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orKing raper series
The E ffe ct o f C o s tly C o n s u m p tio n
A d ju s tm e n t on A s s e t P rice V o la tility
David A. Marshall and Nayan G. Parekh
3
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
December 1994 (WP-94-21)
FEDERAL RESERVE BANK
OF CHICAGO
THE.' EFFECT OF COSTLY CONSUMPTION ADJUSTMENT
ON ASSET PRICE VOLATILITY
David A. Marshall*
and
Nay an G. Parekh**
October 13, 1994
’Federal Reserve Bank of Chicago, and J.L. Kellogg Graduate School of Management,
Northwestern University.
‘‘Department of Economics, Northwestern University.
Discussions with Kent Daniel, Bob McDonald, and Lew Segel are gratefully acknowledged.
All errors remain the responsibility of the authors.
The Effect of Costly Consumption Adjustment
on Asset Price Volatility
Abstract
We investigate Grossman and Laroque’s (1990) conjecture that costs of adjusting
consumption can reconcile the low variance of aggregate consumption growth with the high
variance of asset returns. We incorporate small fixed costs of consumption adjustment into a
consumption-based capital asset pricing model (CCAPM) with uninsurable idiosyncratic shocks.
We calibrate the model to match the first and second moments of asset returns observed in post
war U.S. data, and ask whether the model can replicate observed moments of aggregate
consumption process. We find that the CCAPM’s implications are non-robust to extremely small
adjustment costs. In particular, undetectably small consumption adjustment costs can account for
most of the discrepancy between the observed variance of nondurable consumption growth and
the predictions of the CCAPM. However, the model is unable to match the moments of the
growth rate of the service flow from the stock of durables or housing.
1. INTRODUCTION
The consumption-based capital asset pricing model (CCAPM)1 of Lucas (1978), Breeden
(1979), and Grossman and Shiller (1982) has not fared well empirically. Perhaps the most basic
problem with the CCAPM is that it has difficulty replicating the high volatility we see in asset
price data. (See, e.g., Grossman, Melino, and Shiller (1987), Hansen and Jagannathan (1991).)
The source of this problem is clear. The asset pricing operator delivered by the CCAPM is a
function of aggregate consumption, which is a very smooth series. This pricing operator can only
generate volatile asset prices if it imposes a highly nonlinear transformation on aggregate
consumption data. In practice, the CCAPM must incorporate extreme curvature into the utility
function by assuming high risk aversion2 or a high degree of habit persistence.3
In this paper, we seek to reconcile the high variance of asset returns with the low variance
of aggregate consumption growth
w ith o u t
extreme curvature assumptions on preferences.
Following a conjecture of Grossman and Laroque (1990), we ask whether the empirical failure
of the CCAPM may be due to frictions in consumption transactions. In particular, Grossman and
Laroque (1990) assume that changing the rate of consumption requires payment of a fixed cost.
To see why consumption adjustment costs could affect the implications of the CCAPM, notice
that the CCAPM has the following three properties:
1 By "CCAPM" I refer to models where the stochastic discount factor determining asset
prices is a function of aggregate consumption.
2 For example, the estimation exercises in Grossman, Melino, and Shiller (1987) that yield
the least evidence against their CCAPM deliver estimates of the coefficient of relative risk
aversion of 154 and 185.
3 For example, in Constantinides’s (1990) habit-formation model, agents’ subsistence
point is, on average, 80% of current consumption. In other words, if consumption is constant
a long time and then drops by 20%, the consumer experiences a utility of negative infinity.
1
1.
An asset’s price is determined by the covariance between the asset’s payoff and
investors’ intertemporal marginal rate of substitution (IMRS) in wealth.
2.
The IMRS in wealth equals the IMRS in consumption.
3.
The IMRS in consumption is a function of the aggregate consumption process.
*
The first of these properties is shared by all modem asset-pricing theories. The second is the
e n velo p e p ro p e rty .
It only holds if wealth can be converted into consumption one-for-one.
Consumption adjustment costs drive a wedge between the IMRS in wealth and the IMRS in
consumption, invalidating the envelope property. The third property is the a g g re g a tio n p r o p e r ty .
Models with fixed adjustment costs do not aggregate: aggregate consumption does not resemble
the optimal consumption path of any individual agent.
Rather, an individual’s optimal
consumption follows a discontinuous (S,s) pattern, while aggregate consumption is smooth.4
Caballero (1990) shows that the variance of this aggregate consumption process can be
substantially lower than the variance of individual consumption.
These considerations suggest that the CCAPM may be sensitive to fixed costs of
consumption adjustment.
However, it not known whether
p la u sib le
adjustment costs are
sufficient to reconcile the high variance of asset returns with the low variance of aggregate
consumption growth. In this paper, we address this question. We introduce fixed consumption
adjustment costs into a simple consumption-based model with heterogeneous agents. We then
compute the relative means and variances of asset returns and aggregate consumption growth
implied by the model, and compare these statistics with estimates from post-war US data.
4
Lynch (1994) explores an asset pricing model similar to ours in which the (S,s)
consumption pattern is imposed directly on individual agents.
2
What constitutes a "plausible" adjustment cost? The answer depends critically on what
we mean by "consumption".
Grossman and Laroque (1990) and Reider (1993) interpret
consumption as the service flow from a stock of durable goods or housing: the cost of
consumption adjustment is the cost of changing this stock.
adjustment costs are plausible.
Under this interpretation, high
Consider, for example, the closing costs in a real estate
transaction or the wholesale/retail spread paid when trading in a used car.
Alternatively, one could follow most of the empirical literature on consumption-based
asset pricing, and interpret consumption as expenditures on nondurable goods. The problem with
this interpretation is that there are no major costs to changing the flow rate of nondurable
consumption. However, we argue that costs of adjusting nondurables are plausible p r o v id e d the
costs are undetectably small. That is, we would only consider costs so small that they would
swamped by simple measurement error in collecting aggregate consumption data. These costs
might include the time spent implementing a new consumption/savings plan, search costs in
finding vendors for the new, better (or worse) quality goods to be purchased, or even the
psychological effort of thinking about reoptimizing consumption, rather than continuing with last
period’s consumption plan. (This last category of adjustment costs is similar to what Cochrane
(1989) calls "near-rational behavior".) Marshall (1994) provides evidence that undetectably small
adjustment costs are sufficient to dramatically change the relative variances of consumption
growth and asset returns. Intuitively, there is a relatively small utility gain from period-by-period
adjustment of consumption in response to movements in asset returns. Even a small fixed cost
of consumption adjustment would be sufficient to induce agents to forego this small utility gain,
and to choose a consumption path that is virtually flat. When we evaluate our adjustment cost
model, we will consider both the "durables" and the "nondurables" interpretation for consumption.
3
In the standard formulation of the CCAPM (e.g., Lucas (1978)), aggregate consumption
is taken as an exogenous endowment flow, calibrated to observed aggregate consumption data.
The researcher then asks whether the returns implied by the model are sufficiently volatile to
match the data. In our paper, we reverse this logic. Following Constantinides (1990), we take
the returns process as given, calibrated to observed asset return data. We then compute the
optimal consumption decisions given this returns process, and we ask whether the implied
aggregate consumption process is sufficiently n o n -vo la tile to match observed consumption data.
This strategy is essentially a partial equilibrium approach, although one could interpret this model
as a general equilibrium economy in which the capital assets are in infinitely elastic supply. We
adopt this approach because the fixed adjustment cost introduces a nonconvexity into the standard
CCAPM. To solve for equilibrium prices in such a model would require us to compute the
equilibrium of a nonconvex economy. The theoretical and computational difficulties of such an
exercise are prohibitive in an economy with multiple agent-types.5
Our model does not possess the aggregation property, so it is essential to explicitly model
cross-sectional heterogeneity. The way this is usually done is to assume that each agent receives
an idiosyncratic, uninsurable endowment, which is a proxy for uninsurable labor income.6 In
our model, we take returns as exogenous rather than endowments, so we assume that each agent
has access to an idiosyncratic investment technology, whose return is stochastic and uninsurable.
One can think of this idiosyncratic asset as a proxy for investment in human capital, household
production, or a family business.
5 For a fuller discussion of these issues, see Marshall (1994).
6 See, e.g., Lucas (1990), Marcet and Singleton (1990), Heaton and Lucas (1991), Telmer
(1993), and Constantinides and Duffie (1992).
4
Our results confirm that a very small fixed cost of consumption adjustment sharply
reduces the variance of aggregate consumption growth predicted by the model. In our benchmark
case, adding an adjustment cost equal to ^/lOOO)11 of current consumption cuts the variance of
15
aggregate monthly consumption growth by almost 70%, which is almost enough to match the
variance of the growth rate of nondurable consumption in post-war US data. On the other hand,
we find that even very large adjustment costs cannot reduce the variance of consumption growth
sufficiently to match data measuring the service flow from the stock of durables or housing.
We interpret these result as evidence that (1) the empirical rejections of the CCAPM using
data on nondurable consumption may be due in part to the omission of extremely small
adjustment costs from the model; and (2) reinterpreting consumption as a service flow from a
stock of illiquid durable goods is unlikely to help the empirical performance of the CCAPM.
More generally, our results show that the quantitative implications of the CCAPM are non-robust
to the introduction of extremely small frictions.
The remainder of the paper is organized as follows. Section 2 introduces the model,
which incorporates the consumption/portfolio problem of Grossman and Laroque (1990) into a
heterogeneous-agent economy. In section 3, we simulate the model, characterize the moments
of aggregate consumption growth implied by the model, and compare these implications to post
war US data. Section 4 summarizes our conclusions.
2 THE M O D E L
.
A. Basic Structure
There are N agents. Each agent can invest in two risky assets and one risk-free asset.
The risk-free asset pays an instantaneous rate of return, denoted r, which is constant through time.
5
The first risky asset is available to all agents. We will refer to this asset as the market asset.7
Let bt denote the value of one unit of the market asset at date t. It is assumed that bt is a
diffusion process:
db, “ bt[(p+r)dt + adwt]
(1)
*
where p and a are mean excess return and the standard deviation respectively of the market
return, and wt is a standard Wiener process. Let X{ denote the value of the j* agent’s holdings
of the market asset, and let B{ denote the value of the j* agent’s holdings of the risk-free asset.
The second risky asset is idiosyncratic to the individual agent. Let bjt denote the value
of one unit of agent j ’s idiosyncratic asset. The increment in b}t represents the payoff flow from
the idiosyncratic investment. Its law of motion is analogous to that of the market asset:
dbltJ = bIJ[(p, +r)dt + a,dz. J
t
(2)
where p, and a, denote the mean excess return and the standard deviation for the idiosyncratic
investment, and Z t is a standard Wiener process. It is assumed that p, and a, are the same for
j
all agents, but the shocks to the idiosyncratic returns are uncorrelated with each other and with
the shock to the market return:
cov(Zjt,z^t) = 0 , V j,k
cov(zjt,wt) = 0, Vj.
Let IJ denote the stock of the idiosyncratic investment held by agent j. There are no other assets
available. In particular, there is no market in claims contingent on the realization of zjt, so the
7
Grossman and Laroque (1990) show that two-fund separation holds in this model, so
there is no loss of generality in assuming a single publicly-traded risky asset, rather than a
number of such assets.
6
risk of the idiosyncratic investment cannot be hedged.
It is costless for an agent to maintain the same rate of consumption, but changing the rate
of consumption involves a fixed cost, which is proportional to the previous rate of consumption.
Specifically, if the consumption rate immediately before a consumption change at date t for agent
j is denoted c,^, tile cost at date t of changing consumption is A,ct., where 0 < A < 1.
,
Agent j ’s total wealth, Qj, is given by
Q,j = Btj + X{ + Itj.
(4)
If consumption is not adjusted at date t, the law of motion for Q is:
{
dQtj = Q,jrdt + x{(pdt + adw t) + I ^ d t + a ^ , ) - ctjdt.
(5)
If consumption is adjusted at date t,
Qt = Qr - X.cT
..
(6)
Finally, agents face a no-bankruptcy condition:
Qtj - Xc,j > 0.
Agents are not infinitely-lived.
(7)
Rather, as in Blanchard (1985), death is a Poisson
process, uncorrelated across agents. In particular, in the time interval (t, t+At), each agent faces
a probability of dying equal to pAt + o(At), where p
>
0 is the Poisson parameter. Under this
assumption, the steady state cross-sectional distribution of age and the remaining lifetime of each
agent are both exponentially distributed.
Let an agent’s (random) date of death be denoted tx. Each agent solves the following
7
optimal control problem:
(8 )
subject to constraints (4), (5), (6), and (7), and initial conditions {Qq c0 }. Parameters 8 and a
?, j
\
are assumed to be the same for each agent. As shown in Merton (1971), the objective function
in (8) is equivalent to that of an infinitely-lived agent with a higher discount rate:
8>0, a < l, a^O
(9)
In the event of an agent’s death, the agent is replaced by a single newly-bom agent. All
agents born at a given date t are endowed with the same initial wealth, Q„ where the stochastic
process Qt is defined such that log(Q,) equals the cross-sectional mean of log(Q{). The lagged
consumption ct. of each newborn agent (which determines that agent’s initial consumption
adjustment cost) equates the newborn’s consumption/wealth ratio to the consumption/wealth ratio
of the recently deceased agent that this newborn replaces. These assumptions insure that both
the cross-sectional distribution of log(Qj) around its cross-sectional mean and the cross-sectional
distribution of the consumption/wealth ratio are stationary.
B. Optimality Conditions
When
X =
0 (no adjustment costs) the optimization problem corresponds to Merton
(1971). Optimal consumption and asset holdings are proportional to wealth:
8
Xtj =
j _
It
1-a
1-a
(10)
°
i
where
p s 8+p
When
X>
1
ar - _
2
0, the problem corresponds to that studied by Grossman and Laroque (1990).
The derivation of the optimality conditions is provided in the Appendix. In this section we
summarize the optimality conditions. Let V(Q,c) denote the maximized value of the objective
in (9), starting from initial conditions (Q,c).
V is homogeneous of degree
a,
so the
dimensionality of the state space can be reduced. Define:
c
h(y) = c ‘‘V(Q,c) = V(X+y,l),
M
=
(11)
sup(y+X)'*h(y).
The optimal consumption and asset holdings of an individual agent are functions of the
transformed value function h(y). If h(y) = y*M, then it is optimal to adjust consumption. If h(y)
9
> yaM, it is not optimal to adjust consumption. The optimal asset holdings are given by
v ■ . -h'(y) p
h"(y) a 2
(1 2 )
i-c d iM A .
h//(y) of
(13)
and
(where the superscripts and subscripts in X{ and l| have been suppressed). The function h(y)
satisfies the differential equation
1 h'(y)2
2 h"(y) ° 2 o?
h'(y)[r(y+ ^)-l] - (8+p)h(y) + 1 = 0 .
a
(14)
Following Grossman and Laroque (1990), it can be proved that the optimal consumption
strategy is determined by three points y, < y* < y2, where y, and y2 satisfy
h(ys = y(aM
)
(15)
h'(y,) = ay^'M , i = 1,2,
and y* = argmaxy (y+X)'ah(y). The interval (y^
is the region of inaction for consumers: when
ye (yi,y2 it is optimal not to adjust consumption. Whenever y = y, (i.e., the wealth/consumption
),
ratio is sufficiently low) or y = y2 (i.e., the wealth/consumption ratio is sufficiently high) it is
optimal to adjust consumption so that y = y \ in which case
ct "
Qt
y ’ + a.
(16)
Notice that 8 and p enter the optimal decision rules only as the composite discount rate
10
8
+p.
Also, by substituting (12) and (13) into (5), one can see that p! and c r affect consumption,
wealth, and asset holdings only through the Shaipe ratio p/Gj.
For this reason, we treat p/cr,
as the single parameter characterizing the idiosyncratic return process.
3. SIMULATION RESULTS
For a given set of parameter values, we compute the equilibrium by numerically solving
equation (14) to obtain the function h.
(Details are provided in the Appendix.) Once the
function h has been computed, we use this function in (11) and (15) to compute y„ y2, and y \
We then simulate the returns processes, and we use the optimal consumption and portfolio rules
(12) and (13) to trace the evolution of consumption through time for a population of 1000
individuals. We approximate continuous time by evaluating the consumption and portfolio rules
at time intervals At = (l/120)th of a year. For each parameter set, the model is simulated for
50,000 periods (416.67 years). The choice of initial value for the cross-sectional mean of wealth
is arbitrary, since the model is homothetic in wealth. As described in the appendix, the initial
cross-sectional distribution of the log of wealth around this arbitrary mean is set equal to the
steady-state cross-sectional distribution when X = 0.
.
Aggregate consumption is computed by
summing the 1000 individual consumptions cross-sectionally, then temporally aggregating up to
a monthly or quarterly series.
We calibrate p and
o2
to match the mean and variance of the monthly real return to the
value-weighted portfolio of NYSE stocks from 1959:2 through 1986:12. The risk-free rate r is
calibrated to the mean of the monthly real return to one-month treasury bills over this same
period. In particular, the mean monthly log return on the value-weighted stock portfolio is .0038,
and the standard deviation of this return is .0425. The mean monthly log return on one month
11
T-bills over this period is .00072. The rate-of-return parameters p, a, and r, are annualized rates,
so the mean of the monthly log return is (p + r - a 2
/2)/12, the variance is 0 ^ 12 , and the monthly
risk-free log return is r/12. Setting these expressions equal to the moments in the observed
monthly data, we obtain p = .0484,
a -
. 147, and r = .008.
We must choose values for the other parameters
{a , X,
5,
p,
p/a,}.
The values of
parameter a we consider are -1 and -4, which correspond to a coefficient of relative risk aversion
of 2 and 5 respectively. The parameters 5 (the subjective discount rate) and
p
(the Poisson
parameter governing the probability of death) enter the individual’s decision rule only as the sum
8 + p . (See equation (9).)
In principle, 8 and p could affect the behavior of the economy in
distinct ways because p affects the cross-sectional distribution of wealth. In practice, however,
varying p while keeping 8 + p constant has no detectable effect on aggregate consumption for
the range of parameters we use. Therefore, in reporting our results we treat 8 + p as the
parameter to be varied. We report simulations for 8 + p = .02 and .04.
We think of the idiosyncratic asset as analogous to human capital, home production, or
investment in a small family business. Unfortunately, reliable data for the mean and variance
of the return to these types of investments are unavailable, so we have little empirical guidance
in choosing values for pj/cTj. We proceed by assuming that p,/a, does not differ greatly from p/a,
the Sharpe ratio for the market return. Noting that p/a = 0.33, we consider values of p/Oj
between 0.2 and 0.6.
We wish to simulate the model for reasonable values of the adjustment cost. When
consumption is interpreted as the service flow from the stock of durables or housing, we set X
,
= 0.4 as the upper limit of the adjustment cost parameter. We regard this figure as conservative.
If the fixed cost of changing one’s house is
6%
12
of the value of the house being sold and the
annual service flow from a house is 15% of its value, then
X
would equal 40%. Similarly,
suppose the adjustment cost involved with buying a car is equal to the retail/wholesale spread
that the consumer loses on the trade-in. If this spread equals 8% of the trade-in’s value (a rather
low figure), and if the service flow from a car equals 20% of the car’s value, then
equal 40%.
X
would again
On:the other hand, when consumption is interpreted as the expenditure on
nondurable goods, we consider only trivially small adjustment costs: for this case, we set
X
=
.002 as the upper limit of the adjustment cost parameter. To give an idea of the magnitude of
the fixed costs involved, per capita consumption of nondurables and services in the United States
in 1993 was approximately $14,000. A value of X equal to .002 would correspond to a fixed cost
of consumption adjustment of $28. Costs this small are virtually undetectable.
We take as our benchmark case the parameterization
{a
= -4, 5 + p = .04, p/Gj = .2}.
Results from simulating this benchmark case are displayed in Table 1. Increasing
X
has little
effect on the mean of monthly aggregate consumption growth, but it substantially reduces the
variance of this series. The most rapid variance reduction is for small values of the adjustment
cost. For example, when we increase
X
from 0 to .002 in this benchmark case, we observe a
67% reduction in the variance of monthly aggregate consumption growth, from .00024 to .00008.
This rapid fall-off in variance as X increases is illustrated in Figure 1, which graphs the variance
,
as a function of X e [0, .002]. The convex shape of this graph is typical for all parameterizations
of this model: in all cases, very small adjustment costs have the biggest proportional impact on
consumption-growth variance. Additional increases in the adjustment cost parameter beyond .002
imply relatively smaller additional variance reductions: for the benchmark case, increasing X from
.002 to .01, a five-fold additional increase in
consumption growth from .00008 to .00005.
13
X,
only reduces the variance of monthly
Tables 2 and 3 display how the behavior of the model changes as we move away from
the benchmark case. In Table 2, we set a = -1. This lower value of risk aversion implies both
a higher mean and a higher variance of consumption growth, since agents hold a larger fraction
of wealth in the risky assets. As in Table 1, most of the variance reduction from increasing the
adjustment cost comes at very low values of X. In Table 3 we change p + 8 and p/a, from the
benchmark levels. In all these cases, the variance reduction from increasing X follows the pattern
displayed in Figure 1, so we only report the moments for three values of
X.
In Panels A and
B of Table 3, we reduce the composite discount rate p + 8 to .02. The effect on the moments
of aggregate consumption growth is concentrated on the mean: lower discounting of the future
implies more savings, and therefore a higher mean growth rate of consumption.
There is
virtually no effect on the variance of consumption growth. Finally, Panels C and D of Table 3
show the effect of increasing the productivity of the idiosyncratic asset.
As p/Gj increases, the
idiosyncratic asset becomes more attractive, agents invest more in this risky asset, so both the
mean and variance of aggregate consumption growth are higher. The effect on the variance,
however, is rather small, compared to the effect of decreasing risk aversion.
We now ask how well the benchmark case replicates the moments of consumption growth
and asset returns observed in the data. The risk-free rate and the mean and variance of the
market return are already calibrated to match observed asset returns; the key question is whether
plausible adjustment costs can reduce the variance of aggregate consumption growth to the level
observed in the post-war US economy. In Table 4 we display means and variances of the
monthly growth rate of expenditures on nondurables and on nondurables plus services. We also
display the means and variances of the quarterly growth rate of the service flow from the stock
of durables and of the service flow from the stock of residential housing. These four cases
14
correspond to four different interpretations for consumption in our model. We consider each
interpretation in isolation.
For example, when we interpret consumption as purchases of
nondurable goods, we abstract from consumption of services from durables, and vice versa. This
procedure is essential for tractability: to include multiple types of consumption, each with a
different cost of adjustment, would vastly complicate the model.
Let us first compare the performance of the benchmark model, displayed in Table 1, with
the moments of observed data on consumption of nondurable goods, reported in the first row of
Table 4.
For all values of
X,
the mean of monthly aggregate consumption growth in the
benchmark case is within one standard error of the value of 1.0007 estimated using data on
nondurable consumption. In the absence of adjustment costs, the benchmark model implies a
consumption-growth variance of 2.4x1 O more than four times the variance observed for
'4,
nondurable consumption growth in the data. However, when we increase A to .002, the variance
,
of consumption growth in the model drops to 8xl0'5, which is quite close to the estimated
variance of 5.4x1 O'5. It appears from this evidence that the observed variances of equity returns
and nondurable consumption growth are consistent with a simple CCAPM when extremely small
consumption adjustment costs are introduced.
If it is assumed that consumer services are prefect substitutes for consumption of
nondurable goods, it is appropriate to construct an alternative measure of consumption by adding
these two series. The mean and variance of this measure of consumption are displayed in the
second row of Table 4.8 The mean growth rate of consumption of nondurables and services is
8
Our measure of expenditures on services excludes the service flow from housing, which
is usually included in data on consumption of services as reported by the Bureau of Economic
Analysis. We do so because the adjustment cost on this service flow from housing is likely
to be much larger than the other components of expenditures on services.
15
estimated to be about 1.0013, somewhat higher than the prediction of the benchmark model,
while the estimated variance is 2.4xl0'5, which is substantially lower than that predicted by the
model. The model’s prediction for the mean can be increased by assuming a lower composite
discount rate. (See, for example, Panel A of Table 3.) However, in order to match the observed
variance of consumption of nondurables plus services, one would have to assume a value of
X
around .02. We regard costs of this magnitude as implausibly high. To summarize, plausible
consumption adjustment costs can account for about 75% of the discrepancy between the variance
of consumption growth predicted by the frictionless CCAPM (i.e., X = 0) and the variance of the
observed growth rate of nondurables plus services.
Let us now turn to an alternative interpretation of consumption as a service flow from
durables. Adjustment costs have a natural interpretation in this context as the cost of changing
the stock of durables. Reider (1993) suggests that the empirical performance of the CCAPM may
improve if consumption is interpreted in this way. Somewhat surprisingly, we fine that the
adjustment cost model fares rather poorly for this measure of consumption. Even with values
of
X
as high as 0.4, the model cannot match the low variance of the growth rate of the services
from durables or housing.
Table 4 reports that the mean of the quarterly growth rate of the
service flow imputed from the stock of consumer durables is 1.008, and the variance of this
growth rate is 0.000025. The corresponding moments for the growth rate of the service flow
from the stock of housing are 1.005 and .000016.
In both cases, the benchmark model
underpredicts the mean quarterly growth rate, and, even with
X
set at .40, the variance of
quarterly consumption growth predicted by the model is over four times the observed variance.
The growth rate of service flows from consumer durables or housing seems too smooth to be
explained by this simple model, even when large adjustment costs are assumed.
16
4. CONCLUSIONS
The results of this paper support our conjecture that fixed costs of consumption
adjustment have quantitatively important implications for the co-movements of aggregate
consumption and asset returns. We find that costs too small to be detected empirically can
reduce the variance of aggregate consumption growth by more than 60%.
Costs of this
magnitude can account for most of the discrepancy between the high variance of asset returns
and the low variance of observed aggregate nondurable consumption. Our results do not support
Reider’s (1993) conjecture that illiquidity in
d u ra b le
goods can explain the poor empirical
performance of the CCAPM. We find that even high adjustment costs cannot reconcile the high
volatility of asset returns with the low volatility of consumption services from durables or
housing.
Our results have important implications for the use of the CCAPM. In particular, they
suggest that predictions of consumption-based models for the short-run movements of asset
returns are non-robust to extremely small perturbations. It follows that the consumption-based
approach is not well-suited to modelling high frequency asset-price behavior. On the other hand,
this class of models still may be useful for understanding the co-movements of asset returns and
macroeconomic quantity variables at cyclical or longer frequencies.9
9 For a recent effort in this direction, see Campbell and Cochrane (1994).
17
Appendix: Derivation of Optimality Conditions; Discussion of Simulation Procedure
This sketch of the derivation of the optimality conditions follows Grossman and Laroque
(1990), with appropriate changes to accommodate the features we add.
For a more complete
description, the interested reader can refer to Grossman and Laroque (1990).
Let V(Q,c) denote the maximized value of the objective in (9), starting from initial conditions
(Q,c). If x denotes the first date at which it is optimal to adjust consumption, and c* denotes the
consumption level immediately following that adjustment, then V(Q,c) satisfies
X
V(Q,c) =
sup
e -(8*P |_dt + e -(5,P
)t—
)TV(Qr -Xcr ,c*)
a
E
c\T ,(X t,lt,te 10, tl)
(17)
subject to (4), (5), (6), and (7). Use (11) to transform (17) in terms of the single variable y:
h(y) = supE
J
p. -(5 ♦ p) t
_ ------ dt + e '(5,p)TMy *
a
(18)
where x, = X,/c and T|t = 1,/c, and the maximization is subject to the appropriately transformed
constraints. In particular, the law of motion for the new state variable, y, can be obtained from (5):
dy = [r(y+ ^)-l]dt + x[pdt + adw] + riJpj + ajdzJ
(19)
and the no bankruptcy condition is simply yt > 0.
Grossman and Laroque’s (1990) key insight is that, for fixed M, control problem (18) is an
optimal stopping problem in which the payoff upon stopping in state y is simply My*. Let h(y;M)
denote the function h which solves problem (18) for a particular fixed M. This function can be
characterized using results from the optimal control literature. Once h(y;M) has been determined,
18
h can be computed as the function h(-;M*), where M* is implicitly defined by:
M * = sup (y + X) h(y;M *)
y
(20)
The function h(y;M) has the following properties: h(y;M) > yaM (since it is always feasible to stop
immediately). If h(y;M) > y*M then it is not optimal to stop. In this case, consumption equals c
for a time interval [0, At], and the function h(y;M) must satisfy the Bellman equation
-(8 +p)s
a
(21)
ds + e ~
(5tp)Ath(yA
t)
subject to (19) (where the dependence of h(y;M) on M is suppressed for simplicity). Taking limits
as At
— 0
>
and applying Ito’s Lemma, one obtains the Hamilton-Jacobi-Bellman equation
T
sup i l i ^ . [ x 2C2 + n 2af] + h'(y)[xp + r(y+X)-l + T1P,] -(8+p)h(y) + i .
2
a
x.n
= 0
(22)
The optimal portfolio choices, can be found by maximizing equation (22) with respect to x and
T|.
Doing so, one obtains equations (12) and (13). By substituting (12) and (13) into (22), one obtains
(14). If h(y) = yaM, then it is optimal to stop. At these stopping times, the smooth pasting condition
h'(y) = ay*'1 must hold.
Finally, it can be shown, using a proof strategy analogous to that used in Grossman and
Laroque (1990, Theorem 3.3), that the set {y |h(y) > yaM} is a closed interval [y,, yj, so the optimal
strategy is to keep consumption constant until y = yt or y = y2, at which point consumption is
adjusted to set y = y* = argmax(y +X)'ah(y). Equations (15) immediately follow.
y
To solve the model we choose an initial guess for M and an initial guess for yt, set h(yt) =
y]M and h'fyj) = ay^M, continue h by solving differential equation (14) using a fifth-order Runge-
19
Kutta algorithm, until a value for y >
is found where h'(y) = aya 1
_M. If, at this value of y, it is
also the case that h(y) = yaM, then this value of y corresponds to y2, and the computed path for h(-)
corresponds to h(-;M) for the given M. If not, we adjust yt and repeat the process until we have
an h(*,M) process for which all boundary conditions hold. We then evaluate our initial guess for
M by computing,' as a criterion function,
S(M) = M - max (y+X)" h(y;M)J2. The algorithm
y
searches for a value M* for which S(M*) = 0.
When
X
= 0, the cross-sectional distribution of the log of wealth at date t is a time-invariant
mixture-of-normals with identical means and exponentially-distributed variances.1 (The mean of
0
this distribution is a random variable.) In our simulations, use this time-invariant distribution as the
initial cross-sectional distribution of the log of wealth for all values of
X.
(The initial value of the
cross-sectional mean is arbitrary.)
Time is discretized in units of At = (l/120)th of a year. (Experiments setting At equal to one
day yielded virtually identical results.) The gross return to the market asset between t and t+At
equals expfp+r+cri/lJAt + c(At)'se], where e is a standard normal variate.
(The returns to the
idiosyncratic asset are computed analogously.) The only approximation in discretizing time is that
agents are allowed to adjust their consumption and portfolios only at discrete intervals.
1
0
The normality assumption on the return to the idiosyncratic asset, along with equation (13),
imply that the cross-sectional distribution of the log of wealth for agents of age x is Gaussian
with variance x(pI/[(l-a)aI])2. Death is a Poisson process, so the cross-sectional distribution of
age x is exponential.
20
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Caballero, R.J, 1993, "Durable Goods: An Explanation for their Slow Adjustment,"
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Jou rn a l o f
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of Illiquid Durable Consumption Goods," E con om etrica 58, 25-52.
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Grossman, S.J., and R.J. Shiller, 1982, "Consumption Correlatedness and Risk Measurement in
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E co n om ics 10, 195-210.
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a Consumption-Based Asset Pricing Model," working paper, NBER.
Lucas, D.J., 1990, "Estimating the Equity Premium with Undiversifiable Risk and Short Sales
Constraints", working paper, Department of Finance, Kellogg Graduate School of
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E co n o m e trica
46, 1426-14
Lynch, A.W., 1994, "Staggered Decision-making by Individuals: Pricing Implications and Empirical
Evidence," manuscript, University of Chicago.
Marshall, D.A., 1994, "Asset Return Volatility with Extremely Small Costs of Consumption ReOptimization," working paper, Department of Finance, Kellogg Graduate School of
Management, Northwestern University
Marcet, A., and K.J. Singleton, 1990, "Equilibrium Asset Prices and Savings of Heterogeneous
Agents in the Presence of Portfolio Constraints", working paper, Stanford University.
Merton, R.C, 1971, "Optimal Consumption and Portfolio Rules in a Continuous-Time Model,"
J o u rn a l o f E co n o m ic T h eo ry , 3, 373-413.
Reider, R.L., 1993, "Can Transaction Costs On Durables Explain High Estimates of Risk Aversion?"
manuscript, Wharton School, University of Pennsylvania.
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J o u rn a l o f F in a n ce,
48,1803-
Table 1
Moments of Aggregate Consumption Growth
As Consumption Adjustment Costs Increase:
Benchmark Case
Coefficient of Relative Risk Aversion = 5
p + 5 = .04
P/a i = 1
Reduction
in Variance
of Monthly
(ct/ct.,)
Reduction in
Variance of
Quarterly
(c,/ct.,)
Mean of
Quarterly
(c/c,.,)
Variance
of
Quarterly
(C/Cn)
1.0023
0.00070
84.5%
1.00205
0.00022
69.0%
0.000027
88.3%
1.00204
0.00017
76.1%
1.00065
0.000023
90.2%
1.00198
0.00014
79.8%
0.2000
1.00065
0.000019
91.8%
1.00198
0.00012
83.1%
0.4000
1.00065
0.000016
93.1%
1.00197
0.00010
85.8%
Vs
Mean of
Monthly
(c,/ct.,)
Variance of
Monthly
(Ct/Cn)
0.0000
1.00076
0.000235
0.0002
1.00071
0.000137
41.7%
0.0003
1.00070
0.000127
46.0%
0.0005
1.00069
0.000109
53.5%
0.0007
1.00069
0.000101
56.9%
0.0010
1.00068
0.000092
61.0%
0.0020
1.00068
0.000078
66.8%
0.0050
1.00066
0.000056
76.2%
0.0100
1.00066
0.000046
80.6%
0.0200
1.00067
0.000036
0.0500
1.00067
0.1000
Notes: The model was simulated for a cross-section of 1000 individuals; 50,000 time periods were
simulated, each period corresponding to (l/120)th of a year. The parameters of the market return
process are: r = .008; p = .0484; a = .147. Aggregate consumption ct is computed by summing
individual consumptions cross-sectionally, then temporally aggregating up to a monthly series (for
columns 2 and 3) or a quarterly series (for columns 5 and 6). The columns labelled "Reduction in
Variance of (c jc ^ )" give the percentage reduction in variance, compared to the case with X = 0.
Table 2
Moments of Aggregate Consumption Growth
As Consumption Adjustment Costs Increase:
Lower Risk Aversion
Coefficient of Relative Risk Aversion = 2
p + 8 = .04
Pi/°i = -2
Reduction in
Variance of
Quarterly
(c/c,.,)
Mean of
Quarterly
(c /O
Variance of
Quarterly
(c,/ct.,)
1.0070
0.00444
79.1%
1.0057
0.00181
59.3%
0.00024
84.0%
1.0055
0.00141
68.1%
1.00170
0.00019
87.1%
1.0054
0.00117
73.7%
0.2000
1.00169
0.00016
89.2%
1.0053
0.00097
78.0%
0.4000
1.00167
0.00013
91.1%
1.0052
0.00082
81.6%
Reduction
in Variance
of Monthly
(c,/ct.,)
Vs
Mean of
Monthly
(c/c,.,)
Variance of
Monthly
(ct/ct.,)
0.0000
1.00276
0.00148
0.0002
1.00255
0.00105
28.7%
0.0003
1.00251
0.00098
33.7%
0.0005
1.00247
0.00089
40.0%
0.0007
1.00243
0.00082
44.8%
0.0010
1.00239
0.00075
49.1%
0.0020
1.00233
0.00062
57.8%
0.0050
1.00226
0.00047
67.9%
0.0100
1.00221
0.00038
74.1%
0.0200
1.00178
0.00031
0.0500
1.00174
0.1000
Notes: See notes for Table 1.
Table 3
Moments of Aggregate Consumption Growth
For Alternative Parameterizations of the Model
Panel A
Coefficient of Relative Risk Aversion = 5
p + 8 = .02
p/a, = .2
Reduction in
Variance of
Monthly (c/c,.,)
Vs
Mean of
Monthly
(ct/ct.,)
Variance of
Monthly
(ct/c,.j)
0.0000
1.00110
0.000235
0.0010
1.00103
0.000086
63.2%
0.0100
1.00100
0.000046
80.6%
Panel B
Coefficient of Relative Risk Aversion = 2
p + 5 = .02
p,/a, = .2
Vs
Mean of
Monthly
(c,/ct.,)
Variance of
Monthly
(c,/ct.,)
0.0000
1.00319
0.00148
0.0010
1.00283
0.00074
49.7%
0.0100
1.00266
0.00038
74.1%
Reduction in
Variance of
Monthly (c,/c,.j)
Table 3 (Continued)
Panel C
Coefficient of Relative Risk Aversion = 5
p + 8 = .04
Pi/<Ji = .4
Reduction in
Variance of
Monthly
(c,/ct.,)
Vs
Mean of
Monthly
(c,/ct.,)
Variance of
Monthly
(c,/ct.,)
0.0000
1.00178
0.000236
0.0010
1.00171
0.000103
56.2%
0.0100
1.00169
0.000051
78.3%
Panel D
Coefficient of Relative Risk Aversion = 5
p + 8 = .04
p,/a, = .6
Reduction in
Variance of
Monthly (c,/ct.j)
Mean of
Monthly
(c,/ct.,)
Variance of
Monthly
0.0000
1.00347
0.000239
0.0010
1.00342
0.000123
48.7%
0.0100
1.00340
0.000062
74.1%
Vs
( C ./C ,.,)
Notes: See notes for Table 1.
Table 4
Moments of Growth Rate of Aggregate Consumption
Estimated from Post-War US Data
Mean Consumption
Growth
Variance of
Consumption Growth
Consumer Expenditures on Nondurable
Goods
1.000677
(0.000303)
0.0000588
(0.0000058)
Consumer Expenditures on
Nondurables plus Services
1.001309
(0.000235)
0.0000238
(0.0000020)
Imputed Service Flow from Stock of
Consumer Durables
1.00798
(0.00111)
0.0000251
(0.0000048)
Imputed Service Flow from Stock of
Housing
1.00541
(0.00073)
0.0000163
(0.0000025)
Consumption Definition
Monthly Consumption Measures
Quarterly Consumption Measures
Notes: Monthly data for consumer expenditures on nondurables and on services are from the
consumer expenditure survey of the BEA. Data for monthly expenditures on services excludes
the imputed service flow from housing. Quarterly data for the imputed service flow from the
stock of consumer durables is from the Federal Reserve Board. Quarterly data for the imputed
service flow from the stock of residential housing is from the CITIBASE database. The sample
period for the monthly series is 1959:01 - 1986:12, and the sample period for the quarterly series
is 1959:01 - 1986:04. All data is seasonally adjusted, and is rendered per capita by dividing by a
measure of the noninstitutional civilian population over age 16 from the Bureau of Labor
Statistics. Robust Hansen-White standard errors are in parentheses, and are computed using the
Newey-West estimate of the residual covariance matrix with six lags.
CRRA = 5; Idiosyncratic Sharpe Ratio = 0.2; Delta + p = 0.04
Variance of Aggregate Consumption Growth
xl(H
Figure 1: The variance of aggregate consumption growth implied by the benchmark model is
plotted as a function of X, for
X
ranging from 0 to .002.
Working Paper Series
A series o f research studies on regional econ om ic issues relating to the Seventh Federal
R eserve D istrict, and on financial and econ om ic topics.
REGIONAL ECONOMIC ISSUES
Estimating M onthly R egional V alue Added by C om bining R egional Input
With National Production Data
WP-92-8
P i i R I r i e i h and Kenneth N Kuttner
hlp . salvc
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Local Impact o f Foreign Trade Z one
WP-92-9
David D. Weiss
Trends and Prospects for Rural Manufacturing
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William A Testa
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State and Local G overnm ent Spending--T he B alance
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Richard H. Mattoon
Forecasting with R egional Input-Output Tables
WP-92-20
P R I r i e i h R Mahidhara, and G.J.D. Hewings
.. salvc, .
A Primer on Global A uto Markets
WP-93-1
Paul D. Ballew and Robert H. Schnorbus
Industry Approaches to Environmental P olicy
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David R. Allardie Richard H. Mattoon and William A Testa
c,
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WP-93-8
The M idw est Stock Price Index—Leading Indicator
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WP-93-9
William A Strauss
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Lean M anufacturing and the D ecision to V ertically Integrate
Som e Empirical E vidence From the U .S . A utom obile Industry
WP-94-1
Thomas H. Klier
D om estic C onsum ption Patterns and the M idw est E conom y
WP-94-4
Robert Schnorbus and Paul Ballew
1
W r i gpprsre cniud
o k n ae eis otne
T o Trade or N ot to Trade: W ho Participates in REC LAIM ?
W P -94-11
Thomas H. Klierand Richard Mattoon
Restructuring & W orker D isp lacem en t in the M idw est
W P -94-18
Paul D. Ballew and Robert H. Schnorbus
ISSUES IN FINANCIAL REGULATION
Incentive C onflict in D eposit-Institution Regulation: E vidence from Australia
W P -92-5
Edward J Kane and George G. Kaufman
.
Capital A dequacy and the Growth o f U .S. Banks
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Herbert Baer and John McElravey
Bank Contagion: Theory and E vidence
W P -92-13
George G. Kaufman
Trading A ctivity, Progarm Trading and the V olatility o f Stock Returns
W P -92-16
James T Moser
.
Preferred Sources o f Market D iscipline: D epositors vs.
Subordinated Debt H olders
W P-92-21
Douglas D. Evanoff
An Investigation o f Returns Conditional
on Trading Performance
W P -92-24
James T Moser and Jacky C So
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The E ffect o f Capital on Portfolio Risk at L ife Insurance C om panies
W P -92-29
E i a Brewer H , Thomas H. Mondschean, and P i i E Strahan
ljh
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A Framework for Estim ating the V alue and
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W P -92-30
David E Hutchison, George G. Pennacchi
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Capital Shocks and Bank G row th -1973 to 1991
W P-92-31
Herbert L Baer and John N. McElravey
.
The Impact o f S& L Failures and Regulatory C hanges
on the C D M arket 1987-1991
W P -92-33
E i a Brewer and Thomas H. Mondschean
ljh
2
W r i gpprsre cniud
o k n ae eis otne
Junk Bond H oldings, Premium Tax O ffsets, and R isk
Exposure at L ife Insurance C om panies
W P-93-3
E lija h B r e w e r III a n d T h o m a s H . M o n d s c h e a n
Stock Margins and the C onditional Probability o f Price R eversals
W P -93-5
P a u l K o fin a n a n d J a m e s T. M o s e r
Is There L if(f)e After D TB?
C om petitive A spects o f C ross Listed Futures
Contracts on Synchronous Markets
W P -93-11
P a u l K o fm a n , T o n y B o u w m a n a n d J a m e s T. M o s e r
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W P -93-18
H e r b e r t L. B a e r , V ir g in ia G . F r a n c e a n d J a m e s T. M o s e r
The Ownership Structure o f Japanese Financial Institutions
W P -93-19
H esn a G en a y
Origins o f the Modern Exchange C learinghouse: A History o f Early
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W P -94-3
J a m e s T. M o s e r
The E ffect o f Bank-H eld D erivatives on Credit A ccessib ility
W P -94-5
E lija h B r e w e r III, B e r n a d e t te A . M in to n a n d J a m e s T. M o s e r
Small B usiness Investm ent Com panies:
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W P -94-10
E lija h B r e w e r III a n d H e s n a G e n a y
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An Exam ination o f Change in Energy D ependence and E fficien cy
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W P -92-2
J a c k L. H e r v e y
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W P -92-3
V e fa T a r h a n
Investm ent and Market Im perfections in the U .S. M anufacturing Sector
W P -92-4
P a u la R. W o r th in g to n
3
W orking paper series continued
B usin ess C ycle D urations and Postw ar Stabilization o f the U .S . E conom y
WP-92-6
Mark W. Watson
A Procedure for Predicting R ecessio n s w ith Leading Indicators: E conom etric Issues
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James H. Stock and Mark W. Watson
Production and Inventory Control at the General M otors Corporation
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WP-92-10
Anil K Kashyap and David W. Wilcox
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Lawrence J Christiano and Martin Eichenbaum
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WP-92-17
Kenneth N Kuttner
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T esting Long Run Neutrality
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Robert G. King and Mark W. Watson
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Charles Evans, Steven St o g n and Francesca Eugeni
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Ellen R Rissman
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W age Growth and Sectoral Shifts: Phillips Curve Redux
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Ellen R Rissman
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Steven Strongin
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WP-92-26
Bruce Petersen and Steven Strongin
The Identification o f M onetary P olicy Disturbances:
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WP-92-27
Steven Strongin
4
W orking paper series continued
Earnings L osses and D isplaced W orkers
WP-92-28
Louis S Jacobson, Robert J LaLonde, and Daniel G. Su l v n
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lia
Som e Empirical E vidence o f the E ffects on M onetary P olicy
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WP-92-32
Martin Eichenbaum and Charles Evans
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Mark H'. Watson
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Michael T K Hon ath and Mark H'. Watson
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5
W orking paper series continued
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Lawrence J Christiano and Jonas DM. Fisher
.
Identification and the E ffects o f M onetary P olicy Shocks
WP-94-7
Lawrence J C r s i n , Martin Eichenbaum and Charles L Evans
. hitao
.
Sm all Sam ple B ias in GM M Estim ation o f C ovariance Structures
WP-94-8
Joseph G. A t n i and Lewis M. Segal
loj
Interpreting the P rocyclical Productivity o f M anufacturing Sectors:
External E ffects o f Labor Hoarding?
WP-94-9
Argia M. Sbordone
E vidence on Structural Instability in M acroeconom ic T im e Series R elations
WP-94-13
James H. Stock and Mark W. Watson
The Post-W ar U .S. Phillips Curve: A R evision ist E conom etric History
WP-94-14
Robert G. King and Mark W. Watson
The Post-W ar U .S . Phillips Curve: A C om m ent
WP-94-15
Charles L Evans
.
Identification o f Inflation-U nem ploym ent
WP-94-16
Bennett T McCalliun
.
The Post-W ar U .S. Phillips Curve: A R evision ist E conom etric History
R esponse to Evans and M cC allum
WP-94-17
Robert G. King and Mark W. Watson
6
W orking paper series continued
Estim ating D eterm inistic Trends in the
Presence o f Serially Correlated Errors
WP-94-19
Eugene Canjels and Mark W. Watson
S olvin g N onlinear Rational Expectations
M odels by Parameterized Expectations:
C onvergence to Stationary Solutions
WP-94-20
Albert Marcet and David A Marshall
.
The E ffect o f C ostly Consum ption
Adjustm ent on A sset Price V olatility
WP-94-21
David A Marshall and Nayan G. Parekh
.
1
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