1D28

,M414

),

•..2

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]

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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:

Y Y

Then define e = (^ /C ) - (C*/C* ) where {C } is the consumption

series used in the tests. Substituting in (9):

Vil ^^c^)^-^^ ^it 1 = 1 •

t-1

Vi^ ^-rj ht ^ -'^h-i^^tht^ • ^''^

-12-

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

-13-

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

-14-

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

-15-

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

-16-

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