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merican Economic ssociation
The Cross Section of Foreign Currency Risk Premia and Consumption Growth Risk: CommentAuthor(s): Craig BurnsideSource: The American Economic Review, Vol. 101, No. 7 (DECEMBER 2011), pp. 3456-3476Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/41408746
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2/22
Americanconomic
eview
01December
011):
456-3476
http:/Avww.aeaweb.org/articles.php?doi=10.1257/aer.
01.
.3456
The Cross Section of Foreign Currency Risk Premia and
Consumption
Growth
Risk: Comment*
By
Craig
Burnside*
Hanno
Lustig
and
Adrien Verdelhan
2007)
claim that
aggregate
consumption
growth
isk
xplains
theexcess
returns o
borrowing
S dollarsto finance
ending
n
other urrencies.
hey
reach this conclusion after
stimating
consumption-based
asset
pricing
model
using
data on
the returns f
portfolios
f
short-term
oreign-
currency
enominated
money
market ecurities orted
according
to their nterest
differential
ith he United States. Based on their
vidence and additionalUS
data,
I
argue
that
onsumption
isk
explains
none of the cross-sectional
variation n the
expected
returns f their
ortfolios.
Standard
theorypredicts
that
the
expected
excess return f an
asset,
E(Ret),
s
given
by
-co
v(Re
m,
,
where
m,
denotes some
proposed
stochastic iscount
factor
(SDF).
Therefore,
ny
risk-based
xplanation
f the cross-section
f returns elies
on
significant
pread,
across
portfolios,
n the covariance between the returns nd
the
SDF.
For
the
SDFs
that
Lustig
and Verdelhan
henceforth,
V)
calibrate
and
estimate n their 007
article,
t s
impossible
to
reject
hat here s no
spread
n
these
covariances. n fact, t s impossibletorejectthat hese covariances are all zero.
LV's SDF is linear
n
a vectorof risk
factors,
o
they mplement widely
used
two-passprocedure
o estimate ts
parameters.
he first
ass
is a series of time eries
regressions
f each
portfolio's
xcess return n the risk factors.These
regressions
determine he factor
etas,
.
When there re
n
portfolios
nd risk
factors,
is an
nxk matrix. n LV's case n
=
8 and
=
3. None of the ndividual lements f LV's
estimate,
,
is
statistically
ifferentrom ero. For each of thethree
actors,
we also
cannot
reject
the
hypothesis
hat ll
eight
of the relevant lementsof
are
ointly
zero. Confrontedwith his
vidence, alone,
it would be reasonable to conclude that
LV's model does not
explain currency ortfolios
orted
n
interest ates.
The statisticalnsignificancef the factor etas impliesthatLV's measure of the
SDF is also uncorrelatedwith the
excess returns hat
they study.
To demonstrate
this,
considerthree
alibrations f the
parameters
f the SDF in order o construct
time series for
mt:
(i)
the
SDF
parameters
orresponding
o LV's
two-pass
esti-
mates,
(ii)
LV's
Generalized Method of Moments
(GMM,
Lars
P.
Hansen
1982)
estimates f
the SDF
parameters,
nd
(iii)
Motohiro
Yogo's
(2006)
estimates f the
*Duke
niversity,
epartment
f
conomics,urham,
C
7708,
niversity
f
Glasgow,
nd ationalureau
of
Economic
esearch
e-mail:
thankohn
ochrane,
artin
ichenbaum,
avi
Jagannathan,
ergio
ebelo,
ichael
eber,
nd n
nonymous
eferee
or
elpful
omments,
nd he
ational
Science
oundationor inancial
upport
SES-05
6697).
he sual
isclaimer
pplies.
Toviewdditionalaterials,isithe rticleaget
http://www.aeaweb.org/articles.php?doi=
0. 57/aer.017.3456.
3456
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BURNSIDE:ISK REMIAND ONSUMPTION:OMMENT
3457
SDF
parameters
ased on stockreturns. then un series of time eries
regressions
of each
portfolio's
xcess return n the
resulting
m,
series. n each
case,
I
find hat
estimatedSDF betas are
jointly
statistically
ero.
This,
again, suggests
that LV's
model does notexplainthe returns o their urrency ortfolios.
The second
pass component
of LV's estimation
procedure
s a cross-sectional
regression
f
average portfolio
eturns n the betas. This
regression
etermines he
lambdas, X,
x 1
vectorof factor isk
premia.
There are two
problems
withLV's
estimates f X.
First,
hey
ocus almost
entirely
n standard rrors orX that reat he
betas,
3,
as known
regressors,
ather han
generated egressors.
With hese standard
errors
X
appears
to
be
statistically ignificant,
o LV draw favorable nference
bout
theirmodel.1 But
treating
as
known
eads
to a
misleading
evel of confidence
n
themodel. With
onventionally
alculated standard
rrors
Shanken,GMM)
none
of
the estimated actor isk
premia
n LV's benchmarkmodel and none of the
param-
etersof thecorrespondingstimated DF is statisticallyignificant,xcept n cases
where the model has
verypoor
fit.
Bootstrapped
5
percent
onfidence
egions
for
these
parameters
lways encompass
zero.
Consequently,
draw unfavorable
nfer-
ence where
they
do
not.
Second,
forX to be
identified,
musthave full olumn
rank.Because most
of the
elements f
3
are
statistically
lose to
zero,
statistical ests ndicate hat
herank f
is
very
ow,
perhaps
s low as 0. The identification
roblem
aises two
mportant
ssues.
First,
nd most
mportantly,
t weakens
nferencen the sense that
ests f the
pricing
errors ased on the second
pass regressions
ave little
power
to
rejectmisspecified
models
(Raymond
Kan and Chu
Zhang
1999a;
Burnside
2010).
Second,
confidence
regions or
stimates f thefactor isk
premia,X,generated sing symptotic
tandard
errors,
ecome unreliable
Kan
and
Zhang 1999b).
Using
methods hat
re robust o
weak
dentification,
show that
V's data contain lmost
no informationbout
X. This
reinforcesheunfavorable
nference draw
regarding
heirmodel.
In their
eply,
V
defend
heir
indings
n
fourmain
grounds.
First,
hey
discard
most
of
my
comment
s an obscure discussion
of
sampling
uncertainty
s
opposed
to
point
estimates.
t is true that
do not
dispute
their
point
estimates;
this
com-
ment
s not a trivial
eport
n errors
n LV's code for
ordinary
east
squares
(OLS).
Unfortunately,
owever,
ntil
ataseis are
nfinitely
arge,
nferencewill
nvolveboth
point
estimates nd
standard rrors.
n their
original
article,
LV
clearly
recognize
the
mportance
f statistical
ignificance
or nference.
hey repeatedly
efer o the
statistical ignificance ftheir stimates nd to theresultsof statistical ests.Once
inference
s conducted
properly,
owever,
here s little
upport
or
LV's model.
Second,
they appeal
to
a robustness
heck,
described
n their
rticle,
n which
additional test
assets
(six
equity portfolios
nd five
bond
portfolios)
re
included
in
the
model estimation.
he
inclusionof these
test
ssets,
however,
has little
ffect
on
my
conclusions.
One
still cannot
reject
the
null
hypothesis
hat he
covariances
between the excess
returns
f LV's
currency ortfolios
nd
the SDF
are all zero.
Thus,
regardless
f the statistical
ignificance
f
the
parameters
hatdetermine
he
1 herere evenablesf arameterstimatesn heriginalrticle.he tandardrrorsomputedn he irst
six ables
reat
he etas
s known.
he tandard
rrors
omputed
n heast able
ffectively
reathe etass
unknown,
ut
hey
annot
e
ompared
o he
tandard
rrors
n he
estf
he
rticle,
ecause
hey
re
alculated
for different
odel
one
without
constant).
hey
re
lso alculated
ncorrectly,
s
explain
elow.
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4/22
3458 THE
MERICANCONOMIC
EVIEW DECEMBER011
factor
oadings
n
the
SDF,
I
cannot
reject
hat hemodel
predicts
hat
(Re,)
=
0 for
all of the
currency ortfolios. ncluding
more
testassets leads to modest
mprove-
menton the dentification
ront,
argely
because the
equityportfolios
re
correlated
with one of the model's risk factors: he return o the aggregateUS stock market.
Not
surprisingly,
his eads to some estimates f themodel
parameters eing
statisti-
cally significant.
owever,
the model stilldoes not
explain
the
cross-section f for-
eign currency
isk
premia.
The
R2
for he
currency
ortfolios
lone is at best
roughly
zero,
indicating
hat he
model
cannot
explain why
some
currency ortfolios
ave
significantlyositive
returns
while othershave
significantly egative
eturns.
Third,
LV refer o
empirical
evidence
not
in
their
original
article. As in their
more recent
paper
with Nick Roussanov
(forthcoming),
hey
construct
new set
of seven
portfolios
rom heir
riginal
set of
eightportfolios y
considering
trat-
egies whereby
the investor hort ells the low interest ate
portfolio
while
going
long in one of the seven higher nterest ateportfolios.Evidence regarding he
seven "differenced"
ortfolios
ffectsnone of
my
conclusions. Most
important,
since these
portfolios
re linear combinationsof the
original portfolios,
ne can-
not
reject
the null
hypothesis
hatthe covariances
between the excess returns f
the "differenced"
ortfolios
nd the SDF are all
zero, and, therefore,
ne cannot
reject
the null
hypothesis
hat he model
predicts
E(Ret)
=
0 for ll of the
"differ-
enced"
portfolios.
Also,
because the
"differenced"
ortfolios
re
smaller
n
num-
ber,
nd are formed s linear
combinations f the
originalportfolios,
working
with
these
portfolios
an
only
make the dentification
roblem
worse.
Finally,
LV
bring
o bear additional
vidence based on the recent
financial risis.
They argue
that he financial
risis, lone,
s sufficient
vidence that heir
onsump-
tion-basedmodel works.
ndeed,
thefinancial risis
s a
single
observation
hat uits
their
hypothesis.
Consumptiongrowth
ell,
and
currency
eturnswere
negative,
n
late
2008.
However,
show
that
carry
rade
returns nd
consumptiongrowth
re
uncorrelated ver
the full
post-Bretton
Woods
period.
Carry
rade
returns re cor-
related with stock
returns n the
post-Bretton
Woods
period,
but the
market eta
of the
carry
rade s far
oo small to
explain
ts
high average
return. also
establish
that
here s
only
a
very
weak
tendency
f the market
eta of
carry
radereturns o
increase
during
US
recessions and
periods
of stock
market urmoil. his
casts doubt
on
any simple
explanation
f the
returns o the
carry
radebased on
market isk.
I
conclude
that,
aken
s a
whole,
the
vidence for
LV's
consumption-based
model
is extremelyweak. I cannotrejectthat hemodel-predictedxpectedreturns fthe
currency ortfolios
hey
tudy
re
all zero. In their
eply,
V
conclude
by
arguing
that
had the
researchers f
25
years
ago
been
confrontedwith their
results,
here
would never
have been a
"forward
remium uzzle."
I
am alive
now,
nd I have
read
their
rticle.The
forward
remium
s still
puzzle.
In
Section
I,
I
briefly
eview LV's
model,
data,
and
methodological
pproach.
n
Section
I,
I
present
he
first-pass
stimates f
thebetas
thatunderlie
heir
stimates
of
the factor
isk
premia
and
demonstrate
hat here s
little
vidence of
significant
covariance
between
the
portfolio
eturns
nd the risk
factors. n
Section
III,
I
dis-
cuss the
second-pass
estimates
of the
factorrisk
premia
and
the
interpretation
f
thepricing rrors nd calculate standard rrors orfactor iskpremiathat orrectly
account for
estimation
f the
betas. I
discuss
robustness f
my
negative
findings
n
Section
V. Section
V
concludes.
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5/22
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BURNS1DE:
ISK
REMIAND
ONSUMPTION:OMMENT
3459
I.
Model, Data,
Estimation,
nd Inference
LV work
primarily
with a
log-linearized
version of Motohiro
Yogo's
(2006)
model, nwhich the stochastic iscountfactors given by
(1)
m,
=
[1
-
bc(Ac,
-
)
-
bd{Adt
-
)
-
br(rm
-
/ir)].
Here
c,
represents
he
logarithm
f a
representative
ousehold's
consumption
f
nondurable
goods,
d,
is the
logarithm
f
the household's durable
consumption,
rWt
s the
logarithm
of the
gross aggregate
return to
wealth,
c
=
E(Act),
Hd
=
E(Adt),
and
=
E(rWt).
LV
study
the returns o
borrowing
US dollars
in
the
money
market o finance
short-termecurities enominated n
foreign urrency. hey
form
ightportfolios
f
suchpositions,which re createdby sortinghecurrencies ccording o their nterest
differentialersusthe
United States. refer o these
portfolios
s
PI
, P2,
.
P8 with
the order
running
rom ow interest atecurrencies o
high
nterest atecurrencies.2
LV estimate he model
by exploiting
he null
hypothesis
hatthe
approximated
stochastic discount factor
SDF),
m
prices
the n
x
1 vector of
portfolio
xcess
returns,
f.
The
pricing quation
s
(2) E(Retmt)
=
0.
I rewrite
1)
generically
s
(3)
=
[1
-
(f,
-
|x)'b],
where
f,
s
x
1 vectorof risk
factors,
p,
=
^f,),
b
is
x
1 vectorof coeffi-
cients,
nd
s a scalar
representing
he mean of the SDF.
A.
The
Beta
Representation
nd Two-Pass
Regressions
It follows from
3)
and
(2)
that
(4)
(R0
=
cov(R
8/11/2019 Burnside 2011
6/22
3460
THE MERICANCONOMIC
EVIEW DECEMBER
011
represents
he /th ow n
.
LV
estimate
he
ystem
f
equations represented
by
(5)
using equation-by-equation
LS. Given
(4),
the second
pass
is a cross-sec-
tional
regression
f
average portfolio
eturns n
the estimated etas:
(6)
'
=
(3-
+
a,
i
=
,...,n,
where
Re
-
}^(=1
Reit, ,
is the OLS estimate of
,
obtained
in the first
tage,
and
a,
is a
pricing
rror. et
the
OLS
estimator f X be X
=
(' (3)_1
'
R',
where
Re
is an
n X 1 vectorformed rom he ndividualmean returns. he
model's
pre-
dicted
mean returns re
X
and the
pricing
errors re the
residuals,
6t
=
R*
-
X.
The model's fit s assessed
using
the
following
tatistic:
(i'
U
/?*
=
1
-
('
-
X)'(R'
-
X)
(i'
U
=
1
-
(Re - Re)'(Re - Re)
'
where
Ft
=
iE"=i
is thecross-sectional
verage
of themean returnsn thedata.
The model is testedon the basis of the estimated
ricing
rrors
sing
the statistic
C&
=
Ta.'
l,
where
&
is a consistent stimator or he
asymptotic
ovariance
matrix f
VTol
and the nverse s
generalized.
JohnH. Cochrane
(2005)
discusses
how to form
and shows
that
C
-
>
Xn-k
It is common to include a constant n the
second-pass regression
s follows:
(8)
R'
=
7
+
-X
+
,
i
=
1
...,n.
The
constant,
,
is often
nterpreted
s the model's
pricing
error or
the risk-free
rate,
but this error s shared
by
all assets. The
statistical
rgument
or
running
he
regression
without he
constant s thatwe know with
ertainty
hat heexcess return
to a risk-free
sset,
or
any
other ero-beta
sset,
s zero. One
argument
or
ncluding
the constant s the notion hat he
risk-free ate s
imperfectly
easured as the real
return n T-bills.
B. GMM Estimation
Cochrane
(2005)
describes a GMM
procedure
hat
produces
the same
point
esti-
mates as thetwo-pass regressionmethod butallows forheteroskedasticity-robust
inference.When the
constant s included n the
model the moment
estrictionsre
(9)
E{R%
-
a
-
if,)
=0,
i
=
l,...,
.
(10)
E[(Rl
-
a
-
:fr)ft']
0,
i
=
1,...,.
(11)
E{R%-
7
-
|X)
=
0,
i
=
When the
constant s excluded from
he
model,
the ast set of
moment
estrictions
is replacedby
(12)
tf(eft-iX)=0,
i
=
l,...,
.
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01
NO.
BURNSIDE:ISK REMIAND
ONSUMPTION:
OMMENT
3461
In both
cases,
an
identity
matrix s used to
weight
he moment onditions.
The model can also be estimated
sing
a GMM
procedure
hat
reats he SDF as
the
primary bject
of interest. his
procedure,
escribed n more detail n
Cochrane
(2005), estimates hemodel, 3), usingthe moment onditions:
(13)
{Rf[l
-
(f,
-
|x)'b]}
=
0
(14)
E(
f,
-
ji)
=
0.
The
parameter
s unidentifiednd is set
equal
to 1.
The moment ondition
13)
can
also be
modified o allow for common
pricing
rror
cross assets:
(15)
{ftf[l
-
(f,
-
ji)'b]
-
7}
=
0.
As
described
n
the online
Appendix,
the GMM
procedure
based on
(14)
and
(15)
can be set
up
so that t s
numerically
dentical
o
the
two-passregression
method n
terms f
pricing
rrors.3
II. First-Pass
stimates
f
Betas
Like
LV,
I
compute first-pass
stimatesof
the betas
by running
he east
squares
regressions
escribed
by
(5).
I
compute
standard rrors
sing
standard
ystem
OLS
formulas,
s well as GMM-based
procedures.
also calculate 95
percent
onfidence
regions using
a
bootstrapprocedure. Using any
of the these
procedures,
none of
the24 estimated etas is
individually
tatistically ignificant
t the 5
percent
evel.4
More
important,
hen there s
spread
n
the
expected
returns cross
portfolios,
here
should also be
statistically ignificantpread
n thebetas across
portfolios.
With
his
in
mind,
we can test whether
or each
factor
the
eight
factorbetas are
ointly sig-
nificantly
ifferent
rom ero. As Table
1(A)
indicates,
t conventional
ignificance
levels one cannot
reject
he
hypotheses
hat
y
=
j
V
,
and
y
=
0 V
,
for ach fac-
tor
=
1
k. Since the contribution f factor
to the vectorof
model-predicted
expected
returnss A
the atter
hypothesis
ests
mply
thatone
cannot
reject
the
null that each factor's risk
premium
contributes
othing
to the
model-predicted
expected
returns.
In their eply,LV mistakenly rguethat have looked onlyat individualbetas,
when,
n
fact,
n
every
version
of
my
comment,
ncluding
this
one,
I
have
tested
for
spread
in
the
betas.
Second,
they argue
that should have looked
at univari-
ate betas rather han multivariate
etas. This is
puzzling, given
that multivariate
betas are what
appears
in the beta
representation,4),
and what enters
nto the
second-pass regression.
Certainly,
ne can define the matrix
of univariate
betas,
"
=
cov(Rf,f()Df
'
where
Df
is a
matrixwiththe variances
of the factors
n the
3If
n
stimate
f
X s
computed
s X
=
Xy
,
where
yis
he
ample
ovariance
atrix
f
his
stimate
s
identical
o he
wo-pass
stimatef
X.The
quivalence
f he MM
nd
wo-passrocedures
sdemonstrated
n
the nlineppendix.Due o
pace
imitations
report
ullablesf etasn he nline
ppendix.
heGMM-basedtandardrrors
I
present
re
omputed
sing
variant
f he
ARHAC
rocedure
escribed
y
Wouter. en
aan
nd ndrew
T.Levin
2000).
use
VARHACtandard
rrorso ake
ntoccount
ossible
erial
orrelation
n
GMM
rrors.
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8/22
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THE MERICANCONOMIC
EVIEW
DECEMBER
011
Table Factor
etas ndCovariances:ests
or
pread
nd gainstero
(A)
Betas
(B)
Covariances
Ac Ad
rw
Ac Ad
rw
Standardrror
ype
Tests
or o
pread
p- alues)
System-OLS
0.738
0.563 0.273
- -
GMM-VARHAC 0.838
0.596 0.437 0.611 0.901 0.405
Jointestsersusero
p- alues)
System-OLS
0.813
0.623 0.365
- -
GMM-VARHAC 0.799 0.668
0.511 0.447 0.682 0.510
NotesAnnual
ata,
953-2002.
n
part
A)
the
egressionquation
s
Reit a
+
f'ti eit,
where
%
sthe xcesseturnf
ortfolio
at ime
,
t
(
Act
A
dt
W{)
c sreal
er
ouse-
hold
onsumption
nondurables
nd
ervices)
rowth,
d
s
real
er
ouseholdurableon-
sumptionrowth,
nd
w
s he alue
eighted
S
tock
arket
eturn.he
ortfolios
re
qually
weightedroups
f hort-term
oreign-currency
enominated
oney
arketecuritiesorted
accordingo heirnterestifferentialithhe nitedtates,hereI and 8 re he ortfo-
lios
with,
espectively,
he mallestnd
argest
nterestifferentials.he able
eports-values
for estsf he
ypotheses
hat
=
j
V and
=
0 Vi for ach actor
.
In
part
B)
the
covariances
etween
he
xcesseturnsf he
ortfolios
nd he
actors,
ov(i?f,JJ),
re stimated
by
GMM.
he
able
eports
-
alues
or ests
f he
ypotheses
hat
ov(/?f,
)
=
c;
V/
nd
co
(/?f
fj)
=
0 V
forach
actor
.
diagonal,
and zeros
off-diagonal.
his leads to an alternative eta
representation
in which
(Rf)
=
"X"
and X"
=
D^b.
It
is
perhaps
more
straightforward,
ow-
ever,
to work with the SDF
representation
(Rf)
=
cov(Rf, f,)b.
The columns of
cov(R?,f)
are
proportional
o the columns of
".
We can
directly
stimate
the
elements of
cov(Rf, f,)
by
GMM and then test whether
ov(Rf,
;)
,
V
,
and
cov(Rf
fj,)
=
0
Vi.
As Table
1(B)
indicates,
hese
hypotheses
annotbe
rejected
t
conventional
ignificance
evels.
It
makes littledifferencewhich betas we
use, however,
ecause what matters n
the end is how these
betas are reflected
n
the covariance between
the SDF and the
portfolio
eturns.
sing
(3), (2)
can be rewrittens
(16)
E( Rf)
=
-cov(Rf,
m,)/E(mt).
With henormalization = 1,(3) impliesthat (mt) = 1,so we can rewrite16) as
(17)
E( Rf)
=
-
cov(Rj, mt)
=
m,
where
m
=
-
cov(Rf, m,)/
l,, 'm
=
a2m
nd
a2m
s the
variance of
mt.
Given
the
definition f
m
(is, a.
_
m
8/11/2019 Burnside 2011
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VOL.
01
NO.
BURNSIDE:
ISK REMIAND
ONSUMPTION:
OMMENT
3463
Table
SDF
Betas: ests or preadnd gainst ero
Model
i)
Model
ii)
Model
iii)
Standardrror
ype
Testsor o
pread
p- alues)
System-OLS
0.508
0.477 0.575
GMM-VARHAC
0.688
0.589
0.602
Jointestsersus
ero
p- alues)
System-OLS
0.469 0.443
0.505
GMM-VARHAC
0.306 0.170
0.329
Notes: nnual
ata,
953-2002.
he
egressionquation
s
R%
a
mtim
f
lt,
here
mt
1
-
(f,
f)'b,
sthe
ample
ean
f
,
nd he ectortakesn ne
f he
ollowing
three
alues:
i)
b
=
(-
21.0129.9
.46)' corresponds
o V's
wo-pass
stimateith
con-
stant),ii)
=
(37.0
4.7
.65)'
corresponds
oLV'sGMM
stimateith o
onstant)
nd
(iii)
o
(6.74
3.3
.31)'
the
alibrated
odel).
ere
eit
s
a
portfolio
eturn,
nd
,
s the
vector
f
actors,
escribed
n
able .The able
eports-
aluesorestsf he
ypotheses
hat
im m
nd
im
0 V.
measure the SDF
betas,
we need data for
m
which can be constructed
sing
(3),
and values for he
elements f the vectorb. Here
I
use three
ersionsof b taken
directly
rom V's article.
In Table
2(i)
I
use the b
vector
corresponding
o LV's
two-pass
estimates of
X:
bc
-
-21
,bd
=
130,
and
br
=
4.5.
Table
2(ii)
uses theb vector
orresponding
o
LV's GMM estimates f b:
bc
=
37,
bd
=
75,
and
br
=
4.7. Table
2(iii) repeats
he
exercise
using
the
b vector
orresponding
o the
calibratedmodel discussed
in sec-
tion E of LV's article:bc
=
6.7, bd
=
23,
and
br
=
0.31. As Table
2
indicates,
n
all
of thesecases the
null
hypotheses
hat
im
m
or ll i and
im
=
0
for ll i cannot
be
rejected.
n
other
words,
here s no
spread
n the
betas,
and
they
re
ointly
ero.
Tests based
directly
n
cov(Rf,
m,
rather han
m
each the ame
conclusionbut
re
not
reported
n
the
table.
Given
that the SDF betas
are
jointly
statistically
nsignificant,
conclude
that
LV's
model does not
explain
the cross-section
f the
expected
returns f
their
ort-
folios.
In Section
IV I show that his
finding
s robust o
(i)
estimates
of the SDF
based on an
expanded
set of
test assets
including equities
and
bonds,
(ii)
using
the seven
"differenced"
ortfolios
mphasized
in
their
eply,
nd
(iii)
a
higher
fre-
quency,post-Bretton
Woods,
developed-country
atabase that
xtends
through
he
recentfinancial risis.Thus,themessage of this comment s immune o thepoints
emphasized
by
LV in their
eply.
III. Second
Pass and
GMM Estimates
fthe
Model
The second
pass
and GMM
estimates
of the model
provide
us another
pportu-
nity
o assess
LV's
proposed
explanation
of the cross-section
f returns
o
foreign
currency ortfolios.
Of
particular
nterest re
the
point
estimates
f X and
b,
the
R2
measure
of fit nd the
testsof the
pricing
rrors.
LV's
second-pass
regression,
which ncludes
the onstant
,
is
reproduced
n Table
3.Whenpresentingheir indingshey howOLS standard rrors,which ssumethat
the
first-pass
etas
are known.
Given these
standard
rrors,
hefactor
isk
premia
for
consumption
nd
durables
are both
positive
nd
highly
tatistically
ignificant.
he
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10/22
3464
THE MERICANCONOMIC
EVIEW
DECEMBER
011
Table Second-Pass
egression
ith Constant
Factor
rices
X)
Constant
Nondurables
Durables
Market
R2
p-
alue MAE
() {Ad) (rw)
-2.94
29 4/70
333 (
0.44
(0.86) (0.83)
(0.97) (7.59)
(0.483)
[2.23]
[2.11] [2.42]
[18.8] [0.972]
{2.66} {2.48}
{2.41} {23.1}
{0.994}
Notes
nnual
data,
1953-2002.
he table
eports
esults rom
unning
he cross-sectional
egres-
sion
*
7
+
fX
-
where
e
sthemeanxcess
eturnf
ortfolio
and
,
sthe ectorf actoretas
f
portfolio
estimated
n he
irst-pass
egression.
he
ortfolios
nd actorsre escribed
nTable
.
For he ac-
tor isk
remia
X)
OLS
tandardrrors
re n
parentheses,
hanken
tandardrrorsre
n
quare
rackets,
nd
GMM-VARHAC
tandardrrorsre
n races.
ootstrapped
5
percent
onfidence
egions
re
n
ngled
rackets.
For he ests
f he
ricing
rrors
compute
he esttatistic
orach f he hree ethods
f
omputing
he
ovari-
ancematrix
f
OLS, hanken,
nd
MM-VARHAC)
nd
eport
he ssociated
-value.
he statisticromhe
second-passegressionsreportedlong ithhemeanbsolutericingrrorMAE).
R2 of the model is 0.87 and the
p-value
forthe
test for
significance
f the
pricing
errors
s
0.48. These results
re a
key
basis of LV's
positive
ssessment f themodel.
There are three easons our assessment hould be less
sanguine.
The
main one is
thatOLS standard rrors re
nappropriate iven
heestimation f the
betas,
and this
turns ut to matter
great
deal for nference.Once standard rrors re
computed
appropriately,
stimatesof X and b are
statisticallynsignificant.
he latter
inding
is especially mportant ecause itsuggests hat heconsumption actors o nothelp
price currency
eturns. he second reason to be
skeptical
s that he model
performs
much more
poorly
when we
impose
therestrictionhat heconstant s
equal
to zero.
The b
parameters
emain
nsignificant,
nd the fitof the model deteriorates ub-
stantially.
he third eason to be doubtful
bout the model estimates s thatthere
is a severe
identification
roblem.
Under nonidentificationr weak
identification,
asymptotic
tandard rrors
even arguably ppropriate
nes)
are
likely
o
understate
the
degree
of
uncertainty
bout the model
parameters.
A.
Inference
bout Model Parameters
As Cochrane (2005) pointsout,the fact that he betas are estimated n the first
pass
mattersfor
inference bout the factor risk
premia,
and this remains true
asymptotically.
here are three tandard
ways
to
deal withthis
problem.
One is
to
use the correction
f the standard rrors
uggestedby Jay
Shanken
1992).
Another
is to
compute
tandard rrors
sing
the
first f the two GMM
procedures
described
above,
because it
produces
the
same
point
estimates.A third
s to construct on-
fidence
regions
for the
parameters sing
bootstrap
methods.
By
construction,
he
alternative
pproaches
o
calculating
tandard
rrors o not ffect he
point
stimates
of the
factor isk
premia.
The three
procedures
ead to similar nference
egarding
the
model,
and
using
them,
ather
han OLS
standard
rrors,
matters
oth
qualita-
tively ndquantitatively.
The Shanken
and
GMM-corrected standard
rrorsfor the
model with the con-
stant
Table
3)
are
roughly
wo
to three imes
arger
han the
OLS standard rrors
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VOL 01
NO.
BURNSIDE:ISK REMIAND
ONSUMPTION:OMMENT
3465
that
gnore
estimationof the
betas.
Why
is the Shanken
correction o
big?
Let
0
=
(7
X')',
E
=
E(ete't),
and let
f
be a matrixwith
a
leading
column and row
of
zeros,
and
Ef
in the ower
right
orner.When the betas are
treated s known he
covariancematrix fV (0 - 0) is
(19)
e
=
(+'
V
+'
E+(+' +1
+
f.
Here
+
=
(t )
and l is an n x 1
vector f ones. With he
Shankencorrection he
covariancematrix s
(20)
n
=
(1
+
X'
Ef1
X)(+'
V
0+'
S+(+' +)"'
+
f.
In
some finance
pplications
the Shanken
correction s small. For
example,
forthe
CAPM estimated singthe annual returns f Fama and KennethR. French's 1993)
25
portfolios
ortedon
size and book-to-market alue over the
period
1953-2002,
the Shanken-correction
erm,
1
+
A2/
},
is estimated o be 1.03. In LV's case the
estimate f
1
-I-
X'
Ef1
X is
6.79.
Although
he ndividualAs
in
LV's model are of
the same orderof
magnitude
s for he
CAPM,
the
consumption
actors ave much
smallervariancethanthe market eturn.
his blows
up
the size
of
the Shanken cor-
rection
ubstantially.
Using
either he Shanken or GMM standard
rrors,
none of the estimatedfac-
tor risk
premia
in Table 3 are
statistically ignificant
t the 5
percent
evel. The
bootstrap-based
5
percent
onfidence
egions
for he
parameters
lso
encompass
0.
These resultsdo not
mply
that he
price
of
consumption
isk s zero.
Instead,they
indicate that he
oint
behavior of the
currency
eturns nd
consumption
actors
s
uninformativebout the
price
of
consumption
isk.
LV defend he tatistical
ignificance
f their
indings
n three
rounds.
irst,
hey
appeal
to Ravi
Jagannathan
nd
Zhenyu
Wang
1998)
to defend heuse
of OLS stan-
dard errors ather
han the
Shanken
correction. his is
inappropriate.
agannathan
and
Wang's point
s thatunder
heteroskedasticity,
he Shanken
correction s
inap-
propriate,
nd thatmore
general
GMM errors re
appropriate.
hanken's
proof
hat
corrected tandard rrors re
necessarilybigger
thanOLS standard
rrors oes not
work forGMM standard rrors.GMM
errors ould be smaller
than OLS standard
errors,
ut n LV's case
they
re not.
Second,
in theirfootnote
11,
they rgue
that
theOLS standard rrors re close inmagnitude o theGMM standard rrors. his is
because
they nappropriatelyompare
GMM standard rrors
or he model
without
a
constant,
o the OLS standard rrors
or model with constant.5
he
appropri-
ate
GMM
standard rrors re
actually
close in
magnitude
o the Shanken
standard
errors.
hird,
hey rgue
that tandard
rrors rom
bootstrap rocedure
re small
enough
to make the estimated
isk
premia
significant.
ather
than focus on boot-
strap
tandard
rrors,
use the
entire istribution
f
bootstrapped
stimates o show
that 5
percent
onfidence
egions asily
encompass
0.
It is
especially
important
o know
whether he
consumption
actors
help
to
price
the
currency
eturns. his
requires
us to focus
on the
parameter
ector b.
GMM
5
When
V
eport
hanken
nd
tandard
rrors
orhemodel
ithout
he
onstant,
heyppear
o
se ncor-
rect
ormulas,
sdetailed
n he nline
ppendix.
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12/22
3466 THE MERICANCONOMIC
EVIEW
DECEMBER011
Table
A
GMM
stimates
f heModelwith he
Constant
Model
arameters
b)
Stage
Nondurables
Durables Market
R2
p-value
MAE
(Ac) (Ad) (rw)
First
-21.0
129^9
46 087 0.44
(87.7) (97.4)
(4.83)
8/11/2019 Burnside 2011
13/22
8/11/2019 Burnside 2011
14/22
3468
THE MERICANCONOMIC
EVIEW
DECEMBER
011
Table
Second-Pass
egressionithoutConstant
Factor
rices
X)
Nondurables
urables Market
R2
/?-
alue
MAE
(Ac) (A ) (rw)
0.59
1.10
11.7 0.34
1.17
(0.73)
(1.02) (7.40)
(0.001)
1.01] [1.40]
[10.1]
[0.059]
{1.17}
{1.69}
{10.6}
{0.173}
8/11/2019 Burnside 2011
15/22
VOL. 01
NO.
BURNSIDE:ISK REMIA
ND
ONSUMPTION:OMMENT
3469
Table
GMM
stimatesf
heModel
with o
Constant
Factor
rices
X)
Stage
Nondurables
Durables
Market
(Ac) (Ad) (rw)
First
0.59
1.10
11.7
(1.07)
(1.64)
(8.26)
Second
2.37
3.48
10.2
(1.00)
(1.13)
(7.37)
8/11/2019 Burnside 2011
16/22
3470 THE MERICANCONOMIC
EVIEW DECEMBER011
my
conclusion,
based on betas
alone,
that he model cannot
explain
differencesn
expected
returns cross
currency ortfolios.
Adding
the six Fama-French
quityportfolios
o the set of test ssets
slightly
lle-
viates the dentificationroblembecause equitieshave statistically ignificantetas
with
respect
o the market eturn
actor,
w.
However,
he rank ests
presented
n
the
online
Appendix
ndicate that he
matrix
till
appears
to
have
reduced
rank.The
identification
roblem
does not
go away
withthe furtherdditionof the five
Fama
bond
portfolios.
Estimatesof the model without he constant
sing
the
currency
nd
equityport-
folios as test assets are
presented
n the online
Appendix.
As indicated
there,
with
sufficientterations ver the
weighting
matrix,
he factor isk
premia
for
onsump-
tion
growth
nd durables
growth
re
statistically ignificant.
owever,
he fit f the
model with
respect
o
currency ortfolios
s
verypoor.
When theR2 statistic s cal-
culated ust for urrency ortfoliostrangesfrom .03 (at the firsttageofGMM)
to -0.76
(for
iterated
GMM).
The mean absolute
pricing
errorfor the
currency
portfolios anges
from1.44
(at
thefirst
tage
of
GMM)
to 1.88
(for
terated
GMM).
Why
do I
compute
these statistics
ust
for
currency ortfolios?
First,
the
goal
is to
explain
the cross-section f returns f
currency ortfolios.
he
R2
across all
assets does not tell us whether he model
explains why
some
currency ortfolios
earn
high
returns nd others arn ow returns.
nstead,
he
R2
across
ust
the
currency
portfolios
ells us whether he model
explains why
some
currency
ortfolios
like
P7)
earn
high
returns,
nd other
urrency ortfolios
like PI)
earn ow
returns,
n
average.
Second,
we are not
trying
o
explain
why
the
currency ortfolios
ll have
relatively
ow
returns
the average
excess return
cross LV's
eightcurrency ort-folios is 0.1
percent)
compared
to the
equityportfolios
the
average
excess returns
of the six
Fama-French
portfolios
re all above 6
percent,
nd
they
re centered
near 9
percent).
That s not
a
puzzle, given
that
urrency ortfolios
re
only weakly
correlatedwithrisk
factors hat
price equity portfolios.
he
puzzle
is
why
the ow
interest ate
currency ortfolio,
I,
has an
average
return f -2.3
percent,
nd
why
the
high
nterest ate
currency ortfolios,
7 and
P8,
have
average
returnsn
excess
of
2
percent.
Adding
the fiveFama
bond
portfolios
oes not
mprove
he
situation.At the
first
two
stages
of GMM the
results re
quite
similarto those
obtained
using only
the
currency
nd
equity
portfolios,
lthough
urther
terations ver
the
weighting
matrix
eventually rive utconsumption rowthnd durablesgrowths significantiskfac-
tors.Once
again,
thefit f the
model with
espect
o
currency ortfolios
s
verypoor.
If
theR2
statistic s
calculated
ust
for
urrency ortfolios
t
ranges
from
.03
(at
the
first
tage
of
GMM)
to
-
1.35
(for
terated
GMM).
The mean absolute
pricing
rror
for he
currency
ortfolios
anges
from1.40
(at
thefirst
tage
of
GMM)
to 1.64
(for
iterated
GMM).
B.
Differenced
urrency
ortfolios
In
their
eply,
V
argue
that
hey
an
explain
the
excess returns o
the
strategy
f
holdingPi andshortingI, for = 2, ,8. Theyclaimthat heir rticlesreally bout
these even
"differenced"
ortfolios,
hich
refer o as
D2,
D3,
.
D8,
and not
really
about
he
original
ight ortfolios.
iven
that he
ntire rticle s
about he
P-portfolios
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OMMENT
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this comment s
surprising,specially given
the
following
tatement
n the
original
article:
consumption-based
models can
explain
thecross-section f
currency
xcess
returns
f and
only
f
high
nterest
ate
currencies
ypically epreciate
when real US
consumption rowths low,while ow interest ate urrenciesppreciate."Notice that
these tatementsre not bout whether here s a differenceetween herates f return
of the
portfolios;
t s a statementbout therates f return hemselves.
LV
also
argue
for
working
with he
D-portfolios
n the basis that
large swings
n
thedollarmake t hard o
accurately
stimate he
onstant,"
he onstant
eing
7
in
the
model for he
P-portfolios.
his
argument
s not
persuasive
because the
ntercept
an
be "estimated"with
perfect ccuracy.
We know hat he mean excess return f a zero
beta asset s
zero,
so we can set
7
equal
to zero without ven
having
o estimate t.
Nonetheless,
whatof LV's
point
hat he onstant o
longerplays
an
important
ole
in
explaining
hecross-section nce we consider
he even D
portfolios?
V are
right,
butthispoint an easilybe made without ewtables ofpoint stimates. onsider he
model with heconstant. he estimates
f
the
econd-passregression atisfy:
(21)
Re
=
7
+
,-
X
+
h
i
=
1
This is
equation 8)
with
7
and X
replaced
by 7
and X
(the
two-passestimates)
nd
replaced by
,
the
diosyncratic ricing
rror r
residual).
Now
suppose
we con-
sider henew set of excess
returns,
f
=
R
-
R',
for
>
2. Given thedefinition
f
Rf
and
equation
21),
it follows
that he
sample
mean
of
Rf
is
given by:
(22) f
=
((
-
O'
X
+
-
,
i
=
2,...,.
PI is not
ust
any
asset.
It
happens
to be an asset
forwhich the SDF
beta is
roughly
zero
(
X
=
0)
and the
idiosyncratic ricing
rror s
very
small
(
=
0).
Using
these facts n
equation
22),
we have
(23)
Rf
=
(3,-
+
h
i
=
2
...,n.
Now,
since
=
0 it
also follows thatthe
average
value
of
across
i
=
2,
..,n,
is
roughly
ero.
Thus,
the same X
obtained
by running
he cross-sectional
egres-
sion
for the
P-portfolios
an
approximately
it he cross-sectional
distribution
f
Rfwithout constant.
In their
eply,
t seems that
V concede that
heirmodel
does not
price
the
original
portfolios.
ut this
means
they
have
not dentified
hetrue DF.
All of the
portfolios
(the
P and
D-portfolios)
hould
be
priced
by
the same SDF.
Effectively
his
means
there
mustbe a
missing
factor hat
prices
PI. Since this
factor s
responsible
for he
fit f the
original
portfolios
we are
back to
square
one.
Are the
D-portfolios
riced by
consumption
rowth?
On
thebasis
of factor
etas
theanswer
s
clearly
no."
The beta matrix
or he
D-portfolios
s the ame
transfor-
mation f
the
-portfolios
sed to
testwhether
hey
re
equal
to
a common
constant:
i.e.,
D
=
8/11/2019 Burnside 2011
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3472
THE MERICANCONOMIC
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DECEMBER011
consumption
rowth,
.47 1 fordurables
growth
nd 0. 186 for hemarket
eturn. he
SDF
betas associated with he
D-portfolios
re also
jointly tatisticallynsignificant
(Table 10).
Workingwith heD-portfolios lso does not alleviatethe dentificationroblem.
The identification
roblem
arises
because thereexists at least one
nonzero
x
1
vector
x
such
that
px
=
0,
statistically.
iven
the definition f
D
it follows that
Dx
=
0,
statistically.
n
fact,
the identification
roblem gets
worse,
because the
transformation
is not nvertible.
ny
x such that
px
= 0
implies
that
Dx
= 0.
But there
may
be additional
x
such
that
Dx
=
0 forwhich
px
0.
This is
hardly
surprising,
ince
throwing way
information
s never
ikely
to
improve
dentifica-
tion.
Formal test tatistics
erifying
his re
provided
n the
Appendix.
C. The Post-BrettonWoodsEra
My
comment
mainly
iscusses
the onclusionswe shoulddrawfrom V's evidence.
Additional vidence from he
post-Bretton
Woods
era,
similar o the evidence ntro-
duced
by
LV in
their
eply,
asts furtheroubt
n
a
consumption-basedxplanation
f
carry
radereturns. ere I discuss evidence
gleaned
from
sample
of
21
developed-
country
urrencies verthe
period
1976-2010.
The same
sample
of currenciess used
by
Burnside t al.
(201 1)
for he
period
1976-2009,
and similar
amples
are used
by
Lustig,
Roussanov,
and Verdelhan
forthcoming)
nd Lukas Menkhoff t al.
(forth-
coming),
for he
period
fter
983,
to
study arry
rade
portfolios.
My
dataset
consists of
spot
and forward
xchange
ratesfrom
Reuters/WMR
nd
Barclays,
available on Datastream.The raw data are
daily
observations f
spot
and
one-month orward
xchange
rates. use end of month alues
of thesedata to create
monthly
bservations.The data
span
the
period January
976
to December
2010,
with he
ample
varying
y currency.
s in
Lustig,
Roussanov,
and
Verdelhan
forth-
coming),
n each
period,
the
available currencies n
my
sample
are sorted nto six
bins
according
o their orward iscount
gainst
he
US dollar.The first in ncludes
those currencieswith he
smallestforward iscounts
the
owest nterest
ates),
the
second bin the next
smallest, tc.,
with
he sixthbin
consisting
f those
currencies
with
he
argest
forward iscounts
and,
therefore,
he
highest
nterest
ates).
then
compute
the
payoff
ssociated with
borrowing
ne dollar n
order o invest
qually
in the
riskless securities f the
constituent urrencies f each
bin. This
procedure
producessix currency ortfolios hat refer o as Q 1 Q2, . Q6. Then, followthe
procedure
dvocated
by
LV in
their
eply.
construct he fivedifferenced
ortfolios,
DQ2,DQ3, ...,DQ6,
which nvolve
holding
Qi
and
shorting
l,
for
=
2, ..,6.
1
measure
payoffs
o
these
portfolios
n
a
monthly
asis and
then
omputequarterly
excess
returns. o assess the
model I use data
for
consumptiongrowth,
urables
growth,
nd
the market eturn
hat re available
from1976:11 o
2010:1
(sources
for
these data
are described n
Burnsideet al.
201
1,
and in the online
data
archive).
Using
these
portfolios
find ven
stronger
vidence
against
consumption-based
explanation
of
currency
eturns. s the
detailed results n
the
Appendix
indicate,
none of the
betas of the
differenced
ortfolios
with
respect
o
consumption rowth
and durablesgrowth s individually ignificant. he pointestimates also do not
display
any pattern
f
increasing
with
the nterest
ifferential.
ot
surprisingly,
s
Table 1 1
indicates,
we
cannot
reject
thenull
hypotheses
hat
tj
=
0
for
ll
i,
for he
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MERICANCONOMIC
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011
Table
Factor
etas nd
Covariances
or
uarterly
ifferenced
ortfolios,
Tests or pread
nd gainst
ero
(A)
Betas
(B)
Covariances
Ac Ad w Ac Ad fw
Standardrror
ype
Testsor o
pread
p- alues)
System-OLS
0629 0499 85
- -
~
GMM-VARHAC 0.542 0.615 0.044 0.494 0.402 0.072
Jointestsersusero
/?-alues)
System-OLS
0.763 0.618 0.016
- -
GMM-VARHAC 0.685 0.681 0.049 0.640 0.470
0.133
Notes:
uarterly
ata,
976:11-2010:1.
n
part
A)
the
egression
quation
s
Reit
a
+
f
t
-
eit,
here
eit
s he
xcess
eturnf
ortfolio
at ime
,
nd
,
s he
ector
f
iskactorsescribed
inTable .The
ortfolios
re
Q2,DQ3,..,DQ5,
he eturnso
holding
ong ositions
n
he
five
uarterlyortfolios
2,Q3,..,Q6
while
olding
short
osition
n he
uarterlyortfo-
lioQl.The ableeports- aluesorestsf he ypotheseshattj .V and tj 0 Vfor
each actor
.
In
part
B)
the ovariancesetweenhe xcess
eturns
nd
he
actors,
ov
Rehfj),
are
stimated
y
GMM. he able
eports7-
aluesor
estsf he
ypotheses
hato
(/f,JJ)
=
Cj
V and ov
Rehfj)
0 Vforach actor
.
centers n thefact hat
urrency
eturns nd
consumption rowth
re
approximately
uncorrelatedwith ne another. he
global
financial risis s an observation
hat uits
LV's
hypothesis,
ecause it
s also an
episode
in which
US
consumption rowth
was
low. The other rises
Mexican,
Asian, Russian,
and
Argentinian)
hat
hey
mention
in their
eply
are
not,
because
they
did
not coincide with
periods
of low
US con-
sumption rowth.According
to LV's
data,
durables
growth
was well above
average
through
ll of the atter
pisodes,
while nondurable
onsumption rowth
was well
above
average during
wo and
slightly
elow
average
n two.
Overall, however,
there s little evidence
that
average
carry
trade
returns an
be
explained by
increased
exposure
to stock market isk
during
periods
of
reces-
sion,
market
ownturns,
r
market
olatility.
o
demonstrate
his,
first
egress
he
monthly
eturns o the
DQ6
portfolio
equivalent
o
LV's HML
carry
rade
portfolio)
on the
monthly
xcess returns f
the
value-weighted
US stock market
the
factor).
The
market eta of
DQ6
in the
period
1976:2-2010:12 is 0.18 and
s statisti-
cally significant.
s in
the
quarterly ample,
thisbeta is
much too small
(by
a
factor
ofmore thanfive)toexplaintheaveragereturn f thecarry rade.
I
then
divide
my
data into
recessions
and
expansions
as
defined
by
the
NBER,
periods
of
high
and low stock
market
olatility,
nd
periods
of
high
and low
stock
market eturns.
o measure
volatility,
use
the
daily
standard eviation
of the mar-
ketexcess returnn
each
month.
Months
n
which this
measureof
volatility
s
more
than one
standard
deviation above
its mean are
denoted as
"high
volatility."
o
divide
thedata into
periods
of
high
and low
stock
returns,
denote
months
n
which
the
market xcess
returns
more than ne
standard
eviationbelow its
mean as "low
return"
months. he
market eta
of
DQ6
is
0.14
during xpansions
and 0.26
during
recessions. t is
0.25 in
periods
of
highvolatility
nd
0.
1 1
in other
eriods.
t is 0.33
in periodsof low stockreturns nd 0.15 in otherperiods.In each case, however,
the
difference
etween the
two betas is
not
statistically
ignificant,
nd
the betas
certainly
emain
nsufficientlyarge
to
explain
the
average
returns f the
carry
rade.
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ONSUMPTION:OMMENT
3475
Table 2 SDFBetas or
uarterly
ifferenced
ortfolios,
Tests or preadnd gainst ero
Model
i)
Model
ii)
Model
iii)
Standardrrorype Testsor o preadp- alues)
System-OLS
0.333 0.185 0.273
GMM-VARHAC
0.280 0.103
0.301
Joint
estsersusero
p- alues)
System-OLS
0409
068 0.399
GMM-VARHAC
0.332 0.139 0.431
Notes:
uarterly
ata,
76:11-20
0Q:I.
he
egressionquation
s
Reit
ai
-
mtimeit,
where
t
1
-
(f,
f)'b,
s
he
ample
ean
f
,,
nd he ectortakesn ne f he ol-
lowing
hreealues:
i)
b
=
(-21.0
129.9
.46)'
corresponds
o V's
wo-pass
stimateith
a
constant),ii)
=
(37.0
4.7
.65)' corresponds
o V'sGMM stimateith o
onstant),
and
iii)b (6.74
3.3
.31)' the
alibrated
odel).
ere
eit
nd
,
re he eturnsnd ac-
tors,
escribedn he oteo able The able
eports-values
orestsf he
ypotheses
hat
m ^ia aAm=^i-
V. Conclusion
To
explain
cross-sectional variation
in
expected
returns,
risk-based
story
requires
that ssets withnonzero
returns e correlatedwith the
proposed
SDF. As
Section
II
demonstrates, owever,
LV's
consumption-based
DF is
approximately
uncorrectedwith ll of thereturns
hey tudy.
Given this
fact,
ne cannot
reject
hat
LV's estimates re consistent
with
consumption
isk
explaining
none of the
cross-
sectionalvariation
n
the
expected
returns
hey
tudied.
I
have
argued
hat he tatistical
nsignificance
f the SDF
betas eads to two addi-
tional
problems
with
LV's conclusions.
First,
t
mplies
that ne
cannot
gnore
am-
pling
uncertainty
n
the
betas when
conducting
nference
bout factor isk
premia.
The statistical
ignificance
V
point
o
argely
vanisheswhen standard rrors
ppro-
priately
eflect his
uncertainty.
econd,
the
degree
of
uncertainty
bout the betas
implies
that
hefactor isk
premia
re
veryweakly
dentified.
his makes
asymptotic
inference ess reliable.
Confidence ets
that re robust o weak identification
uggest
thatLV's data
are
approximately
ninformativebout
consumption
isk.
Finally,
have
argued
that
LV are able to
report
trikingly
igh
R2 measures of fit
because theirmodel
includes a constant
ricing
rror,
which
s treated s
part
f the
model's predicted xpectedreturns.When this onstants excludedfrom hemodel,
the
R2 statistics re much
smaller
nd,
n
many
cases,
negative.
A central
oint
of
my
discussion
s that he betas
of
consumption
actors re
very
poorly
stimated.
his is
why
a
consumption-based
model is difficult
o
rejectusing
a
formal est f the
pricing
rrors.
f
a
great
deal moredata
were
collected,
one
might
obtain
ufficientlyrecise
estimates
f thebetas
to enable
sharper
onclusions
about
the
model. But with
he data we
have,
both LV's data
and
highfrequency
ata
from
the
post-Bretton
Woods
period,
there
s no
justification
or
drawing strong
on-
clusion
in
favor
f a
consumption-based
model. Burnside
et al.
(2011)
have
argued
that
out-of-sample
isk s a
potential xplanation
for the
returns o
the
carry
rade.
If thesecurrentlyut-of-sampleventsoccur nthefuture,tmaywell turn ut that
a
consumption-based
model works.
At the
moment,
however,
model
based
on in-
sample
variation
n
consumption
oes
not work.
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011
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