Macroeconomics and Asset Markets: some

Mutual Implications: a note on adding

Epstein-Zin preference features under the

assumption of a unit root in TFP.∗

Harald UhligHumboldt University Berlin

Deutsche Bundesbank, CentER and CEPR

PRELMINARYCOMMENTS WELCOME

First draft: August 15, 2004This revision: September 30, 2005

∗I am grateful to John Campbell for a useful exchange of thoughts, to Kjetil Storeslet-ten for a useful conversation and to Wouter den Haan for useful comments. The viewsexpressed herein are those of the author and not necessarily those of the Bundesbank.This research was supported by the Deutsche Forschungsgemeinschaft through the SFB649 “Economic Risk” and by the RTN network MAPMU. Address: Prof. Harald Uh-lig, Humboldt University, Wirtschaftswissenschaftliche Fakultat, Spandauer Str. 1, 10178Berlin, GERMANY. e-mail: uhlig@wiwi.hu-berlin.de, fax: +49-30-2093 5934, home pagehttp://www.wiwi.hu-berlin.de/wpol/

1 The model

We use capital letters to denote the original variables, and small letters todenote log-deviations from steady state (unless explicitly stated otherwise).

Let the representative agent or the social planner solve

max(Vt,Ct,Lt,Xt,Nt,Kt)

V0 (1)

s.t. Vt = (1 − β)U(Ct,ΦtLt; Φt) (2)

+β exp((1 − η)µΦ)H−1 (Et [H(Vt+1)])

Ct +Xt = Yt = F (Kt−1,ΓtNt) (3)

Kt = (1 − δ)Kt−1 + ΞtG

(

Xt

Kt−1

)

Kt−1 (4)

1 = Nt + Lt (5)

I.e., the social planner maximizes the recursively defined welfare Vt of a rep-resentative agent, given by a concave, differentiable and strictly increasingfelicity (or per-period utility) U(·, ·; ·) in consumption Ct and leisure Lt and astrictly monotone and differentiable transformation H(·; Φt). The maximiza-tion is subject to a feasibility constraint, that consumption and investmentXt add up to output Yt, which is produced according to the concave, differen-tiable and strictly increasing production function F (·, ·) with predeterminedcapital Kt−1 and labor Nt. I assume that the production function has con-stant returns to scale. Capital in turn can be produced by adding investment,subject to the concave adjustment cost function G(·). Initial capital K−1 isgiven, and I assume that H(·) is such that V0(K−1) is concave. As is stan-dard in the literature, I assume that G(δ) = δ and G′(δ) = 1, so that thefirst-order behavior of the capital accumulation is the same as the usual no-adjustment-cost equation. There is one unit of time as endowment, whichcan be split between labor and leisure.

There are three exogenous stochastic processes. The process Γt repre-sents labor-augmenting technological progress. I shall assume that log Γt hasa unit root and a deterministic drift, and hence, that some part of the sur-prise technology technology innovations have a permanent impact. The mostsimple process of this kind would be ∆ log Γt = µΓ + ǫΓ,t, where µG is thenonstochastic growth factor and ǫΓ,t are productivity shocks. The processΞt represents investment-specific technology shifts, which we assume to be

1

stationary, with a nonstochastic steady state X = 1. The process Φt finallyrepresents a preference shifter or - alternatively - a productivity parameterfor turning non-working time Lt into quality leisure ΦtLt. I shall assume thatlog Φt− log Γt is stationary. Thus, the long-run progress to improving overallefficiency of labor in production equals the long-run progress in improvingthe quality of leisure1 and the nonstochastic growth factor µΦ for the pref-erence shifter is the same as for technology. For technical reasons, we writethe discount factor for preferences as a product of an expression involvingthis growth factor and some parameter η > 0 as well as a parameter β.

Note that if H(V ) ≡ V , then the problem can be rewritten in the morefamiliar form with a discounted sum of expected utilities,

maxE

[

∞∑

t=0

(β exp((1 − η)µΦ))t U(Ct,ΦtLt; Φt)

]

Ct +Xt = Yt = F (Kt−1,ΓtNt)

Kt = (1 − δ)Kt−1 + ΞtG

(

Xt

Kt−1

)

Kt−1

1 = Nt + Lt

1.0.1 Recursive Preferences

The formulation above allows for the separation between intertemporal sub-stitution and risk aversion, i.e. allows for Epstein-Zin (1989, 1991) prefer-ences. Here I use and generalize the formulation of Tallarini (2000). Like thisauthor, I apply the recursive formulation to a felicity function U(Ct,ΦtLt; Φt).While Tallarini (2000) assumes U(Ct, Lt) = logCt + θ logLt, I allow moregeneral specifications.

I allow the preference shifter to also appear as a separate argument of thefelicity function U(C,L; Φ). The reasons is largely technical and driven bythe desire to allow for balanced growth in which intertemporal substitutionacross periods, relative risk aversion as well as marginal rates of substitutionbetween consumption and (quality-adjusted) leisure do not shift. To thatend, I shall impose some further parametric restrictions regarding the felicity

1This assumption frees up some functional restrictions on the utility function U(C,L),see subsection 1.0.1.

2

function U(·, ·; ·) as well as the aggregator function H(·; ·). I assume that

U(ΦC,ΦL; Φ) = Φ1−η(

U(C,L) + χ)

− χ (6)

andH(V ) = ((1 − η)(V + χ))

1−ν1−η (7)

for some parameters η > 0 already appearing in the discount factor in (2)and furthermore ν > 0, χ and χ. I assume that these parameters are chosensuch that U(·, ·) + χ ≥ 0 and such that χ and χ satisfy the equation

(1 − β exp((1 − η)µΦ))χ = (1 − β)χ

Define

Ct =Ct

Φt

Vt = Φη−1t (Vt + χ) − χ

Equation (2) can be rewritten as

Vt = (1 − β)U(Ct, Lt) + (8)

β exp ((1 − η)µΦ))H−1

(

Et

[

(

Φt+1

Φt

)1−ν

H(Vt+1)

])

We shall use the nonstochastic steady state as the benchmark, around whichwe loglinearize all equations. At the nonstochastic steady state, the growthof the preference shifter is deterministic, Φt+1/Φt = exp(µΦ). Let V , U , Cand L be the values of Vt, Ut, Ct and Lt at that nonstochastic steady state.It follows directly, that

V = U (9)

We finally assume that

1 =U1

(

C, L)

C

U(10)

This can always be assured by shifting the intercept of the felicity functionU(·, ·), and allows us to calculate random shifts in the value function V interms of equivalent permanent increases in consumption.

3

A particular case would be e.g.

U(C,L) =(C1−αLα)

1−η

1 − η− χ (11)

whereχ =

η

1 − η

(

C1−αLα)1−η

(12)

and thusχ =

η

1 − ηU(C, L) =

η

1 − ηV

in order to achieve (10. With this choice and holding C, L constant,

U(C,L; η) → (1 − α) log(C) + α log(L) + 1 −(

(1 − α) log(C) + α log(L))

as η → 1. This thus delivers the preference specification in Tallarini (2000)except for the constant intercept. Likewise,

H−1 (E[H(Vt+1]) → log

(

ℵ1 − ν

E[

exp(

1 − ν

ℵ Vt+1

)]

)

where

ℵ =1 − β

1 − β exp((1 − η)µΦ)V

Let

ζ = −H′′(V )V

H ′(V )=

be the elasticity of the function H(·), measuring the degree of curvature indeparting from the benchmark expected discounted utility framework. Inthe example (11) and (7) with (12), this calculates to

ζ = (ν − η)

(

1 − η

(

1 − 1 − β

1 − β exp((1 − η)µΦ)

))

−1

and thus measures the difference between the the risk aversion parameterν and the inverse of the intertemporal elasticity of substitution η modulo aterm near unity, due to trend log utility growth. In particular, for µΦ = 1,we have

ζ = ν − η

4

Further, let

ηcc = − U11(C, L)C

U1(C, L)

ηll = − U22(C, L)L

U2(C, L)

ηcl,c =U12(C, L)C

U2(C, L)

ηcl,l =U12(C, L)L

U1(C, L)

which characterize the curvature properties of the felicity function U . Notethat ηcc ≥ 0 is the usual risk aversion with respect to consumption, ηnn ≥ 0is risk aversion with respect to leisure, and ηcn,c as well as ηcn,n are cross-derivative terms.

It is useful to define

κ =ηcl,c

ηcl,l

=U1(C, L)C

U2(C, L)L

Note that this is the ratio of the expenditure shares for consumption toleisure, provided the usual first-order condition relating wages (in units ofconsumption) to the ratio of marginal utilities holds.

A few more restrictions can be derived on the curvature characteristics.First, concavity of U implies that

ηll ≥ηcl,cηcl, l

ηcc

=κη2

cl,l

ηcc

Second, the scaling assumption (6) implies

C O N T I N U E H E R E

1.1 Analysis

1.1.1 First-order conditions

Introduce (shadow) wages and (shadow) dividends as the marginal productof labor and capital,

Wt = F2(Kt−1,ΓtNt)

5

Dt = F1(Kt−1,ΓtNt)

where F1 and F2 denote the partial derivatives of F (·, ·) with respect to thefirst resp. the second argument.

Let βtΩt, βtΛt and βt(Λt + Ψt) be the Lagrange multipliers on the first,

second and third constraint (2), (3), (4). Thus, βtΨt is the difference betweenthe Lagrange multipliers on the third and the second constraint, and is zero,if the adjustment cost function is linear. The first-order conditions are

∂

∂Vt

: Ωt−1

(

H−1)

′

(Et−1 [H(Vt)])H′(Vt) = Ωt

∂

∂Ct

: (1 − β)ΩtU1(Ct,ΦtLt) = Λt

∂

∂Lt

: Φt(1 − β)ΩtU2(Ct,ΦtLt) = ΛtWt

∂

∂Xt

: Λt = (Λt + Ψt) ΞtG′

(

Xt

Kt−1

)

∂

∂Kt

: Λt = βEt [Λt+1Rt+1]

where Rt+1 is the gross return in terms of the consumption good of investingin the capital stock,

Rt+1 = ΞtG′

(

Xt

Kt−1

)

Dt+1 +1 − δ + Ξt+1G

(

Xt+1

Kt

)

Ξt+1G′

(

Xt+1

Kt

) − Xt+1

Kt

This expression will simplify considerably in the log-linearized version below.

1.1.2 Detrending

To analyze this model further, I rewrite the model in terms of detrendedvariables,

Ct =Ct

Φt

Kt =Kt

Γt

6

Xt =Xt

Γt

Yt =Yt

Γt

Wt =Wt

Φt

but no detrending forDt, Nt or Rt. Note that I am detrending Ct andWt withthe preference shifter Φt rather than the production technology parameter Γt.This is a matter of convenience only, as it allows me to keep the discussionof preferences and its first-order conditions in terms of preference-specificvariables only. Let

µΓ,t =Γt

Γt−1

µΦ,t =Φt

Φt−1

be the growth factors for the exogenous stochastic processes Γt and Φt.The feasibility equations become

Vt = (1 − β)U(ΦtCt,ΦtLt) + βH−1 (Et [H(Vt+1)])

Φt

Γt

Ct + Xt = Yt = F (Kt−1

µΓ,t

, Nt)

Kt = (1 − δ)Kt−1

µΓ,t

+ ΞtG

(

µΓ,tXt

Kt−1

)

Kt−1

µΓ,t

1 = Nt + Lt

The first-order conditions are

∂

∂Vt

: Ωt−1

(

H−1)

′

(Et−1 [H(Vt)])H′(Vt) = Ωt

∂

∂Ct

: (1 − β)ΩtU1(ΦtCt,ΦtLt) = Λt

∂

∂Lt

: (1 − β)ΩtU2(ΦtCt,ΦtLt) = ΛtWt

∂

∂Xt

: Λt = (Λt + Ψt) ΞtG′

(

µΓ,tXt

Kt−1

)

7

∂

∂Kt

: Λt = βEt [Λt+1Rt+1]

where Rt+1 can be rewritten in terms of the detrended variables as

Rt+1 = ΞtG′

(

µΓ,tXt

Kt−1

)

Dt+1 +1 − δ + Ξt+1G

(

µΓ,t+1Xt+1

Kt

)

Ξt+1G′

(

µΓ,t+1Xt+1

Kt

) − µΓ,t+1Xt+1

Kt

1.1.3 The nonstochastic steady state

Consider the nonstochastic steady state (or nonstochastic balanced growthpath), where only the constant deterministic drift remains for log Γt andlog Φt and where Ξt ≡ 1. We assume w.l.o.g. that Γt = Φt for the non-stochastic steady state, keeping level normalizations to the specification ofthe functional form of F (·, ·). Thus, Φt = Γt = Γ0 exp tµΓ.

We shall use bars on top of all variables to denote the variables or ratiosof variables to Γt in the nonstochastic steady state. In particular, let C bethe ratio of Ct to Φt, K be the ratio of Kt−1 to Γt and Y be the ratio of Yt

to Γt, while Nt ≡ N and Rt ≡ R. Note already that

R = D + 1 − δ (13)

as usual, where we keep in mind that bars denote the nonstochastic steadystate and not the mean of the stochastic economy. Indeed, for asset pricingpricing implications, this difference is key and we shall explore it furtherbelow.

1.1.4 Normalizations and Log-Linearization

Let

θ =FK(K, N)K

F (K, N)

ϕkk = − FKK(K, N)K

FK(K, N)

ϕnn = − FNN(K, N)N

FN(K, N)

8

which characterize the curvature properties of the production function. Notethat θ is the capital share, while ϕkk ≥ 0 and ϕnn ≥ 0 are the elasticities ofdividends with respect to capital and of wages with respect to labor. Due toconstant returns to scale, it is easy to see (and probably well known that)

ϕkk =FKN(K, N)N

FK(K, N)

ϕnn =FKN(K, N)K

FN(K, N)

For a Cobb-Douglas production function, we have ϕkk = 1 − θ and ϕnn = θ,but in general, this does not have to be the case.

Finally, let

= − 1

G′′(δ)δ> 0

which is the traditional notation, and coincides with the parameter for thespecific cost-of-adjustment functional form

G

(

Xt

Kt−1

)

=a1

1 − 1/

(

Xt

Kt−1

)1−1/

+ a2

with a1 and a2 chosen so that G(δ) = δ,G′(δ) = 1, see also Jermann(1998), Hornstein and Uhlig (2000) and Boldrin et al (2001). The benchmarkcase of no adjustment costs is = ∞.

Let Λt and (Λt + Ψt) be the Lagrange multipliers on the first and secondconstraint, i.e. Ψt is the difference between the Lagrange multipliers on thesecond and the first constraint, and is zero, if the adjustment cost functionis linear.

Loglinearizing all equations around the steady state (and using smallletters to denote the loglinear deviations) leads to

yt =X

Yxt +

(

1 − X

Y

)

ct (14)

yt = θkt−1 + (1 − θ)nt (15)

kt = (1 − δ)kt−1 + δxt (16)

wt = zt + ϕnn(kt−1 − nt) (17)

9

dt = zt − ϕkk(kt−1 − nt) (18)

lt =1 − L

Lnt (19)

vt = (1 − β)(ct +1

κlt) + βEt[vt+1] (20)

ωt = ωt−1 − ζ (vt − Et−1[vt]) (21)

λt = ωt − ηccct + ηcl,llt (22)

λt + wt = ηcl,cct − ηlllt (23)

ψt =1

(xt − kt−1) (24)

rt =R− 1 + δ

Rdt − ψt−1 +

1

Rψt (25)

0 = Et [λt+1 − λt + rt+1] (26)

For convenience, we have collected these equations also as table 3 in theappendix.

Equation (20) is the loglinearization of (2), where I have made use of(??). Further, suppose that we observe a permanent one-percent increasein consumption ct ≡ 1, t ≥ 0, compared to the steady state, while leisure isunchanged. Then (20) implies that vt = 1. Thus, vt measures the change inwelfare in terms of permanent changes in consumption.

Note that (20) does not depend on the specifics of the function H(·).Rather, the elasticity of H(·) matters for the first order conditions (21) and(22). If ζ = 0, which is the benchmark case of welfare as the discounted sumof expected utilities, then ωt ≡ 0, starting at the steady state ω−1 = 0.

1.2 Parameter restrictions

The parameter calibrations and theoretical and numerical restrictions aresummarized in table 1.

10

Restrictionsparameter theoretical economic calibration

θ free capital share 0.36δ free deprec. rate 0.025R free gross cap. return 1.01ϕnn free elast. of wages θ (Cobb-Douglas)ϕkk free elast. of div. 1 − θ (Cobb-Douglas) ≥ 0 free adj. cost 0.23 or ∞L free leisure share 2/3ηcc free cons. risk. avers. [1,∞)ηcl,l free cross derivative (−∞,∞)ζ free risk av. - 1/subst.elas. [0,∞)XY

= δθR−1+δ

investm. share 25.7%

κ =ηcl,c

ηcl,l= (1−L)

L

(1− X

Y )(1−θ)

rel. expend. shares 0.58

ηll ≥ κη2cl,l

ηccleisure risk.av. [0,∞)

Table 1: The list of parameters of the basic model and their restrictions.

2 Asset price implications

2.1 General remarks

Equipped with the utility function above, we can study the asset price im-plications. For convenience, we collect some well-known implications of log-linear asset pricing, see e.g. Lettau and Uhlig (2002) or Campbell (2004).Generally, for any asset with gross return Rt+1 (not just investment in phys-ical capital), the Arrow-Lucas-Rubinstein asset pricing equation has to besatisfied,

1 = Et[βΛt+1

Λt

Rt+1] (27)

or0 = log β + log

(

Et

[

exp(

∆λt+1 + rt+1

)])

(28)

where λt+1 = log Λt+1, etc., and where ∆ denotes the first difference. A“period” here shall be interpreted to be the relevant investment horizon. Forexample, while trading costs (and, in some countries, Tobin taxes) probably

11

are a major friction for short investment horizons such as a few months, theypresumably matter less, if the horizon is several years. Thus, we shall abstractfrom trading costs, despite the considerable attention they have attracted,see e.g. Luttmer (1999), and instead investigate a variety of investmenthorizons. A further reason for considering different investment horizons is thereturn predictability, which has been observed at longer rather than shorterhorizons.

Assume that, conditionally on information at date t (and where we as-sume that Λt is part of that information), Λt+1 and Rt+1 are jointly lognor-mally distributed. Let σ2

·,t denote conditional variances, cov·,·,t conditionalcovariances and ρ·,·,t conditional correlations, given information up to andincluding t. For example

covλ,r,t = Et

[(

λt+1 − Et[λt+1])

(rt+1 − Et[rt+1])]

σ2λ,t

= covλ,λ,t

ρλ,r,t =covλ,r,t

σλ,t σr,t

These variances, covariances and correlations may depend on time, as indi-cated above, although we shall concentrate attention on the homoskedasticcase when studying the macroeconomic model. Note that

covλ,r,t = cov∆λ,r,t

σλ,t = σ∆λ,t

ρλ,r,t = ρ∆λ,r,t

Using the standard formula for the expectation of lognormally distributedvariables, equation (28) can be rewritten as

0 = log β + Et[∆λt+1] + Et[rt+1] +1

2

(

σ2λ

+ σ2r + 2ρλ,rσλσr

)

(29)

This can be simplified further. First, note that for the risk-free rate rft ,

i.e. for an asset with σ2r = 0, we have

rft = − log β − Et[∆λt+1] −

1

2σ2

λ,t(30)

We see that the risk-free rate varies over time either due to variations inthe expected growth rate of the shadow value of wealth, Et[∆λt+1], or its

12

conditional variance, σ2λ,t

. Since the risk-free rate does not fluctuate very

much, either these terms do not fluctuate very much, or their fluctuationsjust offset each other.

Second, for a risky asset, note that

logEt[Rt+1] = Et[rt+1] +1

2σ2

r,t

Let SRt denote the Sharpe ratio of that asset, calculated as the ratio of therisk premium or equity premium and the standard deviation of the log return,

SRt =logEt[Rt+1] − rf

t

σr,t

The Sharpe ratio is the “price for risk”, and generally a more useful numberthan the equity premium itself, see Lettau and Uhlig (2002) for a detaileddiscussion. We find that

SRt = −ρλ,r,tσλ,t (31)

In particular, we see that the maximally possible Sharpe ratio SRmaxt for any

asset isSRmax

t = σλ,t (32)

which depends on preferences only.

2.2 Consumption and leisure

We now apply this standard logic to the preference specification above. Sincethe model was formulated such that there is a steady state, the results abovestay valid, if we replace the logarithms of the Lagrange multiplier with thelog-deviations, etc.., except that for comparison to the data, one ought tokeep in mind (and possibly correct the formulas with) the average expectedconsumption growth rate.

Equation (22) states the log deviation of the Lagrange multiplier to be

λt = ωt − ηccct + ηcl,llt

Consistent but slightly more restrictive than above, we shall assume, thatasset returns, consumption and leisure are jointly lognormally distributed,conditional on information at date t. Thus,

Et[∆λt+1] = −ηccEt[∆ct+1] + ηcl,lEt[∆lt+1]

13

for the expected change in the shadow value of wealth for the risk free rateequation (30), where we have exploited that Et[∆ωt+1] = 0 per (21). Furtherand similar to the derivation of the Sharpe ratio formula above,

SRt = −ρω,r,tσω,t + ηccρc,r,tσc,t − ηcl,lρl,r,tσl,t (33)

In principle, thus, it appears as if nonseparability between consumptionand leisure can help. Without recursive utility, σω,t ≡ 0, and with preferencesseparable in consumption and leisure, ηcl,l = 0, a high relative risk aversionηcc is usually required to explain the observed Sharpe ratio. However, withthese additional terms, in particular with the appropriate value for the cross-derivative term ηcl,l, one can now vary ηcc considerably. This comes at a price.A higher absolute value for ηcl,l requires a higher relative risk aversion inleisure, see equation (??). Furthermore, these choices will have consequencesfor the endogenous choices in the macroeconomic model above.

2.3 On the correlation of stock returns and the La-

grange multiplier on recursive preferences

In order to take the formulas above to the data, one needs to calculate theterm ρω,r,tσω,t in the expression (33) for the Sharpe ratio.

To that end and for any variable x, define the date-t news about xt+j per

ǫx,j,t = Et[xt+j] − Et−1[xt+j]

Next, note with (20), that

ǫv,j,t = Et[vt+j] − Et−1[vt+j]

= (1 − β)(

ǫc,j,t +1

κǫl,j,t

)

+ ǫv,j+1,t

With this and equation (21), note now that

ωt+1 − Et[ωt+1] = −ζ(vt+1 − Et[vt+1])

= −ζǫv,0,t+1

= −ζ(1 − β)∞∑

j=0

βj(

ǫc,j,t+1 +1

κǫl,j,t+1

)

14

where I have telescoped out the previous equation. Hence, for equation (33),use

ρω,r,tσω,t = −ζτc,r,t + 1

κτl,r,t

σr,t

where

τc,r,t = (1 − β)∞∑

j=0

βjEt [ǫc,j,t+1ǫr,0,t+1]

τl,r,t = (1 − β)∞∑

j=0

βjEt [ǫl,j,t+1ǫr,0,t+1]

These quantities can be calculated from the data as follows. Let Yt be avector of variables and their lags up to some maximal lag length, containingin particular consumption ct, leisure lt and returns rt as the first, second andthird variable. Suppose that one can summarize the correlation structure inform of a VAR,

Yt = BYt−1 + ut, Et−1[utu′

t] = Σt−1 (34)

where one lag in the VAR here shall suffice due to “stacking” lags of the vari-ables into the vector Yt. Note that I used the somewhat unconventional datet− 1 for the variance-covariance matrix of the innovation dated t: that way,the VAR notation is consistent with the notation for conditional covariancesabove. Note that I allow for heteroskedasticity in the innovations but not inthe VAR coefficients. The news about consumption and leisure is now givenby

ǫc,j,t+1 =(

Bjut+1

)

1

ǫl,j,t+1 =(

Bjut+1

)

2

The covariance with the return in news is now

Et[ǫc,j,t+1ǫr,0,t+1] =(

BjΣt

)

13

Et[ǫl,j,t+1ǫr,0,t+1] =(

BjΣt

)

23

Summing up, one now obtains

τc,r,t = (1 − β)

∞∑

j=0

βjBj

Σt

13

15

= Q(β, t)13

τl,r,t = Q(β, t)23

whereQ(β, t) = (1 − β) (I − βB)−1 Σt

and where the dependence on the preference parameter β and time t hasbeen indicated via the argument.

The same VAR allows one to easily calculate the standard deviation ofthe news in returns as well as the correlations between consumption newsor leisure news with return news. Substituting this into the Sharpe ratioformula (33), I obtain

SRt

√

(Σt)33 = ζQ(β, t)13 +ζ

κQ(β, t)23 + ηcc (Σt)13 − ηcl,l (Σt)23 (35)

This equation shows the relationship between the data and the preferenceparameters ζ, β, κ, ηcc and ηcl,l.

To calculate the maximal Sharpe ratio per equation (32), write

ǫλ,0,t+1 = ǫω,0,t+1 − ηccǫc,0,t+1 + ηcl,lǫl,0,t+1

= −ηccǫc,0,t+1 + ηcl,lǫl,0,t+1 − ζ(1 − β)∞∑

j=0

βj(

ǫc,j,t+1 +1

κǫl,j,t+1

)

Define the matrix A(β, t) per

A(β, t) =∞∑

j=0

β2jBjΣt(B′)j

The matrix A(β, t) thus solves the equation

A(β, t) = Σt + β2BA(β, t)B′

orvec(A(β, t)) = vec(Σt) + β2(B ⊗B)vec(A(β, t))

where vec(·) denotes columnwise vectorization and ⊗ is the Kronecker prod-uct. The explicit solution is given by

vec(A(β, t)) = (I − β2(B ⊗B))−1vec(Σt)

16

With this, calculate

σ2λ,t

= ~a′Σt~a− 2~a′Σt~b+~b′A(β, t)~b

where the vectors ~a, ~b are defined as

~a =

−ηcc

ηcl,l

0...0

, ~b =

ζ(1 − β)ζ(1 − β)/κ

0...0

(36)

To calculate the maximal Sharpe ratio in equation (32), take the square rootof σ2

λ,t.

2.4 k-Period Asset Holdings

Since there may be frictions and preventive trading costs in adjusting port-folios on a quarterly basis, it may be more sensible to apply the asset pricingformulas to a holding period of k rather than one period.

The derivation is quite similar, and a sketch suffices. The Lucas assetpricing formula is

1 = Et

[

Λt+k

Λt

Rt+1Rt+2 . . . Rt+k

]

Define the compounded log return as

rt,t+k =k∑

i=1

rt+i

where rt+i = logRt+i. Define the normalized Sharpe ratio as

SRk,t =(Et[rt,t+k]) − rf

t,t+k√kσrt,t+k,t

where rft,t+k is the k−period risk free compounded log return at date t. Since

it is the difference of the log returns that matters, it usually does not muchmatter whether both returns are calculated in real terms or in nominal terms.

17

The calculations in nominal terms are usually easier due to the availability ofsuitable data. Obviously, if rf

t,t+k is a safe nominal return, it will not be a safereal return. This matters if unpredictable inflation volatility is substantial:the Sharpe ratio would then not fully reflect the excess return of a risky overa safe asset.

Note that if the expected k-period log excess return can be written as asum of per-period excess returns re

t+i,

(

k∑

i=1

rt+i

)

− rft,k =

k∑

i=1

ret+i

in such a way that the per-period excess returns are iid, then SRk,t = SR1,t =SRt, i.e. the normalized Sharpe ratio is independent of the holding horizon.Since excess returns are known to be somewhat predictable, one would notgenerally expect this independence from the horizon k to hold exactly, butit provides a benchmark for comparison.

For the general case and under joint log-normality, equation (31) nowbecomes

SRt,k = −ρλt+k,rt,t+k,tσλt+k,t/√k (37)

or, alternatively,

√kSRt,kσrt,t+k,t = −covt(λt+k, rt,t+k) (38)

For pricing the asset, it is the conditional covariances that matter. Note that

rt,t+k − rft,t+k − Et[rt,t+k − rf

t,t+k] = rt,t+k − Et[rt,t+k]

i.e., the risk-free rate disappears from the calculations, except for calculatingthe Sharpe ratio itself. I now proceed to calculate these conditional covari-ance.

Note that

λt+k − Et[λt+k] = ωt+k − Et[ωt+k] − ηcc(ct+k − Et[ct+k]) + ηcl,l(lt+k − Et[lt+k])

and furthermore that

ωt+k − Et[ωt+k] =k∑

i=1

(ωt+i − ωt+i−1) −k∑

i=1

Et [ωt+i − ωt+i−1]

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= −ζk∑

i=1

ǫv,0,t+i

= −ζ(1 − β)k∑

i=1

∞∑

j=0

βj(

ǫc,j,t+i +1

κǫl,j,t+i

)

The decomposition of the compounded log returns and the k−period changein log consumption and leisure into news is

ct+k − Et[ct+k] =k∑

i=1

ǫc,k−i,t+i

lt+k − Et[lt+k] =k∑

i=1

ǫl,k−i,t+i

rt,t+k − Et[rt,t+k] =k∑

i=1

(rt+i − Et[rt+i])

=k∑

i=1

k−i∑

j=0

ǫr,j,t+i

As above, suppose that the dynamics of the data can be summarized by theVAR in equation (34) with the first, second and third entry of Yt correspond-ing to log consumption, log leisure and log returns respectively. Calculationof the conditional covariances now yields

covt (ωt+k, rt,t+k) = −ζ(

Q(β, t, k)13 +1

κQ(β, t, k)23

)

where

Q(β, t, k) = (1 − β)(1 − βB)−1k∑

i=1

k−i∑

j=0

Σt+i−1(B′)j (39)

In the homoskedastic case Σt ≡ Σ, this expression simplifies to

Q(β, t, k) = (1 − β)(1 − βB)−1Σk∑

i=1

i(B′)k−i

Define

S(t, k) =k∑

i=1

k−i∑

j=0

Bk−iΣt+i−1(B′)j

19

which is generally not a symmetric matrix except for the trivial case k = 1.Note that

σ2rt,t+k,t =

k∑

i=1

k−i∑

j=0

k−i∑

m=0

BmΣt+i−1(B′)j

33

Combining, I now obtain the version of (35) for the k−period holding periodas

√kSRt,kσrt,t+k,t = (40)

= ζQ(β, t, k)13 +ζ

κQ(β, t, k)23 + ηcc (S(t, k))13 − ηcl,l (S(t, k))23

As in equation (35), this equation provides a relationship between the pref-erence parameters, given the data.

To calculate the maximal Sharpe ratio

SRmaxt,k =

1√kσλt+k,t

define the matrix A(β, t, k) per

A(β, t, k) =k∑

i=1

∞∑

j=0

β2jBjΣt+i−1(B′)j

The explicit solution is given by

vec(A(β, t, k)) = (I − β2(B ⊗B))−1vec(k∑

i=1

Σt+i−1)

Define

Σ(t, k) =k∑

i=1

Bk−iΣt+i−1(B′)k−i

and

R(t, k) =k∑

i=1

Bk−iΣt+i−1(I − βB′)−1

With this, calculate

σ2λt+k,t

= ~a′Σ(t, k)~a− 2~a′R(t, k)~b+~b′A(β, t, k)~b

where the vectors ~a, ~b are defined as before in equation (36).

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2.5 Empirical implications for preferences

CONTINUE HERE ...

We now use these observations to draw out implications for preferences,assuming now that volatilies and correlations stay constant. The standardcase, on which practically the entire asset pricing literature has focussed, isthe case ηcl,l = 0. In that case, (33) implies

ηcc =SR

ρc,rσc

(41)

for the level of relative risk aversion in consumption. Using an annual holdingperiod, k = 4, and the data of the tables above, one obtains

ηcc =0.27

1.64% ∗ 0.39= 42

Even assuming perfectly positive correlation, one needs ηcc = 16.5. Otherauthors typically find even much higher values, see Campbell (2004). Thesevalues seem high on a priori grounds and incompatible with standard macroe-conomic models.

With nonseparabilities between consumption and leisure, however, lowervalues for ηcc are possible, when the value of the cross-derivative is changedsimultaneously as well. To that end, rewrite equation (33) as

ηcl,l =SR− ηccρc,rσc

−ρl,rσl

(42)

For the macroeconomic implications, and since leisure is fairly volatile, it isdesirable to pick the relative risk aversion with respect to leisure as low aspossible. We thus assume that equation (??) holds with equality,

ηll =κη2

cl,l

ηcc

For holding periods of one year, k = 4 and two years, k = 8, table 2 as wellas figures 1 and 2 show the resulting values as a function of the relative riskaversion for consumption, ηcc.

21

0 10 20 30 40 50 60−40

−20

0

20

40

60

80

100

ηcc

η cl,l

k= 4k= 8

Figure 1: The implied value for the cross-derivative ηcl,l, when varying the

relative risk aversion for consumption between 3 and 60.

22

0 10 20 30 40 50 600

200

400

600

800

1000

1200

1400

ηcc

η ll

k= 4k= 8

Figure 2: The implied value for the minimal relative risk aversion in leisure

ηll, when varying the relative risk aversion for consumption between 3 and

60.

23

ηcc ηcl,l ηll

k=4 k=8 k=4 k=83.0 84.7 41.1 1389.2 327.55.0 80.4 38.7 749.8 173.510.0 69.5 32.5 280.0 61.215.0 58.5 26.3 132.6 26.720.0 47.6 20.1 65.8 11.730.0 25.8 7.7 12.9 1.140.0 4.0 -4.7 0.2 0.350.0 -17.9 -17.1 3.7 3.4

Table 2: Implied values for the cross-derivative term ηcl,l and the minimal

relative risk aversion in leisure ηll, when varying the relative risk aversion in

consumption ηcc.

We see that explaining the Sharpe ratio remains hard: low values for therelative risk aversion in consumption require dramatically high values for therelative risk aversion in leisure. It is some progress that one can explainthe observed Sharpe ratio at levels of relative risk aversion below 20, evenwhen taking account the correct correlations, using the calculations basedon a holding period of k = 8 quarters. Obviously, these are still fairly highnumbers.

A The loglinear equations of the basic model

References

[1] Epstein, L.G., Zin, S.E. (1989), ”Substitution, risk aversion and the tem-poral behavior of consumption and asset returns: a theoretical frame-work,” Econometrica 57, 937-969.

[2] Epstein, L.G., Zin, S.E. (1989), ”Substitution, risk aversion and the tem-poral behavior of consumption and asset returns: an empirical analysis,”Journal of Political Economy 99, 263-286.

[3] Tallarini, Thomas D. (2000), ”Risk-sensitive real business cycles,” Jour-nal of Monetary Economics 45, 507-532.

24

feasibility: yt = XYxt +

(

1 − XY

)

ctgoods production: yt = θkt−1 + (1 − θ)nt

cap. production: kt = (1 − δ)kt−1 + δxt

wages: wt = zt + ϕnn(kt−1 − nt)dividends: dt = zt − ϕkk(kt−1 − nt)

time endowment: lt = − 1−LLnt

DEF recursive util: vt = (1 − β)(ct + lt/κ) + βEt[vt+1]FOC recursive util: ωt = ωt−1 − ζ (vt − Et−1[vt])shadow value of wealth: λt = ωt − ηccct + ηcl,lltshadow value of time: λt + wt = ηcl,cct − ηllltadj. cost friction / Tobin’s q: ψt = 1

(xt − kt−1)

return on capital: rt = R−1+δR

dt − ψt−1 + 1Rψt

Lucas asset pricing: 0 = Et [λt+1 − λt + rt+1]

Table 3: List of the log-linearized equations of the basic model.

25