ≥ 2} are
all independent. Moreover, the distribution of the number of observed records N must be
independent of F .
3. Characterization results for a special case
We now turn to characterization results for F . We begin by deriving expressions for the
relevant moments we use, and provide conditions for these moments to exist. We then
consider the special case where Γ (F ) = G0 (F (x)) and provide a characterization result.
Consider record moments that condition on the event N ≥ n. Using the expression for
the likelihood h (r1 , ...rn ∩ N ≥ n) from Bunge and Nagaraja (1991) and integrating out
r2 through rn−2 yields the following expression for the joint likelihood of r1 , rn−1 , and rn
when N ≥ n for n ≥ 3:
h (r1 , rn−1 , rn ∩ N ≥ n) =
·
µ
¶¸n−3
1
1 − qF (rn−1 )
qf (rn−1 )
qg (r1 )
− ln
f (rn ) .
(n − 3)!
1 − qF (r1 )
1 − qF (r1 ) 1 − qF (rn−1 )
Limiting attention to n ≥ 3, we derive the following expression for E (Rn | N ≥ n):
E (Rn | N ≥ n) =
=
1
P (N ≥ n)
Z
0
1
Z
un−1
0
Z
∞
Z
rn−1
Z
∞
rn h (r1 , rn−1 , rn ∩
rn−1
h
³
´in−3
¡
¢
n−1
− ln 1−qu
g F −1 (u1 )
1−qu1
(n − 3)!P (N ≥ n) f (F −1 (u1 ))
−∞
−∞
·φF (un−1 )
where
φF (un−1 ) =
Z
qdu1 qdun−1
1 − qu1 1 − qun−1
N ≥ n) drn dr1 drn−1
(7)
1
F −1 (un ) dun .
un−1
8
(8)
Similarly, the expected record spacing E (Rn − Rn−1 | N ≥ n) when n ≥ 3 can be expressed
as
Z
E (Rn − Rn−1 | N ≥ n) =
1
Z
un−1
0
0
h
³
´in−3
¡
¢
g F −1 (u1 )
(n − 3)!P (N ≥ n) f (F −1 (u1 ))
− ln
1−qun−1
1−qu1
·φF (un−1 )
where
φF (un−1 ) =
Z
1
qdu1 qdun−1
1 − qu1 1 − qun−1
¤
£ −1
F (un ) − F −1 (un−1 ) dun .
un−1
(9)
(10)
In the same fashion, we can use the expression for h (r1 , ..., rn ∩ N = n) derived above and
integrate out r2 through rn−2 to obtain the joint likelihood of r1 , rn−1 , and rn when N = n,
where n ≥ 3:
h (r1 , rn−1 , rn ∩ N ≥ n) =
·
µ
¶¸n−3
1
1 − qF (rn−1 )
qf (rn−1 )
qg (r1 )
− ln
f (rn ) .
(n − 3)!
1 − qF (r1 )
1 − qF (r1 ) 1 − qF (rn−1 )
Using this, one can deduce that
Z
E (Rn | N = n) =
0
1
Z
un−1
h
³
´in−3
¡
¢
n−1
− ln 1−qu
g F −1 (u1 )
1−qu1
(n − 3)! Pr (N = n) f (F −1 (u1 ))
0
·φF (un−1 )
where
φF (un−1 ) =
Z
1
F −1 (un )
un−1
and likewise
E (Rn − Rn−1 | N = n) =
Z
0
1
Z
un−1
0
h
qdu1 qdun−1
1 − qu1 1 − qun−1
(1 − q) dun
,
1 − qun
³
φF (un−1 ) =
Z
1
un−1
(12)
´in−3
¡
¢
g F −1 (u1 )
(n − 3)!P (N = n) f (F −1 (u1 ))
− ln
1−qun−1
1−qu1
·φF (un−1 )
where
(11)
qdu1 qdun−1
1 − qu1 1 − qun−1
¤
1 − q £ −1
F (un ) − F −1 (un−1 ) dun
1 − qun
(13)
(14)
All four moment sequences above can thus be expressed as an integral of a common term
multiplying a function φF (un−1 ) that varies with the particular moment at hand.
We now provide a sufficient condition for the above moments to exist. Here we use the fact
that for any random variable Y , E (Y ) exists if and only if E (|Y |) < ∞.
9
Proposition 3: If E (|X2 |) < ∞, then
a. E (|Rn | | N ≥ n)
b. E (|Rn | | N = n)
exist for all n ≥ 2.
Proof : Suppose first that n ≥ 3. From (7),
E (|Rn | | N ≥ n) =
Z
0
1
Z
0
un−1
Z
1
un−1
h
³
´in−3
n−1
− ln 1−qu
1−qu1
(n − 3)!P (N ≥ n)
¢
¡
¯
g F −1 (u1 ) ¯¯ −1
qdu1 qdun−1
.
·
F (un )¯ dun
−1
f (F (u1 ))
1 − qu1 1 − qun−1
Since u1 and un−1 both lie in [0, 1],the above expression
)
¢
¶2 Z 1 Z 1 (Z un−1 ¡ −1
n−3 µ
¯
¯
g F (u1 )
[− ln (1 − q)]
q
≤
du1 ¯F −1 (un )¯ dun dun−1
−1 (u ))
(n − 3)!P (N ≥ n) 1 − q
f
(F
1
0
0
un−1
¶2 Z 1 Z 1
n−3 µ
¯
¡
¢¯
[− ln (1 − q)]
q
=
G F −1 (un−1 ) ¯F −1 (un )¯ dun dun−1
(n − 3)!P (N ≥ n) 1 − q
0
un−1
¶2 Z 1 Z 1
n−3 µ
¯
¡
¢¯
[− ln (1 − q)]
q
≤
G F −1 (un−1 ) ¯F −1 (un )¯ dun dun−1
(n − 3)!P (N ≥ n) 1 − q
0
0
¶2
n−3 µ
[− ln (1 − q)]
q
≤
E (|X2 |) .
(n − 3)!P (N ≥ n) 1 − q
As long as E (|X2 |) < ∞, then E (|Rn | | N ≥ n) exists. A similar argument applies for the
case where n = 2. Extending the argument for E (|Rn | | N = n) is straightforward. ¥
Remark 4: For n ≥ 2, the existence of the n-th record moment does not depend on whether
E (X1 ) exists. This is because for n ≥ 2, the relevant moment is always conditioned on an
event in which M ≥ n and max {X2 , ..., XM } > X1 . As demonstrated in Nagaraja and Bar-
levy (2003), if M is geometric with success probability p, then E (E (|max {X2 , ..., XM }| | M ))
< p−1 E (|X2 |) . Thus, we are conditioning on the event that X1 is exceeded by a random
variable whose mean is finite. Even if X1 does not have a well-defined unconditional mean,
conditioning on the event that its value is exceeded by a random variable with a finite mean
suffices to ensure that E (X1 | N ≥ n) is finite.
Equipped with these preliminaries, we turn to characterizing F from moment sequences.
However, we first need to impose an additional assumption on the range of Γ.
10
¡
¢
Assumption
3: For any cdf F , there exists an ε > 0 such that g F −1 (u) ; F (·) =
¯
¯
d
is positive for almost all u ∈ (0, ε).
Γ (F )¯¯
dx
x=F −1 (u)
This assumption implies that G−1 (0) = F −1 (0). This assumption is clearly necessary: if
two distributions F1 6= F2 differ only below G−1 (0), they would necessarily yield identical
record moment sequences.
We first focus on a special case, namely where G (x; F (·)) = G0 (F (x)) for some absolutely
continuous cdf G0 :[0, 1] → [0, 1]. Recall from Proposition 1 that this is true if and only if
Pr (N = n) does not depend on F . We return to the more general case in the next section.
Proposition 4: Suppose Assumptions 1-3 are satisfied, E (|X2 |) < ∞, and G (x; F (·)) =
G0 (F (x)) for some absolutely continuous cdf G0 :[0, 1] → [0, 1].
a. If two distributions F1 and F2 give rise to either the same sequence E (Rn | N ≥ nj )
P
= ∞, then F1 (x) = F2 (x) for
or the same sequence E (Rn | N = nj ) where j n−1
j
almost all x;
b. If two distributions F1 and F2 give rise to either the same sequence E (Rn − Rn−1 |N ≥ nj )
P
or the same sequence E (Rn − Rn−1 | N = nj ) where j n−1
j = ∞, then there exists
a c such that F1−1 (x) = F2−1 (x) + c for almost all x.
Proof
: Since
¡
¢ Γ (F ) = G0 (F (x)), it follows that g (x; F (·)) = g0 (F (x)) f (x), which implies
g F −1 (u1 )
= g0 (u1 ). Substituting in, any of the moments above can be expressed as
f (F −1 (u1 ))
·
µ
¶¸n−3
Z 1Z 1
1 − qun−1
qdun−1 qdu1
1
g0 (u1 ) − ln
φF (un−1 )
(n − 3)!P (N ≥ n) 0 u1
1 − qu1
1 − qun−1 1 − qu1
where φF (·) depends on the particular moment in question, i.e. either (8) or (10) in case
(a) and either (12) or (14) in case (b). Changing variables according to
qdun−1
1 − e−t
, un−1 =
1 − qun−1
q
−s
qdu1
1−e
s = − ln (1 − qu1 ) ⇒ ds =
, u1 =
1 − qu1
q
t = − ln (1 − qun−1 ) ⇒ dt =
and setting c = − ln (1 − q) allows us to rewrite the above expression as
µ
µ
¶
¶
Z cZ c
1 − e−s
1 − e−t
1
n−3
g0
dtds.
φF
(t − s)
(n − 3)!P (N ≥ n) 0 t=s
q
q
11
We change variables yet again by setting ω = t − s to rewrite the above as
µ
µ
¶
¶
Z c Z c
1 − e−(t−ω)
1 − e−t
1
g0
φF
ω n−3 dtdω.
(n − 3)!P (N ≥ n) ω=0 t=ω
q
q
Define
η F (ω) =
Z
c
g0
t=ω
µ
1 − e−(t−ω)
q
¶
φF
µ
1 − e−t
q
¶
dt.
Let F1 (·) and F2 (·) denote two continuous cdf’s that give rise to the same subsequence of
moments. Since G (x; F (·)) = G0 (F (x)) implies Pr (N ≥ n) is the same for all F (·), it
follows that for all n ≥ 3,
Z c
ω=0
Z
η F1 (ω) ω n−3 dω =
c
ω=0
ηF2 (ω) ω n−3 dω
By the Müntz-Szász theorem, it further follows that η F1 (ω) = ηF2 (ω) for almost all ω ∈
(0, c), i.e.
µ
µ
µ
µ
¶
¶
¶
¶
Z c
Z c
1 − e−(t−ω)
1 − e−t
1 − e−(t−ω)
1 − e−t
g0
g0
φF1
dt =
φF2
dt (15)
q
q
q
q
t=ω
t=ω
for almost all ω ∈ (0, c).
We next argue that (15) implies φF1
µ
1 − e−t
q
¶
= φF2
µ
1 − e−t
q
¶
for almost all t ∈ (0, c).
It will suffice to prove that if
µ
¶
Z c
1 − e−(t−ω)
g0
φ (t) dt = 0 for almost all ω ∈ (0, c)
q
t=ω
(16)
then φ (t) = 0 for almost all t ∈ (0, c). Appealing to a change in variables w = c − t and
z = c − ω, (16) can be transformed into the following integral equation:
Z z
a (z − w) b (w) dw = 0 for almost all z ∈ (0, c)
(17)
0
µ
¶
1 − e−x
where a (x) = g0
and b (x) = φ (c − x). Applying Theorem VII in Titchmarsh
q
(1926) [or from Theorem 151 in the more accessible reference Titchmarsh (1948, p. 324-5)],
there exists a c∗ such that a (x) = 0 for all x ∈ (0, c∗ ) and b (x) = 0 for all x ∈ (0, c − c∗ ).
But Assumption 3 implies that there exists an ε > 0 such that g0 (z) > 0 for almost all
z ∈ (0, ε), which in turn implies that a (z) > 0 for almost all z ∈ (0, ε). Hence, c∗ must
equal 0, implying b (z) = 0 for almost all z ∈ (0, c). But then φ (t) = b (c − t) = 0 for almost
all t ∈ (0, c), as claimed.
12
Lastly, we need to show that the statement of the proposition follows from the fact
φF1 ((1 − e−t )/q) = φF2 ((1 − e−t )/q) for almost all t ∈ (0, c). Consider case (a); then
Z 1
¤
£ −1
F1 (u) − F2−1 (u) du = 0
−t
1−e
q
or
Z
1
1−e−t
q
·
¸
F1−1 (u) − F2−1 (u)
du = 0
1 − qu
for almost all t ∈ (0, c). But from Taylor (1965, p. 415), this implies that the function inside
the integral is equal to 0 almost surely, which implies F1−1 (u) = F2−1 (u) almost surely as
claimed. In case (b), with φF given in (10), the fact that φF1 (t) = φF2 (t) for almost all
t ∈ (0, c) implies that for almost all u ∈ (0, 1),
Z
Z 1
¤
£ −1
F1 (un ) − F1−1 (u) dun =
u
1
u
¤
£ −1
F2 (un ) − F2−1 (u) dun
or with φF in (14) we get
¸
¸
Z 1 · −1
Z 1 · −1
F1 (un ) − F1−1 (u)
F2 (un ) − F2−1 (u)
dun =
dun
1 − qun
1 − qun
u
u
Nagaraja and Barlevy (2003) already showed that this implies there exists a constant c such
that F1−1 (u) = F2−1 (u) + c. ¥
Remark 5: From the proof above, we can further deduce what happens when we relax
Assumption 3, i.e. when we assume that G−1
0 (0) > 0. By the Titchmarsh convolution
theorem, for any solution b (w) to (17), there exists a value c∗ such that a (x) = 0 for all
x ∈ (0, c∗ ) and that b (x) = 0 for all x ∈ (0, c − c∗ ). However, without Assumption 3, we
¡
¢
can only conclude that c∗ ≤ − ln 1 − qG−1
0 (0) . Consequently, we can deduce that φ (t) =
¢ ¢
¡
¡
−t
b (c − t) = 0 for almost all t ∈ − ln 1 − qG−1
0 (0) , c , and hence that φF1 ((1 − e )/q) =
¡
¡
¢
¢
φF2 ((1 − e−t )/q) for almost all t ∈ − ln 1 − qG−1
0 (0) , c . In case (a) it would therefore
¡ −1 ¢
−1
follow that F1 (x) = F2 (x) for almost all x > F
G0 (0) , and in case (b) it would follow
that F1−1 (u) = F2−1 (u) + c for almost all u > G−1
0 (0). In other words, one can generalize
Proposition 4 to imply that the moment sequences in the statement of the proposition
¢
¢
¡
¡
uniquely characterize the distribution F over the range F −1 G−1
0 (0) , ∞ .
4. Characterization results for the general case
We now move to the general case of any arbitrary function G (x; F (·)) which satisfies Assumptions 1-3. In this case, it may no longer be true that record moments alone characterize
13
the distribution F . However, record moments and the distribution of the number of records
together do characterize F .
Proposition 5: Suppose Assumptions 1-3 are satisfied, and E (|X2 |) < ∞.
a. If two distributions F1 and F2 give rise to the same sequences E (Rn | N ≥ nj ) and
P
Pr (N = nj ) where j n−1
j = ∞, then F1 = F2 and G1 = G2 almost surely
b. If two distributions F1 and F2 give rise to the same sequences E (Rn − Rn−1 | N ≥ nj )
P
= ∞, then F1−1 (x) = F2−1 (x) + c and G−1
and Pr (N = nj ) where j n−1
1 (x) =
j
G−1
2 (x) + c for some constant c.
Proof : Since both F1 and F2 give rise to the same sequence Pr (N = nj ) where
P
j
n−1
j =
∞, from Lemma 1 and (2) we conclude that Pr (N ≥ n) also match for n ≥ 1, and for almost
all u ∈ (0, 1),
¢
¡
g1 F1−1 (u)
¢
¡ −1
f1 F1 (u)
=
¢
¡
g2 F2−1 (u)
¢.
¡ −1
f2 F2 (u)
Let us write this common function as g0 (u). In contrast to the previous section, g0 (u) now
depends on the sequence Pr (N = nj ) as opposed to a stand-alone function.
Recall that all four moment sequences above can be written as
h
³
´in−3
Z 1 Z un−1 − ln 1−qun−1
1−qu1
qdu1 qdun−1
g0 (u1 ) φF (un−1 )
(n − 3)!P (N ≥ n)
1 − qu1 1 − qun−1
0
0
for appropriately defined φF (·). Using the change of variables
qdun−1
1 − e−t
, un−1 =
1 − qun−1
q
−s
qdu1
1−e
s = − ln (1 − qu1 ) ⇒ dt =
, u1 =
1 − qu1
q
c = − ln (1 − q)
t = − ln (1 − qun−1 ) ⇒ ds =
we can rewrite this expression as
µ
¶
Z cZ c
(t − s)n−3
1 − e−s
g0
φF (t)
dtds
q
(n − 3)!P (N ≥ n)
0
t=s
Setting ω = t − s, we can further rewrite this expression as
µ
¶
Z c Z c
1 − e−(t−ω)
ω n−3
g0
φF (t)
dtdω
q
(n − 3)!P (N ≥ n)
ω=0 t=ω
14
Let us define
µ
¶
1 − e−(t−ω)
φF (t) dt
q
t=ω
Let F1 and F2 denote two continuous cdf’s that give rise to the same sequences. Define N1
ηF (ω) =
Z
c
g0
as the number of records when X2 ∼ F1 and N2 as the number of records when X2 ∼ F2 .
If F1 and F2 give rise to the same moment sequences, then
Z c
Z c
1
1
n−3
η (ω) ω
dω =
η (ω) ω n−3 dω
(n − 3)!P (N1 ≥ n) ω=0 F1
(n − 3)!P (N2 ≥ n) ω=0 F2
Since Pr (N1 ≥ n) = Pr (N2 ≥ n), it follows that
Z c
Z
η F1 (ω) ω n−3 dω =
ω=0
c
ω=0
ηF2 (ω) ω n−3 dω
and then by the Müntz-Szász theorem ηF1 (ω) = ηF2 (ω) almost surely, i.e.
µ
µ
¶
¶
Z c
Z c
1 − e−(t−ω)
1 − e−(t−ω)
g0
g0
φF1 (t) dt =
φF2 (t) dt
q
q
t=ω
t=ω
As in the proof of Proposition 4, we rely on the Titchmarsh convolution theorem to establish
that φF1 (t) = φF2 (t) almost surely, from which we conclude that F1 = F2 in case (a) and
F1−1 (u) = F2−1 (u) + c in case (b).
Next, in case (a), we use the fact that F1 (x) = F2 (x) for almost all x and the fact (from
Lemma 1) that
¢
¢
¡
¡
g2 F2−1 (u)
g1 F1−1 (u)
¢ = ¡ −1
¢
¡
f1 F1−1 (u)
f2 F2 (u)
for almost all u to conclude that
for almost all u ∈ (0, 1). Hence,
Z x
G1 (x) =
¢
¢
¡
¡
g1 F1−1 (u) = g2 F2−1 (u)
F1−1 (0)
g1 (x) dx =
Z
x
F2−1 (0)
g2 (x) dx = G2 (x)
as claimed.
In case (b), we use the fact that F1−1 (u) = F2−1 (u) + c for almost all u, to conclude that
¢
¢
¡
¡
g1 F2−1 (u) + c
g2 F2−1 (u)
¢ = ¡ −1
¢
¡
f1 F2−1 (u) + c
f2 F2 (u)
¢
¢
¡
¡
But for almost every u ∈ (0, 1), it must also follow that f1 F2−1 (u) + c = f2 F2−1 (u) .
Hence, we have
¢
¢
¡
¡
g2 F2−1 (u)
g1 F2−1 (u) + c
¢ = ¡ −1
¢.
¡
f2 F2−1 (u)
f2 F2 (u)
15
This implies that for almost every u ∈ (0, 1),
¢
¢
¡
¡
g1 F2−1 (u) + c = g2 F2−1 (u)
and hence that
G1 (x + c) =
Z
x+c
F2−1 (0)+c
g1 (x) dx =
Z
x
F2−1 (0)
g2 (x) dx = G2 (x)
−1
which implies G−1
1 (u) = G2 (u) + c. This completes the proof. ¥
By a similar argument, one can show that the proposition above remains true if we condition
on the event that N = n rather than on the event that N ≥ n.
Remark 6: If Γ is known, then once we identify F , we can also recover G = Γ (F ). But
Proposition 5 implies that G is itself characterized by the sequences E (Rn | N ≥ n) and
Pr (N = n). Hence, we can test certain conjectures on Γ by checking whether the distribution
Γ (F ) at the F we identify is the same as the G directly implied by the moment sequences
and the distribution of N .
5. Characterization results across GRR models
The modified GRR model we study can be summarized by a triple {Γ, F, q}. So far, we
have implicitly focused on results that characterize F within a given model. That is, for
a given q and Γ, we showed that there is at most one F for which the model is consistent
with a given sequence of record moments and a given distribution for N . In this section,
we ask whether it is possible to characterize the model itself as opposed to the distribution
F within a given model. We show that if two models {Γ1 , F1 , q1 } and {Γ2 , F2 , q2 } yield the
same record moments and the same distribution of the number of records, then q1 = q2 ,
F1 = F2 almost surely, and Γ1 (F1 ) = Γ2 (F2 ) almost surely. In other words, the sequences
considered in Proposition 5 characterize not only F and G but also the distribution of M .
Proposition 6: Suppose two models {Γ1 , F1 , q1 } and {Γ2 , F2 , q2 } both satisfy Assumptions
1-3, and E (|X2 |) < ∞ in both models. Let G1 = Γ1 (F1 ) and G2 = Γ2 (F2 ) and assume
P −1
j nj = ∞.
a. If {Γ1 , F1 , q1 } and {Γ2 , F2 , q2 } give rise to the same sequence Pr (N = nj ), then q1 = q2
16
and (2) holds, i.e.,
for almost all u ∈ (0, 1).
¢
¢
¡
¡
g2 F2−1 (u)
g1 F1−1 (u)
¢ = ¡ −1
¢
¡
f1 F1−1 (u)
f2 F2 (u)
b. If, in addition to the condition in (a), the two GRR models give rise to the same
sequence E (Rn | N ≥ nj ), then G1 = G2 almost surely, and F1 = F2 almost surely.
c. If, in addition to the condition in (a), the two GRR models give rise to the same
sequence
−1
E (Rn − Rn−1 | N ≥ nj ), then there exists a c such that G−1
1 (x) = G2 (x) + c almost
surely, and F1−1 (x) = F2−1 (x) + c almost surely.
Proof : We prove (a). The remaining two claims then follow from Proposition 5.
Since both {G1 , F1 , q1 } and {G2 , F2 , q2 } give rise to the same sequence Pr (N = nj ), then
for j = 1, 2, 3, ...
Z
0
1
¢
¶¸nj −1 ¡ −1
· µ
g1 F1 (u)
1 − q1 u
1 − q1
¢ du
¡
ln
1 − q1 u
1 − q1
f1 F1−1 (u)
¢
¶¸nj −1 ¡ −1
· µ
Z 1
g2 F2 (u)
1 − q2 u
1 − q2
¢ du.
¡
=
ln
1 − q2
f2 F2−1 (u)
0 1 − q2 u
Set t = ln ((1 − q1 u)/(1 − q1 )) and ln ((1 − q2 u)(1 − q2 )) respectively on the left hand side
and right hand side above. We can then rewrite the equation above as
³
³
´´
t
−1 1−(1−q1 )e
Z − ln(1−q1 )
g
F
q1
1 − q1 1 1
³
³
´´ tn−1 dt
t
1−(1−q
−1
1 )e
q1 f1 F
0
1
q1
³
³
´´
t
−1 1−(1−q2 )e
Z − ln(1−q2 )
g
F
2
2
q2
1 − q2
³
³
´´ tn−1 dt.
=
t
1−(1−q
)e
−1
2
q
2
0
f F
2
2
q2
Let us rewrite this equation as
Z
0
− ln(1−q1 )
h1 (t) tn−1 dt =
Z
− ln(1−q2 )
0
q1 1 − q2
h2 (t) tn−1 dt.
q2 1 − q1
We now proceed to prove the claim by contradiction. Suppose wlog that q2 > q1 . Then
− ln (1 − q2 ) > − ln (1 − q1 ) .
17
Define
b
h1 (t) =
Then we have
Z
h1 (t) if t ≤ − ln (1 − q1 )
− ln(1−q2 )
0
0
if t ∈ (− ln (1 − q1 ) , − ln (1 − q2 )) .
b
h1 (t) tn−1 dt =
Z
− ln(1−q2 )
0
q1 1 − q2
h2 (t) tn−1 dt
q2 1 − q1
for all n = 1, 2, 3, ... By Müntz-Szász, it follows that
q1 1 − q2
b
h1 (t) =
h2 (t)
q2 1 − q1
for almost all t ∈ (0, − ln (1 − q2 )), which implies h2 (t) = 0 for almost all t in
(− ln (1 − q1 ) , − ln (1 − q2 )). But this implies g2 (x) = 0 for almost all x in
¤
£ −1
F2 (0) , F2−1 ((q2 − q1 )/[q2 (1 − q1 )]) , which violates Assumption 3.
Given q1 = q2 ≡ q, the fact that both {G1 , F1 , q} and {G2 , F2 , q} give rise to the same
sequence Pr (N = nj ) implies, in view of Lemma 1, that (2) holds. ¥
6. Application
Finally, we discuss how the results of this paper can be used to identify the wage offer
distribution in job search models. In particular, we show that the offer distribution is
identified in a larger class of models than was previously demonstrated in work by Nagaraja
and Barlevy (2003) and Barlevy (2005).
Consider the following model of job search, which is frequently used in labor economics.
(The literature on job search is too vast; for a survey of previous work on the identification
and estimation of these models, see Eckstein and Van den Berg (2005).) At any point
in time, a worker can be either employed or unemployed. While unemployed, all workers
receive a fixed dollar amount X ∗ per unit time (which may be zero). This amount can reflect
unemployment benefits, as well as the monetary value of the leisure she enjoys while not
working. Employed and unemployed workers encounter employers at a constant rate λ per
unit time. Each time a worker encounters an employer, the latter offers her a wage of X that
is drawn independently from a continuous offer distribution F . The worker must then choose
whether to stay on her current job (alternatively, remain unemployed) or accept the new offer
and change employers (alternatively, exit unemployment). In addition, a worker can lose her
18
job, an event that occurs at constant rate δ whenever she is employed. When a worker loses
her job, she cannot recall any of her past offers, and instead becomes unemployed. Workers
are assumed to maximize their earnings. Hence, the optimal strategy for an employed
worker is to only accept offers that surpass her current wage. Similarly, while unemployed,
the worker should only accept offers that exceed X ∗ . We assume F −1 (0) ≥ X ∗ , i.e. all
employers offer at least X ∗ (otherwise their offers would never be accepted).
Let M denote the number of job offers a worker receives between intervening spells of
unemployment, and index the offers according to the order in which they arrive so that Xi
denotes the i-th offer since the worker was last unemployed. Barlevy (2005) shows that
−1
M will have a geometric distribution, i.e. Pr (M = m) = q m−1 p where p = δ (δ + λ)
and
q = 1−p. Given the worker’s strategy, the wages on the jobs the worker accepts corresponds
to records from the sequence {Xi , 1 ≤ i ≤ M }. In the typical datasets economists use,
workers are only queried on the jobs they work on, not on job offers they received but
turned down. Thus, we assume that the only available data consists of {Rn , 1 ≤ n ≤ N },
not the original observations {Xi , 1 ≤ i ≤ M } or even the number of observations M . A
question of interest for economists is whether this data can identify the offer distribution F .
Since R1 = X1 , the distribution F is obviously identified from the empirical distribution
of wages of workers on the first job. However, a key obstacle in taking the above model
to data is that empirically a considerable number of workers voluntarily move into lower
wage jobs, in direct violation of the model. To resolve this discrepancy, economists have
argued that wages in the data are a noisy version of wages in the model, i.e. we observe
not Rn but Rn + εn for some random variable εn where E (εn ) = 0. The εn can be viewed
as measurement error, but alternative interpretations for this term have been offered (see
Barlevy (2005) for a discussion). Once we assume that we only observe Rn + εn , we can no
longer identify F from the distribution of X1 . Previous work, as summarized in Eckstein
and Van den Berg (2005), resorted to parametric assumptions on F and the distribution
of ε to proceed with estimation. By contrast, Nagaraja and Barlevy (2003) and Barlevy
(2005) argued that characterization results for the GRR model imply that F is identified
non-parametrically, since one can still recover E (Rn ) from noise-ridden data.
However, in order to apply this identification result, we need to keep track of all jobs
between spells of unemployment so that we can determine which record number n each job
represents. Unfortunately, this is not possible in many datasets. In particular, many surveys
19
collect data on workers that are already employed. For those workers, we have no way of
classifying which record number to assign to the jobs we observe for them. Although we
could wait until the worker is next unemployed, unemployment is often a sufficiently low
probability event that a large part of the data would have to be thrown out. The results
of this paper suggest a way to incorporate data for workers who are already employed. In
particular, we know from previous work on search models, e.g. Burdett and Mortensen
(1998), that the economy described by this model converges in the limit to a steady state
in which the fraction of all employed workers who earn a wage of x or less is equal to
G (x) =
F (x)
1 + λ/δ (1 − F (x))
(18)
Moreover, the number of offers the worker receives starting from any job continues to have
a geometric distribution. Thus, as long as the economy we consider is at its steady state,
the wages on the jobs we observe for a randomly chosen employed worker will correspond to
records from a sequence {Xi , 1 ≤ i ≤ M } where M has a geometric distribution, X1 ∼ G as
defined in (18), and X2 , ..., XM ∼ F . Since G = G0 (F (x)), we can appeal to Proposition 4
to argue that average wages or average wage changes identify the wage offer distribution F .
For example, if the average wage gains of workers is constant regardless of how many jobs
they have changed since the first job we observe them on, the wage offer distribution must
be exponential.
More generally, our results can be applied whenever the distribution of the wage on the
first job we observe for a worker differs from the offer distribution F . For example, some
surveys focus on the poor, and use the initial earnings of a worker as a criterion for selection
into the survey. In this case, even if we could track workers from their very first job out of
unemployment, the distribution of wages on the worker’s first job would correspond to
(
F (x) /z if F (x) < z
(19)
G (x) =
1
else
where z reflects the percentile of the threshold wage workers must earn within the wage
offer distribution to qualify for the survey. Once again, we can appeal to Proposition 4 to
argue that average wages or average wage changes identify the offer distribution F .
Discrepancies between the wage on the first job we observe for a worker and the offer
distribution F (·) are not confined to sampling issues. Suppose we could track workers from
the first job out of unemployment and that no wages were censored. However, suppose
the amount workers earn while unemployed varies across workers. For example, they might
20
enjoy leisure differently, or they might earn different unemployment benefits (which is not
unreasonable given these often depend on what the worker earned on his last job before
becoming unemployed). Let H (x) denote the fraction of workers whose X ∗ is x or less, and
suppose H −1 (0) ≤ F −1 (0). Workers whose X ∗ will hold out for a higher wage before they
accept a job offer. The wage on the first job out of unemployment for a worker chosen at
random from H is now given by
Rx
G (x; F (·)) =
R−∞
∞
−∞
H (w) dF (w)
H (w) dF (w)
=
R F (x)
0
R1
0
¡
¢
H F −1 (u) du
H (F −1 (u)) du
.
(20)
Workers continue to draw offers from F at rate λ, so the wages of a worker chosen at random
between two consecutive unemployment spells will correspond to records from a sequence
{Xi , 1 ≤ i ≤ M } where M has a geometric distribution, X1 ∼ G as defined in (20), and
X2 , ..., XM ∼ F . Since G (x; F (·)) cannot be represented as G0 (F (x)) for some function
G0 (·), we must appeal to Proposition 5 to argue that average wages or average wage changes,
together with the distribution of the number of jobs workers hold between unemployment
spells, identify F . If the distribution of reservation wages H is itself unknown, Proposition
5 implies we can also identify the distribution of wages of workers on their first job G.
It is easily demonstrated that given F and G, one can recover H. Thus, when workers
have different reservation wages, not only is the common offer distribution F they face
still identified, but so is the distribution of X ∗ across workers. Thus, we could infer the
distribution of how much workers value leisure from the extent of job mobility we observe
for them once they become employed. Lastly, Proposition 6 tells us that we do not need to
know the ratio λ/δ in advance to identify F , since we can recover it from data on N , i.e. the
distribution of how many jobs workers hold between consecutive unemployment spells. For
an empirical implementation of these ideas using panel data on young workers, see Barlevy
and Nagaraja (2005).
References:
Arnold, B. C., Balakrishnan N. and Nagaraja H. N. (1998). Records. New York: John Wiley
and Sons.
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22
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Outsourcing Business Services and the Role of Central Administrative Offices
Yukako Ono
WP-02-01
Strategic Responses to Regulatory Threat in the Credit Card Market*
Victor Stango
WP-02-02
The Optimal Mix of Taxes on Money, Consumption and Income
Fiorella De Fiore and Pedro Teles
WP-02-03
Expectation Traps and Monetary Policy
Stefania Albanesi, V. V. Chari and Lawrence J. Christiano
WP-02-04
Monetary Policy in a Financial Crisis
Lawrence J. Christiano, Christopher Gust and Jorge Roldos
WP-02-05
Regulatory Incentives and Consolidation: The Case of Commercial Bank Mergers
and the Community Reinvestment Act
Raphael Bostic, Hamid Mehran, Anna Paulson and Marc Saidenberg
WP-02-06
Technological Progress and the Geographic Expansion of the Banking Industry
Allen N. Berger and Robert DeYoung
WP-02-07
Choosing the Right Parents: Changes in the Intergenerational Transmission
of Inequality Between 1980 and the Early 1990s
David I. Levine and Bhashkar Mazumder
WP-02-08
The Immediacy Implications of Exchange Organization
James T. Moser
WP-02-09
Maternal Employment and Overweight Children
Patricia M. Anderson, Kristin F. Butcher and Phillip B. Levine
WP-02-10
The Costs and Benefits of Moral Suasion: Evidence from the Rescue of
Long-Term Capital Management
Craig Furfine
WP-02-11
On the Cyclical Behavior of Employment, Unemployment and Labor Force Participation
Marcelo Veracierto
WP-02-12
Do Safeguard Tariffs and Antidumping Duties Open or Close Technology Gaps?
Meredith A. Crowley
WP-02-13
Technology Shocks Matter
Jonas D. M. Fisher
WP-02-14
Money as a Mechanism in a Bewley Economy
Edward J. Green and Ruilin Zhou
WP-02-15
1
Working Paper Series (continued)
Optimal Fiscal and Monetary Policy: Equivalence Results
Isabel Correia, Juan Pablo Nicolini and Pedro Teles
WP-02-16
Real Exchange Rate Fluctuations and the Dynamics of Retail Trade Industries
on the U.S.-Canada Border
Jeffrey R. Campbell and Beverly Lapham
WP-02-17
Bank Procyclicality, Credit Crunches, and Asymmetric Monetary Policy Effects:
A Unifying Model
Robert R. Bliss and George G. Kaufman
WP-02-18
Location of Headquarter Growth During the 90s
Thomas H. Klier
WP-02-19
The Value of Banking Relationships During a Financial Crisis:
Evidence from Failures of Japanese Banks
Elijah Brewer III, Hesna Genay, William Curt Hunter and George G. Kaufman
WP-02-20
On the Distribution and Dynamics of Health Costs
Eric French and John Bailey Jones
WP-02-21
The Effects of Progressive Taxation on Labor Supply when Hours and Wages are
Jointly Determined
Daniel Aaronson and Eric French
WP-02-22
Inter-industry Contagion and the Competitive Effects of Financial Distress Announcements:
Evidence from Commercial Banks and Life Insurance Companies
Elijah Brewer III and William E. Jackson III
WP-02-23
State-Contingent Bank Regulation With Unobserved Action and
Unobserved Characteristics
David A. Marshall and Edward Simpson Prescott
WP-02-24
Local Market Consolidation and Bank Productive Efficiency
Douglas D. Evanoff and Evren Örs
WP-02-25
Life-Cycle Dynamics in Industrial Sectors. The Role of Banking Market Structure
Nicola Cetorelli
WP-02-26
Private School Location and Neighborhood Characteristics
Lisa Barrow
WP-02-27
Teachers and Student Achievement in the Chicago Public High Schools
Daniel Aaronson, Lisa Barrow and William Sander
WP-02-28
The Crime of 1873: Back to the Scene
François R. Velde
WP-02-29
Trade Structure, Industrial Structure, and International Business Cycles
Marianne Baxter and Michael A. Kouparitsas
WP-02-30
Estimating the Returns to Community College Schooling for Displaced Workers
Louis Jacobson, Robert LaLonde and Daniel G. Sullivan
WP-02-31
2
Working Paper Series (continued)
A Proposal for Efficiently Resolving Out-of-the-Money Swap Positions
at Large Insolvent Banks
George G. Kaufman
WP-03-01
Depositor Liquidity and Loss-Sharing in Bank Failure Resolutions
George G. Kaufman
WP-03-02
Subordinated Debt and Prompt Corrective Regulatory Action
Douglas D. Evanoff and Larry D. Wall
WP-03-03
When is Inter-Transaction Time Informative?
Craig Furfine
WP-03-04
Tenure Choice with Location Selection: The Case of Hispanic Neighborhoods
in Chicago
Maude Toussaint-Comeau and Sherrie L.W. Rhine
WP-03-05
Distinguishing Limited Commitment from Moral Hazard in Models of
Growth with Inequality*
Anna L. Paulson and Robert Townsend
WP-03-06
Resolving Large Complex Financial Organizations
Robert R. Bliss
WP-03-07
The Case of the Missing Productivity Growth:
Or, Does information technology explain why productivity accelerated in the United States
but not the United Kingdom?
Susanto Basu, John G. Fernald, Nicholas Oulton and Sylaja Srinivasan
WP-03-08
Inside-Outside Money Competition
Ramon Marimon, Juan Pablo Nicolini and Pedro Teles
WP-03-09
The Importance of Check-Cashing Businesses to the Unbanked: Racial/Ethnic Differences
William H. Greene, Sherrie L.W. Rhine and Maude Toussaint-Comeau
WP-03-10
A Firm’s First Year
Jaap H. Abbring and Jeffrey R. Campbell
WP-03-11
Market Size Matters
Jeffrey R. Campbell and Hugo A. Hopenhayn
WP-03-12
The Cost of Business Cycles under Endogenous Growth
Gadi Barlevy
WP-03-13
The Past, Present, and Probable Future for Community Banks
Robert DeYoung, William C. Hunter and Gregory F. Udell
WP-03-14
Measuring Productivity Growth in Asia: Do Market Imperfections Matter?
John Fernald and Brent Neiman
WP-03-15
Revised Estimates of Intergenerational Income Mobility in the United States
Bhashkar Mazumder
WP-03-16
3
Working Paper Series (continued)
Product Market Evidence on the Employment Effects of the Minimum Wage
Daniel Aaronson and Eric French
WP-03-17
Estimating Models of On-the-Job Search using Record Statistics
Gadi Barlevy
WP-03-18
Banking Market Conditions and Deposit Interest Rates
Richard J. Rosen
WP-03-19
Creating a National State Rainy Day Fund: A Modest Proposal to Improve Future
State Fiscal Performance
Richard Mattoon
WP-03-20
Managerial Incentive and Financial Contagion
Sujit Chakravorti, Anna Llyina and Subir Lall
WP-03-21
Women and the Phillips Curve: Do Women’s and Men’s Labor Market Outcomes
Differentially Affect Real Wage Growth and Inflation?
Katharine Anderson, Lisa Barrow and Kristin F. Butcher
WP-03-22
Evaluating the Calvo Model of Sticky Prices
Martin Eichenbaum and Jonas D.M. Fisher
WP-03-23
The Growing Importance of Family and Community: An Analysis of Changes in the
Sibling Correlation in Earnings
Bhashkar Mazumder and David I. Levine
WP-03-24
Should We Teach Old Dogs New Tricks? The Impact of Community College Retraining
on Older Displaced Workers
Louis Jacobson, Robert J. LaLonde and Daniel Sullivan
WP-03-25
Trade Deflection and Trade Depression
Chad P. Brown and Meredith A. Crowley
WP-03-26
China and Emerging Asia: Comrades or Competitors?
Alan G. Ahearne, John G. Fernald, Prakash Loungani and John W. Schindler
WP-03-27
International Business Cycles Under Fixed and Flexible Exchange Rate Regimes
Michael A. Kouparitsas
WP-03-28
Firing Costs and Business Cycle Fluctuations
Marcelo Veracierto
WP-03-29
Spatial Organization of Firms
Yukako Ono
WP-03-30
Government Equity and Money: John Law’s System in 1720 France
François R. Velde
WP-03-31
Deregulation and the Relationship Between Bank CEO
Compensation and Risk-Taking
Elijah Brewer III, William Curt Hunter and William E. Jackson III
WP-03-32
4
Working Paper Series (continued)
Compatibility and Pricing with Indirect Network Effects: Evidence from ATMs
Christopher R. Knittel and Victor Stango
WP-03-33
Self-Employment as an Alternative to Unemployment
Ellen R. Rissman
WP-03-34
Where the Headquarters are – Evidence from Large Public Companies 1990-2000
Tyler Diacon and Thomas H. Klier
WP-03-35
Standing Facilities and Interbank Borrowing: Evidence from the Federal Reserve’s
New Discount Window
Craig Furfine
WP-04-01
Netting, Financial Contracts, and Banks: The Economic Implications
William J. Bergman, Robert R. Bliss, Christian A. Johnson and George G. Kaufman
WP-04-02
Real Effects of Bank Competition
Nicola Cetorelli
WP-04-03
Finance as a Barrier To Entry: Bank Competition and Industry Structure in
Local U.S. Markets?
Nicola Cetorelli and Philip E. Strahan
WP-04-04
The Dynamics of Work and Debt
Jeffrey R. Campbell and Zvi Hercowitz
WP-04-05
Fiscal Policy in the Aftermath of 9/11
Jonas Fisher and Martin Eichenbaum
WP-04-06
Merger Momentum and Investor Sentiment: The Stock Market Reaction
To Merger Announcements
Richard J. Rosen
WP-04-07
Earnings Inequality and the Business Cycle
Gadi Barlevy and Daniel Tsiddon
WP-04-08
Platform Competition in Two-Sided Markets: The Case of Payment Networks
Sujit Chakravorti and Roberto Roson
WP-04-09
Nominal Debt as a Burden on Monetary Policy
Javier Díaz-Giménez, Giorgia Giovannetti, Ramon Marimon, and Pedro Teles
WP-04-10
On the Timing of Innovation in Stochastic Schumpeterian Growth Models
Gadi Barlevy
WP-04-11
Policy Externalities: How US Antidumping Affects Japanese Exports to the EU
Chad P. Bown and Meredith A. Crowley
WP-04-12
Sibling Similarities, Differences and Economic Inequality
Bhashkar Mazumder
WP-04-13
Determinants of Business Cycle Comovement: A Robust Analysis
Marianne Baxter and Michael A. Kouparitsas
WP-04-14
5
Working Paper Series (continued)
The Occupational Assimilation of Hispanics in the U.S.: Evidence from Panel Data
Maude Toussaint-Comeau
WP-04-15
Reading, Writing, and Raisinets1: Are School Finances Contributing to Children’s Obesity?
Patricia M. Anderson and Kristin F. Butcher
WP-04-16
Learning by Observing: Information Spillovers in the Execution and Valuation
of Commercial Bank M&As
Gayle DeLong and Robert DeYoung
WP-04-17
Prospects for Immigrant-Native Wealth Assimilation:
Evidence from Financial Market Participation
Una Okonkwo Osili and Anna Paulson
WP-04-18
Institutional Quality and Financial Market Development:
Evidence from International Migrants in the U.S.
Una Okonkwo Osili and Anna Paulson
WP-04-19
Are Technology Improvements Contractionary?
Susanto Basu, John Fernald and Miles Kimball
WP-04-20
The Minimum Wage, Restaurant Prices and Labor Market Structure
Daniel Aaronson, Eric French and James MacDonald
WP-04-21
Betcha can’t acquire just one: merger programs and compensation
Richard J. Rosen
WP-04-22
Not Working: Demographic Changes, Policy Changes,
and the Distribution of Weeks (Not) Worked
Lisa Barrow and Kristin F. Butcher
WP-04-23
The Role of Collateralized Household Debt in Macroeconomic Stabilization
Jeffrey R. Campbell and Zvi Hercowitz
WP-04-24
Advertising and Pricing at Multiple-Output Firms: Evidence from U.S. Thrift Institutions
Robert DeYoung and Evren Örs
WP-04-25
Monetary Policy with State Contingent Interest Rates
Bernardino Adão, Isabel Correia and Pedro Teles
WP-04-26
Comparing location decisions of domestic and foreign auto supplier plants
Thomas Klier, Paul Ma and Daniel P. McMillen
WP-04-27
China’s export growth and US trade policy
Chad P. Bown and Meredith A. Crowley
WP-04-28
Where do manufacturing firms locate their Headquarters?
J. Vernon Henderson and Yukako Ono
WP-04-29
Monetary Policy with Single Instrument Feedback Rules
Bernardino Adão, Isabel Correia and Pedro Teles
WP-04-30
6
Working Paper Series (continued)
Firm-Specific Capital, Nominal Rigidities and the Business Cycle
David Altig, Lawrence J. Christiano, Martin Eichenbaum and Jesper Linde
WP-05-01
Do Returns to Schooling Differ by Race and Ethnicity?
Lisa Barrow and Cecilia Elena Rouse
WP-05-02
Derivatives and Systemic Risk: Netting, Collateral, and Closeout
Robert R. Bliss and George G. Kaufman
WP-05-03
Risk Overhang and Loan Portfolio Decisions
Robert DeYoung, Anne Gron and Andrew Winton
WP-05-04
Characterizations in a random record model with a non-identically distributed initial record
Gadi Barlevy and H. N. Nagaraja
WP-05-05
7
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