Predictability and habit persistence

ARTICLE IN PRESS

Journal of Economic Dynamics & Control 30 (2006) 2217–2260

0165-1889/$ -

doi:10.1016/j

�CorrespoFrance. Tel.:

E-mail ad

www.elsevier.com/locate/jedc

Predictability and habit persistence

Fabrice Collarda, Patrick Feveb,�, Imen Ghattassic

aUniversity of Toulouse, CNRS-GREMAQ and IDEI, FrancebUniversity of Toulouse, CNRS-GREMAQ and IDEI, Banque de France, Research Division,

manufacture des Tabacs, bat F, 21 allee de Brienne, 31000 Toulouse, FrancecUniversity of Toulouse, GREMAQ, France

Received 13 January 2005; accepted 20 June 2005

Available online 2 November 2005

Abstract

This paper highlights the role of persistence in explaining predictability of excess returns. To

this end, we develop a CCAPM model with habit formation when the growth rate of

endowments follows a first order Gaussian autoregression. We provide a closed form solution

of the price–dividend ratio and determine conditions that guarantee the existence of a

bounded equilibrium. The habit stock model is found to possess internal propagation

mechanisms that increase persistence. It outperforms the time separable and a ‘Catching up

with the Joneses’ version of the model in terms of predictability therefore highlighting the role

of persistence in explaining the puzzle.

r 2005 Elsevier B.V. All rights reserved.

JEL classification: C62; G12

Keywords: Asset pricing; Catching up with the Joneses; Habit stock; Predictability

0. Introduction

Over the last twenty years, the predictability of excess stock returns has attracted agreat deal of attention. The initial piece of evidence in favor of predictability was

see front matter r 2005 Elsevier B.V. All rights reserved.

.jedc.2005.06.016

nding author. GREMAQ-Universite de Toulouse 1, 21 allee de Brienne, 31000 Toulouse,

+33 5 61 12 85 75; fax: +335 61 22 55 63.

dress: [email protected] (P. Feve).

www.elsevier.com/locate/jedc

ARTICLE IN PRESS

F. Collard et al. / Journal of Economic Dynamics & Control 30 (2006) 2217–22602218

obtained by examining univariate time series properties (See e.g. Poterba andSummers, 1988). The literature has also reported convincing evidence that financialand accounting variables have predictive power for stock returns (See Fama andFrench, 1988, 1989; Campbell and Shiller, 1988; Hodrick, 1992; Campbell et al.,1997; Cochrane, 1997, 2001; Lamont, 1998; Lettau and Ludvigson, 2001; Campbell,2003). The theoretical literature that has investigated the predictability of returns bythe price–dividend ratio has established that two phenomena are key to explain it:persistence and time-varying risk aversion (see Campbell and Cochrane, 1999;Menzly et al., 2004 among others). Leaving aside time-varying risk aversion, thispaper evaluates the role of persistence in accounting for predictability.

However, recent empirical work has casted doubt on the ability of theprice–dividend ratio to predict excess returns (see e.g. Stambaugh, 1999; Torouset al., 2004; Ang, 2002; Campbell and Yogo, 2005; Ferson et al., 2003; Ang andBekaert, 2004 for recent references). In light of these results, we first re-examine thepredictive power of the price–dividend ratio using annual data for the period1948–2001.1 We find that the ratio has indeed predictive power until 1990. In thelatter part of the sample, the ratio keeps on rising while excess returns remain stable,and the ratio looses its predictive power after 1990. Our results are in line with Ang(2002) and Ang and Bekaert (2004), and suggest that the lack of predictability isrelated to something pertaining to the exceptional boom of the stock market in thelate nineties rather than the non-existence of predictability. Furthermore, thepredictability of the first part of the sample remains to be accounted for.

To this end, we develop an extended version of the consumption based capitalasset pricing model (CCAPM). The model – in its basic time separable version –indeed fails to account for this set of stylized facts, giving rise to a predictabilitypuzzle. This finding is now well established in the literature and essentially stemsfrom the inability of this model to generate enough persistence. Excess returnsessentially behave as iid stochastic processes, unless strong persistence is added to theshocks initiating fluctuations on the asset market. Therefore, neither do they exhibitserial correlation nor are they strongly related to other variables. But recenttheoretical work has shown that the CCAPM can generate predictability of excessreturns providing the basic model is amended (see Campbell, 2003 for a survey). Thiswork includes models with heterogenous investors (see Chan and Kogan, 2001) oraforementioned models with time varying risk aversion generated by habitformation. This paper will partially pursue this latter route and consider a habitformation model. It should be noted that the literature dealing with habit formationfalls into two broad categories. On the one hand, internal habit formation capturesthe influence of individual’s own past consumption on the individual currentconsumption choice (see Boldrin et al., 1997). On the other hand, external habitformation captures the influence of the aggregate past consumption choices on thecurrent individual consumption choices (Abel, 1990). This latter case is denoted‘Catching up with the Joneses’. Two specifications of habit formation are usuallyconsidered. The first one (see Campbell and Cochrane, 1999) considers that the agent

1The data are borrowed from Lettau and Ludvigson (2005).

ARTICLE IN PRESS

F. Collard et al. / Journal of Economic Dynamics & Control 30 (2006) 2217–2260 2219

cares about the difference between his/her current consumption and a consumptionstandard. The second (see Abel, 1990) assumes that the agent cares about the ratiobetween these two quantities. One important difference between the two approachesis that the coefficient of risk aversion is time varying in the first case, while it remainsconstant in the second specification. This has strong consequences for the ability ofthe model to account for the predictability puzzle, as a time-varying coefficient isthought to be required to solve the puzzle (see Menzly et al., 2004). This thereforeseems to preclude the use of a ratio specification to tackle the predictability of stockreturns. One of the main contribution of this paper will be to show that, despite theconstant risk aversion coefficient, habit formation in ratio can account for a non-negligible part of the long horizon returns predictability. Note that the model is byno means designed to solve neither the equity premium puzzle nor the risk free ratepuzzle, since time varying risk aversion is necessary to match this feature of thedata.2 Our aim is rather to highlight the role of persistence generated by habits inexplaining the predictability puzzle, deliberately leaving the equity premium puzzleaside.

We develop a simple CCAPM model a la Lucas (1978). We however depart fromthe standard setting in that we allow preferences to be non time separable. The modelhas the attractive feature of introducing tractable time non separability in a generalequilibrium framework. More precisely, we consider that preferences are character-ized by a ‘Catching up with the Joneses’ phenomenon. In a second step, we allowpreferences to depend not only on lagged aggregate consumption but also on thewhole history of aggregate consumption, therefore reinforcing both time non-separability and thus persistence. Our specification has the advantage of being simpleand more parsimonious than the specification used by Campbell and Cochrane(1999) while maintaining the same economic mechanisms and their implications forpersistence. We follow Abel (1990) and specify habit persistence in terms of ratio.This particular feature together with a CRRA utility function implies thatpreferences are homothetic with regard to consumption. As in Burnside (1998), weassume that endowments grow at an exogenous stochastic rate and we keep with theGaussian assumption. These features enable us to obtain a closed form solution tothe asset pricing problem and give conditions that guarantee that the solution isbounded. We then investigate the dynamic properties of the model and itsimplications in terms of moment matching and predictability over long horizons.We find that, as expected, the time separable model fails to account for most of assetpricing properties. The ‘Catching up with the Joneses’ model weakly enhances theproperties of the CCAPM to match the stylized facts but its persistence is too low tosolve the predictability puzzle. Conversely, the model with habit stock is found togenerate much greater persistence than the two previous versions of the model.Finally, the habit stock version of the model outperforms the time separable and thecatching up models in terms of predictability of excess returns. Since, risk aversion is

2Habit formation in ratio is known to fail to account for both puzzles. See Campbell et al. (1997,

pp. 328–329) and Campbell (2003).

ARTICLE IN PRESS


held constant in the model, this result illustrates the role of persistence in accountingfor predictability.

The remaining of the paper is organized as follows. Section 1 revisits thepredictability of excess returns by the price–dividend ratio using annual data forthe US economy in the post-WWII period. Section 2 develops a catching up with theJoneses version of the CCAPM model. We derive the analytical form of theequilibrium solution and the conditions that guarantee the existence of boundedsolutions, assuming that dividend growth is Gaussian and serially correlated. InSection 3, we extend the model to a more general setting where preferences are afunction of the whole history of the past aggregate consumptions. We again providea closed form solution for price–dividend ratio and conditions that guaranteebounded solutions. In Section 4, we investigate the quantitative properties of themodel and evaluate the role of persistence in accounting for predictability. A lastsection offers some concluding remarks.

1. Empirical evidence

This section examines the predictability of excess returns using the data of Lettauand Ludvigson (2005).3

1.1. Preliminary data analysis

The data used in this study are borrowed from Lettau and Ludvigson (2005).These are annual per capita variables, measured in 1996 dollars, for the period1948–2001.4 We use data on excess return, dividend and consumption growth, andthe price–dividend ratio. All variables are expressed in real per capita terms. Theprice deflator is the seasonally adjusted personal consumption expenditure chain-type deflator ð1996 ¼ 100Þ as reported by the Bureau of Economic Analysis.

Although we will be developing an endowment economy model whereconsumption and dividend streams should equalize in equilibrium, in the subsequentanalysis we acknowledge their low correlation in the data. This led us to first measureendowments as real per capita expenditures on nondurables and services as reportedby the US department of commerce. Note that since, for comparability purposes, weused Lettau and Ludvigson data, we also excluded shoes and clothing from the scopeof consumption. We then instead measure endowments by dividends as measured bythe CRSP value-weighted index. As in Lettau and Ludvigson (2005), dividends arescaled to match the units of consumption. Excess return is measured as the return onthe CRSP value-weighted stock market index in excess of the three-month Treasurybill rate.

3We are thankful to a referee for suggesting this analysis.4More details on the data can be found in the appendix to Lettau and Ludvigson (2005), downloadable

from http://www.econ.nyu.edu/user/ludvigsons/dpappendix.pdf.

http://www.econ.nyu.edu/user/ludvigsons/dpappendix.pdf

ARTICLE IN PRESS

Table 1

Summary statistics

Sub-sample (1948:1990) Whole sample (1948:2001)

Dct Ddt vt ert Dct Ddt vt ert

Mean 2.12 4.85 0.59 7.79 2.01 4.01 0.74 8.35

Std. Dev. 1.17 12.14 0.24 17.82 1.14 12.24 0.39 17.46

Correlation matrix

Dct 1.00 �0.12 0.42 �0.00 1.00 �0.13 0.18 �0.06

Ddt 1.00 �0.35 0.65 1.00 �0.33 0.65

vt 1.00 �0.16 1.00 �0.09

ert 1.00 1.00

Autocorrelation function

rð1Þ 0.29 �0.26 0.82 �0.09 0.34 �0.25 0.87 �0.07

rð2Þ �0.03 �0.09 0.66 �0.33 0.05 �0.03 0.72 �0.25

rð3Þ 0.01 0.06 0.56 0.17 �0.00 0.02 0.59 0.09

rð4Þ �0.05 0.11 0.47 0.39 �0.04 0.09 0.47 0.32


Table 1 presents summary statistics for real per capita consumption growth ðDctÞ,dividend growth ðDdtÞ, the price–dividend ratio ðvtÞ and the excess return ðertÞ fortwo samples. The first one, hereafter referred as the whole sample, covers the entireavailable period and spans 1948–2001. The second sample ends in 1990 and is aimedat controlling for the trend in the price–dividend ratio in the last part of the sample(see Ang, 2002; Ang and Bekaert, 2004).

Several findings stand out of Table 1. First of all, as already noted by Lettau andLudvigson (2005), real dividend growth is much more volatile than consumptiongrowth, 1.14% versus 12.24% over the whole sample. This remains true when wefocus on the restricted sub-sample. Note that the volatilities are remarkably stableover the two samples except for the price–dividend ratio. Indeed, in this case, thevolatility over the whole sample is about twice as much as the volatility over therestricted sub-sample. This actually reflects the upward trend in the price–dividendratio during the nineties.

The correlation matrix is also remarkably stable over the two periods. It is worthnoting that consumption growth and dividend growth are negatively correlatedð�0:13Þ within each sample. A direct implication of this finding is that we willinvestigate the robustness of our theoretical results to the type of data we use(consumption growth versus dividend growth). Another implication of this finding isthat while the price dividend ratio is positively correlated with consumption growth,it is negatively correlated with dividend growth in each sample. It is interesting tonote that if the correlation between dividend growth and the price–dividend ratioremains stable over the two samples, the correlation between the price–dividend ratioand consumption growth dramatically decreases when the 1990s are brought back inthe sample (0.18 versus 0.42). The correlation between the excess return and the

ARTICLE IN PRESS


price–dividend ratio is weak and negative. It is slightly weakened by the introductionof data pertaining to the latest part of the sample.

The autocorrelation function also reveals big differences between consumptionand dividend data. Consumption growth is positively serially correlated whiledividend growth is negatively serially correlated. The persistence quickly vanishes asthe autocorrelation function shrinks to zero after horizon 2. Conversely, theprice–dividend ratio is highly persistent. The first order serial correlation is about 0.8in the short sample, while it amounts to 0.9 in the whole sample. This suggests that astandard CCAPMmodel will have trouble matching this fact, as such models possessvery weak internal transmission mechanisms. This calls for a model magnifying thepersistence of the shocks. Finally, excess returns display almost no serial correlationat order 1, and are negatively correlated at order 2.

1.2. Predictability

Over the last 20 years the empirical literature on asset prices has reported evidencesuggesting that stock returns are indeed predictable. For instance, Campbell andShiller (1987) or Fama and French (1988), among others, have shown that excessreturns can be predicted by financial indicators including the price–dividend ratio.The empirical evidence also shows that the predictive power of these financialindicators is greater when excess returns are measured over long horizons. Acommon way to investigate predictability is to run regressions of the compounded(log) excess return ðerk

t Þ on the (log) price–dividend ðvtÞ evaluated at several lags

erkt ¼ ak þ bkvt�k þ uk

t , (1)

where erkt �

Pk�1i¼0 rt�i � rf ;t�i with r and rf , respectively, denote the risky and the

risk free rate of return.This procedure is however controversial and there is doubt of whether there

actually is any evidence of predictability of excess stock returns with theprice–dividend ratio. Indeed, following the seminal article of Fama andFrench (1988), there has been considerable debate as to whether or not theprice–dividend ratio can actually predict excess returns (see e.g. Stambaugh, 1999;Torous et al., 2004; Ang, 2002; Campbell and Yogo, 2005; Ferson et al., 2003;Ang and Bekaert, 2004 for recent references). In particular, the recent literaturehas focused on the existence of some biases in the bk coefficients, a lack of efficiencyin the associated standard errors and upward biased R2 due to the use of(i) persistent predictor variables (in our case vt) and (ii) overlapping observa-tions. Stambaugh (1999), using Monte-Carlo simulations, showed that theempirical size of the Newey-West t-statistic for a univariate regression of excessreturns on the dividend yield is about 23% against a nominal size of 5%. Thistherefore challenges the empirical relevance of predictability of stock returns. Inorder to investigate this issue, we generate data under the null of no predictability(bk ¼ 0 in Eq. (1)):

erkt ¼ ak þ ek

t , (2)

ARTICLE IN PRESS

Table 2

Predictability bias

k Short sample Whole sample

bk R2 bk R2

1 �0.054 0.001 �0.077 0.009

ð0:142Þ ð0:101Þ

2 �0.005 0.000 0.033 0.001

ð0:189Þ ð0:130Þ

3 0.013 0.000 �0.023 0.000

ð0:199Þ ð0:140Þ

5 �0.011 0.000 �0.060 0.001

ð0:288Þ ð0:189Þ

7 0.024 0.000 �0.039 0.000

ð0:360Þ ð0:242Þ

Note: bk and R2 are average values obtained from 100,000 replications. Standard errors into parenthesis.


where ak is the mean of compounded excess return, and ekt is drawn from a gaussian

distribution with zero mean and standard deviation se. We generated data for theprice–dividend ratio, assuming that vt is represented by the following AR(1) process:

vt ¼ yþ rvt�1 þ nt, (3)

where nt is assumed to be normally distributed with zero mean and standarddeviation sn. The values for ak, y, r, se and sn are estimated from the data over eachsample.

We then generated 100,000 samples of T observations5 under the null (Eqs. (2)and (3)) and estimated

erkt ¼ ak þ bkvtk

þ ekt

from the generated data for k ¼ 1; 2; 3; 5; 7. We then recover the distribution of theNewey-West t-statistics testing the null bk ¼ 0, the distribution of bk and thedistribution of R2. This procedure allows us to evaluate (i) the potential bias in ourestimations and therefore correct for it, and (ii) the actual size of the test for the nullof non-predictability of stock returns.

Simulation results are reported in Table 2. Two main results emerge. First of all,the bias decreases with the horizon and is larger in the whole sample. Moreimportantly, the regressions do not suffer from any significant bias in our data. Forinstance, the bias is �0:077 for k ¼ 1 in the whole sample with a large dispersion ofabout 0.1, which would not lead to reject that the bias is significant at conventionalsignificance level. The bias is much lower in the short sub-sample. Hence, theestimates of predictability do not exhibit any significant bias and do not call for any

5We actually generated T þ 200 observations, the 200 first observations being discarded from the

sample.

ARTICLE IN PRESS

Table 3

Simulated distributions

k Short sample Whole sample

Emp. size etinf etsup Emp. size etinf etsup1 0.099 �2.692 1.872 0.160 �3.242 1.259

2 0.088 �2.349 2.280 0.097 �2.626 2.001

3 0.092 �2.253 2.410 0.094 �2.543 2.098

5 0.094 �2.405 2.293 0.102 �2.756 1.865

7 0.094 �2.268 2.422 0.096 �2.595 2.107

Note: These data are obtained from 100,000 replications.


specific correction. Second, as expected from the previous results, the R2 of theregression is essentially 0, which confirms that the model is well estimated as vt�k hasno predictive power under the null. Hence, the R2 is not upward biased in oursample.

There still remains one potential problem in our regressions, as the empirical sizeof the Newey-West t-statistic ought to be distorted. Therefore, in Table 3 we report(i) the empirical size of the t-statistic should it be used in the conventional way (using1.96 as the threshold), and (ii) the correct thresholds that guarantee a 5% two-sidedconfidence level in our sample.

Table 3 clearly shows that the size of the Newey-West t-statistics is distorted. Forexample, applying the standard threshold values associated to the two-sided t-statistics at the conventional 5% significance level would actually yield a 10% size inboth samples. The empirical size even rises to 16% in the whole sample for theshortest horizon. In other words, this would lead the econometrician to reject theabsence of predictability too often. But the problem is actually more pronounced ascan be seen from columns 3, 4, 6 and 7 of Table 3. Beside the distortion of the size ofthe test, an additional problem emerges: the distribution are skewed, which impliesthat the tests are not symmetric. This is also illustrated in Figs. 3 and 4 (see AppendixA) which report the cdf of the Student distribution and the distribution obtainedfrom our Monte-Carlo experiments. Both figures show that the distributions aredistorted and that this distortion is the largest at short horizons. Therefore, whenrunning regressions on the data, we will take care of these two phenomena.

We ran the predictability regressions on actual data correcting for theaforementioned problems. The results are reported in Table 4.

Panels (a) and (b) report the predictability coefficients obtained from theestimation of Eq. (1). The second line of each panel reports the t-statistic, tk,associated to the null of the absence of predictability together with the empirical sizeof the test. Then the fourth line gives the modified t-statistics, ck, proposed byValkanov (2003) which correct for the size of the sample ðck ¼ tk=

ffiffiffiffiTpÞ and the

associated empirical size. The empirical size used for each experiments were obtainedfrom 100,000 Monte-Carlo simulations and therefore corrects for the size distorsionproblem. Finally, the last line reports information on the overall fit of the regression.

ARTICLE IN PRESS

Table 4

Predictability regression

k 1 2 3 5 7

(a) Short sample: 1948–1990

bk �0.362 �0.567 �0.679 �1.102 �1.414

tk �3.847 �3.205 �3.683 �5.494 �6.063

½0:003� ½0:006� ½0:002� ½0:000� ½0:000�

ck �0.594 �0.501 �0.582 �0.891 �1.011

½0:002� ½0:003� ½0:001� ½0:000� ½0:000�

R2 0.223 0.314 0.434 0.625 0.680

(b) Whole sample: 1948–2001

bk �0.145 �0.243 �0.275 �0.532 �0.912

tk �1.882 �1.261 �1.133 �0.879 �0.868

½0:169� ½0:183� ½0:193� ½0:312� ½0:266�

ck �0.258 �0.175 �0.159 �0.126 �0.127

½0:169� ½0:183� ½0:193� ½0:309� ½0:261�

R2 0.078 0.104 0.101 0.166 0.296

Note: tk and ck, respectively, denote the t-statistics associated to the null of ak ¼ 0, and Vlakanov’s

corrected t-statistics of the null (ck ¼ tk=ffiffiffiffiTp

where T is the sample size). Empirical size (from simulated

distributions) into brackets.


The estimation results suggest that excess returns are negatively related to theprice–dividend ratio whatever the horizon and whatever the sample. Moreover, thelarger the horizon, the larger the magnitude of this relationship is. For instance,when the lagged price–dividend ratio is used to predict excess returns, the coefficientis �0:362 in the short sample, while the coefficient is multiplied by around 4 and risesto �1:414 when 7 lags are considered. In other words, the price–dividend ratioaccounts for greater volatility at longer horizons. A second worth noting fact is thatthe foreseeability of the price–dividend ratio is increasing with horizon as the R2 ofthe regression increases with the lag horizon. For instance, the one yearpredictability regression indicates that the price–dividend ratio accounts for 22%of the overall volatility of the excess return in the short run. This share rises to 68%at the 7 years horizon. It should however be noticed that the significance of thisrelationship fundamentally depends on the sample we focus on. Over the shortsample, predictability can never be rejected at any conventional significance level,whether we consider the standard t-statistics or the corrected statistics. The empiricalsize of the test is essentially zero whatever the horizon for both tests. The evidence infavor of predictability is milder when we extend the sample up to 2001. For instance,the empirical size of the null of no predictability is about 17% over the short horizon,and rises to 30% at the 5 years horizon. This lack of significance is witnessed by themeasure of fit of the regression which amounts to 29% over the longer run horizon.This finding is related to the fact that while excess return remained stable over thewhole sample, the price–dividend ratio started to raise in the latest part of thesample, therefore dampening its predictive power. Taken together, these findings

ARTICLE IN PRESS


suggest that the potential lack of predictability of the price dividend ratio essentiallyreflects some sub-sample issues rather than a deep econometric problem. The latenineties were marked by a particular phase of the evolution of stock markets whichseems to be related to the upsurge of the information technologies, which may havecreated a transition phase weakening the predictability of stock returns (see Hobijnand Jovanovic, 2001 for an analysis of this issue). This issue is far beyond the scopeof this paper. Nevertheless, the data suggest that the price dividend ratio offered apretty good predictor of stock returns at least in the pre-information technologyrevolution.

2. Catching up with the Joneses

In this section, we develop a consumption based asset pricing model in whichpreferences exhibit a ‘Catching up with the Joneses’ phenomenon. We provide theclosed-form solution for the price–dividend ratio and conditions that guarantee theexistence of a stationary bounded equilibrium.

2.1. The model

We consider a pure exchange economy a la Lucas (1978). The economy ispopulated by a single infinitely lived representative agent. The agent has preferencesover consumption, represented by the following intertemporal expected utilityfunction:

Et

X1s¼0

bsutþs, (4)

where b40 is a constant discount factor, and ut denotes the instantaneous utilityfunction, that will be defined later. Expectations are conditional on informationavailable at the beginning of period t.

The agent enters period t with a number of shares, St – measured in terms ofconsumption goods – carried over the previous period as a means to transfer wealthintertemporally. Each share is valuated at price Pt. At the beginning of the period,she receives dividends, DtSt where Dt is the dividend per share. These revenues arethen used to purchase consumption goods, ct, and new shares, Stþ1, at price Pt. Thebudget constraint therefore writes

PtStþ1 þ CtpðPt þDtÞSt. (5)

Following Abel (1990, 1999), we assume that the instantaneous utility function, ut,takes the form

ut � uðCt;V tÞ ¼

ðCt=VtÞ1�y� 1

1� yif y 2 Rþnf1g;

logðCtÞ � logðV tÞ if y ¼ 1;

8<: (6)

where y measures the degree of relative risk aversion and V t denotes the habit level.

ARTICLE IN PRESS


We assume V t is a function of lagged aggregate consumption, Ct�1, and istherefore external to the agent. This assumption amounts to assume that preferencesare characterized by a ‘Catching up with the Joneses’ phenomenon.6 More precisely,we assume that7

Vt ¼ Cjt�1, (7)

where jX0 rules the sensitivity of household’s preferences to past aggregateconsumption, Ct�1, and therefore measures the degree of ‘Catching up with theJoneses’. It is worth noting that habit persistence is specified in terms of the ratio ofcurrent consumption to a function of lagged consumption. We hereby follow Abel(1990) and depart from a strand of the literature which follows Campbell andCochrane (1999) and specifies habit persistence in terms of the difference betweencurrent and a reference level. This particular feature of the model will enable us toobtain a closed form solution to the asset pricing problem while keeping the mainproperties of habit persistence. Indeed, as shown by Burnside (1998), one of the keysto a closed form solution is that the marginal rate of substitution betweenconsumption at two dates is an exponential function of the growth rate ofconsumption between these two dates. This is indeed the case with this particularform of catching up. Another implication of this specification is that, just alike thestandard CRRA utility function, the individual risk aversion remains time-invariantand is unambiguously given by y.

Another attractive feature of this specification is that it nests several standardspecifications. For instance, setting y ¼ 1 leads to the standard time separable case,as in this case the instantaneous utility function reduces to logðCtÞ � j logðCt�1Þ. Asaggregate consumption, Ct�1, is not internalized by the agents when taking theirconsumption decisions, the (maximized) utility function actually reduces toEt

P1s¼0 b

s logðCtþsÞ. The intertemporal utility function is time separable and thesolution for the price–dividend ratio is given by Pt=Dt ¼ b=ð1� bÞ.

Setting j ¼ 0, we recover a standard time separable CRRA utility function of theform Et

P1s¼0 b

sðC1�y

tþs � 1Þ=ð1� yÞ. In such a case, Burnside (1998) showed that aslong as dividend growth is log-normally distributed, the model admits a closed formsolution.8

Setting j ¼ 1 we retrieve Abel’s (1990) relative consumption case (case B inTable 1, p. 41) when shocks to endowments are iid. In this case, the household valuesincreases in her individual consumption vis a vis lagged aggregate consumption. Inequilibrium, Ct�1 ¼ Ct�1 and it turns out that utility is a function of consumptiongrowth.

At this stage, no further restriction will be placed on either b, y or j.

6Note that had V t been a function of current aggregate consumption, we would have recovered Galı’s

(1994) ‘Keeping up with the Jones’. In such a case the model admits that same analytical solution as in

Burnside (1998).7Note that this specification of the preference parameter can be understood as a particular case of Abel

(1990) specification which is, in our notations, given by V t ¼ ½CDt�1C

1�D

t�1 �g with 0pDp1 and gX0:

8Note that this result extends to more general distribution. See for example Bidarkota and McCulloch

(2003) and Tsionas (2003).

ARTICLE IN PRESS


The household determines her contingent consumption fCtg1t¼0 and contingent

asset holdings fStþ1g1t¼0 plans by maximizing (4) subject to the budget constraint (5),

taking exogenous shocks distribution as given, and (6) and (7) given. Agents’consumption decisions are then governed by the following Euler equation

PtC�yt C

jðy�1Þt�1 ¼ bEt½ðPtþ1 þDtþ1ÞC

�ytþ1C

jðy�1Þt �, (8)

which may be rewritten as

Pt

Dt

¼ Et 1þPtþ1

Dtþ1

� ��Wtþ1 � Ftþ1

� �� Ct, (9)

whereWtþ1 � Dtþ1=Dt captures the wealth effect of dividend, Ftþ1 � b½ðCtþ1=CtÞ�y�

is the standard stochastic discount factor arising in the time separable model. ThisEuler equation has an additional stochastic factor Ct � ðCt=Ct�1Þ

jðy�1Þ whichmeasures the effect of ‘Catching up with the Joneses’. These two latter effects capturethe intertemporal substitution motives in consumption decisions. Note that Ct isknown with certainty in period t as it only depends on current and past aggregateconsumption. This new component distorts the standard intertemporal consumptiondecisions arising in a standard time separable model. Note that our specification ofthe utility function implies that j essentially governs the size of the catching upeffect, while risk aversion, y, governs its direction. For instance, when risk aversion islarge enough – y41 – catching up exerts a positive effect on the time separableintertemporal rate of substitution. Hence, in this case, for a given rate ofconsumption growth, catching up reduces the expected return.

Since we assumed the economy is populated by a single representative agent, wehave St ¼ 1 and Ct ¼ Ct ¼ Dt in equilibrium. Hence, both the stochastic discountfactor in the time separable model and the ‘Catching up with the Joneses’ term arefunctions of dividend growth Dtþ1=Dt

Ftþ1 � b½ðDtþ1=DtÞ�y� and Ct � ðDt=Dt�1Þ

jðy�1Þ.

Any persistent increase in future dividends, Dtþ1, leads to two main effects in thestandard time separable model. First, a standard wealth effect, stemming from theincrease in wealth it triggers ðWtþ1Þ, leads the household to consume more andpurchase more assets. This puts upward pressure on asset prices. Second, there is aneffect on the stochastic discount factor ðFtþ1Þ. Larger future dividends lead to greaterfuture consumption and therefore lower future marginal utility of consumption. Thehousehold is willing to transfer tþ 1 consumption toward period t, which can beachieved by selling shares therefore putting downward pressure on prices. Wheny41, the latter effect dominates and prices are a decreasing function of dividend. Inthe ‘catching up’ model, a third effect, stemming from habit persistence ðCtÞ, comesinto play. Habit standards limit the willingness of the household to transferconsumption intertemporally. Indeed, when the household brings future consump-tion back to period t, she hereby raises the consumption standards for the nextperiod. This raises future marginal utility of consumption and therefore plays againstthe stochastic discount factor effect. Henceforth, this limits the decrease in assetprices and can even reverse the effect when j is large enough.

ARTICLE IN PRESS


Defining the price–dividend ratio as vt ¼ Pt=Dt, it is convenient to rewrite theEuler equation evaluated at the equilibrium as

vt ¼ bEt ð1þ vtþ1ÞDtþ1

Dt

� �1�yDt

Dt�1

� �jðy�1Þ" #

. (10)

2.2. Solution and existence

In this section, we provide a closed form solution for the price–dividend ratio andgive conditions that guarantee the existence of a stationary bounded equilibrium.

Note that up to now, no restrictions have been placed on the stochastic processgoverning dividends. Most of the literature attempting to obtain an analyticalsolution to the problem assumes that the rate of growth of dividends is an iid

gaussian process (see Abel (1990, 1999) among others).9 We depart from the iid caseand follow Burnside (1998). We assume that dividends grow at rate gt � logðDt=Dt�1Þ, and that gt follows an AR(1) process of the form

gt ¼ rgt�1 þ ð1� rÞgþ et, (11)

where et*Nð0;s2Þ and jrjo1. In the AR(1) case, the Euler equation rewrites

vt ¼ bEt ð1þ vtþ1Þ expðð1� yÞgtþ1 � jð1� yÞgtÞ� �

. (12)

We can then establish the following proposition.

Proposition 1. The solution to Eq. (12) is given by

vt ¼X1i¼1

bi expðai þ biðgt � gÞÞ, (13)

where

ai ¼ ð1� yÞð1� jÞgi þ1� y1� r

� �2 s2

2ð1� jÞ2i � 2

ð1� jÞðr� jÞ1� r

ð1� riÞ

�þðr� jÞ2

1� r2ð1� r2iÞ

�,

bi ¼ð1� yÞðr� jÞ

1� rð1� riÞ.

First of all it is worth noting that this pricing formula resembles that exhibited inBurnside (1998). We actually recover Burnside’s formulation by setting j ¼ 0 – i.e.imposing time separability in preferences. Second, when the rate of growth ofendowments is iid over time ðgt ¼ gþ etÞ, and j is set to 1, we recover the solution

9There also exist a whole strand of the literature introducing Markov switching processes in CCAPM

models. See Cecchetti et al. (2000) and Brandt et al. (2004) among others.

ARTICLE IN PRESS


used by Abel (1990) to compute unconditional expected returns:

zt ¼ b exp ð1� yÞ2s2

2þ ð1� yÞðgt � gÞ

� �. (14)

In this latter case, the price–dividend ratio is an increasing (resp. decreasing) andconvex function of the consumption growth if y41 (resp. yo1). Things are morecomplicated when we consider the general model. Indeed, as shown in Proposition 1(see coefficient bi), both the position of the curvature parameter, y, around 1 and theposition of the persistence of dividend growth, r, around the parameter of habitpersistence, j, matter.

The behavior of an agent in face a positive shock on dividend growth essentiallydepends on the persistence of the process of endowments. This is illustrated in Fig. 1which reports the behavior of the price–dividend ratio as a function of the rate ofgrowth of dividends for y below and above 1.

Let us consider the case y41 (see right panel of Fig. 1). As we established in theprevious section, a shock on dividends exerts three effects: (i) a standard wealtheffect, (ii) a stochastic discount factor effect and (iii) a habit persistence effect. Thetwo latter effects play in opposite direction on intertemporal substitution. Whenj4r, the stochastic discount factor effect is dominated by the force of habits, as theshock on dividend growth exhibits less persistence than habits. Therefore, the secondand the third effects partially offset each other and the wealth effect plays a greaterrole. The price–dividend ratio increases. Conversely, when jor habit persistencecannot counter the effects of expected stochastic discounting, and intertemporalsubstitution motives take the upper hand. The price–dividend ratio decreases. Notethat in the limiting case where r ¼ j (plain dark line in Fig. 1) the persistence ofdividend growth exactly offsets the effects of ‘catching up’ and all three effects cancelout. Therefore, just alike the case of a logarithmic utility function, the price–dividendratio is constant. The reasoning is reversed when yo1 (see left panel of Fig. 1).

log

Pt

Dt

Dt

log

Pt

� < 1 � > 1

� < �� = �

� > �

�t�t

Fig. 1. Decision rules.

ARTICLE IN PRESS


It is worth noting that Proposition 1 only establishes the existence of a solution,and does not guarantee that this solution is bounded. Indeed, the solution for theprice–dividend ratio involves a series which may or may not converge. The nextproposition reports conditions that guarantee the existence of a stationary boundedequilibrium.

Proposition 2. The series in (13) converges if and only if

r � b ð1� yÞð1� jÞgþs2

2

ð1� yÞð1� jÞ1� r

� �2" #

o1.

As in Burnside (1998), this proposition shows that, given a 4-uplet ðy;j;r; sÞ, bo1is neither necessary nor sufficient to insure finite asset prices. In particular, thesolution may converge even for b41 when agents are highly risk adverse.Furthermore, the greater the ‘catching up’, the easier it is for the series to converge.Conversely, b should be lower as r approaches unity.

Related to the convergence of the series is the convergence of the moments of theprice–dividend ratio. The next proposition establishes a condition for the first twomoments of the price–dividend ratio to converge.

Proposition 3. The mean and autocovariances of the price– dividend ratio converge to a

finite constant if and only if ro1.

Proposition 3 extends previous results obtained by Burnside (1998) to the case of‘Catching up with the Joneses’. The literature has shown that this representation ofpreferences fails to account for the persistence of the price–dividend ratio and thedynamics of asset returns. In the next section we therefore enrich the dynamics of themodel.

3. Catching up with the Joneses and Habit Stock

In this section, we extend the previous framework to a more general habitformation process. In particular, we allow habits to react only gradually to changesin aggregate consumption. We provide the closed-form solution for the price–dividend ratio and conditions that guarantee the existence of a stationary boundedequilibrium.

3.1. The model

We depart from the previous model in that preferences are affected by the entirehistory of aggregate consumption per capita rather that the lagged aggregateconsumption (see e.g. Sundaresan, 1989; Constantidines, 1990; Heaton, 1995; orCampbell and Cochrane, 1999 among others). More precisely, the habit level, V t,takes the form

Vt ¼ Xjt ,

ARTICLE IN PRESS


where X t is the consumption standard. We assume that the effect of aggregateconsumption on the consumption standard vanishes over time at the constant rated 2 ð0; 1Þ. More precisely, the consumption standard, X t, evolves according to

X tþ1 ¼ Cdt X 1�d

t . (15)

Note that this specification departs from the standard habit formation formulausually encountered in the literature. Nevertheless, in order to provide economicintuition, the evolution of habits (15) may be rewritten as

xt � logðX tÞ ¼ dX1i¼0

ð1� dÞi logðCt�i�1Þ. (16)

The reference consumption index, X t, can be viewed as a weighted geometric average ofpast realizations of aggregate consumption. Eq. (16) shows that ð1� dÞ governs the rateat which the influence of past consumption vanishes over time, or, otherwise states dgoverns the persistence of the state variable X t. Note that in the special case of d ¼ 1, werecover the ‘Catching up with the Joneses’ preferences specification studied in theprevious section. Conversely, setting d ¼ 0, we retrieve the standard time separableutility function as habit stock does not respond to changes in consumption anymore.

The representative agent then determines her contingent consumption fCtg1t¼0 and

contingent asset holdings fStþ1g1t¼0 plans by maximizing her intertemporal expected

utility function (4) subject to the budget constraint (5) and taking the law of habitformation (15) as given.

Agent’s consumption decisions are governed by the following Euler equation:

PtC�yt X

�jð1�yÞt ¼ bEtðPtþ1 þDtþ1ÞC

�ytþ1X

�jð1�yÞtþ1 (17)

which may actually be rewritten in the form of Eq. (9) as

Et

Ptþ1 þDtþ1

Pt

� Ftþ1

� ��Xtþ1 ¼ 1, (18)

where Ftþ1 is the stochastic discount factor defined in Section 2.1 and Xtþ1 �

ðX tþ1=X tÞjðy�1Þ accounts for the effect of the persistent ‘Catching up with the Joneses’

phenomenon. Note that as in the previous model, the predetermined variable Xtþ1

distorts intertemporal consumption decisions in a standard time separable model.

3.2. Solution and existence

In equilibrium, we have St ¼ 1 and Ct ¼ Ct ¼ Dt, implying that X tþ1 ¼ Ddt X 1�d

t .As in the previous section, we assume that the growth rate of dividends follows anAR(1) process of the form (11). It is then convenient to rewrite Eq. (17) as

yt ¼ bEt½expðð1� yÞð1� jÞgtþ1 � jð1� yÞztþ1Þ þ expðð1� yÞð1� jÞgtþ1Þytþ1�,

(19)

where zt ¼ logðX t=DtÞ denotes the (log) habit–dividend ratio and yt ¼ vt

expð�jð1� yÞztÞ.

ARTICLE IN PRESS


This forward looking equation admits the closed form solution reported in thenext proposition.

Proposition 4. The equilibrium price– dividend ratio is given by

vt ¼X1i¼1

bi expðai þ biðgt � gÞ þ ciztÞ, (20)

where

ai ¼ ð1� yÞg ð1� jÞi þjdð1� ð1� dÞiÞ

h iþ

V

2,

bi ¼ ð1� yÞrð1� jÞ1� r

ð1� riÞ þjr

1� d� rðð1� dÞi � riÞ

� �,

ci ¼ jð1� yÞð1� ð1� dÞiÞ,

and

V ¼ ð1� yÞ2s21� j1� r

� �2

i � 2r

1� rð1� riÞ þ

r2

1� r2ð1� r2iÞ

� �(

þ 2jð1� jÞ

ð1� rÞð1� d� rÞð1� dÞ

dð1� ð1� dÞiÞ �

r1� r

ð1� riÞ

��

rð1� dÞ1� rð1� dÞ

ð1� ðrð1� dÞÞiÞ þr2

1� r2ð1� r2iÞ

�þ

j2

ð1� d� rÞ2ð1� dÞ2

1� ð1� dÞ2ð1� ð1� dÞ2i

Þ

��2



1� r2ð1� r2iÞ

�).

This solution obviously nests the pricing formula obtained in the previous model.Indeed, setting d ¼ 1, we recover the solution reported in Proposition 1. As shown inSection 2.2, the form of the solution essentially depends on the position of thecurvature parameter, y, around 1 and the position of the habit persistence parameter,j, around the persistence of the shock, r. In the generalized model, things are morecomplicated as the position of the persistence of habits, 1� d, around j and r is alsokey to determine the form of the solution as reflected in the form of the coefficient bi.Nevertheless, expression (20) illustrates two important properties of our model.First, the price–dividend ratio is function of two state variables: the growth rate ofdividends gt and the habit–dividend ratio zt. This feature is of particular interest asthe law of motion of zt is given by

ztþ1 ¼ ð1� dÞzt � gtþ1. (21)

Therefore, zt is highly serially correlated for low values of d, and the price–dividendratio inherits part of this persistence. A second feature of this solution is that any

ARTICLE IN PRESS


change in the rate of growth of dividend exerts two effects on the price–dividendratio. A first direct effect transits through its standard effect on the capital income ofthe household and is reflected in the term bi. A second effect transits through itseffect on the habit–dividend ratio. This second effect may be either negative orpositive depending on the position of y with regard to 1 and the form of ci. Thisimplies that there is room for pro- or counter-cyclical variations in thedividend–price ratio. This is critical for the analysis of predictability in stock returnsas Section 4.3 will make clear. Finally, note that as soon as do1, the price–dividendratio will be persistent even in the case when the rate of growth of dividends is iid

ðr ¼ 0Þ (see the expression for ci).As the solution for the price–dividend ratio involves a series, the next proposition

determines conditions that guarantee the existence of a stationary boundedequilibrium.

Proposition 5. The series in (20) converges if and only if

r � b exp ð1� yÞð1� jÞgþs2

2

ð1� yÞð1� jÞ1� r

� �2" #

o1.

It is worth noting that the result reported in Proposition 5 is the same as inProposition 2. Hence, the conditions for the existence of a stationary boundedequilibrium are not altered by this more general specification of habit formation.From a technical point of view, this result stems from the geometrical lag structure ofhabit stock, which implies strict homotheticity of the utility function with respect tohabit. From an economic point of view this illustrates that habit formationessentially affects the transition dynamics of the model while leaving unaffected thelong run properties of the economy.

Just like in the previous model, it is possible to establish the convergence of thefirst two moments of the price–dividend ratio.

Proposition 6. The mean and the autocovariances of the price– dividend ratio converge

to a constant if and only if ro1.

Propositions 5 and 6 provide us with a set of restrictions on the deep and forcingparameters of the economy, which can be used to guarantee the relevance of ourquantitative evaluation of the models.

4. Quantitative evaluation

This section investigates the quantitative properties of the model putting emphasison the predictability of stock returns.

4.1. Parametrization

We partition the set of the parameters of the model in two distinct groups. In thefirst group we gather all deep parameters defining preferences. These parameters are

ARTICLE IN PRESS

Table 5

Preferences parameters

Parameter Value

Stochastic discount factor (Ftþ1)

Curvature y 1.500

Constant discount factor b 0.950

Habit formation (Ct;X tþ1)

Habit persistence parameter j [0,1]

Depreciation rate of habits d [0.05,1]


reported in Table 5. The calibration takes advantage of Propositions 2 and 5 to placerestrictions on the parameters that guarantee the existence of a bounded solution.

The two parameters b and y ruling the properties of the stochastic discount factorðFtþ1Þ are set to commonly used values in the literature. More precisely, thehousehold is assumed to discount the future at a 5% annual psychological discountrate, implying b ¼ 0:95. The parameter of risk aversion, y, is set to 1.5. We howevergauge the sensitivity of our results to alternative values of the degree of risk aversionðy ¼ 0:5; 5Þ.

The parameters pertaining to habit formation, j and d, are first set to referencevalues, we will then run a sensitivity analysis to changes in these parameters. Theparameter ruling the sensitivity of preferences to habit formation, j, takes on valuesranging from 0 to 1. We first study the standard case of time separable utilityfunction, which corresponds to j ¼ 0. We then investigate Abel’s (1990) case wherej is set to 1. This latter parametrization implies that, in equilibrium, therepresentative agent values aggregate consumption growth in the case of purecatching up with the Joneses ðd ¼ 1Þ. We also consider intermediate values for j inour sensitivity analysis. Several values of the habit stock parameter, d, areconsidered. We first set d to 1, therefore focusing on the simple ‘Catching up withthe Joneses’ model. We then explicitly consider the existence of a habit stockformation mechanism by allowing d to take on values below unity. FollowingCampbell and Cochrane (1999), the value for d is selected such that the model cangenerate the first order autocorrelation of the price–dividend ratio. This leads us toselect a value such that the effect of current consumption on the consumptionstandard vanishes at a 5% annual rate ðd ¼ 0:05Þ. This result is in line with previousstudies which have shown that high persistence in habit stock formation is requiredto enhance the properties of the asset pricing model along several dimensions (seeHeaton, 1995; Li, 2001; Allais, 2004, among others). We however evaluate thesensitivity of our results to higher values of d and therefore to less persistent habitformation.

The second group consists of parameters describing the evolution of the forcingvariables. The latter set of parameters is obtained exploiting US postwar annual databorrowed from Lettau and Ludvigson (2005) (see Section 1.1 for more details on thedata). The values of these parameters are reported in Table 6.

ARTICLE IN PRESS

Table 6

Forcing variables

Dividend growth Consumption growth

1948–1990 1948–2001 1948–1990 1948–2001

Mean of dividend growth g 4.85% 4.01% 2.12% 2.01%

Persistence parameter r �0.26 �0.25 0.29 0.34

Std. Dev. of innovations s 11.30% 11.50% 1.10% 1.00%


As aforementioned in Section 1.1, endowments can be either measured relying onconsumption or dividend data. We therefore investigate these two possibilities andestimate the parameters of the forcing variable fitting an AR(1) process on bothconsumption and dividend growth data. Moreover, as suggested by empirical resultsin Section 1.1, we consider two samples: the first one covers the whole period runningfrom 1948 to 2001, the second one ends in 1990.

Several results emerge from Table 6. First of all, the choice of a particular sampledoes not matter for the properties of the forcing variables. Both the persistence andthe volatility of the forcing variable remains stable over the two samples. Wetherefore use results obtained over the whole sample in our subsequent evaluation.Second consumption and dividend data yield fairly different time series behaviors.For instance, dividend growth data exhibit much more volatility than that found inconsumption data. Furthermore, dividend growth is negatively serially correlatedwhile consumption growth displays positive persistence. But it is worth noting thatboth dividend and consumption data yield weak persistence (respectively, �0:25 and0.34). Then, provided our model possesses strong enough internal propagationmechanism, differences in the persistence of endowment growth ought not toinfluence much the dynamic properties of the model. This finding will be confirmedin the impulse response analysis (see Section 4.2). Therefore, in order to save spaceand unless necessary, we mainly focus on results obtained relying on dividend data.

4.2. Preliminary quantitative investigation

This section assesses the quantitative ability of the model to account for a set ofstandard unconditional moments characterizing the dynamics of excess returns andthe price–dividend ratio.

The model is simulated using the closed-form solution (20). Since it involves aninfinite series, we have to make an approximation and truncate the infinite sum at along enough horizon (5000 periods). We checked that additional terms do not altersignificantly the series. Each experiment is conducted by running 1000 draws of thelength of the sample size, T . We actually generated T þ 200 observations, the 200first observations being discarded from the sample.

We begin by reporting the impulse response analysis of the model in face a positiveshock on endowment growth. Fig. 2 reports impulse response functions (IRF) of

ARTICLE IN PRESS

(a)

(b)

Fig.2.Im

pulseresponse

functions:(a)Consumptiondata;(b)Dividenddata.

No

te:TS:timeseparable

preferencesðj¼

0Þ,CJ:

CatchingupwiththeJoneses

preferences(j¼

1andd¼

1),HS:habitstock

specifications(j¼

1andd¼

0:05).


ARTICLE IN PRESS


endowments, habit/consumption ratio, excess return and the price–dividend ratio toa standard deviation shock on dividend growth when the endowment growth processis estimated with both dividend and consumption data. IRF are computed taking thenon-linearity of the model into account (see Koop et al., 1996). Panel (a) of the figurereports IRFs when the forcing variable is measured relying on consumption data,Panel (b) provides results obtained with dividend data. Three cases are underinvestigation: (i) the time separable utility function (TS), (ii) the ‘Catching up withthe Joneses’ (CJ) and (iii) habit stock (HS).

Let us first consider the time separable case (TS). In order to provide a betterunderstanding of the internal mechanisms of the model, it is useful to first investigatethe iid case.10 A one standard deviation positive shock on dividend growth thentranslates into an increase in the permanent income of the agent. Therefore,consumption instantaneously jumps to its new steady state value. Since the utilityfunction is time separable, the discount factor Ftþ1 is left unaffected by the shock.Asset prices then react one for one to dividends. The price/dividend ratio is leftunaffected. Since the excess return is a function of the discount factor, the shockexerts no effect whatsoever on its dynamics. As soon as the iid hypothesis is relaxed,stock returns and the price–dividend ratio do react. The choice of data used tocalibrate the endowment growth process affects the response of the variables ofinterest as serial correlation is either positive or negative. Let us first focus onconsumption data. In this case (see Panel (a) of Fig. 2), the autoregressive parameter,r, is positive. An increase in endowments leads the agent to expect an increase in herfuture consumption stream. Hence, the marginal utility of future consumptiondecreases and so does the discount factor. Therefore, as long as y41, this effectdominates the wealth effect and the price/dividend ratio decreases. Consequently,excess return raises. Note that, given the rather low value of r (0.34), theprice–dividend ratio quickly converges back to its steady state level.

When endowment growth is estimated on dividend data, r takes on a negativevalue ð�0:25Þ. Therefore, a current increase in dividend growth is expected to befollowed by a relative drop (See Panel (b) of Fig. 2). Hence, the wealth effectdecreases while the marginal utility of future consumption increases and so does thediscount factor. Since y41, the latter effect takes the upper hand over the wealtheffect and the price–dividend ratio raises. This effect reverses in the next period andso on. Once again, given the low value of r, the price–dividend ratio quicklyconverges back to its steady state level. Excess returns display the oppositeoscillations.

Note that the impact effect of a one standard deviation positive shock on theprice–dividend ratio is higher when exogenous endowment growth are estimatedusing dividend data as consumption growth is less volatile. For instance, the effecton the price–dividend ratio does not exceed 0.002% when the endowment growthprocess is estimated with consumption data compared to 0.02% when the process isestimated with dividend data.

10In order to save space we do not report the IRF for the iid case. They are however available from the

authors upon request.

ARTICLE IN PRESS


Bringing the ‘Catching up with the Joneses’ phenomenon into the story affects thebehavior of price–dividend ratio and excess returns at short run horizon, especiallywhen the endowment process is persistent. Since endowments are exogenouslydetermined, the consumption path is left unaffected by this assumption. The onlymajor difference arises on utility and asset prices, since they are now driven by theforce of habit in equilibrium. Consider once again the iid case. The main mechanismsat work in the aftermaths of the shock are the same as in the TS version of the model.The only difference arises on impact as the habit term, Ct, shifts and goes back to itssteady state level in the next period. As the force of habit plays like the wealth effect,the price–dividend ratio raises. When the shock is not iid, the ‘Catching up with theJoneses’ phenomenon can reverse the behavior of the price–dividend ratio when r ispositive (see Panel (a) of Fig. 2). Indeed, the force of habit then counters thestochastic discount factor effect. When j is large enough, habit reverses the impactresponse of the price–dividend ratio. However, the latter effect is smoothed until theconsumption goes back to its steady state level. When ro0, the negative serialcorrelation of the shock shows up in the dynamics of asset prices and excess returns(see Panel (b) of Fig. 2).

Asset prices and stock returns are largely affected by the introduction of habitstock formation. Both the size and persistence of the effects are magnified. The mainmechanisms at work on are the same as in the CJ version of the model: habits playagainst the discount factor effect and reverse the behavior of price–dividend ratio.But, in the short run, the magnitude of the effect is lowered by habit stock formation.More importantly, habit stock generates greater persistence than the TS and CJversions of the model. For example, as shown in Fig. 2, the initial increase inendowments leads to a very persistent increase in habits even when dividend growthis negatively serially correlated.11 A direct implication of this is that the effect ofhabits ðXtþ1Þ on the Euler equation is persistent. This long lasting effect shows up inthe evolution of the price–dividend ratio that essentially inherits the persistence ofhabits. A second implication of this finding is that the persistence of the forcingvariable does not matter much compared to that of the internal mechanismsgenerated by habit stock formation. Note that stock returns are less persistent thanthe price–dividend ratio and that they both respond positively on impact.

The preceding discussion has important consequences for the quantitativeproperties of the model in terms of unconditional moments. Table 7 reports themean and the standard deviation of the risk premium and the price–dividend ratiofor the three versions of the model and the two calibrations of the endowmentprocess.

The top panel of the table reports the unconditional moments when we calibratethe model with dividend data. It is important to note that since we are focusing onthe role of persistence on predictability, we chose to shut down one channel that isusually put forward to account for predictability – time-varying risk aversion – byspecifying habit formation in ratio term. By precluding time-varying risk aversion,this modeling generates no risk premium in the model. It should therefore come at

11This persistence originates in the low depreciation rate of habits, d ¼ 0:05.

ARTICLE IN PRESS

Table 7

Unconditional moments

TS CJ HS

Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.

Dividend data

r� rf 1.40 17.81 1.52 30.12 1.49 20.30

p� d 0.66 0.01 0.74 0.06 0.74 0.14

Consumption data

r� rf 0.01 0.45 0.01 1.53 0.01 0.67

p� d 0.69 0.00 0.74 0.01 0.74 0.01

Note: TS: time separable preferences (j ¼ 0), CJ: Catching up with the Joneses preferences (j ¼ 1 and

d ¼ 1), HS: habit stock specifications (j ¼ 1 and d ¼ 0:05).


no surprise that the model does not generate any risk premium both in the case oftime separable utility function and habit formation model. This is confirmed by theexamination of Table 7. For example, the average equity premium is low in the timeseparable model, 1.4%, and only rises to 1.49% for the habit stock model and 1.52%for the ‘Catching up with the Joneses’ model when dividend data are used. This is atodds with the 8.35% found in the data (see Table 1). The results worsens whenconsumption data are used as consumption growth exhibits very low volatility. Oneway to solve the risk premium puzzle is obviously to increase y. In order to mimic theobserved risk premium, the time separable model requires to set y to 4.5, while lowervalues for y can be used in the habit stock model ðy ¼ 3:6Þ when the endowmentprocess is calibrated with dividend data. Hence, the habit stock model canpotentially generate higher risk premia despite risk aversion is not time-varying. It ishowever worth noting that the model performs well in terms of excess returnvolatility (17.5 in the data). The habit stock model essentially outperforms the othermodels in terms of price–dividend ratio, although it cannot totally match itsvolatility. As suggested in the IRF analysis, the time separable version of the modelgenerates very low volatility (1.2%). The Catching up with the Joneses modeldelivers a slightly higher volatility (around 6%). Only the habit stock model cangenerate substantially higher volatility of about 14%. It is also worth noting thathabit persistence is crucial to induce greater volatility in the price–dividend ratio.This is illustrated in Table 8 which reports the volatility of the price–dividend ratiofor various values of j in the CJ model and d in the HS model.

As can be seen from the table, larger values for j – i.e. greater habit formation –leads to greater volatility in the price–dividend ratio, as it magnifies the force ofhabit and therefore increases the sensitivity of the price–dividend ratio to shocks.Likewise, the more persistent habit formation – lower d – the more volatile is theprice–dividend ratio.

As can be seen from Table 9, introducing habit stock formation enhances theability of the model to account for the correlation between the price–dividend ratio

ARTICLE IN PRESS

Table 9

Unconditional correlations

Dividend data Consumption data

TS CJ HS TS CJ HS

corr(p� d; r� rf ) 0.99 0.98 0.25 �0.81 0.94 0.15

corr(r� rf ; g) 0.99 0.98 0.99 0.81 0.94 0.98

corr(p� d; g) 1.00 1.00 0.26 �1.00 1.00 0.11



Table 8

Price to dividend ratio volatility

Catching up (d ¼ 1)

j 0 0.1 0.2 0.5 0.7 0.9 1

sðp� dÞ 0.01 0.02 0.02 0.04 0.05 0.06 0.06

Habit stock (j ¼ 1)

d 0.05 0.10 0.20 0.50 0.70 0.90 1

sðp� dÞ 0.14 0.11 0.08 0.06 0.06 0.06 0.06


and, respectively, excess return and endowment growth. For example, the correlationbetween the ratio and excess return is close to zero in the data. The time separablemodel generates far too much correlation between the two variables (�0:81 forconsumption data and 0.99 for dividend data). Likewise, the catching up modelproduces a correlation close to unity. Conversely, the habit stock model lowers thiscorrelation to 0.25 with dividend data and 0.15 with consumption data.

The same result obtains for the correlation between the price–dividend ratio andendowment growth. Indeed, the habit stock model introduces an additional variablethat accounts for past consumption decisions and which therefore disconnects theprice–dividend ratio from current endowment growth.

The model major improvements are found in the ability of the model to matchserial correlation of the price–dividend ratio (see Table 10). The first panel of thetable reports the serial correlation of the price–dividend ratio for the three versionsof the model when the endowment process is calibrated with dividend data. The timeseparable model totally fails to account for such large and positive persistence, as theautocorrelation is negative at order 1 ð�0:25Þ and is essentially 0 at higher orders.This in fact reflects the persistence of the exogenous forcing variable as it possessesvery weak internal propagation mechanism. The ‘Catching up with the Joneses’model fails to correct this failure as it produces exactly the same serial correlations.This can be easily shown from the solution of the model, as adding catching upessentially re-scales the coefficient in front of the shock without adding anyadditional source of persistence. The habit stock model obviously performsremarkably well at the first order as the habit stock parameter d was set in order

ARTICLE IN PRESS

Table 10

Serial correlation in price–dividend ratio

Order 1 2 3 5 7

Dividend data

HS 0.88 0.86 0.81 0.74 0.67

CJ �0.25 0.05 �0.03 �0.01 �0.01

TS �0.25 0.05 �0.03 �0.01 �0.01

Consumption data

HS 0.98 0.93 0.88 0.79 0.70

CJ 0.30 0.07 �0.00 �0.03 �0.04

TS 0.30 0.07 �0.00 �0.03 �0.04




to match the first order autocorrelation in the data (0.87 in the whole sample). Moreimportantly, higher autocorrelations decay slowly as in the data. For instance, themodel remains above 0.67 at the fifth order. Therefore, although simpler and moreparsimonious, the model has similar persistence properties as Campbell andCochrane’s (1999). The second panel of Table 10 reports the results fromconsumption data. The main conclusions remain unchanged. Only the habit stockmodel is able to generate a very persistent price–dividend ratio. As consumption dataare more persistent, the model generates greater autocorrelations coefficients. In thiscase, setting d ¼ 0:2, we would also recover the first order autocorrelation. Shouldendowment growth be iid, setting d ¼ 0:12 would have been sufficient to generate thesame persistence. This value is similar to that used by Campbell and Cochrane (1999)to calibrate the consumption surplus process with an iid endowment growth process.

As a final check of the model, we now compute the correlation between theprice–dividend ratio in the data and in the model when observed endowment growthare used to feed the model. Table 11 reports the results.

As can be seen from the table, both the time separable and catching up model failto account for the data as the correlation between the price–dividend ratio asgenerated by the model and its historical time series is clearly negative. This obtainsno matter the sample period nor the variable used to calibrate the rate of growth ofendowments. As soon as habit stock formation is brought into the model the resultsenhance, although not perfect, as the correlation is now clearly positive. This result isfundamentally related to habit stock formation, and more precisely to persistence.Indeed, when persistence is lowered by increasing d, the correlation between themodel and the data decreases dramatically. For instance, when d ¼ 0:1 it falls downto zero (0.04) when endowment growth is measured using dividend data.

4.3. Long horizon predictability

In this section we gauge the ability of the model to account for the long horizonpredictability of excess return. Table 12 reports the predictability tests on simulated

ARTICLE IN PRESS

Table 12

Predictability: benchmark experiments

k TS CJ HS

bk R2 bk R2 bk R2

Dividend data

1 �4.69 0.11 �2.18 0.20 �0.44 0.04

2 �3.74 0.06 �1.70 0.11 �0.56 0.06

3 �4.20 0.06 �1.88 0.11 �0.73 0.08

5 �4.41 0.05 �1.92 0.09 �1.00 0.12

7 �4.77 0.05 �2.03 0.09 �1.20 0.16

Consumption data

1 �1.30 0.64 �0.09 0.01 �0.04 0.03

2 �1.68 0.32 �0.17 0.02 �0.16 0.05

3 �1.75 0.20 �0.26 0.02 �0.29 0.08

5 �1.66 0.10 �0.40 0.03 �0.53 0.13

7 �1.50 0.06 �0.57 0.04 �0.75 0.17



Table 11

Correlation between the model and the data

Dividend growth Consumption growth

1948–1990 1948–2001 1948–1990 1948–2001

TS �0.35 �0.35 �0.43 �0.22

CJ �0.29 �0.32 �0.05 �0.07

HS 0.35 0.37 0.42 0.42




data. More precisely, we ran regressions of the (log) excess return on the (log)price–dividend ratio evaluated at several lags (up to 7 lags)

erkt ¼ ak þ bkvt�k þ uk

t

where erkt �

Pk�1i¼0 rt�i � rf ;t�i. The table reports, for each horizon k, the coefficient

bk and the R2 of the regression which gives a measure of predictability of excessreturns.

The time separable model (TS) fails to account for predictability. Although theregression coefficients, bk, have the right sign, they are too large compared to thosereported in Table 4 and remain almost constant as the horizon increases. Thepredictability measure, R2, is higher when regression horizon is limited to one periodand then falls to essentially 0 whatever the horizon. This should come as no surprise

ARTICLE IN PRESS

Table 13

Predictability: sensitivity analysis

k TS: y CJ: j HS: ðj; dÞ

0.5 5 0.1 0.5 (0.5,0.5) (0.5,0.05) (1,0.5)

bk R2 bk R2 bk R2 bk R2 bk R2 bk R2 bk R2

1 1.61 0.05 �1.91 0.24 �3.79 0.12 �2.60 0.16 �1.96 0.11 �0.93 0.05 �1.34 0.10

2 1.37 0.03 �1.49 0.14 �3.01 0.07 �2.04 0.09 �1.75 0.07 �1.07 0.06 �1.27 0.08

3 1.61 0.03 �1.64 0.14 �3.37 0.06 �2.26 0.09 �2.02 0.08 �1.36 0.08 �1.47 0.08

5 1.82 0.03 �1.66 0.12 �3.52 0.06 �2.33 0.07 �2.22 0.08 �1.81 0.11 �1.63 0.09

7 2.12 0.03 �1.74 0.11 �3.79 0.05 �2.48 0.07 �2.48 0.08 �2.23 0.15 �1.82 0.09




as the impulse response analysis showed that the price–dividend ratio and excessreturns both respond very little and monotonically to a shock on dividend growth. Afirst implication of the little responsiveness of the price–dividend ratio is that themodel largely overestimates the coefficient bk in the regression (around �4). Asecond implication is its tiny predictive power especially at long horizon, as the R2 isaround 0. This obtains whatever the data we consider. One potential way to enhancethe ability of the model to account for predictability may be to manipulate the degreeof risk aversion. This experiment is reported in the first panel of Table 13. When thedegree of risk aversion is below unity, the model totally fails to match the data as allcoefficients have the wrong sign and the R2 is essentially nil whatever the horizon.When y is raised toward 5, we recover the negative relationship between excessreturns and the price–dividend ratio, but the predictability is a decreasing function ofthe horizon which goes opposite to the data (see Table 4).

As shown in the IRF analysis, the ‘Catching up with the Joneses’ model possessesslightly stronger propagation mechanisms. This enhances its ability to account forpredictability. However, the price–dividend ratio and the excess returns respond onlyat short run horizon to a positive shock to endowments. In Tables 12 and 13, wereport predictability tests for this version of the model for several values of the habitpersistence parameter j. The first striking result is that allowing for ‘catching up’indeed improves the long horizon predictive power of the model. More precisely, thecoefficients of the regression are decreasing with the force of habit. For instance,when j ¼ 1, the coefficient b1 drops to �2:18, to be compared with �3:79 whenj ¼ 0:1 – low habit persistence – and �4:69 in the time separable model whenendowment corresponds to dividend. It should however be noted that, as the horizonincreases, bk remains constant which is at odds with the empirical evidence (seeTable 4). Moreover, the patterns of R2 is reversed as it decreases with horizon.Hence, the catching up model cannot account for predictability for any value of jalthough the results improve with larger j. This comes at no surprise as this versionof the model cannot generates greater persistence than the TS model.

ARTICLE IN PRESS


In the last series of results, we consider the habit stock version of the model.Results reported in Table 12 show that our benchmark version of the habit stockmodel enhances the ability of the CCAPM to account for predictability compared tothe TS and CJ versions of the model. First of all, estimated values of bk are muchcloser to those estimated on empirical data (see Table 4). For instance, in the shortsample where predictability is really significant, b1 is �0:36 to be compared with�0:44 found in the model. Likewise, at longer horizon, b7 is �1:01 in the data versus�1:2 in the model. Therefore, the model matches the overall size of the coefficients,and reproduces their evolution with the horizon. It should also be noted that, asfound in the data, the R2 is an increasing function of the horizon. Table 13 showsthat what really determines the result is persistence. Indeed, lowering the force ofhabit – setting a lower j – while maintaining the same persistence (experimentðj; dÞ ¼ ð0:5; 0:05Þ) does not deteriorate too much the results. The shape of thecoefficients and the R2 is maintained. Conversely, reducing persistence – higher d –while maintaining the force of habit (experiment ðj; dÞ ¼ ð1; 0:5Þ) dramaticallyaffects the results. First of all, the model totally fails to match the scale of the bk’s.Second, predictability diminishes with the horizon.

5. Concluding remarks

This paper investigates the role of persistence in accounting for the predictabilityof excess return. We first develop a standard consumption based asset pricing modela la Lucas (1978) taking ‘Catching up with the Joneses’ and habit stock formationinto account. Providing we keep with the assumption of first order Gaussianendowment growth and formulate habit formation in terms of ratio, we are able toprovide a closed form solution for the price–dividend ratio. We also provideconditions that guarantee the existence of bounded solutions. We then assess theperformance of the model in terms of moment matching. In particular, we evaluatethe ability of the model to generate persistence and explain the predictability puzzle.We then show that the habit stock version of the model outperforms the timeseparable and the catching up versions of the model in accounting for predictabilityof excess returns. Since risk aversion is held constant in the model, this result stemsfrom the greater persistence habit stock generates.

Acknowledgements

We would like to thank P. Ireland and two anonymous referees for helpfulcomments on the previous version of the paper.

Appendix A. Distributions

Distorsion of distributions is given in Figs. 3 and 4.

ARTICLE IN PRESS

-6 -4 -2 0 2 4

0.2

0.4

0.6

0.8

1k=1

-5 0 5

0.2

0.4

0.6

0.8

1k=2

0.2

0.4

0.6

0.8

1k=3

0.2

0.4

0.6

0.8

1k=5

0.2

0.4

0.6

0.8

1k=7

-5 0 5 -5 0 5

-5 0 5

Fig. 3. Distorsion of distributions (short sample). Note: The dark plain line corresponds to the empirical

distribution obtained from 100,000 Monte-Carlo simulations. The gray plain line is the student

distribution. The two dashed lines report the actual thresholds of a two-sided test for a 5% size.


ARTICLE IN PRESS

-6 -4 -2 0 2 4

0.2

0.4

0.6

0.8

1k=1

-5 0 5

0.2

0.4

0.6

0.8

1k=2

-5 0 5

0.2

0.4

0.6

0.8

1k=3

-5 0 5

0.2

0.4

0.6

0.8

1k=5

-5 0 5

0.2

0.4

0.6

0.8

1k=7

Fig. 4. Distorsion of distributions (whole sample). Note: The dark plain line corresponds to the empirical

distribution obtained from 100,000 Monte-Carlo simulations. The gray plain line is the student

distribution. The two dashed lines report the actual thresholds of a two-sided test for a 5% size.


ARTICLE IN PRESS


Appendix B. Proof of Propositions

Proposition 1. See proof of Proposition 4.



Proposition 4. First of all note that setting d ¼ 1 in this proof, we obtain a proof for

Proposition 1. Let us denote vt ¼ Pt=Dt the price– dividend ratio, and zt ¼ logðX t=DtÞ

the habit to dividend ratio. Finally, letting yt ¼ vt expð�jð1� yÞztÞ, the agent’s Euler

equation rewrites

yt ¼ bEt½expðð1� yÞð1� jÞgtþ1 � jð1� yÞztþ1Þ þ expðð1� yÞð1� jÞgtþ1Þytþ1�

Iterating forward, and imposing the transversality condition, a solution to this forward

looking stochastic difference equation is given by

yt ¼ Et

X1i¼1

bi exp ð1� yÞð1� jÞXi

j¼1

gtþj � jð1� yÞztþi

!. (22)

Note that, the definition of zt and the law of motion of habits imply that zt evolvesas

ztþ1 ¼ ð1� dÞzt � gtþ1 (23)

which implies that

ztþi ¼ ð1� dÞizt �Xi�1j¼0

ð1� dÞjgtþi�j (24)

Plugging the latter result in (22), we get

yt ¼ Et

X1i¼1

bi exp ð1� yÞXi

j¼1

ðð1� jÞ þ jð1� dÞi�jÞgtþj � jð1� yÞð1� dÞizt

!.

Let us focus on the particular component of the solution

G � Et exp ð1� yÞXi

j¼1

ðð1� jÞ þ jð1� dÞi�jÞgtþj

!.

Since we assumed that dividend growth is normally distributed, making use ofstandard results on log-normal distributions, we have that

G ¼ exp EþV

2

� �,

ARTICLE IN PRESS


where

E ¼ Et ð1� yÞXi

j¼1


!

and

V ¼ Vart ð1� yÞXi

j¼1


!.

Since gt follows an AR(1) process, we have

gtþj ¼ gþ rjðgt � gÞ þXj�1k¼0

rketþj�k

such that

E ¼ Et ð1� yÞXi

j¼1

ðð1� jÞ þ jð1� dÞi�jÞ gþ rjðgt � gÞ þ

Xj�1k¼0

rketþj�k

!" #

¼ ð1� yÞXi

j¼1

ðð1� jÞ þ jð1� dÞi�jÞðgþ rjðgt � gÞÞ

" #

¼ ð1� yÞð1� jÞXi

j¼1

ðgþ rjðgt � gÞÞ þ ð1� yÞjXi

j¼1

ð1� dÞi�jðgþ rjðgt � gÞÞ

¼ ð1� yÞg ð1� jÞi þjdð1� ð1� dÞiÞ

h iþ ð1� yÞ

rð1� jÞ1� r

ð1� riÞ

�þ

jr1� d� r

ðð1� dÞi � riÞ

�ðgt � gÞ.

The calculation of the conditional variance is a bit more tedious

V ¼ Vart ð1� yÞXi

j¼1

ð1� jþ jð1� dÞi�jÞXj�1k¼0

rketþj�k

" #,

which after some accounting rewrites as

¼ Vart ð1� yÞXi�1j¼0

1� j1� r

ð1� ri�jÞ þj

1� d� rðð1� dÞi�j

� ri�jÞ

� �etþjþ1

" #

¼ ð1� yÞ2s2Xi�1j¼0

1� j1� r

ð1� ri�jÞ þj

1� d� rðð1� dÞi�j

� ri�jÞ

� �2

ARTICLE IN PRESS


¼ ð1� yÞ2s21� j1� r

� �2Xi

k¼1

ð1� rkÞ2þ 2

jð1� dÞð1� rÞð1� d� rÞ

"

�Xi

k¼1

ð1� rkÞðð1� dÞk � rkÞ þj

1� d� r

� �2Xi

k¼1

ðð1� dÞk � rkÞ2

#.

Calculating all the infinite series, we end-up with

V ¼ ð1� yÞ2s21� j1� r

� �2

i � 2r

1� rð1� riÞ þ

r2

1� r2ð1� r2iÞ

� �(

þ 2jð1� jÞ

ð1� rÞð1� d� rÞð1� dÞ

dð1� ð1� dÞiÞ �

r1� r

ð1� riÞ

��



1� r2ð1� r2iÞ

�þ

j2

ð1� d� rÞ2

�ð1� dÞ2

1� ð1� dÞ2ð1� ð1� dÞ2i

Þ � 2rð1� dÞ

1� rð1� dÞð1� ðrð1� dÞÞiÞ

�þ

r2

1� r2ð1� r2iÞ

�).

Therefore, the solution is given by

yt ¼X1i¼1

bi expðai þ biðgt � gÞ þ eciztÞ,

where

ai ¼ ð1� yÞg ð1� jÞi þjdð1� ð1� dÞiÞ

h iþ

V

2

bi ¼ ð1� yÞrð1� jÞ1� r

ð1� riÞ þjr

1� d� rðð1� dÞi � riÞ

� �eci ¼ � jð1� yÞð1� dÞi,

recall that yt ¼ vt expð�jð1� yÞztÞ, such that the price–dividend ratio is finally givenby

vt ¼ expðjð1� yÞztÞX1i¼1

bi expðai þ biðgt � gÞ þ eciztÞ

or

vt ¼X1i¼1

bi expðai þ biðgt � gÞ þ ciztÞ,

where ci ¼ jð1� yÞð1� ð1� dÞiÞ.

ARTICLE IN PRESS



Proposition 2. Let us define

wi ¼ bi expðai þ biðgt � gÞ þ ciztÞ,

where ai, bi and ci are obtained from the previous proposition. Then, the price– dividend

ratio rewrites

vt ¼X1i¼1

wi.

It follows that

wiþ1

wi

�� ¼ b expðDaiþ1 þ Dbiþ1ðgt � gÞ þ Dciþ1ztÞ,

where

Daiþ1 ¼ ð1� yÞg½ð1� jÞ þ jð1� dÞi� þ ð1� yÞ2s2

2

1� j1� r

� �2

ð1� 2riþ1 þ r2ðiþ1ÞÞ

(

þ 2jð1� jÞ

ð1� rÞð1� d� rÞðð1� dÞiþ1 � riþ1 � ðrð1� dÞÞiþ1 þ r2ðiþ1ÞÞ

þj2

ð1� d� rÞ2ðð1� dÞ2ðiþ1Þ � 2ðrð1� dÞÞiþ1 þ r2ðiþ1ÞÞ

),

Dbiþ1 ¼ ð1� yÞ ð1� jÞriþ1 �jr

1� d� rðð1� rÞri � dð1� dÞiÞ

� �,

Dciþ1 ¼ jð1� yÞdð1� dÞi.

Also note that provided jrjo1 and d 2 ð0; 1Þ, we have

limi!1

Daiþ1 ¼ ð1� yÞgð1� jÞ þ ð1� yÞ2s2

2

1� j1� r

� �2

,

limi!1

Dbiþ1ðgt � gÞ ¼ 0,

limi!1

Dciþ1zt ¼ 0.

Therefore

limi!1

wiþ1

wi

�� ¼ r � b exp ð1� yÞgð1� jÞ þ ð1� yÞ2s2

2

1� j1� r

� �2 !

.

Using the ratio test, we now face three situations:

(i)
When ro1, then limi!1jwiþ1
wijo1 and the ratio test implies that

P1i¼1 wi

converges. P
(ii) When r41, the ratio test implies that 1
i¼1 wi diverges.

ARTICLE IN PRESS


(iii)
When r ¼ 1, the ratio test is inconclusive. But, if r ¼ 1, we know that
exp ð1� yÞð1� jÞgþð1� yÞð1� jÞ

1� r

� �2 s2

2

!¼

1

b

and the parameter ai rewrites

ai ¼ ð1� yÞð1� jÞgþð1� yÞð1� jÞ

1� r

� �2 s2

2

!i

þ1� y1� r

� �2 s2

2

ðr� jÞ2

1� r2ð1� r2iÞ � 2


ð1� riÞ

� �¼ � logðbÞi þ

1� y1� r

� �2 s2

2

ðr� jÞ2

1� r2ð1� r2iÞ � 2


ð1� riÞ

� �After replacement in wi, we get

wi ¼ expð ~ai þ biðgt � gÞ þ ciztÞ,

where

eai ¼1� y1� r

� �2 s2

2

ðr� jÞ2

1� r2ð1� r2iÞ � 2


ð1� riÞ

� �.

Since

limi!1jeaij ¼1� y1� r

� �2 s2

2

ðr� jÞ2

1� r2� 2ð1� jÞðr� jÞ

1� r

� ��40,

then the series vt ¼P1

i¼1 wi diverges.

Therefore, ro1 is the only situation where a stationary bounded equilibrium exists.


Proposition 3. We first deal with the average of the price/dividend ratio. We want to

compute

EðvtÞ ¼ EX1i¼1

bi expðai þ biðgt � gÞ þ cizt

!¼X1i¼1

biEðexpðai þ biðgt � gÞ þ ciztÞ.

By the log-normality of gt, we know that

Eðexpðai þ biðgt � gÞ þ ciztÞ ¼ exp Ei þVi

2

� �,

where

Ei ¼ Eðai þ biðgt � gÞ þ ciztÞ ¼ ai þ ciEðztÞ

and

Vi ¼ Varðai þ biðgt � gÞ þ ciztÞ ¼ b2i s

2=ð1� r2Þ þ c2i VarðztÞ þ 2cibiCovðzt; gtÞ

ARTICLE IN PRESS


Recall that zt ¼ ð1� dÞzt�1 � gt, therefore

EðztÞ ¼ �gd

and

Covðzt; gtÞ ¼ Covðð1� dÞzt�1 � gt; gtÞ ¼ ð1� dÞrCovðzt�1; gt�1Þ � VarðgtÞ.

Hence

Covðzt; gtÞ ¼ �s2

ð1� rð1� dÞÞð1� r2Þ

Furthermore, we know that

VarðztÞ ¼ ð1� dÞ2VarðztÞ þ VarðgtÞ � 2ð1� dÞCovðzt�1; gtÞ

¼s2

ð1� r2Þð1� ð1� dÞ2Þ

1þ rð1� dÞ1� rð1� dÞ

� �.

Therefore

Vi ¼ b2i

s2

1� r2þ c2i

s2



� �� 2cibi

s2

ð1� rð1� dÞÞð1� r2Þ

Hence we have to study the convergence of the seriesX1i¼1

bi exp ai � ci

gdþ

s2

2ð1� r2Þb2

i þ1þ rð1� dÞ

ð1� rð1� dÞÞð1� ð1� dÞ2Þc2i

��

2bici

1� rð1� dÞ

��Defining

wi ¼ bi exp ai � ci

gdþ

s2

2ð1� r2Þb2

i þ1þ rð1� dÞ

ð1� rð1� dÞÞð1� ð1� dÞ2Þc2i

��

2bici

1� rð1� dÞ

��,

the series rewritesP1

i¼1 wi, whose convergence properties can be studied relying onthe ratio test.

wiþ1

wi

�� ¼ b exp Daiþ1 � Dciþ1gdþ

s2

2ð1� r2Þ

�� Db2

iþ1 þ1þ rð1� dÞ

ð1� rð1� dÞÞð1� ð1� dÞ2ÞDc2iþ1 �

2Dðbiþ1ciþ1Þ

1� rð1� dÞ

� ��.

ARTICLE IN PRESS


Given the previously given definition of ai, bi and ci, we have

Daiþ1 ¼ ð1� yÞg½ð1� jÞ þ jð1� dÞi� þ ð1� yÞ2s2

2

1� j1� r

� �2

ð1� 2riþ1 þ r2ðiþ1ÞÞ

(

þ 2jð1� jÞ

ð1� rÞð1� d� rÞðð1� dÞiþ1 � riþ1 � ðrð1� dÞÞiþ1 þ r2ðiþ1ÞÞ

þj2

ð1� d� rÞ2ðð1� dÞ2ðiþ1Þ � 2ðrð1� dÞÞiþ1 þ r2ðiþ1ÞÞ

),

Db2iþ1 ¼ ð1� yÞ2

rð1� jÞ1� r

� �2

ð2rið1� rÞ � r2ið1� r2ÞÞ

"

þjr

1� d� r

� �2

ð2ðrð1� dÞÞið1� rð1� dÞÞ � ð1� dÞ2ð1� ð1� dÞ2Þ

� r2ið1� r2ÞÞ þ2r2jð1� dÞ

ð1� rÞð1� d� rÞðrið1� rÞ � dð1� dÞi

þ ðrð1� dÞÞið1� rð1� dÞÞ � r2ið1� r2ÞÞ

#,

Dc2iþ1 ¼ ðjð1� yÞÞ2ð2dð1� dÞi � ð1� dÞ2ið1� ð1� dÞ2ÞÞ

DðbiciÞ ¼ jð1� yÞ2 ð1� jÞriþ1 þjr

1� d� rðrið1� rÞ � dð1� dÞiÞ

��

rð1� jÞ1� r

ðrið1� rÞ þ dð1� dÞi � ðrð1� dÞÞið1� rð1� dÞÞÞ

�jr

1� d� rððrð1� dÞÞið1� rð1� dÞÞ � ð1� dÞ2i

ð1� ð1� dÞ2ÞÞ�.

Then note that

limi!1

Daiþ1 ¼ ð1� yÞgð1� jÞ þ ð1� yÞ2s2

2

1� j1� r

� �2

,

limi!1

Dciþ1 ¼ 0,

limi!1

Db2iþ1 ¼ 0,

limi!1

Dc2iþ1 ¼ 0,

limi!1

Dðbiþ1ciþ1Þ ¼ 0.

This implies that

limi!1

wiþ1

wi

�� ¼ ð1� yÞgð1� jÞ þ ð1� yÞ2s2

2

1� d1� r

� �2

� r.

ARTICLE IN PRESS


Therefore, following Proposition 5, the average of the price–dividend ratio convergesto a constant if and only if ro1.

We now examine the autocovariances of the ratio. As just proven, theprice–dividend ratio is finite for ro1. Therefore, it is sufficient to show thatEðvtvt�kÞ is finite for all k. The idea here is to provide an upper bound for thisquantity. If the process is stationary it has to be the case that Eðvtvt�kÞpEðv2t Þ.

We want to compute

Eðv2t Þ ¼ EX1i¼1

X1j¼1

biþj expððai þ ajÞ þ ðbi þ bjÞðgt � gÞ þ ðci þ cjÞzt

!

¼X1i¼1

X1j¼1

biþjEðexpððai þ ajÞ þ ðbi þ bjÞðgt � gÞ þ ðci þ cjÞztÞ.

By the log-normality of gt, we know that

Eðexpððai þ ajÞ þ ðbi þ bjÞðgt � gÞ þ ðci þ cjÞztÞ ¼ exp Ei;j þVi;j

2

� �,

where

Vi ¼ Varðai þ aj þ ðbi þ bjÞðgt � gÞ þ ðci þ cjÞztÞ

¼ ðbi þ bjÞs2

1� r2þ ðci þ cjÞVarðztÞ þ 2ðbi þ bjÞðci þ cjÞCovðzt; gtÞ.

From the first part of the proof, we know that

EðztÞ ¼ �gd,

Covðzt; gtÞ ¼ �s2

ð1� rð1� dÞÞð1� r2Þ,

VarðztÞ ¼s2



� �.

Using the definition of ak, bk and ck, k ¼ i; j, it is straightforward – although tedious– to show that

Ei;j ¼ ð1� yÞð1� jÞgþ1� y1� r

� �2 s2

2ð1� jÞ2

" #ði þ jÞ þ ð1� yÞ2

s2

2Cij,

where

Cij �r2

1� r21� j1� r

þj

1� d� r

� �2

ð2� r2i � r2jÞ þj2ð1� dÞ2

ð1� d� rÞð1� ð1� dÞ2Þ

�ð2� ð1� dÞ2i� ð1� dÞ2j

Þ �2rjð1� dÞ

ð1� rð1� dÞÞð1� d� rÞ1� j1� r

þj

1� d� r

� �

ARTICLE IN PRESS


�ð2� ðrð1� dÞÞi � ðrð1� dÞÞjÞ � 2r

1� r1� j1� r

1� j1� r

þj

1� d� r

� ��ð2� ri � rjÞ þ 2

1� dd

jð1� jÞð1� rÞð1� d� rÞ

ð2� ð1� dÞi � ð1� dÞjÞ

and

Vi;j ¼ ð1� yÞ2s2r2

1� r2V1

i;j þj2ð1� yÞ2s2



� �V2

i;j

�jð1� yÞ2s2

ð1� rð1� dÞÞð1� r2ÞV3

i;j,

where

V1i;j � 4

1� j1� r

� �2

þj

1� d� r

� �2

ðð1� dÞ2iþ 2ð1� dÞiþj

þ ð1� dÞ2jÞ

þ1� j1� r

þj

1� d� r

� �2

ðr2i þ 2riþj þ r2jÞ � 41� j1� r

�1� j1� r

þj

1� d� r

� �ðri þ rjÞ þ 4

jð1� jÞð1� rÞð1� d� rÞ

�ðð1� dÞi þ ð1� dÞjÞ � 2j

1� d� r1� j1� r

þj

1� d� r

� ��ððrð1� dÞÞi þ ðrð1� dÞÞj þ rið1� dÞj þ rjð1� dÞiÞ

and

V2i;j � 4þ ð1� dÞ2i

þ ð1� dÞ2j� 4ðð1� dÞi þ ð1� dÞjÞ þ 2ð1� dÞiþj

and

V3i;j � 4

1� j1� r

þ 2j

1� d� rðð1� dÞi þ ð1� dÞjÞ � 2

1� j1� r

þj

1� d� r

� �ðri þ rjÞ

� 21� j1� r

ðð1� dÞi þ ð1� dÞjÞ �j

1� d� rðð1� dÞ2i

þ 2ð1� dÞiþjþ ð1� dÞjÞ

�1� j1� r

þj

1� d� r

� �ððrð1� dÞÞi þ ðrð1� dÞÞj þ rið1� dÞj þ rjð1� dÞiÞ.

Using the triangular inequality, we have CijpC where

C � 4r2

1� r21� j1� r

þj

j1� d� rj

� �2

þ 4j2ð1� dÞ2

j1� d� rjð1� ð1� dÞ2Þ

þ8rjð1� dÞ

ð1� rð1� dÞÞj1� d� rj1� j1� r

þj

j1� d� rj

� �þ 8

r1� r

1� j1� r

1� j1� r

þj

j1� d� rj

� �þ 8

1� dd

jð1� jÞð1� rÞj1� d� rj

.

ARTICLE IN PRESS


Likewise, the triangular inequality implies that

V1i;jp 4

1� j1� r

� �2

þ 4j

j1� d� rj

� �2

þ 41� j1� r

þj

j1� d� rj

� �2

þ 81� j1� r

1� j1� r

þj

j1� d� rj

� �þ 8

jð1� jÞð1� rÞðj1� d� rjÞ

þ 8j

j1� d� rj

�1� j1� r

þj

j1� d� rj

� �p16

1� j1� r

þj

j1� d� rj

� �2

similarly

V2i;jp16

and

V3i;jp4

1� j1� r

þ 4j

j1� d� rjþ 4

1� j1� r

þj

j1� d� rj

� �þ 4

1� j1� r

þ 4j

j1� d� rj

þ 41� j1� r

þj

j1� d� rj

� �p16

1� j1� r

þj

j1� d� rj

� �.

Therefore, we have Vi;jpV, where

V � 16ð1� yÞ2s2r2

1� r21� j1� r

þj

j1� d� rj

� �2

þ 16j2ð1� yÞ2s2


�1þ rð1� dÞ1� rð1� dÞ

� �þ 16

jð1� yÞ2s2

ð1� rð1� dÞÞð1� r2Þ1� j1� r

þj

j1� d� rj

� �.

Hence, we have that

Eðv2t ÞpX1i¼1

X1j¼1

biþj exp ð1� yÞð1� jÞgþ1� y1� r

� �2 s2

2ð1� jÞ2

" #ði þ jÞ

!

� exp ð1� yÞ2s2

2Cþ

V

2

� �p exp ð1� yÞ2

s2

2Cþ

V

2

� �X1i¼1

X1j¼1

riþj

p exp ð1� yÞ2s2

2Cþ

V

2

� �rX1k¼1

nrn.

As long as ro1, the seriesP1

k¼1 nrn converges, such that in this case Ev2t o1.

ARTICLE IN PRESS


We can now consider the autocovariance terms

Eðvtvt�kÞ ¼X1i¼1

X1j¼1

biþj exp ai þ aj � ðci þ cjÞgdþ

1

2ðb2

i VarðgtÞ þ b2j Varðgt�kÞ

�þ c2i VarðztÞ þ c2j Varðzt�kÞ þ 2bibjCovðgt; gt�kÞ þ 2biciCovðgt; ztÞ

þ 2bicjCovðgt; zt�kÞ þ 2bjciCovðgt�k; ztÞ þ 2bjcjCovðgt�k; zt�kÞ

þ 2cicjCovðzt; zt�kÞÞ

�,

where12

Covðgt; gt�kÞ ¼ rk s2

1� r2¼ rkVarðgtÞ,

Covðzt; zt�kÞ ¼s2

1� d� rð1� r2Þð1� dÞkþ1 � ð1� ð1� dÞ2Þrkþ1

ð1� rð1� dÞÞð1� r2Þð1� ð1� dÞ2Þ

!Fzz;kVarðztÞ,

Covðzt; gt�kÞ ¼s2

1� d� rrkþ1

1� r2�ð1� dÞkþ1

1� rð1� dÞ

!¼ Fzg;kCovðzt; gtÞ,

Covðgt; zt�kÞ ¼ �s2

ð1� r2Þð1� dð1� dÞÞrk ¼ rkCovðgt; ztÞ.

with

Fzz;k ¼ð1� dÞð1� r2Þð1� dÞk � rð1� ð1� dÞ2Þrk

ð1� r2Þð1� dÞ � ð1� ð1� dÞ2Þr,

Fzg;k ¼ð1� dÞð1� r2Þð1� dÞk � rð1� rð1� dÞÞrk

ð1� r2Þð1� dÞ � rð1� rð1� dÞÞ.

Note that, by construction, we have jFzz;kjo1 and jFzg;kjo1. Eðvtvt�kÞ can then berewritten as

Eðvtvt�kÞ ¼X1i¼1

X1j¼1

biþj exp ai þ aj � ðci þ cjÞgdþ

1

2Vi;j;k

� �,

where

Vi;j;k � ðb2i þ b2

j þ 2bibjrkÞVarðgtÞ þ ðc2i þ c2j þ 2cicjFzz;kÞVarðztÞ

þ 2ðbici þ bjcj þ bicjrk þ cjciFzg;kÞCovðgt; ztÞ.

12These quantities can be straightforwardly obtained from the Wold representations of gt and zt:

gt ¼ gþX1i¼0

riet�i and zt ¼X1i¼0

riþ1 � ð1� dÞiþ1

1� d� ret�i .

ARTICLE IN PRESS


Since bibj40 and jrjo1, we have b2i þ b2

j þ 2bibjrkpðbi þ bjÞ2. Likewise cicj40

and jFzz;kjo1, so that ðc2i þ c2j þ 2cicjFzz;kÞpðci þ cjÞ2. Finally, we have bnc‘40,

ðn; ‘Þ 2 fi; jg � fi; jg and both jrjo1 and jFzg;kjo1, such that ðbici þ bjcj þ bicjrkþ

cjciFzg;kÞpðbi þ bjÞðci þ cjÞ. This implies that

jVi;j;kjpðbi þ bjÞ2VarðgtÞ þ ðci þ cjÞ

2VarðztÞ þ ðbi þ bjÞðci þ cjÞjCovðzt; gtÞj.

Hence Ev2t is an upper bound for Eðvtvt�kÞ. Therefore, as Ev2t is finite for ro1, so isEðvtvt�kÞ.

References

Abel, A., 1990. Asset prices under habit formation and catching up with the Joneses. American Economic

Review Papers and Proceedings 80, 38–42.

Abel, A., 1999. Risk premia and term premia in general equilibrium. Journal of Monetary Economics 43,

3–33.

Allais, O., 2004. Local substitution and habit persistence: matching the moments of the equity premium

and the risk-free rate. Review of Economic Dynamics 7, 265–296.

Ang, A., 2002. Characterizing the ability of dividend yields to predict future dividends in log-linear present

value models. Mimeo, Columbia University.

Ang, A., Bekaert, G., 2004. Stock Return Predictability: is it there? Mimeo, Columbia University.

Bidarkota, P.V., McCulloch, J.H., 2003. Consumption asset pricing with stable shocks: exploring a

solution and its implications for mean equity returns. Journal of Economic Dynamics and Control 27,

399–421.

Boldrin, M., Christiano, L.J., Fisher, D.M., 1997. Habit persistence and asset returns in an exchange

economy. Macroeconomic Dynamics 1, 312–332.

Brandt, M.W., Zeng, Q., Zhang, L., 2004. Equilibrium stock return dynamics under alternative rules of

learning about hidden states. Journal of Economic Dynamics and Control 28, 1925–1954.

Burnside, C., 1998. Solving asset pricing with Gaussian shocks. Journal of Economic Dynamics and

Control 22, 329–340.

Campbell, J., 2003. Consumption-based asset pricing. In: Constantinides, G., Harris, M., Stulz, R. (Eds.),

Handbook of the Economics of Finance. North-Holland, Amsterdam, forthcoming.

Campbell, J.Y., Cochrane, J.H., 1999. By force of habit: a consumption-based explanation of aggregate

stock market behavior. Journal of Political Economy 107, 205–251.

Campbell, J.Y., Shiller, R.J., 1987. Cointegration and tests of present value models. Journal of Political

Economy 95, 1062–1087.

Campbell, J.Y., Shiller, R.J., 1988. The dividend–price ratio and expectations of future dividends and

discount factors. Review of Financial Studies 1, 195–227.

Campbell, J.Y., Yogo, 2005. Efficient tests of stock return predictability. Forthcoming in Journal of

Financial Economics.

Campbell, J., Lo, A., MacKinlay, C., 1997. The Econometrics of Financial Markets. Princeton University

Press, Princeton, NJ.

Cecchetti, S.G., Lam, P.S., Mark, N.C., 2000. Asset pricing with distorted beliefs: are equity returns too

good to be true? American Economic Review 90, 787–805.

Chan, Y.L., L. Kogan, 2001. Heterogeneous preferences and the dynamics of asset prices. Technical

Report 8607, NBER.

Cochrane, J.H., 1997. Where is the market going? Uncertain facts and novel theories, Economic

perspectives. Federal Reserve Bank of Chicago 11, 1–37.

Cochrane, J.H., 2001. Asset Pricing. Princeton University Press, Princeton, NJ.

Constantidines, G.M., 1990. Habit formation: a resolution of the equity premium puzzle. Journal of

Political Economy 98, 519–543.

ARTICLE IN PRESS


Fama, E.F., French, K.R., 1988. Dividend yields and expected stock returns. Journal of Financial

Economics 22, 3–25.

Fama, E.F., French, K.R., 1989. Business conditions and expected returns on stocks and bonds. Journal

of Financial Economics 25, 23–49.

Ferson, W.E., Sarkissian, S., Simin, T.T., 2003. Spurious regressions in financial economics? Journal of

Finance 58, 1393–1413.

Galı, J., 1994. Keeping up with the Joneses: consumption externalities, portfolio choice and asset prices.

Journal of Money, Credit, and Banking 1–8.

Heaton, J., 1995. An empirical investigation of asset pricing with temporally dependent preference

specifications. Econometrica 63, 681–717.

Hobijn, B., Jovanovic, B., 2001. The information-technology revolution and the stock market: evidence.

American Economic Review 91, 1203–1220.

Hodrick, R., 1992. Dividend yields and expected stock returns: alternative procedures for inference and

measurement. Review of Financial Studies 5, 357–386.

Koop, G., Pesaran, M.H., Potter, S.M., 1996. Impulse response analysis in nonlinear multivariate models.

Journal of Econometrics 74, 119–147.

Lamont, O., 1998. Earnings and expected returns. Journal of Finance 53, 1563–1587.

Lettau, L., Ludvigson, S.C., 2005. Expected returns and expected dividend growth. Journal of Financial


Lettau, M., Ludvigson, S., 2001. Consumption aggregate wealth and expected stock returns. Journal of

Finance 56, 815–849.

Li, Y., 2001. Expected returns and habit persistence. Review of Financial Studies 14, 861–899.

Lucas, R., 1978. Asset prices in an exchange economy. Econometrica 46, 1439–1445.

Menzly, L., Santos, T., Veronesi, P., 2004. Understanding predictability. Journal of Political Economy

112, 1–47.

Poterba, J., Summers, L.H., 1988. Mean reversion in stock prices. Journal of Financial Economics 22,

27–59.

Stambaugh, R.F., 1999. Predictive regressions. Journal of Financial Economics 54, 375–421.

Sundaresan, S.M., 1989. Intertemporally dependent preferences and the volatility of consumption and

wealth. The Review of Financial Studies 2, 73–89.

Torous, W., Valkanov, R., Yan, S., 2004. On predicting stock returns with nearly integrated explanatory

variables. Journal of Business 77, 937–966.

Tsionas, E., 2003. Exact solution of asset pricing models with arbitrary shock distributions. Journal of

Economic Dynamics and Control 27, 843–851.

Valkanov, R., 2003. Long-horizon regressions: theoretical results and applications. Journal of Financial


Documents

Predictability and habit persistence