Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns Author(s): Hansen and . : Journal of Political Economy, Vol. 91, No. 2 (Apr., 1983), pp. 249-265 Published by: The University of Chicago Press
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Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns
Hansen UTulversity (f Chicago
. – iz)(ls it’V
This paper stu(lies the time-series behavior of asset returns an(l aggregate constlilption. Using a representative consumer model
and imposing restrictions on preferences and the joint distribution
of consurnlption andI returns, we cleduce a restricted log-linear time- series representation. Preference parameters for the representative
agent are estimated and the implied restrictions are tested using postwar data.
I. Introduction
In the asset pricing models of Rubinstein (1976b), Lucas (1978), Breeden (1979), and Brock (1982), among others, agents effect their consumption plans by trading shares of ownership of firms in a com- petitive stock market. An implication. of this trading is that the serial correlation properties of stock returns are intimately related to the stochastic properties of conssumption and the degree of risk aversion of investors. The purposes of this paper are to characterize explicitly
This research was supported in part bv NSF grant SES-8007()016. Ilelpftul comments
on earlier drafts of this p)lper were provided b)v members of the workshops at Car- negie- , the Uni versity of’ Washington, and the Summer NBER Insti-
tute. We wish, in particular. to acknowledge the helpful suggestions received from ainId anil anonymous referee. and assisted with
the COmpUtLtionS.
1 o u I ol P/ I’it I(l L(OI I .,,cotT 1 983, vol. ( 1 )I 1. 21
5 1983 [ Ty [l ‘li\’t isN o)1′ (lli(aigo. All igi tl st rscrtytd. ()()922-38()8/8’,j 1]()2-‘()()11 $()15()
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2 50 JOURNAL OF POLITICAL ECONOMY
the restrictions on the joint distribution of’ asset returns and consump- tion implied by a class of’ general equilibrium asset pricing models and to obtain maximum likelihood estimates of’ the parameters describing preferences and the stochastic consumption process.
‘Tihe motivation for this analysis derives from two considerations. First, in general equilibrium models of’ stock price behavior with risk- neutral agents (i.e., linear utility), share prices will be set so that the expected return on each asset is constant. Thus, asset returns will be serially uncorrelated and, in particular, past values of’ consumption will be uncorrelated with current-period asset returns. LeRoy and Porter (1981) and Shiller (1981) have recently conducted tests of’ the linear present-value formula ftOr stock prices, implied by this result, and in both studies the model was rejected. As Grossman and Shiller (1981) have emphasized, these rejections suggest that agents do con- sider consumption risk when making portfolio decisions. Second, if’ agents are risk averse, then the temporal covariance structure of con- sumption and asset returns will be nontrivial, except under very strong restrictions on the underlying production technology (see, e.g., Rubinstein [1976b], johnsen [1978], and Sec. II below). It is this tem- poral covariation that we attempt to characterize here.
The framework for this analysis is a production-exchange economy of’ identical agents who choose consumption and investment plans so as to maximize the expected value of a time-additive von Neumann-
Morgenstern utility function. In order to derive the restrictions on the joint distribution of consumption and stock returns implied by this optimizing behavior, it is necessary to specify a distribution func- tion and to paramneterize preferences. The joint distribution of con- surnption an1d returns is assumed to be lognormal, and preferences are assumed to exhibit constant relative risk aversion (CRRA). This particular f’ormn of’ utility was chosen in part because of its preeminent role in many previous theoretical studies of’ asset pricing (e.g., Merton
1973; Rubinstein 1976a). In addition, the assumptions of CRRA util- ity and lognormality together lead to an empirically tractable, closed- form characterization of’ the restrictions implied by the model.’
More precisely, these assumptions lead to a restricted linear time- series representation of’ the logarithms of’ consumption and asset re- turns. The restrictions imply that the predictable components of’ the
I A similar interplay among the CRRA utility function and lognormal returns was exploited by Merton (1973, 1980), Rulhinstein (1976b), Breeden (1977), and Grossman
and Shiller (1981), among others, to obtain closed-form solutions to their models.
Breeden and Litzenberger (1978) derive a version of the CAPM for a model with
CRRA utility and lognormal returns and consuLmtioll.
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STOCHASTIC CONSUMPTION 2 51
logarithms of asset returns are proportional to the predictable com- ponent of the change in the logarithm of consumption, with the pro- portionality factor being minus the coefficient of relative risk aver- sion. Maximum likelihood estimates of the coefficient of relative risk aversion, the subjective discount factor, and the parameters that de- scribe the temporal evolution of consumption are obtained using this closed-form characterization of the restrictions. The model is es- timated for returns on stocks listed on the Stock Exchange and for returns on Treasury bills, using monthly data for the period 1959:2 through 1978: 12. Then likelihood ratio tests of the joint hypothesis underlying the model are conducted.
The remainder of this paper is organized as follows. In Section II the model is described and the implied time-series representation for consumption and returns is derived. In Section III, maximum likeli- hood estimation is discussed and estimates of the parameters are pre- sented. Concluding remarks are presented in Section IV.
II. The Model of Stock Market Returns
Consider a single-good economy of identical consumers, whose utility functions are of the CRRA type:
U(c,) = c,/y; y < 1, (1)
where c, is aggregate real per capita consumption and U() is the period utility function. The representative consumer in this economy is assumed to choose a stochastic consumption plan so as to maximize the expected value of' his time-additive utility function,
EOLL f3'U(c,)1 0 < 13 < 1. (2)
In (2), 3 is a discount factor and U( ) is given by (1). The m expectation E,() is conditioned on information available to agents at
time t, I,. Current and past values of real consumption and asset
returns are assumed to be included in I,.
Consumers substitute present for future consumption by trading the ownership rights of' N financial and capital assets. These assets include default-free, multiperiod bonds that agents issue for the pur- pose of' borrowing or lending among themselves and shares of' own- ership of firms in the economy. If firms rent capital from consumers, as in Brock's (1982) model, then the stocks of' capital leased to the firms by the representative consumer will also be included among the traded assets. Let w, denote the holdings of the N assets at the date t, q, denote the vector of prices of' the N assets in w, net of' any distribu-
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252 JOURNAL OF POLITICAL ECONOMY
tions, and q* denote the vector of values of these distributions during period t. Then a feasible consumption and investment plan {c, w,} must satisfy the sequence of budget constraints,
c, + q, , w, + I - (qt + q*8) * w, + A(3)
where y, is the level of (real) labor income at date t. 2
The first-order necessary conditions for the m-naximnization of (2) subject to (3), that involve the equilibrium prices of the ta assets, are
(Lucas 1978; Brock 1982)
U'(c,) = PEI[ ((,, -)rt1) r ]; i 1 . , (4)
where r-,. I is the return on the ith asset expre consumption good. Substituting (1) into (4) and rearranging gives
EIl|( ; )rt i+I = 1; i = 1, .N. (5)
with ( =-y - 1. Breeden (1979) has derived an intertemporal capital asset pricing representation in a continuous-time environment. In his representation, expected excess returns on risky assets are linked to covariances of aggregate consumption and returns. Grossman and Shiller (1981) have shown how to obtain an analogous representation for the discrete time model studied here. Their representations are useful for studying the riskiness of a cross section of asset returns. The focus of this paper is instead on the link between forecastable movements in consumption and forecastable movements in asset re- turns. Accordingly, we proceed to derive a relation among these fore- castable components implied by (5).
For the analysis of (5) that follows, it is not necessary to examine explicitly firms' production decisions, since it is not our goal to solve for an explicit representation of equilibrium prices in terms of the underlying shocks to technology. By assuming that the joint distribu- tion of consumption and returns is lognormal, we are implicitly im- posing restrictions on the production technology, however. As in many previous theoretical and empirical studies of asset pricing (see n. 1 above), we leave unspecified the exact nature of these restrictions.
A formal Justification of the assumption of lognormiality can be pro vided for some economic environments for which closed-form equi-
librium pricing functions have been derived. Such a justification has
not been provided at the level of generality at which our empirical
analysis is conducted, however. We adopt our general representation
2 Ihe inclusion of y, (toes not affect our analysis if labor is supplied inelastic Alternatively, we can introduce a period t labor supply variable, L, into the specifica-
tion of U" anid let VT(,, L,) = UI(c,) - U,(L,), where L, is a choice variable of the
constimer. For this case, Y, = LW,, where W, is the real wage rate at (late t.
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STOCHASTIC CONSUMPTION 253
to accommodate a rich temporal covariance structure which might emerge when the investment environment faced by firms is more complicated than the environments in the models of Lucas (1978) and Brock (1982) (e.g., serially correlated production shocks, costly ad- justment in altering capital stocks, and gestation lags in producing new capital).
From (5) and the accompanying assumptions, a restricted linear time-series representation of the logarithms of consumption and asset returns can be derived. Suppose that observations on the first a of the N assets traded by economic agents are to be used in the econometric analysis. Let x, - /c,c l and ua, n-,, i = 1, . Then (5) can be rewritten as
El -I(u,-) =1/3, i n. (6)
Next, let X, -log x,, R,, = log r11 Y. = (X,, R I.., ,)', Ui, = log Ua (i = 1, . . . a), and W,- 1 denote the information set {Y, _: s ? 1}. Further, assume that {Y.} is a stationary Gaussian process. This distri-
butional assumption implies that the distribution of Ud, conditiona 1 is normal with a constant variance (T. and a mean Pi,- 1 that is a
linear function of past observations on Y,. Hence,
) = exp + + (&/2)]. (7)
Since got 1 C I, 1, we can take expectations of both sides of (6) conditional on tJI, 1 to obtain
E(uijI - 1) 1/13. (8)
Equating the right-hand sides of equations (7) and (8) and solving for Li _ yields j, I -log 13 - Define
V,- U., - I= cX, + R,, + log 1 + (cr /2),
i= 1.=. (9)
Then, E (Vi, t, W1) = (and
E (Ri 1-l) = -x (X, I t, -X)- log13 - (o('2/9), 1 it.(I0
Equations (9) and (1(0) summarize the relationships among serial cor- relation of consumption, the level of risk aversion, and serial correla- tion of asset returns implied by the first-order conditions (5). Risk neutrality, for example, corresponds to the case of (Y = 0, which implies that R., is equal to a constant plus the serially uncorrelated error Vi, and hence that Rd, is serially uncorrelated, i = 1, n. Alternatively, if ' = -1, then agents have logarithmic utility func- tions. In this case, R,, -X, = -log 13 - (o/2) + Vi,. Thus, the slope
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2 54 JOURNAL OF POLITICAL ECONOMY
coefficients in the projections of R, and X, onto a subset 4, 1 of' I,_ must be the same, and this equality must hold for the returns on all assets. More generally, (10) implies that (ignoring constant terms) the coefficients in the projection of' R, onto tj, -I are equal to the coeffi- cients in the projection of X, onto 4,,- multiplied by -a.
To translate these observations into statements about the predict- ability of asset returns, it is useful to derive an expression for the coefficient of determination (Ri) from the projection of R-, onto 4,_, implied by (10). By definition,
2i = var [E (RI1.,, )] (11)
var (Ritlj,_ i) + var [E(R
where var is the variance operator. From (10) it follows that the vari-
ances of' the predictable components of' log rd, and log (c,/c, i) are related by the expression:
var [E (Ri, W1 J, )] = ai var [E(X,|,_)]. (12) Substituting (12) into (11) gives
2'9 = otx var [E(X, +tj- I)]
var (R.,lt , -) + ox2 var [E(X, 1W,- I)](
From (13) it follows that a necessary condition for asset returns to have predictable components is that agents be risk averse (a # 0).
Risk aversion is not a sufficient condition for predictability, how-
ever. For the special case in which the projection E (XI t, I) is con- stant, the R.'s are equal to zero or, equivalently, the projections of the
Rd, onto 4,_ I are constants. This implication of our model is consisten with the conclusion of' Rubinstein (1976b) that asset returns will be
serially uncorrelated when consumption follows a logarithmic ran-
dom walk and agents have CRRA preferences. When there are non-
trivial predictable components in X, and a 7# 0, then real asset returns will also have predictable components.
The assumption that the vector process {Y,} is stationary and Gauss- ian implies that the conditional expectations in (10) have linear, time- invariant representations and that the conditional variances are con- stant (a fact that we have exploited above). Thus, the movements in the conditional distributions of' the logarithms of consumption and asset returns are completely summarized by movements in the condi- tional means. This distributional specification leads to a very conve- nient representation of' the intertemporal behavior of' consumption and asset returns for the purposes of empirical analyses. Once the
projection E(X,1W,_i) is parameterized as a linear function of values of' Y, the free parameters of (10) can be estimated by the
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STOCHASTIC CONSUMPTION 255
method of maximum likelihood and the overidentifying restrictions can be tested using the likelihood ratio statistic. Since our characteri- zation of the overidentifying restrictions relies on an assumption about the joint distribution of consumption and returns, rejection of these restrictions may result from misspecifying that distribution rather than from the empirical failure of the time-additive CRRA preference form of the asset pricing model.
Other authors have studied this asset pricing model by relying on the same distributional assumption as that employed here. Grossman and Shiller (1981) have shown how to identify preference parameters under a joint lognormality assumption on consumption and returns. They abstain from studying the intertemporal correlations of these variables and express the estimators of their preference parameters as functions of the first and second unconditional moment of two re- turns. Hall (1981) has independently adopted an approach that is very similar to the one employed here to estimate at for different data sets.3 Neither of these studies considers tests of overidentifying restrictions.
III. Maximum Likelihood Estimates of the Parameters To proceed with estimation, we assume that
L(XI, J = a(L)'Y,l + jx, (14)
where a(L) is an n + 1 dimensional vector of finite order polynomials in the lag operator L. Combining equations (14) and (9) gives
A (Y, = AI(L)YI - + p. + V,, (15)
where V, = (W., V, . V1,)' and W.-, = - E(X,|+,_ i). The matrix A() is given by
A'0 a 1~,1
with a = (ox a, . . ., a)' and I an n X n identity matrix; the matrix lag polynomial A,(L) is given in partitioned form by
Al(L) = a(L)'
- There are two differences in the estimation strategy employed by Hall (198 1 he assumes that economic agents do not know the true parameter values in the forecast-
ing equation for asset returns. Instead, they use Bayesian updating formulas as they
accumulate new information over time about these parameters. Second, he expands the
vector Y, to include variables other than asset returns and consuMption.
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256 JOURNAL OF POLITICAL ECONOMY and the vector of constants p. is given by
p. = [jy, log 13 ? (2/2), . . .,log 13 ? (4/2)]'.
From equation (9) it follows that {(W,_ ,, V,_ ,,...,V,,, ); s - O} spans the space 4,. Hence, the autoregressive representation of Y, is ob- tained by premultiplying both sides of (15) by A(7 -1
Now let 0 denote the vector of unknown parameters containing a,
13, vx, the parameters of a(L), and the elements of the covariance matrix of V,, denoted by E. It is assumed that I is nonsingular. Sup-
pose that T observations on Y are available for estimation of 0. Then,
in view of the relation (15), the joint density function of the sample, conditioned on the initial values of the variables, is given by
f(,,., YT; 0) (2iTr) 1+)T/211 -TT/2 (16)
exp -(4) E [AOY,- AI(L)Yl - I ]' [AOY, - AI(L)Yl - I j.
Note that (16) is also the joint density function of (V1, . VT), since
the Jacobian of the transformation AO I that transforms (15) into the autoregressive representation is unity. The logarithm of the condi-
tional likelihood function (16) is, up to a constant term,
L(Y1 .... , YT; 0) T
-(T12) log III - (1) > [AWY, -A1(L)Y, 1 – p]’Z 1 (17)
x [AOY, -AI(L)Y,/ – I t].
The maximum likelihood (ML) estimate of O is obtained by maximiz-
ing (17). Unfortunately, unless n = 1, the conditional log-likelihood function cannot be concentrated, because FL is a function of the pa- rameters in
Estimates were obtained using monthly data for the period 1959 :2 through 1978: 12. The monthly, seasonally adjusted real consump- tion series, dating back to January 1959, were obtai
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