'&)
CJeWey
HD28
.M41A
iHI
MIT LIBRARIES
3 9080 00932 7856
Econometric Evaluation of Asset Pricing Models
August 1993
Revised July 1994
WP
Sloan
3606
Lars Peter Hansen
University of Chicago, NBER and NORC
John Heaton
and NBER
M.I.T.
,
Erzo Luttmer
Northwestern University
Correspondence and requests for reprints to:
John Heaton
Sloan School of Management
E52-435, MIT
50 Memorial Drive
Cambridge, MA 02142
(617) 253-7218
We thank Craig Burns ide, Bo Honore, Andrew Lo, Marc Roston, Whitney Newey,
Jose Scheinkman, Jean-Luc Vila, Jiang Wang, Amir Yaron and especially Ravi
Jagannathan, Richard Green (the editor) and an anonymous referee for helpful
comments and discussions.
We also received valuable remarks from seminar
participants at the 1992 meetings of the Society of Economic Dynamics and
Control, the 1992 NBER Summer Institute and at Cornell, Duke, L.S.E. and
Waterloo Universities.
Finally, we gratefully acknowledge the financial
assistance of the International Financial Services Research Center at M.I.T.
(Heaton), the National Science Foundation (Hansen and Heaton) and the Sloan
Foundation (Heaton and Luttmer).
Abstract
In
this
paper
we
provide
econometric
tools
for
evaluation
the
of
intertemporal asset pricing models using specification-error and volatility
bounds.
We formulate analog estimators of these bounds, give conditions for
consistency and derive the limiting distribution of these estimators.
analysis
incorporates market
frictions such as
proportional transactions costs.
use
the
methods
to
assess
The
short-sale constraints and
Among several applications we show how to
specific
asset
pricing
models
and
nonparametric characterizations of asset pricing anomalies.
JUU281995
LlBRAR,ES
to
provide
In this paper we provide statistical methods for assessing asset-pricing
using
models
procedures
specification-error
can
account
for
volatility
and
market
frictions
short-sale constraints,
and are easier to
interpret
For
these
implement,
the
most
part
transactions
to
asset-pricing models.
statistical
The
bounds.
due
costs
or
tests of
than standard
methods are quite easy
to
They are designed to
even when market frictions are considered.
failures of some popular
provide a better understanding of the statistical
asset-pricing models and to offer guidance in improving these models.
Models of asset pricing with frictionless markets imply that asset prices
represented by a stochastic discount factor or pricing kernel.
can be
stochastic discount
and,
For example,
"discounts"
payoffs
in
in the Capital Asset Pricing Model
given by a constant plus a scale multiple of
portfolio.
each state of
world
the
adjusts the price according to the riskiness of the
as a consequence,
payoff.
is
factor
A
the discount factor
return on the market
the
In the Consumption-Based CAPM the discount factor is given by the
intertemporal marginal rate of substitution of an investor.
The
implications of particular models with observable
(up
to
a
finite
number of parameters) stochastic discount factors are often tested by looking
directly at the average "pricing errors" of the models.
Formal statistical
tests are performed using a time series of portfolio payoffs and prices by
examining whether the sample analogs of the average and predicted prices are
significantly
procedure,
different
see
Hansen
from
and
each
other.
Singleton
examples
For
(1982),
Brown
this
of
and
Gibbons
type
of
(1985),
MacKinlay and Richardson (1991) and Epstein and Zin (1991).
While tests such as these can be informative,
interpret the resulting statistical rejections.
directly applicable
costs
or
short-sale
when
there
constraints.
are
market
Extending
it
is often difficult
Further,
frictions
the
such
models
to
these tests are not
to
as
transactions
allow for
such
frictions entails inequalities instead of the pricing equalities that prevail
in frictionless market models
1993).
can not
tests
these
Finally,
see Prisman
(e.g.,
1986 and Jouini and Kallal
used when
be
candidate discount
the
factor depends on variables unavailable to the econometrician.
As an alternative to considering the average pricing errors of a model,
consider
we
different
a
specification-error
bounds
volatility
set
bounds
of
Hansen
tests
of
Hansen
of
Jagannathan
Jagannathan
and
diagnostics
and
and
using
the
and
the
(1993),
We
(1991).
also
consider
extensions of these tests and diagnostics, developed by He and Modest (1993)
and Luttmer
(1994),
that handle transactions costs,
short-sale restrictions
We develop an econometric methodology to provide
and other market frictions.
consistent estimators of the specification-error and volatility bounds,
and
an asymptotic distribution theory that is easy to implement and that can be
used to make statistical inferences.
Among other
whether
a
volatility
information
specific model
bounds
results
the
things,
the
of
implicit
about
the
in
means
in
this paper allow one
stochastic
returns;
deviations
standard
and
factor
discount
asset-market
to:
test
satisfies
the
compare
the
(ii)
of
(i)
discount
factors
contained in different sets of asset returns; and (iii) test hypotheses about
the size of possible pricing errors of misspecified asset pricing models.
While Burnside
and Cecchetti,
(1994)
Lam and Mark
tests of models of the discount factor along the lines of
are
based
on
a
parameterization
different
yields
accommodate market
(ii),
our
results
parameterization
tests
frictions
permit
in
these
that
a
are
of
the
simpler
to
among
data
(iii)
is
to
shift
the
focus of
statistical
analyses
Our
bound.
and
implement
In
sets
independent of a specific stochastic discount factor model.
for
their tests
(i),
volatility
straightforward manner.
comparisons
have devised
(1993)
can
regards
to
be
to
made
Our motivation
of
asset
pricing
away
models
whether
from
the
models
correctly
are
specified
and
towards
measuring the extent to which they are misspecif ied.
The rest of the paper is organized as follows.
the
1993),
volatility
and
specification
He and Modest
(1993)
bounds
and Luttmer
In Section
Hansen
of
(1994).
we review
1
Jagannathan
and
(1991,
We show formally that
the
volatility bound can be viewed as a special case of the specification-error
bound.
This permits us
develop the underlying econometric
to
tools
in
a
unified way.
In Section 2 we provide consistency and asymptotic distribution
results
estimators
for
applications
our
of
results
Section
independently.
the
of
3.
of
bounds.
In
Section
2,
A shows how to use
models of the discount factor.
In
Section
each
of
present
we
3
which
be
can
the volatility bounds
Section 3.B we extend
to
two
read
test
distribution
the
theory of specification-error bound to the case where there are parameters of
the discount factor proxy that are unknown and must be estimated.
Section
discusses
4
the
distribution
limiting
parameters
the
of
underlying the bounds both with and without market frictions.
things,
is
these results can be used to determine whether the volatility bound
degenerate
sharpens
Among other
the
or
more
bound.
generally
whether
Finally,
Section
additional
5
describes
security
some
market
extensions
data
and
provides some concluding remarks.
1.
General Hodel and Bounds
Our starting point is a model in which asset prices are represented by a
stochastic discount factor or pricing kernel.
pricing
subject
to
transactions
costs,
we
constraints for a subset of the securities.
is an extreme version of a transactions cost,
To accommodate security market
permit
there
to
be
short-sale
Although a short-sale constraint
other proportional transactions
.
costs such as bid-ask spreads can also be handled with this formalism.
in Foley
is done as
constructing
according
A short-sale constraint
sold.
and Kallal
Jouini
(1970),
payoffs
two
to
whether
(1993 ^
and Luttmer
security
a
is
(1994)
by
purchased
or
securities to
imposed on both artificial
is
This
and a bid-ask spread is
enforce the distinction between a buy and a sell,
modeled by making the purchase price higher than the sale price.
Suppose
vector of security market payoffs used
the
analysis is denoted
The vector x
x.
is
used
to
in
an econometric
generate a collection of
payoffs formed using portfolio weights in a closed convex cone C of R
P
=
{p
:
to
C
[x
'
R
is
,x ']
where
constraints
constraints.
More
.
and
the
contains
x
If
partition
generally,
contains
x
(1)
incorporate all of the short-sale constraints
imposed in the econometric investigation.
then
:
p = a'x for some a € C}
The cone C is constructed
n
n
k
the
there are no market frictions,
x
components
two
not
subject
components
I
components:
into
subject
to
to
x'
=
short-sale
short-sale
Then the cone C is formed by taking the Cartesian product of R
and the nonnegative orthant of R
I
.
Let q denote the random vector of prices corresponding to the vector x
of securities payoffs.
assets
are
traded
and
These prices are observed by investors at
are
permitted
to
be
random because
reflect conditioning information available to the investors.
the
the time
prices
may
In the absence
of short sales constraints, prices can be represented by:
where
m
q
=
E(mx|?)
is
a
stochastic
(2)
discount
factor
and
?
is
the
information
set
available to investors at the time of trade.
Since it is difficult to model
empirically the conditioning information available to investors,
we
instead
work with the average or expected value of (2):
Eq
Some
-
Emx =
conditioning
(3)
.
information
can
be
incorporated
in
the
usual
way
by
multiplying the original set of payoffs and prices by random variables in the
conditioning information of economic agents.
More generally in the case of market frictions, x is partitioned in the
manner described previously and the pricing is represented by:
Eq
Eq
n
-
s
-
=
Emx"
Emx
s
£
(4)
.
For notational simplicity we write (4) as:
Eq
-
Emx € C
(5)
where the elements of C
are of
absence of frictions we take C
encompasses
(3).
The
vector of payoffs x
s
the
to contain only the zero vector so that
(5)
pricing
the
inequality
subject
to
nonnegative.
In
the
form
(0,0')',
restriction
(3
emerges
because
short-sale constraints must allow for the
possibility that these constraints bind and hence contribute positively to
the market price vector.
Haintained Assumptions
l.A
There are three restrictions on the vector of payoffs and prices that
are central to our analysis.
the second
The first is a moment restriction,
to the absence of arbitrage on the space of portfolio payoffs,
is equivalent
and the third eliminates redundancy in the securities.
For pricing relation (5) to have content, we maintain:
Assumption 1.1:
Assumption 1.2:
Further for a €
<«, £|q|<
£|x|
For any a € C,
C,
a'
Eq £
a.'
Eq
if a'x £
>
Recall that C is the Cartesian product of R
I
IR
,
which captures
Assumption 1.2
is
the
a
and Probia' x
>
0}
>
0.
if a'x = 0.
short-sale
statement of
expected prices and modified
to
and the nonnegative orthant of
restrictions on some
the
of
the
securities.
Principle of No-Arbitrage applied
account
for
the
fact
that
C need not
to
be
linear [e.g., see Kreps (1981), Prisman (1986), Jouini and Kallal (1993), and
Luttmer
(1994)].
It
guarantees that there exists a non-negative stochastic
discount factor m with finite second moment such that (5) holds.
Kallal
(1993)
discuss additional assumptions that
imply
Jouini and
the existence of
a
positive discount factor satisfying (5), however in our analysis we consider
only nonnegative discount factors.
Next we limit the construction of x by ruling out redundancies
in the
securities:
Assumption 1.3:
a = a
.
If a'x = a 'x and a' Eg = a
'
Eq for some a and a
in C,
then
In the absence of transaction costs,
Assumption 1.3 precludes the possibility
that the second moment matrix of x is singular.
a
nontrivial
probability one.
linear
combination of
linear
light of
In
combination would
(5),
have
to
Otherwise,
there would exist
the
payoff vector x
the
(expected) price of
be
zero,
violating
that
is
zero with
this nontrivial
Assumption
To
1.3.
accommodate securities whose purchase price differs from the sale price, we
permit
second moment matrix of
the
Assumption
requires
then
1.3
composite vector x to be singular.
the
distinct
that
portfolio
l.B
used
weights
construct the same payoff must have distinct expected prices.
to
2
Minimum-Distance Problems
There are two problems that underlie most of our analysis.
M denote
Let
the set of all random variables with finite second moments that satisfy (5),
and let
M
be the set of all nonnegative random variables in M.
Assumption
implies
1.2
satisfies
so
(5)
that
there
that
is
a
nonnegative
both sets are nonempty.
variable for a stochastic discount factor that,
satisfy relations (5).
Let
discount
y denote
Recall that
factor
some
2
=
"proxy"
strictly speaking, does not
Following Hansen and Jagannathan (1993), we consider
the following two ad hoc least squares measures of misspecif ication:
8
that
min £[(y
mzM
- m)
min £[(y
meM
- m)
3
2
]
,
]
.
(6)
and
2
(7)
Clearly, the specification-error bound implied by (7) is no smaller than that
implied by
(6)
since
solutions to (6) and
it
(7)
is
obtained using a smaller constraint
set.
The
are the objects we are interested in estimating and
s
inferences
making
Sections
about.
and
2
provide
3
large-sample
justifications for the solutions to sample counterparts to these optimization
problems.
setting
By
collapse
the
finding
to
proxy
to
y
bounds
on
the
zero,
the
second
error
problems
stochastic
discount
specification
moment
of
factors as constructed by Hansen and Jagannathan (1991), He and Modest (1993)
and
Luttmer
(1994).
In
particular,
derived
bounds
the
in
Hansen
Jagannathan (1991) are obtained by setting y to zero and solving (6) and
when there are no short-sale constraints imposed (when C is set to R
n
);
and
(7)
the
bound derived in He and Modest (1993) is obtained by solving (6) for y set to
and the bound derived by Luttmer (1994)
zero;
y set
to
These second
zero.
deriving feasible
regions
for
means
is obtained by solving
and
(7)
subsequently be used
bounds will
moment
for
in
standard deviations
of
stochastic
and
in
developing
discount factors.
In
solving
the
squares problems
least
(6)
and
econometric methods associated with those problems,
study the conjugate maximization problems.
5
2
2
=
max{Ey
aeC
=
max {£y
-
£[(y
-
x'a)
2
]
it
(7)
is most
convenient to
They are given by
- 2a'
Eg}
(8)
,
and
S
2
where the notation h
2
-
£[(y
-
x'a)
+2
denotes max{h,0}.
by introducing Lagrange multipliers on
]
- 2a' Eg}
(9)
The conjugate problems are obtained
the
pricing constraints
exploiting the familiar saddle point property of
in
the Lagrangian.
(5)
and
The
a'
then have interpretations as the multipliers on the pricing constraints.
The conjugate problems in (8) and (9) are convenient because the choice
variables are finite-dimensional vectors whereas the choice variables in the
original least squares problems (6) and (7) are random variables that reside
possibly
in
problem
infinite-dimensional
constraint
sets.
the conjugate problems are justified formally
of
1993) and Luttmer
(1991,
and S
estimators of S
and
so
In
(9).
solutions
In
is a standard quadratic programming problem.
(8)
doing
problems
to
in
fact,
optimization
The specifications
Hansen and Jagannathan
In Section 2 we develop the properties of
(1994).
based on time series sample analogs to problems (8)
rely
we
and
(8)
on
(9)
several
important
and
an
on
proprieties
additional
of
the
identification
assumption.
Notice that the criteria for the maximization problems are concave in a
and that the first-order conditions for the solutions are given by:
Eq
-
-
£[(y
x'a)x]
e C
(10)
in the case of problem (8) and
Eq
-
+
EUy
- x'a) x)
€
in the case of problem (9),
Interpreting
conditions.
C
(11)
along with the respective complementary slackness
the
first-order
conditions
for
these
problems,
observe that associated with a solution to problem (8) is a random variable m
=
(y
-
x'oc)
in
M and associated with
nonnegative random variable m =
the unique
(up
to
(y - x' a)*
a
solution
in M*
.
to
problem
(9)
is
a
These random variables are
the usual equivalence class of random variables
that are
equal with probability one) solutions to the original least squares problems
(6)
and (7).
Consistency of the estimators of 5
and 5
relies upon the fact that the
Compactness in the case of (8)
sets of solutions to (8) and (9) are compact.
is easily established.
Since Assumption 1.3 eliminates redundant securities
and the random variable
-
(y
x'
is uniquely determined,
a.)
conjugate problem (8) is also unique.
criterion must be the same for all
however.
case
this
In
x' a)
is uniquely determined,
hand,
the random variable (y - x'a)
as
solutions,
implying that
they all must
The solution to conjugate problem (9) may
have the same expected price a' Eg.
not be unique,
the solution a to
This follows because the value of the
is
the
truncated random variable
-
(y
the expected price a' Eg.
On the other
is not necessarily unique,
so we can not
exploit Assumption 1.3 to verify that the solution a is unique.
The set of
solutions is convex due to the concavity of the criterion and the convexity
of the constraint set.
in asymptotic distribution
As is typical
need an
identification restriction that
a sufficient
is also compact.
it
theory,
there
is
a
in Section 2 we will
unique solution to
except when all prices are constant.
conjugate problems,
solutions is convex,
the
As is shown in Appendix A,
Since the set of
local uniqueness implies global uniqueness.
condition for local uniqueness,
let x
the
To display
denote the component of
composite payoff vector x for which the pricing relation is
satisfied
with equality:
=
Emx*
where g
x
Eg*
(12)
is the corresponding price vector.
may contain elements of x
event {m>0>.
s
.
Let 1,~ „,
Notice that in addition to x
Ex x 'l/~
>0 \
is nonsingular.
10
,
be the indicator function for the
A sufficient condition for local uniqueness is that
Assumption 1.4:
n
To
why
see
this
complementary
is
valid
a
sufficient
conditions
slackness
we
condition,
know
observe
that
multipliers
the
a
from
are
the
zero
whenever the pricing constraints are satisfied with a strict inequality.
a result m is given by (y - x
E«
£
=
^
{
'/3)
for some vector
Eixx,1
S>o}
As
and consequently,
J3,
(13)
Cm >o}^
When the matrix £(x x 'l/~ sn i) is nonsingular, we can solve (13) for
{m>0>
l.C
Volatility Bounds and Restrictions on Means
The
second moment
bounds described
in
the
previous subsection can be
converted into standard deviation bounds via the formulas:
a
o-
~2
where 6
2
=
IS
=
r
-
~2
[8
-
{Em)
,„
W2
2
(14)
]
,2,1/2
(Em)
]
~2
and 6 are constructed by setting the proxy to zero.
Em is equal to
the average price of the unit payoff when trade in this payoff is not subject
to
transaction costs.
there
If
are
transactions
costs
or
if
no
data
is
available on the price of a conditionally riskless payoff then Em cannot be
identified.
In these circumstances,
volatility bounds can still be obtained
for each choice of Em by adding a unit payoff to P (augmenting x with a
and assigning a price
of
Em to
that
payoff
(augmenting Eq with Em).
1)
In
forming the augmented cone, there should be no short-sale constraints imposed
on
the
additional
obtained using (8),
security.
(9)
Mean-specific
volatility bounds
can
then be
and (14).
Although Em may not be identified,
the Principle of No-Arbitrage does
ooo
put bounds on the admissible values of Em: Em € [A ,v
]
where A
is the lower
bound
arbitrage
and
v
arbitrage
upper
the
is
bounds
These
bound.
are
computed using formulas familiar from derivative claims pricing:
infia'Eq
A
=
-
v
s
infia'Eq
a e C and a'x 2 -1}
:
a € C and a'x 2 1}
:
is always well defined via
While X
(15)
(15),
v
(16)
may not be because there may not
exist a payoff in P that dominates a unit payoff.
define v
v
to be
In Section 2 we show how to consistently estimate X
+00.
Consistent estimation of these bounds is important
.
deviation bound
we
and
since the standard
is infinite for choices of Em outside these bounds.
<r
Estimation of the Bounds
2.
section
this
In
results for
the
we
develop
consistency
prices
are
associated
(x.
'
,q
'
,y
and
replicated
with
the
)' }
whose
over
time
composite
sequence
that
is
some
in
vector
of
joint distribution of (x'.q'.y)'.
the data on asset
(x'.q'.y)'
empirical
is
a
(Borel
measurable)
is
distribution
in Section
stationary
payoffs and
That
fashion.
a
stochastic
distributions
A key
1.
is,
process
approximate
the
We denote integration with respect to the
empirical distribution for sample size T as T.
that
asymptotic
specification-error bounds presented
presumption underlying our analysis
{
In such circumstances,
function
of
More precisely,
(x' ,q' ,y)
with
a
for any z
finite
first
moment, we will approximate £z by T.z where
j;z
=
H/T)£ =i z t
(17)
.
\?.
Among other things,
we require
approximation becomes arbitrarily
this
that
That is we presume that {z
good as the sample size T gets large.
Law of Large Numbers.
A sufficient condition for this is:
Assumption 2.1:
composite
The
process
i(x
'
,q
'
,y
)}
}
obeys a
stationary and
is
ergodic.
Under this assumption, we can think of (x'.q'.y) as (x
4
'
,q
'
,y
).
To estimate the specification-error bounds, we suppose that a sample of
size T is available and that the empirical distribution implied by this data
is used
the population distribution.
in place of
[Thus we are applying the
Analogy Principle of Goldberger (1968) and Manski (1988)].
random functions
4>
We introduce two
and #:
>(a)
=
y
i(a)
=
y
2
-
(y - a' x)
-
(y - a'x)
2
- 2a'q,
(18)
and
2
(d
)
T
=
2
max
+2
-
2a'q
y[i(a)]
.
(19)
(20)
tt€C
and
13
2
(d
2.
T [0(a)]
max
aeC
=
)
T
(21)
.
Consistent Estimation of the Specification-Error Bounds
A
establish
first
We
sequences {d
and {d
>
the
statistical
consistency
estimator
the
of
}:
almost surely to 5 and
<5,
respectively.
The proof of this proposition is given in Appendix A and is not complicated
by
the
presence
population and
of
short-sale
sample
constraints.
criterion
functions
basic
The
for
concave and the sets of maximizers are convex.
idea
conjugate
the
that
the
problems
are
is
By Assumptions 1.1 and 2.1,
the criterion functions converge pointwise (in a and 6) almost surely to the
population criterion functions introduced in Section l.B.
concavity of the criterion functions,
sets almost surely [for example, see Rockafellar (1970)].
sets
maximizers
of
the
of
limiting
light
In
the
of
this convergence is uniform on compact
criterion
Finally, since the
functions
are
compact,
for
sufficiently large T one can find a compact set such that the maximizers of
the
sample and population criteria are contained
see Hildenbrand
example,
(1974)
that
in
compact
set
[for
Hence the conclusion
and Haberman (1989)].
follows from the uniform convergence of the criteria on a compact set.
2.B
Asymptotic Distribution of the Estimators of the Bounds
We
consider
sequences
objects
of
of
the
next
the
limiting
specification-error
interest
as
solutions
to
distribution
bounds.
the
14
Our
conjugate
of
the
analog
ability
problems
to
estimator
express
permits us
the
to
obtain results very similar to those in the literature on using likelihood
ratios
devices
as
selection
model
for
in
environments
models
when
are
possibly misspecified [for example, see Vuong (1989)].
We show that when the
specification error bounds are positive,
limiting distribution
we obtain a
that is equivalent to the one obtained by ignoring parameter estimation,
when
specification
the
error
bound
is
zero
the
and
distribution
limiting
is
[See Theorem 3.3 of Vuong (1989) page 307 for the corresponding
degenerate.
result for likelihood ratios.]
be a maximizer of T $>
Let a
T
(j>,
a
and
maximizer
a
of
a a maximizer of
study
To
£0.
the
E<p,
limiting
a
a
maximizer of
behavior
of
the
estimators, we use the decompositions:
\/T[(d
2
)
T
2
- I
)
= •T^.[0(a
)
-
i(a)] +
/T^liu)
-
)
-
$(£)] +
/T^tfU)
- £0(a)]
T
E*U)]
(22)
,
and
2
•T[(d
)
- 5
2
T
]
=
/I^[$(Z
T
.
(23)
We make the following assumptions:
Assumption 2.2:
*(a) - E*(a)
**,
converges in distribution to a
[{mx - q) - Eimx - q)]
normally distributed random vector with mean zero and covariance matrix
Assumption
a
2. 3:
y%
^(a) - £0(a)
V.
converges in distribution to
[(mx - q) - E(mx - q)]_
normally distributed random vector with mean zero and covariance matrix
More
primitive
assumptions
that
imply
the
central
limit
V.
approximations
underlying Assumptions 2.2 and 2.3 are given by Gordin (1969) and Hall and
Heyde (1980).
Let
denote
u
followed
by
selection
a
+
k
one
a
in
its
first
position
distributions
limiting
The
zeros.
I
with
vector
the
for
specification-error bound estimators are given by Proposition 2.2:
Proposition 2.2:
-
2.1
2.3,
As
Suppose that
WT[d
2.2,
WT[d
-
8]}
-
other
maximizers by
negligible.
converges
indicates
values depend only on
In
6*0.
and
to
Under Assumptions 1.1
-
1.3,
normally distributed random vector
a
5]> converges in distribution to a normally distributed random
Proposition 2.2
(23).
5*0
the
second
the
words,
sample
the
the
limiting distributions
terms
impact
maximizers
replacing
in
the
the
in
unknown
criterion
sample
maximized
the
decompositions
the
of
of
for
(22)
and
population
functions
is
As a consequence the presence of short-sale constraints does not
complicate the limiting distribution.
To
see
why
the
asymptotic
distribution
in
Proposition
2.2
is
not
affected by sampling error in the estimation of the multipliers, consider the
~
case of the sequence {(d)
~
2
}.
By the concavity of
<f>,
we have the following
gradient inequalities:
(24)
[(mx - q) - E(mx
+
However,
it
follows
from
E(mx
the
- q)
(a
-
q)]-(a^
- a)
first-order
16
- a)
.
conditions
(including
the
complementary slackness conditions) for the population conjugate problem that
(25)
constrained to be in
Since,
a
Combining (24) and (25) we have that
£
/TEjtfU^
£
VT^Umx
if
Limit
the
-
0(a)]
- q)
by Assumption 2.3,
Central
zero
C.
Theorem,
(26)
qU-(i
E(mx -
-
the sample counterparts of the pricing errors obey
{v'TY [0(a
-
)
0(a)
maximizers can be chosen so
surely to zero.
- a)
]>
that
converges
{(a
-
in
a)}
probability to
converges almost
This latter convergence can be demonstrated by exploiting
the concavity of the population criterion function and the convexity of the
constraint
[for
set
example,
see
the
discussion on page
1635
of
Haberman
(1989) and Appendix A].
To
u'Vu or
use
Proposition 2.2
Consider
u'Fu.
sequence {0 (a
):
t=l,2,3,
the
...
practice
in
case
T>
of
requires
u'Vu.
For
consistent
each
T
estimation of
form
the
scalar
and use one of the frequency zero spectral
density estimators described by Newey and West (1987) or Andrews (1991), for
example.
As is shown in Appendix
A,
when the price vector q is a vector of real
numbers (degenerate random variables), the asymptotic distribution for
{v^Tld
- 6]> remains valid even when the population version of the conjugate maximum
problem fails to have a unique solution
this case,
the
lack
of
identification of
(Assumption 1.4
the
is
violated).
parameter vector a does not
alter the distribution theory for the specification-error bound.
17
In
While this
special
case
is
of
conditioning
using
Section
interest,
considerable
information
it
rules out
synthetic
form
to
the
payoffs
possibility of
described
as
in
1.
Notice that if 5 =
Proposition 2.2 breaks down.
or 8 = 0,
This occurs
if y is a valid stochastic discount factor in which case the solutions to the
population conjugate problems are a = a =
4>
are
(a)
both
"2
distribution for Wl(d
convergence of
the
giving
zero
identically
~
and Wl(d^)
2
As a consequence,
0.
rise
to
a
Our results
<f>
degenerate
(a)
and
limiting
in Section 4 on
the
parameter estimators can be used to establish that
the
}
)
"2
rate of convergence of {(d)
~
>
}.
and {(d)
2
}
and is given by a weighted
is T,
utions (see Vu
converge at the rate
v'T,
although the limiting distribution is not normal.
Consistent Estimation of the Arbitrage Bounds
2.C
As we discussed in Section
into standard deviation bounds
1,
the second moments bounds can be converted
the mean of m
if
estimated using the price of a risk-free asset.
be prespecif ied.
not
is
is
known or
if
it
can be
When Em is not known it must
Let v be the hypothesized mean of m when a risk free asset
available.
Proposition
2.1
can
be
applied
to
establish
the
consistency of the second moment bound estimators for each admissible price
assignment
v.
In the case of 5
,
for the price assignment to be admissible,
must not induce arbitrage opportunities onto the augmented collection of
it
asset payoffs and prices.
(A ,d
o
)
o
Any price (mean) assignment in the open interval
is admissible in this sense,
The final question we explore in this section is whether the arbitrage
bounds, A
and u
given in (15) and (16), can be consistently estimated using
the sample analogs:
-infia'Yq
=
t
a € C and
:
a'x
(27)
* -1 for all
t-1,2,
.
.
,T>
.
and
infia'Yq
=
u
:
a € C and
a'x
The
estimated
upper
arbitrage
i
1
bound
(28)
for all t-1,2
u
is
T>
always
.
when
finite
there
is
a
payoff on a limited liability security that is never observed to be zero in
Our estimated range of the admissible values for the
the sample.
price of a unit payoff and hence mean of n is
[I
,u
Notice
].
(average)
these
that
linear programming problems.
bounds can be computed by solving simple
In
Appendix A we prove:
almost surely.
If v
Consistent
2.D
is finite,
Estimation
then {u
the
of
}
converges to v
Feasible Region
of
almost surely; and
Means
and
Standard
Deviations
We now consider consistent estimation of the set of feasible means and
standard deviations of
that
for
given
a
mean
stochastic discount
of
the
stochastic
factors.
discount
deviation bound can be consistently estimated.
stochastic
discount
factor
typically
is
not
Previously we
factor,
However,
known.
As
the
a
the
showed
standard
mean of
result
it
the
is
important to understand the sense in which the entire feasible region can be
approximated.
Such a region can be computed with or without
no-arbitrage restriction that
Let S
the
stochastic discount
imposing the
factors be positive.
denote the feasible region without positivity and S* the (closure) of
Similarly,
the feasible region with positivity
The question we now turn
sample counterparts.
and S
S* good approximations to S
let S
fco
and the S* denote the
is in what sense are S
and
?
When there is a unit payoff, all four feasible regions are vertical rays
in
standard deviation space because
mean and
rays S
o
and S
the more usual
.
case when data on the price of a unit payoff is not
matters are a little more complicated.
longer vertical
instead are unions
but
rays
represented
(possibly
as
(hypothetical
and
mean),
convergence
counterparts.
of
extended)
our
the
real-valued
previous
sample
The feasible regions are
such rays
of
The boundaries of
convex sets with nonempty interiors.
mean)
this
of
o
available,
no
price
(average)
and S* can be estimated consistently by the points of origin of the
corresponding rays S
In
the
In this case the points of origin of the
payoff is the mean discount factor.
analysis
analog
functions
to
in
these sets can be
functions
implies
resulting
of
the
ordinate
pointwise
their
the
(in
population
This result implies uniform convergence of the sample analog
functions in following sense.
Since
estimated,
the
lower
and
upper
for large enough T,
arbitrage
are finite on any compact subset of
(A ,v
).
functions are finite on any compact subset of
convex
functions
result
[see Theorem 10.8 of Rockafellar
of
converge uniformly,
the
bounds
can
be
consistently
the sample analog functions under positivity
hypothetical
almost surely,
When positivity is ignored the
R.
mean of
(1970)]
Further these functions are
the
discount
factor.
As
the sample analog functions
on any compact
subset of
(A ,v
)
in
case of positivity and on any compact set when positivity is ignored.
difficulty is that the approximations deteriorate as the mean assignment,
approaches the arbitrage bounds
large when positivity is ignored.
in
a
the case of positivity,
the
One
v,
or when v gets
.
arbitrage bound turns out not
be problematic.
to
To
see
this,
instead of
viewing the boundaries of the feasible regions as functions of the ordinate,
we explore the approximation error from a set-theoretic vantage point in
Consider first the case in which v
<
(R
.
Associated with a sample of size T
+00.
is an approximation error as measured by the Hausdorff metric:
max<n(S ,S
(29)
,S )>
TO ),tt(STO
where:
n(K ,K
12
Measuring
ordered
=
)
the
pairs
sup
)eK
approximation
get
to
measures of distance,
as
)-lv,w)\
\(v ,w
222
)€K
(v ,w
via
error
1
(30)
1
Hausdorff
the
restricting
without
close
them
metric
have
to
allows
same
the
we no longer confine our attention to "vertical"
In other words,
ordinate.
inf
111
(v ,w
the
is
when we view
case
the
boundaries
of
the
feasible regions as functions of the hypothetical (expected) prices of a unit
The added flexibility in the Hausdorff metric permits us to exploit
payoff.
better
consistent
the
estimation of
the
upper
and
arbitrage
lower
bounds
(Proposition 2.3)
When v
is
infinite,
the approximation error
t>
defined by (29) will be
As a remedy, we replace n by
infinite.
(C ,C
re
P
*
)
2
=
sup
(..
...
inf
\~v
Osv sp
(.,
...
|(v ,w )-(v ,w
\^v
1
1
2
)|
2
(31)
where p is any arbitrary positive number greater
than
Under Assumptions 1.1-1.3 and 2.1,
Proposition 2.4.
{y
arbitrage
lower
the
converges to zero
>
almost surely.
proposition
This
estimated
follows
This
and
the
arbitrage
the
lower
bounds
can
boundaries of
consistently
be
{S
approach
}
the
uniformly on any compact interval within the arbitrage
lower boundary of S
bound.
since
(Proposition 2.3)
latter
convergence
follows
from
Proposition
and
2.1
the
convexity of the boundary.
3.
Applications
two applications of the analysis of Section
In this section we discuss
First
2.
we
show
standard-deviations
how
can be
feasible
the
used
to
test
a
regions
for
specific model
of
and
means
the
the
discount
factor.
Burnside (1994) and Cecchetti, Lam and Mark (1993) have developed a
version
of
constraints
implemented
Section
there
2.
are
this
test
when
or
transactions
in
a
there
are
costs.
We
relatively
simple
no
assets
demonstrate
manner
by
subject
how
to
this
exploiting
the
short-sale
can
be
results
of
test
Further we formulate the test so that it is also applicable when
assets
subject
to
short-sale
constraints.
As
a
result
this
provides (large-sample) statistical foundation to the tests of asset pricing
models suggested by He and Modest (1993), and Luttmer (1994).
Second we outline an extension of the specification-error bound analysis
that
is
useful
when the discount factor proxy under consideration depends
upon a vector of unknown parameters.
We consider how these results can be
used to select between two nonnested models by comparing the minimized values
of the specification-errors.
Testing a Specific Hodel
3. A
of
Factor using Volatility
Discount
the
Bounds
Suppose that in addition to asset-market data,
a model
of the discount
factor is posited and a time series of observations of the discount factor is
available:
whether
{m
satisfies
it
One way to test
T}.
t=l
:
the model
volatility bounds discussed in Sections
the
a
unit payoff can be estimated by the mean of m.
the
original
vector
with a
payoffs
of
and
1
Specifically,
unit
form x by
form q by
payoff;
augmenting the original vector of prices with the random variable
and form
m;
C by constructing the the Cartesian product of the original cone with
effect,
we have added a
subject
to
results
Section
of
functions
short-sale constraint.
a
and
4>
subtracting m
4>
1
2.B
and
R.
In
is
not
can apply
the
payoff with an average price m that
unit
forming a
In
with
one
minor
test,
we
modification.
are now constructed by setting
2.
the average price of
Since observations of the discount factor are available,
augmenting
examine
to
is
The
the proxy y
to
random
zero and
2
:
2
i(ct)
=
-
(-a'x)
0(a)
=
-
(-a'x)*
-
-
2a'q
m
(32)
,
and
Subtracting m
2
2
does not
- 2a'
alter
population maximization problems.
values
of
the
criteria
q
functions.
-
m
2
solutions
the
It
(33)
.
does,
The
to
however,
either
volatility bounds
sample
the
change
the
for
or
maximized
Em will
be
satisfied,
and only if
if,
=
£
max
aeC
s
E<p(a)
0,
(34)
0,
(35)
when positivity is ignored, or
max
aeC
when
positivity
Proposition 2.2
£
E<t>(cc)
limiting
The
imposed.
is
distribution
reported
of these hypotheses using sample analog estimators of £ and £.
have
formulated
the
plays
estimation
in
(appropriately modified) can be applied to construct a test
no
problem
role
so
that
due
to
distributions
for
these
limiting
the
in
we
Again,
approximation error
parameter
sample
analogs.
In practice we find the solutions for
estimate
particular,
Then
asymptotic
the
-
£•]
mean zero and variance
the
statistic.
tests.
In
This variance can be estimated in the manner
u'l^u.
Since £ is not specified under the null hypothesis
"conservative"
choice of
£,
=
is
used
in
constructing the test
7
Similar approaches
applied when a
instead
one-sided
form
converges in distribution to a normal random variable with
described in Section 2.B.
(35),
and
errors,
be the maximized value of F (0) over the constraint set C.
let c
Wl[c
standard
the sample maximization problems,
of
time
actual
to
series
data.
testing a model
for
In
m can be
this
case
of
the
discount
constructed
the
from
randomness
of
factor
can
simulated
0(a)
be
data
can
be
decomposed additively into two components, one due to the randomness of the
security market payoffs and prices and the other due to the simulation of
ZA
m.
As in the work of McFadden (1989),
Duffie
and
(1991)
Singleton
and
Pakes and Pollard
(1993),
(1989),
variance
asymptotic
the
Lee and Ingram
limiting distribution will now have an extra component due to
error
induced by simulation.
For
an example
of
this
the
in
the
sampling
approach and a more
extensive discussion see Heaton (1993).
When
the
first
moments of m can be computed numerically with an
two
arbitrarily high degree of accuracy, we can proceed as follows.
price vector with Em instead of m and subtract Em
of m
2
as in (32) and (33).
2
Augment the
from the criteria instead
This same strategy can be employed to assess the
accuracy of the estimated feasible region for means and standard deviations
of stochastic discount factors.
For any hypothetical mean-standard deviation
pair for m, one can compute the corresponding test statistic and probability
value.
3.B
Minimizing the Specification-Error Bound for Parameterized Families of
Models
Recall that the specification-error bounds provide a way to assess the
usefulness
In
pricing
asset
an
of
misspecif ied.
many
unknown parameters.
situations
For
example,
model
the
in
a
even
discount
when
technically
is
it
depends
proxy
factor
on
representative consumer model with
constant relative risk aversion preferences, the pure rate of time preference
and the coefficient of relative risk aversion are typically unknown.
case
one way
specification
discount
to
estimate
error.
the
parameters
Alternatively,
factor proxy depends on a
unknown coefficients.
As
in
in
linear
model
of
the
an
observable
is
(1987)
minimize
factor
combination of
the work of Shanken
to
the
In this
model,
the
the
factors with
one could
selecting factor coefficients to minimize the specification error.
imagine
We now
sketch how the results of Section 2.B extend in a straightforward manner to
25
obtain
theory
distribution
a
for
minimized
the
value
In Section 4 we discuss distribution
specification-error bound.
the
of
theory for
the resulting estimators of the parameters of the discount factor.
Suppose that the discount factor proxy y depends on the parameter vector
€
The population optimization problems of
B where B is a compact set.
interest are now:
5^
5
2
=
min
0eB
max
aeC
=
min
BeB
max
aeC
2
-
|£y(/3)
£y(p)
2
E{{y{B) - x'a)
- £{[(y(/3)
2
- 2a' Eq\
}
- x'a)*]
2
}
(36!
,
- 2a' Eq]
I
(37)
.
>
When 5 and 6 are strictly positive and the parameterized family of stochastic
discount
satisfies
factors
appropriate
the
smoothness
moment
and
restrictions, an extended version of Proposition 2.2 can be obtained for the
sample analog estimators of 5 and
the
same as
the
if
solutions
Again the limiting distribution will be
8.
the population optimization problems were
to
known a priori.
The
approach
can
extended
be
compare
to
specification-errors for two nonnested families of models.
smallest
the
Such a comparison
potentially can be used as a device for selecting between the two families of
Vuong
models.
large-sample
(1989)
behavior
of
examined
very
a
likelihood
ratios
similar
for
two
problem
using
by
nonnested
families
misspecified models (in particular, see the discussion in Section
1989);
and
we
can
imitate
and
adapt
26
his
analysis
to
our
the
5
of
of Vuong
problem.
More
Take the null hypothesis to be:
with two such families.
(38)
Under the null hypothesis,
the smallest specification-error associated with
As a consequence the performance of
each parameterized family is the same.
the
parameterized
two
accounted for.
This
families
can
not
ranked
be
hypothesis can be
tested
once
using
by
error
sampling
the
corresponding
distribution theory for the difference between the analog estimators of
4.
is
S
Asymptotic Distribution of the Multipliers
Recall that the asymptotic distribution f
discussed in Sections 2 and 3 depends only on the population solutions to the
conjugate
problems
(8)
and
other
In
(9).
words,
sampling
error
in
the
estimated multipliers does not contribute to the limiting distribution of the
specification error bounds.
However,
as elsewhere
in econometric practice,
the magnitude of the multipliers remain interesting in their own right as a
measure
of
the
importance
Consequently,
bounds.
components
of
is
it
the
of
advantageous
to
pricing
be
able
to
relations
make
to
the
statistical
inferences about their magnitude.
To amplify this point,
multiplier
estimators
one use of the asymptotic distribution for the
test
to
is
volatility bounds remain the same when
analysis.
question
whether
a
A special case of such a test
of
interest
is
whether
given
the
specification-error
is a region subset
an
initial
set
of
test where the
asset
returns,
additional asset returns result in an increase in the volatility bounds.
is
problematic
to
construct
a
test
or
subset of assets is omitted from the
directly
in
terms
of
the
It
difference
between
measured
a
corresponding bound
with
computed
bound
with
calculated
full
the
more
the
securities
set
limited
array
of
and
the
securities
because the resulting statistic has a degenerate distribution under the null
hypothesis.
This degeneracy follows from the fact that sampling error in the
multipliers does not contribute to the limiting distribution for the bounds
and is circumvented by instead basing the statistical
estimated multipliers.
That is,
is advantageous
it
to
test directly on the
reformulate the null
hypothesis to be:
Ra
=
0,
or R a
=
(39)
where R is an appropriately constructed selection matrix (see below), and to
use the limiting distribution for the estimated multipliers in constructing
an asymptotic chi-square test.
Several
literature.
g
versions
region
of
Snow
For example,
subset
have
tests
been
used
the
in
considered the small firm effect by
(1991)
examining whether the returns on small capitalization stocks have incremental
importance in determining the volatility bounds over and above the returns of
large capitalization stocks.
Other examples can be found in Braun
(1991),
Cochrane and Hansen (1992), De Santis (1993) and Knez (1993).
The
limiting
applications
as
distribution
well.
specification-error
For
analysis
important contributors
to
for
testing
instance,
is
model
multipliers
the
helpful
in
(39)
shows
in
ascertaining
misspecif ication.
Also,
up
in
other
conjunction
which
assets
with
are
when a researcher
uses the specification-error bounds to select among a parameterized family of
discount
factor
proxies,
parameter vector chosen.
it
is
desirable
As we will
see
to
in
make
this
inferences
section,
the
about
the
asymptotic
distribution for the parameter estimator selected in this fashion interacts
ZS
A
with
distribution
limiting
the
for
estimated
the
multipliers.
As
a
consequence, our characterization of the multiplier limit distribution is an
component
important
in
derivation
the
the
of
distribution
limiting
for
estimators of parametric stochastic discount factor proxies.
The remainder of this section is organized as follows.
the distribution theory for the estimated multipliers
In Section 4.
we
comment
briefly on how
the
theory
can
extended
be
assuming
is developed
In Section 4.B
that there are no assets subject to short-sale constraints.
the
to
case
where
short-sale constraints are imposed on some of the assets.
4.
A
Distribution without Market Frictions
In
the
consequence
absence
the
of
short-sale
estimation
problem
constraints,
for
the
the
cone
multipliers
C
is
is
R
n
.
posed
As
as
a
an
unconstrained maximization problem and the limiting covariance matrices for
the asymptotic distribution of the coefficient estimators have a form that is
familiar both from M estimation [for example, see Huber (1981)] and from GHH
estimation
[for example,
see Hansen
(1982)].
The only complication occurs
when considering the bounds in which the no-arbitrage restriction is fully
exploited because of the kink induced by the nonnegativity restriction.
The
moment
conditions
of
interest
are
given
by
the
first
order
conditions (10) and (11) for the conjugate maximum problems:
E[x{y-x'a) - q] =
(40)
0,
and
E[x(y-x'a)
+
- q)
=
(41)
.
29
Y^[x(y-x'a
- q]
)
T
=
(42)
and
Ej.txCy-x'a^)* - ql =
(43)
.
While the equations for a are linear, those for a are nonlinear.
latter
we
case,
use
linear
a
approximation
the
to
In the
conditions
moment
in
deriving the central limit approximation for the parameters:
x(y-x'a)
-
~
q
x(y-x'a)
x(y-x'a)l
Notice
that
the
-
q
- xx' 1
{y _ x ,^ 0>
function of a on the
(44)
q
side
left
=0.
except at values of a such that y-x'a
-
.(a-a)
,~
.
of
is
(44)
differentiable
We assume that such sample points
are "unusual":
Assumption
4.1:
Pr{y-x' a = 0} =
As is shown in Appendix C,
0.
Assumptions 1.1 and 4.1 are sufficient for us to
study the asymptotic behavior of the estimator {a
£[x(y-x'ct)l.
,~ ..
{y-x'aiO}
To
use
linear
£(xx'l. _ /~ >n y)
must
>
using the linearization
9
on the right side of (44)
- q]
M
equation
be
=
system
nonsingular.
(45)
.
(45)
to
Given
identify
Assumption
condition is equivalent to Assumption 1.4 because x and x
no
short-sale
constraints
are
imposed.
30
The
the
a,
4.1,
matrix
this
rank
must coincide when
counterpart
to
this
rank
condition for a is that
second moment matrix E{xx'
the
)
be nonsingular
as
required by Assumption 1.3.
Working
with
the
moment
linear
two
conditions,
we
obtain
the
approximations:
•T(a
/T(a
- a)
*
-[Eixx'l^.-^r^T^lxiy-x'*)*
- a)
~
-lEUx'
T
T
- q)
(46)
))~ 1 VT£ lxiy-x'a) q]
i
where the notation ~ is used to denote the fact that the differences between
the
left and right sides of
[01].
(46)
converge in probability to zero.
Let w a
Combining approximations (46) with Assumptions 2.2 and 2.3 gives us
the asymptotic distribution of the analog estimators.
Proposition 4.1:
Then {v^T(a -a)}
Suppose Assumptions
converges
in
1.1-1.3,
and 2.2 are satisfied.
2.1
distribution to a normally distributed random
Suppose Assumptions 1.1-1.4, 2.1, 2.3 and 4.1 are satisfied.
Wl{a
Then
-a)}
converges in distribution to a normally distributed random vector with mean
zero and covariance matrix:
To apply
these
[Eixx'l. _ /~
>n »)l
limiting distributions
in
wVw' [E{xx' 1. _ /~
>n \)l
practice requires consistent
estimators of the asymptotic covariance matrices.
can be estimated using one of
Under
assumptions
maintained
the
in
The
spectral methods
Proposition
4.1,
terms wVw'
vW
and
referenced previously.
the
matrices
Eixx'
)
and
E(xx'\. _ /~ ») can be estimated consistently by their sample analogs, where
>n
the
estimator a
is
used
in place
of
matrices.
31
a
in
estimating the second of
these
We now briefly consider two extensions of Proposition 4.1:
(i)
Estimation of the parameters of a discount factor proxy.
that the discount factor proxy depends on the parameter vector
be
the specification error when positivity
the parameter vector that minimizes
is
Suppose
and let
Assume that the parameterized family satisfies the appropriate
imposed.
smoothness and moment
parameter space.
restrictions,
and
that
in
is
the
interior
of
the
The population moment conditions are given by:
E{x[y(0)-x'a]
- q>
=
(47)
0;
and
e[— (pXyW
-
[y(£) - a'x]
^30
+
=
}}
(48)
0.
>
The distribution theory for the analog estimators
respectively can be deduced by taking
linear
{b
>
and
{a
approximations
>
to
and a
of
the
sample
moment conditions (47) and (48) and appealing to the results of Appendix
(ii)
assets
Region subset tests.
under
consideration
C.
Let z denote an (n-1 )-dimensional vector of
with
price
vector
s,
and
let
f
be
the
k-dimensional subvector of z including the k-1 asset payoffs that are to be
used to construct the bound augmented by a unit payoff.
Formally,
the region
subset test can be represented as the hypothesis:
£[z(f'9)
El(f'e)*
+
- s]
-
v
]
=0,
=
vector of securities z correctly.
for a prespecified v
,
(49)
0.
One possibility is to test this hypothesis
and the other is to test whether it is satisfied for
our previous setup,
form the n-dimensional vector x by augmenting z with a
unit payoff and form q by augmenting s with the "price" v
(49)
can be
.
Then hypothesis
interpreted as a zero restriction on the coefficient vector a
employed in Sections
1,
2
and
The components of coefficient vector a
4. A.
set to zero correspond to the entries of z that are omitted from f.
sample Wald
test
this
of
restriction
zero
can
be
formed
by
A large
applying
the
based on
the
limiting distribution in Proposition 4.1.
Alternatively,
we
could
estimate
the
overidentif ied system (49) using GMM.
parameter vector
The analysis leading up to Proposition
4.1 can be easily modified to show that the minimized value of the criterion
function is distributed as a chi-square random variable with n - k degrees of
freedom [see Hansen (1982)].
When the hypothesis of interest is altered to
freedom.
4.B
Distribution with Market Frictions
We now briefly describe how
some short-sale constraints are
distribution theory
the
imposed
(C
is
a
is
modified when
proper subset of
IR
)
.
We
will focus on the limiting behavior of Vlia -a), but the results for Vt(a -a)
are very similar.
As in Section
the payoffs with equality or not,
Emx
=
Eq
,
or Emx
<
1,
we partition x by whether or not m prices
that is by whether
Eq
.
(50)
inequality will equal zero with arbitrarily high probability as the
strict
Hence the
sample size gets large.
limiting distribution is degenerate for
these component estimators.
Consider next the estimator of the remaining subvector of
denote
limiting distribution of the estimator of y separately.
the
lower-dimensional
point of C,
to
deduce
However,
be
cone associated with estimating
a
distribution
normal
limiting
if y is at
for
the boundary of the cone C,
function
of
Haberman (1989), page 1645].
5.
which we
y.
If
the
Let C be
interior
an
is
y
then the argument leading up to Proposition 4.1 can be imitated
nonlinear
a
a,
Because of the degeneracy just described, we can, in effect, treat
7.
a
normally
the
estimator.
parameter
the limiting distribution may
distributed
vector
random
[see
10
Conclusions and Extensions
In
we
paper
this
provided
statistical
methods
assessing
for
asset-pricing models based on specification-error and volatility bounds.
Two
significant advantages of the statistical methods are that they are easy to
interpret
and
they
that
are
simple
to
implement
transactions costs and short-sales constraints.
variety of ways.
For example,
the discount factor,
to examine
even
in
the
presence
of
The results can be used in a
they can be used to test specific models of
the
information contained in different sets
of asset-market data, and to assess misspecified asset-pricing models.
There are several
interesting extensions of the econometric methods in
this paper including the following:
(i)
The
short-sale
constraint
"solvency constraints"
formulation could be generalized
whereby portfolios are restricted so
34
that
to
include
portfolio
payoffs
nonnegative.
are
This
As
in
Section
imposing
to
form
a
of
borrowing
Hindy (1993) and Luttmer (1994)].
constraints on portfolio weights
the
1,
amounts
for example,
constraint on consumers [see,
the
in
presence of
solvency constraints can be formulated as a closed convex cone.
However,
without knowledge of the distribution of the payoffs on primitive securities
the cone would be subject to estimation error and hence not directly covered
The consistency proof in Section 2.C for the
by the results in this paper.
arbitrage bounds might well
more generally.
It
errors
constraint
for
the
is
adaptable
be
then
interest
of
sets
approximating constraint
to
how
understand
to
impact
on
approximation
distribution
the
sets
of
the
for
the
specification and/or volatility bounds.
(ii)
difficult
A
feature
of
limiting
the
distribution
theory
Lagrange multipliers in the presence of short-sales constraints (Section 4.C)
is
the
manner
associated
harder
compute
to
in
which
it
depends
discontinuities.
use
in
example Wolak
bounds
on
in
other
parameter
true
makes
has
test
of
vector
and
distribution
the
settings,
probabilities
and Boudoukh,
(1991)
the
feature
This
practice and,
approximate
on
led
Richardson and Smith
theory
researchers
statistics
(1992)].
the
[see,
to
for
Perhaps
similar probability bounds could be derived for the region subset tests with
frictions.
(iii)
Using tools similar to those described in Section
1,
(1995)
have
integration.
should
be
incorporate
developed
possible
nonparametric
to
transactions
extend
costs
the
in
measures
econometric
the
propose.
35
market
of
market
methods
in
integration
Chen and Knez
this
paper
measures
It
to
they
Appendix
Consistency
A:
In this appendix we demonstrate formally the results of Section 2. A,
We maintain Assumptions 1.1-1.3, and 2.1 throughout.
2.D.
Let
n
denote a compact set in
11
denote the closure of
subsets of
11.
t)(/i
For any subset h of
.
K denote
Let
h.
IR
tj
-
a
,/i
)
=
max{ sup
a eh
inf
I
a
on
|,
2
X given by
sup
a eh
2
to define notions of convergence of compact sets.
use
construct
the
of
a
lim
we let cl(h)
11,
the collection of all nonempty closed
We use the Hausdorff metric
a eh
112
will
and
sup
of
a
inf
a eh
I
-
a
a
|>
.
(A.l)
11
2
For some of our results we
sequence
in
We
K.
follow
Hildenbrand (1974) and define:
Definition A.l:
For a sequence {h
>
in
11,
lim sup h
Since
the
sets,
it is closed and not empty.
lim sup
is
the
=
n cl
»
J
J
(
u h
J2*>
)
J
intersection of a decreasing sequence of
closed
An alternative way to characterize the Jim
sup is to imagine forming sequences of points by selecting a point from each
h
.
and,
All of the limit points of convergent subsequences are in the lim sup,
in
fact,
all
of
elements
of
the
lim sup can be
represented
in
this
manner.
We shall make reference to an implication of a Corollary on page 30 of
Hildenbrand
(1974)
that
characterizes
the
set
"approximating" function over an "approximating set."
3(3
of
minimizers
of
an
Suppose
Lemma A.l:
{0
(i)
is
}
sequence of continuous functions mapping
a
converges uniformly to
ip
1/
into R that
;
and
{h
(ii)
converges to h
}
Then Jim sup g
and iim min
°°
=
</<
=
{u e h
:
(u)
\p
)
for all u'
€ h
J
.
CO
°°
denote the set of positive integers augmented by
usual metric for a one-point
sequence
in
{</>
of
(ii),
the
sequence
{h
in
}
conjunction with h
then
Turning
1
follows
from
the
Corollary
of page 22 in Hildenbrand.
to
the
result
in
Sections
the
(i),
defines a continuous function on }
continuous compact correspondence mapping } into
Proposition
let }
and endow £ with the
+oo,
Then in light of
compactif ication.
in conjunction with
}
light
Al
},
J
To verify that this follows from the Corollary in Hildenbrand,
Proof:
Lemma
(u'
\p
h
J
and
i
J
J
J
min
Jt
h
where g
c g
J
x
11;
defines a
The conclusion of the
Ii.
together
with
part
(ii)
of
Q.E.D.
1
and
2. A,
we
first
establish the
compactness of the set of solutions to the conjugate problems:
Lemma
(9)
A. 2:
The set of solutions to conjugate maximization problems
are compact.
37
(8)
and
consider
Proof:
We
problem
(8)
constraint
is
set
the
case
The
similar.
set
and
closed
is
problem
of
The proof
(9).
solutions
of
criterion
the
for
closed
is
function
the
case
because
of
the
continuous.
is
Boundedness of the set of solutions can be demonstrated by investigating the
We consider two cases: directions
tail properties of the criterion function.
< for which
which
(9)
<;'x
£'
x is negative with positive probability and directions
is nonnegative.
and divide it by
1
+
for
C,
To study the former case we take the criterion in
|a|
2
For large values of
.
|a|
the scaled criterion
is approximately:
-
Hence
|<|
payoff
in
E[(-x'0
+2
]
where <
Consequently,
P.
Consider
next
1.2 we
identically zero.
When
that
|a|
2 1/2
]
(A. 2)
.
Moreover,
|a|.
unsealed criterion will
the
x is a
C,'
decrease
(to
-co)
|a|.
directions
have
a/[l +
large values of
is approximately one for
quadratically for large values
Assumption
=
C,'
<;'x = 0,
C,
for
Eq must
it
which
be
C,'
x
is
nonnegative.
From
strictly positive unless
C,'
x
is
follows from Assumptions 1.2 and 1.3 that
C' Eq is again strictly positive.
For directions ^ for which the payoff
C,'
x is nonnegative,
tail behavior of the criterion after dividing by
approximately
-
<'£q for large values of
|a|.
(1
+
|a|
we study the
2 1/2
)
,
which yields
Hence in these directions the
the unsealed criterion must diminish (to -») at least linearly in
|a|
.
Thus
in either case, we find that the set of solutions to conjugate problem (9)
bounded.
Q.E.D.
We now formally establish Proposition 2.1:
is
Proof of Proposition 2.1:
We treat only the
for
{d
very
is
>
cc
Assumptions
similar.
converges almost surely to E$(a)
concave as
is
s N>
|ot|
set
in
2.1
imply
and D
=
C
{a e
IR
from Lemma
Further
.
:
I
a
=
I
Then C
N>.
maximizers of
and D
N
we have
surely,
in D
be the maximized value of
5
<
By
the
statistical
arbitrage
{£>} converges
the
A. 2,
are
of
set
{a € C
:
By
compact.
the maximizers of T
T,
turn
results
the
to
consistency of
bounds
A
4>
Lf
v
<p}
on C
that
compact
explore
"directions
set
for
to
the
choice
infinity."
variables
study
We
on C
almost
are also not
follows
that
for
}
to 5 follows from the
Q.E.D.
Section
in
Recall
.
.
it
2.C
and
{£
>
in
investigate
and
arbitrage
the
representable as solutions to linear programming problems.
natural
E<j>
coincide with those over
over C
analog estimators
sample
and
Then by
3 choice of N
.
N
the maximizers of £_0 over C
almost sure uniform convergence of {T
now
is
contains all of the
converges uniformly to
{Y ^}
the almost sure convergence of {d
Consequently,
We
over D
convexity of C and concavity of
sufficiently large
C.
Since
6.
for sufficiently large T,
.
Ed>
r
N
that
70
over the constraint set C and that none of the maximizers
E4>
Let 5
.
{Yiioc)}
T,
define C
choosing N to be sufficiently large we can ensure that C
are in D
that
Since for each
For a positive number N,
is bounded.
E4>
and
Theorem 10.8 of Rockafellar implies that
E$,
uniformly on any compact
maximizers of
1.1
each aeC.
for
these
these
{u
>
for
bounds
the
the
are
Since there is no
problems,
"directions"
we
must
using
a
compactif ication of the parameter space.
First consider any a e C such that
with probability one a'x
£
-1
for
all
a'
t
x £ -1 with probability one.
with probability one and
Then
(aTq)
<
I
u-'Tji'
^
follows
sup
lim
that
*
I
To construct a compact parameter space,
we map
space for each problem into the closed unit ball in R
We consider explicitly the case of u
with
A
the
n
completely analogous to the case for u
one.
parameter
original
which we denote as
the case of
The proofs for
.
probability
I
II.
are
and are omitted.
Notice that the constraint set used in defining v
can be represented as
the set of all a € C satisfying the equation:
£[(1 - a'x)
+
]
=
(A. 3)
.
As in the proof of Lemma A. 2,
we map the parameter space into the unit ball
(with a slightly different transformation).
maps R
n
into
open unit
the
ball.
The transformation < = a/(l+|a|)
compact if y
To
the
transformed
space, we consider adding the boundary points of the unit ball.
we
can
recover
the
original
parameterization
by
considering
parameter
Notice that
the
inverse
mapping:
a
for |<| <
1.
=
C/(l
This
(A. 4)
|C|)
Using the transformation in
a's that satisfy
D*
"
=
(A. 3)
{
we consider:
< e llnC
transformation
instead of considering those
(A. 4),
|
£{[(1 - |<|) -
potentially
adds
x'0
solutions
40
+
=
>
to
}
(A. 3)
(A. 5)
.
by
including
the
boundary of the unit ball.
for which x'
C,
£ 0,
The potentially problematic values of C are those
and
C,eC
=
|<|
We rule this out by limiting attention
1.
to values of < in
(A. 6)
effect
by
focusing
on
£'s
in
payoffs with "high" prices.
D
we
are
eliminating
concerned with estimated upper arbitrage bounds
Also,
high.
any < in D for which
is nonpositive.
Lemma
Proof:
T)(D ,D)
First
|<|
)
=
|<|
notice
that
<
1
corresponding
Fq
since
are
that
- x'C,]*
on U.
1)
Note that
-
x'cj
+
-
s
n D
converges
too
to
D
c
Eq
n
almost
We next establish that
0.
show that T [(1
e||[(1 - IC
1
low,
not
too
-
11
|^|)
-
x'
C,]
Hence,
from Lemma A.l,
the
.
D.
surely,
lim D
then
= D
.
converges uniformly to
n C is compact and that:
[d
-
K2 D
(1
+
(£|x|
+
-
*'V
(A7)
l|
1/2
2
)
)K r
C z |.
This is sufficient for the Uniform Law of Large Numbers of Hansen (1982)
apply.
to
must have an (average) price that
Then lim sup D
oo.
converges almost surely to
To do this we first
£[(1 -
s
This eliminates the troublesome points (directions) from D
Suppose that u
A. 3:
C,'
This does not cause us problems because we are
to
lim sup of the sequence of minimizers of
-
T [(1
-
ICU
- x'
- x' <;]*
I<l)
over
C.)
11
Since v
.
minimizers of £[(1
must also be in D
-
n C is contained in the set of minimizers of £[(1
<
oo,
*'<]*.
-
|<|)
the set D
for all Til.
is not empty
and D
With probability one,
Since D
is separable,
the set of
is
any point
in D
a common probability
First note that
Proof:
= mini
u
C'Ej-g/d-lCN
< € D
I
n D
t
>
t
for sufficiently large T,
(A. 8)
and
in
smaller
values of
constraint set
Lemma
A. 2
to D
n
Lemma
A. 5:
Proof:
D.
the
maximized criterion.
For
instance,
augmented to include all of the points in D
is
suppose
n
the
Then
D.
implies that this sequence of augmented constraint sets converges
The conclusion then follows from Lemma A.l.
Suppose that v
Since v
=
o
probability one.
oo,
=
oo.
Then iu
>
Q.E.D.
diverges with probability one.
there are no values of a € C such that a'x £
Consequently,
the only values of £ in D
1
with
are ones for which
Kl
=
First suppose that D
We consider two cases.
1.
-
convergence of ^[(1
sufficiently large
T,
-
|<M
to £[(1
x' C,)*
= a and u
D
=
<x>.
-
|<|)
-
= a.
+
Next suppose that D
there are no arbitrage opportunities (Assumption 1.2), C'Eq
D
such that
ll^'xll
>
e
,
it
=
inf{C,'Eq
follows
large T,
CI_<7
C,'
Eq
for any <
>
:
C,
€ D* } >
from Lemma
>
A.
*
n
=
II<'jcII
0.
(A. 9)
converges almost surely to
for
The convergence of {D
.
>
sufficiently
to D
coupled
have norm one then implies that
{u
>
Q.E.D.
Taken together. Lemmas
In
such that
with probability one
that
1
c/2 for all £ e D
diverges almost surely.
B:
D*
in
.
with the fact that all elements of D
Appendix
for any C
>
is closed implying that
Since {Y q} converges to Eq almost surely and D
D
Since
* 0.
Also, Assumption 1.2 together with the no-redundancy
0.
Assumption 1.3 imply that
Furthermore, D
The uniform
implies that for
x' <]
A. 4
A. 3,
and
A. 5
imply Proposition 2.3.
Asymptotic Distribution of the Bounds Estimators
this appendix we show that
in
the
case
in which
the prices of
the
payoffs are constant, the asymptotic distribution of the estimated bounds can
be demonstrated
even when
the
parameter vector
is
not
uniquely
identified
(even when Assumption 1.4 is not satisfied).
Proof of Proposition 2.2:
We consider the case of d
set of maximizers of
Y
4>
an d
l
.
et h
The case of d
is similar.
Let h
be the set of maximizers of £#.
be the
For each
let
T,
be a measurable selection from h
a
Since lim sup h
page 54).
(1974),
there is a sequence {a
a € h
h
,
q = £{y(y-a'x)
a'
+
(y-a'x)
-
consider the decomposition of /IT
v'T[(5
2
- I
)
+2
-a
A.
=0
\
2
}
t
=
Hi ldenbrand
/IX^U^
[
(d
compact,
is
almost surely (see
of Hansen and Jagannathan
1
+
(y-a'x)
.
implies that for
(9)
so that a' q is the same for all a €
>,
the random variable 0(a)
As a result,
.
\a
the complementary slackness condition for problem
Also,
of
result in the same random variable m =
that all a e h
is
1
almost surely and h
Further, an implication of Lemma
Appendix A).
(1991)
= h
such that lim
in h
}
Theorem
(see
- 5
)
- 4>(a )]
+
r
is the same for all a € h
Now
.
as in (23):
]
VT^[4>U
-
)
t
E4>U
r
(B.l)
)].
As in relation (26), we have:
s
s
VTj^lila^
-
4>U
t
(B.2)
)}
,
V TX [(mx - q) - E[mx - q))-{a^
T
Since
\a
-a
Appendix
C:
In
|
this
converges almost surely to
0,
-
aj
.
the result follows.
Q.E.D.
Asymptotic Distribution of the Multipliers
appendix
consider
we
the
asymptotic
distribution
of
our
estimator of the Lagrange multipliers when there are no transaction costs.
We
begin by demonstrating
that
restrictions used
in
Hansen
can
be
(1989)
to
(1982)
extended along the lines of Pollard (1985) and Pakes and Pollard
accommodate "kinks" in the functions used to represent the moment conditions.
We then show how to use this result to prove Proposition 4.1.
The
notation
used
in
our
initial
proposition
for
GMM
estimators
conflicts with some of the notation used elsewhere in the paper.
denote the parameter vector of interest and
underlying parameter space T.
We let
any hypothetical point in the
The parameter space is restricted to satisfy:
Assumption C.l: T contains an open ball in
IR
about
We will use the construct of a random function.
.
A random function
ip
maps the
set of sample points into the space of vector-valued continuous functions on
T.
We require that ^(0) be an n-dimensional random vector for each
in T.
We also consider an approximating function
The composite random function satisfies:
that is linear 0.
Assumption C.2:
{
(i/»
'
,\Jj*'
)
'
}
is stationary and ergodic and has finite first
moments.
We now specify the sense in which
approximation error induced by using is
r
t
(0)
=
|0 (0)
t
-
0*O)I
is required to approximate
\p
\p
in place of
\p
ip
.
The
is
•
Define:
Note that dmodA-)
is
monotone
in
6.
45
Therefore,
we
can
take
almost
sure
We impose the following restrictions on mod
limits as 5 declines to zero.
Assumption C.3:
lim dmod (8) =
5*0
Assumption C.4:
Eldmod AS)]
The approach adopted
<
in
almost surely.
<*>
for some 5
Hansen
continuity of the derivative of
.
>
(1982)
0.
is
to
restrict
the
modulus of
to converge almost surely to zero and to
ifi
have a finite expectation for some neighborhood of the parameter.
It
follows
from the Mean-Value Theorem that restrictions imposed in Hansen (1982) on the
local behavior of
We
use
ip
imply Assumptions C.3 and C.4.
Assumptions
-
C.3
stochastically differentiable.
c(8)
=
sup
<\Zj?lt(P)
C.4
to
study
the
sense
in
which
T^
is
Hence look at the approximation error
-
E^'OH/I/HM
:
ie-3 l<5,0*0
o
>
.
o
By the Triangle Inequality we have that
c(5)
£
^.dmod(5)
Thus by Assumptions C.1-C.2, we have that
lim lim sup c (5)
5* o
t-x»
^
lim
Edmod{8)
5* o
4b
(C.l)
0.
This
in
turn implies
stochastic differentiability condition
because the counterpart to c (5)
(1985)
•TI0-0 |/(1
/TI0-0
+
o
in
the
implies
(C.l)
monotone in
in Pollard's
which is less than one.
|),
Also,
taken
limit
in
Pollard's
Therefore,
70
~
E,P
because
c
is
stationary and
{Y A-£A>
ergodic,
satisfies the stochastic
given by 7A-EA.
differentiability condition with derivative at
is
condition
The differentiability of limiting moment function £0 follows
6.
directly from Assumption C.4.
{*/!.}
iterated limit
the
o
the
Pollard
in
condition is scaled by
converges
surely
almost
Since
to
zero
hence the derivative is asymptotically negligible.
Next we
function
can
also
function
This
rank
.
be
by 0*
Since the approximation of
viewed
as
a
identification
local
is
this condition
local,
condition
on
the
original
.
condition
differentiability
condition
identification condition on the approximating
impose a global
(iii)
on
the
conditions
Theorem
in
derivative
already
3.3
of
together
established
Pakes
and
with
imply
Pollard
discussion on page 1043 of Pakes and Pollard).
»A* (b T
)
=
o
combination of moment conditions to be used in estimation.
47
the
the
stochastic
equicontinuity
(1989)
(see
the
Assumption C.6:
{b
Assumption C.7:
}
{a
converges in probability to
6.
probability
to
}
converges
in
a
nonrandom matrix
a
where a £A is nonsingular.
Finally,
to obtain a limiting distribution for {b
Assumption
C.8:
{/T£_.0O
)>
converges
}
we assume:
distribution
in
to
a
normally
distributed random vector with mean zero and nonsingular covariance matrix
Sufficient
conditions
for
Assumption C.8
described
Gordin
approximations
as
Hansen (1985).
This condition implies that
by
can
be
and
Hall
(1969),
E^O
obtained using martingale
)
Heyde
is equal
(1980)
and
to zero.
The following extension of Theorem 3.1 in Hansen (1982)
is now a direct
consequence of Theorem 3.3 and Lemma 3.5 in Pakes and Pollard (1989).
Proposition C.l:
,
{V T(b_-£ )>
Suppose
Assumptions
that
C.1-C.8
satisfied.
are
Then
converges in distribution to a normally distributed random vector
with mean zero and covariance matrix
[a
o
£(A,)]
t
aV
a
ooo
'
[E(A
to
'
)a
]
Estimation of £A follows as in Hansen (1982) as long as A can be expressed in
finite first moment for some 5
probability to £A.
>
0.
In
this case,
{F D(b )>
converges
in
Proof of Proposition 4.1:
In
light
satisfies
(44)
we now verify that
of Proposition C.l,
Assumption C.3.
r(a)
Let
denote
our
approximation
in
random approximation
the
error:
r(a)
It
|x(y-x'a)(l.
J
in ,-l,{y-x ,-\.
n JI
aiO}
{y-x'aiO>
=
,
(c2)
•
follows from the Cauchy-Schwarz Inequality that
r(«)
Vio}
*
l'(ri'«)llli^ (l0)
s
#~ „»)l
|xx'a - xx'a||(l,
ny -l,
{y-x'aaO>
{y-x'a^O}
y-x'a
-
,
where the second inequality follows because
whenever
(c 3)
)l
and
y-x'a
have
opposite
|x'a - x'a|
dominates |x(y-x'a)|
Therefore,
signs.
the
random
approximation error satisfies:
r(a)/|a-a|
2
s
(C.4)
\x\
for a * a implying that the modulus of differentiability
dmodic)
is dominated by
|x|
positive value of
=
2
c,
sup{r(oc)/|a-a|
|a-a|
<
e
|a-a|<e,
for a*a>
(C.5)
Combined with Assumption 1.1 this implies that for any
£[dmod(E)]
~ _, =
except
when 1,
e
{y-x a=0}
that
:
and 1, _
.
=
finite.
this case
In
1.
,
is
1
As c
it
so that r(a)
49
is
=
*
0,
dmod(c) goes to zero
possible to choose a such
|xx'|.
However Assumption
4.1
with probability zero
implies that this occurs
to zero.
converges almost surely
Q.E.D.
50
Formally C
2
is the dual cone of C.
A weaker version of
Assumption
1.3
this
more
does
restriction would replace Eq by
than
eliminate
redundant
In
q.
securities.
effect,
It
also
precludes cases in which distinct portfolio weights give rise to the same
payoff, with possibly different prices but the same expected prices.
3
Hansen and Jagannathan (1993) showed that the least squares distance between
a proxy and the set
an
alternative
M
of (possibly negative) stochastic discount factors has
pricing-error
interpretation:
Formally,
the
interpretation for the least squares problem (6) is
inf
mzM
sup
peP
2
£p =l
\Emp - Eyp\
and for (7) is
inf sup
mzM* peH
\Emp - Eyp\
2
£ P =1
where H is the set of payoffs on hypothetical derivative claims.
51
pricing-error
'
Assumption 2.1 could be weakened in a variety of ways, but it is maintained
process
{
(x.
,q.
More
simplicity.
pedagogical
for
'
,y. )}
might
we
generally,
is asymptotically stationary,
the stationary distribution
of
(x'.q'.y)
given
is
that
the
where the convergence to
sufficiently fast to ensure that the Law of
is
In this case,
Large Numbers applies to averages of the form (17).
distribution
imagine
by
the
stationary
the joint
point
limit
of
the
The Hausdorff metric is usually employed for compact sets to ensure that the
resulting distance
is
Because
finite.
of
vertical
the
character
of
the
regions and the existence of finite arbitrage bounds,
the Hausdorff distance
will be finite even though the sets are not bounded.
The Euclidean distance
could be replaced by the square root of a
quadratic form in the
in
(30)
differences between two points as long as a positive weight is given to both
dimensions.
Even
making
if
hypothesis
(35)
is
satisfied,
implementation problematic.
the
This
outside the estimated arbitrage bounds.
sample analog may be
happens
when
infinite,
sample
the
mean
is
This phenomenon does not arise for
hypothesis (34).
7
Burnside
(1994)
alternative
and Cecchetti,
versions
costs are introduced.
from positivity and
of
the
Lam and Mark
volatility
bounds
(1993)
tests
developed and studied
when no
transactions
The test used by Cochrane and Hansen (1992) abstracted
can be formulated equivalently using
<p
in
(34).
Burnside (1994) for a Monte Carlo comparison of various volatility tests.
52
See
g
The impetus for this work was the econometric discussion in an unpublished
precursor to this paper: Hansen and Jagannathan (1988).
9
Our formal derivation of
and Pollard
(1989).
A
the distribution
byproduct
theory uses a result from Pakes
from our analysis
in
the
appendix
is
a
(modest) weakening of the assumptions imposed in Hansen (1982) to accommodate
kinks in the moment conditions used in estimation.
Haberman characterized this nonlinear function as a particular projection
onto a closed convex set formed by translating C by -y.
(1989)
only
considers
the
case
in
which
the
data
Although Haberman
are
iid,
his
characterization of the limiting distribution applies more generally with a
covariance matrix replaced by a spectral density matrix at frequency zero.
References
D.W.K.,
Andrews,
"Heteroskedasticity
1991,
Autocorrelation
and
Consistent
Covariance Matrix Estimation," Econometrica, 59, 817-858.
Bansal,
R.
Hsieh,
D.A.
,
and
Viswanathan,
S.
1992,
working paper,
International Arbitrage Pricing,"
"A
Approach
New
Fuqua School
to
Business,
of
Duke University.
Boudoukh,
J.
,
Richardson,
M.
from
Implied
Restrictions
and
Smith,
1992,
Asset
Pricing
T.
Conditional
Inequality
"Testing
working
Models,"
paper, New York University.
Braun,
"Tests
1992,
P. A.,
working paper,
of
Kellogg School
Consistent
Graduate
School
Pricing
of
and
Arbitrage,"
No
Northwestern
Management,
University.
Brown,
and
D.P.
M.R.
Gibbons,
1985,
"A
Simple
Econometric
Approach
for
Utility-based Asset Pricing Models," Journal of Finance 40, 359-381.
Burnside,
C.
,
1994,
"
Hansen- Jagannathan Bounds as Classical Tests of Asset
Pricing Models," Journal of Business and Economic Statistics, 12, 57-79.
Cecchetti,
S.G.,
P.
Lam,
and
N.C.
Mark,
1993,
"Testing
Volatility
Restrictions on Intertemporal Marginal Rates of Substitution Implied by Euler
Equations and Asset Returns," Journal of Finance, 49, 123-152.
54
Chen,
Zhiwu and Peter
Knez,
J.
1995,
"Measurement of Market
Integration and
Arbitrage," forthcoming, Review of Financial Studies.
Cochrane
and
J.H.,
Hansen,
L.P.
"Asset
1992,
Macroeconomics," NBER Macroeconomics Annual
1992,
Pricing
Explorations
MIT Press,
Cambridge,
for
MA,
115-165.
De
Santis,
Tests
G.
"Volatility Bounds
1993,
,
Implications
and
from
for
International
Stochastic
Stock
Discount
Returns,"
Factors:
University
of
Chicago Ph.D. Dissertation.
Duffie,
D.
,
and K.J. Singleton,
1993,
"Simulated Moments Estimation of Markov
Models of Asset Prices," Econometrica, 61, 929-952.
Epstein, L.G., and S.E. Zin,
1991,
"Substitution, Risk Aversion, and Temporal
Behavior of Consumption and Asset Returns: An Empirical Analysis," Journal of
Political Economy, 99, 263-286.
Foley,
D.K.,
1970,
Economic Theory,
Goldberger,
Gordin,
A.
M.I.,
,
2,
"Economic Equilibrium with Costly Marketing," Journal of
276-291.
1968,
1969,
Topics in Regression Analysis, Macmillan, New York.
"The
Soviet Mathematics Doklady,
Central
10,
Limit
1174-1176.
55
Theorem for Stationary Processes,"
Statistics,
Hall,
P.,
Estimation,"
and
"Concavity
1989,
S.J.,
Haberman,
Annals
The
of
1631-1661.
17,
and C.C. Heyde,
1980,
Martingale Limit Theory and Its Application,
New York, Academic Press.
Hansen, L.P.,
1982,
"Large Sample Properties of Generalized Method of Moments
Estimators," Econometrica, 50,
Hansen,
L.P.,
Method
"A
1985,
1029-1054.
for
Calculating
Bounds
on
the
Asymptotic
Covariance Matrices of Generalized Method of Moments Estimators," Journal of
Econometrics, 30, 203-238.
Hansen,
and
L.P.,
Rates
Marginal
Jagannathan,
R.
Substitution
of
"Restrictions
1988,
Implied
by
Asset
on
Intertemporal
Prices,"
working
paper, University of Chicago.
Hansen,
Data
for
L.
and
P.,
Models
of
Jagannathan,
R.
1991,
Dynamic Economies,"
"Implications of Security Market
Journal
of
Political
Economy,
99,
225-262.
Hansen,
L.P.,
and R.
Jagannathan,
1993,
"Assessing Specification Errors
in
Stochastic Discount Factor Models," manuscript.
Hansen,
L.P.,
and K.J.
Singleton,
Estimation of Nonlinear Rational
1982,
"Generalized Instrumental Variables
Expectations Models,"
56
Econometrica,
50,
1269-1286.
He,
H.
and D.M. Modest,
,
1993,
"Market Frictions and Consumption-Based Asset
Pricing," working paper, University of California at Berkeley.
Heaton,
J.
1994,
,
Hildenbrand,
W.
1974,
,
Investigation
Empirical
"An
Temporally Dependent Preference
of
Asset
Pricing
with
Specification," forthcoming, Econometrica.
Core and Equilibria of a Large Economy, Princeton NJ,
Princeton University Press.
Hindy,
1993,
A.,
"Viable
Prices
Markets
in
with
Solvency
Constraints,"
forthcoming Journal of Mathematical Economics.
Huber, P.,
1981,
Jouni,
and
E.
,
Securities
Robust Statistics, Wiley, New York.
H.
Markets
Kallal,
with
1993,
"Martingales,
Transaction
Costs,"
Arbitrage and Equilibrium in
working
paper,
University
of
Chicago.
Knez,
P.J.,
1993,
"Pricing Money Market Securities with Stochastic Discount
Factors," working paper, University of Wisconsin.
Kreps,
D.M.,
1981,
"Arbitrage and Equilibrium in Economies with
Many Commodities," Journal of Mathematical Economics, 15-35.
57
Infinitely
iiiiiifnlMiiiii
3 9080 00932 7856
Prisman,
E.Z.
,
"Valuation of Risky Assets in Arbitrage Free Economies
1986,
with Frictions," The Journal of Finance, 41, 545-559.
Rockafellar,
R.T.,
1970,
Convex
Analysis,
Princeton,
Princeton
University
Press.
Shanken,
J.
,
1987,
"Multivariate Proxies and Asset Pricing Relations:
with the Roll Critique," Journal of Financial Economics,
Snow, K.N.,
18,
Living
91-110.
"Diagnosing Asset Pricing Models Using the Distribution of
1991,
Asset Returns," Journal of Finance, 46, 955-984.
Vuong, Q.H.,
1989,
"Likelihood Ratio Tests for Model Selection and Non-Nested
Hypotheses," Econometrica, 57, 307-333.
Wolak,
F.
A.,
1991,
"The
Local
Nature
of
Hypothesis
Tests
Involving
Inequality Constraints in Nonlinear Models," Econometrica, 59, 981-995.
)I97
59
Date Due