Terry A Marsh.

On Euler-equation restrictions on the temporal behavior of asset returns online

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Oev with strict inequality for at
t-1 It —

least one outcome.

Replacing the state prices '''^.i in (8) by a specific utility
function U(»,»), as in (5), does little to boost its content. Rubinstein
[1976], Breeden and Litzenberger [1978], and Grossman and Shiller [1982] have
shown that if (8) holds for all investors with power utility, then tt ,
will be proportional to aggregate consumption changes. Bhattacharya [1979]
showed that, as the time interval becomes infinitesimal, this proportionality

will hold for all concave utility functions as long as asset prices have

* *
continuous sample paths. Suppose that we define C /C as that

'" Y Y "*

measure of consumption for which (5) holds when U(C _ ,C ) = C _ /y + B C /y

That is:


Then define e = (^ /C ) - (C*/C* ) where {C } is the consumption
series used in the tests. Substituting in (9):

Vil ^^c^)^-^^ ^it 1 = 1 •


Vi^ ^-rj ht ^ -'^h-i^^tht^ • ^''^


That is, if tests show that (11) holds when e and Z. are not
' t It

orthogonal, the Euler equation condition for rational asset pricing will be
rejected. Alternatively, the estimate of the risk aversion parameter in (9)
will differ from that obtained by assuming that the RHS of (11) is zero.

What can we conclude about risk aversion, asset pricing, or consumption
behavior? So long as util-prob models exist in which {e } is
correlated with asset returns, and these models are at least as plausible as
that in which the RHS of (11) is set to zero, the answer is nothing. Thus,
from now on, it will be assumed that asset prices and consumers are rational
to see what can be concluded, if anything, about the joint behavior of asset
prices and nonspeculative price variable such as consumption. By reasoning
which now seems to me to be in sympathy with that of Sims [1980], we may be
able to make meaningful statements about the joint behavior of these
speculative price and non-speculative-price series from their reduced form
distributions. Note that such an approach, not coincidentally , has all the
hallmarks of the portfolio-based approaches which, as discussed earlier, reach
all the way back to Markowitz's work.

2 .3 Reduced Form Analysis of Asset Prices and Consumption
It is interesting that in the empirical work cited above on Euler equation
cum "general equilibrium" specification of asset pricing behavior, a good deal
of the specification is already in reduced form. True, the form of the
utility function is spelled out, but it is unlikely that this functional form
"matters much." Observed changes in U.S. per capita consumption expenditures
have a standard deviation below 5% per annum, and it is unlikely that
nonlinearity in utility functions is important over this range, a conjecture


which Mehra and Prescott's [1983] simulations can be interpreted as bearing
out. Bhattacharya [1979] further pointed out that nonlinearity of utility
functions won't matter in this context in the limit of continuous time, so
that empirical estimates of utility parameters obtained by Euler equation
methods will tend to reflect the differencing interval of the data (typically
monthly or quarterly). Alternatively, for a given "small" differing interval
(e.g., monthly), estimates of utility parameters obtained by "different"
empirical procedures (e.g., Hansen and Singleton [1982,1983] use a nonlinear
generalized method-of-moments procedure) will reflect nothing other than the
relative properties of the procedures.

These observations suggest that comovements between marginal utility
changes and asset returns in (5) can probably be safely replaced by
comovements between consumption changes and assets returns, a corollary of the
observations being that the forfeited estimate of risk aversion parameter will
be poor anyhow. Of course, if utility is not time separable, or if
consumption services differ markedly from consumption expenditures (e.g., Dunn
and Singleton [1984]), "all bets are off" on the influence of nonlinearity,
but it will be argued below that this only furthers the case for abandoning
the utility-based analysis.
II With the Euler equation test no longer tied to a specific form of the
utility function, the comovements between consumption changes and asset
returns will depend on shocks to utilities and/or wealth and technology. If
utility shocks, which are generally assumed away, are important, then there is
virtually no alternative to using the "reduced form" analysis. A "general
equilibrium" explanation of comovements in asset returns, consumption, and


other instrumental variables in terms of direct utility shocks "...of
unexplained origin in unmeasurable influence" (Hall [ ] , quoted in Sims
11980, p. 29]) is really no explanation at all.

This leaves the probablity-technological component of the util-prob
specification to breathe testable content into the Euler equation (5). But,
for example, Hansen and Singleton [1983] suppress the technological
specification in making the assumption that asset returns and aggregate
consumption changes are joint lognormal, while Hansen and Singleton [1982]
assume that conditional moments of the distribution of asset returns and
consumption changes are a stable function of an arbitrarily specified set of
instrumental variables.

How rich a variety of joint behavior of asset returns and consumption can
be explained by technological specification? That is, how much weight does
the "prob" specification likely have in empirical versions of Euler equation
restrictions on asset returns and consumption? In the following brief
discussion, I argue that, on a priori grounds, the answer is a great deal.

Consider first Lucas's [1978] exchange economy. There, capital consists
of (say) fruit trees. Output each period consists of the fruit which falls
from the trees. The output is stochastic, following a stationary Markov
process. Output is nonstorable, so nonsatiated investors consume all output
each period. However, when time (t - 1) output and consumption are (say)
high, investors would like to carry some of their "feast" into period t
where expected marginal utility of consumption is higher. Because output is
nonstorable, their collective efforts to postpone consumption fail, but those
attempts force up the prices of the assets (the fruit trees), and so force
down expected asset returns. As Lucas shows, the extent of variation in


expected asset returns occasioned by variation in output and consumption
realizations over time follows completely from the curvature of the utility
function and the probability distribution generating those realizations.

Now suppose that a stochastic-constant-returns-to-scale production
technology is introduced. Consumption will take place instantaneously, but
producer and consumer durables have a production cycle of at least one period,
and the payoff on investment is uncertain. If output in any period is high
because of a transitory shock, permanent income and hence consumption will be
only negligibly affected. Consumption as a fraction of output will fall,
investment as a fraction of income will rise, the ex-dividend price of assets
per dollar of investment (or per unit of consumption goods) — Tobin's q — will
remain unchanged, as will expected asset returns, which will always equal the
state-conditional expected marginal product of capital. If output is instead
temporarily low, the same conclusions will follow so long as capital can be
transformed into consumption goods as readily as consumption goods can be
transformed into capital. Although the lack of movement in expected asset
returns follows solely from the technological side in this new economy, it
could also be obtained by making utility linear in consumption.

Intermediate cases will be those where utility is not linear and the
production technology is not constant-stochastic-returns-to-sale. The latter
could arise because production itself exhibits diminishing returns to scale,
or because consumption goods cannot be instantantly transformed into any
desired capital goods. If investment takes time, there is a cost of foregone
consumption and, in an uncertain world, possibly foregone investments which
would have been preferable ex post. In this case, temporarily high output
should still be accompanied by an increase in investment, but now the shadow


price of finished capital goods in terms of consumption goods would increase,

thus forcing down the conditional expected return on any asset i,

E (Z. ) (i.e., Tobin's q would exceed unity). The unconditional

expectation of E , (Z. ) would equal the steady-state marginal product
t-1 It ^ ' e> f

of capital, but E . (Z. ) would be stochastically "shocked" away from

that long-run tendency. As was Lucas's point in the exchange economy without

risk neutrality, E ,(Z. ) need not be serially independent of Z. ,.

■' t-1 It J r l,t-l

The more general case involves serially dependent shocks in output, though
it seems that many of the features of a constant-stochastic returns to scale
model with serially dependent shocks could alternately be derived in a model
with uncorrelated shocks in which it takes time to turn consumption goods into
capital. A priori , the serially dependent shocks scenario might be most
interesting. Corporate managers spend a considerable amount of time trying to
estimate which current events and demands can be expected to "persist" and so
generate cash flows in the future if they make investments. Current

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