# An exploration of asset returns in a production economy with relative habits.

IntroductionHabit forming preferences have been notably successful in solving the equity premium puzzle. One of their most popular representations is due to Abel (1990, 1999, 2006), who shows that an asset pricing model with relative habits can account for financial regularities that can not be explained by standard models. However, this finding is confined to a fruit-tree framework in which equilibrium consumption follows a stochastic process. To date, we still do not know how Abel's habits fare when households are allowed to take consumption decisions. Referring to his model of asset pricing, Abel (1999, p. 27) argues that "the next step is to extend the model to allow the economy to transfer goods across time by capital investment" This paper takes that step by asking whether a model with relative habits and consumption choice can be consistent with the empirical equity premium and business cycle facts simultaneously.

After Mehra and Prescott's (1985) statement of the equity premium puzzle, researchers attempted to come up with a solution by allowing for habit forming preferences. A first body of research showed that introducing habits in an endowment economy framework can go a long way towards accounting for the empirical equity premium (Abel 1990; Constantinides 1990; Boldrin et al. 1997; Campbell and Cochrane 1999). Habits were modeled in two ways: as survival habits (agents derive utility from the gap between current consumption and their habit stock) and as relative habits (agents derive utility from the ratio between current consumption and their habit stock). The endowment economy framework nevertheless has its limitations, and some authors have extended the analysis to production economies. This new scenario challenges models of habit formation by offering the possibility of studying business cycle and financial phenomena simultaneously. Christiano and Fisher (1998), Jermann (1998) and Boldrin et al. (2001) develop versions of the real business cycle model (RBC model, henceforth) that can generate a sizable premium when short term rigidities are introduced in the capital market. However, these advances are confined to the survival representation of habits and, up to date, we do not know whether relative habits are consistent with business cycle and asset pricing facts simultaneously. This is unfortunate, as relative habits have become popular by providing explanations to a variety of economic problems (1).

The goal of this paper is, therefore, to undertake the modest task of documenting the financial and macroeconomic implications of relative habits. It must be noted that the paper does not attempt to compare the relative performance of the different utility functions that have been proposed in the asset pricing literature. Conducting a thorough comparison should include not only the survival and relative specification of habits, but also other popular utility functions such as Epstein and Zin's (1989) preferences and their more elaborated versions in Tallarini (2000), Hansen et al. (2005) and Bansal et al. (2007). Though promising, performing such a scrutiny is a task well beyond of the scope of this paper.

We develop a model with a nontrivial production sector and consumption choice. Households in the artificial economy are very reluctant to changes in consumption levels, but, unlike in endowment economies, they can take consumption decisions to insure against consumption risk. In order to mitigate this smoothing channel, we introduce capital adjustment costs. New capital is accumulated according to a concave technology so that changing the capital stock rapidly is more costly than changing it slowly. After a positive shock, firms react allocating less resources to investment or, to put it different, very risk averse individuals end up following volatile consumption paths despite the habit formation process. Thus, in equilibrium, households face high consumption risk, and require a substantial premium on stocks to be compensated by the risk borne.

In spite of their popularity, the macroeconomic predictions of relative habits have not been documented. To fill this gap, we report business cycle statistics and show that when combined with capital adjustment costs, relative habits are consistent with salient business cycle features.

The rest of the paper is organized as follows. First, we describe the model economy and define its equilibrium. Second, we discuss the rationale behind the choice of functional forms and parameters. Third, we report the main findings upon financial and macroeconomic variables. Finally, we present the concluding remarks and outline some directions for further research.

The Model

The model presented is a modification of the one sector stochastic growth model. The economy is populated by a representative firm and an arbitrarily large number of identical households, each endowed with a fixed amount of labor which is supplied to the firm. The firm produces a single consumption/investment good and finances 1its investment through retained earnings. Two assets in this economy are traded in complete financial markets: a perfectly divisible equity share of the firm and a one-period riskless bond. The equity is a claim to a infinite stream of firm's dividends. The bond is a right to perceive one unit of consumption one period ahead.

Households

Households' tastes over streams of consumption are given at time t by

[V.sub.t] = [E.sub.t] [[infinity].summation over (j=0)] [[beta].sup.j] U ([C.sub.t+j, [H.sub.t + j)] (1)

where [beta] is the subjective discount factor, [C.sub.t] is the consumption rate, and [H.sub.t] is the habit stock. Current habits are determined by yesterday's consumption,

[H.sub.t] = [C.sub.t - 1]. (2)

Labor time is exogenous and normalized to unity, [L.sub.t] = 1. At each period, the budget constraint is

[W.sub.t] [L.sub.t] + [S.sub.t] ([P.sup.e.sub.t] + [D.sub.t]) + [B.sub.t] = [C.sub.t] + [S.sub.t+1] [P.sup.e.sub.t] + [B.sub.t + 1] [P.sup.b.sub.t] (3)

where [L.sub.t] is labor, [W.sub.t] is the wage rate, [S.sub.t] and [B.sub.t] are the quantities of equities and bonds the investor holds from period t - 1 to period t, and [P.sup.sub.t] and [p.sup.b.sub.t] denote the price of the risky asset and the price of the riskless bond, respectively. The dividend on equity is [D.sub.t]. The bond pays one unit of the consumption good at time t + 1 and then expires. Households maximize Eq. 1 subject to Eqs. 2 and 3.

The Firm

The representative firm produces according to a constant returns to scale production function, with labor and capital as inputs

[Y.sub.t] = [Z.sub.t]F([K.sub.t, [L.sub.t]) (4)

where [Z.sub.t] is a technology shock. At each period, the firm has to decide on the amount of investment and how much labor to hire. The objective of the firm manager is to maximize the discounted present value of a infinite sequence of cash flows to the owners,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [M.sub.t + j] is the shareholders' marginal valuation in terms of today's consumption of one additional unit of consumption at time t + j. More formally, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the derivative of Eq. 1 with respect to [C.sub.t].

The firm accumulates capital according to the following equation

[K.sub.t + 1] = (1 - [delta]) [K.sub.t] + [psi]([I.sub.t]/[K.sub.t])[K.sub.t] (6)

where [delta] is the depreciation rate and [psi](*) is a positive, concave function. Dividends to shareholders are defined as the net cash flow--the residual of output value after labor wages have been paid and investment has been financed--of the firm at every period,

[D.sub.t] = [Y.sub.t] - [W.sub.t][L.sub.t] - [I.sub.t]. (7)

Equilibrium

Let [X.sub.t] = ([K.sub.t], [Z.sub.t], [H.sub.t]) be the vector of state variables. An equilibrium is a sequence of households plans {[C.sub.t],([X.sub.t]), [S.sub.t],([X.sub.t]), [B.sub.t],([X.sub.t])}.sup.[infinity].sub.t=1], a sequence of farm aggregates {[Y.sub.t]([X.sub.t]), [I.sub.t]([X.sub.t]), [D.sub.t]([X.sub.t])}.sup.[infinity].sub.t=1] and a sequence of prices, {[W.sub.t]([X.sub.t]), [P.sup.e.sub.t]([X.sub.t]), [P.sup.b.sub.t]([X.sub.t])}.sup.[infinity].sub.t=l] such that: (a) Aggregate consumption and investment equals output, [C.sub.t]([X.sub.t]) + [I.sub.t]([X.sub.t]) = [Y.sub.t]([X.sub.t]), the aggregate holdings of the equity is a fixed amount normalized to one, [S.sub.t]([X.sub.t]) = 1, and the riskless asset is in zero net supply, [B.sub.t]([X.sub.t]) = 0; (b) for all t, factor inputs are determined by the firm's first order conditions, [W.sub.t] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the labor input and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for capital; and (c) asset prices are determined by the first order conditions of households, [P.sup.e.sub.t] = [E.sub.t][[M.sub.t+l]([P.sup.e.sub.t+1] + [D.sub.t+l])] and [P.sup.b.sub.t] = [E.sub.t]([M.sub.t+l]).

Aggregate quantities and asset prices are computed from the solution to the related social planner's problem. The rate of return on equity and the riskfree asset are, respectively,

[R.sup.e.sub.t+1] = [P.sup.e.sub.t+1] + [D.sub.t+1]/[P.sup.e.sub.t] - 1 and [R.sup.f.sub.t] = 1/[P.sup.b.sub.t] - 1 (8)

The premium on equity is then

[EP.sub.t] = [R.sup.e.sub.t+1] - [R.sup.f.sub.t]. (9)

Calibration

We calibrate the model on a quarterly basis. First, we choose explicit functional forms for the utility function, the production function, the technological process, and the capital accumulation equation. Then, we assign values to the parameters involved so that the model reproduces some features of US macroeconomic and financial data.

Functional Forms

The representative household has relative habits. We use the formulation used in Abel (1990, 1999),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

According to this specification, the enjoyment of consumption depends on the proportion of current consumption relative to the habit stock (2). Increasing today's purchases has two effects. On the one hand, it increases instantaneous utility. On the other hand, it reduces the enjoyment of future consumption through the induced increase in the habit stock. We assume that households are aware of this second effect, i.e., habits are internal (3).

Output in the economy is given by a Cobb-Douglas production function,

[Y.sub.t] = [Z.sub.t][K.sup.[alpha].sub.t][L.sup.1-[alpha].sub.t] (11)

where [Z.sub.t] is a random productivity parameter that introduces uncertainty in this economy. At the steady state, [Z.sub.t] grows at a rate g. The state of technology evolves according to

[z.sub.t+1] = [bar.z] + [rho][z.sub.t-1] + [[epsilon].sub.t] [[epsilon].sub.t] ~ N(0, [[sigma].sup.2.sub.[epsilon]]) (12)

for [z.sub.t] = log([Z.sub.t]), where [epsilon] is a random disturbance term that is normally distributed with zero mean and standard deviation [[sigma].sub.[epsilon]. Finally, the cost of adjustment function is

[psi]([I.sub.t]/[K.sub.t] = a/1 - 1/[xi] [([I.sub.t]/[K.sub.t]).sup.1-1/[xi]]+b. (13)

The above capital equation can be related to the q-theory. Tobin's q, defined as the derivative of tomorrow's capital stock with respect to investment, is given by (4)

[q.sub.t] = [psi]'([I.sub.t]/[K.sub.t]) = a([I.sub.t]/[K.sub.t]).sup.-1/[xi]]. (14)

When [xi] = [infinity] the marginal rate of transformation between consumption and capital is constant and equal to one, that is, there are not capital adjustment costs. When [xi] < [infinity] [q.sub.t] is decreasing in investment and fluctuates endogenously over the cycle.

Parameters

The model is calibrated to replicate a set of long-run averages computed from US data. We target the investment-capital ratio, the capital share, the riskfree rate and the average consumption growth. These conditions give [delta] = 0.025, [alpha] = 0.40, [beta] = 0.99, g = 0.005. The standard deviation of the shock innovation is set to [[sigma].sub.[epsilon]] = 0.01, a value that replicates in the benchmark model the US postwar output growth volatility of 1%. The autocorrelation parameter is set to [rho] = 0.95. Finally, the elasticity of the cost of adjustment function is set to [xi] = 0.25, as in Jermann (1998), Hornstein and Uhlig (2000) and Boldrin et al. (2001).

Once the standard RBC parameters and the cost of adjustment function have been calibrated, we turn to the utility function. The preference parameter [tau] is set to 5. Then, the size of habits is chosen to replicate the empirical equity premium. We take as a target a 6.63% annual premium, which taking an arithmetic average gives 1.66% in quarterly terms. This condition yields [gamma] = 0.80, a value that is in line with previous parameterizations (5,6].

This is our benchmark calibration. In an earlier version of the paper we performed some sensitivity analysis by changing the values of the elasticity of Tobin's q and the size of habits. These results are available upon request.

Findings

This section presents a set of financial and macroeconomic observations for the artificial economy, and compares them to those of US postwar data. To compute quantities and asset prices, in a first stage we calculate nonlinear functions of the state variables using the nonlinear approach described in Judd (1992). Then, in a Returns are at quarterly frequency and in percent terms. The US numbers are estimates for the sample period 1947:1-2004:4. The simulated data corresponds to arithmetic averages of 500 replications of sample size 200. EP Equity premium, SR Sharpe ratio, [R.sup.f] riskfree rate, [[sigma].sub.x] standard deviation of variable x

second stage, we simulate the model by obtaining a random vector of technology shocks and calculating the series of asset prices and macroeconomic variables (7).

The Equity Premium and the Sharpe Ratio

Table I reports a set of financial statistics for different models. We start by focusing on the equity premium. A model without habits and without capital adjustment costs ('Standard') predicts an almost zero premium, thus giving rise to the equity premium puzzle. As opposite, the benchmark economy ('Benchmark') matches the empirical 1.66% premium on equity. To understand the channels of this success, consider the following formula (8),

[E.sub.t](1 + [EP.sub.t+1]) = - [Cov.sub.t]([M.sub.t+1], 1 + [R.sup.e.sub.t+1])/[E.sub.t]([M.sub.t+1]). (15)

The premium on any asset depends on how its returns commove with the marginal valuation of consumption. If the asset pays little in a recession--i.e., it fails to deliver wealth precisely when wealth is more valuable to investors--agents require a higher interest rate on it to compensate for the risk borne. This suggests that to induce large premia, a model should display (a) large upturns of the marginal utility of consumption at recessions, and (b) procyclical and volatile asset returns. Clearly, this is not the case of the standard model.

According to the above formula, a switch from power utility to habit persistence has two primary effects on the equity premium. First, it widens the spread across states of nature of the one-period-ahead marginal utility of consumption. For a given process of the return on equity, this raises its mean excess return. This effect is accounted for by the first term in the covariance formula and, following Boldrin et al. (1997), we will call it the curvature channel. The second term of the formula captures the capital gains channel: it measures the increase in the equity premium induced by the change of the structure in equity returns when we switch from power utility to habit forming preferences and hold the sequence of marginal rates of substitution constant.

Consider a model without capital adjustment costs. In this case, switching from power utility to habit forming preferences ('Habit') has a negligible impact on the risk premium. Under habit formation, sudden peaks in consumption are very harmful for future utility and, consequently, households adjust their savings decisions to avoid consumption risk. This smoothing channel minimizes the impact of the curvature channel, and the resulting equity premium is close to zero.

When we introduce capital adjustment costs in a model with standard preferences ('CAC'), the predicted premium rises to 0.19%, a value that is still too far from its empirical counterpart. With frictions in the capital market, adjusting the capital stock slowly costs fewer resources than doing so rapidly. Therefore, firms choose smooth investment paths, which translates into high consumption volatility. Then, it may seem surprising the failure of this model to generate a premium. However, it should not be so under the light of the covariance formula. Recall that it is the combination of the capital gains channel with the curvature channel that boosts the risk premium. Even though consumption is fairly more volatile under capital adjustment costs, the lack of curvature in the utility function removes a large fraction of the stocks' risk.

Now consider the benchmark economy. With frictions in the capital market consumption is very volatile, in spite of the attitude of households towards risk. This has two effects on the equilibrium risk premium. On the one hand, the marginal valuation of consumption becomes less predictable and more volatile, which ceteris paribus increases the risk premium through the curvature channel. On the other hand, households with habits have strong incentives to allocate resources in financial markets after a positive innovation. This drives stock prices up at the expansions and makes returns highly procyclical. As a consequence, the risk premium increases through the capital gains channel. Overall, the model's ability to generate a sizable premium comes from its ability to combine the curvature and the capital gains channel.

We can obtain further insights on the determinants of equity premia by analyzing Table 2. It contains information on the cyclical behavior of the two components of stock returns: prices and dividends. The first thing to note is that in all models returns are basically driven by variations in prices. In the benchmark economy, the fluctuation of prices generates large variations in the returns, thus making stocks a bad hedge against risk. In contrast, stocks are not sufficiently risky in the other models. In the 'Habit' and 'Standard' models the volatility of stock prices is too low.

In the 'CAC' model, prices are volatile and dividends are clearly procyclical. In this case, however, the lack of curvature in the utility function removes the riskiness of stocks.

The Sharpe ratio--defined as the ratio of the mean and standard deviation of the excess return on stocks--is an additional measure to test the consistency of financial models. The empirical Sharpe ratio (around 0.22) cannot be replicated by the 'Standard', 'Habit' and 'CAC' models (Table 1). In contrast, the benchmark economy hits upon the empirical value. To some extent, this comes as a surprise. The model economy was not tailored to match the risk-return trade-off in financial markets, but the equity premium. Interestingly, we find that by matching the equity premium, the model economy also matches the market price of risk or, at least, comes close to its empirical estimates (9).

The Riskfree Rate

Here, we focus on the ability of the model to match the first and second moments of the real interest rate. Following Mehra and Prescott's (1985) equity premium puzzle, Weil (1989) stated the riskffee rate puzzle: models with high consumption risk aversion face serious difficulties in reconciling historical consumption growth rates with (low) riskfree rates. However, as illustrated by Kocherlakota (1996), habit forming preferences can overcome this puzzle. At the core of this success is the fact that habits increase the household's demand for savings relative to the reference specification, thus decreasing the equilibrium interest rate.

In line with this evidence, we find that the benchmark economy can match the historical riskfree rate. Unfortunately, intertemporal variations in this rate are much larger in the artificial economy than in the data. As Table 1 shows, the predicted 3.70% standard deviation is too high as compared to the empirical 0.89%. As opposite, the standard model underpredicts variations of the riskfree rate by almost one order of magnitude. The intuition underlying these results is simple. In the standard case, the curvature of utility is low. Therefore, fluctuations in consumption levels do not translate into large fluctuations in the marginal valuation of consumption. This keeps the return on the riskless asset practically unchanged. In contrast, the curvature of utility is high in the benchmark economy and, thus, changes in consumption levels translate into large fluctuations in the riskfree rate.

Business Cycle Statistics

Given the relative success of the model economy on financial market predictions, it is worth investigating its macroeconomic implications. To that purpose, Table 3 reports a set of popular business cycle statistics.

Again, we can obtain useful insights by analyzing different scenarios. Consider a model without capital adjustment costs. In this case, switching from standard preferences to habit forming preferences reduces the volatility of consumption from 0.34 to 0.24. This change reflects the nature of habit forming households: they optimally choose smoother consumption plans to reduce the negative impact that current purchases have on the enjoyment of future consumption. To illustrate this, Fig. 1 reports the impulse of consumption to a positive, one-standard deviation shock to technology. While in the standard case consumption responds immediately after a shock, it peaks after some periods in the habit case. This clearly reflects the willingness of consumers to adjust their habit stock gradually over time.

Consider now the 'CAC' model. As Fig. 1 shows, with a concave capital accumulation equation firms choose flatter investment plans, and consumption absorbs most of the change in output caused by a shock to technology. As a consequence, the model's performance on the relative volatilities of aggregate variables is poor. It overstates the volatility of consumption--which is more volatile than output--and understates the volatility of investment--which is less volatile than output.

[FIGURE 1 OMITTED]

Finally, consider the benchmark case. The model economy can replicate salient business cycle features, such as the pro-cyclicality of consumption and investment as well as the ordering of volatilities: investment is more volatile than output, which in turn is more volatile than consumption. The reason for these findings has to do with the interaction between risk aversion and rigidities in the capital market. On the one hand, habit households use savings as a buffer against consumption risk, which translates into large fluctuations in the investment rate. On the other hand, the firm's decision is to avoid large fluctuations in the investment rate, due to the capital adjustment costs. Thus, part of the change in output is absorbed by consumers, who end up following volatile consumption paths despite their attitude towards risk. These two effects can be seen in Fig. 1.

As an additional finding, the model economy generates positive autocorrelation in consumption growth. The mechanism underlying this result is simple. Households are very averse to short term fluctuations in consumption and, therefore, today's purchases are influenced by yesterday's consumption. This dependence increases the autocorrelation of consumption growth from 0.02 in the standard model to 0.39, a value that comes close to the 0.31 found in the data. Unfortunately, the model economy lacks a mechanism to account for the persistence of output.

The match between the model economy and the observed data is good but clearly not perfect. Due to the capital adjustment costs, investment is almost 1.8 times less volatile in the model than in the data, while consumption is too volatile. It should be noted, however, that the consumption data comes from US consumption of services and nondurables, as usual. If we consider instead personal expenditure or durables, the observed volatility rises to 0.97 and 3.73, respectively (10). Thus, if we interpret the consumption good of the model as a combination of different types of consumption, then the model's prediction comes closer to the data. As a second shortcoming, investment and consumption are too procyclical. Still, the benchmark economy does better than the standard model.

The Correlation Between Consumption Growth and Stock Returns

In the data, consumption growth is positively correlated with past stock returns and negatively correlated with future returns. As Table 4 shows, the benchmark economy is consistent with this evidence. In contrast, the alternative models are not. The ability of the model economy to generate this pattern comes from the willingness of habit forming households to allocate resources in financial markets after a positive shock. This drives stock prices up, which ceteris paribus decreases future stock returns. Moreover, in the subsequent periods consumption rises gradually and households' demand for asset falls. This drives prices down, and returns on equity drop. Indeed, this effect is particularly strong and, as a consequence, the correlations between consumption growth and future returns are more decisively negative in the benchmark economy than in the data.

The contemporaneous correlation between returns and consumption growth is relatively high in the model economy. This suggests that to some extent households fear stocks because they pay bad when consumption is already low. Unfortunately, this explanation is much less evident in the data, insofar as indeed consumption and returns are poorly correlated. Notwithstanding this, the predicted 0.63 correlation constitutes a small improvement over endowment economies, where the joint determination of consumption and dividends generates a correlation close to unity.

Summary and Conclusions

In this paper, we explored asset returns in a production economy with habit forming households and capital adjustment costs. We found that the model economy can account for salient financial facts and business cycle phenomena simultaneously.

We used a formulation of habits that is novel among RBC models. We found that relative habits can account for the empirical equity premium, the Sharpe ratio, and the riskfree rate in a context of consumption choice. The challenge came from the fact that previous work by Abel (1990, 1999, 2006) was confined to a fruit-tree framework in which equilibrium consumption and dividends were exogenous stochastic processes.

As a shortcoming, the model overpredicts the volatility of the riskfree rate. This is a typical problem in models with habit formation. Campbell and Cochrane (1999) have shown that a more elaborated representation of habits can contribute to solve this anomaly. However, their achievements are based on a process for consumption that is exogenous and, therefore, can not be easily transferred to RBC models (11). Extending these advances into the endogenous consumption framework is a compelling task for future research.

As a second shortcoming, the model assumes that hours worked are fixed. Previous evidence suggests that allowing for labor decisions damages the financial predictions of these types of models. The reason is that agents can use labor movements to insure against consumption fluctuations. However, it is not clear how relevant this problem is. Uhlig (2007) gives reasons for optimism by showing that imposing an exogenous law of motion on wages can contribute to explain macroeconomic and financial facts simultaneously.

In the literature, there exist several and competing representations of time-nonseparable preferences. This paper focused on one of their most popular representations: relative habits. Our next step is to explicitly compare the financial and macroeconomic implications of the different specifications that have been proposed in order to disentangle which utility function can mimic more accurately the empirical facts.

The author thanks Antonia Diaz for fruitful discussions and Jose Victor Rios-Rull, Tim Kehoe, Xavier Raurich, Javier Diaz-Jimenez, Juan Carlos Conesa, Patrick Toche and seminar participants at BEMAD, the Carlos III Macroworkshop, Villa Mondragone Workshop, and the Universities of Modena, Malaga, and Basque Country for their comments

Published online: 22 July 2008

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S. Budria (mail) Department of Economics, University of Madeira, Rua Penteada, 9000-390 Funchal, Portugal e-mail: sbudria@uma.pt

S. Budria Applied Economics Center of the Atlantic (CEEAplA), Funchal, Portugal

(1) Drawing on this specification, Carroll et al. (2000) construct a growth model that accounts for the observed positive correlation between savings and growth, Fuhrer (2000) shows that a monetary policy model can capture the gradual hump-shaped response of real spending and inflation to various shocks, Diaz et al. (2003) ask whether Aiyagari's model of precautionary savings can account for the empirical wealth distribution when habits are incorporated, and Ravn et al. (2006) explore the demand function faced by habit forming individuals.

(2) See Carroll (2001) and Caballe et al. (2005) for a characterization of the optimal solution to the consumer's problem under this type of utility functions.

(3) Sometimes habits are said to be external. This corresponds to a "keeping up with the Joneses" utility function in which the reference consumption level depends on the economy-wide consumption. In this case, the habit stock is viewed by the household as evolving exogenously. Although the financial implications of internal and external habits are similar, they are associated to different levels of risk aversion. In particular, Boldrin et al. (1997) have shown that the coefficient of risk aversion in wealth is counterfactually high with "keeping up with the Joneses" utility functions, while it is reasonably low for internal specifications.

(4) See Boldrin et al. (2001) for the details.

(5) For instance, Fuhrer (2000) reports that the best achievements of his model occur when [gamma] =0.80. Diaz et al. (2003) use [gamma],=0.75 in their benchmark calibration. In Carroll et al. (2000), [gamma] ranges from 0.25 to 0.75.

(6) It is convenient to note that while in the standard utility case the curvature on utility, as measured by the Arrow-Pratt coefficient, is equal to the preference parameter [tau], with habit forming preferences it is jointly determined by [tau] and the size of habits [gamma]. Thus, for a given [tau], higher values of [gamma] imply higher curvature and, consequently, higher risk aversion. For our benchmark calibration, the Arrow-Pratt coefficient amounts to 33, a value that is 6.6 times larger than in the standard case. See Boldrin et al. (1997) and Budria and Diaz (2006) for closed-form solutions of alternative measures or curvature and risk aversion under habit forming preferences.

(7) A detailed description of the simulation procedure is available upon request.

(8) See Campbell and Cochrane (1995) for the details of the calculation.

(9) Estimations of the Sharpe ratio may differ, depending on the sample period and the characterization of the true market portfolio. Differences nevertheless are small. See Lettau and Uhlig (2002) for a set of alternative estimates.

(10) The numbers are calculated from NIPA data for the period 1959:1-2004:4.

(11) AS an additional drawback, the Campbell--Cochrane specification has anomalous implications that need to be solved. See Ljungqvist and Uhlig (2004).

Table 1 Financial statistics [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE EP SR [R.sup.f] IN ASCII] Data 1.66 0.22 0.25 0.89 Standard 0.00 0.01 3.78 0.11 Habit 0.00 0.03 2.11 0.09 CAC 0.19 0.05 3.66 0.64 Benchmark 1.66 0.22 0.25 3.70 Returns are at quarterly frequency and in percent terms. The US numbers are estimates for the sample period 1947:1-2004:4. The simulated data corresponds to arithmetic averages of 500 replications of sample size 200. EP Equity premium, SR Sharpe ratio, [R.sup.f] riskfree rate, [[sigma].sub.x] standard deviation of variable x Table 2 Moments of stock prices and dividends [MATHEMATICAL [MATHEMATICAL EXPRESSION EXPRESSION NOT NOT REPRODUCIBLE REPRODUCIBLE EP IN ASCII] IN ASCII] Standard 0.00 0.11 0.17 Habit 0.00 0.12 0.18 CAC 0.19 4.01 3.98 Benchmark 1.66 7.70 7.66 [[sigma]. [rho] (Y, sub.D] [R.sup.e]) Standard 3.28 -0.18 Habit 4.95 0.44 CAC 1.07 0.99 Benchmark 2.35 0.90 [rho] (Y, [P.sup.e]) [rho](Y D) Standard -0.01 -0.99 Habit 0.48 -0.96 CAC 0.99 0.99 Benchmark 0.90 -0.44 Returns are at quarterly frequency and in percent terms. Prices and dividends are filtered with the first-difference filter. The simulated data corresponds to arithmetic averages of 500 replications of sample size 200. EP Equity premium, [R.sup.e] equity return, [P.sup.e] equity price, D dividends, Y output, [[sigma].sub.x] standard deviation of variable x, [rho](x, k) the correlation between variable x and variable k Table 3 RBC statistics [[sigma]. [[sigma]. sub.C]/ sub.I]/ [[sigma]. [[sigma]. [[sigma]. [rho] sub.Y] sub.Y] sub.Y] (Y, C) Data 1.00 0.55 2.64 0.49 Standard 1.30 0.34 2.68 0.99 Habit 1.01 0.24 3.02 0.69 CAC 1.01 1.01 0.98 0.99 Benchmark 1.00 0.76 1.91 0.91 [rho] (Y, I) [rho](C) [rho](Y) Data 0.71 0.31 0.26 Standard 0.99 0.02 -0.02 Habit 0.99 0.70 -0.01 CAC 0.99 0.02 -0.02 Benchmark 0.92 0.39 -0.01 The US numbers are calculated from the NIPA and cover the sample period 1959:1--2004:4, GDP for output, consumption of nondurables and services for consumption, and fixed investment for investment. Variables have been filtered with the fast-difference filter. The simulated data corresponds to arithmetic averages of 500 replications of sample size 200. Y Output, C consumption, I investment, [[sigma].sub.x] the standard deviation of variable x, [rho](x, k) the correlation between variable x and variable k, [rho](x) the autocorrelation of variable x Table 4 Cross-correlations, consumption and returns [rho] ([DELTA][C.sub.t], [R.sup.e.sub.t+j]) -2 -1 0 +1 +2 Data 0.15 0.19 0.12 -0.10 -0.05 Standard 0.07 0.07 -0.09 0.39 0.36 Habit 0.35 0.52 0.77 0.61 0.55 CAC -0.02 -0.02 0.99 -0.03 -0.03 Benchmark 0.11 0.26 0.63 -0.46 -0.20 The Data row covers the sample period 1947:1--1995:4 and is taken from Campbell and Cochrane (1999). The reported statistics are based on raw, unfiltered data. The simulated data corresponds to arithmetic averages of 500 replications of sample size 200. [rho] ([DELTA][C.sub.t], [R.sup.e.sub.t+j]) The correlation between consumption growth at time t and stock returns at time t+j

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Author: | Budria, Santiago |
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Publication: | Atlantic Economic Journal |

Geographic Code: | 1USA |

Date: | Sep 1, 2008 |

Words: | 6362 |

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