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orKing raper series
C o n s u m p t i o n - B a s e d Modeling
of Long-Horizon Returns
K e n t D . D a n ie l a n d D a v id A . M a r s h a ll
3
■)
W o r k in g P a p e r s S e r ie s
R e s e a r c h D e p a r tm e n t
F e d e r a l R e s e r v e B a n k o f C h ic a g o
D e c e m b e r 1 9 9 8 (W P -9 8 -1 8 )
FEDERAL RESERVE B A N K
OF CHICAGO
December 24, 1998
Consumption-Based M odeling of Long-Horizon
Returns*
K e n t D . D a n ie l* a n d D a v id A . M arshall**
Abstract
Numerous studies have documented the f i u e of consumption-based pricing mod
alr
e s to explain observed patterns in stock and bond returns. This f i u e has some
l
alr
times been attributed to f i t o s transaction costs or d r b l t . I such f i t o s
rcin,
uaiiy f
rcin
axe important, they should primarily a f c the higher frequency components of
fet
asset returns. The long-swings, or lower-frequency comovements should be l s a
es f
fe t d Consequently i transaction costs are important, t s s of the consumption
ce.
f
et
based asset pricing model which concentrate on lower-frequency components may
be more s c e s u .
ucsfl
We investigate t i hypothesis using a variety of diagnostic t s s We f r t use
hs
et.
is
coherence analysis and bandpass f l e i g analysis to show t
itrn
hat, while there i
s
a complete lack of correlation between asset returns and consumption growth at
frequencies higher than about 0.7 years-1 (swings longer than 1 4 ye r ) the coher. as,
ence/correlation between the two s r e at lower frequencies i above 60%. We per
eis
s
form Hansen and Jagannathan (1991) bounds t s s x 2 t s sofmoment r s r c i n ,
et,
et
etitos
and Hansen and Jagannathan (1997) specification t s softhree consumption-based
et
models of the asset-pricing k r e : Time-separable preferences with power u i i y
enl
tlt,
the Abel (1990) “
Catching up with the Joneses” preferences, and Constantinides
(1990) habit-formation preferences. While none ofthese models perform well at the
quarterly horizon, the performance ofthe Abel and Constantinides models improves
strikingly at the two-year horizon.
tW e thank Steve C ecchetti, Larry C hristiano, John C ochrane, G eorge C onstantinides, Larry E pstein,
Lars H ansen W alter Torous and participants o f the N BER A sset P ricing Group M eeting and UCLA
F inance Sem inar for helpful discussions, and we acknow ledge D enise Duffy and G lenn M cAfee for
superlative research assistance. D aniel gratefully acknowledges research support from the C enter for
R esearch in Security Prices (C R SP) at th e U niversity o f C hicago. T he opinions expressed in th is paper
are th ose o f th e authors, and do not reflect the view s o f the Federal R eserve Bank o f C hicago or the
Federal R eserve System .
’ K ellogg G raduate School o f M anagem ent, N orthw estern U niversity, E vanston, IL 60208,
847-491-4113, kent@ kent.kellogg.nw u.edu, h ttp ://k en t.k ellogg.n w u .ed u /
’ ’ Federal R eserve Bank of C hicago, 312-322-5102, dm arshall@ frbchi.org
1
Introduction
Most research on consumption-based a
sset pricing focuses on short-horizon returns. The start
ing point i the familiar intertemporal Euler equation
s
1=
(
1)
E t [m T+Tr l +T]
t
where rJ+T denotes the gross real cumulative asset return from date t to date t
+ r
and m J+T
denotes the intertemporal marginal rate of substitution (IMRS) between wealth at date t and
wealth at date t + r The unconditional version of th s Euler equation i
.
i
s
1=
(
2)
E [mJ+TrJ+T] .
In the literature on consumption-based asset p i i g m j i modeled as a function of aggregate
rcn,
s
consumption, and the implications of ( or ( are tested f r return horizons r equal to one
1)
2)
o
month or one quarter. The l t r t r typically r j c s the conditional Euler equation ( )
ieaue
eet
1.
Presumably, the unconditional Euler equation ( should be easier to ftto the data. Equa
2)
i
tion ( only requires that the IMRS correctly price equity returns ”on average”. Yet, Cochrane
2)
and Hansen (1992) show how d f i u t i i to even ft t i unconditional Euler equation. The
ifcl t s
i hs
problem i that, to ft equation ( , one needs substantial covariation between m J+T and r[+ r .
s
i
2)
Formally, equation ( ) implies
2
E [rj+ r
~
rf t ] E
=
-cm
(m[+T,rt =
T
+T)
(3)
- o m a r p m>r
where r f ( denotes the gross re l r s - r e rate from date t to date t + r a m and
a ikfe
,
ar
denote
the standard deviations ofm l +T and rJ+T r s e t v l , the correlation c e f c e t between m[+T
epciey
ofiin
and rJ+T i denoted pm>r,and we use the implication of equation ( ) that £[mj+T]=
s
1
E [ l/ r f{ \ .
The left-hand side of equation ( ) i the mean equity premium, ’discounted” by the mean
3 s
’
inverse r s - r e r t . To generate a large mean equity premium, a model of m j+ r must display
ikfe ae
substantial negative covariation with equity returns. Within the context ofconsumption-based
asset pricing, th s would require substantial positive covariation between equity returns and
i
consumption growth.
Empirically, the contemporaneous correlation between quarterly returns and quarterly con
sumption growth i small ( e s than 0 1 ) and the la g s correlation at any lead/lag (when
s
ls
.5,
ret
1
returns lead consumption growth by two quarters) i l s than O.2.1 Cochrane and Hansen
s es
(1992) c l t i low correlation between the return on market proxies and consumption growth
al hs
the “
correlation puzzle.”2 A number of factors have been proposed to account for the low
correlations between stock returns and aggregate consumption growth at short-horizons, in
cluding uninsurable cross-sectional heterogeneity,3 fixed costs ofadjusting consumption,4 costs
of portfolio adjustment,5 and even small deviations from perfect rationality.6 While these fac
tors could substantially a f c the co-movements of asset returns and aggregate consumption
fet
at high frequencies, they should be l s disruptive to the theory at longer horizons. Simple
es
correlations between consumption growth and the V W index return suggest that there may be
merit in this argument. While the contemporaneous correlation between consumption growth
and returns at the one-year horizon i about the same as f the one-quarter horizon, the
s
or
correlation between one-year consumption growth and one-year returns lagged by two quarters
i 0 3 , almost twice as high as the maximal correlation found between quarterly returns and
s .5
quarterly consumption growth
I consumption-based pricing operators perform better at pricing long-horizon a s t , t i
f
ses hs
would provide indirect evidence that the basic intuitionunderlying the equilibrium asset pricing
theories of Lucas (1978), Breeden (1979), and Grossman and S i l r (1982) i sound; the wel hle
s
l
documented f i u e of this theory at matching high-frequency data could be attributed to
alrs
transaction c s s market imperfections, and uninsurable heterogeneity. This analysis would
ot,
not be a substitute f r formal modeling of these f i t o s but would at l a t suggest that
o
rcin,
es
t i approach i on the right t a
hs
s
r ck. I, however, consumption-based pricing proves useless
f
at a l frequencies, th s would represent a substantial challenge to equilibrium pricing theory.
l
i
In a l such models, financial assets are vehicles f r transferring consumption across time and
l
o
1 T hese num bers were calculated using real non-durable and services consum ption d ata, and returns
on the CRSP value-w eighted index, deflated by the ND& S deflator.
2Even if the correlation betw een consum ption grow th and equity returns is sm all, equation (3)
suggests an alternative way o f generating a large m ean equity premium: increase th e variability o f the
IM RS (for exam ple, by assum ing a high degree of risk aversion). C ecchetti, Lam , and M ark (1993) try
th is second strategy. W hile they can m atch the m ean equity prem ium , they have difficulty m atching
both the first and second m om ents o f the equity and risk-free return data.
3See, e .g ., C onstantinides and D uffie (1996), Lucas (1994), H eaton and Lucas (1995)
4G rossm an and Laroque (1990), M arshall (1994), M arshall and Parekh (1998).
®Luttmer (1996), He and M odest (1995).
6See C ochrane (1989).
2
random states. It is not clear what sort of model can capture this intuition without implying
a theoretical linkage between aggregate consumption and asset returns at some horizon.
In this study, we investigate the a i i y of of consumption growth to explain asset returns
blt
at low frequencies using a variety of diagnostic t s s We f r t use multivariate spectral analysis
et.
is
to characterize to co-movements of consumption growth and excess equity returns. We find
that, while the coherence between these variables i i s g i i a t at high frequencies (above 0.7
s ninfcn
years-1) at lower frequencies th s coherence i high and s a i t c l y s g i i a t We confirm
,
i
s
ttsial infcn.
the coherence analysis using bandpass f l e i g analysis similar to that suggested by Baxter
itrn
and King (1994). We find no s a i t c l y s g i i a t correlation between the two s r e at high
ttsial infcn
eis
frequencies. However, at lower “
business-cycle” frequencies, we find a correlation of over 50%.
We then turn to more formal t s s of consumption-based asset pricing at longer horizons.
et
We look at standard time-separable power u i i y the Abel (1990) “Catching up with the Jone
tlt,
ses” preferences, and the Constantinides (1990) habit-formation preferences. Our investigation
uses three diagnostic t o s f r t we apply the Hansen and Jagannathan (1991) mean-variance
o l : is,
analysis, modified to take into account the unconditional correlation between the pricing kernel
and asset returns; second, we apply the standard x 2 ofthe moment r striction in ( ) The third
e
2.
diagnostic we use follows Hansen and Jagannathan (1997). They note that some preference
specifications may result in an extremely v l t l m t, but one which does not r a l “ i ” the
oaie
e l y ft
data better. However, because of the increased v l t l t , such a preference specification may
oaiiy
s i lyield a low x 2- Hansen and Jagannathan (1997) and Hansen, Heaton, and Luttmer (1995)
tl
suggest a specification t s which i immune to t i problem. We employ t i t s as a further
et
s
hs
hs et
diagnostic t o .
ol
According to our empirical r s l s none of the consumption-based models performs well at
eut,
the quarterly horizon. However, when the horizon i lengthened to two years, versions of the
s
Abel (1990) and Constantinides (1990) models ofpreferences perform f i l w l . In particular,
ary el
no signi i a t violations of the Hansen-Jagannathan r s r c i n are found, the x 2 s a i t c are
fcn
etitos
ttsis
insignificantly d f e e t from z r , and the Hansen and Jagannathan (1997) diagnostic reveals
ifrn
eo
rather small discrepancies from equation ( .
2)
Other papers in the l t rature have asked whether the r s r
ie
e t ictions of economic theory are
3
better s t s i d with longer-horizon returns than with monthly or quarterly returns. Brainard,
aife
Shapiro, and Shoven (1991) find that stock returns correspond more closely to the theoretical
return to invested capital at longer horizons. Cochrane and Hansen (1992), however, find that
the performance of time-separable preferences deteriorates as the horizon lengthens. F
inally,
Daniel and Marshall (1997) use a vector A R C H model to estimate the conditional covariance
between equity returns and the pricing kernel implied by various models. All models do
poorly at the quarterly horizon. At the two-year horizon, however, they find that the mean
and variance of this conditional covariance s r e approximate the corresponding moments of
eis
the conditional equity premium as long as s f i i n time-nonseparability i incorporated into
ufcet
s
preferences. Furthermore, the variation through time in th s conditional equity premium i to
i
s
some extent reflected in variation in the conditional covariance s r e .
eis
The conditional evidence i important. Although we find evidence ofa strong unconditional
s
relationship here, the high unconditional correlations between consumption growth and asset
returns at business cycle frequencies could be caused by co-movement of the expected growth
and return s r e as opposed to co-movements of the innovations. There i strong evidence
eis
s
of variation in expected returns across the business cycle.7 Thus a t s of the conditional
et
model might f i when the business cycle variables are used as instruments, even though the
al
unconditional t s does not r j c the model. The r sults in Daniel and Marshall (1997) suggest
et
eet
e
that this should not be the c s .
ae
The remainder of the paper i organized as f l o s In section 2 we present evidence that
s
olw:
,
although consumption-growth i virtually uncorrelated with asset returns at high frequencies,
s
i displays substantial covariation with asset returns at business-cycle frequencies. In section 3
t
we characterize the preference specifications to be explored in the remainder of the paper; we
discuss the model diagnostics to be used; and we evaluate particular parameterizations of these
models at various horizons. In that s c ion, we also discuss problems in modeling long horizon
et
returns that were pointed out by Cochrane and Hansen (1992). We argue that these problems
can be resolved, in p i c
r n iple, by the use of time-nonseparable preferences as advocated by Abel
7For exam ple, Fam a and French (1989) show th a t th e term spread and th e default spread forecast
future long horizon returns, and Chen (1991) show s th at th ese sam e variables forecast future econom ic
activity. If th ese sam e variables forecast future consum ption grow th, th is could result in an uncondi
tion al correlation, even if the conditional correlation is zero.
4
(1990) and Constantinides (1990). In section 4 we provide a more formal analysis of the Abel
,
and Constantinides models in which the model parameters are chosen to minimize the Hansen
and Jagannathan (1997) specification c i e i n Section 5 summarizes.
rtro.
2
L o w - F r e q u e n c y Correlation of Asset R e t u r n s a n d C o n s u m p
tion G r o w t h
In th s section, we ask whether the Cochrane and Hansen (1992) correlation puzzle i a phe
i
s
nomenon at a ll frequencies, or just at high frequencies. We use two techniques to explore t i
hs
question: multivariate spectral or coherence ana y i , and an analysis of the correlations of
lss
bandpass f l e e consumption growth and a
itrd
sset returns.
2.1
Coherence Analysis
To assess the frequency-by-frequency breakdown ofthe correlation between consumption growth
and a market proxy, we perform coherence analysis of the consumption growth and excess V W
index return s r e . Ess n i l y what the coherence analysis does i to s l t each of the two
eis
etal,
s
pi
s r e up into a s t of Fourier components at d f e e t frequencies, and then to determine
eis
e
ifrn
the correlation of a s t of Fourier components f r the two s r e around each frequency. The
e
o
eis
method we use f r generating the coherence s r e i described in the appendix. In addition,
o
eis s
the method yields the phase relation between the two s r e , which i a measure of how f r the
eis
s
a
s r e must be shifted to maximize the correlation of the s t of Fourier components.
eis
es
In Figure 1, the upper panel gives the coherence between the two s r e , as a function of
eis
the frequency. Approximate 95% confidence i t r a s f r both coherence and phase are plotted
nevl o
as dashed l n s in the figure.8 We also plot the 95% confidence interval that the coherence i
ie
s
above z r , which i the dot-dashed l n i the f g r .
eo
s
ie n
iue
Figure 1 shows that the coherence i low a high frequencies. However, f r lower frequencies,
s
t
o
but not the very lowest, the coherences are r l t v l l r e and are considerably more than
eaiey ag,
two standard errors away from z r . This suggests that any correlation that a i e between
eo
rss
8T hese are calculated follow ing Bloom field (1976). For the coherence series, th e standard error is
inversely proportional to the level of the coherence, and in calculating these confidence intervals we
have assum ed th a t th e true coherence is equal to the estim ator.
5
Coherence — Consumption Growth and excess V W returns —
Phase —
Consumption Growth and excess V W returns —
19 47-1997
1947— 1 997
T his figure show s, in th e upper panel, the coherence betw een quarterly real non-durable and services
consum ption grow th and quarterly real returns on th e CRSP VW index over th e period 1947:1-1994:1.
T he low er panel show s th e phase relationship, in degrees, betw een th e tw o series a t th e sta ted frequency.
A ham m ing w indow and a sm oothing param eter ( n / m ) o f 16 is used. D etails o f th e coherence and phase
calculations are given in the A ppendix.
Figure 1 Coherence and Phase plots for consumption Growth and excess V W index
:
Returns
the consumption growth and asset return s r e i due to the co-movement at these lower
eis s
frequencies.
The lower panel of Figure 1 displays the phase as a function of frequency. Notice that at a
frequency of z r , the phase i z r , and then decreases approximately linearly with frequency
eo
s eo
up to a frequency of about 1 year""1. This suggests a constant-length lead/lag relationship
between the two variables, since the approximate lead/lag length equals the phase multiplied
by
(1 / f r e q u e n c y
* 360). Performing t i calculation, Figure 1 t l s us that return s r e leads
hs
el
eis
the consumption growth s r e by approximately two quarters. To verify t i , in Figure 2 we
eis
hs
lag the V W return s r e by 2 quarters and rerun the coherence a a ysis. The coherence values
eis
nl
increase sli h l at most frequencies, except a frequency zero where i increases dramatically.
gty
t
t
The phase i now approximately zero f r frequencies l s than one year”1.
s
o
es
6
Coherence — Consumption Growth and excess V W returns —
Phase —
Consumption Growth and excess V W returns —
19 47— 1997
1947— 1 997
T he construction o f th is figure was identical to th at for Figure 1, except th a t the VW index return was
lagged by tw o quarters before calculating the coherence.
Figure 2 Coherence and Phase plots for Consumption Growth and Lagged excess
:
V W index Returns
In summary, the coherence analysis suggests while there i l t l relation between con
s ite
sumption growth and asset returns at high frequencies, there i a strong relationship at low
s
frequencies, but that the relationship i not contemporaneous: consumption growth lags the
s
market return by about two quarters. This might t e inwell with a f i t o s s o y I there i an
i
rcin tr: f
s
extra cost of adjusting consumption quickly rather than s o l , then th s i exactly the sort of
lwy
i s
relationship one might expect to observe. Alternatively, i agents’preferences exhibited habit
f
formation ( . . Constantinides (1990)) and there were a u i i y cost to rapid consumption
eg,
tlt
adjustment, consumption would respond slowly to change in asset p i e .
rcs
7
Figure 3 Bandpass Filter Frequency Response
:
T his figure show s th e gain o f the three filters used in the bandpass filtering analysis over the frequency
range from 0 to 1 years” 1. The gain from 1 to 2 years " 1 is not show n, but is approxim ately 1 for th e
high filter and 0 for th e other tw o.
1.4
M
Tr©ndMf l e
itr
-- "Cycle- f l e
itr
----High-Pass- fl e
itr
1.2
0.8
0.6 -
\ /
\/
\‘
/
/
'
/'
•/
.
04 .-
/*
.
0.2 -
/
\
/
\
/
*
/ ....
✓.■’
***
' •.
* \
0.4
O.S
0.6
0.7
Frequency (1/years)
.
0.1
2 .2
0.2
0.3
. .\
0.8
0.0
Bandpass Filtering Analysis
To verify the coherence r s l s we also conduct a time-series bandpass f l e i g analysis, in
eut,
itrn
which we break the consumption growth and asset returns s r e into d f e e t components
eis
ifrn
using a s of moving-average (MA) f l e s To do t i , we u i i e a bandpass f l e i g analysis
et
itr.
hs
tlz
itrn
similar to that suggested by Baxter and King (1994), and u i ized by Baxter (1994).
tl
The M A f l e swe u i i ehere are each designed to pick out a certain range offrequencies in
itr
tlz
the data. The f l e s are symmetric two-sided f l e s and consequently do not introduce phase
itr
itr,
distortions into the data. Though the design of the f l e s uses spectral analysis techniques,
itr
the f l e s themselves are simple M A f l e s The process of f l e i g a data s r e i conducted
itr
itr.
itrn
eis s
by simply convolving the data s r e with the s t of f l e weights, as with any M A f l e .
eis
e
itr
itr
The f l e design i done using the FIRLS ( i i e impulse response l a t squares) function
itr
s
fnt
es
in M
ATLAB?
This function takes as an input a desired piecewise linear spectral response
function, and then finds the length n (here 31 quarters) M A f l e which provides a spectral
itr
response which i as close as possible to the desired spectral shape ( n a weighted least-squares
s
i
9T his function is taken from M A T L A B s th e Signal P rocessing T oolbox.
8
sn
e se).
We constructed three f l e s To conform to the business cycle l t r t r , we c l these the
itr.
ieaue
al
tre n d
, c y c le and
h ig h
f l e s The
itr.
tren d
f l e i designed to eliminate a l swings in the data
itr s
l
shorter than 8 years ( . . with a frequency of higher than 0.125 years-1.. The
te,
)
cy cle
fle i
itr s
designed to pass a l components with wavelength between 1 5 and 8 years, and the h ig h f l e
l
.
itr
i designed to eliminate a l swings longer than 1 5 years. The spectral gain obtained f the
s
l
.
or
three f l e s are shown in Figure 3.10 While we would l k these f l e s to have a gain of 1
itr
ie
itr
over the bandpass region and zero elsewhere, t i i not achievable with a f n t length f l e .11
hs s
iie
itr
Instead, th s figure shows that the f l e frequency responses are about as good as i achievable,
i
itr
s
given the constraints.
The motivation f r eliminating high frequencies from the data has already been discussed.
o
The motivation f r eliminating the correlations at very low frequencies i that there may be
o
s
long-run changes in the structure of the economy that a f c the co-movement of consumption
fet
and returns on financial a s t . (Changes in the l g l and institutional framework of the
ses
ea
pension-fund industry i one example.) At a more basic l v l since other macroeconomic
s
ee,
variables move together at business cycle frequencies, and not elsewhere, th s may also be true
i
f r consumption.
o
The correlations between various measures of consumption growth and V W return s r e ,
eis
f l e e using each of the three f l e s in turn, are presented in Table 1 We consider s x
itrd
itr
.
i
alternative measures of consumption: consumer durables, consumer non-durables, consumer
s r i e , nondurables plus s r i e , consumer services excluding the implied service flow from
evcs
evcs
owner-occupied housing, and nondurables plus services excluding the service flow from owneroccupied housing. The procedure f r constructing each of these correlations was to f l e each
o
itr
of the two s r e , and then calculate the maximum correlation p max between the two s r e
eis
eis
over a lead-lag range of eight quarters, that i
s
10We also demean the set of M A c e f c e t output from the FIRLS procedure f r the cycle and high
ofiins
o
f l e weight s r e . Since these are f n t length f l e s i the sum of the M A c e f c e t i zero than
itr
eis
iie
itr, f
ofiins s
the f l e s w l eliminate any integrated (1(1)) component present in the data.
itr i l
11 Baxter and King (1994) discuss the reasons that an “ideal spectral shape” i not achievable in
s
practice.
9
Table 1 The Correlation of Bandpass Filtered Returns and Consumption Growth
:
Series
T he quarterly real C R SP VW index returns and the real non-durable and services consum ption grow th
series (1947:1-1997:4), were each filtered using one o f the three filters, as described in th e te x t. T he
m axim um correlation coefficient betw een th e tw o filtered series, pmax, is com puted as in equation (4 ).
T he value o f p max as w ell as the m axim izing value o f r are given in the Table for each filter. T he M onteC arlo determ ined p-values are the probability o f obtaining a number at least as large as th e sam ple value
assum ing th a t th e returns are i.i.d. norm al, and independent o f consum ption grow th. Six m easures o f
consum ption are used: consum er durables (”D ur.”); consum er non-durables (”N D ”); consum er services
(”Serv.”); nondurables plus services (”ND& S”); consum er services excluding th e im plied service flow
from ow ner-occupied housing (”SxH”); and nondurables plus services excluding th e im plied service flow
from ow ner-occupied housing (”ND&SxH”).
Dur.
Pmax
Tmax
ND
Pmax
Tmax
Serv.
Pmax
Tmax
ND&S
Pmax
Tmax
SxH
Pmax
Tmax
ND&SxH
Pmax
Tmax
/
Pmax
1
Fle
itr
trend
cycle
-0.1053 (0.718) 0.5343 (0.0 2
0)
0
2
-0.0521 (0.657) 0.5092 (0.003)
2
5
-0.0148 (0.619) 0.4733 (0.020)
8
2
-0.0884 (0.703) 0.5497 (0.0 2
0)
2
8
-0.1427 (0.747) 0.4248 (0.052)
2
8
-0.1504
0.5360 ( 002)
0.
8
2
\ / 1
high
0.2053 (0.192)
-3
0.2735 (0.019)
1
0.1449 (0.634)
4
0.1662 (0.443)
1
0.1344 (0.698)
4
0.1739 ( . 8
03)
1
\ m ra{T,T+r}
min{ r+T}
T-
— max
£
T (a(r)a(Ac)) (t - t )
t= m a x {l,l+ r}
^
8 < r < 8.
(4)
where r and Ac* are demeaned returns and consumption growth over period t respectively,
*
,
and a ( r )
a (A c)
are the corresponding sample standard deviations f r the s r e .
o
eis
We use Monte-Carlo analysis to determine the significance l v l ofthe correlations in Table
ees
1 both in order to account f r the f c that we axe reporting the maximum of 17 c e f c e t ,
o
at
ofiins
and to account f r the s r a correlation induced in the two s r e by the f l e i g operations.
o
eil
eis
itrn
The p -v a lu e s reported in Table 1 are therefore the probabilities ofobtaining a c e f c e t at l a t
ofiin
es
as large as the one found i the quarterly returns s r e were i.i.d . normal, and independent of
f
eis
consumption growth.
10
As predicted by the coherence a a y i , the maximum correlation between the two s r e
nlss
eis
a t r f l e i g using the cycle f l e i high. For example, the maximized correlation c e f c e t
fe itrn
itr s
ofiin
i 54.97% f r ND&S consumption, with a one-tailed p-value of 0.
s
o
2%. Also, as predicted by
the coherence analysis, the correlation i maximized when the returns s r e i lagged by two
s
eis s
quarters.
We find l t l correlation between the returns s r e and any of the high-pass f l e e con
ite
eis
itrd
sumption growth s r e . The maximized correlation c e f c e t i about 20% f r most measures
eis
ofiin s
o
of consumption, and i ins g i i a t y d f e e t from z r . An exception i the non-durable s
s infcnl ifrn
eo
s
e
r e , which i signi i a t at the 1.9% (one-tailed) value. To t s whether the correlation i
is
s
fcn
et
s
d f e e t at the high and business cycle frequencies, we conduct an additional Monte-Carlo
ifrn
simulation. Note that the correlation between the business-cycle f l e e s r e i approxi
itrd eis s
mately 0.5 f r a l measures of consumption. In our simulation e e c s , we impose the null
o l
xrie
hypothesis that the correlation between the high-pass f l e e s r e i also 0 in population.
itrd eis s
.5
We then compute the probability of obtaining a maximum observed correlation of 0.27 (the
highest correlation in the l s column of Table 1 . We find that th s probability i l s than
at
)
i
s es
0.001. We conclude that the correlations between high-pass f l e e s r e are si n f c n l l s
itrd eis
g i i a t y es
than the correlations between business-cycle f l e e s r e .
i trd eis
I t r s i g y the maximum correlation of the trend components of the two s r e i also
neetnl,
eis s
i s g i i a t at - 8% (one-tailed p-value of 70.3%) f r non-durable and s r i e . A Monteninfcn,
8.
o
evcs
Carlo simulation shows that the probability ofgetting a measured trend-correlation ofl s than
es
zero i 4.8%, when the true correlation between the two s r e equal to 0 at a l frequencies.
s
eis
.5
l
Thus, i appears that the correlation i indeed lower at trend frequencies than a business cycle
t
s
t
frequencies.
The normalized,
cycle
-itrd a
f l e e sset returns and consumption growth s r e s r plotted
e i s ie
in Figure 4.12 We have also lagged the V W index return s r e by two quarters. The plot
eis
shows the very strong relationship between the two s r e in t i frequency range from the 50’
eis
hs
s
through the early 80’. The l t 80’ are the only period in the plot where the relation i not
s
ae s
s
extremely strong, perhaps because of the 1987 crash.
12To normalize the s r e , we divide each s r e by i s sample standard deviation.
eis
eis
t
11
Figure 4 Filtered Asset Returns and Consumption Growth Series
:
This figure plots the business-cycle filtered real consumption growth (Total) and filtered, two-quarterlagged, VW returns series. The filtering method is described in the text. The period is 1947:1-1994:1,
but the first and last four years are truncated in the filtering process. Each series is normalized by
dividing by its standard deviation.
Business Cycle Components
3
Unconditional M o m e n t s Tests of the Asset Pricing Relation
Section 2 documents that consumption growth i more highly correlated with equity returns at
s
business cycle horizons than at short horizons, and indeed that there i in i n f c n correlation
s sgiiat
at frequencies of l s than 0.67 years-1. This suggests that there may be f i t o s which
es
rcin
“
de-link” consumption and asset price movements at high frequencies. I s , t s s of the
f o et
consumption based asset pricing model may be considerably more successful at pricing longerhorizon returns. For the remainder ofthe paper, we explore th s conjecture at various horizons.
i
3.1
Preference Models
3.1.1
Time-Separable Preferences
The most widely-studied (and widely-rejected) preference specification in the consumptionbased pricing literature i time-separable power u i i y In this s e i i a i n agents solve the
s
tlt.
pcfcto,
12
following maximization problem:
oo
max U
{* j J o
c+ ) i
1-7
(5)
= E t Y B j -^ ~
jjtS
Zo
A _T
1 '
subject to the usual budget constraint. The r-period IMRS i:
s
(
6)
An important treatment of long-horizon returns with time-separable preferences can be found
in Cochrane and Hansen (1992). They use the method of Hansen and Jagannathan (1991) to
look at the implications of time-separable u i i y at horizons ranging from one quarter to f v
tlt
ie
years. They find that the performance of time-separable u i i y actually
tlt
d e te rio ra te s as
the
horizon lengthens. This f i u e of time-separable preferences at long horizons i largely caused
alr
s
by a high implied r s - r e r t . Aggregate consumption i a (
ikfe ae
s stochastically) growing s r e . In
eis
the time-separable model, agents seek to transfer some of the high future consumption to the
present by borrowing. A counterfactually high r s - r e rate i needed to discourage t i bor
ikfe
s
hs
rowing. (Recall that net borrowing must equal zero in equilibrium.) However, in a stochastic
model this e f c i partially countered by the precautionary motive f r saving: agents might
fet s
o
wish to insure against the possi i i y of consumption downturns. As the horizon lengthens,
blt
th s precautionary demand becomes weaker, since the empirical probability of a consump
i
tion downturn i smaller f r the longer horizons. (For example, Cochrane and Hansen (1992)
s
o
note that there i no fiv - e r period in post-wax US data over which aggregate consumption
s
eya
d c i e . As a r s l , the equilibrium r s - r e rate implied by the time-separable model i
elns)
eut
ikfe
s
counterfactually high f r longer-horizon data.
o
What i needed, then, i a reason why agents would be willing to save at low i t r s r t s
s
s
neet ae,
even though they know that their future consumption i l k l to grow. One possible reason
s iey
i that agent’ within-period utility-of-consumption changes through time. Preference s e i i
s
s
pcf
cations with t i property include the Constantinides (1990) habit-formation preferences and
hs
Abel (1990) “
catching-up-with-the-Joneses” preferences. In these spe i i a i n , the marginal
cfctos
u i i y of a given l v l of consumption grows through time. Agents refrain from borrowing to
tlt
ee
increase current consumption because they know that they w l need the consumption more
il
in the future. (As Weil (1989) has pointed out, the same e f c could be induced by having a
fet
13
subsistence point that grows deterministically in time.) To put t i another way, the problem
hs
discussed by Cochrane and Hansen (1992) cam be resolved, in p i ciple, i agents are f a f l
rn
f
eru,
not of a decline in consumption, but of a decline in consumption
p o in t,
re la tiv e to so m e re fe re n ce
where the reference point i s l grows at the same rate that consumption grows. To
tef
resolve the r s - r e rate puzzle in t i way, we w l u i i e the Abel (1990) and Constantinides
ikfe
hs
il tlz
(1990) models in our empirical tests.13 In the following sect o s we formalize these preference
in,
specif c t o s
iain.
3.1.2 The Abel (1990) ” Catching-Up-With-the-Joneses” Preferences
Let ct denote the per-capita consumption at date t. The agent s
olves
ma x
U
=
E t y
{c*+>)r=o
()
7
]
1 -7
fr'o
subject to the usual budget constraint, where
ht =
f >Vi,
r, >
0, 0 <
S <
1.
(
8)
i= l
The interpretation i that agents compare their consumption to the consumption of their
s
neighbors (the "Joneses”) in the recent past. In the formal model, the neighbors’consumption
i represented by c , and agents behave as i they have a subsistence point equal to t] times a
s
*
f
weighted average of the per-capita consumption l v l over the past m periods. Notice that
ees
agents treat h t as exogenous: the marginal u i i y of a fixed l v l of consumption inherits the
tlt
ee
upward trend in ct, but agents cannot a t r the ht process by t
le
heir own actions. (Of course,
in equilibrium c t
= ct.
)
With Abel preferences,
(ct+r h t+ r)
mr _
t+T " ( t - h t r*
c
Let the value function
V ( W ,h )
^
(9)
be defined as the maximum value of the objective function
that can be attained given i i i l wealth W and an i i i l subsistence point h . We define the
nta
nta
c e f c e t of relative r s aversion (denoted R R A t ) by
ofiin
ik
rV W W { W ,h )
R R A t = -W V w ( W ,h )
•
(
10)
13 We n ote th a t we have also investigated preferences in which u tility is based on consum ption relative
to a d eterm inistic trend w ith sim ilar results for th e unconditional m om ent tests.
14
For Abel preferences,
RRAt
=
(1
1)
ct — ht
Notice that the c e f c e t of r l t v r s aversion i time-varying, and everywhere exceeds 7 .
ofiin
eaie ik
s
The r s - r e rate puzzles w l be partially resolved by the Abel s e i i a i n s n e empirically,
ikfe
il
pcfcto, ic,
declines in ( * —
c
3.1.3
h t)
are observed more frequently than declines in c.
*
Constantinides (1990) Habit-Formation Preferences
Constantinides (1990) models agents as maximizing an objective function of the same form as
( ) with the following alternative specification f r h t :
7
o
^ =
7 > 0 >0 < ^ < 1 ?
i=l
The difference between ( and ( 2 i that in ( 2 the stochastic subsistence point
8)
1) s
1)
function of the agent’ own consumption
s
ct,
(12)
ht
i a
s
rather than the per-capita consumption. The
marginal rate of substitution i now
s
m
r
_ o r M U t+ r
t+ T
~
P
‘M
(3
1)
U t
where the marginal u i i y of consumption M U t i defined by
tlt
s
M U t = ( c t ~ h tH
-
£ W
t
=1
* E t [ct+i ~
fti
i + )]-7
(4
1)
With habit-formation preferences, agents consider the e f c of their current consumption on
fet
future values of h t . The presence of conditional expectations in equation (
14) r f e t t i
elcs hs
f c . These conditional expectations must be computed when we construct rri[. We do t i as
at
hs
f l o s F r t define the variable D t by:
olw. is,
„ _,
D,
»(1 -
= 1-
S)
E2.1
W *
W---- '
s ^
n
(
15)
The variable D t behaves as a stationary stochastic process. Equations (1 ) and (15) imply
3
that, in the Constantinides model,
m,t+T
i c t+ r
h t+ T)
(c t — h t)
15
7 D t+ r D t+ r + m
7 E tD t+ m
(6
1)
Since D t is stationary, we can fit an autoregressive time-series model for this variable, and use
the fitted values as our estimate of E t D t + m . For most models, the likelihood ratio statistics
testing
n
lags against n —1 lags in the autoregression for
four lags. We estimate a fourth-order autoregression in
(for
Dt
D t,
n
between 1 and 5) favor
and project the fitted regression
m periods into the future.
We consider Constantinides preferences separately from Abel preferences for two reasons.
First, it is possible that the behavior of
in (13) may differ substantially from that implied
by (9). A second, and more important reason, is that habit formation preferences do not
accentuate risk aversion in the way that Abel’s preferences do:14 the coefficient of relative
risk aversion implied by habit-formation for a given specification of {7 , r?, <, /?} is smaller than
5
that given in (11). From the perspective of the individual agent, Abel preferences are timeseparable, since a change in an individual’s ct does not affect his marginal utility with respect
to
c t+ i
for
i
/ 0. For such preferences, the coefficient of relative risk aversion equals the
curvature of the per-period utility function U , as measured by
.
In contrast, habit formation preferences incorporate true time nonseparability: The marginal
utility at date
t
+
t
is affected by changes in the state-contingent consumption plan for dates
i = l, ...,m. In response to a wealth shock at date
i,
t,
the agent with habit-formation
preferences adjusts her state-contingent plans for future consumption so as to optimally adjust
h t+ i,
i = l , ..., m . This attenuates the impact of a given wealth shock on the objective function,
as compared to the Abel specification. In particular, higher 77 does not increase R R A t as much
as in (1 1 ). While
RRAt
cannot be computed analytically for our model of habit-formation,
Constantinides (1990) and Person and Constantinides (1991) obtain closed-form solutions for
RRAt
in the context of a simpler model. They show that, for preference parameters similar to
ours, the mean coefficient of relative risk aversion is not too far above
7 . 15
14This point is extensively discussed by Constantinides (1990), Person and Constantinides (1991),
and Boldrin, Christiano, and Fisher (1995).
15Boldrin, Christiano, and Fisher (1995) report a similar result.
16
3 .2
Tests of Long-Horizon Pricing Using Unconditional M o ments
In Section 3.3 we redo the long-horizon Hansen-Jagannathan analysis of Cochrane and Hansen
(1992) using Abel (1990) and Constantinides (1990) preferences, and find that there i indeed
s
considerable support f r consumption based asset pricing models at long horizons, and very lt
o
i
t esupport at short horizons. In th s section we describe the long-horizon Hansen-Jagannathan
l
i
t s s and t s - t t s i s we u i i e
et
etsaitc
tlz.
3 .2.1
The Hansen-Jagannathan Analysis
Cochrane and Hansen (1992) derive the Hansen and Jagannathan (1991) mean-variance bounds
using a linear projection of a candidate stochastic discount factor on the space of portfolio
p y f s In th s s c i n we describe an alternative derivation that i helpful in pointing out the
aof.
i eto,
s
importance of the unconditional correlation between the pricing kernel and the mean-variance
e f c e t p r f l o In vector form, Equation ( i:
fiin o t o i .
1) s
E [ m T R*T]=
t
1.
(7
1)
where R [ denotes the vector consisting of the gross-returns between period t and period t +
r
on a s t of n a s t , and 1 i an n-vector of ones. Equation (1 ) holds f r a l r s , in the
e
ses
s
7
o l , o
remainder of th s s c i n we suppress the superscript r Equation (17) implies tha :
i eto,
.
t
co v (m ,
R) =
B[mR] — E[m]E[R]
1 - E[m]E[R]
=
(8
1)
Rearranging, we obtain f r any return r
o
:
E [r)
-
E lm } - 1 =
c o v (m , _
r
E [m ]
~ ~ P m ’r
/a mo r \
V^h J ‘
(9
1)
where oy and crm are the standard deviations ofr and r , resp c i e y and p m r i the correlation
a
etvl,
s
c e f c e t between r and m. Since the correlation c e f c e t — pm r cannot be l s than — 1,
ofiin
ofiin
es
we see tha :
t
or
)
17
~ E [m ]
(
20)
where the ”< ” must hold f r every asset and every f a i l p r f l o Let us define the l f o
esbe otoi.
et
hand side ofequation (2 ) as the asset’ S h a rp e
0
s
ra t io .16
Hansen and Jagannathan (1991) derive
the strongest bound implied by ( : in e f c , they evaluate the left-hand side of ( at the
20)
fet
20)
portfolio return with the highest Sharpe r t o which i simply the unconditional mean-variance
ai,
s
e f c e t p r f l o This portfolio has a Sharpe ratio of
fiin o t o i .
s r
*
= v/(^[R] - 1 .E M - i y r r ^ R ] - 1 •%]->) <
where f i the covariance matrix ofreturns. For any given value ofE [ m \ , this gives a bound on
l s
am
which i dependent only on the asset return f r t and second moments, and i independent
s
is
s
of preferences.
Equation (19) also t l sus something about the importance ofthe correlation between asset
el
returns and the pricing ke n l This equation can be rewritten:
re.
&m
E [m ]
_
~
SRj
pm,Rj'
where S R j denotes the Sharpe ratioofassetj . This equation says t
hat, f rany asset or portfolio
o
of assets with
SRj >
0 the lower bound on em —► oo as p f 0-: i the pricing operator m
,
r
f
has an arbitrarily small negative correlation with some asset with a positive Sharpe r t o this
ai,
implies an
a r b it r a r ily
t
ight bound on cm . This c early t e in with our results in Section 2:
r
l
is
the positive correlation between equity returns and consumption growth suggests a negative
correlation between equity returns and the marginal rate of substitution in consumption that
serves as the consumption-based proxy f r m. I the correlation between short-horizon equity
o
f
returns and short-horizon consumption growth i positive but small, the implied bound on <jm
s
i l k l to be much tighter than the standard Hansen-Jagannathan bound.
s iey
3.2.2
Projected Hansen-Jagannathan Bounds Tests
Following Cochrane and Hansen (1992), we note that i m s t s i s ( 7 , then so w l the
f
aife 1 )
il
projection (denoted m*) of m onto the s t of asset returns and a constant:
e
m* = a +
R' •b,
(21)
16The risk-free rate equals Et[mt+i]-1, so our definition of the Sharpe ratio differs from the usual
definition by a "Jensen’s inequality” correction.
If the risk-free rate were constant, the two definitions would coincide.
18
where
m
= a + R/ •b + e
,
and i?[eR] = 0 . If the correlation of m with the elements of R is very small, then the variance
of m* will be very small relative to that of m . As in equation (20), this means that a higher
value of
crm
will be required to satisfy the HJ bounds when implemented using m*. This
test therefore provides a tighter restriction than the standard HJ test. We present HansenJagannathan tests both for
m
and
m *.
We also use the procedure described in section 3.1
of Hansen, Heaton, and Luttmer (1995) to test formally whether the Hansen-Jagannathan
mean-variance restrictions are satisfied.
3.2.3
A x2 test of the moment restriction B[mR] = 1
Because we are not estimating model parameters in our test, it is straightforward to directly
test the moment restriction £[m R —1] = 0. for the n assets being considered. If we define
gt
=
(mtRf —1), the sample moment estimator is g r = (1 /T) Y j T g - We can construct a test
t
statistic
J t = T g tp S o 1 gr>
(2 2 )
where So is a consistent estimator of the spectral density matrix of g* at frequency zero, and
where this spectral density matrix is estimated using the Newey-West procedure. 17 As shown
in Hansen (1982),
Jt
is asymptotically distributed x 2 with n degrees of freedom. In our case,
we use two distinct asset returns, so n = 2 .
For the time-separable model, Cochrane and Hansen show that this x 2 test (for these two
assets) strongly rejects the time-separable power utility model for 7 ’s of less than 50.18
3.2.4
The Hansen-Jagannathan (1997) Specification Test
As pointed out in Hansen and Jagannathan (1997), x 2 tests of the moment restriction (17),
like that discussed in Section 3.2.3, can be misleading in comparing specifications. If a model
17The number of lags used equals 8 , 11, 15, and 19 lags for the quarterly, yearly, two-year, and
three-year horizons, respectively.
18See Cochrane and Hansen (1992), Table 2, p. 129.
19
produces a highly v l t l m, the eigenvalues ofthe spectral-density matrix may be huge, so the
oaie
weighting matrix Sq 1 in equation (22) may be close to singular. This would imply a “
small”
X2 s a i t c even when the pricing errors are large.19
ttsi,
To help to assess whether observed low x2 t s s a i t c r s from a superior f t of a
e t t t s i s e ult
i
given model, we u i i e a distance measure derived by Hansen and Jagannathan (1997).2 We
tlz
0
denote th s measure the ”HJ97 s a i t c This t s constructs an estimator the mean squared
i
t t s i ”.
et
distance between the candidate stochastic discount factor
and the set ( M
)
of valid pricing
kernels (that i, the s t of y’ that s t s y E ( y R — 1) = 0 . Formally, this distance measure,
s
e
s
aif
)
<2,i defined by
5 s
52
= min E ( y t — m t)2
(23)
y& M
As stated, t i minimization i over a s t of random variables M
hs
s
e
.
Equation ( in Hansen,
8)
Heaton, and Luttmer (1995) show that 8 can be computed more simply as the solution to the
conjugate maximization problem. We follow t
heirprocedure in computing
(the finite-sample
analogue to S2) f the various models of m t we study.21
or
HJ97 s a i t c axe not directly comparable across d f e e t investment horizons without
ttsis
ifrn
some normalization; here we derive a modified HJ97 s a i t c with appropriate normalization
ttsi
to allow such a comparison. The r period long horizon misspecification measure ( n our
i
notation) i:
s
%
= min
E(yJ
yr 6 M r
where /iT =
E (y [
— ) and <2 =
r
va r(yl
- m [)2 = p2 + a2
—
mj"), where y is the
a rg m in
(
24)
of (24). If we assume that
the “best-fit” pricing kernel y T will not change with horizon, then the r period pricing kernels
19For example, the x 2 test discussed in 3.2.3 (and presented by Cochrane and Hauisen (1992)), strongly
rejects for small 7 ’s ( J t = 5.1 for 7 = 40 for our sample), but fails to reject the time-separable model
for a 7 ’s over 200 ( J t is equal to 0 .8 8 for 7 = 240). However, the HJ97 statistic 6 2 (defined in equation
(23)) is equal to 0.257 for 7 = 0 and 0.416 for 7 = 240. In other words, despite the low x 2 statistic, the
m for the 7 = 240 model is fu rth e r from M than the naive (and obviously sub-optimal) specification
m t = 1 , for all t.
20See also Hansen, Heaton, and Luttmer (1995) (HHL), Section 1 .2 ).
21 Hansen, Heaton, and Luttmer (1995) show that if m t G M (that is, if S = 0 ), then the limiting
distribution of y / T d r is degenerate. For this reason, we do not use the HJ97 statistic to test whether
mt is a valid pricing kernel.
20
in (24) are simply products of the single period pricing kernels:
IIi=1mJ+i_1. Since, for small r, both
yl —
II[_1yt1
+i_1 and
=
and m should be close to one, and since:
y
N
« 1+
n
j)Li(i +
^ 2 5i
i= 1
we have that:
r
til - m l
= nf=1tit+i-i - n U
i m l + i - i « J 2 (tit+ i-1
- W +i-i)
ti
=
Thus, under this approximation,
« T/Xi
,
Hr
and
a2 —
where the variance ratio
( y T — m T)
V R (t )
t
• a f • V R ( t ),
is defined in the usual way as the ratio of the variance of
to the variance of (y1 —m1), divided by r:
V R ( r ) , ^
.
It follows that the HJ97 statistic can be written
— n l + a*
We now define the
^ r 2 • Ai + r • a i
*
as
m o d ifie d H J 9 7 s t a t is t i c
(5 1 )2 = £
T z
(25)
’ V R ( t ).
+ £ =
h
(26)
+ <2 -V R (t )
t
T
For r = 1, (££)2 is equal to the standard HJ97 statistic in (24). To implement (26), we compute
y
as the
a r g m in
of the sample version of equation (24) ,22, we calculate the sample flT and dT,
and form the square-root of the sample version of our modified statistic,
<vT.
In the empirical
section, we use the modified HJ97 statistic S T to assess the fit of the model at different horizons.
The modified HJ97 statistic tells us exactly what we want to know about how the char
acteristics of
(y
—m ) change as the test horizon lengthens: if
(y t
—m$) is characterized by
22HHL show that the y € M which minimizes the sample version of equation (24) is y t = m t — R 'a,
where a is the vector of Lagrange multipliers of the original minimization problem, and where the
F.O.C. defining a is given in their equation (10). Therefore, (yt —m t) = a'R. /zr and a T are the sample
mean and standard deviation of this series.
21
rapid, negatively autocorrelated movements which are “
washed-out” at longer horizons, then
the variance-ratio in (25) w l be small and d\. w l be small f r long horizons. This would t l
il
il
o
el
us that eliminating high frequency components improves the ft of the model. I, on the other
i
f
hand,
(yt
— m*) i characterized by positively correlated, long-swings, then the variance-ratio
s
in ( 5 , and d . will be large f r long horizons, and the modified HJ97 s a i t c will indicate a
2)
j
o
ttsi
poorer ft to the data at longer-horizons.
i
3.3
Empirical Results
We now present the r
esults from our empirical analysis of the models presented in Section
3 1 using the t
.,
ests presented in Section 3 2 We use quarterly data from 1952-1997. Detailed
..
description of the data can be found in the Data Appendix. We follow the standard practice
in the literature in using consumer expenditures on nondurables and services(”N D & S ”) as our
primary measure of consumption. We also perform the analysis f r consumer expenditures on
o
nondurables ( D ”) F n lly, The data compiled by the Bureau of Economic Analysis data
”N
. ia
f r consumer expenditures on services includes a s r e measuring the imputed rental value
o
eis
of owner-occupied housing. Rather than being compiled from observed surveys on consumer
expenditure, th ss r e i a construct. I has rather d f e e t properties from other consumption
i eis s
t
ifrn
s r e . In particular, Marshall and Parekh (1998) note that the covariance between the growth
eis
rate of this s r e and equity returns i negative. To allow f r the possibility that this s r e i
eis
s
o
eis s
substantially mis-measured, our third measure of consumption i N D&S minus this imputed
s
rental value of owner-occupied housing.
3.3.1
Time-Separable Preferences
We examine time-separable preferences f r four horizons: one quarter, one year, two years, and
o
three years. We s t (3 equal to 1 0 and we vary the curvature parameter 7 . The behavior of
e
.,
this model f r N D & S consumption i displayed in Figure 5.23 The s l d “
o
s
o i U-shaped” curves in
Figure 5 plot the Hansen-Jagannathan bounds, calculated using two returns: the real CRSP
23We also conducted this analysis using non-durable consumption and using N D & S minus the service
flow from owner-occupied housing. The results are very similar to those fore ND&S, and are not
displayed.
22
F ig u re 5: H a n s e n a n d J a g a n n a t h a n b o u n d s t e s t s fo r T im e - S e p a r a b le P r e f e r e n c e s ,
N o n - D u r a b le a n d S e r v ic e s C o n s u m p t io n
The four solid curves in both panels are the H ansen-Jagannathan bounds on the quarterly standard
deviation of th e pricing kernel, inferred from the asset return data at the four horizons: one quarter,
one year, tw o years, and three years. In the top panel, th e dotted lines p lot th e m eans/ standard
d eviations for th e tim e-separable m odel m arginal rate o f substitution for th e follow ing four horizons:
one quarter ( “+ ”), one year ( “*” ), two years ( “circle” ), and three years ( “x ”). W e set /? = 1 and we
le t th e value o f 7 range from 0 to 200; for each line th e spacing betw een the sym bols is 7 = 5. In
the bottom panel, th e dashed lines are generated by running a regression o f th e candidate discount
factors on th e set o f real returns (in this case th e VW index and th e T -B ill rate). The m eans and
standard d eviations o f th e fitted regressions are plotted for 7 = 0 , ..., 200. C onsum ption is m easured
as expenditures on consum er nondurables plus services; th e tim e period is 1947:1-1997:4.
Normalized Mean (Quarterly)
23
VW index return and the real 3-month T-Bill return. To facilitate comparison across different
investment horizons, we normalize the plots to put both mean and variance of the pricing
kernel in quarterly terms. To do this, we divide the resulting lower bound on a m by y /r , where
r is the horizon length in quarters (1, 4,
8
or 12 in this plot). The mean is normalized by
plotting .E[m]1/T. In the upper panel of Figure 5 we plot the mean-standard deviation loci
of m t implied by time-separable preferences with ND&S consumption, for the four horizons
and for values of 7 ranging from 0 to 200. In the lower panel, we project the time-separable
model’s pricing kernel onto the set of asset returns as described in Section 3.2.2, and plot the
mean and standard deviation of m* (defined in equation (21) against the same H-J bounds.
Figure 5 confirms the failure of the time-separable model that has been noted by many
previous studies. At every horizons from one quarter through three years, the (mean, stan
dard deviation) loci of the pricing kernel implied by this model are substantially outside of the
Hansen-Jagannathan bound. As in Cochrane and Hansen (1992), the longer horizons are fur
ther from the bound than the quarterly horizon. We test formally these Hansen-Jagannathan
mean-variance restrictions using the procedure described in Hansen, Heaton, and Luttmer
(1995).242 The failure of the time-separable model is generally confirmed. In particular, the
6
5
Hansen-Jagannathan restrictions for the projected kernel are rejected at the 1% marginal sig
nificance level for all horizons except when 7 is near zero25 or, for the quarterly horizon, for
values of 7 exceeding 20 0.26
One should not, however, take these results as indicating that time-separable preferences
can fit the data with 7 ’s near zero or (for the quarterly horizon) near 200. When
x2
statistics
axe calculated using equation (2 2 ). The statistics reject the model for all values of 7 . In
particular, the smallest values of these x 2 statistics are 24.8, 16.8, 14.9, and 12.32 for horizons
of one quarter, one year, two years, and three years, respectively. All of these values exceed
10.60, which is the 0.5% critical value for a
x2
statistic with two degrees of freedom. The
modified HJ97 statistics, given in equation (26), tell the same story. These statistics range
between 0.273 and 0.437, implying that the root-mean-squared deviation of the model’s pricing
24See Hansen, Heaton, and Luttmer (1995), section 3.1.
25When 7 = 0 (risk-neutrality), this test has p-values of 0.073, 0.048, 0.074, and 0.056 for horizons
of one quarter, one year, two years, and three years, respectively.
26The p-value for the quarterly horizon when 7 = 200 is 0.087.
24
kernel from the set of valid kernels is between 2730 bp and 4370 bp per quarter.
3.3.2
Time-Nonseparable Preferences: A n Overview of the Empirical Results
In this section, we conduct an analysis of the Abel and Constantinides models analogous to
that done in section 3.3.1 for time-separable preferences. That is, we look at the models’
performance as curvature increases. While the model with time-separable preferences has only
two parameters,
(3
(the subjective discount rate) and
and Constantinides models have five parameters:
depreciates),
77
7
(3, 7
(the curvature parameter), the Abel
, S (the rate at which the habit stock
(the ratio of mean consumption to mean habit stock) and
m
(the number of
lagged consumptions that enter the habit stock). In this subsection, we proceed somewhat
informally: We simply fix
{f3 ,8 , 77, m}
at values where the models perform fairly well27 and
we vary 7 , as in figure 5. In section 4, below, we consider the models’ performance when the
parameters are chosen optimally.
Note that equation (14) for the marginal utility of consumption in the Constantinides model
only makes sense if this marginal utility is positive. If the second term in equation (14) exceeds
the first term, the expected effect of additional consumption on the future habit stocks would
outweigh the effect on current period utility. In that case, it would be optimal for the consumer
to reduce consumption via free disposal. In addition, the right-hand side of equations (9) or
(14) could be negative or complex if (cj —ht) is negative. We discard any parameterizations
of these models where either of these anomalies occurred.
3.3.3
The Abel Model
For the Abel model, we set
77
= 0.8;
8
= 0.9;
(3
= 1 ; and
m
= 20. Figure
6
is analogous to
Figure 5. The upper panel plots the mean-standard deviation loci of m t implied by Abel with
ND&S consumption, for the four horizons, and for various values of the curvature parameter
7
. The lower panel is the analogous plot for the projection of Abel model pricing kernel onto
the set of asset returns.
The upper panel shows that the model pricing kernels with ND&S consumption satisfy the
27We conduct a preliminary search in which we set (3 — 1.0 and we vary 77 and 8 from 0.5 through
0.9, we vary 7 from 0 through 30, and we vary m from 2 through 20.
25
Figure 6:
Hansen and Jagannathan bounds tests for Abel Preferences, for
Non-Durable and Services Consumption
The four solid curves in both panels are th e H ansen-Jagannathan bounds on th e quarterly standard
deviation o f th e pricing kernel, inferred from th e asset return data a t the four horizons: one quarter,
one year, tw o years, and three years. In th e top panel, th e d otted lines p lot th e m eans/standard
deviations for the A bel m odel m arginal rate o f su b stitu tion , as given in equation (9, for th e follow ing
four horizons: one quarter ( “+ ”), one year ( “*”), tw o years ( “circle”), and three years ( “x ”). T he
m odel param eters are rj = 0.8, 5 = 0 .9 , m = 20, fi = 1 . T he value o f 7 ranges from 0 to 30; for each
line the spacing betw een the sym bols is 7 = 2. In the bottom panel, the dashed lines are generated by
running a regression o f th e candidate discount factors on th e set o f real returns (in th is case th e V W
index and the T -B ill rate). The m eans and standard d eviations o f th e fitted regressions are p lotted for
7 = 0, ..., 30. C onsum ption is m easured as expenditures on consum er nondurables plus services; th e
tim e period is 1947:1-1997:4.
26
HJ bounds at the yearly (“*”) two-year (“circle”), , and three-year (“x”) horizons when 7 =
10-14. The quarterly horizon (“+ ”) requires a somewhat higher value of
(over 20). The
7
lower panel, where the pricing kernels are projected onto the space of asset returns, give a
somewhat different story. While the two-year horizon comes closest to the HJ bounds, none
of the point estimates implied by the model actually satisfy these bounds. The reason is that
the correlation between consumption growth and returns is small. However, when the HansenJagannathan mean-variance restrictions are tested formally using the Hansen, Heaton, and
Luttmer (1995) procedure, the distance between the HJ bound and the (mean,variance) loci
of the pricing kernels is significantly different from zero only for the quarterly horizon (and
perhaps for the 3-year horizon) at low values of 7 . In particular, for 7 ’s in the range of 6-24,
p-values of above
7
10 %
are achieved for the three longer horizons. (For the quarterly horizon,
’s greater than 10 are required.) This means that, for moderately high values of 7 , none of
the models are significantly outside the HJ bounds according to this test.
The x 2 statistics, calculated using equation (22) give similar results. These statistics are
graphed in Figure 7. For each of the four return horizons, the dashed line displays the value
of the x 2 statistic as a function of 7 . The scale for this statistic is given at the left-hand side
of each graph. The horizontal dotted line indicates the value of 5.99, which is the 5% critical
value for a x 2 random variable with two degrees of freedom. This critical value is given only as
a point of reference. If the parameters were chosen before looking at the data, the x 2 (2) would
be the appropriate distribution under the null hypothesis that the model fits in population,
since equation (2) must hold for each of two assets. However, the parameters used in this
section were chosen after an informal grid search, so the x 2 (2 ) critical values should only be
used as an informal guide.
In figure 7, the value of the x 2 statistic drops rapidly for all horizons as
7
increases. For
high enough values of 7 , there seems to be little evidence against the model. Note in particular
the rapid monotonic decline for the quarterly horizon (panel A). Results like these lead some to
argue that models with time non-separabilities are consistent with the behavior of short-term
asset returns.28 An alternative interpretation, put forth by Hansen and Jagannathan (1997), is
28See,
for example, Ferson and Constantinides (1991) and Campbell and Cochrane (1994).
27
Figure 7: x 2 statistics and H ansen-Jagannathan (1997) statistics for Abel Prefer
ences.
The figure plots the x 2 s a i t c (dashed l n ) and the HJ97 s a i t c ( o i f n ) implied by the Abel
ttsi
ie
ttsi sld ie
model at the quarterly (panel A), one-year (panel B), two-year (panel C), and three-year (panel D)
horizon fo values of 7 ranging from 0 to 30 (horizontal a i ) The scale for the x 2 s a i t c i on the
r
xs.
ttsi s
left-hand side of the graph; the scale for the HJ97 s a i t c i on the right-hand side of the graph.
ttsi s
The horizontal dotted l n indicates the value of 5.99, which i the 5% c i i a value for a x 2 random
ie
s
rtcl
variable with two degrees of freedom. The model parameters are 77 = 0 8 S = 0 9 m = 20, / = 1
.,
.,
?
.
Consumption i measured as expenditures on consumer nondurables plus s r i e ; the time period i
s
evcs
s
1947:1-1997:4.
Panel A: Quarterly Horizon
Panel B One-Year Horizon
:
28
that the x 2 statistic lacks power. In figure 7, the solid lines plot the modified HJ97 statistics as
7
increases. The scale for these statistics, given at the right-hand side of each graph, is chosen
so that the HJ97 statistic and the x 2 statistic are at the same level for 7 = 0 . According to
these results, only the longer horizons with 7 around 16-20 perform particularly well. A useful
point of comparison for these statistics is the value when 7 = 0 , which implies a constant IMRS
of one. The HJ97 statistics for 7 = 0 are therefore the root-mean square (RMS) distance from
the set
M.
of valid pricing kernels to mj = 1. The HJ97 statistics for the quarterly test of the
Abel model show that the RMS distance from the model to
distance to m* = 1 for
any
M
is not much smaller than the
value of 7 . However, for the 1 and 2 year horizons, the statistics
are about a factor of six smaller than for the model with constant m*: this indicates that m
is indeed getting close to the set
M ..
The minimal value of the modified HJ97 statistic at the
two-year horizon (figure 7, panel C) is 0.037, occurring at
7
= 18. That is, the minimal RMS
error is on the order of 3.7%. The RMS error for the three-year horizon is approximately 4.5%,
when 7 equals 16.
We have conducted a similar analysis of the Abel model using the alternative consumption
measures.29 When consumption is measured as purchases of consumer nondurables, the HJ97
statistics tend to be less favorable to the model. Higher values of 7 are required to get a
reasonable fit at the two-year horizon, and the three-year horizon does poorly. In contrast, when
the service flow from owner-occupied housing is omitted from the standard ND&S measure of
consumption, the model tends to perform somewhat better. In particular, rather low values
of the HJ97 statistic are found for the two- and three-year horizon when
7
equals 14.30 This
is of interest because the service flow from owner-occupied housing is not an observed series,
but is a construct that may be fraught with measurement error.
3.3.4
Constantinides Habit Persistence
We conclude this section by performing an analogous study of a particular parameterization
of Constantinides preferences. We measure consumption as ND&S, and set
5
= 0.9,
77
= 0.8,
29Detailed tabulations of these results are available from the authors.
30With this measure of consumption, the HJ97 statistic when 7 = 14 is 0.038 for the two-year horizon
and 0.053 for the three-year horizon.
29
Figure 8 Hansen-Jagannathan bounds tests for Constantinides Preferences, for
:
Non-Durable and Services Consumption
The description for figure 6,applies here a s , except that the preferences here are Constantinides; the
lo
model parameters are 17 = 0 8 6 = 0 9 m = 2, / = 1 ;the value of 7 ranges from 0 to 18 and the
.,
.,
?
;
spacing between the crosses i 7 = 1 This figure i for Non-Durable and Services Consumption.
s
.
s
30
and m = 2. The Hansen-Jagannathan bounds plots for are presented in Figure
the Abel model displayed in Figure
6
8
. Unlike
, the point estimates for this parameterization of the
Constantinides model satisfy the HJ bounds only at the two-year horizon. When the pricing
kernels are projected onto the space of asset returns (lower panel), none of the point estimates
satisfy the HJ restrictions. As with the Abel model, the two-year horizon comes closest to
the HJ bounds. However, the Hansen-Jagannathan mean-variance restrictions not rejected
statistically by the Hansen, Heaton, and Luttmer (1995) test except at the three year horizon.
Figure 9 is the analogue for Constantinides preferences to figure 7. The x 2 test appears
less favorable to the quarterly horizon than with Abel preferences: In panel A of figure 9
the x 2 statistics exceed the 5.99 critical value for all values of 7 . The three-year horizon does
somewhat more poorly according to this criterion with Constantinides preferences, as compared
to Abel preferences. However, both the one- and two-year horizons do quite well for values
of
7
exceeding 4. Moreover, the modified HJ97 statistics (solid lines) indicate that the x 2
statistics are small because the distance between m and M is indeed shrinking. For example,
the RMS distance between the model’s IMRS and the set of valid pricing kernels is only 0.018
for the two-year horizon with 7 equal to 11. As with the Abel model, the Constantinides model
does less well when consumption is measured by nondurables only. When housing services are
omitted from the data, the results are similar to those for ND&S.
To summarize the evidence from this section, the graphical H-J analysis, the x 2 statistics,
and the HJ97 statistics suggest that the Abel model performs best at the two-year horizon and
the Constantinides model does fairly well at both the one- and two-year horizons.
4
Evaluation of T i m e N o n - S e p a r a b l e Preferences W h e n M o d e l
Parameters A r e Ch o s e n Optimally
In the previous section, we simulate the Abel and Constantinides models over a grid of pa
rameters. We found that, broadly speaking, these models appeared to perform poorly at the
quarterly horizon, fairly well at the two-year horizon, and with intermediate performance at
the one- and three-year horizons. In this section, we ask how well these models perform when
the parameters are chosen optimally. In particular, for each model we wish to see whether
31
Figure 9 x2 statistics and Hansen-Jagannathan (1997) statistics for Constantinides
:
Preferences.
The figure plots the x2 statistic (dashed line) and the HJ97 statistic (solid line) implied by the Con
stantinides model at the quarterly (panel A), one-year (panel B), two-year (panel C), and three-year
(panel D) horizon for values of 7 ranging from 0 to 30 (horizontal axis). The scale for the x 2 statistic
is on the left-hand side of the graph; the scale for the HJ97 statistic is on the right-hand side of the
graph. The horizontal dotted line indicates the value of 5.99, which is the 5% critical value for a x2
random variable with two degrees of freedom. The model parameters are r) — 0.8, S = 0.9, m = 2 ,
f = 1. Consumption is measured as expenditures on consumer nondurables plus services; the time
3
period is 1947:1-1997:4.
Panel A: Quarterly Horizon
Panel B: One-Year Horizon
Panel C: Two-Year Horizon
Panel D: Three-Year Horizon
32
there are parameter configurations that set the HJ97 statistic to zero (indicating that the
mean residuals from the asset pricing Euler equations equal zero in sample). If a zero value
of this criterion is achieved, we wish to see whether the parameter that achieve this value are
’’plausible” in some intuitive sense.31
In conducting this exercise, we note that the model is under-determined. We have only
two unconditional Euler equations (one for the 3-month T-bill return, and one for the equity
return). However, in each of the two models we have five parameters to choose:
We set the quarterly subjective discount factor
0
{/3 , 7 , 77, 8, m } .
= .99 (implying a yearly discount rate
of approximately 4%). We treat m, the number of lagged consumption in the habit-stock
formation, as an approximation to an infinite lag. We do so by letting m depend on
set
m
as the smallest number such that
Sm <
0.05. That is, we choose an
m
8:
we
sufficiently big
that the discarded lags (those greater than m) all have discount factors less than 5%. This
still leaves us with three parameters {7 , 77, <} and only two model restrictions. Since this is a
5
highly nonlinear model, one could find no parameter combinations that set the HJ97 statistic
to zero or one could find multiple parameter combinations that achieve this goal. To explore
this question we fix 8 at three values {0.6,0.7,0.8), and for each value we minimize the HJ97
statistic over parameters {7 , 77 }. When 8 = 0.8 (the maximum value we use), m = 14, implying
that the habit stock incorporates three-and-a-half years of lagged consumptions.
Before describing our general results, we illustrate the behavior of a particular model. In
figure 10 we consider how well the Constantinides model performs for two-year returns when
8
= 0.6. We consider values of 7 from zero to 40 (horizontal axis), and for each value of 7 we
find the value of 7 that minimizes the HJ97 statistic. Note that both 7 and 7 are curvature
7
parameters. One might think that there is a trade-off between these two parameters: a high
value of 7 acting similarly to a high value of 77. In the bottom panel of figure 10 we see that
this is indeed the case: The optimal choice of 7 declines as 7 increases. It does not then follow,
however, that 7 and 7 are not separately identified. As one can see in the top panel of figure
7
10, there is a unique {7 , 77} combination that minimizes the HJ97 criterion. This pattern holds
31
This exercise is analogous to that performed by Campbell (1998) for time-separable preferences.
Using a variety of data sets from different countries, he computes the risk aversion parameter needed
to match the mean equity premium. His test of the model is whether the requisite level of risk aversion
is plausible from an economic standpoint.
33
Figure 1 : Optimal Choice of Curvature Parameters for Constantinides Model, T w o
0
Year Horizon
For a grid of 7 *s ranging from zero to 40, the Hansen-Jagannathan (1997) statistic (”HJ97”) is min
imized by choice of parameter t j . The Constantinides model of preferences is used with consumption
measured by ND&S. Other model parameter are set as follows: /? = 0.99; 8 = 0.6, m = 6 . The top
panel of this figure plots the minimized HJ97 statistic for each value of 7 . The bottom panel gives the
minimizing choice of t) for each value of 7 . The time period is 1947:1-1997:4.
34
generally: In all model variants and all return horizons, for each < there is a uniquely-identified
5
optimal {7 , 77} combination.
Table 2 gives the best-performing configuration of {5,7 , 77} for each model and each hori
zon. (When multiple values of
5
achieve a zero value for the HJ97 statistic, we display the
parameterization with the lowest value of 7 .) The results broadly confirm the patterns dis
cussed in the previous section. First, for no model can we find a parameter combination that
performs acceptably at the quarterly horizon. In the Abel model, the minimum HJ97 statis
tic for quarterly returns is 0.214; for the Constantinides model, this minimum HJ97 value is
0.115. Varying the number of lags in the habit stock does not improve the performance of
these models, nor does using alternative measures of consumption. 32 We conclude that these
models are inconsistent with the unconditional moments of short-horizon asset returns.
Second, for each model and each measure of consumption there exist parameter combina
tions {7 , 77, <} that set the HJ97 statistic to zero for the two-year horizon. For the Constan
5
tinides model, this good fit to the data does not require an excessive value of the curvature
parameter 7 . In particular, when < = 0.8, the Constantinides model requires 7 of only 7.4.
5
The Abel model requires somewhat higher values of 7 to set the HJ97 statistic to zero. The
best fit for the Abel model at the two-year horizon occurs when
6
= 0.6. For this value of S,
the optimal 7 equals 17.9. In that sense, one can argue that the Abel model has more difficulty
than the Constantinides model in fitting the unconditional moments of the asset-return data.
Furthermore, in the Abel model the coefficient of relative risk aversion is time-varying, but
always exceeds 7 , often by a good deal. (See equation (11).) For example, the mean coefficient
of relative risk aversion implied by the parameters that achieve a zero HJ97 statistic for the
two-year horizon (5
=
0.6,7 = 17.9,7 = 0.85) is 114. Thus, the best-fitting parameterizations
/
of this model do imply extremely high (perhaps implausible) risk aversion. In contrast, it has
been noted by Constantinides (1990), Ferson and Constantinides (1991), and Boldrin, Christiano, and Fisher (1995) that habit formation in the Constantinides model does not accentuate
risk aversion to the extent that it does in the Abel model. While a closed-form expression for
32The minimal HJ97 statistics obtained for the quarterly horizon using nondurable consumption only
are 0.232 and 0.136 for the Abel and Constantinides models, respectively. When housing services are
omitted from ND&S, the corresponding statistics are 0.215 and 0.117.
35
the coefficient of relative risk aversion is not available for the Constantinides model, numerical
analyses of simpler versions of the model by these authors suggest that the mean coefficient of
relative risk aversion is only slightly higher than 7 .We conclude that the Constantinides model
can match the unconditional properties of these asset returns at the two-year horizon without
imposing extreme levels of risk aversion.
Third, in both models we can find parameter combinations {7 , 7 ,5} that set the HJ97 cri
terion to zero for the one-year horizon. This generally holds for other measures of consumption
as well. 33 However, the requisite value of 7 needed at this one-year horizon is always sub
stantially higher than that needed for the two-year horizon. Specifically, the best-performing
parameter configuration for the Abel model at the one-year horizon sets 7 equal to 72. As with
the two-year horizon, the Constantinides model requires a substantially lower value of 17.2.
Still, this value of 7 exceeds that required to fit the data at the two-year horizon. Fourth, the
Constantinides model does not set the HJ97 statistic to zero at the three-year horizon for any
value of 8. The Abel model can achieve a zero value for the HJ97 statistic at the three-year
horizon (when
8
equals 0.7 or 0.8), if a high value of 7 (in excess of 68) is assumed.
Finally, the results do seem sensitive to the number of lagged consumptions (”m”) used to
construct the habit stock. While the Constantinides model performs fairly well at the two-year
horizon when 6 = 0.6 or 0.8, it does rather poorly when 8 = 0.7. The problem is not with the
value of < p e r
5
m
but with the value of m = 9 implied by 8 = 0.7. When we fix 8
= 6, the model behaves much like the case in Table 2 with 8
(but
5
se,
8
=
=
0.7 but set
0.6. Similarly, when m = 10
remains fixed at 0.7), the model resembles the case in Table 2 with 8
=
0.8.
Conclusions
In this paper, we ask whether consumption-based pricing models work better at longer hori
zons than at the quarterly horizon. Our motivation is that if frictions, transactions costs,
or durability affect the comovements of consumption growth and asset returns, they should
primarily affect the higher frequency components.
3 3 An exception is the Abel model using nondurable consumption, where the minimal value of the
HJ97 statistic is 0.015 (when 8 is set to 0.8).
36
Table 2: Minim um H J97 statistics for A b el and Constantinides Models
For each model, < is fixed at 0.6,0.7, or 0.8, m is set to the minimum value such that 6m < 0.05, and
5
the HJ97 statistic is minimized over {7 , 7 These tables report the minimal value of the test statistic
?}.
obtained, for horizons of one quarter and 1, 2, and 3 years. Consumption is measured as ND&S. When
the minimized HJ97 statistic equals zero for more than one value of <, the case with the lower value of
5
7 is displayed.
Horizon
(years)
Abel Model
HJjntn
S
0.25
1
2
3
0.214
0.015
0.000
0.000
0.8
0.8
0.6
0.6
7
23.0
72.1
17.9
68.2
Constantinides Model
i)
m
0.79
0.59
0.85
0.79
14
14
6
6
HJmin
S
6.115
0.8
0.8
0.8
0.6
0 .0 0 0
0 .0 0 0
0.014
7
28.9
17.2
7.4
58.8
V
m
0.77
0.80
0.87
0.85
14
14
14
6
We first show that consumption-growth and equity returns are virtually uncorrelated at
high frequencies. However, we find that at lower frequencies, corresponding to swings longer
than one and one-half years, the two series are highly correlated. We then test three models
of the pricing kernel: time-separable power utility; the Abel (1990) “Catching up with the
Joneses” preferences; and the Constantinides (1990) habit-formation preferences.
We find
while all models perform poorly at quarterly horizons, the Abel (1990) and Constantinides
(1990) models perform well at longer-horizons. The Hansen and Jagannathan (1991) bounds,
modified to take account of correlation the correlation between the pricing kernel and asset
returns, are satisfied. A x 2 moment restriction test is not rejected, and the modified Hansen,
Heaton, and Luttmer (1995) specification test suggests that our inability to reject the model
is not due high volatility of the pricing kernel.
Our results raise numerous questions for further research. First, it suggests that correlation
puzzle may at least partially be due to frictions that disrupt the high-frequency co-movements
of marginal utility growth and returns. However, it also necessary to first understand why past
attempts at modeling frictions to explain the equity-premium puzzle have not been entirely
successful.
The Abel and Constantinides models motivate agents to save at relatively low interest
rates, even though consumption grows, because the marginal utility-value of a given level of
consumption also grows. There may be other ways to motivate saving, such as by looking
37
carefully at life-cycle models of saving and investment.
Finally, we consistently find that the models with time-nonseparabilities perform better at
the two-year horizon than at the three-year horizon. (The Constantinides model does rather
poorly at the three-year horizon.) This represents somewhat of a puzzle. If the only reason for
the poor performance of consumption-based models at short horizons is transient, short-term
frictions, then one would expect the models’ performance to improve monotonically as the
horizon lengthens. One possible reason is that consumers are particularly averse to risk at
business cycle frequencies. At the three year horizon, much of the business cycle behavior of
consumption growth and asset returns is filtered out. However, it is unclear why consumers
would be relatively insensitive to lower-frequency risk. Whether this reflects a degree of myopia
or a hitherto un-modeled aspect of preferences remains to be determined.
38
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40
Jo u rn a l of
Appendices
A
C o n s t r u c t i o n of t h e D a t a
The total quarterly real non-durable, durable, services, and total consumption series, the
deflators for each of the three components series, and the population series (GPOP) were
extracted from CITIBASE for the 1947:1-1997:4 period.
Monthly VW index returns were obtained from CRSP, and were cumulated to obtain quar
terly returns. One month T-Bill returns were taken from the CRSP RISKFREE file. One,
two, and three year nominally risk-free rates were computed as the returns to one-, two-, and
three-year zero-coupon bonds, computed from the Fama-Bliss data in CRSP.
B
Constr u c t i o n of the C o h e r e n c y a n d P h a s e Estimates a n
C o n f i d e n c e Intervals
To construct the coherency and phase estimates plotted in Figures 1 and 2, the two series x and
y (log consumption growth and returns) and are first each subdivided into N non-overlapping
subsamples of length n (here, n = 16 quarters). Each of these subsamples is then detrended
and windowed using a Hamming window. The detrended, windowed subsamples are then
fast-fourier transformed to generate J x (u>) and J y (w), which are equal to, for x,
n- 1
Jx{u )
= n- 1
^ 2 xte*™ 1.
t= o
where each of the subsamples is indexed from t = 0, ...,n —1 . Note that J x and J y will be
complex. The x and y power-spectral densities and the cross-spectral densities are then defined
as:
k=1
a ,M
= ^
where J^(w) denotes the Fourier transform of the fc’th subsample, and * denotes the complex
conjugate.
Although Pxx and p yy are real valued, in general the cross- spectral density will not be.
The coherency between the two series is defined as:
S .J , M
=
ia » i
[PXX(w)Pyy (u>)]1//2
41
and the phase is defined as:
< x,y(u )
j>
=
a rc ta n
( Irn (P xy(u ))
\R e{P xy(u > )) y
where I m (-) and i?e(-) denote the imaginary and real components, respectively. With these
definitions,
PXy(w ) = (Pxx((j)P yy(u >)) ^ SXty(l^)ei^x'v^ °\
Finally, confidence intervals for the coherency and phase were calculated using the method
described in Bloomfield (1976, Section 9.5). The upper and lower bounds of the 95% confidence
intervals are therefore:
ta n k ^a rc ta n h {P xy(u;)) ±
where g 2 — (2/3)(n/T), is a constant based on the Hamming window (see p. 224 of Bloomfield
(1976).) The 95% confidence intervals for the phase are:
42
@FedFRASER