Measuring Intratemporal and Intertemporal Substitutions When
Both Income and Substitution Effects Are Present:
The Role of Consumer Durables
MICHAL PAKOŠ1
First Draft: April 2003
Revised Manuscript: September 2007
1
Michal Pakoš is Assistant Professor at the Tepper School of Business, Carnegie Mellon University,
5000 Forbes Avenue, Pittsburgh PA 15213 (email: misko@andrew.cmu.edu)
Abstract
Hall (1988) estimates the intertemporal substitutability for nondurable goods economically
insignificant. Ogaki and Reinhart (1998) introduce the service flow from durable goods using
nonseparable homothetic preference specification, and estimate (i) the intertemporal substitution large (around 0.4) and significant, and (ii) the intratemporal substitution above one. I
show that homotheticity induces a striking statistical bias of the estimates of the intratemporal and intertemporal substitutions. Using aggregate data, I find that the estimate of the
intertemporal substitutability is economically negligible, on the order of 0.06, close to Hall’s
(1988) original estimate. In addition, I estimate the intratemporal substitutability between
nondurables and service flow also relatively small, around 0.46, but economically significant.
Finally, I find strong support in favor of nonhomotheticity, with nondurables being necessary
goods and durables luxury goods. However, due to the rapid decline of the rental cost, the
budget share of consumer durables declines secularly.
JEL Classif ication : D11, D12, D13, D91, E01, E13, E21
Keywords : Durable Goods, Nonhomotheticity, Intertemporal Substitution, Intratemporal
Substitution
1
Introduction
The elasticities of intertemporal (across time) and intratemporal (across goods) substitutions
are two central parameter inputs into most modern macroeconomic models. They significantly
affect the quantitative implications of various economic policy changes implied by these models (Hall 1988). Firstly, the intertemporal substitution is perhaps one of the most important
determinants of the response of saving and consumption to the predictable changes in the real
interest rate. If expectations of real interest rates shift, there should be a corresponding shift
in the rate of change of consumption. The magnitude of this intertemporal substitution is
measured by the response of total consumption expenditures to a change in real interest rate
expectations. Furthermore, in a two-good economy with consumer durables, the magnitude of
the intratemporal substitution directly affects the measure of the intertemporal substitution.
The real interest rate positively affects the user cost of consumer durables. Therefore, a rise
in interest rates leads to an increase in the user cost and consumers rationally substitute from
durables service flow to nondurable consumption. Intratemporal substitution is another channel that links consumption growth with real interest rates and indirectly affects the magnitude
of the elasticity of intertemporal substitution. Secondly, the magnitude of the elasticity of
intratemporal substitution is important for asset pricing. It, in addition to the coefficient of
risk aversion, determines the variability of the marginal utility and asset risk premia. In fact,
low substitutability between consumption goods means that a small variation in nondurable
consumption translates into dramatic fluctuations in marginal utility. This raises the intriguing question whether the consumption risk of the stock market is really so small. In a recent
paper, Pakoš (2005) empirically evaluates this effect and its impact on the asset risk premia,
in particular the equity premium.
A large literature focuses on the estimation of these two parameters. In a seminal paper,
Hall (1988) presents estimates of the elasticity of the intertemporal substitution ”... that are
small. Most of them are also quite precise, supporting the strong conclusion that the elasticity
1
is unlikely to be much above 0.1, and may well be zero.”. Using improved inference methods,
Hansen and Singleton (1983) find that there is less precision and even obtain estimates that are
negative. Using international data, Campbell (2003) estimates the elasticity of intertemporal
substitution statistically and economically insignificant. However, these studies assume that
the felicity function is separable over nondurables and durables (see Lewbel 1987 for another
criticism of Hall’s model). In response, Mankiw (1982, 1985) and Ogaki and Reinhart (1998)
enrich the model by explicitly introducing the service flow from consumer durables. They argue
that ignoring this consumption good induces a bias in favor of finding a low or even negative
magnitude of intertemporal substitution. The real interest rate directly affects the user cost of
consumer durables. For instance, a rise in the real interest rates leads to a higher user cost and
consumers substitute toward nondurable goods. This channel is completely missing in the one
good economy and hence the estimate of the intertemporal substitution is biased, compared
to a one-good economy. Secondly, Mankiw (1985) finds that the service flow from consumer
durables is itself more responsive to the interest rates and estimates a large elasticity of substitution for durable goods. Focusing on non-separability across goods, Ogaki and Reinhart
(1998) find that there is quite a large intertemporal substitutability when both nondurable
and durable goods are considered.
One important criticism of the Ogaki and Reinhart’s (1998) empirical results is that they
work with homothetic preferences1 and thus their demand function for durables is free from
potentially significant income effects. A correct modeling of nonhomotheticity in the demand
function for durables is essential to obtain unbiased estimates of the magnitudes of the intratemporal and intertemporal substitutions. Empirically, the ratio of durables over nondurables has
been trending up secularly (see Figure 1). The mainstream interpretation is that investors
optimally substituted (in the sense of Hicks) consumer durables for nondurables in response
to their falling price, with income effects playing no role whatsoever (Eichenbaum and Hansen
1990, Ogaki and Reinhart 1998). However, durables may not have an easy substitute, and
therefore we would expect a priori the Hicksian price effects to be relatively small. I illustrate
2
this point with an example. Suppose the consumer faces the choice of commuting to work by
either taxi or by renting a car. If the car rental cost increases relative to the cost of taxi and
the consumer gets a compensation in order to hold his real income constant, and he does alter
his consumption of car services little relative to the taxi, the Hicksian price effect is small.
Note that it is important to compensate the consumer so that we isolate the pure price effect.
It is common to use the word ”substitute” in the sense of ”serving in place of another” (Oxford
Dictionary). However, that is not the technical meaning of the word. It confuses income versus
substitution effects precisely because in both cases consumers use another consumption good
in place of the original one. But these two effects are distinct reaction channels to a price
change. The empirical results in the paper appear to indicate that the Hicksian substitution
effects are smaller than we thought, and thus the income effects should be important to fit the
demand function for durable goods.
In order to see the effects of this misspecification, I use the two-step cointegration-Euler equation approach pioneered by Cooley and Ogaki (1996) and Ogaki and Reinhart (1998), and
allow for (i) nonseparability across goods (nondurables and the service flow from consumer
durables) and (ii) a very general form2 of nonhomotheticity in the relative demand function
for durables by generalizing the constant-elasticity-of-substitution (CES) felicity function. In
the first step, using cointegration techniques, I obtain super-consistent estimates of the elasticity of intratemporal substitution and the ratio of within-period expenditure elasticities. In
the second step, I invoke the Hansen’s (1982) Generalized Method of Moments to estimate the
rest of the preference parameter vector, and formally test the model. Then, following Atkeson
and Ogaki (1996), I define the elasticity of intertemporal substitution IES as an inverse of the
elasticity of the marginal utility of the total consumption expenditure. This step requires a
numerical solution of the second stage of two-stage budgeting to construct the indirect felicity
function and its partial derivatives.
The results of the paper are interesting. First, consistent with microeconomic studies, I find
3
that the Engel curves are not linear and hence the preferences exhibit nonhomotheticity, with
durable goods being luxury goods and nondurable goods necessary goods. Second, the budget
share of durable goods is on the order of 10% - 14% and in fact goes down because of the
declining rental cost of durables, despite the fact that they are luxury goods. This in turn
implies that nondurable goods compose increasingly larger share of the total consumption
expenditures. Hall (1988) estimates the IES for nondurables close to zero whereas Mankiw
(1982) estimates the IES for durables economically significant. Because the consumption basket is a weighted average of nondurable goods (negligible IES) and durable goods (significant
IES), and the budget share of durable goods is relatively small, I confirm the intuitive result
that the elasticity of intertemporal substitution IES for the total consumption expenditures is
economically negligible.
It is well-known that adjustment costs play an essential role in the consumption of durable
goods (Bernanke 1984, Lam 1989, Grossmand and Laroque 1990, Eberly 1994). Mankiw (1985)
concludes that ”...future work should pay closer attention to the role of adjustment costs. The
model that took account of the adjustment process would be better suited for examining the
effects of shorter term fluctuations in the real interest rate.” Ogaki and Reinhart (1998) correctly point out that one of the big advantages of the two-stage cointegration-Euler equations
approach is that ”... it is robust to various specifications of adjustment costs, relying on the
cointegration properties between the observed and the desired stock of durables in the presence
of adjustment costs, which is discussed in Caballero (1993).”. The inference based on the cointegrating regressions yields consistent estimates as long as adjustment costs do not affect the
long-run behavior of the service flow from consumer durables. Furthermore, it can be shown
that the Euler equation for nondurable consumption is robust to various forms of adjustment
costs for durable goods consumption. Unfortunately, many exciting studies such as Dunn and
Singleton (1986) or Eichenbaum and Hansen (1990), among others, do not allow implicitly nor
explicitly in their estimation for adjustment costs.
4
2
2.1
Theoretical Framework
Model
I refine the standard neoclassical macroeconomic model where the consumer maximizes the Von
Neuman-Morgenstern expected utility function defined over nondurable goods ct by explicitly
introducing the service flow st from the stock of consumer durables dt as follows
E0
(
∞
X
t=0
t
)
β U [C (ct , st )]
(1)
I define the felicity function U (C) = (1 − 1/σ)−1 C 1−1/σ to be an isoelastic function. Many
studies implicitly use durable goods by assuming that nondurable consumption and the service flow from durables enter the felicity function separably (Hansen and Singleton 1982, 1983).
Eichenbaum and Hansen (1990) find empirical evidence in favor of nonseparability. Fleissig,
Gallant and Seater (2000) derive a seminonparametric utility function with both nondurable
goods and consumer durables, and find that CRRA preferences defined only over nondurable
goods are severely misspecified. They also discover evidence in favor of significant nonseparabilities across the consumption goods3 . Therefore, I specify the consumption index C(ct , st )
as a generalization of the constant-elasticity-of-substitution (CES) index over the nondurable
goods ct and the service flow st from consumer durables as follows
θ
1− 1θ
1− θη θ−1
C (ct , st ) = (1 − ã) × ct
+ ã × st
(2)
where ã ∈ (0, 1) is the preference weight. Observe that the consumption index C(ct , st )
exhibits nonhomotheticity. Dunn and Singleton (1986) assume that the consumption index
C(ct , st ) is homothetic and Cobb-Douglas; 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. As it will become clear later on, nonhomotheticity is dictated
by the data to fit the relative demand for durables, rather than an appetite for an additional
degree of freedom. Using the second stage of two-stage budgeting, I interpret η as the ratio
5
of within-period expenditure elasticities of nondurables and services flow. Empirically, I find
η < 1 and therefore durables are luxury goods and nondurables necessary goods. Homothetic
preferences correspond to η = 1. On the technical side, it is also to be noted that this preference specification easily accommodates variable within-period expenditure elasticities of both
goods as long as their ratio remains constant.
I interpret the parameter θ as a yardstick of the intratemporal substitution, which is precisely
measured by the elasticity of intratemporal substitution, denoted ES. The two, however, are
related by the equation from Appendix C, ES = θ + ε∗dd × (η − 1), where ε∗dd is the Hicksian
own-price elasticity of the demand for the service flow from durable goods. I shall discuss this
point deeper hereafter.
Consumers accumulate the stock of durables dt dictated by the following law of motion
dt = (1 − δ) dt−1 + it
(3)
where it denotes durable goods investment, and δ is the depreciation rate. The flow of services
st is produced by a linear household production function (Stigler and Becker 1977, Mankiw
1985), which is time- and state-independent
st = A × dt
(4)
The budget constraint is standard and is not displayed here.
I take a monotonic transformation of the utility function4 and reparametrize the consumption index as
θ
1− 1θ
1− ηθ θ−1
C := C (ct , dt ) = ct
+ a dt
where I denote a = ã A1−η/θ / (1 − ã).
6
(5)
2.2
First-Order Conditions
The intertemporal marginal rate of substitution mt+1 is given by
mt+1
β Uc (ct+1 , dt+1 )
:=
= β
Uc (ct , dt )
ct+1
ct
where ft is defined as
−1/σ ft+1
ft
−
θ−σ
σ(θ−1)
(6)
1− ηθ
ft = 1 +
a dt
1− θ1
(7)
ct
The intertemporal first-order condition is the standard Euler equation
1 = Et [mt+1 Rt+1 ]
(8)
where Rt+1 is the (gross) return on test asset(s). The intratemporal first-order condition states
that the marginal rate of substitution between durables and nondurables equals the rental cost
Ud (ct , dt )
= qt − (1 − δ) Et {mt+1 qt+1 }
Uc (ct , dt )
(9)
Intuitively, suppose we purchase one unit of durables at price qt , which after one period depreciates to 1 − δ. We can sell it for (1 − δ) qt+1 . The implicit rental cost rct is the net present
value of this transaction, that is,
rct := qt − (1 − δ) Et {mt+1 qt+1 }
(10)
Divide the intratemporal first-order condition (9) by qt to obtain the following
Ud (ct , dt )
qt+1
= 1 − (1 − δ) Et mt+1
qt Uc (ct , dt )
qt
7
(11)
After some algebra (see Appendix B), imposing (at this point the assumed) stationarity of the
right-hand side, I obtain
log(ct ) = intercept + θ log(qt ) + η log(dt ) + εt
(12)
If the variables log(ct ), log(dt ) and log(qt ) are co-integrated, I can estimate the preference parameters θ and η superconsistently as the cointegrating vector by running the above regression
in levels.
2.3
Interpretation of the Parameters θ and η
The preference parameters θ and η are easily interpreted in case of deterministic setup using
the second stage of two-stage budgeting, which holds due to the weak separability of the consumer preferences (Deaton and Muellbauer 1980). I here summarize the results and refer the
interested reader to Appendix C for formal derivations.
Each period the consumer optimally chooses his consumption expenditure et . Then, depending
on the rental cost of consumer durables he chooses his allocation of nondurable consumption
and the service flow from renting consumer durables. The elasticity of intratemporal substitution ES - substitution across goods within a given time-period holding the real income constant,
gauges how much the relative Hicksian demand for durables changes in response to a percent
change in the rental cost of consumer durables rc. In the model, its yardstick, but not a precise
measure, is the parameter θ. The two are related by the equation ES = θ + ε∗dd × (η − 1),
where ε∗dd is the Hicksian own-price elasticity of the demand for the service flow of durable
goods. Furthermore, the expenditure elasticity measures how much the demand for a consumption good changes in a response to a 1% rise in expenditures, ceteris paribus. For future
reference, I denote the expenditure elasticity of nondurable consumption ηc and that of the
service flow ηd . It turns out that the parameter η is a ratio of these two elasticities, namely,
η = ηc / ηd . That these claims hold true may be easily verified by comparing the slopes in the
8
cointegrating regressions based on (i) the intratemporal first-order condition (12), and (ii) the
first-order condition from the second stage of two-stage budgeting (53), derived in Appendix C.
When both income and substitution effects are present, the relative demand function takes
the form
d log (ct / dt )
d log rct
= |ES × {z
}
+
substitution ef f ect
(η − ηd ) × d log êt
| c
{z
}
(13)
income ef f ect
The parameter êt denotes the real expenditure on both consumption goods, measured in terms
of the composite good (see also Appendix C). The equation (13) shows that the relative demand
changes are either due to substitution effect or due to income effect. It offers a framework to
understand the secular rise in the consumption of durables relative to nondurables. The typical
interpretation is that consumers substituted (in the sense of Hicks) to durable goods in response
to their falling relative price, with income effects playing no role at all. This corresponds to the
case where elasticity of substitution ES is large5 and the preferences are homothetic, which
automatically eliminates the income effect. Formally,
d log (ct /dt ) = ES × d log rct
(14)
As the empirical results in later sections indicate, after correcting for the income effects, the
substitutability ES between the services flow and nondurables is smaller than estimated by
Hansen and Eichenbaum (1990) and Ogaki and Reinhart (1998), and hence equation (13) dictates significant nonhomotheticity. Formally, (ηc − ηd ) × d log êt is not negligible, and thus
affects the estimates of both the intra- and inter-temporal substitutions.
The real income in the U.S. economy has been rising steadily and thus the previous equation suggests that ηc − ηd < 0. Because the average income elasticity must be one, we get
the plausible result that durables are luxury goods and nondurables are necessary goods, i.e.
9
ηc < 1 < ηd . Empirically, I estimate6 ηd ∈ [1.25, 1.27] and ηc ∈ [0.95, 0.97]. For comparison, Costa (2001) estimates the income elasticities for food at home 0.47 in 1960-94 and 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 (which I also count as nondurable goods) are 1.37 in 1972-94, 1.41 in 1917-35,
and 1.82 in 1888-1917.
3
3.1
Empirical Section
Consumption and Asset Return Data
Quarterly consumption data are from the U.S. national accounts as available from the Federal
Reserve Bank of St. Louis. I measure nondurable consumption ct as the sum of real personal
consumption expenditures (PCE) on nondurable goods and services. The measure of the stock
of consumer durables dt is constructed iteratively using the real PCE data on durable goods
expenditures it as follows
dt+1 = (1 − δ) dt + it
(15)
with the depreciation rate δ = 6% per quarter. This corresponds to about 21.9% per annum,
which is consistent with Wykoff (1970) estimates of a depreciation rate from resale values of
automobiles. Ogaki and Reinhart (1998) use value of 22% per annum. As the initial stock
of consumer durables d1947:Q1 is not known, my sample period is 1956:Q1 - 2001:Q4. This
restriction implies that the unknown stock of consumer durables d1947:Q1 has depreciated from
100% to about 8.5% (= 0.9440 ), and hence is negligible. Furthermore, I construct the durables
price as the ratio of the price index for PCE on consumer durables over the price index for
PCE on nondurable goods and services. All consumption data are converted to per-capita
basis by dividing by the population size at the end of the quarter.
The asset return data is the monthly return on the U.S. Treasury Bills, converted to quarterly
values. I thank Kenneth French for providing these data on his web page. The real return
was computed by deflating the nominal return with the consumer price index (CPI), obtained
10
from the web page of the Federal Reserve Bank of St. Louis.
3.2
Estimating the Measures of Intratemporal Substitution θ and Nonhomotheticity η: Cointegration Approach
3.2.1
Methodology
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 and including a
constant and a linear time trend, I cannot reject the hypothesis that the data are difference
stationary. Table 1 summarizes the results. Therefore, the marginal rate of substitution mt+1
and the ratio qt+1 / qt are stationary. Because it is reasonable to assume that the conditional
expectation of a stationary variable is itself stationary, I assume that the conditional expectation Et {mt+1 qt+1 / qt } is stationary.
The theoretical section derives the precise log-linear intratemporal first-order condition as
log(ct ) = intercept + θ log(qt ) + η log(dt ) + εt
(16)
where εt is the cointegrating regression disturbace term. If the variables log(ct ), log(dt ) and
log(qt ) are cointegrated, I can estimate the preference parameters θ and η super-consistently
as the cointegrating vector by running the above regression in levels. This follows Ogaki and
Reinhart (1998) who estimate the elasticity of intratemporal substitution. They focus on the
homothetic case η = 1, in which case ES = θ, and their regression is
log(ct ) − log(dt ) = intercept + θ log(qt ) + εt
(17)
The presence of significant nonhomotheticity may bias the yardstick of the intratemporal substitution θ and the elasticity ES itself. In fact, imposing the homotheticity assumption in
the case of durable goods biases upward these parameters. Intuitively, the relative demand
11
may change either due to income effect or due to substitution effect. Homotheticity dictates
that it was Hicksian substitution in response to a secular change in the relative price that
led consumers to purchase more durable goods. Neglecting income effects forces the substitution effects to be larger to fit the secular trend in the consumption of durables relative
to nondurables. In general, considering income effects may be important, in particular for
goods with large expenditure shares and/or no easy substitutes such as housing or durables, as
Slutsky equation says that the income effects are proportional to the expenditure shares. Empirically, Eichenbaum and Hansen (1990) use CES aggregator over nondurables and durables
but impose homotheticity. They estimate both intratemporal and intertemporal equations
jointly using GMM and get the elasticity of intratemporal substitution ES = 0.91. Using the
same preference specification but estimating the elasticity of substitution using cointegrating
regression, Ogaki and Reinhart (1998) cannot reject the null hypothesis H0 : θ > 1. As argued
before, neglecting income effects biases upward the estimate of the elasticity of substitution
ES, and thus the parameter θ. This observation is forcefully confirmed empirically hereafter.
I test for cointegration using the likelihood ratio test of Johansen (1988, 1991). In detail,
I use VAR(1) for likelihood ratio test, suggested by Akaike Vector Information Criterion, and
AR(5), suggested by both Bayesian Information Criterion and Schwartz Criterion, for the
cointegrating residual to create confidence intervals and t-statistics. The eigenvalues of the
Johansen matrix are λ̂1 = 0.13, λ̂2 = 0.06 and λ̂3 = 0.001, with
= 25.49
− T × log 1 − λ̂1
= 11.32
− T × log 1 − λ̂1
= 0.18
− T × log 1 − λ̂1
(18)
(19)
(20)
The likelihood ratio test of the null hypothesis of h = 0 cointegrating relations against the
alternative of h = 3 cointegrating relations is LR = 25.49 + 11.32 + 0.18 = 35.99. Since
35.99 > 31.26, the 5% critical value, the null hypothesis of no cointegration is rejected at
12
the 5% level. In order to exclude the possibility of h = 2 and h = 3 cointegrating relations,
I perform additional statistical tests. That is, I consider the likelihood ratio test of the null
hypothesis of h = 1 co-integrating relation against the alternative of h = 3 cointegrating relations: LR = 11.32 + 0.18 = 11.50. Since 11.50 < 17.84, the 5% critical value, I cannot reject
the null hypothesis of a single cointegrating relation. Furthermore, I perform the likelihood
ratio test of the null hypothesis of h = 1 cointegrating relation against the alternative of h = 2
cointegrating relations: LR = 11.32. Since 11.32 < 14.60, the 5% critical value, I again cannot
reject the hypothesis of a single cointegrating relation.
Stock and Watson (1993) and Wooldridge (1991) suggest to augment the regression (16) with
leads and lags of the right hand side variables to correct for the correlation between the innovations in dt and qt and the cointegrating residual εt . This is important for the construction
of confidence intervals and hypothesis testing. I therefore estimate such a regression with the
number of leads/lags suggested by Akaike Vector Information Criterion N = 1. The sample
period is 1956:Q1 - 2001:Q4.
3.2.2
Interpretation of the Empirical Results
Table 2, row 2, reports the point estimates of the yardstick of the intratemporal substitution
θ̂ = 0.36 with s.e.(θ̂) = 0.05. The estimate has the correct sign and is statistically significant.
However, we have to be cautious at this point and anticipate some results as Appendix C shows
that θ is not a precise measure of intratemporal substitution. In fact, that is the elasticity
of intratemporal substitution, denoted ES, and the two are related (see Appendix C and D)
by means of ES = θ + ε∗dd × (η − 1). Based on the analysis presented in Appendix D, I
estimate7 ES = 0.46, marginally higher than θ. Compared to previous studies, the economic
magnitude is economically significantly smaller. I attribute the difference to the simultaneous
modeling of income and substitution effects in the relative demand function for durables.
Small intratemporal substitution has several implications. Firstly, it indirectly affects the
13
estimate of the intertemporal substitutability. As real interest rates rise, the rental cost of
consumer durables rises but consumers substitute less from durables to nondurables compared8 to Hansen and Eichenbaum’s (1990) and Ogaki and Reinhart’s (1998) world. Mankiw
(1982) estimates that the IES for pure durables is economically and statistically significant,
in contrast to Hall (1988) who estimates the EIS for nondurable goods to be economically
negligible. This suggests that if consumer durables have relatively small budget share in the
total consumption expenditure, which they do, the low intratemporal substitutability coupled
with the negligible IES for (only) nondurables should yield relatively small, if not zero, IES
for the total consumption expenditure. Empirically, the budget share of durable goods is on
the order of 10% - 14% and declines secularly in spite of the fact that they are luxury goods
(see the discussion later in the paper). We therefore, in addition, expect the IES to go down
over time precisely because (i) nondurables have negligible IES, (ii) durable goods have high
IES, and (iii) the budget share of consumer durables sd,t declines secularly. This conclusion is
strongly confirmed in the data in the subsequent section. Secondly, the intratemporal substitutability, in addition to the coefficient of risk aversion, controls the variability of the marginal
rate of substitution and thus has asset pricing ramifications. In a related paper, Pakoš (2005)
empirically evaluates the ability of the presented model to explain the equity premium puzzle.
Table 2, row 3, reports the point estimates of the ratio of the expenditure elasticities η =
ηc / ηd , where ηc is the expenditure elasticity of nondurable consumption and services, and
ηd is the expenditure elasticity of the service flow from the stock of consumer durables. The
superconsistent estimate is η̂ = 0.76 with s.e.(η̂) = 0.03. The estimate is statistically and
economically significant and has a positive sign, in light of which both nondurables and service
flow from durables are normal goods. Furthermore, homotheticity corresponds to η = 1. I
formally test this restriction; the t-statistics t = −8.42, and I thus reject the null hypothesis
of homothetic preferences in favor of nonhomotheticity at 1% significance level.
14
3.2.3
Robustness Check: Bootstrapping the Cointegrating Regression
It is well-known that the cointegrating vector (1, −θ, −η) is super-consistent but is biased.
This is important especially in smaller samples. I therefore apply the sieve bootstrap to
the cointegrating regression (Chang, Park and Song 2005), and construct percentile confidence
intervals. The empirical distributions (histograms) based on 1,000,000 Monte Carlo simulations
are displayed in Figure 2. The mean of the distribution for the intratemporal substitutability
θ is 0.34, only marginally lower than the point estimate in Table 2. In fact, the 5% confidence
interval is CIθ = [0.12, 0.56], wider than the asymptotic confidence interval. Furthermore,
the mean of the distribution for the nonhomotheticity measure η is 0.75, also only marginally
lower than the point estimate in Table 2. The 5% confidence interval is CIη = [0.63, 0.88],
wider than the asymptotic confidence interval. As before, we again are unable to accept
the hypothesis of homotheticity at the 5% level. In summary, the point estimates from the
cointegrating regressions are slightly biased upward relative to the mean of the bootstrapped
empirical distribution, and the asymptotic confidence intervals underestimate the sampling
variability of the parameters θ and η.
3.3
Estimation of the Rest of the Parameter Vector: Euler Equations Approach
3.3.1
Methodology
My approach to estimation and inference of the rest of the preference parameter vector p =
(σ, β, a) follows that of Dunn and Singleton (1986), Eichenbaum, Hansen and Singleton (1988),
and Ogaki and Reinhart (1998). These authors show how to modify the analysis of Hansen
and Singleton (1982) to allow for multiple consumption goods. The primary testable asset
pricing implications of the model are the set of intertemporal Euler equations
Et [mt+1 Ri,t+1 ] = 1
15
(21)
where mt+1 is the intertemporal marginal rate of substitution,
mt+1 = β Uc (ct+1 , dt+1 ) / Uc (ct , dt )
(22)
and Ri,t+1 is the gross return on an asset i.
In addition, we have the intratemporal first-order condition (9). Recall from the previous
section that the cointegrating regression yields superconsistent estimates for parameters θ and
η but does not pin down the preference weight a that enters the formula (9). Therefore, in
order for all parameters σ, β and, in particular, a to be identified, it is essential to include the
intratemporal first-order condition in the GMM estimation9 , that is, to include
Et {mt+1 (1 − δ) (qt+1 / qt ) + Ud (ct , dt ) / (qt Uc (ct , dt ))} = 1
(23)
Let zt be a vector of variables in the consumers’ information set at time t. Using the components
of zt as instruments, I form the vector function







mt+1 Ri,t+1 − 1


gT (p) := ET 
⊗
z

t


 mt+1 (1 − δ) (qt+1 / qt ) + Ud (ct , dt ) / (qt Uc (ct , dt )) − 1

and use the Hansen’s (1982) notation ET = T −1
PT
t=1 .
I calibrate the parameters θ and η
using the super-consistent estimates obtained by the cointegration argument as θ̂ = 0.36 and
η̂ = 0.76. I estimate the rest of the preference parameter vector p̂ by the choice of p that
makes gT (p) close to zero in the sense of minimizing the quadratic form
min gT′ (p) WT gT (p)
p
where WT is a symmetric positive definite distance matrix. Hansen (1982) shows how to choose
WT optimally so that it minimizes the asymptotic covariance matrix of p̂. I follow Hansen,
Heaton and Yaron (1996) and use the continuous-updating GMM.
16
In my analysis I consider several instrumental variables. First, I use the lagged nondurable
consumption growth rate, the lagged growth rate of the stock of durables and the lagged growth
rate in the durables price. Secondly, a natural choice for the instrumental variables are both
the growth rate of the number of civilians unemployed less than 5 weeks and the growth rate
of the number of civilians unemployed more or equal 15 weeks. Of course, the instruments are
lagged twice to take care of the time aggregation problem. Finally, I also use the lagged real
return on the U.S. Treasury Bills as an instrument.
3.3.2
Interpretation of the Empirical Results
Table 3 summarizes the GMM output for 4 different sets of instruments when the test asset
is the U.S. Treasury Bill. The subjective discount factor β is estimated quite precisely and
marginally above one. Kocherlakota (1990) shows that β > 1 is still consistent with equilibrium if there is a growth in the economy. The preference weight a for the service flow from
consumer durables is estimated also precisely and marginally above 10 (see also Appendix A).
The preference parameter σ is estimated from 0.043 up to 0.060, and significantly different
from zero. The JT statistics does not reject the model at the 5% significance level irrespective
of the choice of instrumental variables.
3.3.3
Robustness Check: Weak Identification of Parameters
It is well-known that asymptotic normality may provide a rather poor approximation to the
sampling distribution of GMM estimators (Tauchen 1986, Kocherlakota 1990, Neeley 1994,
West and Wilcox 1994, Hansen, Heaton and Yaron 1996). For example, the sampling distribution of GMM estimators can be skewed and can have heavy tails, and tests of overidentifying
restrictions can exhibit substantial size distortions. One source of these problems as suggested
by Stock and Wright (2000) may be weak identification of parameters, that is to say, the instruments are only weakly correlated with the relevant first-order condition so that the parameters
are poorly identified. To check the robustness of my results, I follow Stock and Wright (2000)
17
who develop nonstandard asymptotic approximations to the distributions of GMM estimators
when some or all parameters are weakly identified. The reason is that the nonstandard asymptotic approximations generally are, in contrast to the common normal approximations, found
to match well the finite sample distributions.
A common symptom of weak identification is that the point estimates vary across different
sets of instruments, and/or the estimated standard errors are extreme. As Table 3 indicates,
none of the preference parameters σ, β or a exhibits this behavior. However, as a robustness
check, I treat (σ, a) as weakly identified, and the subjective discount factor β as strongly identified. Theorem 3 [Stock and Wright (2000), p. 1063] enables us to construct a confidence
set for the weakly identified parameters (σ, a) directly by inverting the objective function;
these sets are called S-sets. These sets consist of parameter values (σ, a) at which one fails to
reject the joint hypothesis that the parameters equal the true values and the over-identifying
restriction holds. Table 3, columns 8 and 9, display the one-dimensional 95% S-set for σ and a,
obtained as the projections of the joint 2-dimensional 95% S-set for (σ, a) onto the respective
axes. It appears that neither σ nor a are weakly identified as the respective S-sets are narrower
than the asymptotic confidence intervals. Figures 3, 4 and 5 portray the 2-dimensional S-sets
graphically along with the asymptotic 95% confidence ellipses.
4
The Budget Share, the Rental Cost and the Expenditure
Elasticity of Durable Goods
Tables 2 (column 2) and 3 (row 4) allow us to construct the time series of (i) the implied rental
cost of consumer durables rct , (ii) the budget share of durable goods sd,t in total expenditures
et = ct + rct × dt , and (iii) the nondurables and the durables expenditure elasticities ηc,t and
ηd,t . First, I construct the implied rental cost of durables as the intratemporal marginal rate
18
of substitution between the service flow from durable goods and nondurable goods,
rct =
Ud (ct , dt )
Uc (ct , dt )
(24)
It is displayed in Figure 7 as a dotted line. Empirically, it turns out that the rental cost is
about 6% - 8% of the durable goods price per quarter (26%-36% annually), which appears
to be a plausible magnitude. Furthermore, it is not surprising to find that the ratio rct / qt
is stationary10 , and hence both the real rental cost rct and the real durables price qt decline
secularly in the post-war U.S. economy (see also Figure 1). This has an essential implication
for the budget share of durable goods sd,t , which in fact goes down despite the fact that durable
goods are luxury goods. Formally, one may easily show by log-differentiating the formula for
the budget share sd,t that
d log sd,t = (1 + εdd,t ) × d log rct + (ηd,t − 1) × d log et
(25)
Back of the envelope calculation, along with the Slutsky equation εdd,t = ε∗dd,t − sd,t × ηd,t ,
suggest that indeed the budget share of durable goods declines over time. In detail, the Marshallian price elasticity εdd,t is small and negative as (i) the Hicksian price elasticity ε∗dd,t is
small (i.e. no easy substitutes for durable goods) and negative11 , (ii) the goods are normal
with expenditure elasticities positive, and (iii) the budget share of durable goods is relatively
small; hence, the term (1 + εdd,t ) ≈ 1. The term (ηd,t − 1) ≈ 0.26, which is negligible relative to one, 0.26 ≪ 1. This suggests, and Figure 7 (thick line) confirms empirically, that
despite the nonhomotheticity and the secular increase in the total consumption expenditures
(i.e. d log et > 0), the budget share of durable goods goes down precisely because the rental
cost of durables declines so steeply, that is, d log rct < 0.
The intraperiod budget constraint implies that the weighted average of expenditure elasticities
must be one
sc,t × ηc,t + sd,t × ηd,t = 1
19
(26)
where sc,t and sd,t are the shares of nondurables and durables, respectively, in total withinperiod consumption expenditures et = ct + rct × dt . I find that ηd ∈ [1.25, 1.27] and
ηc ∈ [0.95, 0.97], see also Figure 8. Clearly, durables are luxury goods and nondurables are
necessary goods.
5
The Intertemporal Elasticity of Substitution IES
The elasticity of intertemporal substitution IES is an essential input into many dynamic
macroeconomic models (Hall 1988). Because the preference specification features nonhomotheticity, the elasticity of intertemporal substitution does not equal the parameter12 σ. In fact,
what we are interested in is how much the total consumption expenditure, or the saving, if
you like, changes in response to predictable changes in the real interest rate. Appendix C
shows that, under deterministic setup, the intratemporal first-order condition is a solution of
the second-stage of two-stage budgeting, and the indirect felicity function V (e, rc) is the value
function of the following concave program
V (e, rc) =
max
{(c, d) ∈ R2+ }
U [C (c, d)]
(27)
subject to the budget constraint
c + rc × d ≤ e
(28)
where e is the total consumption expenditure within the period and rc is the rental cost of
consumer durables.
Following Atkeson and Ogaki (1996), I define13 the intertemporal elasticity of substitution
IES as the inverse of the elasticity of the marginal utility Ve (e, rc) of the total consumption
expenditures e,
IES = −
20
∂ log Ve (e, rc)
∂ log e
−1
= −
Ve (e, rc)
e × Vee (e, rc)
(29)
Intuitively, suppose the elasticity of the marginal utility Ve of the total consumption expenditures e is large. That means that a small change in e leads to a large change in the associated
marginal utility Ve . Because consumers strive to spread their consumption expenditures e over
time, depending on their forecasts of the real interest rate, in order to maximize their welfare,
they will alter their spending plans minimally; otherwise, there will be a dramatic variation in
marginal utility Ve across time that cannot be possibly optimal. As a result, the intertemporal
elasticity of substitution IES is small; that is, a predictable change in the real interest rate will
give occasion to a relatively small change in the consumption expenditures E.
Figure 9 portrays the time-series of the IES (thick line) along with the 95% confidence bounds
(dotted lines) for the preference parameter calibrations based on the super-consistent estimates
from the intratemporal first-order condition and the GMM estimates from Table 3, row 414 .
As may be observed, the IES is economically negligible and in fact declines marginally from
about 0.061 in 1956:Q1 down to 0.06 in 2001:Q4. The economically inconsequential magnitude
of the IES for the total consumption expenditures makes intuitive sense. In fact, the budget
share of durable goods is only about 10% - 15%. As the bulk of consumption is composed predominantly of nondurable goods with practically zero IES (Hall 1988), it comes as no surprise
to discover that considering durable goods that themselves in fact do have high IES (Mankiw
1982) does not raise the overall IES for the total consumption. The time-variation in the IES is
a direct consequence of the preference nonhomotheticity; the IES slopes down due to declining
budget share of durable goods.
6
Conclusion
This paper argues that ignoring the income effects (nonhomotheticity) in the relative demand
function for durables strikingly biases the estimates of the magnitude of the intertemporal and
intratemporal substitutions. When I correct for nonhomotheticity, I find the magnitude of the
elasticity of intertemporal substitution IES to be negligible, on the order of magnitude 0.06.
21
In addition, I find compelling evidence against the separability across consumption goods in
the felicity function. However, in contrast to Eichenbaum and Hansen’s (1990) value of 0.91
and Ogaki and Reinhart’s (1998) value greater than one, my estimate of the elasticity of
intratemporal substitution, after careful correction for nonhomotheticity, is around 0.46. As
suggested by Pakoš (2005), weak consumption substitutability across nondurable and durable
goods may have essential implications for the asset risk premia and perhaps even lead to
a resolution of the equity premium puzzle. Last, I have found strong evidence in favor of
nonhomotheticity in the aggregate consumption data. Nondurable consumption is a necessary
good and the service flow from durables is a luxury good, with the within-period expenditure
elasticity greater than one. This is consistent with the exciting results in Ogaki (1992) and
Costa (2001).
22
Acknowledgements
This paper is an outgrowth of my Ph.D. Dissertation at the Graduate School of Business, University of Chicago. I am grateful for the advice and encouragement from my thesis committee:
John Cochrane, John Heaton (chair), Ľuboš Pástor, Monika Piazzesi, and Pietro Veronesi.
My thanks also go to Gary Becker and Kevin Murphy from whom I have learned so much.
In addition, I have benefitted from helpful comments by Peter Bossaerts, John Campbell,
George Constantinides, George-Levi Gayle, Rick Green, Lior Menzly, Masao Ogaki, Jonathan
Parker, Lukasz Pomorski, Ruy Ribeiro, Nick Roussanov, Sergei Sarkissian, Fallaw Sowell, Chris
Telmer, Stan Zin, and the Editor, the anonymous Associate Editor in charge and two anonymous referees. Research support from the Oscar Mayer Foundation is gratefully acknowledged;
any opinions expressed herein are the author’s and not necessarily those of the Oscar Mayer
Foundation.
23
A
Identification and Back of the Envelope Calculation of the
Plausible Values of the Preference Parameter a
In the empirical part, I reparametrize the consumption index so that all the parameters are identified. For
example, we cannot separately identify the preference weight ã and the coefficient A from the household production function (see equation 4). In detail, suppose we keep the parametrization of the consumption index
as
n
o θ
η
1
θ−1
C (c, d) = (1 − ã) c1− θ + ã (A × d)1− θ
(30)
Clearly, ã ∈ (0, 1) and A ∈ (0, 1]. ”Taking out” the number 1 − ã from the consumption index, and rearranging
the terms yields
θ
C (c, d) = (1 − ã) θ−1
(
η
1
c1− θ +
ã A1− θ 1− ηθ
d
1 − ã
)
θ
θ−1
(31)
Because applying a strictly increasing operator (i.e. multiplying by a positive number in our case) does not
θ
change the preference ordering, I may drop the term (1 − ã) θ−1 . In addition, I denote
η
a =
ã A1− θ
1 − ã
(32)
to obtain
n
o θ
η
1
θ−1
C (c, d) = c1− θ + a × d1− θ
,
(33)
the preference specification that I actually use in the GMM estimation.
There is a deep reason for using this ”simpler” consumption index (33). In fact, as may be easily inferred
from the analysis above, the indices (30) and (33) are observationally equivalent and generate exactly the same
preference orderings. However, as econometricians, we clearly cannot separately identify the preference weight
ã and the household production function coefficient A; only the parameter a is econometrically identified. Unfortunately, a hurried reader tends to have a strong prior on the plausible magnitudes of the parameter a, which
seemingly looks like a preference weight, and thus expects magnitudes a ∈ (0, 1). That this is not necessarily
so I shall show hereafter. Specifically, I shall show that a ∈ [0.60, 1, 463.00] is still quite a plausible magnitude,
consistent with a reasonable preference weight ã and a household production function parameter A that determines the service flow from consumer durables!
Back of the envelope calculation suggests, using the super-consistent estimates θ = 0.36 and η = 0.76, that
”
”
“
“
ã
0.76
ã
= 0.95
= 0.05
= 19, minã ∈ [0.05, 0.95] 1−ã
≈ 0.05, 1 − ηθ = 1 − 0.36
≈ −1.11,
maxã ∈ [0.05, 0.95] 1−ã
0.05
0.95
”
“
”
“
η
η
−1.11
1− θ
−1.11
1− θ
= 0.02
≈ 77, and minA ∈ [0.02, 0.10] A
= 0.10
≈ 12. In other words,
maxA ∈ [0.02, 0.10] A
24
we obtain the following plausible, but admittedly surprising even for the author, bounds
η
0.60 ≈
ã A1− θ
1 − ã
min
ã ∈ [0.05, 0.95], A ∈ [0.02, 0.10]
!
η
≤ a ≤
max
ã ∈ [0.05, 0.95], A ∈ [0.02, 0.10]
ã A1− θ
1 − ã
!
≈ 1, 463.00
(34)
I choose ã ∈ [0.05, 0.95] as an economically plausible magnitude for the true preference weight. I justify
A ∈ [2%, 10%] as follows. It is reasonable to assume that the service flow from consumer durables as measured
by the (unnormalized) parameter A is related to the depreciation rate δ = 6% per quarter. In fact, the depreciation rate of 6% tells us that in a sense we use up 6% of consumer durables, which economically means that
the service flow from consumer durables is 6% of the stock; that is, A should be equal δ. I allow more generous
bounds and choose 2% ≤ A ≤ 10%.
As a result, in my GMM estimation (routine fmincon in M atlabT M ), I set the parameter space Θ for the
preference vector (σ, β, a) to be the compact set Θ = [0.01, 20.00] × [0.80, 1.20] × [0.60, 1, 463.00]. These
bounds turn out not to be binding in the GMM estimation.
B
Derivation of the Cointegrating Regression
The utility function is
U [C(ct , dt )] =
1
1 − 1/σ
„
1
1− θ
ct
η
1− θ
+ a dt
«
θ (1−1/σ)
θ−1
.
(35)
The intratemporal marginal rate of substitution is
−
η
a dt θ
Ud (t)
=
−1
Uc (t)
ct θ
Dividing by the durables price yields
−η
a dt θ
1 Ud (ct , dt )
=
−1
qt Uc (ct , dt )
qt ct θ
The left-hand side of this equation is stationary and positive as the statistics of the empirical part convincingly
demonstrate. Let us denote it as Const × eǫt . Then, we may rewrite the equation as
η
−θ
a dt
−1
qt ct θ
= Const × eǫt
Taking logs, and multiplying by θ yields
log(ct ) − η log(dt ) − θ log(qt ) + θ ln (a) = θ log(Const) + θ ǫt
25
Finally, re-arranging the terms, denoting intercept ≡ θ log(Const) − θ ln (a) and εt ≡ θ ǫt gives the desired
result. Observe that the cointegrating regression does not pin down the parameter a, but only θ and η. Therefore,
we will have to add the intratemporal first-order condition, calibrated with the superconsistent estimates of θ
and η, to the intertemporal Euler equation in the GMM estimation. This will guarantee that the parameter a
is strongly identified.
C
Interpretation of the Preference Parameters in a Deterministic Setup
The preference parameters θ and η are most easily interpreted in the deterministic setup. Let us think of
consumers as renting their stock of durables dt in a perfect rental market, with the rental cost rct given by the
right-hand side of equation (9), namely,
rct = qt − (1 − δ) Et {mt+1 qt+1 }
(36)
In a deterministic setup, the risk-free interest rate 1 + rtf = m−1
t+1 and the rental cost of capital satisfies
rct = qt − (1 − δ) (1 + rtf )−1 qt+1 . Viewed this way, the preferences over nondurables and durables are weakly
separable. Weak separability is a necessary and a sufficient condition for the second-stage of two-stage budgeting
to hold (Deaton and Muellbauer 1980). Intuitively, suppose the consumer has already chosen his optimal withinperiod t total consumption expenditure et , expressed in terms of nondurable consumption, which is a numeraire
throughout unless stated otherwise. The level of the nondurable consumption ct and the stock of durable goods
to be rented dt are then chosen so that the second-stage optimization holds, formally,
(ct , dt ) = arg
h “
”i
max
U C c̃t , d˜t
{(c̃t , d˜t ) ∈ R2
+}
(37)
subject to the budget constraint
c̃t + rct × d˜t ≤ et
(38)
Note that the expenditures et in other periods t are unaffected and the consumer thereby maximizes his lifetime
well-being.
The first-order condition associated with the second stage is helpful to interpret the preference parameters.
26
The Marshallian demands are functions of the relative price rct and the expenditure et ,
ct = c (rct , et )
(39)
dt = d (rct , et )
(40)
d log ct = εcd × d log rct + ηc × d log et
(41)
d log dt = εdd × d log rct + ηd × d log et
(42)
Log-differentiating yields
The parameters ηc and ηd are the expenditure elasticities associated with the within-period t expenditure level
et . They tell us how much the demands ct and dt change (in percentage terms) in response to a 1% rise in
the total within-period t expenditure level et , ceteris paribus. Formally, they are defined as percentage changes
in the Marshallian demands in response to a percentage change in expenditures, ηc = ∂ log ct / ∂ log et and
ηd = ∂ log dt / ∂ log et . The budget constraint (38) implies that the weighted average of these elasticities must
be one. The parameters εcd and εdd are Marshallian price elasticities. Subtracting one equation from the other,
using Slutsky equation εij = ε∗ij − ηi sj , where sj is the share of good j ∈ {c, d} in expenditures et , and ε∗ij is
the compensated Hicksian price elasticity, implies that the relative demand function satisfies
d log (ct / dt )
=
ES × d log rct
{z
}
|
substitution ef f ect
+
(ηc − ηd ) × d log êt
|
{z
}
(43)
income ef f ect
where ES ≡ ε∗cd − ε∗dd denotes the elasticity of intratemporal substitution (see the discussion below). The
variable et is the expenditure on both consumption goods, expressed in terms of nondurable goods whereas
what I call the real expenditure êt is expressed in terms of the composite good, with the implicitly defined price
index. In fact, it is defined as d log êt = d log et − sd d log rct . The appropriate price index is pt = 1sc × rcst d ,
and hence the real expenditure satisfies d log êt = d log (et / pt ).
The elasticity of substitution ES is defined as a percentage change in the relative Hicksian demand in response
to a percentage change in the relative price,
ES = ∂ log(c∗t /d∗t ) / ∂ log rct = ε∗cd − ε∗dd
(44)
and it is a measure of the concavity of the indifference curves. For instance, ES = 0 for Leontief preferences
and thus the indifference curves are extremely concave. It may be shown by combining the equations in this
27
appendix that
ES = θ + ε∗dd × (η − 1)
(45)
where in anticipation of a future result, I define
θ
=
(ε∗cd − ε∗dd η)
(46)
η
=
ηc / ηd .
(47)
Based on a back of the envelope calculation, the parameter θ underestimates the true elasticity of intratemporal
substitution ES. That is, ε∗dd is non-positive because the substitution matrix is negative semidefinite, and the
parameter η is less than one, empirically. As a result,
ES ≥ θ
(48)
but I still occasionally refer to θ as a yardstick of intratemporal substitutability.
I now derive the deterministic equivalent of the intratemporal first-order condition (16). Eliminate the expenditure et in the system of the Marshallian demands to get the conditional demand
d log ct = (ε∗cd − ε∗dd η) × d log rct +
„
ηc
ηd
«
× d log dt
(49)
or, using the definition of θ and η in equations (46) and (47),
d log ct = θ × d log rct + η × d log dt
(50)
One may integrate the equation - this is perfectly consistent with the model which is log-linear (see the intratemporal condition in the main text) - and hence the parameters are constant, to obtain
log ct = intercept + θ × log rct + η × log dt
(51)
Let us link the rental cost of durables rct to the price of durables qt
ff«

„
qt+1
log rct = log qt + log 1 − Et mt+1
qt
(52)
where the second term on the right hand side is assumed to be stationary to obtain the intratemporal first-order
condition
log ct = intercept + θ × log qt + η × log dt + error
If stochastic setting is considered explicitly, all stochastic terms go into the error term.
28
(53)
D
Details of the Estimation of the Elasticity of Intratemporal
Substitution ES
The Marshallian demand function for durable goods (see Appendix C) implies the result
d log dt = εdd × d log rct + ηd × d log et ,
(54)
that allows to estimate the Marshallian price elasticity εdd,t as
εdd,t =
(1 − L) log dt − ηd,t × (1 − L) log et
(1 − L) log rct
(55)
where L is the lag operator. Then, I use Slutsky equation ε∗dd,t = εdd,t + ηd,t × sd,t to compute the own
Hicksian own-price elasticity of durable goods. Finally, using the super-consistent estimates of θ and η, and the
equation
ESt = θ + ε∗dd,t × (η − 1)
(56)
from Appendix C enables the computation of the time-series of the elasticity of intratemporal substitution ESt .
E
The Value Function V (e, rc) of the Second-Stage of the TwoStage Budgeting
In order to find the intertemporal elasticity of substitution IES, one needs first to compute the value function
V (e, rc), that is, to solve the following optimization problem
V (et , rct ) =
max
{(ct , dt ) ∈ R2
+}
U [C (ct , dt )]
(57)
subject to the budget constraint
ct + rct × dt ≤ et
(58)
The IES parameter is then defined (Atkeson and Ogaki (1996)) as
IES = −
∂ V (e, rc) / ∂ e
e × ∂ 2 V (e, rc) / ∂ e2
(59)
Technically, I substitute the budget constraint into the objective function and then use M AT LAB T M ’s function
fminbnd applied to the minus of the objective function. I impose the constraints that d > 0 and d < e/rc so
29
that the implied c > 0. In addition, I construct the rental cost of consumer durables as
rct =
Ud (ct , dt )
Uc (ct , dt )
(60)
To evaluate the partial derivatives necessary to compute IES, I use the following finite-difference approximations
∂ V (e, rc)
∂e
∂ 2 V (e, rc)
∂ e2
=
=
`
´
V (e + ∆e, rc) − V (e − ∆e, rc)
+ O (∆e)2
2× ∆e
`
´
V (e + ∆e, rc) − 2 × V (e, rc) + V (e − ∆e, rc)
+ O (∆e)2 ,
2
(∆ e)
and I choose ∆ e = 10−5 .
30
(61)
(62)
(63)
Notes
1
While I was working on this draft I learned about two recent papers by Okubo Masakatsu (2004a, 2004b)
which also deal with the role of nonhomotheticity in a model with consumer durables. However, his preference
specification does not allow for an easy interpretation of the preference parameters.
2
In fact, my proposed preference specification allows for time-varying expenditure elasticities of nondurable
goods and the service flow from consumer durables as long as their ratio is a constant.
3
See also Chen and Ludvigson (2004) for a different way of modeling nonseparabilities in consumers’ prefer-
ences.
4
This doesn’t alter the preference ordering but is necessary for the preference parameters to be identified.
See Appendix A for details.
5
Eichenbaum and Hansen’s (1990) estimate is ES = 0.91. Ogaki and Reinhart’s (1998) estimate is greater
than one, ES > 1.
6
Calibration based on the super-consistent estimates from the intratemporal first-order condition, and Table
3, row 4.
7
Technically, ES is time varying but the amplitude is quite negligible. Also, the construction of asymptotic
confidence interval for ES is desirable but problematic because we need to know the correlation between θ̂, η̂
and the GMM-estimated parameter â. But these two subsets of the preference vector are estimated using two
different methods.
8
Hansen and Eichenbaum’s (1990) point estimate is ES = 0.91. Ogaki and Reinhart’s (1998) point estimate
is statistically greater than one. Both papers impose homotheticity.
9
In fact, if we miss this condition, the parameter a is not identified, with its projection of the 2-dimensional
S-set for (σ, a) being at least [0, 100], an implausibly wide interval.
10
In fact, the intratemporal first-order condition stipulates exactly that.
11
Recall that the substitution matrix is negative semidefinite and hence all diagonal terms are non-positive.
12
In case of homotheticity, IES = σ. I also verified numerically that this is the case.
13
As the preferences are nonhomothetic, a closed-form solution for V (e, rc) may not exist. However, a nu-
merical solution, described in Appendix E, is rather straightforward; I evaluate also the partial derivatives
numerically in the standard way. See Okubo (2004a, 2004b) and Hanoch (1977) for a different treatment.
14
The graphs of the IESt are practically identical for the calibrations from Table 3 across all 4 rows.
31
References
[1] Atkeson, A., and Ogaki, M. (1996), ”Wealth Varying Intertemporal Elasticities of Substitution: Evidence from Panel and Aggregate Data”, Journal of Monetary Economics, 38,
507-534.
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35
Durables Stock and Relative Price
2
Relative
Price
of
Durables
1.5
Durables Stock
Over
Nondurables Flow
1
0.5
1960
1970
1980
Quarter
1990
2000
Figure 1: The plot portrays the quarterly time-series of the ratio of the constructed real
consumer durables stock over real nondurable goods and services, and the relative price of
consumer durables, computed as the ratio of the price index for PCE on consumer durables
over the price index for PCE on nondurable goods and services. Bars represent recessions as
classified by the National Bureau of Economic Research (NBER). Sample period 1956:Q1 2001.4.
36
Table 1: Test for the Null of Difference Stationarity
Phillips-Perron Test
zρ
zt
log(Ct )
-4.09
-1.35
log(Dt )
-5.11
-1.47
log(Qt )
-1.78
-0.87
NOTE: Critical value for zρ is −20.70 (5% level) and −17.50 (10% level), zt is −3.45 (5% level) and
−3.15 (10% level). The number of lags used in the Phillips-Perron test is four. Sample period 1956:Q1
- 2001:Q4.
37
Table 2: Estimated Cointegrating Vector
Point
Estimate
Asymptotic
Standard Error
Bootstrap
Confidence Intervals
intercept
-1.83
(0.13)
N/A
θ
0.36
(0.05)
[0.12,0.56]
η
0.76
(0.03)
[0.63,0.88]
NOTE: The table reports the estimated cointegrating vector (1, −θ, −η) and their asymptotic standard errors, using Stock and Watson (1993) regression. The Akaike information criterion suggests the
use of 1 lag. The empirical distribution is constructed using the sieve bootstrap of Chang, Park and
Song (2005); 1,000,000 Monte Carlo simulations were used. The last column shows the 95% confidence
intervals. Sample period 1956:Q1 - 2001:Q4
38
Empirical Distribution of the Parameter θ
Frequency
1500
1000
500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
*
θ
Empirical Distribution of the Parameter η
Frequency
1500
1000
500
0
0.55
0.6
0.65
0.7
0.75
*
η
0.8
0.85
0.9
0.95
Figure 2: The picture displays the empirical distributions (histograms) of the parameters θ
and η obtained by 1,000,000 Monte Carlo simulations of the sieve bootstrap (Chang, Park and
Song 2005), applied to the cointegrating regression as implied by the intratemporal first-order
condition. Sample period 1956:Q1 - 2001:Q4.
39
Table 3: GMM Estimation Results
Continuous-Updating GMM
Instruments (Euler Eq.)
40
Row
Instruments (Intratemporal F.O.C.)
σ
β
a
JT
CIσ95%
Sσ95%
Sa95%
Const, CG, DG, QG, U1, U2, T-Bill
Const
0.060
(0.025)
1.092
(0.039)
10.185
(0.147)
5.215
(0.390)
(0.011, 0.109)
(0.037, 0.062)
(9.8, 10.6)
1
Const
Const, CG, DG, QG, U1, U2, T-Bill
0.051
(0.016)
1.118
(0.039)
10.150
(0.143)
8.381
(0.137)
(0.020, 0.082)
(0.033, 0.047)
(9.7, 10.4)
2
Const, CG, DG, QG, U1, U2, T-Bill
Const, CG, DG, QG
0.043
(0.013)
1.136
(0.041)
10.136
(0.141)
8.811
(0.359)
(0.018, 0.069)
(0.023, 0.040)
(9.8, 10.5)
3
Const, CG, DG, QG, U1, U2, T-Bill
Const, U1, U2, T-Bill
0.057
(0.023)
1.101
(0.039)
10.128
(0.138)
14.875
(0.061)
(0.012, 0.102)
(0.044, 0.050)
(10, 10.3)
4
NOTE: The symbol CIσ95% denotes the 95% asymptotic GMM confidence interval for the parameter σ. The symbols Sσ95% and Sa95% stand for the
projections of the 95% joint S-set for (σ, a), based on the objective function JT concentrated with respect to β, and as projections, they are of course
conservative. The instruments, except Rf , are lagged twice to take account of the time aggregation. Variable definitions: CG = nondurable consumption
growth rate, DG = durables stock growth rate, QG = durables price growth rate, T-Bill = U.S. Treasury Bill deflated with CPI, U1 = growth rate of the
number of civilians unemployed less than 5 weeks, U2 = growth rate of the number of civilians unemployed more than or equal 15 weeks. The test asset
is the nominal return on the U.S. Treasury Bill, deflated with the consumer price index (CPI). Multiple local minima were encountered. I also restrict
a ∈ [0.60, 1, 463.00], a rich enough set (see also Appendix A). Standard errors are in the parentheses, except for JT column where the p-value is in the
parenthesis. Quarterly data 1956:Q1 - 2001:Q4.
S−Set for the Preference Vector (σ,a), Calibration: Table 3, Row 1
12
11.5
11
a
10.5
10
9.5
9
8.5
8
0
0.02
0.04
0.06
0.08
0.1
σ
0.12
0.14
0.16
0.18
0.2
Figure 3: The picture displays the 2-dimensional 95% S-set for the preference vector (σ, a)
based on the objective function JT , concentrated with respect to β, and the 2-dimensional
asymptotic confidence ellipse for the preference vector (σ, a). Calibration is based on the
superconsistent estimates of θ and η from the intratemporal first-order condition (Table 2)
and the GMM estimates of the rest of the preference parameter vector (Table 3, row 1).
Sample period 1956:Q1 - 2001:Q4.
41
S−Set for the Preference Vector (σ,a), Calibration: Table 3, Row 2
11
10.8
10.6
10.4
a
10.2
10
9.8
9.6
9.4
9.2
9
0
0.01
0.02
0.03
0.04
0.05
σ
0.06
0.07
0.08
0.09
0.1
Figure 4: The picture displays the 2-dimensional 95% S-set for the preference vector (σ, a)
based on the objective function JT , concentrated with respect to β, and the 2-dimensional
asymptotic confidence ellipse for the preference vector (σ, a). Calibration is based on the
superconsistent estimates of θ and η from the intratemporal first-order condition (Table 2)
and the GMM estimates of the rest of the preference parameter vector (Table 3, row 2).
Sample period 1956:Q1 - 2001:Q4.
42
S−Set for the Preference Vector (σ,a), Calibration: Table 3, Row 3
11
10.8
10.6
10.4
a
10.2
10
9.8
9.6
9.4
9.2
9
0
0.02
0.04
0.06
0.08
0.1
σ
0.12
0.14
0.16
0.18
0.2
Figure 5: The picture displays the 2-dimensional 95% S-set for the preference vector (σ, a)
based on the objective function JT , concentrated with respect to β, and the 2-dimensional
asymptotic confidence ellipse for the preference vector (σ, a). Calibration is based on the
superconsistent estimates of θ and η from the intratemporal first-order condition (Table 2)
and the GMM estimates of the rest of the preference parameter vector (Table 3, row 3).
Sample period 1956:Q1 - 2001:Q4.
43
S−Set for the Preference Vector (σ,a), Calibration: Table 3, Row 4
11
10.8
10.6
10.4
a
10.2
10
9.8
9.6
9.4
9.2
9
0
0.02
0.04
0.06
0.08
0.1
σ
0.12
0.14
0.16
0.18
0.2
Figure 6: The picture displays the 2-dimensional 95% S-set for the preference vector (σ, a)
based on the objective function JT , concentrated with respect to β, and the 2-dimensional
asymptotic confidence ellipse for the preference vector (σ, a). Calibration is based on the
superconsistent estimates of θ and η from the intratemporal first-order condition (Table 2)
and the GMM estimates of the rest of the preference parameter vector (Table 3, row 4).
Sample period 1956:Q1 - 2001:Q4.
44
0.15
0.14
0.13
R, sD
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.05
1960
1965
1970
1975
1980
Time
1985
1990
1995
2000
Figure 7: The picture displays the time-series of the ratio of the implied rental cost rc over
the consumer durables price q (dotted line), and the budget share sd of the rented consumer
durables rc × d in the total consumption expenditure e (thick line). Calibration is based on
the super-consistent estimates from the intratemporal first-order condition (Table 2) and the
GMM estimates of the rest of the preference parameter vector (Table 3, row 4). Bars represent
recessions as classified by the National Bureau of Economic Research (NBER). Sample period
1956:Q1 - 2001:Q4.
45
1.27
Income elasticity of Consumer Durables, η
D
1.268
1.266
1.264
1.262
1.26
1.258
1.256
1.254
1.252
1.25
1960
1965
1970
1975
1980
Time
1985
1990
1995
2000
Figure 8: The picture displays the time-series of the expenditure elasticity of consumer durables
ηd . The computation is based on the equations η = ηc,t / ηd,t , 1 = sc,t ηc,t + sd,t ηd,t and
sc,t + sd,t = 1. Calibration is based on the super-consistent estimates from the intratemporal
first-order condition (Table 2) and the GMM estimates of the rest of the preference parameter
vector (Table 3, row 4). Bars represent recessions as classified by the National Bureau of
Economic Research (NBER). Sample period 1956:Q1 - 2001:Q4.
46
Time−Series of the IES (Calibration: Table 3, Row 4)
0.062
0.0615
0.061
IES
0.0605
0.06
0.0595
0.059
0.0585
1960
1965
1970
1975
1980
Time
1985
1990
1995
2000
Figure 9: The picture displays the time-series of the elasticity of intertemporal substitution
IESt , computed as IESt = −Ve (et , rct ) / (et × Vee (et , rct )), where et denotes the total consumption expenditure et = ct +rct ×dt at time t, and rct is the implied rental cost of consumer
durables, equal to the intratemporal marginal rate of substitution between the service flow st
from durables dt and the nondurable goods ct . Calibration is based on the super-consistent
estimates from the intratemporal first-order condition (Table 2) and the GMM estimates of
the rest of the preference parameter vector (Table 3, row 4). The dotted lines portray the
asymptotic confidence intervals for the IESt , computed numerically using the ∆-method for
the preference parameter vector p = (σ, β, a). Bars represent recessions as classified by the
National Bureau of Economic Research (NBER). The IESt is time-varying due to the nonhomotheticity of the consumers’ preferences. Sample period 1956:Q1 - 2001:Q4.
47