»(st ) =

¯ 1 ’ 2 (st ’ s) ’ 1

¯

Sr

γ

¯ (21.351)

S=σ

1’φ

The extra complication of (21.349) rather than (21.347) means consumption is always above

habit, since S = es > 0. Other habit models can give consumption below habit which leads

to in¬nite or imaginary marginal utility.

St becomes the single state variable in this economy. Time-varying expected returns,

price-dividend ratios, etc. are all functions of this state variable.

Marginal utility is

uc (Ct , Xt ) = (Ct ’ Xt )’γ = St Ct .

’γ ’γ

The model assumes an external habit “ each individual™s habit is determined by everyone

else™s consumption, as in Abel™s (1990) “keeping up with the Joneses” speci¬cation. This

is mostly a technical simpli¬cation, since it allows us to ignore terms by which current con-

sumption affect future habits; the opposite speci¬cation gives very similar results (see prob-

lem 2).

With marginal utility, we now have a discount factor.

µ ¶’γ

uc (Ct+1 , Xt+1 ) St+1 Ct+1

Mt+1 ≡ δ =δ .

uc (Ct , Xt ) St Ct

Since we have a stochastic process for S and C, and each is lognormal, we can evaluate the

conditional mean of the discount factor to evaluate the riskfree rate

1

f

(352)

rt = ’ ln Et (Mt+1 ) = ’ ln(δ) + γg ’ γ(1 ’ φ).

2

We gave up on analytic solutions and evaluated the price-dividend ratio as a function of the

state variable by iteration on a grid:

· µ ¶¸

Pt Ct+1 Pt+1

(st ) = Et Mt+1 1+ (st+1 )

Ct Ct Ct+1

With price-dividend ratios, we can calculate returns, expected returns, etc.

How does it work “ equity premium and predictability

We choose parameters, simulate 100,000 arti¬cial data points, and report standard statis-

tics and tests in arti¬cial data. The parameters g = 1.89, σ = 1.50, rf = 0.94 match their

values in postwar data. The parameter φ = 0.87 matches the autocorrelation of the price-

¯

dividend ratio and the choice γ = 2.00 matches the postwar Sharpe ratio. δ = 0.89, S =

0.57 follow from the model.

426

SECTION 21.2 NEW MODELS

Table 2cc presents means and standard deviations predicted by the model. The model

replicates the postwar Sharpe ratio, with a constant 0.94% risk free rate and a reasonable

subjective discount factor δ < 1. Of course, we picked the parameters to do this, but given

the above equity premium discussion it™s already an achievement that we are able to pick any

parameters to hit these moments.

Some models can replicate the Sharpe ratio, but do not replicate the level of expected

returns and return volatility. E = 1% and σ = 2% will give the right Sharpe ratio, but this

model predicts the right levels as well. The model also gets the level of the price-dividend

ratio about right.

Table 2cc. Means and standard deviations of simulated and historical data.

Consumption Dividend Postwar

Statistic claim claim data

0.50 0.50

E(R ’ R )/σ(R ’ Rf )

f

6.64 6.52 6.69

E(r ’ rf )

15.2 20.0 15.7

σ(r ’ rf )

18.3 18.7 24.7

exp[E(p ’ d)]

0.27 0.29 0.26

σ(p ’ d)

The model is simulated at a monthly frequency; statistics are calculated from arti-

¬cial time-averaged data at an annual frequency. Asterisks (*) denote statistics that

model parameters were chosen to replicate. All returns are annual percentages.

Table 5cc shows how the arti¬cial data match the predictability of returns from price-

dividend ratios. The paper goes on, and shows how the model matches the volatility test result

that almost all return variation is due to variation in expected excess returns, the “leverage

effect” of higher volatility after a big price decline, and several related phenomena.

Table 5cc. Long-horizon return regressions

Horizon Cons. claim Postwar data

(Years) 10—coef. R2 10—coef. R2

1 -2.0 0.13 -2.6 0.18

2 -3.7 0.23 -4.3 0.27

3 -5.1 0.32 -5.4 0.37

5 -7.5 0.46 -9.0 0.55

7 -9.4 0.55 -12.1 0.65

How does it work?

How does this model get around all the equity premium - riskfree rate dif¬culties de-

scribed above, and explain predictability as well?

When a consumer has a habit, local curvature depends on how far consumption is above

427

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

the habit, as well as the power γ,

’Ct ucc (Ct ’ Xt ) γ

·t ≡ =.

uc (Ct ’ Xt ) St

As consumption falls toward habit, people become much less willing to tolerate further falls

in consumption; they become very risk averse. Thus a low power coef¬cient γ can still mean

a high, and time-varying curvature. Recall our fundamental equation for the Sharpe ratio,

f

Et (r) ’ rt

= · t σt (∆c)corrt (∆c, r).

σ t (r)

High curvature ·t means that the model can explain the equity premium, and curvature ·t that

varies over time as consumption rises in booms and falls toward habit in recessions means that

the model can explain a time-varying and countercyclical (high in recessions, low in booms)

Sharpe ratio, despite constant consumption volatility σt (∆c) and correlation corrt (∆c, r).

So far so good, but didn™t we just learn that raising curvature implies high and time-

varying interest rates? This model gets around interest rate problems with precautionary sav-

ing. Suppose we are in a bad time, in which consumption is low relative to habit. People want

to borrow against future, higher, consumption, and this force should drive up interest rates.

(In fact, many habit models have very volatile interest rates.) However, people are also much

more risk averse when consumption is low. This consideration induces them to save more, in

order to build up assets against the event that tomorrow might be even worse. This desire to

save drives down interest rates. Our »(s) speci¬cation makes these two forces exactly offset,

leading to constant real rates.

The precautionary saving motive also makes the model more plausibly consistent with

variation in consumption growth across time and countries. Adding (21.351) to (21.352), we

can write

1 ³ γ ´2 2

f

r = ρ + γg ’ ¯σ

2S

The power coef¬cient γ = 2 controls the relation between consumption growth and inter-

est rates, while the curvature coef¬cient γ/St controls the risk premium. Thus this habit

model allows high “risk aversion” with low “aversion to intertemporal substitution,” and it is

consistent with the consumption and interest rate data.

As advertised, this model explains the equity premium and predictability by fundamen-

tally changing the story for why consumers are afraid of holding stocks. The k’ period

stochastic discount factor is

µ ¶’γ

St+k Ct+k

k

Mt’t+k = δ .

St Ct

covariances with S shocks now drive average returns as well as covariances with C shocks.

S = (C ’ X)/C is a recession indicator “ it is low after several quarters of consumption

428

SECTION 21.2 NEW MODELS

declines and high in booms.

While (Ct+k /Ct )’γ and (St+k /St )’γ enter symmetrically in the formula, the volatility

of (Ct+k /Ct )’γ with γ = 2 is so low that it accounts for essentially no risk premia. There-

fore, it must be true, and it is, that variation in (St+k /St )’γ is much larger, and accounts for

nearly all risk premia. In the Merton language of (21.345) and (21.346), variation across as-

sets in expected returns is driven by variation across assets in covariances with recessions far

more than by variation across assets in covariances with consumption growth.

At short horizons, shocks to St+1 and Ct+1 move together, so the distinction between a

recession state variable and consumption risk is minor; one can regard S as an ampli¬cation

mechanism for consumption risks in marginal utility. dS/‚C ≈ 50, so this ampli¬cation

generates the required volatility of the discount factor.

At long horizons, however, St+k becomes less and less conditionally correlated with

Ct+k . St+k depends on Ct+k relative to its recent past, but the overall level of consumption

may be high or low. Therefore, investors fear stocks because they do badly in occasional

serious recessions, times of recent belt-tightening. These risks are at the long run unrelated

to the risks of long-run average consumption growth.

As another way to digest how this model works, we can substitute in the s process from

(21.349) and write the marginal rate of substitution as

µ ¶’γ

St+1 Ct+1

Mt+1 = δ

St Ct

ln Mt+1 = ln δ ’ γ (st+1 ’ st ) ’ γ(ct+1 ’ ct )

= {ln δ ’ γ(1 ’ φ)¯} + {γ (1 ’ φ) st + γg» (st )} ’ γ [» (st ) + 1] (ct+1 ’ ct )

s

ln Mt+1 = a + b(st ) + d(st )(ct+1 ’ ct )

Up to the question of logs vs. levels, this is a “scaled factor model” of the form we studied in

Chapter 8. It still is a consumption-based model, but the sensitivity of the discount factor to

consumption changes over time.

The long-run equity premium is even more of a puzzle. Most recession state variables,

such as recessions, labor, and instruments for time-varying expected returns (“shifts in the

investment opportunity set”) are stationary. Hence, the standard deviation of their growth

rates eventually stops growing with horizon. At a long enough horizon, the standard deviation

of the discount factor is dominated by the standard deviation of the consumption growth term,

and we return to the equity premium puzzle at a long enough run.

Since this model produces predictability of the right sign, it produces a long run equity

premium puzzle. How it manages this feat with a stationary state variable St is subtle (and

’γ

we didn™t notice it until the penultimate draft!) The answer is that while St is stationary, St

’γ

is not. St has a fat tail approaching zero so the conditional variance of St+k grows without

bound.

While the distinction between stationary S and nonstationary S ’γ seems initially minor, it

429

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

is in fact central. Any model that wishes to explain the equity premium at long and short runs

by means of an additional, stationary state variable must ¬nd some similar transformation so

that the volatility of the stochastic discount factor remains high at long horizons.

This model does have high risk aversion. The utility curvature and value function cur-

vature are both high. Many authors require that a “solution” of the equity premium puzzle

display low risk aversion. This is a laudable goal, and no current model has attained it. No

current model generates the equity premium with a low and relatively constant interest rate,

low risk aversion, and the right pattern of predictability “ high prices forecast low returns, not

high returns, and consumption is roughly a random walk. Constantinides (1990) and Boldrin,

Christiano and Fisher (1997) are habit models with a large equity premium and low risk aver-

sion, but they don™t get the pattern of predictability right. Boldrin, Christiano and Fisher have

highly variable interest rates to keep consumption from being predictable. Constantinides

(1990) has a constant interest rate, but consumption growth that is serially correlated, so con-

sumption rises to meet i.i.d. wealth growth. The long-run equity premium is solved with

counterfactually high long-run consumption volatility.

To get a high equity premium with low risk aversion, we need to ¬nd some crucial char-

acteristic that separates stock returns from wealth bets. This is a dif¬cult task. After all, what

are stocks if not a bet? The answer must be some additional state variable. Stocks must pay

off badly in particularly unfortunate states of the world.

Again, the trouble with predictability is that stocks pay off well in particularly bad states

of the world “ states with low future returns. This makes stocks even more desirable, requir-

ing even higher risk aversion to explain the equity premium. The alternative, not yet found,

is to ¬nd some measure of the state of the world that is particularly bad when stocks pay

off badly, enough to explain not only the standard equity premium, but the long run equity

premium resulting from the fact that stocks are less risky at longer horizons.

I write this not to say that such a model is impossible. The point is to show the hurdle that

must be overcome, in the hope that someone will overcome it.

21.2.3 Heterogeneous agents and idiosyncratic risks

A long, increasing, and important literature in the equity premium attacks the problem instead

with relatively standard preferences, but instead adds uninsured idiosyncratic risk. As with

the preference literature, this literature is interesting beyond the equity premium. We are

learning a lot about who holds stocks and why, what risks they face. We are challenged to

think of new assets and creative ways of using existing assets to share risks better.

Constantinides and Duf¬e (1996) provide a very clever and simple model in which id-

iosyncratic risk can be tailored to generate any pattern of aggregate consumption and asset

prices. It can generate the equity premium, predictability, relatively constant interest rates,

smooth and unpredictable aggregate consumption growth and so forth. Furthermore, it re-

quires no transactions costs, borrowing constraints or other frictions, and the individual con-

430

SECTION 21.2 NEW MODELS

sumers can have any nonzero value of risk aversion. Of course, we still have to evaluate

whether the idiosyncratic risk process we construct to explain asset pricing phenomena are

reasonable and consistent with microeconomic data.

A simple version of the model

I start with a very simpli¬ed version of the Constantinides-Duf¬e model. Each consumer

i has power utility,

X 1’γ

e’δt Cit

U =E

t

Individual consumption growth Cit+1 is determined by an independent, idiosyncratic normal

(0,1) shock ·it ,

µ ¶

Cit+1 12

(353)

ln = ·it+1 yt+1 ’ yt+1

Ci,t 2

where yt+1 is, by construction since it multiplies the shock ·it , the cross-sectional standard

deviation of consumption growth. yt+1 is dated t + 1 since it is the cross-sectional standard

deviation given aggregates at t+1. The aggregates are determined ¬rst, and then the shocks

·it are handed out.

Now, yt+1 is speci¬ed so that people suffer a high cross-sectional variance of consump-

tion growth on dates of a low market return Rt+1 ,

¸s

· µ ¶¯

p

Cit+1 ¯ 2

¯ Rt+1 = (354)

yt+1 = σ ln δ ’ ln Rt+1 .

Cit ¯ γ(γ + 1)

Given this structure, the individual is exactly happy to consume {Cit } without further

trading in the stock. (We can call Cit income Iit , and prove the optimal decision rule is to

consume income Cit = Iit .) His ¬rst-order condition for an optimal consumption-portfolio

decision

" #

µ ¶’γ

Cit+1

1 = Et e’δ Rt+1

Cit

holds, exactly.

To prove this assertion, just substitute in for Cit+1 /Cit and take the expectation:

· ¸

12

1 = Et exp ’δ ’ γ· it+1 yt+1 + γyt+1 + ln Rt+1

2

Since · is independent of everything else, we can use E [f (·y)] = E [E(f (·y|y)] . Now,

431

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

with · normal (0,1),

· ¸

¡ £ ¤ ¢ 122

E exp ’γ·it+1 yt+1 | yt+1 = exp γ yt+1 .

2

Therefore, we have

· ¸

122 12

1 = Et exp ’δ + γ yt+1 + γyt+1 + ln Rt+1 .

2 2

Substituting in from (21.354),

· µ ¶ ¸

1 2

1 = Et exp ’δ + γ(γ + 1) (δ ’ ln Rt+1 ) + ln Rt+1

2 γ(γ + 1)

1 = Et 1!

The general model

In the general model, Constantinides and Duf¬e de¬ne

s r

2 Ct+1

(355)

yt+1 = ln mt+1 + δ + γ ln

γ(γ + 1) Ct

where mt is a strictly positive discount factor that prices all assets under consideration,

pt = Et [mt+1 xt+1 ] for all xt+1 ∈ X. (356)

By starting with a discount factor that can price a large collection of assets, where I used the

’1

discount factor Rt+1 to price the single return Rt+1 in (21.354), idiosyncratic risk can be

constructed to price exactly a large collection of assets. We can exactly match the Sharpe

ratio, return forecastability, and other features of the data.

Then, they let

µ ¶

vit+1 12

ln = ·it+1 yt+1 ’ yt+1

vit 2

Cit+1 = vit+1 Ct+1 .

yt+1 is still the conditional standard deviation of consumption growth, given aggregates “

returns and aggregate consumption. This variation allows uncertainty in aggregate consump-

tion. We can tailor the idiosyncratic risk to and consumption-interest rate facts as well.

432

SECTION 21.2 NEW MODELS

Following exactly the same argument as before, we can now show that

" #

µ ¶’γ

Cit+1

1 = Et e’δ Rt+1

Cit

for all the assets priced by m.

A technical assumption

Astute readers will notice the possibility that the square root term in (21.354) and (21.355)

might be negative. Constantinides and Duf¬e rule out this possibility by assuming that the

discount factor m satis¬es

Ct+1

(357)

ln mt+1 ≥ δ + γ ln

Ct

in every state of nature, so that the square root term is positive.

We can sometimes construct such discount factors by picking parameters a, b in mt+1 =

h ³ ´γ i

max a + b0 xt+1 , eδ CCt to satisfy (21.356). However, neither this construction nor a

t+1

discount factor satisfying (21.357) is guaranteed to exist for any set of assets. The restriction

(21.357) is a tighter form of the familiar restriction that mt+1 ≥ 0 that is equivalent to the

absence of arbitrage in the assets under consideration. Ledoit and Bernardo (1997) show that

the restriction m > a is equivalent to restrictions on the maximum gain/loss ratio available

from the set of assets under consideration. Thus, the theorem really does not apply to any set

of arbitrage-free payoffs.

The example m = 1/R is a positive discount factor that prices a single asset return

1 = E(R’1 R), but does not necessarily satisfy restriction (21.357). For high R, we can have

very negative ln 1/R. This example only works if the distribution of R is limited to R ¤ eδ .

How the model works

As the Campbell-Cochrane model is blatantly (and proudly) reverse-engineered to sur-

mount (and here, to illustrate) the known pitfalls of representative consumer models, the

Constantinides-Duf¬e model is reverse engineered to surmount the known pitfalls of idiosyn-

cratic risk models.

Idiosyncratic risk stories face two severe challenges, as explained in section 1.2. First,

the basic pricing equation applies to each individual. If we are to have low risk aversion and

power utility, the required huge volatility of consumption is implausible for any individual.

Second, if you add idiosyncratic risk uncorrelated with asset returns, it has no effect on

pricing implications. Constantinides and Duf¬e™s central contribution is to very cleverly solve

the second problem.

In idiosyncratic risk models, we cannot specify individual consumption directly as we

do in representative agent endowment economies, and go straight to ¬nding prices. The

endowment economy structure says that aggregate consumption is ¬xed, and prices have

to adjust so that consumers are happy consuming the given aggregate consumption stream.

433

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

However, individuals can always trade consumption with each other. The whole point of

assets is that one individual can sell another some consumption, in exchange for the promise

of some consumption in return in the next period. We have to give individuals idiosyncratic

income shocks, and then either check that they do not want to trade away the idiosyncratic

shock, or ¬nd the equilibrium consumption after they do so.

Early idiosyncratic risk papers found quickly how clever the consumers could be in get-

ting rid of the idiosyncratic risks by trading the existing set of assets. Telmer (1993) and

Lucas (1994) found that if you give people transitory but uninsured income shocks, they

respond borrowing and lending or by building up a stock of savings. As in the classic per-

manent income model, consumption only responds by the interest rate times the change in

permanent income, and at low enough interest rates, not at all. “Self-insurance through stor-

age” removes the extra income volatility and we are back to smooth individual consumption

and an equity premium puzzle.

Constantinides and Duf¬e get around this problem by making the idiosyncratic shocks

permanent. The normal ·it shocks determine consumption growth. In an evaluation in

microeconomic data, this makes us look for sources of permanent shocks.

This, at a deeper level, is why idiosyncratic consumption shocks have to be uncorrelated

with the market. We can give individuals idiosyncratic income shocks that are correlated

with the market. Say, agent A gets more income when the market is high, and agent B gets

more income when it is low. But then A will short the market, B will go long, and they will

trade away any component of the shock that is correlated with the returns on available assets.

I argued above that this effect made idiosyncratic shocks hopeless as candidates to explain

the equity premium puzzle. Shocks uncorrelated with asset returns have no effect on asset

pricing, and shocks correlated with asset returns are quickly traded away.

The only way out is to exploit the nonlinearity of marginal utility. We can give people in-

come shocks that are uncorrelated with returns, so they can™t be traded away. Then we have

a nonlinear marginal utility function turn these shocks into marginal utility shocks that are

correlated with asset returns, and hence can affect pricing implications. This is why Con-

stantinides and Duf¬e specify that the variance of idiosyncratic risk rises when the market

declines. If marginal utility were linear, an increase in variance would have no effect on the

average level of marginal utility. Therefore, Constantinides and Duf¬e specify power utility,

and the interaction of nonlinear marginal utility and changing conditional variance produces

an equity premium.

As a simple calculation that shows the basic idea, start with individuals i with power

utility so

"µ #

¶’γ

i

Ct+1

Re

0=E t+1

i

Ct

434

SECTION 21.2 NEW MODELS

PN

Now aggregate across people by summing over i, with EN = 1

i=1

N

" Ãµ ¶’γ ! #

i

Ct+1

Re

0 = E EN t+1 .

i

Ct

If the cross-sectional variation of consumption growth is lognormally distributed,

·µ ¶ ¸

γ2 2

i i

0 = E e’γEN ∆ct+1 + 2 σN ∆ct+1 Rt+1 e

As you see, the economy displays more risk aversion than would a “representative agent”

with aggregate consumption ∆ca = EN ∆cit+1 . That risk aversion can also vary over time

t+1

if σ N varies over time, and this variation can generate risk premia.

Microeconomic evaluation and risk aversion

Like the Campbell-Cochrane model, this could be either a new view of stock market

(and macroeconomic) risk, or just a clever existence proof for a heretofore troubling class

of models. The ¬rst question is whether the microeconomic picture painted by this model is

correct, or even plausible. Is idiosyncratic risk large enough? Does idiosyncratic risk really

rise when the market falls, and enough to account for the equity premium? Are there enough

permanent idiosyncratic shocks? Do people really shy away from stocks because of stock

returns are low at times of high labor market risk?

This model does not change the ¬rst puzzle. To get power utility consumers to shun

stocks, they still must have tremendously volatile consumption growth or high risk aversion.

The point of this model is to show how consumers can get stuck with high consumption

volatility in equilibrium, already a dif¬cult task.

More seriously than volatility itself, consumption growth variance also represents the

amount by which the distribution of individual consumption and income spreads out over

time, since the shocks must be permanent and independent across people. The 50% or larger

consumption growth volatility that we require to reconcile the Sharpe ratio with risk aversion

of one means that the distribution of consumption (and income) must also spread out by 50%

per year. The distribution of consumption does spread out, but not this much.

For example, Deaton and Paxson (1994) report that the cross-sectional variance of log

consumption within an age cohort rises from about 0.2 at age 20 to 0.6 at age 60. This

√

estimate means that the cross sectional standard deviation of consumption rises from 0.2 =

√

. 45 or 45% at age 20 to 0.6 = . 77 or 77% at age 60. (77% means that an individual one

standard deviation better off than the mean consumes 77% more than the mean consumer.)

We are back to about 1% per year.

Finally, and most crucially, the cross-sectional uncertainty about individual income must

not only be large, it must be higher when the market is lower. This risk-factor is after all the

central element of Constantinides and Duf¬e™s explanation for the market premium. Figure

51 shows how the cross-sectional standard deviation of consumption growth varies with the

435

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

market return and risk aversion in my simple version of Constantinides and Duf¬e™s model.

If we insist on low (γ = 1 to 2) risk aversion, the cross-sectional standard deviation of

consumption growth must be extremely sensitive to the level of the market return. Looking

at the γ = 2 line for example, is it plausible that a year with 5% market return would show a

10% cross-sectional variation in consumption growth, while a mild 5% decline in the market

is associated with a 25% cross-sectional variation?

Figure 51. Cross-sectional standard deviation of individual consumption growth as a

function of the market return in q simple version of the Constantinides-Duf¬e model.

the q

The plot is the variable yt = ln Rt + δ + γ ln CCt . Parameter values are

2 1

γ(γ+1) t’1

ρ = 0.05, ln Ct /Ct’1 = 0.01.

All of these empirical problems are avoided if we allow high risk aversion rather than

a large risk to drive the equity premium. The γ = 25 line in Figure 51 looks possible; a

γ = 50 line would look even better. With high risk aversion we do not need to specify highly

volatile individual consumption growth, spreading out of the income distribution, or dramatic

sensitivity of the cross-sectional variance to the market return.

As in any model, a high equity premium must come from a large risk, or from large risk

aversion. Labor market risk correlated with the stock market does not seem large enough to

account for the equity premium without high risk aversion.

436

SECTION 21.3 BIBLIOGRAPHY

The larger set of asset pricing facts has not yet been studied in this model. It is clearly

able to generate return predictability, but that requires a pattern of variation in idiosyncratic

risk that remains to be characterized and evaluated. It can generate cross-sectional patterns

such as value premia if value stocks decline at times of higher cross-sectional volatility; that

too remains to be studied.

Summary

In the end, the Constantinides-Duf¬e model and the Campbell-Cochrane model are quite

similar in spirit. First, both models make a similar, fundamental change in the description of

stock market risk. Consumers do not fear much the loss of wealth of a bad market return per

se. They fear that loss of wealth because it tends to come in recessions, in one case de¬ned as

times of heightened labor market risk, and in the other case de¬ned as a fall of consumption

relative to its recent past. This recession state-variable or risk-factor drives most variation in

expected returns.

Second, both models require high risk aversion. While Constantinides and Duf¬e™s proof

shows that one can dream up a labor income process to rationalize the equity premium for

any risk aversion coef¬cient, we see that even vaguely plausible characterizations of actual

labor income uncertainty require high risk aversion to explain the historical equity premium.

Third, both models provide long-sought demonstrations that it is possible to rationalize

the equity premium in their respective class of models. This existence proof is particularly

stunning in Constantinides and Duf¬e™s case. Many authors (myself included) had come

to the conclusion that the effort to generate an equity premium from idiosyncratic risk was

hopeless because any idiosyncratic risk that would affect asset prices would be traded away.

21.3 Bibliography

Shiller (1982) made the ¬rst calculation that showed either a large risk aversion coef¬cient

or counterfactually large consumption variability was required to explain means and vari-

ances of asset returns. Mehra and Prescott (1985) labeled this fact the “equity premium

puzzle.” However, they described these puzzles in the context of a two-state Markov model

for consumption growth, identifying a stock as a claim to consumption and a risk free bond.

Weil (1989) emphasized the interaction between equity premium and risk-free rate puzzles.

Hansen and Jagannathan (1991) sparked the kind of calculations I report here in a simpli¬ed

manner. Cochrane and Hansen (1992) derived many of the extra discount factor moment re-

strictions I surveyed here, calculating bounds in each case. Luttmer (1996), (1999) tackled

the important extension to transactions costs.

Kocherlakota (1996) is a nice summary of equity premium facts and models. Much of the

material in this Chapter is adapted from a survey in Cochrane (1997). Campbell (1999) and

(2000) are two excellent recent surveys. Ferson (1995) is a nice survey of consumption-based

model variations as well as some of the beta pricing models discussed in the last chapter.

The general picture of all solutions based on changing preferences is that they introduce

437

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

non-separabilities. If the marginal utility of consumption depends on z as well as c, uc (c, z),

then expected returns depend on covariances with z as well. In turn, this happens if we cannot

write the utility function as u(c, z) = v(c) + w(z), the separable form.

Habit persistence introduces a non-time-separable utility function, since u(c, x) and x de-

P

pends on past c“you can™t write a utility function with habits as a sum t vt (ct ), so one pe-

riod™s consumption affects another period™s marginal utility. The Campbell-Cochrane model

I presented here is a tip of an iceberg of habit research, including prominent contributions by

Constantinides (1990), Ferson and Constantinides (1991), Heaton (1995), Abel (1990).

Models can be nonseparable across goods as well. Leisure is the most natural extra

variable to add to a utility function. It™s not clear a priori whether more leisure enhances the

marginal utility of consumption (why bother buying a boat if you™re at the of¬ce all day and

can™t use it) or vice versa (if you have to work all day, it™s more important to come home to

a really nice big TV). However, we can let the data speak on this matter. Explicit versions

of this approach have not been very successful to date. (Eichenbaum, Hansen and Singleton

1989). On the other hand, recent research has found that adding labor income as an extra ad-

hoc “factor” can be useful in explaining the cross section of average stock returns, especially

if it is scaled by a conditioning variable (Jagannathan and Wang 1996, Reyfman 1997, Lettau

and Ludvigson 2000).

The non-state separable utility functions following Epstein and Zin (1989) are a major

omission of this presentation. The expectation E in the standard utility function sums over

states of nature, e.g.

U = prob(rain) — u(C if it rains) + prob(shine) — u(C if it shines).

“Separability” means one adds across states, so the marginal utility of consumption in one

state is unaffected by what happens in another state. But perhaps the marginal utility of a little

more consumption in the sunny state of the world is affected by the level of consumption in

the rainy state of the world. Epstein and Zin and Hansen, Sargent and Tallarini (1997) propose

recursive utility functions of the form

£ ¤

1’γ

+ βf Et f ’1 (Ut+1 ) .

Ut = Ct

If f (x) = x this expression reduces to power utility. These utility functions are not state-

separable. As with habits, these utility functions distinguish risk aversion from intertemporal

substitution“one coef¬cient can be set to capture the consumption-interest rate facts, and a

completely separate coef¬cient can be set to capture the equity premium. So far, this style

of model as in Epstein and Zin (1989), Weil (1989), Kandel and Stambaugh (1991), and

Campbell (1996) does not generate time-varying risk aversion, but that modi¬cation should

not be too dif¬cult, and could lead to a model that works very much like the habit model I

surveyed here.

Habit persistence is the opposite of durability. If you buy a durable good yesterday, that

lowers your marginal utility of an additional purchase today, while buying a habit-forming

good raises your marginal utility of an additional purchase today. Thus the durability of goods

438

SECTION 21.3 BIBLIOGRAPHY

should introduce a non-time-separability of the form u(ct + θxt ), xt = f(ct’1 , ct’2 , ...)

rather than the habit persistence form u(ct ’ θxt ). Since goods are durable, and we have

a lot of data on durables purchases, it would be good to include both durability and habit

persistence in our models. (In fact, even “nondurables” contain items like clothing; the truly

nondurable purchases are such a small fraction of total consumption that we rely on very little

data.) One must be careful with the time horizon in such a speci¬cation. At a suf¬ciently

small time horizon, all goods are durable. A pizza eaten at noon lowers marginal utility

of more pizza at 12:05. Thus, our common continuous time, time separable assumption

really cannot be taken literally. Hindy and Huang (1992) argue that consumption should

be “locally substitutable” in continuous time models. Heaton (1993) found that at monthly

horizons, consumption growth displays the negative autocorrelation suggestive of durability

with constant interest rates, while at longer horizons consumption is nearly unforecastable

after accounting for time-aggregation.

There is also a production ¬rst order condition that must be solved, relating asset prices

to marginal rates of transformation. The standard here is the q theory of investment, which is

based on an adjustment cost. If the stock market is really high, you issue stock and make new

investments. The trouble with this view is that f 0 (K) declines very slowly, so the observed

price volatility implies huge investment volatility. The q theory adds adjustment costs to

damp the investment volatility. The q theory has had as much trouble ¬tting the data as

the consumption-based model. Cochrane (1991d) reports one success when you transform

the data to returns “ high stock returns are associated with high investment growth. The

more recent investment literature has focused on specifying the adjustment cost problem

with asymmetries and irreversibilities, for example Abel and Eberly (1996).

There is an important literature that puts new utility functions together with production

functions, to construct complete explicit economic models that replicate the asset pricing

facts. Such efforts should also at least preserve if not enhance our ability to understand the

broad range of dynamic microeconomic, macroeconomic, international and growth facts that

the standard models were constructed around. Jermann (1998) tried putting habit persistence

consumers in a model with a standard technology Y = θf (K, L) from real business cycle

models. The easy opportunities for intertemporal transformation provided by that technol-

ogy meant that the consumers used it to dramatically smooth consumption, destroying the

prediction of a high equity premium. To generate the equity premium, Jermann added an

adjustment cost technology, as the production-side literature had found necessary. This mod-

i¬cation resulted in a high equity premium, but also large variation in riskfree rates.

Boldrin, Christiano and Fisher (1997) also added habit-persistence preferences to real

business cycle models with frictions in the allocation of resources to two sectors. They gen-

erate about 1/2 the historical Sharpe ratio. They ¬nd some quantity dynamics are improved

over the standard model. However, they still predict highly volatile interest rates and persis-

tent consumption growth.

To avoid the implications of highly volatile interest rates, I suspect we will need repre-

sentations of technology that allow easy transformation across time but not across states of

439

CHAPTER 21 EQUITY PREMIUM PUZZLE AND CONSUMPTION-BASED MODELS

nature, analogous to the need for easy intertemporal substitution but high risk aversion in

preferences. Alternatively, the Campbell-Cochrane model above already produces the equity

premium with constant interest rates, which can be interpreted as a linear production func-

tion f(K). Models with this kind of precautionary savings motive may not be as severely

affected.

Tallarini (1999) uses non-state separable preferences similar to those of Epstein and Zin in

a general equilibrium model with production. He shows a beautiful observational equivalence

result: A model with standard preferences and a model with non-state-separable preferences

can predict the same path of quantity variables (output, investment, consumption, etc.) but

differ dramatically on asset prices. This result offers one explanation of how the real busi-

ness cycle and growth literature could go on for 25 years examining quantity data in detail

and miss all the modi¬cations to preferences that we seem to need to explain asset pricing

data. It also means that asset price information is crucial to identifying preferences and cal-

culating welfare costs of policy experiments. Finally, it offers hope that adding the deep

modi¬cations necessary to explain asset pricing phenomena will not demolish the success of

standard models at describing the movements of quantities.

The Constantinides and Duf¬e model has roots in a calculation by Mankiw (1986) that

idiosyncratic risk could make the representative consumer seem more risk averse than the

individuals. Work on evaluating the mechanisms in this model in microeconomic data is

starting. Heaton and Lucas (1996) calibrate idiosyncratic risk from the PSID, but their model

explains at best 1/2 of the sample average stock return, and less still if they allow a net supply

of bonds with which people can smooth transitory shocks. More direct tests of these features

in microeconomic consumption data are underway, for example Brav, Constantinides and

Geczy (1999), Storesletten, Telmer and Yaron (1999).

Kandel and Stambaugh (1986) present a model in which a small amount of time-varying

consumption volatility and a high risk aversion coef¬cient generate the large time-varying

discount factor volatility we need to generate returns predictability.

Aiyagari and Gertler (1991), though aimed at the point that the equity premium might be

explained by a “too low” riskless rate, nonetheless was an important paper in specifying and

solving models with uninsured individual risks and transactions costs to keep people from

trading them away.

21.4 Problems

1. Derive the analogue to the Hansen-Jagannathan bound in continuous time for an “excess

return,” i.e. considering a self-¬nancing portfolio, rather than a single return less the risk

free rate.

2. Suppose habit accumulation is linear, and there is a constant riskfree rate or linear

440

SECTION 21.4 PROBLEMS

technology equal to the discount rate, Rf = 1/δ. The consumer™s problem is then

∞ ∞

X X X X

’ Xt )1’γ

t (Ct t t

φj Ct’j

max δ s.t. δ Ct = δ et + W0 ; Xt = θ

1’γ t t

t=0 j=1

where et is a stochastic endowment. In an internal habit speci¬cation, the consumer

considers all the effects that current consumption has on future utility through Xt+j .

In an external habit speci¬cation, the consumer ignores such terms. Show that the two

speci¬cations give identical asset pricing predictions in this simple model, by showing

that internal-habit marginal utility is proportional to external-habit marginal utility, state

by state.

3. Suppose a consumer has quadratic utility with a constant interest rate equal to the

subjective discount rate, but a habit or durable consumption good, so that utility is

1

u(ct ’ θct’1 ) = ’ (c— ’ ct + θct’1 ).

2

Show that external habit persistence θ > 0 implies positive serial correlation in

consumption changes. Show that the same solution holds for internal habits, or

durability. Show that durability leads to negative serial correlation in consumption

changes.

4. Many models predict too much variation in the conditional mean discount factor, or

too much interest rate variation. This problem guides you through a simple example.

Introduce a simple form of external habit formation,

u = (Ct ’ θCt’1 )1’γ

and suppose consumption growth Ct+1 /Ct is i.i.d. Show that interest rates still vary

despite i.i.d. consumption growth.

5. We showed that if m satis¬es the Hansen-Jagannathan bound, then proj(m|X) should

also do so. Hansen and Jagannathan also compute bounds with positivity, solutions to

min σ(m) s.t. p = E(mx), m ≥ 0, E(m) = µ.

Does proj(m|X) also lie in the same bound?

6. One most often compares consumption-based models to Hansen-Jagannathan bounds.

Can you compare the CAPM discount factor m = a ’ bRem to the bound? To the bound

with positivity?

441

Chapter 22. References

Abel, Andrew B. 1988, “Stock Prices under Time-Varying Dividend Risk: An

Exact Solution in an In¬nite-Horizon General Equilibrium Model,” Journal of Monetary

Economics 22, 375-93.

Abel, Andrew B., 1990, “Asset Prices Under Habit Formation and Catching Up

With the Jones,” American Economic Review 80, 38-42.

Abel, Andrew B., 1994, “Exact Solutions for Expected Rates of Return under

Markov Regime Switching: Implications for the Equity Premium Puzzle,” Journal of

Money, Credit, and Banking 26, 345-61.

Abel, Andrew B., 1999, “Risk Premia and Term Premia in General Equilibrium,”

Journal of Monetary Economics 43, 3-33.

Abel, Andrew B. and Janice C. Eberly, 1996, “Optimal Investment with Costly

Reversibility.” Review of Economic Studies 63, 581-593.

Abel, Andrew B. and Janice C. Eberly, 1999, “The Effects of Irreversibility and

Uncertainty on Capital Accumulation,” Journal of Monetary Economics 44, 339-77.

Aiyagari, S. Rao and Mark Gertler, 1991, “Asset Returns with Transactions Costs

and Uninsured Individual Risk: A Stage III exercise,” Journal of Monetary Economics

27, 309-331.

Andrews, Donald W. K., 1991, “Heteroskedasticity and Autocorrelation Consistent

Covariance Matrix Estimation,” Econometrica 59, 817-58.

Atkeson, Andrew, Fernando Alvarez, and Patrick Kehoe, 1999, “Volatile Exchange

Rates and the Forward Premium Anomaly: A Segmented Asset Market View,” Working

paper, University of Chicago.

Bachelier, L. 1900, “Theory of Speculation,” in Cootner, P. (ed), The Random

Character of Stock Prices, Cambridge, MA: MIT press 1964.

Balduzzi, Pierluigi., Giuseppe Bertola and Silverio Foresi, 1996, “A Model of Target

Changes and the Term Structure of Interest Rates,” Journal of Monetary Economics 39,

223, 49.

Banz, Rolf W. 1981, “The Relationship Between Return and Market Value of

Common Stocks,” Journal of Financial Economics, 9, 3-18.

Barsky, Robert and Bradford J. DeLong 1993, “Why Does the Stock Market

Fluctuate?” Quarterly Journal of Economics 108, 291-311.

Bekaert, Geert and Robert J. Hodrick, 1992, “Characterizing Predictable

Components in Excess Returns on Equity and Foreign Exchange Markets,”Journal of

Finance 47, 467-509.

Becker, Connie, Wayne E. Ferson, Michael Schill, and David Myers 1999,

“Conditional Market Timing with Benchmark Investors,” Journal of Financial

Economics 52, 119-148.

Berk, Jonathan, 1997, “Does Size Really Matter?” Financial Analysts Journal,

September/October 1997, 12-18.

442

Bernardo, Antonio and Olivier Ledoit. “Gain Loss and Asset Pricing.” (1999)

Journal of Political Economy 108, 144-172.

Black, Fischer, Michael Jensen and Myron Scholes, 1972, “The Capital Asset

Pricing Model: Some empirical Tests,” in Michael Jensen, Ed., Studies in the Theory of

Capital Markets (Praeger, New York NY)

Black, Fischer and Myron Scholes, 1973, “The Valuation of Options and Corporate

Liabilities.” Journal of Political Economy 81, 637-654.

Bollerslev, Tim, R. Chou and K. Kroner 1992, “ARCH Modeling in Finance: A

Review of Theory and empirical Evidence,” Journal of Econometrics,52, 5-59.

Boudoukh, Jacob, Matthew Richardson 1994 “The Statistics of Long-Horizon

Regressions Revisited,” Mathematical Finance 4, 103-119.

Boudoukh, Jacob, Matthew Richardson, Robert Stanton and Robert Whitelaw,

1998, “The Stochastic Behavior of Interest Rates: Implications from a Nonlinear,

Continuous-time, Multifactor Model,” Manuscript, University of California at Berkeley.

Brav, Alon, George Constantinides and Christopher Geczy, 1999, “Asset Pricing

with Heterogeneous Consumers and Limited Participation: empirical Evidence,”

Manuscript Duke University.

Breeden, Douglas T., 1979, “An Intertemporal Asset Pricing Model with Stochastic

Consumption and Investment Opportunities,” Journal of Financial Economics 7,

265-296.

Breeden, Douglas T., Michael R. Gibbons and Robert H. Litzenberger, 1989,

“empirical Tests of the Consumption-Oriented CAPM,” Journal of Finance 44, 231-262.

Brown, Stephen, William Goetzmann and Stephen A. Ross, 1995, “Survival”

Journal of Finance 50, 853-873.

Buraschi, Andrea and Alexei Jiltsov, 1999, “How Large is the In¬‚ation Risk

Premium in the U.S. Nominal Term Structure,” Manuscript London Business School.

Burnside, Craig, Martin Eichenbaum, and Sergio Rebelo, 1993, “Labor Hoarding

and the Business Cycle,” Journal of Political Economy 101, 245-73.

Campbell, John Y. 1991, “A Variance Decomposition for Stock Returns,” Economic

Journal 101, 157-179.

Campbell, John Y. 1995, “Some Lessons from the Yield Curve,” Journal of

Economic Perspectives, 9, 129-152.

Campbell, John Y. 1996. “Understanding Risk and Return.” Journal of Political

Economy 104, 298-345.

Campbell, John Y. 1999. “Asset Prices, Consumption, and the Business Cycle.” in

John B. Taylor and Michael Woodford eds. Handbook of Macroeconomics. Amsterdam:

North-Holland.

Campbell, John Y., 2000, “Asset Pricing at the Millennium,” Journal of Finance

August

Campbell, John Y., and John H. Cochrane, 1999, “By Force of Habit: A

Consumption-Based Explanation of Aggregate Stock Market Behavior” Journal of

443

CHAPTER 22 REFERENCES

Political Economy, 107, 205-251.

Campbell, John Y., Andrew W. Lo and A. Craig MacKinlay 1997, The Econometrics

of Financial Markets Princeton NJ: Princeton University Press.

Campbell, John Y. and Robert J. Shiller. 1988a. “The Dividend-Price Ratio and

Expectations of Future Dividends and Discount Factors,” Review of Financial Studies 1,

195-227.

Campbell, John Y. and Robert J. Shiller. 1988b. “Stock Prices, Earnings, and

Expected Dividends.” Journal of Finance 43: 661-676.

Campbell, John Y. and Robert J. Shiller 1991, “Yield Spreads and Interest Rates: A

Bird™s Eye View” Review of Economic Studies 58, 495-514.

Carhart, Mark M. 1997, “On Persistence in Mutual Fund Performance,” Journal of

Finance 52, 57-82.

Chamberlain, Gary and Michael Rothschild, 1983, “Arbitrage, Factor Structure, and

Mean-Variance Analysis on Large Asset Markets,” Econometrica 51, 1281-1304

Chen, Nai-Fu, Richard Roll and Steven Stephen A. Ross, 1986, “Economic Forces

and the Stock Market,” Journal of Business 59, 383-403.

Christiano, Lawrence, Martin Eichenbaum, and Charles Evans, 1999, “Monetary

Policy Shocks: What Have we Learned and to What End?” Forthcoming in John Taylor,

ed., Handbook of Monetary Economics

Cochrane, John H 1988.,“How Big is the Random Walk in GNP?”, Journal of

Political Economy 96, 893-920.

Cochrane, John H., 1991a, ”Explaining the Variance of Price-Dividend Ratios” 5,

243-280.

Cochrane, John H., 1991b, “A Simple Test of Consumption Insurance,” Journal of

Political Economy 99, 957-976.

Cochrane, John H., 1991c “Volatility Tests and Ef¬cient Markets: A Review Essay”

Journal of Monetary Economics 27, 463-485.

Cochrane, John H., 1991d “Production-Based Asset Pricing and the Link Between

Stock Returns and Economic Fluctuations,” Journal of Finance 46, 207-234.

Cochrane, John H., 1994, “Permanent and Transitory Components of GNP and

Stock Prices” Quarterly Journal of Economics 109, 241-266.

Cochrane John H. 1994, “Shocks,” Carnegie-Rochester Conference Series on

Public Policy 41, 295-364.

Cochrane, John H., 1996, “A Cross-Sectional Test of an Investment-Based Asset

Pricing Model,” Journal of Political Economy 104, 572-621.

Cochrane, John H., 1997, “Where is the Market Going? Uncertain Facts and Novel

Theories,” Economic Perspectives 21: 6 (November/December 1997) Federal Reserve

Bank of Chicago.

Cochrane, John H., 1999a, “New Facts in Finance,” Economic Perspectives Federal

Reserve Bank of Chicago 23 (3) 36-58.

Cochrane, John H., 1999b, “Portfolio Advice for a Multifactor World” Economic

444

Perspectives Federal Reserve Bank of Chicago 23 (3) 59-78.

Cochrane, John H., 2000, “A Resurrection of the Stochastic Discount Factor/GMM

Methodology,” Manuscript, University of Chicago.

Cochrane, John H. and Lars Peter Hansen, 1992, “Asset Pricing Explorations

for Macroeconomics” In Olivier Blanchard and Stanley Fisher, Eds.,1992 NBER

Macroeconomics Annual 115-165.

Cochrane, John H. and Jesús Saá-Requejo 2000, “Beyond Arbitrage: Good Deal

Asset Price Bounds in Incomplete Markets” Journal of Political Economy 108, 79-119.

Cochrane, John H. and Argia M. Sbordone, 1988, “Multivariate Estimates of the

Permanent Components in GNP and Stock Prices” Journal of Economic Dynamics and

Control, 12, 255-296.

Constantinides, George M., 1989, “Theory of Valuation: Overview and Recent

Developments,” in Sudipto Bhattacharya and George M. Constantinides, eds., Theory of

Valuation Totwa NJ: Rowman & Little¬eld

Constantinides, George M. 1990. “Habit Formation: A Resolution of the Equity

Premium Puzzle.” Journal of Political Economy 98: 519-543.

Constantinides, George M., 1992, “A Theory of the Nominal Term Structure of

Interest Rates,” Review of Financial Studies 5, 531-52.

Constantinides, George M. 1998, “Transactions Costs and the Volatility Implied by

Option Prices.” Manuscript, Graduate School of Business, University of Chicago.

Constantinides, George M. and Darrell Duf¬e. 1996, “Asset Pricing with

Heterogeneous Consumers.” Journal of Political Economy 104, 219“240..

Constantinides, George M. and Thaleia Zariphopoulou, 1997, “Bounds on Option

Prices in an Intertemporal Setting with Proportional Transaction Costs and Multiple

Securities,” Manuscript, Graduate School of Business, University of Chicago.

Cox, John C. and Chi-fu Huang, 1989, “Optimal Consumption and Portfolio

Policies when Asset Prices Follow a Diffusion Process,” Journal of Economic Theory 39,

33-83.

Cox, John C., Jonathan E. Ingersoll, and Stephen A. Ross. 1985. “A Theory of the

Term Structure of Interest Rates.” Econometrica 53: 385-408.

Cox, John C., Stephen A. Ross, and Mark Rubinstein, 1979, “Option Pricing: A

Simpli¬ed Approach,” Journal of Financial Economics 7, 229-63.

Cox, John C., and Mark Rubinstein, 1985, Options Markets Englewood Cliffs, NJ:

Prentice Hall.

Dai, Qiang and Kenneth J. Singleton, 1999, “Speci¬cation Analysis of Af¬ne Term

Structure Models,” forthcoming, Journal of Finance

Daniel, Kent, David Hirshleifer and Avanidhar Subrahmanyam, 1998, “Investor

Psychology and Security Market Under- and Overreactions,” Journal of Finance 53,

1839-1885.

Debreu, Gerard 1959, The Theory of Value New York: Wiley and Sons.

DeBondt, Werner F.M. and Thaler, Richard H., 1985, “Does the Stock Market

445

CHAPTER 22 REFERENCES

Overreact?” Journal of Finance 40, 793-805.

Dixit, Avinash and R. Pindyck. Investment Under Uncertainty. Princeton NJ:

Princeton University Press, 1994.

Duarte, Jefferson 2000 “The Relevance of the Price of Risk in Af¬ne Term-Structure

models” Manuscript, University of Chicago

Duffee, Gregory, 1999, “Forecasting future interest rates: Are af¬ne models

failures?” Manuscript, University of California at Berkeley.

Duf¬e, J. Darrel and Rui Kan, 1996, “A Yield Factor Model of the Term structure of

interest rates” Mathematical Finance 6, 379-406

Dybvig, Philip H. and Jonathan E. Ingersoll Jr., 1982, “Mean-Variance Theory in

Complete Markets,” Journal of Business 55, 233-51.

Dybvig, P., and Stephen Ross, 1985, “Yes, the APT is testable,” Journal of Finance

40, 1173-1188.

Dybvig, P., J. Ingersoll Jr. and Stephen Ross 1996, “Long Forward and Zero-Coupon

Rates Can Never Fall,” Journal of Business, 69, 1-25.

Eichenbaum, Martin, Lars Peter Hansen and Kenneth Singleton, 1988, “A

Time-Series Analysis of Representative Agent Models of Consumption and Leisure

Choice under Uncertainty,” Quarterly Journal of Economics 103, 51-78.

Engel, Charles, 1996, “The Forward Discount Anomaly and the Risk Premium: a

Survey of Recent Evidence,” Journal of empirical Finance 3, 123-192.

Engle, Robert F. and Clive W. J. Granger, 1987, “Cointegration and Error

Correction: Representation, Estimation, and Testing,” Econometrica 55, 251-276.

Epstein, Larry G. and Stanley E. Zin. 1989. “Substitution, Risk Aversion and the

Temporal Behavior of Asset Returns.” Journal of Political Economy 99: 263-286.

Fama, Eugene F., 1965, “The Behavior of Stock Market Prices, Journal of Business

38, 34-105.

Fama, Eugene F., 1970, “Ef¬cient Capital Markets: A review of Theory and

empirical Work,” Journal of Finance 25, 383-417.

Fama, Eugene F., 1984, “Forward and Spot Exchange Rates,” Journal of Monetary

Economics 14, 319-338.

Fama, Eugene F. 1991, “Ef¬cient Markets II,” Journal of Finance 46, 1575-1618.

Fama, Eugene F. and Robert R. Bliss, 1987, “The information in Long-Maturity

Forward Rates,” American Economic Review, 77, 680-92.

Fama, Eugene F. and Kenneth R. French. 1988a. “Permanent and Temporary

Components of Stock Prices.” Journal of Political Economy 96: 246-273.

Fama, Eugene F. and Kenneth R. French. 1988b. “Dividend Yields and Expected

Stock Returns.” Journal of Financial Economics 22: 3-27.

Fama, Eugene F. and Kenneth R. French, 1989, “Business Conditions and Expected

Returns on Stocks and Bonds,” Journal of Financial Economics 25, 23-49.

Fama, Eugene F. and Kenneth R. French, 1993, “Common Risk Factors in the

Returns on Stocks and Bonds,” Journal of Financial Economics 33, 3-56.

446

Fama, Eugene F., and Kenneth R. French, 1996, “Multifactor Explanations of

Asset-Pricing Anomalies,” Journal of Finance 47, 426-465.

Fama, Eugene F., and Kenneth R. French, 1997a, “Size and Book-to-Market Factors

in Earnings and Returns,”Journal of Finance 50, 131-55.

Fama, Eugene F., and Kenneth R. French, 1997b, “Industry Costs of Equity,”

Journal of Financial Economics 43, 153-193.

Fama, Eugene F., and Kenneth R. French, 2000, “The Equity Premium,” Working

paper, University of Chicago.

Fama, Eugene F., and James D. MacBeth, 1973, “Risk Return and Equilibrium:

empirical Tests,” Journal of Financial Political Economy 71, 607-636.

Ferson, Wayne E., 1995, “Theory and Empirical Testing of Asset Pricing Models,”

in R. A. Jarrow, V. Maksimovic, and W. T. Ziemba, eds., Handbooks in OR & MS,

Volume 9, Finance Amsterdam: Elsevier Science B.V.

Ferson, Wayne E. and George Constantinides. 1991. “Habit Persistence and

Durability in Aggregate Consumption: empirical Tests.” Journal of Financial Economics

29: 199“240.

Ferson, Wayne E. and Foerster, Stephen R., 1994, “Finite Sample Properties of the

Generalized Method of Moments in Tests of Conditional Asset Pricing Models,” Journal

of Financial Economics 36, 29-55.

Ferson, Wayne E. and Campbell R. Harvey, 1999, “Conditioning Variables and

Cross-section of Stock Returns”, Journal of Finance 54, 1325-1360.

French, Kenneth, G. William Schwert and Robert F. Stambaugh, 1987, “Expected

Stock Returns and Volatility,” Journal of Financial Economics 19, 3-30.

Friedman, Milton, 1953,“The Methodology of Positive Economics,” in Essays in

Positive Economics Chicago: University of Chicago Press

Friend, I. and M Blume, 1975, “The Demand for Risky Assets,” American

Economic Review 65, 900-922.

Fuhrer, Jeffrey C., George R. Moore, Scott D. Schuh, 1995, “Estimating the

Linear-Quadratic Inventory Model: Maximum Likelihood versus Generalized Method

of Moments,” Journal of Monetary Economics;35 115-57.

Gallant, A. Ronald, Lars Peter Hansen and George Tauchen, 1990, “Using

Conditional Moments of Asset Payoffs to Infer the Volatility of Intertemporal Marginal

Rates of Substitution,” Journal of Econometrics 45, 141-79.

Gallant, A. Ronald, and George Tauchen, 1997, “Estimation of Continuous-Time

Models for Stock Returns and Interest Rates,” Macroeconomic Dynamics 1, 135-68.

Gibbons, Michael, Stephen A. Ross, and Jay Shanken, 1989, “A Test of the

Ef¬ciency of a Given Portfolio,” Econometrica 57, 1121-1152.

Glosten, Lawrence, Ravi Jagannathan and David Runkle, 1993, “On the Relation

Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks”

Journal of Finance 48, 1779-1801.

Grinblatt Michael and Sheridan Titman, 1985, “Factor Pricing in a Finite Economy,”

447

CHAPTER 22 REFERENCES

Journal of Financial Economics 12, 497-507.

Grossman, Sanford J. and Robert J. Shiller, 1981, “The determinants of the

Variability of Stock Market Prices” American Economic Review 71, 222-227.

Grossman, Sanford J. and Joseph E. Stiglitz, 1980, “On the Impossibility of

Informationally Ef¬cient Markets,” American Economic Review 70, 393-408.

Hamilton, James, 1994 Time Series Analysis, Princeton NJ: Princeton University

Press.

Hamilton, James, 1996, “The Daily Market for Federal Funds,” Journal of Political

Economy 104, 26-56.

Hansen, Lars Peter, 1982, “Large Sample Properties of Generalized Method of

Moments Estimators,” Econometrica 50, 1029-1054.

Hansen, Lars Peter, 1987, “Calculating Asset Prices in Three Example Economies,”

in T.F. Bewley, Advances in Econometrics, Fifth World Congress, Cambridge University

Press.

Hansen, Lars Peter, John Heaton and Erzo Luttmer, 1995, “Econometric Evaluation

of Asset Pricing Models.” The Review of Financial Studies 8, 237-274.

Hansen, Lars Peter, John Heaton and Amir Yaron, 1996, “ Finite-Sample Properties

of Some Alternative GMM Estimators,” Journal of Business and Economic Statistics 4,

262-80.

Hansen, Lars Peter and Robert J. Hodrick, 1980, “Forward Exchange Rates as

Optimal Predictors of Future Spot Rates: An Econometric Analysis,” Journal of Political

Economy 88, 829-53.

Hansen, Lars Peter and Ravi Jagannathan 1991, “Implications of Security Market

Data for Models of Dynamic Economies,” Journal of Political Economy 99, 225-62.

Hansen, Lars Peter and Ravi Jagannathan 1997, “Assessing Speci¬cation Errors in

Stochastic Discount Factor Models,” Journal of Finance 52, 557-90.

Hansen, Lars Peter and Scott F. Richard, 1987, “The Role of Conditioning

Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing

Models.” Econometrica 55 (1987): 587-614.

Hansen, Lars Peter and Kenneth J. Singleton, 1982, “Generalized Instrumental

Variables Estimation of Nonlinear Rational Expectations Models,” Econometrica 50,

1269-1288.

Hansen, Lars Peter and Kenneth J. Singleton 1984, “Errata” Econometrica 52,

267-268

Hansen, Lars Peter and Kenneth J. Singleton, 1983, “Stochastic Consumption, Risk

Aversion, and the Temporal Behavior of Asset Returns,” Journal of Political Economy

91, 249-268.

Harrison, J. Michael and David M. Kreps, 1979, “Martingales and Arbitrage in

Multiperiod Securities Markets,” Journal of Economic Theory 20, 381-408.

Hayek, Friedrich A. 1945, “The Use of Knowledge in Society,” American Economic

Review 35, 519-530.

448

He, Hua and Neil Pearson. “Consumption and Portfolio Policies with Incomplete

Markets: The In¬nite Dimensional Case.” Journal of Economic Theory 54 (1992):

259-305.

Heaton, John C., 1993, “The Interaction Between Time-Nonseparable Preferences

and Time Aggregation,” Econometrica 61, 353-385.

Heaton, John C., 1995, “An empirical Investigation of Asset Pricing with

Temporally Dependent Preference Speci¬cations.” Econometrica 63: 681“717.

Heaton, John and Deborah Lucas, 1996, “Evaluating the Effects of Incomplete

Markets on Risk-Sharing and Asset Pricing,” Journal of Political Economy 103, 94-117.

Heaton, John and Deborah Lucas, 1997, “Market Frictions, Saving Behavior and

Portfolio Choice,” Macroeconomic Dynamics 1, 76-101.

Heaton, John and Deborah Lucas, 1997 “Portfolio Choice and Asset Prices: The

Importance of Entrepreneurial Risk,” Manuscript, Northwestern University.

Hendricks, Darryl, Jayendu Patel and Richard Zeckhauser, 1993, “Hot Hands in

Mutual Funds: Short-Term Persistence of Performance,” Journal of Finance 48, 93-130.

Hindy, Ayman, and Chi-fu Huang, 1992, “Intertemporal Preferences for Uncertain

Consumption: A Continuous-Time Approach,” Econometrica 60, 781-801.

Ho, Thomas S. Y. and Sang-bin Ho Lee, 1986, “Term Structure Movements and

Pricing Interest Rate Contingent Claims,” Journal of Finance 41, 1011-1029.

Hodrick, Robert, 1987, The empirical Evidence on the Ef¬ciency of Forward and

Futures Foreign Exchange Markets Chur, Switzerland: Harwood Academic Publishers.

Hodrick, Robert, 1992, “Dividend Yields and Expected Stock Returns: Alternative

Procedures for Inference and Measurement,” Review of Financial Studies 5, 357-386.

Hodrick, Robert, 2000, International Financial Management, Forthcoming,

Englewood Cliffs, NJ: Prentice-Hall.

Hsieh, David and William Fung, 1999, “Hedge Fund Risk Management,” Working

paper, Duke University.

Jacquier, Eric, Nicholas Polson and Peter Rossi 1994, “Bayesian Analysis of

Stochastic Volatility Models,” Journal of Business and Economic Statistics 12, 371-418.

Jagannathan, Ravi and Zhenyu Wang, 1996 “The Conditional CAPM and the

Cross-Section of Expected Returns,” Journal of Finance 51, 3-53.

Jagannathan, Ravi, and Zhenyu Wang, 2000, “Ef¬ciency of the Stochastic Discount

Factor Method for Estimating Risk Premiums,” Manuscript, Northwestern University.

Jegadeesh, Narasimham, and Sheridan Titman, 1993, “Returns to Buying Winners

and Selling Losers: Implications for Stock Market Ef¬ciency,” Journal of Finance 48,

65-91.

Jensen, Michael C., 1969, “The pricing of capital assets and evaluation of

investment portfolios,” Journal of Business 42, 167-247.

Jermann, Urban, 1998, “Asset Pricing in Production Economies,” Journal of

Monetary Economics 4, 257-275.

Johannes, Michael, 2000, “Jumps to Interest Rates: A Nonparametric Approach, ”

449

CHAPTER 22 REFERENCES

Manuscript, University of Chicago.

Jorion, Philippe, and William Goetzmann, 1999, “Global Stock Markets in the

Twentieth Century,” Journal of Finance 54, 953-980.

Kandel, Shmuel and Robert F. Stambaugh. 1990. “Expectations and Volatility of

Consumption and Asset Returns,” Review of Financial Studies 3: 207-232.

Kandel, Shmuel and Robert F. Stambaugh. 1991. “Asset Returns and Intertemporal

Preferences.” Journal of Monetary Economics 27, 39-71

Kandel, Shmuel and Robert F. Stambaugh, 1995, “Portfolio Inef¬ciency and the

Cross-Section of Expected Returns.” Journal of Finance 50, 157-84

Kennedy, Peter, 1994, “The Term Structure of Interest Rates as a Gaussian Random

Field,” Mathematical Finance 4, 247-258.

Keim, Donald and Robert F. Stambaugh, 1986, “Predicting Returns in Stock and

Bond Markets,” Journal of Financial Economics 17, 357-390.

Kleidon, Allan, 1986, “Variance Bounds tests and Stock Price Valuation Models,

Journal of Political Economy, 94, 953-1001.

Kocherlakota, Narayana R., 1990, “On the ™Discount™ Factor in Growth Economies,”

Journal of Monetary Economics 25, 43-47.

Kocherlakota, Narayanna, 1996, “The Equity Premium: It™s Still a Puzzle,” Journal

of Economic Literature 34, 42-71.

Kothari, S. P., Jay Shanken and Richard G. Sloan, 1995, “Another Look at the

Cross-Section of Expected Stock Returns, Journal of Finance, 50, 185-224.

Knez, Peter, Robert Litterman, and Jos© Scheinkman, 1994,“Explorations into

Factors Explaining Money Market Returns,” Journal of Finance 49, 1861-82.

Knez, Peter J. and Mark J. Ready, 1997, “On the Robustness of Size and

Book-to-Market in Cross-Sectional Regressions,” Journal of Finance, 52, 1355-1382.

Kuhn, Thomas 1970, The Structure of Scienti¬c Revolutions, 2nd Ed. Chicago:

University of Chicago Press

Kydland, Finn and Edward C. Prescott, 1982, “Time to Build and Aggregate

Fluctuations,” Econometrica 50, 1345-1370.

Lakonishok, Josef, Andrei Shleifer and Robert W. Vishny, 1992, “The Structure

and Performance of the Money Management Industry,” Brookings Papers on Economic

Activity: Microeconomics 1992, 339-391.

Lamont, Owen, 1998, “Earnings and Expected Returns,” Journal of Finance 53,

1563-1587.

Ledoit, Olivier. “Essays on Risk and Return in the Stock Market.” Ph. D.

dissertation, Massachusetts Institute of Technology, 1995.

Ledoit, Olivier, and Antonio Bernardo, 1999, “Gain, loss and asset pricing,” Journal

of Political Economy 108, 144-172.

Leland, Hayne E. 1985, “Option Pricing and Replication with Transactions Costs”

Journal of Finance 40, 1283-1301.

LeRoy, Stephen F., 1973, Risk Aversion and the Martingale Property of Stock

450

Prices, International Economic Review 14 436-46.

LeRoy Stephen and Richard Porter, 1981, “The Present Value Relation: Tests Based

on Variance Bounds,” Econometrica 49, 555-557.

Levy, Haim, 1985, “Upper and Lower Bounds of Put and Call Option Value:

Stochastic Dominance Approach.” Journal of Finance 40, 1197-1217.

Lewis, Karen K., 1995, in G. Grossman and K. Rogoff, eds., “Puzzles in

International Financial Markets,” Handbook of International Economics, Volume III,

Elsevier Science B.V, 1913-1971.

Lettau, Martin, and Sydney Ludvigson 2000, “Consumption, Aggregate Wealth and

Expected Stock Returns,” Manuscript, Federal Reserve Bank of New York

Lettau, Martin, and Sydney Ludvigson, 1999, “Resurrecting the (C)CAPM: A

Cross-Sectional Test When Risk Premia are Time-Varying„” Manuscript, Federal

Reserve Bank of New York.

Liew, Jimmy, and Maria Vassalou, 1999, “Can Book-to-Market, Size and

Momentum be Risk Factors that Predict Economic Growth?” Working paper, Columbia

University.

Lintner, John, 1965, “The Valuation of Risky Assets and the Selection of Risky

Investment in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics

47, 13-37.

Lintner, John, 1965, ”Security Prices, Risk and Maximal Gains from

Diversi¬cation,” Journal of Finance 20

Longstaff, Francis, 2000, “Arbitrage and the Expectations Hypothesis,” Journal of

Finance 55, 989-994.

Lucas, Robert E., Jr., 1978, “Asset Prices in and Exchange Economy,” Econometrica

46, 1429-1446.

Lucas, Robert E. Jr., 1987, Models of Business Cycles, London and New York:

Blackwell.

Lucas, Robert E., Jr., 1988, “Money Demand in the United States: A Quantitative

Review,”Carnegie-Rochester Conference Series on Public Policy 29, 137-67.

Luttmer, Erzo G. J. 1996, “Asset Pricing in Economies with Frictions.”

Econometrica 64, 1439-67.

Luttmer, Erzo G. J., 1999, “What Level of Fixed Costs Can Reconcile Consumption

and Stock Returns? ” Journal of Political Economy 107, 969-97.

Mace, Barbara, 1991, “Full Insurance in the Presence of Aggregate Uncertainty,”

Journal of Political Economy 99,

Mankiw, N. Gregory. 1986. “The Equity Premium and the Concentration of

Aggregate Shocks.” Journal of Financial Economics 17: 211-219.

Mace, Barbara, 1991, “Full Insurance in the Presence of Aggregate Uncertainty,”

Journal of Political Economy 99, 928-56.

MacKinlay, A. Craig, 1995, “Multifactor Models Do Not Explain Deviations from

the CAPM.” Journal of Financial Economics 38, 3-28.

451

CHAPTER 22 REFERENCES

Malkiel, Burton, 1990, A Random Walk Down Wall Street, New York: Norton

Mankiw, N. Gregory 1986, “The Equity Premium and the Concentration of

Aggregate Shocks,” Journal of Financial Economics, 17, 211-219.

Mankiw, N. Gregory and Stephen Zeldes, 1991, “The Consumption of Stockholders

and Non-Stockholders,” Journal of Financial Economics, 29, 97-112.

Markowitz, Harry, 1952, “Portfolio Selection,” Journal of Finance, 7, 77-99.

McCloskey,Donald N., 1983, “The Rhetoric of Economics” Journal of Economic

Literature 21, 481-517.

McCloskey, Deirdre N., 1998, The rhetoric of economics Second edition. Madison

and London: University of Wisconsin Press

Mehra, Rajnish and Edward Prescott. 1985. “The Equity Premium Puzzle,” Journal

of Monetary Economics 15, 145-161.

Merton, Robert C., 1969 “Lifetime Portfolio Selection Under Uncertainty: The

Continuous Time Case,” Review of Economics and Statistics 51, 247-257.

Merton, Robert C., 1971a, “Optimum consumption and Portfolio rules in a

Continuous Time Model,” Journal of Economic Theory 3, 373-413.

Merton, Robert C., 1973a, “An Intertemporal Capital Asset Pricing Model,”

Econometrica 41, 867-87.

Merton, Robert C., 1973, “The Theory of Rational Option Pricing,” Bell Journal of

Economics and Management Science 4, 141-83.

Miller, Merton and Myron Scholes, 1972, “Rate of return in Relation to Risk: A

Reexamination of Some Recent Findings.” in Michael C. Jensen ed., Studies in the

Theory of Capital Markets New York: Praeger

Moskowitz, Tobias and Mark Grinblatt, 1998, “Do industries explain momentum?”

CRSP working paper 480, University of Chicago.

Moskowitz, Tobias and Mark Grinblatt, 1999, “Tax Loss Selling and Return

Autocorrelation: New Evidence,” Working Paper, University of Chicago.

Merton, Robert C. “Theory of Rational Option Pricing,” 1973, Bell Journal of

Economics and Management Science 4, 141-83.

Newey, Whitney K and Kenneth D. West, 1987a, “Hypothesis Testing with Ef¬cient

Method of Moments,” International Economic Review 28, 777-87.

Newey, Whitney K. and Kenneth D. West, 1987b, “A Simple, Positive Semi-de¬nite,

Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica

55, 703-08.

Ogaki, Masao, 1992, “Generalized Method of moments: Econometric

Applications,” In G. Maddala, C. Rao, and H. Vinod (eds.) Handbook of Statistics,

Volume 11: Econometrics, Amsterdam: North-Holland.

Piazzesi, Monika, 1999, “An Econometric model of the Yield Curve With

Macroeconomic Jump Effects” Manuscript, Stanford University

Popper, Karl 1959, The Logic of Scienti¬c Discovery, New York:: Harper

Poterba, James and Lawrence H. Summers, 1988, “Mean Reversion in Stock

452

Returns: Evidence and Implications.” Journal of Financial Economics 22, 27-60.

Rietz, Thomas A. 1988. “The Equity Risk Premium: A Solution,” Journal of

Monetary Economics 22, 117-131.

Reyfman, Alexander, 1997, “Labor Market Risk and Expected Asset Returns,”

Ph.D. Thesis, University of Chicago.

Rietz, Tom, 1988, The equity Risk Premium: A Solution? Journal of Monetary

Economics 21 117-132

Ritchken, Peter H., 1985, “On Option Pricing Bounds,” Journal of Finance. 40,

1219-1233.

Roll, Richard, 1977, “A Critique of the Asset Pricing Theory™s Tests: Part I,”Journal

of Financial Economics 4, 129-176

Roll, Richard, 1984, “Orange Juice and Weather” The American Economic Review,

74, 861-880.

Roll, Richard and Stephen A. Ross, 1995, “On the Cross-sectional Relation between

Expected Returns and Betas,” Journal of Finance 49, 101-121

Ross, Stephen A., 1976a, “The Arbitrage Theory of Capital Asset Pricing,” Journal

of Economic theory 13, 341-360.

Ross, Stephen A., 1976b, “Options and Ef¬ciency.” Quarterly Journal of

Economics, 90, 75-89.

Ross, Stephen A., 1976c, “Risk, Return and Arbitrage.” in Risk and Return in

Finance, Volume 1, edited by I. Friend and J. Bicksler. 189-218. Cambridge: Ballinger,

Ross 1978 on beta and linear discount factors?

Samuelson, Paul A., 1965, “Proof that Properly Anticipated Prices Fluctuate

Randomly,” Industrial Management Review, 6, 41-49.

Samuelson, Paul A., 1969, “Lifetime Portfolio Selection by Dynamic Stochastic

Programming,” Review of Economics and Statistics 51, 239-246.

Santa Clara, Pedro and Didier Sornette, 1999, “The Dynamics of the Forward

Interest Rate Curve with Stochastic String Shocks,”Forthcoming Review of Financial

Studies

Sargent, Thomas J., 1993, Bounded Rationality in Macroeconomics, Oxford:

Oxford University Press.

Sargent, Thomas J. 1989, “Two Models of Measurements and the Investment

Accelerator,” Journal of Political Economy 97, 251-287.

Schwert, William, 1990, “Stock Market Volatility,” Financial Analysts Journal,

May-June, 23-44.

Shanken, Jay, 1982, “The Arbitrage Pricing Theory: Is it Testable?” Journal of

Finance, 37, 1129-1140.

Shanken, Jay, 1987,Multivariate Proxies and Asset Pricing Relations: Living with

the Roll Critique,” Journal of Financial Economics 18, 91-110.

Shanken, Jay, 1992a, “The Current State of the Arbitrage Pricing Theory.” Journal

of Finance, 47 1569-74.

453

CHAPTER 22 REFERENCES

Shanken, Jay, 1992b, “On the Estimation of Beta Pricing Models,” Review of

Financial Studies 5, 1-34.

Sharpe, William, 1964, “Capital Asset Prices: A Theory of Market Equilibrium

Under Conditions of Risk,” Journal of Finance 19, 425-442.

Shiller, Robert J., 1982.,“Consumption, Asset Markets, and Macroeconomic

Fluctuations.” Carnegie Rochester Conference Series on Public Policy 17, 203-238

Shiller, Robert J., 1981 “Do Stock Prices Move too Much to be Justi¬ed by

Subsequent Changes in Dividends?” American Economic Review 71, 421-436.

Shiller, Robert J, 1989, Market Volatility Cambridge MA: MIT Press.

Stambaugh, Robert F., 1982, “On the Exclusion of Assets from Tests of the Two-

Parameter Model: A Sensitivity Analysis,” Journal of Financial Economics 10, 237-68.

Stambaugh, Robert F., 1988, “The Information in Forward Rates: Implications for

Models of the Term Structure,” Journal of Financial Economics 10, 235-268.

Sundaresan, Suresh M. 1989. “Intertemporally Dependent Preferences and the

Volatility of Consumption and Wealth.” Review of Financial Studies 2: 73-88.

Tallarini, Thomas, 1999, “Risk-Sensitive Real Business Cycles” Manuscript,

Carnegie Mellon University. Forthcoming Journal of Monetary Economics

Kjetil Storesletten, Christopher Telmer and Amir Yaron, 1999, Asset Pricing

with Idiosyncratic Risk and Overlapping Generations, Manuscript, Carnegie Mellon

University.

Taylor, John B. (ed.) 1999, Monetary Policy Rules, Chicago: University of Chicago

Press.

Thompson, Rex, 1978, “The Information Content of Discounts and Premiums on

Closed-End Fund Shares,” Journal of Financial Economics 6, 151-86

Tobin, James, 1958, “Liquidity Preference as a Behavior Towards Risk,” Review of

Economic Studies 25, 68-85.

Vasicek, Oldrich, 1977, “An Equilibrium Characterization of the Term Structure,”

Journal of Financial Economics 5, 177-188.

Vassalou, Maria, 1999, “The Fama-French Factors as Proxies for Fundamental

Economic Risks” Working paper, Columbia University

Vuoltennaho, Tuomo, 1999, “What Drives Firm-Level Stock Returns?” Working

paper, University of Chicago.

Weil, Philippe, 1989. “The Equity Premium Puzzle and the Risk-Free Rate Puzzle.”

Journal of Monetary Economics 24: 401-421.

Wheatley, Simon 1988a, “Some Tests of the Consumption-Based Asset Pricing

Model,” Journal of monetary Economics 22, 193-218.

Wheatley, Simon, 1988b, “Some Tests of International Equity Integration,” Journal

of Financial Economics 21, 177-212.

White, Halbert 1980, “A Heteroskedasticity-Consistent Covariance Matrix

Estimator and a Direct Test for Heteroskedasticity,” Econometrica 48, 817-38.

Yan, Shu, 2000, Ph.D. dissertation, University of California at Los Ageles.

454

PART V

Appendix

455

Chapter 23. Continuous time

This chapter is a brief introduction to the mechanics of continuous time stochastic processes;

i.e. how to use dz and dt. I presume the reader is familiar with discrete time ARMA models,

i.e. models of the sort xt = ρxt’1 + µt , and draw analogies of continuous time constructs to

those models.

The formal mathematics of continuous time processes are a bit imposing. For example,

the basic random walk zt is not time-differentiable, so one needs to rethink the de¬nition

Rt

of an integral and differential to write obvious things like zt = s=0 dzt . Also, since zt is

a random variable one has to specify not only the usual measure-theoretic foundations of

random variables, but their evolution over a continuous time index. However, with a few

basic, intuitive rules like dz 2 = dt, one can use continuous time processes quite quickly, and

that™s the aim of this chapter.

23.1 Brownian Motion

zt , dzt are de¬ned by zt+∆ ’ zt ∼ N (0, ∆).