Asset Pricing with Durable Goods and Non-Homothetic
Preferences
Michal Pakoš∗
First Draft: Sep 5th 2002
This Draft: April 25th, 2003
Abstract
The purpose of this paper is to understand the pattern of the demand for the durable goods
over time and how that relates to and helps explain asset pricing puzzles. The conclusion of the
previous literature is that the relative demand changed a lot in response to a pure substitution
effect, that the marginal rate of substitution is not volatile enough to pass Hansen-Jagannathan
bounds and that there is not enough cross-sectional variation in the covariances between the excess
returns on assets and the discount factor. I show how a correct modelling of income vs. substitution
effects changes all that. The substitutability between non-durable consumption and the services
flow from the stock of durables is very low, in stark contrast to results of the previous literature,
which magnifies risk premia. It was the falling price of durables that raised the real income,
which lead to a rise in the consumption of durables - there is an important non-homotheticity.
The resulting marginal rate of substitution passes Hansen-Jagannathan bounds and also helps
account for the cross-section of expected returns on 25 Fama-French portfolios. I interpret the
stochastic discount factor of Fama and French (1992,1993) as a projection of the marginal rate of
substitution of the C-CAPM with durables on the closed subspace of returns. I also show that the
market price of risk varies and that the co-integrating residual tracks the longer horizon returns
on 25 Fama-French portfolios with R2 s as high as 44%.
∗
Working Paper, Graduate School of Business, University of Chicago. This is a substantionally revised
version of my paper submitted at the end of spring quarter 2002 to satisfy finance requirements at GSB, U of
Chicago. I am grateful to my advisors John Cochrane and John Heaton for their valuable comments. I have
also benefitted from helpful comments of George Constantinides, Max Gillman, Lubos Pastor, Ruy Ribeiro,
Tano Santos and Brown Bag seminar participants at GSB, University of Chicago. Remaining errors are mine.
E-mail: [email protected]
1
1
Introduction
This paper proposes a novel mechanism to explain the quantitative asset pricing puzzles. It
is built around the blend of two ideas not considered in the asset pricing literature before,
namely, non-homotheticity and low substitutability between non-durable consumption and another good, or possibly a multiplicity of goods. It offers a very simple and intuitive alternative
to the various generalizations of the canonical C-CAPM, which ranges from relaxing the timeseparability of the preferences (Constantinides (1990), Heaton (1993, 1995), Campbell and
Cochrane (1999)), expected utility assumption (Kreps and Porteus (197x), Epstein and Zin
(19xx)), market incompleteness (Constantinides and Duffie (1996)), contract enforceability
(Alvarez and Jermann (2000), Lustig (2002)), prospect theory (Barberis, Huang and Santos
(2001)), among others. In this paper I develop this idea by considering the other good to be
the flow of services from the stock of durables.
Preferences that exhibit low substitutability between non-durables and services flow are Leontief preferences. By their nature they translate the small nondurable consumption risk into
large risk premia. Intuitively, in recessions the non-durable consumption and durables investment both fall, but the stock of durables rises. As a result, the nondurable consumption and
the flow of services depart from the optimal proportion which is very costly for the consumer.
An influential study by Lucas (1987) of the welfare cost of the variability of non-durable consumption estimated the cost to be very small. Consumers are not afraid of recessions because
their non-durable consumption falls and not surprisingly in the canonical consumption-based
CAPM they demand low risk premia on assets. It is the close-to-perfect complementarity
between non-durable consumption and the flow of services and the distortion in the optimal
consumption basket in recessions what consumers are afraid of. Therefore, stocks that pay off
badly in recessions have to offer higher equilibrium risk premia. This mechanism also generates
a time-varying market price of risk. I show that the cointegrating residual from the stochastic
cointegrating regression forecasts returns on Fama-French portfolios (especially small stocks).
Non-homotheticity explains why the ratio of flow of services over nondurables increases over
time. The old view has been that consumers optimally substituted to ever-cheaper durables
as their relative price was falling. The new view is that the substitution effect (in the sense
of Hicks) is very small. Consumers are buying more durables because the fall in the relative
price of durables increased their real income. This result is very natural as durables do not
have a substitute and thus the Hicksian price effects are small. Most studies1 have imposed
1
See Ait-Sahalia, Parker and Yogo (2003) for an exciting exception
1
homotheticity2 . Models with homothetic preferences neglect income effects and counterfactually ascribe all changes in the relative demand to a pure substitution effect.
Furthermore, because non-durables and services flow are held in nearly fixed proportions,
one may eliminate the services flow from the preferences. This is essentially what most researchers have been doing implicitly when they wrote down a preferences specification defined
over nondurables only. I conjecture that this near-observational equivalence (durables are still
in the budget constraint) explains how macroeconomics could have gone on for 20 years getting
the right quantities and still miss the asset prices due to the low volatility of the Lucas-Breeden
stochastic discount factor.
The ambition of this paper is also to try to unify the equity premium puzzle literature and
the cross-section of expected returns literature. Traditionally, models that claim to have explained the equity premium puzzle do not provide evidence as to how their discount factor
would price any subset of portfolios. The cross-section of expected returns literature cannot
generate quantitatively high risk premia in the time-series.
The Lucas-Breeden stochastic discount factor of the canonical C-CAPM is not volatile enough
to get inside Hansen-Jagannathan bounds. The cross-sectional variation in covariances between the discount factor and the returns on Fama-French portfolios is also tiny, evidenced
by the basically flat plot of the fitted vs. the realized average returns. I show that the CCAPM with durables is capable to explain the equity premium puzzle and the risk-free rate
puzzle with a low γ. The marginal rate of substitution is volatile enough to get inside the
Hansen-Jagannathan bounds. In addition, although the JT statistics rejects the model in
quarterly data, the plot of average vs. the actual returns is significantly better compared to
the canonical C-CAPM. Nearly all studies that claim to have explained the equity premium
puzzle do not impose upon themselves the additional restriction that their discount factor actually price some subset of portfolios in addition to the risk-free rate and the value-weighted
market return. The results in annual data are also interesting. The discount factor gets inside
HJ bounds. The plot of fitted vs. average returns is better than any in the literature. In fact,
R2 s of that plot is about 91%. However, it is still rejected statistically because it is estimated
very precisely. In contrast, the pricing errors of the canonical C-CAPM are very noisy, they
and their standard errors are an order of the magnitude larger than for CCAPM with durables.
I differ from the cross-section of expected returns literature in that my factor is the marginal
2
Some imposed subsistence level but it is not clear a priori how general non-homotheticity such an assumption
allows for.
2
rate of substitution - I have a one factor model capable to account for the cross-sectional variation in expected returns on Fama-French portfolios. I do not allow for arbitrary loadings on
the discount factor. The only free parameters are preference parameters. In contrast, papers
such as Fama and French (1992,1993), Lettau and Ludvigson (2001), Lustig and Van Nieuwerburgh(2002), Piazzesi, Scheider and Tuzel (2003), Santos and Veronesi (2001), and other, are
multifactor (often 3 or 4) models, where the factor is not the marginal rate of substitution.
I interpret Fama-French (1992, 1993) linear stochastic discount factor as a projection of the
marginal rate of substitution onto the closed subspace of returns. I argue that CCAPM with
durables offers a macroeconomic rationalization of the exciting comovement of stock returns
unveiled by Fama and French (1992, 1993).
Previous studies such as Eichenbaum and Hansen (1990), Ogaki and Reinhart (1998) already
introduced the services flow from durables. Their agenda however was not asset pricing.
Rather, they were interested in estimating the elasticity of substitution between non-durable
consumption and the services flow and worked with homothetic preferences (CES aggregator).
Pakoš (2003) argues that homotheticity biases the estimate of the elasticity of substitution up.
Dunn and Singleton (1986) assume that the preferences over nondurables and durables are
Cobb-Douglas and investigate the term-structure implications of the durability. However, in
addition to homotheticity they restrict the elasticity of substitution to be one. Another related
paper is Heaton(1995) who uses one good, called services which is a weighted average of past
nondurable consumption and is a high-tech treatment of durability (at short horizons many
goods are durable) vs. habit persistence.
Other related recent papers are Piazzesi, Schneider and Tuzel (PST) (2003) and Lustig and
Van Nieuwerburgh(2002) who instead of durables use housing. They impose homotheticity
but as I show later on, their demand equations are misspecified - income effects are crucial
for a correct modelling of housing. Recall from the Slutsky equation that the income effects
are proportional to the expenditure shares which is largest for housing, and moreover, housing
has no substitutes. This is reflected in a biased estimate of the elasticity of substitution (PST
estimate it to be 1.27). Furthermore, PST do not test the Euler equations. They linearize
their stochastic discount factor. It seems that they actually argue that elasticity of substitution
close to one, Cobb-Douglas preferences, is capable to explain quantitative asset pricing puzzles,
the result with which I disagree. Lustig and Van Nieuwerburgh set up a heterogenous-agent
economy, but do not solve it and instead postulate a linear discount factor. None of these two
papers tests neither the restrictions on the coefficients in their discount factor nor perform a
GRS test (or JT test) whether the pricing errors are jointly significantly different from zero.
3
2
Preferences and Asset Prices
2.1
Preferences
The preferences of the representative consumer are defined over nondurable consumption ct
and the imputed services flow st
(∞
)
X
t
U ({ct , st }) = E0
β u (Ω(ct , st ))
(1)
t=0
where u (Ω) is an iso-elastic felicity function defined over the consumption index
u (Ωt ) =
1
Ω(ct , st )1−γ
1−γ
(2)
The consumption index Ω(ct , st ) is a generalized -elasticity of substitution (GES) sub-utility
function (Pakoš (2003))
n
o θ
η
1
θ−1
2
, (θ, η, a) ∈ R+
× (0, 1)
Ω(ct , st ) = (a ct )1− θ + ((1 − a) st )1− θ
(3)
and displays non-homotheticity. The canonical case of iso-elastic felicity function defined over
CES sub-utility index is subsumed as a special case3 .
I assume that the consumer produces the flow of services by a time- and state-independent
linear household production function
s t = k dt
(4)
I normalize k = 1 and substitute the services flow st with the stock of durables4 dt . I then
write the consumption index as Ω(ct , dt ).
The elasticity of substitution is defined as a percentage change in the relative Hicksian de∂ log(c∗t /d∗t )
mand in response to a percentage change in the relative price, ES =
and the
∂ log qt
expenditure elasticity is defined as a percentage change in the Marshallian demand in response
log ct
∂ log dt
to a percentage change in expenditures, η1 = ∂∂ log
et and η2 = ∂ log et , where et is the withinperiod expenditure on the consumption goods. In a related paper, Pakoš (2003) interprets
the parameter θ as the elasticity of substitution5 and the ratio η as the ratio of expenditure
3
For example, Dunn and Singleton (1986) assume that the consumption index Ω(ct , st ) is Cobb-Douglas and
their implied θ = 1 and η = 1. Eichenbaum and Hansen (1990) and Ogaki and Reinhart (1998) relax the
restriction θ = 1 but still keep the homotheticity assumption η = 1. However, that eliminates the income
effect from the demand equation and biases the estimate of θ as we ascribe all changes in the relative demand
to the substitution effect.
4
I occasionally refer to dt as the flow of services instead of saying that the flow of services is a linear function
of dt .
5
This is not exactly correct. See the discussion that follows.
4
Figure 1: Indifference Curves and the Income Expansion Path: Prais-Houthakker (1955) model
100
90
Services Flow From Durables
80
70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
Nondurables Flow
7
8
9
10
NOTE - The graph is plotted for the parameter η = 0.5.
elasticities between nondurable consumption ct and durables dt .
The special case of θ → 0 delivers the non-homothetic case of Leontief sub-utility function,
so-called Prais-Houthakker model, first proposed by Prais-Houthakker (1955)
Ω(ct , st ) = min {a ct , (1 − a)η sηt }
(5)
with the income expansion path st = st (ct ) defined implicitly by
a ct = (1 − a)η sηt
(6)
I plot the indifference curves and the income expansion path in Figure 1. The preference
specification has the feature that both goods are normal and services flow is a luxury good
(i.e. income elasticity η2 is greater than one) and non-durable consumption is a necessity good
with income elasticity η1 less than one. Consumers cannot substitute from non-durables to
services flow (in the sense of Hicks) but as their real income rises they choose to consume more
services flow from the stock of durables. This view is consistent with the empirical results
discussed in subsequent sections.
5
2.2
Homotheticity vs. Non-Homotheticity
At first sight it may seem that allowing for non-homotheticity is not important and just introduces another parameter. In this section I argue how non-homotheticity leads to a substantial
reinterpretation of the results in Eichenbaum and Hansen (1990), Ogaki and Reinhart (1998),
and partly Piazzesi, Schneider and Tuzel (2003) and Lustig and Van Niewerburgh (2002).
There is also huge empirical evidence against homotheticity - the income elasticities vary
across categories of goods and they probably also depend on income and prices themselves as
suggested by their time variation. Houthakker (1957) and Houthakker and Taylor (1970), and
Ogaki (1992), using cross-sectional and time-series data, respectively, find empirical support
for the Engel’s law that the budget share of food declines with the level of wealth. Costa (2001)
estimates the income elasticities for food at home 0.47 in 1960-94, 0.62 in 1917-35. Those for
total food are 0.65 in 1960-94 and 0.68 in 1917-35 and in 1888-1917. Those for recreation are
1.37 in 1972-94, 1.41 in 1917-35, and 1.82 in 1888-1917.
Homotheticity of the felicity function u(ct , dt ) eliminates scale effects in that the relative demand ct /dt depends only on the relative price and hence the Engel curves are straight lines.
It is an analog to the constant-returns-to-scale (CRS) production function, often used in the
theory of the firm. In the context of the parametrization proposed in this paper, it corresponds
to the case η = 1. Eliminating income effects results in ascribing all changes in the relative demand to the pure substitution effect, which biases the estimate of the elasticity of substitution
(Pakoš (2003)). Slutsky equation decomposes the price effect into a substitution effect and
an income effect and it implies for example that the relative demand for durables depends on
the relative durables price and the real expenditure (see Deaton and Muellbauer (1980), Pakoš
(2003) and the discussion at the end of the section dealing with the intratemporal first-order
condition). Formally
d log (ct /dt ) = θ d log qt + (η1 − η2 ) d log êt
(7)
where θ is the elasticity of substitution, and η1 and η2 are income elasticities. Setting η1 =
η2 = 1 yields
d log (ct /dt ) = θ d log qt
(8)
It is an empirical fact that the ratio of durables to nondurables has been increasing steadily
and the relative price has been decreasing. Is it because consumers substituted to ever-cheaper
durables or because declining relative price increased real income? Homothetic preferences do
not even allow you to ask this question. By writing down a homothetic felicity function (i.e.
constant-elasticity of substitution) the answer is immediately that it was due to pure substitution. This is often claimed to be ’confirmed’ by estimating a big elasticity of substitution. The
6
argument is however misleading. The relative demand may change in response only either due
to income effect or substitution effect. If we don’t allow for non-homotheticity, by construction
it can’t be income effect. But the relative demand changed a lot and not surprisingly we find
a large elasticity. But this is not a proof that the relative demand changed in response to
substitution effect. In fact, substitution effect is about compensated changes in demand. It
answers the question - what would a consumer do if the relative price changed, holding real
income constant - it is about moving along an indifference curve. The language is often misused
in practice with people arguing that relative price changed and consumers substituted away but was it a substitution in the sense of Hicks or was it a response to a change in real income.
In fact, in what sense may a consumer substitute from apples to CDs? Even by introspection
it is not so unreasonable to think that the Hicksian price effects are very small - people want
to buy durables and non-durable in nearly fixed proportions (think of the example of food and
refrigerator). They certainly may substitute within a category (i.e. food or refrigerator).
Allowing for non-homotheticity (i.e. η 6= 1) is an important generalization, not yet considered in the asset pricing literature (see Ait-Sahalia, Parker and Yogo (2003) for an exciting
exception). The most often used approach to impose non-homotheticity in the macroeconomics
has been to consider subsistence levels. The approach advocated in this paper has the advantage that it allows to specify expenditure elasticities explicitly rather than implicitly through
subsistence levels. It is also not clear how general non-homotheticity subsistence levels actually
allow for. The specification in this paper is very general, the only restriction is that the ratio
of expenditure elasticities be kept constant and it is likely to be found useful in future research
in macroeconomics in general. In this sense, the paper makes a methodological contribution of
finding a particularly convenient mathematical form of non-homothetic felicity function with
easily interpretable parameters.
It turns out that the answer to the question why the ratio of durables has been changing
steadily is absolutely crucial for asset pricing. Introducing income effects produces an unbiased
estimate of the elasticity of substitution, we no longer force it to pick the effect of missing
income effects. Low substitutability and thus high complementarity between nondurables and
durables as reflected in a small elasticity of substitution - the felicity function close to Leontief
- has the potential to explain quantitative asset pricing puzzles. The feature of the Leontief
preferences is that consumers want to keep durables and non-durables in fixed proportion,
holding real income constant. I show that this mechanism enables to translate a small variation in non-durable consumption into a large change in consumer’s well-being and thus amplify
the risk premia. This also shows why linearization (or log-linearization) of the model may not
be such a good idea. Leontief preferences are not differentiable - they have a kink - and hence
7
the parameters estimated from the linearized version are nearly impossible to interpret.
2.3
Consumer’s Optimization Problem
I allow the consumer to trade in i ∈ I types of Lucas (1978) trees, with the number of each type
normalized to one, at price pit . The trees yield each period dividends divit . The consumer’s
problem is
(∞
)
X
max E0
β t u (Ω(ct , dt ))
(9)
t=0
subject to the budget constraint
ct + qt xt +
X
pit ait+1 =
i∈I
X
ait (pit + divit ) + wt
(10)
i∈I
where wt is the labor income (I do not model the labor-leisure choice) and the law of motion
for the stock of durables
dt+1 = (1 − δ) dt + xt
(11)
with δ denoting the depreciation rate of durables.
The proposed model of consumer durables is neoclassical in that I do not impose non-negativity
constraints on durables investment, so-called irreversible investment. It is an empirical fact
that the per-capita durables purchases are always positive and therefore that constraint is never
binding in equilibrium. I also do not introduce adjustment costs, gestation lags, transaction
costs etc.
In equilibrium, it must be true that
ait = 1
(12)
and that the demand for the consumption goods ct and xt is such that the goods markets clear.
2.4
Intertemporal First-Order Condition and Asset Prices
The marginal utility of non-durable consumption is
uc (ct , dt ) = a
θ−1
θ
−1
ct θ
n
o 1−θ γ
θ−1
1− θ1
1− ηθ
(a ct )
+ ((1 − a) dt )
(13)
The intertemporal marginal rate of substitution (or stochastic discount factor (SDF)) is
Mt+1 = β
uc (ct+1 , dt+1 )
uc (ct , dt )
8
(14)
The first-order condition for the optimal portfolio choice is given by the dynamic Euler equation
1 = Et {Mt+1 Rit+1 }
(15)
where Rit+1 = (pit+1 + divit+1 ) /pit is an asset’s gross return.
Fama and French (1992, 1993) propose an empirically determined SDF of the form
FF
Mt+1
= b0 + b1 Rmkt,t+1 + b2 SM Bt+1 + b3 HM Lt+1
(16)
to understand the risk premia on assets. I interpret their SDF as a projection of Mt+1 onto
the closed subspace of returns Rt+1 , namely,
FF
Mt+1
= proj(Mt+1 | Rt+1 )
(17)
©
ª
1 = Et {Mt+1 Rit+1 } = Et proj(Mt+1 | Rt+1 ) Rit+1
(18)
which follows from
2.5
Intratemporal First-Order Condition
One has to be a bit careful to derive the first-order condition for the optimal choice between
the durables and non-durables. Specifically, I distinguish two cases. Firstly, if the elasticity
of substitution is non-zero (really significantly different from zero from the statistical point of
view), then it must be true that nondurables ct and the services dt satisfy
½
µ
¶¾
ud (ct+1 , dt+1 )
qt = Et Mt+1
+ (1 − δ)qt+1
(19)
uc (ct+1 , dt+1 )
Intuitively, in perfect rental market the value of durables qt is determined as any other asset
value by simply discounting the next-period payoff with the marginal rate of substitution Mt+1 .
The payoff consists of two terms. The first one is the dividend ud (ct+1 , dt+1 ), expressed in terms
of ’utils’, or, ud (ct+1 , dt+1 )/uc (ct+1 , dt+1 ) units of nondurable consumption. The second is the
next-period market value of the depreciated durable (1 − δ)qt+1 . Reshuffling the previous
equation
½·
¸
¾
qt uc (ct , dt )
qt+1 uc (ct+1 , dt+1 )
ud (ct+1 , dt+1 )
= β Et (1 − δ)
+1
(20)
ud (ct , dt )
ud (ct , dt )
ud (ct , dt )
and computing the derivatives yields
η
η − θ (1 − a)1− θ
1
1−θ
a1− θ
µ
sηt
ct
¶− 1
θ
= β Et
½·
¸
¾
qt+1 uc (ct+1 , dt+1 )
ud (ct+1 , dt+1 )
(1 − δ)
+1
ud (ct , dt )
ud (ct , dt )
9
(21)
It is plausible to assume that the right-hand side is stationary and thus this equation implies
that if the series log qt , log ct and log dt are co-integrated, then we can take logs (the model is actually log-linear) on the left and estimate the preference parameters θ and η super-consistently
by running a regression in levels (Ogaki(1992), Ogaki and Reinhart (1998))
log ct = constant + θ log qt + η log dt + ²t
(22)
I interpret this equation as a conditional Marshallian demand function ct = ct (qt , dt ). This
specification differs substantially from Ogaki and Reinhart (1998). They impose the restriction
that η = 1 and rewrite the previous regression as
log (ct /dt ) = constant + θ log qt + ²t
(23)
However, such a regression is misspecified because it neglects income effects and yields a biased
estimate of the elasticity of substitution θ > 1. The same problem is apparent in Piazzesi,
Schneider and Tuzel (2003) who also impose homotheticity to estimate the demand for housing
(relative to nondurable consumption) and obtain θ > 1 as well. Neglecting income effects is
especially perilous for goods with large expenditure shares6 and no substitutes, i.e. housing,
durables etc.
Secondly, if the elasticity of substitution θ is zero, then the felicity function u(ct , dt ) is not
differentiable and we have to use a different argument. Specifically, as θ → 0 then the felicity
function converges to a CRRA utility defined over the Leontief consumption index, namely,
u(ct , dt ) =
1
1−γ
(min {a ct , (1 − a)η sηt })
1−γ
(24)
From the non-satiation7 of u(ct , dt ) and the properties of the Leontief consumption index we
obtain that in the optimum,
a ct = (1 − a)η sηt
(25)
In reality, the world is stochastic and hence ct and dt may depart for a while after a shock. The
previous equation implies that ct and dt are cointegrated and we can estimate η by running a
regression (in levels)
log ct = constant + η log dt + ²t
(26)
Notice that this is basically the previous specification obtained by imposing the restriction
θ = 0.
6
7
The Slutsky equation says that the income effects are proportional to the expenditure shares.
That is, uc (ct , dt ) > 0 and ud (ct , dt ) > 0.
10
Pakoš (2003) interprets θ as the elasticity of substitution and η as the ratio of income elasticities. The intuition for this interpretation is simple. For simplicity and without loss of generality,
assume that the setting is deterministic. If we think of consumers as renting durables in a
perfect rental market and paying the user cost of capital, then the preferences are weakly separable. Weak separability is necessary and sufficient for the second-stage of two-stage budgeting
to hold (Deaton and Muellbauer (1980)). We can therefore think of the previous equation as
a conditional Marshallian demand function. More specifically, the second-stage of two-stage
budgeting program yields that the Marshallian demands for the non-durables and services flow
satisfy after log-differentiating (Pakoš (2003))
d log ct = ε∗12 d log qt + η1 d log êt
d log dt = ε∗22 d log qt + η2 d log êt
where ε∗ij denote Hicksian price elasticities and ηi the income elasticities, êt is the real expenditure on both consumption goods. Eliminating the real expenditure yields
d log ct = (ε∗12 − ε∗22 η) d log qt + η d log dt
where I denoted η = η1 / η2 . For example, η < 1 means that services flow has income elasticity
greater than one and non-durable consumption less than one8 . This is intuitively plausible and
is also consistent with the Engel’s law. The elasticity of substitution ES between nondurables
and services flow is defined as ES = ε∗12 − ε∗22 . We see that, up to the multiple η, ES = θ.
Although they are not exactly equal, I interpret θ as a yardstick of substitutability between
the goods. For θ → 0 we get that ES is very close to θ and they are exactly equal for θ = 0,
in which case there is no substitutability between the goods. One may integrate the previous
equation - this is perfectly consistent with the model which is log-linear (see the intratemporal
condition above) - and hence the parameters are constant,
log ct = constant + (ε∗12 − ε∗22 η) log qt + η log dt
8
Recall the restriction that the average income elasticity must be one. Intuitively, a percent increase in
expenditures must be matched by a percent increase in real consumption.
11
3
Empirical Section
Figure 2 portrays 4 macroeconomic series - durables purchases, nondurables, durables stock
and the ratio of the stock of durables over nondurables, all quarterly. There is a clear secular
rise in the consumption of services flow (from durables) over non-durables. The interpretation
advanced and empirically supported in this paper is that the rising real income in the U.S.
economy enabled consumers to buy more durables. Furthermore, although durables purchases
and nondurables are procyclical the durables purchases are always positive. This provides
empirical support for not imposing non-negativity constraints on the durables investment xt
in the theoretical section.
3.1
Testing the Euler Equations: Generalized Method of Moments Approach
5 off the asset prices by applying
I firstly estimate the parameter vector ( θ , η , γ , β, a ) ∈ R+
Generalized Method of Moments to the dynamic Euler equation. I perform a first-stage GMM
with the weighting matrix W = inv(diag(cov(R))). I estimate the spectral density matrix Ŝ
using the Bartlett weight (with 7 lags, sample size T = 152). I plot the realized vs. the fitted
average returns by using the formula
ET (R) =
1 − covT (m, R)
ET (m)
(27)
The results for quarterly data are reported in Table 1. As a benchmark, I firstly estimate the
canonical C-CAPM (Breeden (1979), Lucas (1978)). It is well-known that the Lucas-Breeden
stochastic discount factor (ct+1 /ct )−γ does not do a very good job pricing risky assets (Hansen
and Singleton (1992, 1993), Mehra and Prescott (1985) and others). My results are consistent
2 . I statiswith this literature. The parameter vector (γ, β) is outside the parameter space R+
tically reject the model. I plot the fitted vs. realized returns in Figure 8. The plot is flat there is practically no cross-sectional variation in covariances of the nondurable consumption
growth with excess return and the model predicts tiny (i.e. ²) risk premia. The volatility of
Lucas-Breeden discount factor is small and it does not pass into Hansen-Jagannathan bounds
(Figure 5).
The results for the C-CAPM with durables are more encouraging. The parameter vector lies in
the parameter space. I estimate the elasticity of substitution θ = 0.0023 with std(θ) = 0.0023
and therefore cannot reject the hypothesis that there is zero substitutability between nondurables and the services flow - case of Leontief preferences. This is already a novel result (see
Eichenbaum and Hansen (1990), Ogaki and Reinhart (1998) for estimates with homothetic
preferences and Pakoš (2003) for how non-homotheticity overturns their result). It illuminates
12
the importance of allowing for non-homotheticity - income effects are crucial and substitution effects are negligible. In fact, I estimate the ratio of income elasticities η = 0.5339
(std(η) = 0.019), which implies that the income elasticity of durables is greater than nondurables. On average, the income elasticity must be one and thus we obtain the very intuitive
result that durables are luxury goods, with income elasticity greater than one, and nondurables
necessary goods, with income elasticity positive but less than one. This is also consistent with
the Engel’s law (Ogaki (1992)). Furthermore, I estimate γ = 1.523 with quite a precision, its
standard error is std(γ) = 0.019. This contrasts the estimate in annual data which is very
noisy (see Table 2 and the section below for more).
Although γ is not a yardstick of risk-aversion, I conjecture (but not prove, to be done) that
the value function will inherit the concavity of the felicity function and thus
−
W JW W
≈ γ
JW
(28)
The intuition for this is as follows. The estimate θ̂ is not significantly different from zero and
thus the consumption index Ω is Leontief. In addition, I provide additional evidence based on
the intratemporal first-order condition in favor of this. Therefore, from the theoretical section
we obtain that
Ω(ct , dt ) = min {a ct , (1 − a)η dηt }
(29)
In deterministic setup, we get that a ct = (1 − a)η dηt . In reality, the world is stochastic and
hence there may be some error ²t . Eliminating durables dt from the preferences yields
Ω(ct , dt ) ≡ Ω(ct ) = a ct + ²t
(30)
and the felicity function becomes
u (Ω(ct )) =
1
1
Ω(ct )1−γ =
(a ct + ²t )1−γ
1−γ
1−γ
(31)
This already looks like a standard model with CRRA preferences, in which case there are
already results that the value function will inherit the concavity of the felicity function under
certain conditions. But the durables still remain in the budget constraint, that’s why the
conjecture. The ambition is to show that the coefficient of the relative risk aversion defined in
terms of atemporal gamble as a concavity of the value function will be tightly related to the
estimated γ and hence small. This with the additional result that the intertemporal marginal
rate of substitution Mt+1 gets inside Hansen-Jagannathan bounds suggests that the proposed
generalization of the canonical C-CAPM may be capable to explain the equity premium and
risk-free rate puzzles. Interestingly, whereas most studies claim to have explained the equity
premium puzzle by calibrating their model and getting inside the Hansen-Jagannathan (HJ)
13
bounds, I get inside HJ bounds (Figure 6) with the estimated parameter vector. And the
HJ bounds are constructed using 25FF portfolios and hence are tighter than the ones usually
considered in the literature (i.e. using only aggregate market return).
The JT test of overidentifying restriction (an analogue to GRS test) rejects the hypothesis
that the pricing errors are zero. However, this is still informative. The plot of the actual vs.
fitted average returns on 25 Fama-French portfolios and the risk-free rate Rf looks substantially
better than for the canonical C-CAPM. The plot actually shows that the pricing errors are
smaller and have lower standard errors than the benchmark model. There is also a statistical
issue of whether the asymptotic standard errors are correct for such highly non-linear model. I
emphasize that my stochastic discount factor is the marginal rate of substitution. I do not linearize it, in stark contrast to most cross-section of expected returns literature (i.e. Lustig and
Van Nieuwerburgh (2002), Piazzesi, Schneider and Tuzel (2003), Santos and Veronesi (2001)
and others)
3.2
Longer-Horizon Results: Annual Data
In this section I estimate the canonical C-CAPM and C-CAPM with durables at the yearly
frequency. Annual data have the advantage that they are not seasonally adjusted and not
surprisingly the model fits a lot better. Using longer horizon returns is also consistent with
findings of Marshall and Daniel(19xx), Parker and Julliard (2003) and is actually the frequency
I used in the original version of this paper (before Parker and Julliard came out).
Because annual data leaves me with 38 observations, I estimate the model using both all
25 Fama-French portfolios and a subset of them. I cannot test the model in the first case as
I have too many moments relative to the time series (26 moments vs. T=38 observations).
Table 2 reports estimated parameter vector for all 25 Fama-French portfolios and for a subset,
so that I can test the model. The estimates based on all 25 FF portfolios are more precise.
The estimates for the ratio of income elasticities η = 0.56 with std(η) = 0.02 and thus are
consistent with the results from quarterly data. The same is true for θ. One parameter that
is estimated differently than in quarterly data is γ which is above 100. However, the estimate
γ̂ is very noisy and is estimated with a huge standard error. As a result, it is statistically
indistinguishable from its quarterly counterpart.
The model with durables is estimated very precisely. In Table 3 I report the pricing errors uT = ET (Mt+1 Rt+1 ) − 1 and their asymptotic standard errors for canonical C-CAPM
and C-CAPM with durables. The pricing errors for C-CAPM with durables are an order of
magnitude smaller and have an order of magnitude smaller standard errors. The fact that I
14
still reject the model with durables (p-value 3.15%) but not the standard C-CAPM (p-value
50%) reflects the very high precision of the pricing errors of C-CAPM with durables. The
canonical model is very noisy! This is evident also from Figure 8. The implied R2 for the CCAPM with durables (annualy) is above 90% and so if I follow many papers in the literature
and just compare the models based on R2 I would reject all other models and accept this one.
But, of course, R2 is not a test of an asset pricing model. As i quarterly data, I display the
Hansen-Jagannathan bounds for both models in Figure 5.
15
3.3
3.3.1
Reconciliation with the Intratemporal First-Order Condition
Tests for Non-Stationarity
I test the null hypothesis that the series log(ct ), log(dt ) and log(qt ) are difference stationary
against the alternative of trend stationarity using Phillips-Perron test (I included a constant
and a linear time trend). I cannot reject the hypothesis that the data may have been generated
by a random walk with drift at 5% significance level. The results are summarized in the Table
xx.
3.3.2
Tests for Deterministic Cointegration
Under deterministic cointegration, the same vector which removes stochastic trends also has
to remove deterministic trends (Engle and Granger (1987)).
A. Quarterly Data
Following the discussion in the theoretical section, I distinguish two cases. Firstly, I consider
the case when the elasticity of substitution θ is non-zero. Then the intra-temporal equation
implies that log ct , log dt and log qt are co-integrated and the co-integrating vector (θ, η) can
be estimated super-consistently by running the following regression (in levels)
log ct = constant + θ log qt + η log dt + ²t
(32)
I obtain θ = 0.0959, η = 0.599 (Table 7). I compute the Phillips-Ouliaris zρ and zt statistics. The results are summarized in Table 5. I cannot reject the hypothesis of deterministic
co-integration at 10% significance level (for lags q = 5, 6 in Newey-West correction), see Table 5.
Secondly, if the elasticity of substitution θ is zero, then the ratio of expenditure elasticities η
may be estimated super-consistently by running the following regression (in levels)
log ct = constant + η log dt + ²t
(33)
I obtain η = 0.542 (Table 7). I cannot reject the hypothesis of deterministic co-integration at
10% significance level (for lags q = 4, 5, 6 in Newey-West correction), see Table 5.
It seems that at the same time (log ct , log dt , log qt ), and (log ct , log dt ) are co-integrated.
This is suggestive that θ → 0 and the Phillips-Ouliaris test does not have power to distinguish between the case of θ very small and θ = 0. This justifies why I estimated the
parameter vector including θ and η off the asset prices. The conventional approach in the literature (Ogaki (1992), Ogaki and Reinhart (1998)) would be to estimate the parameters (θ, η)
16
super-consistently from the intratemporal condition, substitute them into the GMM objective
function and estimate the rest of the parameter vector off the asset prices. The co-integration
results however produce a bit conflicting results - should I use (θ̂, η̂) = (0.09, 0.599) based on
the first regression specification or set θ = 0 and use η̂ = 0.54? Note however that the parameters as estimated off the asset prices in the GMM section (θ̂, η̂) = (0.0023, 0.5339) in quarterly
data (Table 1), and (θ̂, η̂) = (0.0022, 0.5649) using all 26 assets and (θ̂, η̂) = (0.0015, 0.561)
using the subset of assets, both in annual data (Table 2). This is consistent with the case
where θ is small and η ∈ (0.53.0.56).
Furthermore, if it is really true that (log ct , log dt , log qt ) are co-integrated then it shouldn’t
matter if we estimate the regression as
log qt = const0 + const1 log ct + const log dt + ²̃t
(34)
instead and apply the Phillips-Ouliaris co-integration test to the residual ²̃t . However, as Table
5 shows, this specification accepts the null of no co-integration!
I interpret these results as providing evidence in favor of the hypothesis that θ is very small (of
the order 0.002) and η is about 0.5. I therefore conclude that the parameter vector as estimated
in the GMM section off the asset prices is consistent with the intratemporal condition.
B. Annual Data
As in quarterly data, I distinguish between the two cases. The estimates of the co-integrating
vectors is reported in Table 7. I reject the hypothesis of deterministic co-integration at 10%
level. This is surprising given that the series seem to be co-integrated at 10% level in quarterly data. Either the critical values are inappropriate as they are for sample size T = 500
whereas the actual one is T = 39. Or, the series are stochastically co-integrated with a small
trend in the preference parameters, the Phillips-Ouliaris test does not have power against this
alternative, and the trend becomes more important in annual data where the time variation of
the series is significantly lower.
17
3.4
Time-Series Predictability of Long-Horizon Returns on 25 Fama-French
Portfolios
In this section I unveil the novel result that the co-integrating residual ²t from the co-integrating
regression
log ct = constant + η log dt + ²t
(35)
forecasts returns on 25 FF portfolios, mostly the small stocks. Technically, for each year I
use the 4th quarter residual from the regression estimated in quarterly data as those are cointegrated at 10% level (see the discussion above also) and thus ²t is stationary9
I run predictive regressions for horizons (in years) h=1,2,3,4,5
f
f
rt+1 + ... + rt+h − rt+1
− ... − rt+h
= α + β ²̂t + errort+h
(36)
where r denotes the log of gross return and rf is the log of risk-free rate.
I report the results in the Table 8 in the Appendix. The R2 s in general rise with the horizon.
The most predictable seem to be the smallest stocks S1B1-S1B5 where R2 s reach levels above
30%. Secondly, the distressed stocks (stocks with a large book value relative to the market
capitalization) S1B5, S2B5 and S3B5 are predictable with R2 s as high as 44.8%, 43.4% and
29.7% and in fact for the two smallest categories of stocks the distressed stocks S1B5 and
S2B5 seem to be the most predictable. Interestingly, the largest stocks S5B1-S5B5 are not
very predictable which implies that after value-weighting the results do not contradict the
predictability literature where R2 s for the aggregate market above 15% are rare.
This result uncovers an exciting evidence that the equilibrium risk-premia on small stocks
and distressed stocks are related to the distortion of the investor’s consumption basket in recessions as measured by the residual ²t . As shown above, the data are suggestive of Leontief
preferences in which case consumers want to choose a ct = (1 − a)η sηt but because of unanticipated shocks these two quantities depart, with ²t measuring by how much.
It is well-known that there are econometric problems with interpreting long-horizon regressions. Firstly, the regressor ²̂t is persistent, but although it is predetermined it is not exogenous. Nelson and Kim (1993) and Stambaugh (1999) warn that this creates a small sample
bias in favor of finding predictability. Secondly, Valkanov (2001) shows that t-statistics do not
converge to a well-defined distribution when the forecasting horizon is a non-trivial fraction of
the sample size. I address this issue by plotting time-series of the actual 4-year long-horizon
9
If you run the regression in annual data, the results will be about the same. See also the discussion in the
intratemporal first-order condition section where I discuss co-integration tests.
18
return and the predicted one α̂ + β̂ ²̂t in addition to relying on t-statistics and R2 s. Note that
even though we may worry that the |t| > 2 test and R2 magnitude comparison may be inappropriate, the forecast tracks the long-horizon return rather closely, especially for the stocks
S1B1-S1B5. I interpret this result as another evidence in favor of predictability. Even though
we may not believe the R2 s and worry that t-statistics are uninformative, it is unlikely that a
spurious result would deliver such a good predictive plot.
19
4
Model Intuition
The paper proposes a novel mechanism to generate risk premia in line with the empirical
evidence. Investors care about the composition of their consumption basket and dislike when
their consumption of non-durables and the services flow from durables get out of line from the
optimal proportion. Specifically, non-durable consumption ct provides utility only together
with the services flow. Preference specification to deliver this is Leontief preferences, which
naturally generate a recession factor. This is in line with the empirical evidence found in the
paper, namely, that the consumption index has the form
Ω(ct , st ) = min {a ct , (1 − a)η sηt }
Recessions are times when both the durables investment xt and non-durables ct fall. However, investment xt does not fall to zero, that would imply zero sales for U.S. manufacturing - really a super-great depression. From the law of motion for the stock of durables
dt+1 = (1 − δ) dt + xt+1 we see that as long as xt is large enough to replace depreciated stock
(1 − δ) dt , the stock of durables dt+1 increases. This is easily visible in Figure xx where I plot
the non-durable consumption against the stock of durables. In recessions, ct falls and dt rises
and thus they get out of line. This implies large welfare losses for the consumer. Consumers are
afraid of stocks because they do badly in recessions which are times when their consumption
basket is already distorted.
There are two features of the model that are crucial. Firstly, it is the multiplicity of goods. Most
studies focus only on non-durable consumption. However, that doesn’t fluctuate much and this
gives rise to the equity-premium puzzle. Secondly, consumers cannot substitute (in fact, only
very little) from ct to dt . However, the relative Marshallian demand dt /ct changed a lot. This
implies that the income effects are very important - preferences are non-homothetic. Studies
that did introduce another good such as leisure, durables, housing etc. assumed homotheticity.
I emphasize that it is the interplay between multiplicity of goods and non-homotheticity that
is crucial.
20
5
Conclusion
This paper uncovers a novel mechanism to explain quantitative asset pricing puzzles. I show
that the substitutability between nondurables and services flow from durables is very low consumers prefer their consumption of nondurables and the flow of services to be in nearly
constant proportions. In other words, recessions are costly not because the consumption of
nondurables falls per se but because such a fall distorts the consumption basket. Preferences in
addition exhibit non-homotheticity - the relative demand for durables has been rising because
the declining relative price of durables raised the real income. The distortion of consumption
basket in recessions also implies a time-varying market price of risk.
I contribute to, and unify, the equity-premium puzzle and risk-free rate literature, and the
cross-section of expected returns literature. Specifically, I show that the marginal rate of substitution passes Hansen-Jagannathan bounds for both quarterly and annual data. I explain
the cross-section of expected returns on 25 Fama-French portfolios with just the marginal rate
of substitution, in stark contrast to most other studies that log-linearize the model and use
multiple factors. The paper also contributes to the predictability literature and shows that
the cointegrating residual tracks the long-horizon excess returns on 25 Fama-French portfolios.
In addition, I contribute to the empirical macroeconomics literature dealing with the substitutability of goods. I show that a correct modelling of income and substitution effects leads
to a substantial reinterpretation of the time pattern of consumption data in the postwar U.S.
economy. Finally, the paper makes a methodological contribution to dynamic economics of
finding a particularly convenient mathematical form of non-homothetic felicity function with
easily interpretable parameters.
21
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24
A
Data Description
Nondurable Consumption: Real and nominal purchases of nondurables plus services ct from the U.S.
National Income and Product Accounts. Quarterly from 1947,1-2001,4. Converted from quarterly to
annual by summing for all 4 quarters. Corrected for SAAR by dividing by 4.
From http://www.stls.frb.org/fred/data/gdp.html
Durables: Real and nominal purchases of durables xt from the U.S. National Income and Product
Accounts. Quarterly from 1947,1-2001,4. Converted from quarterly to annual by summing for all 4
quarters. Corrected for SAAR by dividing by 4. Stock of durables dt constructed as a weighted average
t
P
of past purchases dt =
(1−δ)t−i xi with the depreciation rate δ = 20% p.a. This is consistent with
i=1947
Wykoff(1970) estimates of a depreciation rate from resale values of automobiles. Ogaki and Reinhart
(1998) use value of 22 percent. I start in the year 1947 but perform tests since 1964 because of two
reasons. One is that the initial stock of durables as of 1947 is unknown but may be safely assumed
to have depreciated away by 1964. The other is that the portfolios used tend to be rather thin before
1964.
Population: The consumption is converted to a per capita basis using yearly U.S. population from
1947-2001 obtained from http://www.FreeLunch.com
Risk-Free Return: Annualized three-month Treasury bill. Converted to ex-post real returns by the
implicit price deflator for the total consumption. Obtained from Ken French’s web site. Yearly 19642001.
Asset Returns: Returns on 25 Fama-French portfolios formed according to the same criteria as in Fama
and French (1992,1993). These data are value-weighted returns for the intersections of five size portfolios
and five book-to-market equity (BE/ME) portfolios on the New York Stock Exchange, the American
Stock Exchange, and NASDAQ stocks in Compustat. The portfolios are constructed at the end of June,
and market equity is market capitalization at the end of June. The ratio BE/ME is book equity at the
last fiscal year end of the prior calendar year divided by market equity at the end of December of the
prior year. All asset returns are converted to ex-post real rates by the implicit price deflator for the
total consumption. Obtained from Ken French’s web site. Yearly 1947-2001.
B
Technical Details of the Estimation
Without loss of generality, but for the sake of numerical accuracy, which is important especially when
θ is close to zero, I re-scale the consumption series ct with mean(ct ), and dt with mean(dt ) as
o θ
n
1− η θ−1
1− 1
Ω(ct , dt ) = (ct /mean(c)) θ + a (dt /mean(d)) θ
(37)
and keep only the preference weight a on the services flow. Notice that I can do it even if the series
trend as it is just a convenient normalization of the series. I am not arguing that mean(c) converges in
probability to E(c), it doesn’t.
25
Table 1: GMM Estimates for Quarterly Data (No Trend in the Preference Parameters)
M odel
C-CAPM
C-CAPM with Durables
θ
η
γ
β
First-Stage GMM, W = inv(diag(cov(R)))
−5.6165
0.9632
(117.1334) (0.6503)
0.0023
0.5339
1.5230
0.2113
(0.0023) (0.0185)
(0.0190)
(0.7073)
a
JT
0.4858
(2.8163)
44.63
(0.006)
114.33
(0.000)
NOTE - Standard errors are in parentheses. Spectral density matrix is estimated using Bartlett weight (with 7
lag). In JT column, p-value is in parentheses. For the canonical CCAPM the preference parameter is outside
2
the parameter space R+
.
Table 2: Longer-horizon GMM Estimates: Annual Data, No Trend in the Preference Parameters
M odel
C-CAPM
C-CAPM with Durables
C-CAPM
C-CAPM with Durables
θ
η
γ
β
a
JT
f
Moments: R and all 25 FF portfolios
109.316
4.076
(63.639)
(2.491)
0.0022
0.5649
95.6765
1.2109
0.9560
(0.0018) (0.014)
(145.185)
(1.518)
(4.020)
f
Moments: R and 8 FF portfolios (11,12,13,14,15 and 54,55)
113.2
4.194
5.319
(101.415) (4.3094)
(0.504)
0.0015
0.561
138.912
0.784
0.861
8.838
(0.0028) (0.0202) (492.2957) (3.3523) (12.4345) (0.0315)
NOTE - The table reports 1st-stage GMM results (annual data, W = inv(diag(cov(R)))) obtained by minimizing
5
the quadratic form of pricing errors over the parameter space R+
. Standard errors are in parentheses. Spectral
density matrix is estimated using Bartlett weight (with 3 lag). In JT column, p-value is in parentheses. No
trend in the preference parameters introduced.
26
Table 3: Pricing Errors for Longer-horizon GMM Estimates: Annual Data, No Trend in the Preference
Parameters
Asset
Rf
11
uT
std(uT )
−0.0041
0.1691
uT
std(uT )
−0.0000
0.0626
12
54
55
0.0986
0.0093
13
14
15
Canonical C-CAPM
0.1148
0.1164
0.1040 0.1239
0.0211
0.0186
0.0158 0.0299
0.0416
0.0430
0.0082
0.0480
0.0032
0.0010
C-CAPM with durables
−0.0138 −0.0035 0.0035 0.0084
0.0015
0.0054
0.0081 0.0027
0.0059
0.0007
−0.0042
0.0032
NOTE - The table reports the pricing errors, uT = ET (Mt+1 Rt+1 ) − 1, for 1st-stage GMM results (annual
data, W = inv(diag(cov(R))), 8 moments used) obtained by minimizing the quadratic form of pricing errors
5
over the parameter space R+
. No trend in the preference parameters introduced.
Table 4: Tests for the trend properties of the data
log ct
log dt
log qt
log ct
log dt
log qt
Tests for the null of difference stationarity
Phillips-Perron Test
q=2
q=3
q=4
zρ
zt
zρ
zt
zρ
Quarterly Data
-9.705 -3.055 -9.467 -3.054 -10.484
-4.985 -2.369 -5.610 -2.352 -6.181
1.551
0.561
1.357
0.477
1.244
Annual Data
-9.232 -3.159 -8.972 -3.172 -8.554
-9.298 -2.493 -9.650 -2.524 -9.487
2.092
0.650
2.759
0.938
3.219
zt
q=5
zρ
zt
-3.070
-2.364
0.430
-10.829
-6.699
1.160
-3.081
-2.387
0.396
-3.204
-2.510
1.170
-8.086
-9.029
3.834
-3.262
-2.470
1.541
NOTE - Critical value for zρ (quarterly data) is −20.7 (5% level) and −17.5 (10% level), zt (quarterly data) is
−3.45 (5% level) and −3.15 (10% level). Critical value for zρ (annual data) is −19.8 (5% level) and −16.8 (10%
level), zt (annual data) is −3.5 (5% level) and −3.18 (10% level).
The parameter q denotes number of lags in Newey-West estimator (Bartlett weight used).
27
Table 5: Phillips-Ouliaris Residual Test for Deterministic Cointegration: Quarterly Data
q=2
q=3
q=4
q=5
q=6
log ct = const + θ log ct + η log dt + ²t
zρ
zt
−20.349
−3.234
−19.068
−3.134
−22.955
−3.430
−24.196
−3.519
−24.473
−3.539
log ct = const + η log dt + ²t
zρ
zt
−17.899
−2.984
−16.728
−2.884
−20.254
−3.175
−21.248
−3.253
−21.380
−3.263
log qt = const0 + const1 log ct + const2 log dt + ²t
zρ
zt
−3.639
−1.332
−3.394
−1.286
−3.845
−1.370
−4.094
−1.415
−4.244
−1.441
NOTE - Critical value for the case of 1 regressors: zρ −21.5 at 5% level and −18.1 at 10% level, for zt it is
−3.42 at 5% level and −3.13 at 10% level. Critical value for the case of 2 regressors: zρ −27.1 at 5% level and
−23.2 at 10% level, for zt it is −3.8 at 5% level and −3.52 at 10% level. Source: Hamilton (1994), Case 3 Tables
B8 and B9. These are for sample size T = 500, my sample size is T = 152.
The parameter q denotes number of lags in Newey-West estimator (Bartlett weight used).
28
Table 6: Phillips-Ouliaris Residual Test for Deterministic Cointegration: Annual Data
q=2
q=3
q=4
q=5
q=6
log ct = const + θ log ct + η log dt + ²t
zρ
zt
−13.01
−2.215
−13.081
−2.222
−12.717
−2.183
−12.239
−2.131
−11.952
−2.099
log ct = const + η log dt + ²t
zρ
zt
−7.203
−1.424
−6.935
−1.385
−6.680
−1.347
−6.529
−1.324
−6.597
−1.335
NOTE - Critical value for the case of 1 regressors: zρ −21.5 at 5% level and −18.1 at 10% level, for zt it is
−3.42 at 5% level and −3.13 at 10% level. Critical value for the case of 2 regressors: zρ −27.1 at 5% level and
−23.2 at 10% level, for zt it is −3.8 at 5% level and −3.52 at 10% level. Source: Hamilton (1994), Case 3 Tables
B8 and B9. These are for sample size T = 500, my sample size is T = 39.
The parameter q denotes number of lags in Newey-West estimator (Bartlett weight used).
29
Table 7: Co-integration Results
log ct
log qt
log dt
T8
T 20
Quarterly Data
(1)
(2)
(3)
0.0959
(0.0348)
0.0214
(0.0616)
(4)
(5)
0.0485
(0.0386)
(6)
0.542
(0.004)
0.599
(0.021)
0.563
(0.031)
0.5526
(0.0052)
0.575
(0.022)
0.5482
(0.0040)
-0.024
(0.018)
-0.029
(0.009)
-0.023
(0.010)
-0.033
(0.007)
Annual Data
(1)
(2)
(3)
0.1572
(0.0470)
-0.0154
(0.0647)
(4)
(5)
(6)
0.0576
(0.0342)
0.552
(0.008)
0.645
(0.026)
0.564
(0.031)
0.5719
(0.0057)
0.596
(0.020)
0.5639
(0.0049)
-0.053
(0.019)
-0.050
(0.009)
-0.043
(0.008)
-0.052
(0.007)
NOTE - The table presents the results of the regression log ct = α0 + α1 T 8 + α2 T 20 + θ log qt + η log dt .
The Newey-West corrected errors (nlags=3) are reported in parentheses.
30
Figure 2: Quarterly Consumption Spending
−3
3.5
x 10
Quarterly Durables Purchases
Quarterly Nondurables + Services
0.02
3
0.018
2.5
0.016
2
0.014
1.5
0.012
1
0.01
0.5
1960
1970
1980
1990
2000
0.008
1960
2010
1970
Quarter
1980
1990
2000
2010
Quarter
Quarterly Durables Stock
Durables Stock over Nondurables + Services
0.07
3.5
0.06
3
0.05
0.04
2.5
0.03
2
0.02
0.01
1960
1970
1980
1990
2000
1.5
1960
2010
Quarter
1970
1980
1990
Quarter
31
2000
2010
Figure 3: Income expansion path, Quarterly Data
0.07
0.06
Quarterly Durables Stock
0.05
0.04
0.03
0.02
0.01
0.008
0.01
0.012
0.014
Quarterly Nondurables + Services
NOTE -
32
0.016
0.018
0.02
Figure 4: Income expansion path, Annual Data
−3
14
x 10
12
Annual Durables Stock
10
8
6
4
2
0.008
0.01
0.012
0.014
Annual Nondurables + Services
NOTE -
33
0.016
0.018
0.02
Figure 5: Volatility bounds for First-Stage GMM parameters (W = inv(diag(cov(R))) and no
trend in preference parameters)
Canonical C−CAPM, Quarterly Frequency
C−CAPM with Durables, Quarterly Frequency
2
5
1.5
4
x
std(m)
std(m)
1
0.5
2
x
0
1
−0.5
−1
3
0
0.5
1
1.5
0
2
0
0.5
E(m)
1
1.5
2
E(m)
Canonical C−CAPM, Annual Frequency
C−CAPM with Durables, Annual Frequency
3
3
2.5
2.5
x
2
x
std(m)
std(m)
2
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
0
2
E(m)
0
0.5
1
E(m)
34
1.5
2
Figure 6: Marginal Rate of Substitution
Canonical C−CAPM, Quarterly Frequency
C−CAPM with Durables, Quarterly Frequency
50
Marginal rate of substitution
Marginal rate of substitution
1.2
1.1
1
0.9
1960
1970
1980
1990
2000
40
30
20
10
0
1960
2010
1970
1980
Time
Canonical C−CAPM, Annual Frequency
2010
8
Marginal rate of substitution
Marginal rate of substitution
2000
C−CAPM with Durables, Annual Frequency
8
6
4
2
0
1960
1990
Time
1970
1980
1990
2000
6
4
2
0
1960
2010
1970
1980
Time
1990
2000
2010
Time
NOTE - The graphs display the marginal rate of substitution mrst+1 = β
and the C-CAPM with durables, at quarterly and annual frequency.
35
uc (t+1)
uc (t)
for the canonical C-CAPM
Figure 7: 1st Stage GMM Results (W = inv(diag(cov(R))): Cross-Sectional Results
C−CAPM with Durables, Quarterly Frequency
6
5
5
Fitted Return (in %)
Fitted Return (in %)
Canonical C−CAPM, Quarterly Frequency
6
4
3
2
1
35
34
45 25
12
32 44
21 54 33
24 15
41 43
23
22
55
13
1151
42
53
14
31
3
2
52
1
Rf
0
4
145
324
251415
35
1152
22
12
34
23
5543
33
32 44
54
53
4241
21
31
51
0
2
Rf
4
0
6
0
2
Realized Return (in %)
15
15
55
5
Rf
0
0
5
4445
53
54
14
24
35
52
42
33
34 25 15
23
43
22 32
51
13
12
41
21
31
10
6
C−CAPM with Durables, Annual Frequency
20
Fitted Return (in %)
Fitted Return (in %)
Canonical C−CAPM, Annual Frequency
20
10
4
Realized Return (in %)
15
14 15
10
2142
11
53
3151
41
52
5
1213 25
23
45 24
35
34
55
33
32 44
43
54
22
Rf
0
20
11
Realized
Return (in %)
0
5
10
15
20
Realized Return (in %)
NOTE - The graphs display realized vs. fitted average returns for the canonical C-CAPM and the C-CAPM with
durables, at quarterly and annual frequency. The stochastic discount factor is the marginal rate of substitution
mt+1 = β ucu(t+1)
. The parameters are estimated using GMM with the weighting matrix W = inv(diag(cov(R))).
c (t)
36
Table 8: Results of Predictive Regressions (Annual Frequency)
Asset
S1B1
S1B2
S1B3
S1B4
S1B5
S2B1
S2B2
S2B3
S2B4
S2B5
S3B1
S3B2
S3B3
S3B4
S3B5
S4B1
S4B2
S4B3
S4B4
S4B5
S5B1
S5B2
S5B3
S5B4
S5B5
h=1
12.60
(-2.64)
19.83
(-2.96)
23.12
(-3.22)
21.44
(-3.03)
27.47
(-3.81)
11.36
(-2.08)
14.54
(-1.93)
16.73
(-2.63)
18.65
(-2.57)
25.93
(-3.51)
8.22
(-1.81)
17.60
(-2.53)
22.42
(-2.88)
17.15
(-2.49)
19.42
(-2.94)
4.55
(-1.35)
15.04
(-1.92)
19.62
(-2.85)
18.87
(-2.28)
17.14
(-2.17)
2.85
(-1.28)
4.30
(-0.94)
3.20
(-1.02)
9.97
(-1.81)
5.73
(-1.46)
h=2
16.98
(-2.65)
26.40
(-3.25)
27.38
(-3.14)
30.57
(-3.25)
40.99
(-4.41)
14.70
(-2.00)
18.77
(-2.04)
21.78
(-2.55)
26.35
(-2.66)
40.75
(-3.76)
12.43
(-1.78)
23.80
(-2.34)
32.81
(-3.03)
26.08
(-2.35)
33.63
(-3.32)
5.44
(-1.11)
18.68
(-1.81)
33.41
(-2.64)
24.46
(-2.00)
29.32
(-2.29)
1.48
(-0.67)
3.10
(-0.67)
7.44
(-1.14)
17.78
(-1.79)
14.06
(-1.57)
h=3
13.41
(-2.55)
19.28
(-3.30)
20.89
(-3.08)
24.68
(-3.51)
34.26
(-4.60)
9.47
(-1.79)
12.51
(-1.93)
15.67
(-2.54)
19.45
(-2.55)
33.41
(-3.83)
7.26
(-1.51)
14.79
(-2.12)
24.39
(-3.24)
15.77
(-2.10)
23.01
(-3.00)
1.18
(-0.60)
8.17
(-1.43)
20.96
(-2.60)
12.16
(-1.67)
13.77
(-1.79)
0.32
(0.38)
0.04
(0.11)
2.54
(-0.83)
5.95
(-1.43)
3.86
(-0.97)
h=4
24.09
(-3.33)
30.41
(-4.55)
30.60
(-4.81)
37.30
(-5.61)
44.74
(-8.70)
17.90
(-2.52)
19.95
(-2.80)
28.41
(-3.64)
32.11
(-4.82)
42.51
(-7.50)
13.61
(-2.08)
22.00
(-2.88)
31.48
(-4.71)
23.73
(-3.02)
28.70
(-4.73)
2.90
(-0.86)
9.57
(-1.70)
27.35
(-4.75)
20.27
(-2.33)
17.08
(-2.33)
0.04
(0.14)
0.04
(-0.13)
3.05
(-1.12)
6.89
(-2.03)
4.81
(-1.37)
h=5
29.12
(-3.29)
31.35
(-4.30)
30.98
(-4.95)
38.07
(-5.68)
44.83
(-8.71)
22.58
(-2.51)
21.74
(-2.80)
28.52
(-3.51)
34.46
(-5.22)
43.41
(-7.62)
17.00
(-2.12)
23.20
(-2.82)
32.20
(-4.67)
28.42
(-3.30)
29.65
(-4.98)
5.69
(-1.07)
11.81
(-1.77)
31.78
(-4.61)
22.82
(-2.76)
16.46
(-2.39)
0.14
(-0.22)
1.44
(-0.61)
6.38
(-1.39)
9.21
(-2.15)
5.73
(-1.51)
NOTE - The first digit after F F refers to the size quintiles (1 indicating the smallest, 5 the largest), and the
37
second digit refers to book-to-market quintiles (1 indicating the portfolio with the lowest book-to-market ratio, 5
with the highest). The first line contains R2 s, t-stats are in parentheses (Newey-West, Bartlett weight, nlags=4)
Figure 8: Realized vs. Forecasted Excess Log Returns: Fama-French Portfolios (S1B1, S1B2,
S1B3, S1B4, S1B5, S2B1, from left to right and then down).
2
2
1
1
0
0
−1
−1
−2
0
10
20
30
40
2
−2
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
2
1
1
0
0
−1
−2
0
10
20
30
40
2
−1
2
1
1
0
0
−1
−1
0
10
20
30
40
−2
NOTE - Each plot corresponds to one portfolio and portrays the time series of the actual and predicted excess
log return (dotted line).
38
Figure 9: Realized vs. Forecasted Excess Log Returns: Fama-French Portfolios (S2B2, S2B3,
S2B4, S2B5, S3B1, S3B2, from left to right and then down).
2
2
1
1
0
0
−1
−2
0
10
20
30
40
2
−1
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
1.5
1
1
0.5
0
−1
0
0
10
20
30
40
−0.5
1
2
0
1
−1
0
−2
0
10
20
30
40
−1
NOTE - Each plot corresponds to one portfolio and portrays the time series of the actual and predicted excess
log return (dotted line).
39
Figure 10: Realized vs. Forecasted Excess Log Returns: Fama-French Portfolios (S3B3, S3B4,
S3B5, S4B1, S4B2, S4B3 from left to right and then down).
1
1.5
0.5
1
0
0.5
−0.5
0
−1
0
10
20
30
40
2
−0.5
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
1
0.5
1
0
0
−1
−0.5
0
10
20
30
40
1
−1
1
0.5
0.5
0
0
−0.5
−1
0
10
20
30
40
−0.5
NOTE - Each plot corresponds to one portfolio and portrays the time series of the actual and predicted excess
log return (dotted line).
40
Figure 11: Realized vs. Forecasted Excess Log Returns: Fama-French Portfolios (S4B4, S4B5,
S5B1, S5B2, S5B3, S5B4, S5B5 from left to right and then down).
1
1.5
2
1
0.5
1
0.5
0
−0.5
0
0
0
20
40
−0.5
0
20
40
−1
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
0
20
40
−0.5
0
20
40
1
1
0.5
0.5
0
0
−0.5
0
20
−0.5
40
0
−0.5
10
0
20
40
0
20
40
20
30
NOTE - Each plot corresponds to one portfolio and portrays the time series of the actual and predicted excess
log return (dotted line).
41
40