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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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8/11/2019 Burnside 2011

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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:

[email protected]).

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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8/11/2019 Burnside 2011

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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.


8/11/2019 Burnside 2011

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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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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

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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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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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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

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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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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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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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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)

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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}

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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)

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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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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

=

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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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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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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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Burnside 2011