HD28
.M414
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ECONOMETRIC EVALUATION OF
ASSET PRICING MODELS
by
Lars Peter Hansen
John Heaton
Erzo Luttmer
WP#
3606-93
August 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
ECONOMETRIC EVALUATION OF
ASSET PRICING MODELS
by
Lars Peter Hansen
John Heaton
Erzo Luttmer
WP#
3606-93
August 1993
M.I.T.
OCT
LIBRARIES
1
1993
RECEIVED
Econometric Evaluation of Asset Pricing Models
August 1993
Lars Peter Hansen
University of Chicago, NBER and NORC
John Heaton
and NBER
M.I. I.
Erzo Luttmer
Northwestern University
We thank Craig Burnside, Bo Honore, Andrew Lo, Marc Roston, Whitney Newey,
Jose Scheinkman, Jean-Luc Vila, Jiang Wang, Amir Yaron and especially Ravi
Jagannathan 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) and the National Science Foundation
(Hansen) and the Sloan Foundation (Luttmer).
Abstract
In
this
paper
we
provide
econometric
tools
for
the
evaluation
of
intertemporal asset pricing models using specification-error and volatility
bounds.
We formulate analog estimators of these bounds,
consistency and derive
analysis
the
limiting distribution of
incorporates market
frictions
proportional transactions costs.
use
the
methods
to
assess
such as
give conditions for
these
short-sale
estimators.
The
constraints and
Among several applications we show how to
specific
asset
pricing
models
nonparametric characterizations of asset pricing anomalies.
and
to
provide
Introduction
I.
Frictionless market models of asset pricing imply that asset prices can
be
represented
example,
given
stochastic
a
constant
a
portfolio.
discount
factor
Capital Asset Pricing Model
the
in
by
by
the
In
plus
a
scale
(CAPM)
multiple
Consumption-Based
CAPM
pricing
or
the
(CCAPM)
the
For
discount factor
return
the
of
kernel.
on
the
discount
is
market
factor
is
given by the intertemporal marginal rate of substitution of an investor.
r
is
the net return on an asset and m is the marginal
If
rate of substitution,
then the CCAPM implies that:
(l.n
=
1
£[m(l+r)|^]
where ? is the information set of the investor today.
is
the stochastic discount factor,
p,
tomorrow is given by:
=
n(p)
(1.2)
£(mp|^)
today's price,
More generally,
nip),
if
m
of an asset payoff,
.
Thus a stochastic discount factor m "discounts" payoffs in each state of the
world and, as a consequence, adjusts the price according to the riskiness of
From the vantage point of an empirical analysis, we envision the
the payoff.
stochastic
discount
factor
the
as
vehicle
linking
the
stochastic
a
theoretical
model
to
observable implications.
Given
implications
a
particular
of
(1.2)
expectations, yielding
model
can
be
for
assessed
by
first
discount
taking
factor,
the
unconditional
=
Enip)
(1.3)
Eimp).
When m is observable (at least up to a finite-dimensional parameter vector)
by the econometrician,
test of
a
can be performed using a time series
(1.2)
of a vector of portfolio payoffs and prices by examining whether
the sample
analogs of the left and right sides of (1.2) are significantly different from
Examples of this type of procedure can be found in Hansen and
each other.
Singleton (1982),
Brown and Gibbons
(1985),
MacKinlay and Richardson (1991)
and Epstein and Zin (1991).
While tests such as these can be informative,
Further,
interpret the resulting statistical rejections.
directly
when
applicable
are
there
(1.2)
frictions
to
these tests are not
such
as
transactions
For example, when an asset cannot be sold
costs or short-sale constraints.
short,
market
is often difficult
it
is replaced with the pricing inequality:
(1.4)
7t(p)
Finally,
these
2:
E{mp\9)
can
tests
.
not
used
be
when
the
discount
candidate
factor
depends on variables unavailable to the econometrician.
As
an alternative
consider
a
different
specification-error
volatility
to
bounds
bounds
of
testing directly pricing errors using
set
of
Hansen
of
Hansen
tests
and
and
and
diagnostics
Jagannathan
Jagannathan
(1991).
using
(1993),
We
(1.3),
also
and
we
the
the
consider
extensions of these tests and diagnostics, developed by He and Modest (1992)
and Luttmer
(1993),
that handle
and other market frictions.
transactions costs,
short-sale restrictions
We develop an econometric methodology to provide
consistent
estimators
of
the
specification-error
and
volatility
bounds.
Further, we develop asymptotic distribution theory that is easy to implement
and
that
can
be
used
to
make
statistical
inferences
about
pricing
The specification-error and
models and asset market data using the bounds.
along with the econometric methodology
volatility bounds,
asset
that we develop,
can be applied to address several related issues.
The specification-error bounds of Hansen and Jagannathan
can be
(1993)
used to examine a discount factor proxy that does not necessarily correctly
price the assets under consideration (see also Bansal,
1992 for an application).
of
Hsieh and Viswanathan
This is important since formal statistical tests
many particular models of asset pricing
imply
that
the
hypothesis
their pricing errors are zero is a very low probability event.
models are typically very simple,
it
is
that
Since these
perhaps not surprising that they do
not completely capture the complexity of pricing
in financial markets.
The
specification-error bounds give measures of the maximum pricing error made by
This provides a way to assess the usefulness of a
the discount factor proxy.
model
even when
it
is
technically
misspecif led.
Further,
this
tool
can
easily accommodate market frictions such as transactions costs and short-sale
constraints.
Given a vector of asset payoffs and prices,
uniquely determine
work.
m.
Instead
there
is
a
(1.3)
typically does not
whole family of
m' s
that
will
Any parametric model for m imposes additional restrictions on that
family,
often sufficient
to
identify a unique
stochastic discount
factor.
Rather than imposing these extra restrictions, Hansen and Jagannathan (1991)
showed how asset market data on payoffs and prices can be used to construct
feasible
factors.
sets
for
means
and
standard
deviations
of
stochastic
discount
The boundary points of these regions provide lower bounds on the
volatility (standard deviation)
and Luttmer (IS
indexed by the mean.
He and Modest
(1992)
showed how to extend this analysis to the case where some
3)
of the assets are subject to transactions costs or short-sales constraints.
of means and standard deviations of
Ue sets
These fea-
the stochastic
discount factor can be used to isolate those aspects of the asset market data
informative about the stochastic discount factor.
that are most
One way to
do this is to ask whether the volatility bound becomes significantly sharper
market data is added
as more asset
stochastic
without
analysis
security market data in an
limit
to
generally,
More
factors.
discount
having
This would help one
the analysis.
importance of additional
assess the incremental
econometric
to
a
priori
is
it
the
valuable
family
of
have
to
a
characterization of the sense in which an asset market data set is puzzling
without having to take a precise stand on the underlying valuation model.
When testing a particular model of asset pricing in which the candidate
m is specified,
feasible region.
it
is
valuable
sufficiently
Moreover, when diagnosing the failures of a specific model,
to
determine whether
volatile
distribution of
to examine whether the candidate is in the
is often useful
it
wh
or
payoffs
asset
\er
and
candidate
the
the
other
is
it
candidate
discount
factor
aspects
of
discount
the
factor
is
not
Joint
that
are
problematic.
As we
remarked previously,
direct observations of
it may still
sometimes
it
is
not
possible to construct
making pricing-error tests infeasible.
m,
However,
be possible to calculate the moments of a stochastic discount
factor
implied
bounds.
For
by
a
example
model
in
which
Heaton
can
then
(1993)
a
be
compared
to
the
volatility
consumption-based CAPM model
is
examined in which the cous-umption lata is time averaged and preferences are
such that a simple linearization of the utility function can not be done to
consistency
result
uninterested
in
estimators
for
consistency
the
arbitrage
the
of
who
but
results,
calculations necessary for conducting statistical
Sections III. A.,
In
Section
are
bounds.
Those
interested
inference,
the
in
need only read
III.C and III.D before moving to Section IV.
IV
we present
applications and extensions of our
several
results each of which can be read independently after reading Sections
III. A,
III.C and III.D.
entire
feasible
In Section
we discuss the sense in which the
IV. A
and
standard
discount factor can be estimated.
Section
means
of
set
II,
deviations
IV.
for
B provides
the
stochastic
discussion of
a
tests of whether the volatility bound becomes sharper with additional asset
Section
market data.
models
of
the
discount
specification-error
shows how to use the volatility bounds to test
IV. C
Finally
factor.
bound
to
case
a
in
where
Section
there
we
IV. D
extend
parameters
are
discount factor proxy that are unknown and must be estimated.
of
the
the
Section V
contains some concluding remarks.
II.
General Model and Bounds
Our starting point is a model
a
stochastic discount
market
pricing
short-sale
factor
subject
to
constraints
short-sale constraint
for
is
or
in which asset prices are represented by
pricing kernel.
transactions
costs,
subset
the
a
of
an extreme version of a
To
we
accommodate
permit
security
there
securities.
to
Although
transactions cost,
be
a
other
proportional transactions costs such as bid-ask spreads can also be handled
with this formalism.
This
is
done as
in
Foley
(1970),
Jouni
and Kallal
(1992) and Luttmer (1993) by constructing two payoffs according to whether a
security is purchased or sold.
A short-sale constraint
is
imposed on both
artificial securities to enforce the distinction between a buy and a sell,
and a bid-ask spread is modeled by making the purchase price higher than the
sale price.
Suppose the vector of security market payoffs used in an econometric
analysis is denoted x with a corresponding price vector
The vector of
q.
X is used to generate a collection of payoffs formed using portfolio weights
in a closed convex cone C of
P
(2.1)
=
{p
:
IR
:
p = a' X for some a e C>.
incorporate all of the short-sale constraints
The cone C is constructed to
If there are no price
imposed in the econometric investigation.
two
components:
=
x'
More generally,
then C is r".
induced by market frictions,
[x"',x°']
where
contains
x"
prices are not distorted by market frictions and x
subject to short-sale constraints.
partition x into
components
contains the
I
whose
components
Then the cone C is formed by taking the
and the nonnegative orthant of R
Cartesian product of R
k
the
distortions
.
Let q denote the random vector of prices corresponding to the vector x
of
These prices are observed by investors at
securities payoffs.
assets
reflect
are
traded
and
conditioning
are
permitted
to
be
information available
random
to
the
because
investors.
the
the
time
prices
Since
it
may
is
difficult to model empirically this conditioning information, we instead work
with the average or expected price vector Eq.
While information may be
lost
in our
conditioning information of investors,
failure to model explicitly the
some conditioning information can be
incorporated in the following familiar ad hoc manner.
security payoffs used
in an
Suppose some of the
econometric analysis are one-period stock or
bond-market returns with prices equal
to
one by construction.
Additional
synthet ic payoffs can be formed by an econometrician by taking one of
original returns, say x
the
and multiplying it by a random variable,
,
information
conditioning
set
economic
of
constructed payoff is then x z with
a
price of
synthetic payoff
is
random even though
constant.
is
subject
x
If
to
a
agents.
the
the
say z,
in
corresponding
The
Hence the price of the
z.
security
price of original
short-sale constraint,
is
then z should be
nonnegative.
The vehicle linking payoffs to average prices is a stochastic discount
To represent formally this link and provide a characterization of a
factor.
stochastic discount factor, we introduce the dual of
This dual consists of all vectors in
More generally,
zero vector.
described previously,
if
the elements of C
.
whose dot product with every element
For instance, when C is all of R
of C is nonnegative.
the
IR
which we denote C
C,
x
•
n
,
C
consists only of
can be partitioned
the
in
are of the form {0,^')'
manner
where g is
nonnegative.
A stochastic discount factor m is a random variable that satisfies the
pricing relation:
Eq
(2.2)
-
Emx e C
.
To interpret this relation,
there
are no market
relation
element,
(2.2)
is
first consider the case in which C is r".
frictions and we have
the
familiar
namely the zero vector.
partitioned
into
the
two
linear
pricing.
pricing equality because
C
In
has
this
only
Then
case
one
Consider next the case in which x can be
components
comparably, and relation (2.2) becomes:
described
previously.
Partition
q
Eflix"
=
Eq^ - Emx^
a
£q" -
(2.3)
0.
The inequality restriction emerges because pricing the vector of payoffs x^
subject to short-sale constraints must allow for the possibility that these
constraints bind and hence contribute positively to the market price vector.
I I.
Maintained Assumptions
A:
There are three restrictions on the vector of payoffs and prices that
The first is a moment restriction,
are central to our analysis.
the second
is equivalent to the absence of arbitrage on the space of portfolio payoffs,
and the third eliminates redundancy in the securities.
For pricing relation (2.2) to have content, we maintain:
2
Assumption 2.1:
E\x\
Assumption 2.2:
There exists an m
<
a,
E\q\
<
oo.
>
satisfying (2.2) such that Em
2
<
oo.
The positivity component of Assumption 2.2 can often be derived from
the Principle of No-Arbitrage (e.g.,
Kallal 1992 and Luttmer 1993).
the
smallest
identically
(2.2),
(2.4)
a
see Kreps 1981, Prisman 1986, Jouni and
The Principle of No-Arbitrage specifies that
associated with any payoff
cost
equal
zero
to
must
be
strictly
that
positive.
stochastic discount factor m satisfies:
a'
Eimx)
s
a'
Eq
for any a e C,
is
nonnegative and not
Notice
that
from
which
Assumption
that
shows
implies
2.2
Principle
the
of
No-Arbitrage
(applied to expected prices).
Next we limit the construction of x by ruling out redundancies in the
securities:
•
•
If a'x = a 'x and
Assumption 2.3:
a = a
In
Eq = a
•
'
Eq for some a and a
in C,
then
precludes
the
.
absence
the
possibility
transaction
of
that
the
there would exist
a
second moment
nontrivial
nontrivial
this
linear
costs,
Assumption
matrix of x
is
In light of
combination would
(2.2),
have
2.3
singular.
combination of
linear
that is zero with probability one.
of
oc'
Otherwise,
the
payoff vector x
the
(expected) price
be
to
zero,
violating
To accommodate securities whose purchase price differs from
Assumption 2.3.
the sale price, we permit the second moment matrix of the composite vector x
to
be
used
weights
prices.
2.3
Assumption
singular.
to
construct
then
same
the
requires
payoff
distinct
that
must
have
portfolio
distinct
expected
analysis.
Let
1
Minimum Distance Problems
II. B:
There
denote
the
are
set
two
of
problems
all
that
random
underlie most
variables
with
of
our
finite
second
moments
M
that
satisfy (2.2), and let M* be the set of all nonnegative random variables in
M.
both sets are nonempty.
Let y denote some
variable for a stochastic discount factor that,
strictly speaking,
In light of Assumption 2.2,
"proxy"
does not satisfy relations (2.2).
Following Hansen and Jagannathan (1993),
10
consider
we
following
the
two
hoc
ad
squares
least
measures
of
misspecif ication:
8^
(2.5)
min £[(y
=
- m)^]
,
meM
and
5^
(2.6)
minjliy
=
- m)^]
meM
When
the
proxy
bounds
finding
constructed
on
the
Hansen
by
zero,
to
second
and
minimization
the
moment
of
Jagannathan
In particular,
Luttmer (1993).
(1991)
set
is
y
problems
He
to
factors
as
discount
stochastic
(1991),
collapse
and
Modest
and
(1992)
the bounds derived in Hansen and Jagannathan
are obtained by setting y
to
zero and solving
(2.5)
there are no short-sale constraints imposed (when C is set to
and
IR
)
(2.6)
when
the bound
;
derived in He and Modest is obtained by solving (2.5) for y set to zero; and
the bound derived by Luttmer (1993) is obtained by solving (2.6) for y set to
These
zero.
feasible
second
regions
for
moment
means
bounds
and
moment bound implied by (2.6)
it
will
subsequently
be
standard deviations.
is no smaller than that
used
Clearly,
in
the
second
implied by (2.5) since
is obtained using a smaller constraint set.
Next consider the case in which the proxy y is not degenerate.
and
deriving
Jagannathan
(1993)
showed
that
the
least
squares
distance
Hansen
between
a
proxy and the set M of (possibly negative) stochastic discount factors has
an alternative
norm of payoffs
interpretation of being the maximum pricing error per unit
'"
P,
where the norm of a payoff is the square root of its
11
s
When the constraint set is shrunk to M
second moment.
dual
the
derivative
hypothetical
abstract
from
account
takes
interpretation
constraints
short-sale
potential
of
Hansen
While
claims.
pricing
and
their
in
interpretations are applicable more generally.
errors
Jagannathan
analysis,
for
(1993)
pricing-error
2
Conjugate Maximization Problems
II. C:
In
solving
development
of
convenient
most
as in problem (2.6),
the
squares
least
problems
(2.5)
and
econometric methods associated with
to
study
those
in
problems,
maximization problems.
conjugate
the
and
(2.6)
our
is
it
They
are
given by
(2.7)
5^ =
maxiEy^
aeC
S^ =
max {Ey^
aeC
-
Eliy
-
- x'a.)^]
Zol'
Eq)
,
and
(2.8)
where the notation h
by
-
£[(y - x'a)*^]
Eg}
The conjugate problems are obtained
denotes max{h,0}.
introducing Lagrange multipliers
- 2a'
on
the
pricing constraints
exploiting the familiar saddle point property of the Lagrangian.
(2.2)
The
and
a'
then have interpretations as the multipliers on the pricing constraints.
The
conjugate problems
in
(2.7)
and
(2.8)
are
convenient
because
the
choice variables are finite-dimensional vectors whereas the choice variables
in
the original
possibly
least squares problems are random variables
infinite-dimensional
constraint
12
sets.
The
that reside
specifications
of
in
the
conjugate problems are justified formally in Hansen and Jagannathan
and Luttmer
problems
maximization
the
Of particular
(1993).
interest
concave
are
in
to
a
us
is
and
that
that
(1993)
the criteria for
the
first-order
conditions for the solutions are given by:
-
Eq
(2.9)
£[(y
-
x'a)x]
e C*
in the case of problem (2.7) and
Eq
(2.10)
-
£[(y
-
x'a)*x]
e
C*
along with the respective complementary slackness conditions
(2.11)
a'Eq
-
a'£[(y
- x'a) x]
=
a'Eq
-
a'£[(y
-
=
0,
and
(2.12)
In
fact,
optimization
x'a)*xl
problem
.
(2.7)
is
a
standard
that associated with a solution to problem (2.7)
in
M
to
is a random variable m =
{y
and associated with a solution to problem (2.8) is a nonnegative
random variable m = (y
(up
programming
Interpreting the first-order conditions for these problems, observe
problem.
- x'a)
quadratic
the usual
-
x'a)
in
M
These random variables are the unique
.
equivalence class of random variables that are equal with
probability one) solutions to the original least squares problems.
Since
Assumption
2.3
eliminates
redundant
13
securities
and
the
random
variable
-
(y
problem
(2.7)
x'a)
uniquely
is
criterion must be
This
unique.
also
is
determined,
same for all
the
uniquely determined,
hand,
the random variable
a
because
the
implying
that
conjugate
to
value
of
they all
as
the
must
The solution to conjugate problem (2.8)
In this case the truncated random variable
may not be unique, however.
is
follows
solutions,
have the same expected price a'Eq.
x'a)
solution
the
is
(y - x'a)
(y -
the expected price a'Eq.
On the other
is not necessarily unique,
so we can not
exploit Assumption 2.3 to verify that the solution a is unique.
As we will
now demonstrate, the set of solutions is convex and compact.
immediately from the concavity of
The convexity follows
function and
solutions
the
convexity of
closed
be
must
constraint
the
because
the
Similarly,
set.
constraint
set
is
criterion
the
the
closed
set
and
of
the
criterion function is continuous.
Boundedness of the set of solutions can be demonstrated by investigating
the
properties
tail
directions
9
the
of
which
for
criterion
9'x
(2.8)
and divide it by
1
two
cases:
probability
and
To study the former case we take
directions 9 for which 9'x is nonnegative.
the criterion in
consider
positive
with
negative
is
We
functions.
+
2
|a|
For
.
large values of
|a|
the scaled criterion is approximately:
-
(2.13)
Hence
|9|
payoff
in
£[(-x'9)*^]
where 9
=
large values of
is approximately one for
P.
Consequently,
Consider
next
directions
.
Moreover, 9'x is a
Ia|.
unsealed criterion will
the
quadratically for large values
a/[l + |a|^]^'^^
decrease
(to
-co)
|a|.
9
for
which
Assumption 2.2 and relation (2.4) we have that
14
9'x
is
nonnegative.
From
(2.14)
s
£fli(G'x)
e'Eq
for some m that is strictly positive with probability one.
be strictly positive unless G'x is
identically zero,
it
identically zero.
Hence e'Eq must
when G'x is
However,
follows from Assumption 2.3 and inequality (2.14)
that
e'Eq is still strictly positive.
we study the
For directions 9 for which the payoff e'x is nonnegative,
tail behavior of the criterion after dividing by
approximately
e'Eq for large values of
-
the unsealed criterion must diminish
+
2 1/2
|al
)
which yields
,
Hence in these directions the
|a|.
(to -m)
(1
at
least linearly in
Thus
|a|.
we find that the set of solutions to conjugate problem (2.8)
in either case,
is bounded.
For some but not all of the results in the subsequent sections, we will
need for there to exist a unique solution to conjugate problem (2.8).
the set of solutions
To display a
is
sufficient
convex,
local uniqueness
condition for
local
Since
implies global uniqueness.
uniqueness,
let
x
denote
the
component of the composite payoff vector x for which the pricing relation is
satisfied with equality:
(2.15)
where q
=
£mx*
is
the
Eq'
corresponding price vector.
indicator function for the event
{fli>0>.
Ex x '^i~yn\ is nonsingular.
15
let
A sufficient
uniqueness is that
Assumption 2.4:
Also,
lf~>Q»
be
the
the
condition for local
To
why
see
this
valid
a
is
sufficient
condition,
observe
complementary slackness condition (2.12), m is given by (y
vector
- x '3)
from
the
for some
Consequently,
p.
=
Eq'
(2.16)
that
£yl^~^Q^ -
£(xV'
1^-^^^ )3
When the matrix £(x x '1,~^„.) is nonsingular, we can solve (2.16) for
\in>vr
S.
Volatility Bounds and Restrictions on Meains
II. D:
The second moment bounds described in the previous subsection can be
converted into standard deviation bounds via the formulas:
(2.17)
0-
=
[5^ -
(£m)^]^^^
?
=
[6^ -
(£m)^]^^^
"2
where
5
and
~2
5
constructed
are
by
setting
the
proxy
to
When
zero.
P
contains a unit payoff, Em is also equal to the average price of that payoff
and hence
is
unit payoff.
available,
restricted to be between the sale and purchase prices of
However,
so that
data on the price of a riskless payoff is often not
is difficult
it
the
to determine Em.
In these circumstances,
bounds can be obtained for each choice of Em by adding a unit payoff to P
(augmenting x with a
£q with v).
constraints
1)
and assigning a price of v to that payoff (augmenting
In forming
imposed
on
the augmented cone,
the
additional
distortions should be introduced.
mean assignment for
m.
there should be no short
security
and
hence
no
new
sale
price
The price assignment v is equivalent to a
Mean-specific volatility bounds can then be obtained
16
(2.8) and
using (2.7).
(2.17).
The Principle of No-Arbitrage puts a limit on the admissible values of
v.
V e
,v
\
where X
]
lower arbitrage
the
is
is
i;
the
upper
'^
computed using formulas familiar from
These bounds are
arbitrage bound.
bound and
derivative claims pricing:
(2.18)
X
s
-
V
=
infia'Eq
infia'Eq
:
a e C and a'x i -1>
and
(2.19)
While A
not
:
a.
C and a'x i 1>
e
is always well defined via
exist
payoff
a
in
P
circumstances, we define v
(2.18),
dominates
that
to be
may not be because there may
v
unit
a
payoff.
+oo.
Econometric Issues
III.
In
this
section
we
develop
consistency
and
asymptotic
distribution
results for the specification-error bounds presented in Section
presumption underlying our analysis is that
prices
are
replicated
associated with
{(x ',q ',y )'}
the
over
time
composite
some
in
vector
whose sequence of
joint distribution of (x',q',y)'.
the data on asset
stationary
(x',q',y)'
empirical
is
a
(Borel
measurable!
payoffs and
a
That
stochastic
is,
process
approximate
the
We denote integration with respect to the
function
moment, we will approximate Ez by
is
A key
II.
fashion.
distributions
empirical distribution for sample size T as Tthat
such
In
Yz
of
where
17
More precisely,
(x',q',y)
with
a
for any z
finite
first
.
j;z
(3.1)
Among other
=
.
(i/T)Z^^iZ,
we require
things,
this approximation becomes
that
good as the sample size T gets large.
That is we presume that {z
Law of Large Numbers.
A sufficient condition for this is:
Assumption
composite
The
3.1:
arbitrarily
process
iix
'
,q
'
,y
)}
obeys a
}
stationary
is
and
ergodic.
Under
assumption,
this
we
can
think
(x',q',y)
of
as
{x
'
,q
'
,y
)
Assumption 3.1 could be weakened in a variety of ways, but it is maintained
for
pedagogical
process iix
to
'
,q
simplicity.
'
))
,y
is
More
generally,
we
might
asymptotically stationary,
the stationary distribution is sufficiently fast
of Large Numbers applies to averages of
joint distribution of
the process
{(.x
'
,q
'
,y
ix'.q'.y)
is
the form
imagine
that
the
where the convergence
to ensure
(3.1).
In
that
the Law
this case,
the
given by the stationary limit point of
)}
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
in place of the population distribution.
Analogy Principle of Goldberger
random functions
(3.2)
^(a)
=
<p
and
1968
and Manski
4>'
y^ - iy - a'x)^ - 2a'q,
and
18
(Thus we are applying the
1988).
We
introduce
two
y^ - iy - a.'x)*^ - 2a'q
=
$(a)
(3.3)
The sample analog estimators of interest are given by
=
(d )^
(3.4)
max
aeC
"^
y[<p{a)]
and
(d )^
[3.5)
=
max
Consistent Estimation of the Specification-Error Bovinds
III. A:
establish
first
We
sequences id
}
and {d
T
Proposition 3.1:
The proof of
the
the
consistency
statistical
5,
{d
}
and {d
this proposition is given in Appendix A.
and
sample
criterion
functions
The basic
for
the criterion functions converge pointwise
the
of
the
concavity
of
the
criterion
functions,
uniform on compact sets almost surely (for example,
Finally,
since the sets of maximizers of
19
the
idea
conjugate
(in a and P)
this
is
By Assumptions
surely to the population criterion functions introduced in Section
light
converge
}
respectively.
problems are concave and the sets of maximizers are convex.
2.1 and 3.1,
estimator
the
>:
Under Assumptions 2.1-2.3 and 3.1,
population
of
T
almost surely to 5 and
that
T^4><^oi]
almost
II. C.
convergence
In
is
see Rockafellar 1970).
limiting criterion functions
)
for sufficiently large T one can find a compact set such that
are compact,
the maximizers of the sample and population criteria are contained in that
compact set
the
(for
1974 and Haberman
see Hildenbrand
example,
conclusion follows from the uniform convergence of
the
1989).
Hence
criteria on a
compact set.
Asymptotic Distribution of the Estimators of the Bounds
III.B:
consider
We
sequences of
the
next
distribution
limiting
specification-error bounds.
interest
objects of
the
solutions
as
the
to
of
analog
the
ability
Our
to
estimator
express
conjugate problems permits us
the
to
obtain results very similar to those in the literature on using likelihood
ratios
devices
as
selection
model
for
possibly misspecified (for example,
see Vuong 1989).
specification error bounds are positive,
when
environments
in
models
are
We show that when the
we obtain a limiting distribution
that is equivalent to the one obtained by ignoring parameter estimation, and
when
bound
specification error
the
zero
is
the
is
the corresponding
(See Theorem 3.3 of Vuong 1989 page 307 for
degenerate.
distribution
limiting
result for likelihood ratios.
Let
Yi,
be a maximizer of
a
and
a
a
maximizer
of
a
r(/>,
E4>.
a
maximizer of
study
To
the
a
E<p,
limiting
a
behavior
estimators, we use the decompositions:
(3.6)
VT[(d )^
- 5^]
=
- iCa)]
VTY^liu^)
+
VlY^l^U)
-
E^(a)]
and
(3.7)
A((d
)^ - 5^]
= VTY^[4>U^)
-
0(a)] + ^^.[^(a) - £^(a)]
20
maximizer of
of
the
now demonstrate,
As we will
limiting distributions
the
for
the
maximized
values depend only on the second terms of these decompositions.
words,
impact
the
population
unknown
the
replacing
of
In
maximizers
other
by
the
sample maximizers in the sample criterion functions is negligible.
~
Take the case of the sequence {(d)
2
Then by the concavity of
>.
^,
we
have the following gradient inequalities:
5(a
(3.8)
- 0(a)
)
=
[
•
{a
- a)
for
- a)
.
(including
conditions
first-order
the
conditions)
slackness
complementary
£(mx - q)
from
follows
it
- a)
(mx - q) - Eimx - q)]-(.a
+
However,
U^T
(mx - q)
£
T
conjugate
population
the
the
problem
that
(3.9)
The
£(mx
-
inequality
while a
is
(3.10)
Therefore,
q)-U
in
- a)
(3.9)
=
is
E[mx - q) -a^
s
VTY^[4>U^) - ^(a)]
s
VTY^Hmx
WTT[4>(a
)
sample counterparts to
-
.
obtained because £(q
constrained to be in
- q)
^
mx)
is
-
q]]-{a^
converges
- a)
the
dual
C
in
.
probability
the pricing errors obey a Central
the maximizers can be chosen so
in
Combining (3.8) and (3.9) we have that
C.
- E(.mx
0(a)]}
-
that
-
{(a
T
21
a)}
to
zero
if
the
Limit Theorem and
converges almost surely to
]
This
zero.
concavity
convergence
latter
of
demonstrated
be
criterion
population
the
can
.
function
and
by
the
exploiting
convexity
the
of
the
constraint set (for example, see the discussion on page 1635 of Haberman 1989
and Appendix A).
0(a) - £<^(a)
Assumption 3.2:
converges in distribution to a
l^'^T[
imx - q) - E(mx - q)]
normally distributed random vector with mean zero and covariance matrix
(p(a)
Assumption 3.3:
- E(f>{a)
imx - q)
-
E(mx - q)
a normally distributed random vector
More
converges in distribution to
l^'^Y^
[
primitive
assumptions
that
V.
with mean zero and covariance matrix
imply
central
the
limit
V.
approximations
underlying Assumptions 3.2 and 3.3 are given by Gordin (1969) and Hall and
Heyde (1980).
Let
followed
denote
u
by
k
a
+
selection
vector
a
one
limiting
The
zeros.
£
with
in
its
first
distributions
position
for
the
specification-error bound estimators are:
Proposition 3.2:
2.3,
3.1
-
3.2,
5*0
Suppose that
WTld
-
converges
5]}
vector with mean zero and variance
3.1
and
3.3,
{VT[d
-
and
u'
to
Vu/(Ad)
converges
5]>
5*0.
a
.
in
Under Assumptions 2.1 -
normally distributed
Under Assumptions
distribution
to
2.
a
1
random
- 2.4,
normally
distributed random variable with mean zero and variance u'Vu/iiS)
To use
u'Vu
or
Proposition 3.2
u'i^'u.
Consider
the
in
practice requires consistent estimation of
case
of
22
u'Vu.
For
each T form
the
scalar
sequence
(a
{4>
t=l,2,3,
):
...
the frequency zero spectral
T} and use one c
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
the asymptotic distribution for
numbers (degenerate random variables),
problem fails
lack
the
case,
have a unique solution
to
(Assumption 2.4
identification of
of
violated).
is
=pecial
case
considerable
of
is
conditioning
using
Section
the
payoffs
synthetic
form
to
rules out
it
While this
possibil: y of
described
as
that
y
if
solutions
to
degenerate
if
the
(a)
valid
results
or
and
5
=
^ (a)
the
discount
both
are
^ce
are
which
in
=
a
=
a
*
/
{vT(d
of
the
establish that the rate of convergence of {(d)
2
{vT(d
parameters
~
2
can
and {(d)
>
~
,
and
}
)
T
2
>
This
case
identically zero giving
for
conver,^
factor
problems
conjugate
distribution
on
Proposition 3.2 breaks down.
0,
stochastic
population
limiting
suDsequent
=
5
a
is
consequence,
the
0.
As
a
rise
to
a
2
)
used
be
is
Our
>.
id
}
and
{d
>
converge
at
the
rate
VT
,
although
to
and is
T,
T
given by a weighted sum of chi-squared distributions (see Vuong 1989).
result
in
II.
Notice
occurs
information
interest,
In
parameter vector a does not
the
alter the distribution theory for the specification-error bound.
As a
limiting
the
distribution is not normal.
III.C:
In
Asymptotic Distribution of the Parameter Estimators
several
situations
conjugate problems
of
T
remains valid even when the population version of the conjugate maximum
- 6l>
this
Wild
interest
to
it
is
useful
to
examine
:ed in constructing the bounds.
the
solutions
For example,
to
the
it may be
examine whether a particular asset or group of assets are
23
important
whether
in
the
degenerate.
the
coefficient
vector
inference,
assets
are
subject
estimation problem
is
covariance
short
to
posed
as
matrices
in
consider
interest
of
which
to
case
the
case
determine
the
where
constraints.
sales
bound
there
other
In
no
words,
we
unconstrained maximization problem,
an
is
are
in the absence of market frictions,
for
asymptotic
the
coefficient estimators have a form that
(e.g.,
may
limit approximation to do this type of
Since,
.
it
zero,
is
initially
we
assume that the cone C is R
limiting
or
In developing a central
statistical
that
bound
determining
is
distribution
of
the
the
the
familiar both from H estimation
see Huber 1981) and from GMM estimation (e.g.,
see Hansen 1982).
Our
formal derivation of this distribution theory is given in Appendix C and uses
a result from Pakes and Pollard
A byproduct from our analysis in the
(1989).
appendix is a (modest) weakening of the assumptions imposed in Hansen (1982)
to accommodate kinks in the moment conditions used in estimation.
The population moment conditions of interest are:
(3.11)
for
£[x(y-x'a)
- q]
= 0,
the specification-error bound
in which
the no-arbitrage
restriction is
not fully exploited, and
(3.12)
£[x(y-x'a)*
- ql
= 0,
Equalities (3.11) and (3.12) are simply
when the no-arbitrage is exploited.
the
first-order
problems when
conditions
short-sale
(2.9)
and
constraints
(2.10)
are
estimators satisfy:
24
not
for
the
imposed.
conjugate
The
sample
maximum
analog
'
Y^[x(y-x'a^) - q] =
(3.13)
and
(3.14)
Lj.[x(y-x'a^)* - q] =
.
While the equations for a are linear,
the latter case,
those for a are nonlinear.
In
we use a linear approximation to the moment conditions in
deriving the central limit approximation for the parameters:
xiy-x'a)*
(3.15)
-
q
«
xiy-x'a.)* - q - xx' 1^^_^, -^^^ (a-a)
=
x(y-x'a)l,
^
aaO}
{y-x i~^n\
-
<J
•
Notice that the function of a on the left side of (3.15)
except
at
values of a such that y-x'a =
0.
.
We assume
is dif ferentiable
that
such sample
points are "unusual":
Assumption 3.4:
Pr{y-x' a = 0> =
0.
To evaluate further the quality of the approximation in (3.15),
denote the random approximation error:
(3.16)
It
r(a)
=
l^(y-^' «)
^^y-^' a^O>-^y-x' a^0>
follows from the Cauchy-Schwarz Inequality that
25
let r(a)
r(a)
(3.17)
\-^y--' -^
.
\
\
^Uy-x' a.orUy-x'a.O)
{y-x aiO>
{y-x aiO}
£
)\
U|^|a-a|
where the second inequality follows because |x'a - x'a| dominates |x(y-x'a)|
whenever
y-x'a
opposite
have
y-x'a
and
Therefore,
signs.
the
random
approximation error satisfies:
r(a)/|a-a|
(3.18)
\x\^
rs
for a * a implying that the modulus of differentiability
(3.19)
is
dmodic)
dominated by
|x|
positive value of
when
except
'^
that
|a-a|
sup{r(a)/ |a-a|
=
2
Combined with Assumption 2.1 this implies that for any
£[dmod(E)]
c,
,~ „,
1,
=
{y-x a=0>
<
for a*a>
|a-a|<c,
:
e and 1,
,
1.
,„,
{y-x a<0>
is
this case
In
=
finite.
1
ks c
it
is
so that r(a) =
^
0,
dmodic)
possible to choose a such
Ixx'l-
However Assumption
3.4 implies that this occurs with probability zero so that as c
converges
almost
surely
to
As
zero.
is
goes to zero
shown
in
^^
Appendix
0,
C,
dmodic)
these
restrictions are sufficient for us to study the asymptotic behavior of the
estimator {a
(3.20)
>
using the linearization on the right side of (3.15):
£[x(y-x'a)l
_^,~^Q^
- q]
=
26
0.
To
use
matrix E{xx'l,
condition
is
/''>n\^
this
rank
*'°
equivalent
coincide when no
system
equation
linear
(3.20)
one
Assumption
in
short-sale constraints
condition
for
a
moment
second
the
this rank
and
x
must
x
counterpart
The
imposed.
are
the
that
is
because
2.4
need
we
a,
Given Assumption 3.4,
^® nonsingular.
to
identify
to
matrix
E{xx'
to
be
)
nonsingular as required by Assumption 2.3.
with
Working
the
conditions,
moment
linear
two
we
obtain
the
approximations:
(3.21)
~^'^'^^^^^^"''' ""^^ "
•T(a^ - a)
«
" t^^^^'
- a)
«
-[£(xx' )l"VT^[x(y-x'a)
•T(a
^{y-x'
aaO
^ ^
-
'^^
q]
where the notation ~ is used to denote the fact that the differences between
the left and right
app roximations
sides of
justified
are
Combining approximations
(3.21)
formally
(3.21)
These
converge in probability to zero.
in
Appendix
Let
B.
w
=
[0
/
n
]
with Assumptions 3.2 and 3.3 gives us the
asymptotic distribution of the analog estimators.
Proposition 3.3:
2.1-2.3,
Suppose Assumptions
3.1
and 3.2 are satisfied.
Then {v^T(a-a)> converges in distribution to a normally distributed random
T
vector
with
mean
zero
and
covariance
[£(xx')]
matrix:
Suppose Assumptions 2.1-2.4, 3.1 and 3.3-3.4 are satisfied.
wVw'(E(xx')]
Then {v'T(a^-a)}
converges in distribution to a normally distributed random vector with mean
zero and covariance matrix:
To apply
these
[£(xx'
1
,
_^, ~^q^
limiting distributions
) ]
wVw' [£(xx' 1^^_^, -^^^
in practice
estimators of the asymptotic covariance matrices.
27
) ]
.
requires consistent
The terms wWw'
and wWw'
can be estimated using one of
assumptions
Under
£(xx'l, _
/~>oy)
the
maintained
can
where the estimator a
Proposition 3.3,
in
estimated
be
used
is
spectral methods
consistently
in place
of a
referenced previously.
matrices
the
by
their
E(.xx'
sample
in estimating
)
and
analogs,
second of
the
these matrices.
We now briefly describe how the distribution theory
some short-sale constraints are
will
focus
on
the
behavior
limiting
VTia -a) are very similar.
imposed
(C
of
in Section
As
is
II,
Emx
=
£q
or Emx
,
<
modified when
proper subset of r").
VT{a -a),
not m prices the payoffs with equality or not,
(3.22)
a
is
but
results
the
We
for
we partition x by whether or
that is by whether
Eq
The component coefficient estimators that multiply x 's for which there is
strict inequality will equal zero with arbitrarily high probability as the
sample size gets large.
Hence the limiting distribution is degenerate for
these component estimators.
Consider next the estimator of the remaining subvector of
denote
Because of
p.
degeneracy just described,
the
we
can,
lower-dimensional
interior point of C,
imitated
to
estimator.
cone
associated
with
estimating
p.
effect,
in
treat the limiting distribution of the estimator of p separately.
the
which we
a,
If
Let C be
3
is
an
then the argument leading up to Proposition 3.3 can be
deduce
a
However
if
limiting
3
is
at
normal
the
distribution
boundary of
the
for
cone C,
the
the
parameter
limiting
distribution may be a nonlinear function of a normally distributed random
vector (see
As
Haberman 1989, page 1545).
3
in any econometric estimation problem with
28
inequality constraints,
the problematic feature of
in which
limiting distribution theory is the manner
iepends on the true parameter vector p including the associated
1
This feature makes the distribution theory harder to use
discontinuities.
in
this
practice
in
and,
other
has
settings,
approximate bounds on probabilities
led
statistics
test
of
researchers
Wolak 1991 and Boudoukh, Richardson and Smith 1992).
in
our
derivation
of
the
distribution
theory
(see,
Recall,
for
to
compute
for
example
however,
that
specification-error
the
bounds, we were able to circumvent the need for a distribution theory for the
parameter
estimators.
parameter
estimators
frictions,
Thus
becomes
even
more
though
the
complicated
distribution
the
in
theory
presence
for
the
market
of
the distribution theory for the specification-error bounds remains
simple.
III.D:
As
Consistent Estimation of the Arbitrage Bounds
we
discussed
in
Section
the
II
second
moments
bounds
can
be
converted into standard deviation bounds if the mean of m is known or if it
can be estimated using the price of a risk-free asset.
it must
be prespecif led.
free asset
is
price assignment
admissible,
it
Let v be the hypothesized mean of m when
not available.
In
v.
must not
the
)
risk
case
bound estimators for each admissible
~2
5
of
,
for
the
price
induce arbitrage opportunities
collection of asset payoffs and prices.
interval (A ,u
open
*^
h
Proposition 3.1 can be applied to establish
the second moment
the consistency of
When Em is not known
assignment
onto
the
to
be
augmented
Any price (mean) assignment in the
is admissible in this sense.
The final question we explore in this section is whether the arbitrage
bounds,
A
and v
given in (2.18) and (2.19),
using the sample analogs:
29
can be consistently estimated
(3.23)
i^
-infia'Y^q
=
a € C and
:
a'x
i -1 for all
t=1.2
T}
and
u^
(3.24)
s
infia'Y^q
:
a e C and
a'x
The
i
1
for all t=l,2
arbitrage bound u
estimated upper
always
is
T>
finite when
there
is
a
payoff on a limited liability security that is never observed to be zero in
the sample.
Our estimated range of the admissible values for the (average)
price of a unit payoff and hence mean of m is
bounds can be computed by solving simple
[I
,u
]
Notice that these
.
linear programming problems.
In
Appendix A we prove:
Under Assumptions 2.1-2.3 and 3.1,
Proposition 3.4:
almost surely.
and if u
IV.
=
+oo,
If
u
then {u
is
}
then <u
diverges to
+oo
>
}
converges
converges to v
to
X
almost surely;
almost surely.
Applications and Extensions
In
this section we discuss several
analysis of Section III.
of
finite,
{i
feasible means and
applications and extensions of the
First we discuss consistent estimation of the set
standard deviations
of
stochastic discount
factors.
Previously we showed that for a given mean of the stochastic discount factor,
the
standard deviation bound can be consistently estimated.
mean of the stochastic discount factor typically is not known.
it
is
However,
the
As a result
important to understand the sense in which the entire feasible region
30
can be approximated.
examine extensions of
We next
are useful
that
whether,
First we extend
given
initial
an
set
analysis
the
portfolio
stocks
of
firms
of
information about
additional
and asset
to
examine
asset
returns
III
additional
III
We call this set of tests region
Snow (1991) used this type of test to examine whether returns
subset tests.
a
the bounds
Section
of
returns,
asset
of
result in a change in the volatility bound.
on
theory of Section
answering several questions about
in
pricing models.
distribution
the
contained
stochastic discount
volatility of
the
capitalization
small
with
factors
over and above that found in the return on the market portfolio; Cochrane and
Hansen
used
(1992)
determine
to
it
conditioning
whether
information
is
(1993) used it in his investigation of the links between the
important; Knez
markets for Treasury bills, certificates of deposit and commercial paper; and
De Santis
used
(1993)
vis-a-vis
securities
volatility bounds.
particular
A
whether
a
domestic
the
significance
securities
the
in
of
returns
on
construction
foreign
of
the
4
example
of
region
the
factor would
constant discount
consideration.
study
to
it
subset
test
occurs
correctly price
when
checking
assets under
the
This is a test of whether the volatility bounds have content
and is an important initial hypothesis to examine since,
if
true,
the bounds
do not preclude constant discount factors (risk-neutral pricing).
We
then
show
standard-deviations
how
can be
feasible
the
used
to
test
a
regions
for
the
specific
model
of
means
the
and
discount
factor.
Burnside (1992) and Cecchetti, Lam and Mark (1992) have developed a
version
of
this
test
when
there
constraints or transactions costs.
in
a
relatively
simple
manner
by
are
no
assets
subject
to
short-sale
We show how this test can be implemented
exploiting
31
the
results
of
Section
III.
Further we formulate the test so that
it
also applicable when there are
is
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
outline
we
Finally
analysis that
and
(1992).
Luttmer
(1993).
extension
an
the
of
bound
useful when the discount factor proxy under consideration
is
depends upon a vector of unknown parameters.
parameter
specification-error
vector
that
minimizes
We consider an estimator of the
specification-error
the
bound
and
briefly
describe how to develop an asymptotic distribution for this estimator and for
the implied bound.
Some of the formal discussion in this section focuses on the case when
positivity
is
imposed
the
in
specification-error bounds.
construction
volatility
and
data on the prices of a unit payoff are
not used in the econometric analysis.
payoff
the
Moreover, when considering volatility bounds, we
study the more usual case in which
or with a unit
of
require
the
Comparable results without positivity
obvious modifications and are sometimes
computationally simpler.
Consistent Estimation of the Feasible Region of Means and Standard
IV. A:
Deviations
As discussed in Section
I
and
it
II,
is
often of interest to construct
approximations to the feasible region of ordered pairs of means and standard
deviations of stochastic discount factors
Such
a
region
can
be
computed
with
or
implied by security market data.
without
imposing
the
restriction that the stochastic discount factors be positive.
the
region
positivity.
without
positivity
Similarly,
let
S
and
S
the
and
the
S
32
(closure)
denote
the
of
the
no-arbitrage
Let S
denote
region
with
sample counterparts.
The
results
From our
in
Section
know
we
III
and
S
when
there
price of
the points of origin
and S
.
In the more usual case when data on a unit payoff and its price
no
instead are unions of
rays but
represented
(possibly extended)
as
convergence
mean)
and
mean),
(hypothetical
sample
the
of
real-valued
previous
our
such rays resulting
The boundaries of
convex sets with nonempty interiors.
analysis
implies
in
these sets can be
functions
functions
analog
is not
The feasible regions are
matters are a little more complicated.
longer vertical
unit
a
can be estimated consistently by the points of origin
and S
of the corresponding rays S
available,
good
S
is
(average)
the
In this case
this payoff is the mean discount factor.
of the rays S
are
that
rays because
regions are vertical
four
all
sense
what
in
is
to
and S ?
approximations to S
payoff,
turn
now
we
question
of
the
pointwise
to
their
ordinate
the
(in
population
This result implies uniform convergence of the sample analog
counterparts.
functions in following sense.
Since
estimated,
lower
the
for
and
large enough
upper
T,
(A ,u
).
functions are finite on any compact subset of
functions
result
(see
of
Theorem
converge uniformly,
the
10.8
hypothetical
of
can
be
consistently
the sample analog functions under positlvity
are finite on any compact subset of
convex
bounds
arbitrage
R.
mean of
Rockafellar
almost surely,
When positivity is ignored the
1970)
Further these functions are
the
discount
the
sample analog functions
on any compact
subset of
factor.
('^-•^q)
As
iri
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
in
the
large when positivity is ignored.
33
case
of
positivity,
or when
a
the
One
v,
v gets
deterioration
The
arbitrage
finite
instead
viewing
of
point
in R
'^
the
analog
not
be
to
of
the
S
to
T
regions
as
vicinity
To
of
see
functions
a
this,
of
the
approximation error from a set-theoretic vantage
an approximation
is
the
in
problematic.
Consider first the case in which u
.
size
sample of
out
boundaries
the
we explore
ordinate,
turns
bound
sample
the
of
error
as
<
Associated with a
+00.
measured by
the
Hausdorff
metric:
(4.1)
max{7iS*,S*).7iS*.S*)}
=
T}
T
T
T
where:
(4.2)
y(/C ,K
sup
=
)
{v
Since
the
boundaries
of
{S
compact
interval
sequence
{tj
}
ordinate.
can
approach
within
[v
the
be
)eK
consistently
lower
arbitrage
the
)|
222
,v
estimated
boundary
of
bound,
the
S
and
the
uniformly
lower
on
approximation
any
error
converges to zero almost surely.
Measuring
ordered
>
)eK
bounds
arbitrage
\{v ,w )-{v ,w
inf
111
,w
pairs
the
approximation
get
to
close
In other words,
measures of distance,
as
error
without
via
the
restricting
Hausdorff
them
to
metric
have
allows
same
the
we no longer confine our attention to "vertical"
is
the
case when we view
the
boundaries of
the
regions as functions of the hypothetical (expected) prices of a unit payoff.
The added flexibility in the Hausdorff metric permits us to exploit better
the
consistent
estimation
of
the
upper
(Proposition 3.5).
34
and
lower
arbitrage
bounds
When V
P
^
=
)
{v ,w
11
is
bound A
W 2 ,w 2
)eK
1
finite
for
)e/C
2
O^v SO
^
2
any arbitrary positive number greater
Then
.
l(v.w)-(v,v)|
inf
sup
^
Q*v So
^
1
where p
will
(4.1)
T
(C ,C
y
defined by
t}
As a remedy, we replace r by
be infinite.
(4.3)
the approximation error
infinite,
is
than
modified approximation error will
the
sufficiently
large
will
and
T
converge
be
arbitrage
lower
the
defined and
well
surely
almost
to
zero.
Thus we still get uniform convergence as long as the ordinate is restricted
to a finite interval.
Region Subset Tests
IV. B:
The first set of tests we consider are whether the volatility bounds can
be constructed using a smaller vector of
subject
to
uniquely
under
vector
initially consider
we
II I. C,
short-sale
the
case where
and
constraints,
identified.
Let
consideration with
denote
z
price
including the k-1
an
vector
payoffs
ass
bound augmented by a unit payoff.
security payoffs.
we
there
assume
are
that
the
)
that
and
let
are
to
Formally,
f
be
be used
section
in
assets
no
(n-1 -dimensional
s,
As
that
are
parameters
are
vector
the
to
the hypothesis of
of
assets
k-dimensional
construct
the
interest can
be represented as:
(4.4)
E[z{.f'9)* - s]
£[(f'e)
- V
One possibility is
]
to
=
0,
=
0.
test
this hypothesis
35
for
a
prespecified v
,
and
the
other is to test whether it is satisfied for some u
Consider first the case in which v
the
setup of Section
payoff
with a unit
III,
and
form
the
is
>
0.
prespecif ied.
To map this
into
n-dimensional vector x by augmenting z
form q by augmenting s with
the
"price"
v
Then
.
hypothesis (4.4) can be interpreted as a zero restriction on the coefficient
vector a employed
in
Sections
II
and
The
III.
components of
coefficient
vector a set to zero are just the entries of z that are omitted from
A
f.
large sample Wald test of this zero restriction can be formed by applying the
Alternatively, we could construct
limiting distribution in Proposition 3.3.
a test by solving the GMM optimization problem:
..„
.4.5,
where
[wWw'
^Tj;[^;f;^j:-j-„„v„-,^i-VTj;flj;^j>j
is
)
one of
the
.
estimators referenced in Section III.
spectral
Since this test is embedded within a GMM estimation problem,
Section III.C.
the analysis in
can be easily modified to show that the minimized value of the
criterion function is distributed as a chi-square random variable with n
degrees of freedom
(see
Hansen
modified to be for some v
>
0,
When the hypothesis of
1982).
interest
-
k
is
this GMM approach is modified by minimizing
the criterion in (4.5) by choice of 9 and v
with a corresponding loss in the
degrees of freedom.
One special case of this setup is a test for risk-neutral pricing.
this case k
is
one and f contains only a unit payoff.
In
other words,
In
a
constant discount factor prices the securities correctly on average and the
volatility bound is zero.
the
volatility
bound
Furthermore,
estimator
the central
(without
imposing
limit approximations for
risk-neutral
pricing)
degenerate since the second moment of the discount factor is a constant.
36
is
For
both of these reasons,
test for risk-neutral pricing is a useful
a
starting
point in an empirical investigation.
Conducting such tests can proceed as described with one modification.
sampling error associated with the second moment condition in
There
is
(4.4)
implying that wVw'
be
no
imposed
= v
(6
)
conditions.
moment
omitted
and
9
Instead this second condition should
singular.
is
and testing should be based only on the initial set of
When v
should
not
is
be
known,
restricted
this
second condition should be
nonnegative
be
to
when
solving
minimization problem (4.5).
Region subset tests without positivity turn out to be closely connected
to
factor
of
tests
intersection tests
This
connection
frontier
for
structure
(e.g.,
follows
stochastic
see Braun 1992 and Knez
from
the
duality
factors
discount
deviation
mean-standard
and
boundary
1993 for an elaboration).
of
the
mean-standard
and
the
comparable
returns (see Hansen and Jagannathan 1991 for an elaboration).
deviation
frontier
Also,
for
the Wald
and GMM test statistics coincide because the moment conditions are linear in
the parameter vector 9.
The tests using the criterion (4.5) rely upon the distribution theory of
the estimator of a.
To use the theory of Section II
are no securities subject to short-sale constraints.
the
subsection,
presence
of
the
inequality
I.
C requires
that
there
As we discussed in that
restriction
on
the
parameter
vector a can complicate the distribution theory of the parameter estimators.
As a result,
remaining
testing zero restrictions on a subvector of a when some of the
coefficients
problematic.
are
against
On the other hand,
a
nonnegativity
constraint
can
be
the results of Haberman (1989) could be used
to develop such a test when the zero restrictions are imposed on at least all
of the securities to which the short-sale constraints apply.
37
Testing
IV. C:
Specific Model
a
the
of
Discoiint
Factor
using Volatility
Bounds
Suppose that in addition to asset market data,
a model
the discount
of
factor is posited and a time series of observations of the discount factor is
available:
{m
One way to
T>.
t=l
:
test
the model
whether it satisfies the volatility bounds discussed in Sections
Since observations of the discount factor are available,
a
unit payoff can be estimated by the mean of
augmenting
the
original
vector
of
m.
payoffs with
a
examine
to
is
and III.
II
the average price of
Specifically,
form x by
unit
form
payoff;
augmenting the original vector of prices with the random variable
we
had
subject
to
a
results
of
Section
functions
(p
and
II
constraint.
forming a
In
III.B with
and
minor
one
are now constructed by setting
4>
R.
In
is
not
can apply
the
payoff with an average price m
an unit
short-sale
subtracting m
(4.6)
added
test,
we
by
and form
m;
C by constructing the the Cartesian product of the original cone with
effect,
q
that
modification.
The
the proxy y
to
random
zero
and
:
^(a)
s
-
5(a)
s
-
(-a'x)^
-
2a'q
-
m^
,
and
(4.7)
Subtracting m
2
(-a'x)
does
not
-
2a'g
alter
population maximization problems.
values
of
the
criteria
functions.
-
m
.
solutions
the
It
does,
The
38
to
however,
either
change
volatility bounds
sample
the
the
for
or
maximized
Em will
be
satisfied,
(4.8)
if,
and only if
max
aeC
=
^
£^(a)
s
0,
when positivity is ignored, or
(4.9)
when
s
?
max
aeC
positivity
Proposition 3.2
of
E<t>{a)
s
0,
limiting
The
imposed.
is
distribution
formulated
estimation
problem
the
plays
role
no
in
so
approximation
that
the
in
can be applied to construct a test
(appropriately modified)
these hypotheses using sample analog estimators of
have
reported
limiting
E,
and
~ 7
Again,
^.
error
due
to
distributions
for
these
we
parameter
sample
analogs.
In practice we find the solutions for the sample maximization problems,
estimate
the
asymptotic
particular,
let c
Then {^T[c
-
in
and
errors,
form
one-sided
tests.
In
be the maximized value of 7 (0) over the constraint set C.
converges in distribution to a normal random variable with
£,]
mean zero and variance
described
standard
Section
hypothesis (4.9),
u'l^u.
II I. B.
This variance can be estimated in the manner
Since
the "conservative"
is
|
not
specified
choice of ^ =
under
the
null
is used in constructing
the test statistic.
Finally when there are no transactions costs, short sales-constraints or
other
constraints
estimators
can
Likelihood
Ratio
be
to
be
used
test)
considered,
to
that
construct
exploits
39
asymptotic
the
a
different
the
distribution
test,
inequality
of
(analogous
restriction
the
to
in
a
the
Considering the case when positivity is
first-stage estimation of the bound.
imposed,
vector
parameter
the
a
problem
solves
that
satisfies
(4.9)
the
moment conditions:
Elx(x'a)*
(4.10)
=
- q]
.
The inequality restriction (4.9) implies the further
E{[0(a) - |]} =
(4.11)
moment
condition:
0.
for some ^ s 0.
Without a constraint on the parameter
(4.11)
exactly identify a and
we
nonpositive,
using
can set
moment
the
up
GMM criterion
a
conditions
subject to the constraint that ^ £
the
boundary
of
appropriately
the
feasible
scaled
and
(4.10)
GMM
and
(4.10)
the
parameter vector
and
minimize
(a,^)
function
this
the population moments of m are on
region S
minimized
in
(4.11)
If
0.
conditions
with the restriction that ^ be
However,
^.
moment
£,,
(that
criterion
=
when |
is,
function
then
0),
has
the
limiting
a
distribution that has probability one-half of being zero and the remaining
half
of
probability
the
is
according
allocated
distribution with one degree of freedom.
bounds
the
hypothesis)
distribution
for
other
of
negative
test
values
of
statistics
0,
it
one
(p
(under
the
positivity
null
of
the
random function is used in
can be shown that the resulting test statistic coincides with
the test statistic based solely on (4.6).
Similar
When
^.
stochastic discount factor is ignored and the
place of
chi-square
a
This chi-square distribution then
GMM
the
to
approaches
to
testing
a
g
model
40
of
the
discount
factor
can
be
applied
when a
instead
of
series
time
actual
for
data.
decomposed additively
can
m
case
this
In
be
from
randomness
the
simulated
of
$(a)
data
can
security market payoffs and prices and the other due to the simulation of
As in the work of McFadden
Duffie
and
(1991)
Pakes and Pollard (1989),
(1989),
Singleton
and
be
one due to the randomness of the
wo components,
int
constructed
asymptotic
the
(1993),
m.
Lee and Ingram
variance
in
the
limiting distribution will now have an extra component due to the sampling
error induced by simulation.
When
the
first
moments of m can be
two
computed numerically with an
arbitrarily high degree of accuracy, we can proceed as follows.
price vector with Em instead of m and subtract Em
of m
2
the
as
in
deviations
and
(4.7).
of
the
estimated
stochastic
of
from the criteria instead
This same strategy can be employed to assess
(4.6)
accuracy
2
Augment the
feasible
discount
mean-standard deviation pair for
m,
region
for
factors.
one
means
any
For
can compute
and
standard
hypothetical
corresponding test
the
statistic and probability value.
Minimizing the Specification-Error Bound for Paurameterized Families of
IV. D:
Models
Recall that the specification-error bounds provide a way to assess the
usefulness
asset
an
of
misspecif led.
In
many
unknown parameters.
pricing
model
situations
For
example,
the
in
a
even
discount
when
it
factor
is
proxy
technically
depends
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.
way
to
estimate
specification
er
;r.
case
,;ne
the
parameters
Alternatively,
in
41
model
of
the
an
observable
is
to
factor
In this
minimize
model,
the
the
discount factor proxy depends on a
unknown coefficients.
As
linear
combination of
the work of Shanken
in
(1987)
factors with
the
one
could
selecting factor coefficients to minimize the specification error.
imagine
We now
sketch how the results of Section III.B extend in a straightforward manner to
obtain a distribution theory for the minimized value of specification-error
bound.
Suppose that the discount factor proxy y depends on the parameter vector
P €
S where
B
is
a
compact
The population optimization problems
set.
of
interest are now:
(4.12)
5^
- Eiiyifi)
max [syO)^
a&C
min
=
fi€B
-
x'a)^>
-
Za'Eql,
and
(4.13)
When
5^
=
5 and 5 are
discount
min
an
satisfies
the
extended version of
sample analog estimators of 5 and
the
same as
if
.
strictly positive and the parameterized family of stochastic
factors
restrictions,
max |£y(|3)^ - £{[(yO) - x'a)*]^} - Za'Eql
the
solutions
to
5.
the
smoothness
appropriate
Theorem 3.2
can
be
and
obtained
moment
for
the
Again the limiting distribution will be
population optimization problems were
known a priori.
The
approach
can
be
extended
to
compare
specification-errors for two nonnested families of models.
the
smallest
Such a comparison
potentially can be used as a device for selecting between the two families of
models.
Vuong
(1989)
examined a very similar
42
problem by using
the
large
behavior
sample
ratios
likelihood
of
misspecified models (in particular,
we
and
1989);
precisely,
let
can
and
5
5=5
1
parameterized
This
accounted for.
our
to
specification-error
the
the smallest
problem.
bounds
More
associated
can
families
hypothesis
specification-error associated with
As a consequence the performance of
each parameterized family is the same.
two
analysis
his
2
Under the null hypothesis,
the
of
Take the null hypothesis to be:
with two such families.
(4.14)
denote
5
families
see the discussion in Section 5 of Vuong
adapt
and
imitate
nonnested
two
for
not
can
ranked
be
tested
be
by
sampling
once
using
error
is
corresponding
the
distribution theory for the difference between the analog estimators of
5
scaled by the square root of the sample size.
and 5
Finally,
we
sketch
the
distribution
theory
for
the
coefficient
estimators when there is a parameterized family of discount factor proxies.
Suppose that the unique solution p of (4.13) is contained in the interior of
B,
the
moment
parameterization
restrictions,
family
and
satisfies
short-sale
no
the
constraints
population moment conditions are given by:
(4.15)
E{x[y(.^)-x'a]* - q}
=
0;
and
(4.16)
£[^(p){y(p)
-
[y(p)
-
a'xl*}1
43
=
appropriate
0.
are
smoothness
and
imposed.
The
the analog estimators
The distribution theory for
{b
and
}
T
taking
respectively can be deduced by
{a
approximations
linear
of
>
and a
8
T
sample
the
to
moment conditions (4.15) and (4.16) and appealing to the results of Appendix
C.
V.
Concluding Remarks
In
pricing
paper
this
provided
we
developing these procedures,
represent
and
conjugate
the
measurements
maximization
computations
procedures
and
can
it
was advantageous
of
This
problems.
market
for
duality
asset
bounds.
In
due
unconstrained
to
approach
simplifies
resulting
The
frictions
assessing
exploit duality theory
to
solutions
as
inferences.
statistical
account
interest
for
volatility
and
specification-error
using
models
methods
statistical
statistical
transactions
to
both
costs
or
short-sale constraints, and are often easier to interpret than standard tests
For the most part these methods are quite easy to
of asset pricing models.
implement,
They are designed to
even when market frictions are considered.
provide a better understanding of the statistical failures of some popular
asset pricing models and to offer guidance in improving these models.
Among
following:
other
(i)
to
things,
the
results
in
this
test whether a specific model
paper
of
allow
the
one
to
do
stochastic discount
factor satisfies the volatility bounds implicit in asset market returns;
compare
to
the
information
about
the
means
and
standard
discount factors contained in different sets of asset returns;
test
hypotheses about
asset pricing models.
the
size
of
the
(ii)
deviations
and
(iii)
of
to
possible pricing errors of misspecified
An advantage
of
(i)
is
that
the
resulting
test
is
robust to misspecif ication of the Joint distribution of asset returns and the
44
stochastic discount
factor.
In
regards
to
(ii),
our
results permit
these
comparisons among data sets to be made independent of a specific stochastic
discount factor model.
Our motivation for
(iii)
is
to
shift
the
focus of
statistical analyses of asset pricing models away from whether the models are
correctly
specified
and
towards
measuring
misspecif led.
45
the
extent
to
which
they
are
Appendix
Consistency
A:
this appendix we demonstrate formally the results of Section
In
denote a compact set in
11
denote the closure of
subsets of
(A. 1)
11.
K denote
on K given by
-
a
I,
)
=
maxi sup
a eh
inf
112
I
a
a €h
sup
construct
the
of
a
2
2
sup
Jim
inf
a eh
a eh
to define notions of convergence of compact sets.
use
cHh)
we let
the collection of all nonempty closed
t)
,h
Ti(h
Let
h.
For any subset h of U,
r".
We use the Hausdorff metric
^
will
II. A,
We maintain Assumptions 2.1-2.3, and 3.1 throughout.
and III.D.
Let
I
of
-
a
|>
.
11
2
For some of our results we
sequence
a
a
I
in
We
K.
follow
Hildenbrand (1974) and define:
Definition A.l:
For a sequence {h
>
in
11,
lim sup h
Since
the
sets,
it
lim sup
is
is
the
=
J
J
intersection of
a
decreasing
n cl
m
(.
u h
J^
sequence
)
.
J
of
closed
An alternative way to characterize the lim
closed and not empty.
sup is to imagine forming sequences of points by selecting a point from each
h
.
and,
All of the
in
fact,
limit points of convergent subsequences are in the Jim sup,
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."
46
of
minimizers
of
an
Suppose
Lemma A.l:
(i)
{^
is
>
continuous functions mapping
a sequence oi
converges uniformly to
11
into R that
i/»
and
{h
(ii)
converges to h
)
J
Then lim
si.
and lim min
h
where
c g
i
=
v^
min
h
'
J
t//
"
.^
.
{u
=
e
h
:
i/(
(u)
^
(u'
i//
{\b
)
"
in conjunction with
i/»
J
in
light
of
(ii),
the
Al
then
Proposition
Turning
1
follows
the
{h
in
>
Corollary
of page 22 in Hildenbrand.
to
the
result
in
in
light
of
conjunction with h
the
(i),
defines a continuous function on J
"
sequence
from
let ^
and endow } with the
Then
continuous compact correspondence mapping } into
Lemma
},
.
metric for a one-point compactif ication.
sequence
h
€
"
denote the set of positive integers augmented by +«,
and
for all u'
To verify that this follows from the Corollary in Hildenbrand,
Proof:
usual
)
x
Ii;
defines a
The conclusion of the
Ii.
together
with
part
(ii)
of
Q.E.D.
Section
III. A,
we
formally
establish
Proposition 3.1:
Proof of Proposition 3.1:
We treat only the consistency of {d
47
}
because the corresponding argument
for
{d
very
is
}
surely to E^(a)
converges almost
concave as
is
Assumptions
similar.
uniformly on any compact set in R
i
|a|
:
implies
that
=
{a e C
=N}.
\a\
:
Then C
and D
we have
.
be the maximized value of E0 over D
5
<
Since
5.
in
D
the
By
convexity of
sufficiently large
C.
T,
concavity of 7
C and
arbitrage
E4>
on C
turn
to
results
the
bounds
X
in
v
id
to 5
}
III.D
that
and
{t
compact
explore
"directions
set
for
to
the
choice
these
in
these
study
investigate
{u
arbitrage
the
variables
We
infinity."
follows from the
and
)
the
for
}
bounds
the
are
Since there is no
representable as solutions to linear programming problems.
natural
for
Q-E.D.
•
Section
Recall
.
that
coincide with those over
analog estimators
sample
and
almost
are also not
follows
it
^,
the maximizers of T ^ over C
consistency of
statistical
to
the maximizers of V.^ over C
almost sure uniform convergence of {7 ^^ °^ ^
now
Then by choice of N
.
N
the almost sure convergence of
Consequently,
We
contains all of the
"T
for sufficiently large T,
.
By
N
converges uniformly
{T ^^
N
surely,
are compact.
over the constraint set C and that none of the maximizers
E4>
Let 5
that
= {a e C
N
choosing N to be sufficiently large we can ensure that C
are in D
is
the set
II,
define C
N
N
maximizers of
7 ^
T,
{7^} converges
Further as argued in Section
.
iY4>(.<x.)}
N
and D
N>
that
Since for each
For a positive number N,
is bounded.
E4>
imply
3.1
each aeC.
for
Theorem 10.8 of Rockafellar
E4>,
of maximizers of
and
2.1
problems,
we
"directions"
must
using
a
compactif ication of the parameter space.
First consider any a € C such that a'x £ -1 with probability one.
with probability one a'x
>
-1
converges almost surely to a'Eq.
f
s
a'y
g,
it
follows
that
for
all
Define
liin
sup
48
t
with probability one and
~
-
£
f
£
(
X
and X
with
=
-X
Then
{a'T g>
.
probability
Since
one.
Similarly,
I
T
if u
and lim inf u
<
lim sup a
oo,
^ v
Hence our interest is in the lim
.
i
nf
.
T
To construct a compact parameter space,
we map
the original
which we denote as
space for each problem into the closed unit ball in R
We consider explicitly the case of u
The proofs
.
completely analogous to the case for u
parameter
for
the
case of
i
\L.
are
and are omitted.
can be represented as
Notice that the constraint set used in defining v
the set of all a e C satisfying the equation:
E((l - a'x)
(A. 2)
=
]
.
Consider now a transformation of the parameter space by mapping the parameter
The mapping
space into the unit ball.
= (x/(l + |a|)
maps r"
To compactify the transformed parameter space,
unit ball.
the
C,
points
boundary
of
the
unit
Notice
ball.
that
into
the open
we consider adding
we
can
recover
the
original parameterization by considering the inverse mapping:
a
(A. 3)
C/(l -
=
K|)
Using the transformation in
for
1^1
a' s
that satisfy (A. 2) we consider:
(A. 4)
This
<
1.
D
s
{
^ e
transformation
Vf^
I
£{[(1 - Kl)
potentially
boundary of the unit ball.
adds
(A. 3),
-
instead of considering those
=
x'C]*>
solutions
to
>
(A. 2)
by
including
the
The potentially problematic values of ^ are those
49
for which x'
and
<€C
s 0,
C,
I
(^
=
I
1
We rule this out by limiting attention
.
to values of ^ in
D H
(A. 5)
< e line
{
Notice that any
effect
in D for which
C,
focusing
by
C'£q s (I-ICDCUq+D
I
on
satisfies
1
we
D
in
^' s
*
Kl
>
eliminating
are
concerned with estimated upper arbitrage bounds
Also,
any ^ in D for which
Suppose that u
First
Proof:
notice
<
to
are
that
too
low,
not
too
must have an (average) price that
.
We first
:
T
Yq
since
that
D
r\
T
Then iim sup D
oo.
converges almost sure to
7)(D ,D)
T
corresponding
be the sample analog of D.
and D
be the sample analog of D
A. 2:
=1
\C,\
consider the limiting
^ behavior of D
Lemma
<' s
In
This eliminates the troublesome points (directions) from D
is nonpositive.
Let D
Eq/[l-\<;\) s (u +1).
This does not cause us problems because we are
payoffs with "high" prices.
high.
C.'
r\
converges
D
to
D
c
Eq
r\
almost
D.
surely,
We next establish that Iim D
0.
= D
then
.
To
T
converges
uniformly
to
This is sufficient for the Uniform Law of Large Numbers of Hansen (1982)
to
do
we
this
£[(1 -
ICI
)
first
- x'C]*
-
(A. 6)
£J|[(1
show
on
that
Note that
IZ.
KJ)
-
V [(1
-
x'Cj*
-
1^1)
li
Hence from Lemma A.l,
the
x'<]*
n C is compact and that:
[(1
-
^ (1 .
apply.
-
KJ)
-
^'<2^*l}
(£|x|^)^^^)K^-
C3I
Iim sup of the sequence of minimizers of
50
y
-
[(1
-
£[(1
Kl)
-
x' C,]*
- x' C,]*
ICI)
over
U n C
Since v
.
of minimizers of £[(1 -
<
-
|<|)
for all Tsl.
must also be in D
C,]
in
the
set
Lemma
= D
Since D
is separable,
The conclusion follows.
.
T
Suppose
that u
'^^
A. 3:
<
co.
minimizers of
is the set
With probability one, any point in
.
Then lim inf u
t?
a common probability
DSD
measure one set of sample points can be selected so that
As a result lim D
of
is not empty and D
the set D
m,
x'
contained
is
for all Tsi.
Q.E.D.
TO
2 u
.
First note that
Proof:
(A. 7)
u
= mini C'L.q/(l-lCl)
V
= mini C'Eq/(l-|Cl)|
C e D* a D
I
}
for sufficiently large T,
and
C,
€ D*
r\
D
}
Hypothetical expansions of the constraint set D
in
smaller
values
constraint set
Lemma
A. 2
to D
r\
D.
of
the
maximized
Proof:
A. 4:
n D
For
for u
can only result
instance,
augmented to include all of the points in D
is
suppose
r\
the
Then
D.
implies that this sequence of augmented constraint sets converges
The conclusion then follows from Lemma A.l.
Finally, we consider the case in
Lemma
criterion.
.
Suppose that v
Since u
=
oo,
= a.
li
Then {u
=
Q.E.D.
oo.
>
diverges with probability one.
there are no values of a € C such that a'x s
51
1
with
Consequently,
probability one.
1^1
=
the only values of
convergence of J^[(l -
sufficiently large
-
Kl)
=
D
T,
x'<]* to £[(1
=
and u
there are no arbitrage opportunities
such that
D
0.
>
ll^'xll
^'£g
Furthermore, D
is closed implying that
infiCEq
s
:
for
>
< e D*}
>
any
(^
,
it
follows
from Lemma
A.
x'^]*
implies that for
C,'
Eq
C'IL<7
with the fact
>
c/2 for all
that
all
diverges almost surely.
1
in D
such
that
In
B:
C,
e
D
ll^'xll
=
0.
converges almost surely to
with probability one
that
for
The convergence of {D
.
sufficiently
•
}
to D
•
coupled
have norm one then implies that
elements of D
{u
>
Q.E.D.
Taken together. Lemmas
Appendix
Since
0.
for any ^ in
>
•
large T,
*
.
Since {7 q) converges to Eq almost surely and D
D
-
(Assumption 2.2),
imply
c
Kl)
The uniform
Next suppose that D
oo.
Assumption 2.3
(A. 8)
-
= a.
Assumption 2.2 together with the no-redundancy
Also,
that
are ones for which
First suppose that D
We consider two cases.
1.
in D
C,
A. 2,
A. 3
and
A. 4
imply Proposition 3.4.
Asymptotic Distribution of 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
(even when Assumption 2.4 is not satisfied).
52
is
not
uniquely
identified
Proof of Proposition 3.2:
We consider the case of d
The case of d
.
T
let
be a measurable selection from h
a
T
Since Jim sup h
page 54).
(1974),
{a
there is a sequence
^
T
result
in
Theorem
1
A.
1
|
T
=0
For each
of Hildenbrand
almost surely and h
00
that all a e h
is
h
(see
T
Further an implication of Lemma
Appendix A).
(1991)
=
be the
T
such that lim |a -a
in h
>
T
Let h
be the set of maximizers of £^.
set of maximizers of T4> and let h
T,
similar.
is
T
is
compact,
almost surely (see
of Hansen and Jagannathan
the same random variable m =
(y-a'x)*.
00
Also (2.12) implies that for a e h 00
a' a
^
Ul
is the same for
e h
<x
a'
,
q = E{y(y-a' x)* -
(y-a'x)*
},
so that
As a result the random variable 0(a)
.
is
the
00
ash
same for all
.
Now consider the decomposition of
v^TF [(d
)
- 5
]
as in
(3.7):
•T[(d
(B.l)
)^ - 5^]
= /T^.[0(a^) - ^(a^)]
+ •JVQ.~4>Coi^'\
-
£0(a^)]
As in relation (3.10), we have:
(B.2)
Since
|a
-a
Appendix
C:
In
parameter
£
•JVi^\.~^{'k^^
£
/TT [(mx
|
- 0(a^)]
- g)
-
£(mx
- q)\-Ca.^ - a^]
converges almost surely to
0,
the result follows.
Q.E.D.
Asymptotic Distribution
this
appendix
estimator.
we
We
consider
the
asymptotic
begin by demonstrating
that
distribution
of
restrictions used
our
in
Hansen (1982) can be extended along the lines of Pollard (1985) and Pakes and
53
.
in the functions used
Pollard (1989) to accommodate "kinks"
it
is
from
Then
conditions.
moment
straightforward
show
to
discussion
the
represent the
Section
in
Proposition 3.3
that
to
follows
.
from
III.C,
the
main
result in this appendix.
The notation used in this appendix conflicts with some of the notation
used
elsewhere
We
paper.
the
in
let
interest and p any hypothetical point
denote
3
in
the
parameter
vector
the underlying parameter
of
space T
The parameter space is restricted to satisfy:
Assumption C.l: T contains an open ball in R
about S
We will use the construct of a random function.
.
A random function
i/»
maps the
set of sample points into the space of vector-valued continuous functions on
T
.
We require that
be an n-dimensional random vector for each p in
i//0)
T
We also consider an approximating function
=
0!'O)
that is linear 3.
Assumption C.2:
^,0
+ A. (3-3
)
)
The composite random function satisfies:
{ (i/»,
'
,i/'f
'
)
'
}
is stationary and ergodic and has finite first
moments.
We now specify the sense
in which
i/»
The approximation error induced by using is
r^O)
=
I0^O)
-
0^O)I
.
54
is
required to approximate ^
in place of
i/»
is
.
Define:
dmodA5)
= sup{r.
t
t
Note
that
dmodA-)
is
0)/l3-3 o
monotone
in
ip-p l<6. P*P
:
I
o
Assumption C.3:
lim dmod AS) =
5^0
Assumption C.4:
Eldmod AS]]
behaved
for
<
oo
for some 5
when
at 3
sample
other
the
>
almost
take
sure
in
.
0.
is typically taken to be the matrix of
is differentiable at P
i/»
points.
The
random
interpreted as the modulus of differentiability for
The approach adopted
can
almost surely.
To satisfy Assumptions C.3 and C.4, A
i/»
.
We impose the following restrictions on mod
limits as 5 declines to zero.
partial derivatives of
we
Therefore,
5.
}
o
Hansen
continuity of the derivative of
i/»
(1982)
is
to
variable
to
i//.
and be well
at p
mod (5)
is
.
restrict
the
modulus of
converge almost surely to zero and to
to
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.
G^(5)
=
sup
{\Y^<p{fi)
C.4
to
study
the
sense
in
which
Hence look at the approximation error
- Y^iP^i!i)\/\p-pJ
By the Triangle Inequality we have that
55
:
||3-^J<5.p*p^>
JAP
^^
(5)
:
s
l^dmod(S)
Thus by Assumptions C.1-C.2, we have that
(CD
s
lim lim sup c (5)
5^
=
This
(1985)
implies
turn
in
+
o
in
0.
stochastic differentiability condition in Pollard
the
because the counterpart to
v^TI3-3 1/(1
VTO-P o I), which
implies
(C.l)
Edmodid)
lim
5^
T^oo
the
limit
c
(5)
in Pollard's
less than one.
is
taken
5.
condition
because
The differentiability of limiting moment function
directly from Assumption C.4.
7^
Therefore,
-
Ei/i
differentiability condition with derivative at p
is
iterated limit
the
is
e
T
monotone in
{i/(^}
Also,
Pollard's
in
condition is scaled by
stationary
and
ergodic,
follows
satisfies the stochastic
given by r.A-£A.
converges
{7 A-£A>
£i/»
almost
surely
Since
to
zero
hence the derivative is asymptotically negligible.
Next we
function
can
i/»^.
also
function
be
tp.
impose a global
Since the approximation of
viewed
rank
as
a
local
i//
by
i//
identification
is
this condition
local,
condition
on
the
original
.
Assumption C.5: £|A
This
identification condition on the approximating
|<oo
condition
differentiability
has full rank k.
and £A
on
the
conditions
derivative
already
together
established
56
with
imply
the
the
stochastic
equicontinuity
condition
Theorem
in
(iii)
of
3.3
Pakes
Pollard
and
(1989)
(see
the
discussion on page 1043 of Pakes and Pollard).
We study the behavior of an estimator b
^^^(b.
that solves the equations:
=
)
for sufficiently large T.
The
(k
x
n)
random matrix a
selects the
linear
combination of moment conditions to be used in estimation.
Assumption C.6:
Assumption
{b
C.7:
}
{a
converges in probability to ^
converges
}
in
probability
.
to
a
nonrandom
matrix
a
where a £A is nonsingular.
Finally,
to obtain a limiting distribution for {b_} we assume:
Assumption
{/TV
C.8:
1
00 o )>
converges
in
distribution
to
a
normally
distributed random vector with mean zero and nonsingular covariance matrix
V
.
Sufficient
conditions
for
Assumption C.8
described
approximations
as
Hansen (1985).
This condition implies that
by
Gordin
can
be
Hall
(1969),
£i//0
obtained using martingale
)
and
is equal
Heyde
(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).
57
Theorem
Wl{h
Suppose
C.l:
that
Assumptions
C.1-C.8
satisfied.
Then
-& )> converges in distribution to a normally distributed random vector
with mean zero and covariance matrix
[a £(A. )]
t
Estimation of
terms
are
of
a
EL.
a
'
to
[£(A,' )a ]~
.
follows as in Hansen (1982) as long as A can be expressed in
random matrix function D
continuous at p
aV
that
satisfies A =
DO
)
where D
is
with probability one and has a modulus of continuity with a
finite first moment for some 5
>
In
0.
probability to EL.
58
this case,
{TDCb
)>
converges
in
Footnotes
A weaker version of
Assumption
does
2.3
this restriction would
more
than
eliminate
replace Eq by
redundant
q.
In
securities.
effect,
It
also
precludes cases in which distinct portfolio weights give rise to the same
payoff, possibly different prices but the same expected prices.
2
Formally,
S
the pricing-error interpretation for least squares problem (2.6)
=
sup
meM p&P
Ep^ = l
inf
\Emp - Eyp\
is
,
and for (2.7) is
S
where
3
//
=
sup \Emp
m^M* peH
£p^=l
inf
- Eyp\
is a complete set of derivative claims on the payoffs in P.
Haberman characterized this nonlinear function as a particular projection
onto a closed convex set formed by translating C by -p.
(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.
59
4
The impetus for this work was the econometric discussion in an unpublished
precursor to this paper: Hansen and Jagannathan (1988).
The Hausdorff metric
is
usually employed for compact sets to ensure that
Because of the vertical character of the
the resulting distance is finite.
regions and the existence of finite arbitrage bounds,
the Hausdorff distance
will be finite even though the sets are not bounded.
The Euclidean distance in
(4.2)
could be replaced by the square root of a
quadratic form in the differences between two points as long as a
positive weight is given to both dimensions.
7
Even if hypothesis
making
(4.9)
is
satisfied,
implementation problematic.
This
the
outside the estimated arbitrage bounds.
sample analog may be
happens
when
the
sample
infinite,
mean
is
This phenomenon does not arise for
hypothesis (4.8).
Q
Burnside
(1992)
alternative
and Cecchetti,
versions
of
the
Lam and Mark
volatility
bounds
(1992)
tests
developed and studied
when
from positivity and can be formulated equivalently using
(1992)
for
transactions
The test used by Cochrane and Hansen (1992) abstracted
costs are introduced.
Burnside
no
a
Monte
Carlo
comparison
including the ones proposed here.
60
of
various
<(>
(4.6).
See
volatility
tests
in
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2935
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63
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3
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UBBARIES
0Qfi3ST32
5
Date Due
SEP.
3
JUN.
019941
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18195
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