Habit formation as a resolution to the equity premium puzzle: what is in the data, what is not.

1. Introduction

   The consumption based asset pricing model of Lucas |14~ defines a theoretical relationship between streams of consumption and equilibrium asset prices. Since data of both aggregate consumption and asset prices are available, the theory can be tested. Empirical tests of the Lucas model using standard time separable utility functions indicate mismatches between the theory and the data. For example, in the GMM (Generalized Method of Moments) estimation of Hansen and Singleton |9~, overidentifying constraints implied by the model were rejected. Mehra and Prescott |16~ demonstrated the difficulty of explaining a particular statistic: the theoretical expected equity premium (the yield differential between stocks and risk-free bonds) is much higher than the observed one if a standard utility function is used. They called the mismatch "the equity premium puzzle".

   To improve the performance of the model, several authors have relaxed the time-separability of preferences. In a stimulating paper, Constantinides |1~ argued that the equity premium puzzle can be resolved through the assumption of "habit formation". The idea is that consumption in the past reduces utility in the present because it establishes habits. His model can match the observed mean and the variance of both the equity premium and the consumption growth rate. The match of the low moments is consistent with the bound tests of Heaton |11~, Gallant, Hansen, and Tauchen |7~, and Hansen and Jagannathan |10~.

   On the other hand, GMM estimations using monthly aggregate consumption of nondurable goods and leisure by Eichenbaum, Hansen, and Singleton |3~, and that using consumption of nondurable goods and durable goods by Dunn and Singleton |2~ and Eichenbaum and Hansen |4~, showed that if current utility depends on current and past consumption, then current utility increases in past consumption, i.e., consumption has "local durability," the opposite of that implied by habit formation. A particularly strong result was obtained by Gallant and Tauchen |6~. They estimated a general form of utility functions with a general law of motion of data and found that the source of the time-non-separability is local durability. Since it is known that local durability produces a smaller equilibrium equity premium, these results imply that in order to match other moments well, the match for the equity premium has to be sacrificed.

   However, the GMM estimations on the nature of time-non-separability are not yet conclusive. Ferson and Constantinides |5~ illustrated that the fit of the model with multi-asset portfolios can be improved by either introducing habit formation or durability, and the chi-square statistics are not dramatically different in these two cases. They also found that the sample estimates are influenced by the choice of instrumental variables. Using the instruments they considered plausible, the GMM estimate indicates that the utility function displays habit persistence.

   There are many factors that affect outcomes of finite sample GMM estimations. One of them is the scaling factor. The actual level of consumption exhibits an upward trend, therefore it is not stationary. As a result, marginal utility of consumption is also non-stationary. Since the asymptotic theory can be expected to give a reasonable approximation under stationarity, researchers use a scaling factor to offset the trend of the marginal utility if the utility function to be estimated is homogeneous. The scaling factors are similar to ordinary instrumental variables except that by construction scaling factors explicitly depend on the parameters to be estimated, whereas ordinary instrumental variables do not. According to the consistency theorem of Hansen |8~, the instruments and scaling factors should not matter asymptotically. But like instruments, the scaling factors can affect finite sample estimates. The various scaling factors used by different authors may have a nontrivial impact, but such an impact has not been drawn out.

   The contribution of the present paper is two fold. First, GMM estimations are conducted using different scaling factors. The paper shows that scaling factors are important to finite sample estimates. Using a plausible scaling factor, our estimated utility function is locally durable. This implies that habit formation assumption cannot explain some moments other than the equity premium. Secondly, it demonstrates that even the equity premium puzzle is not solved by the introduction of habit formation because the correlation between aggregate consumption and the equity premium is too small. It is held that the model equity premium is large when the detrended marginal utility is volatile and when the correlation between marginal utility and equity premium is large. The calculation in the paper shows that comparing with the locally durable and time-separable preferences, the habit persistent preferences yield larger volatility in the marginal utility. So the habit formation assumption implies larger equity premium. But the paper also finds the correlation embodied in the data is so small that in order to match the observed equity premium, the volatility of marginal utility has to be much larger than that implied by strongly habit persistent utility functions.

   The remainder of the paper is organized as follows. Section II specifies the class of utility functions to be studied. Section III reports the results of GMM estimation using the equilibrium conditions on the asset returns and the marginal utility. Section IV demonstrates that in actual data, the correlation between the marginal utility and the equity premium is small under time separable preferences and is even smaller with habit persistent utility functions. Finally, section V concludes.

2. The Utility Function

   The representative agent's lifetime utility is assumed to be

   E |summation of~ ||Beta~.sup.t~ |(|c.sub.t~ + |s.sub.t~).sup.1-|Gamma~~ where t = 1 to |infinity~/(1 - |Gamma~) |Gamma~ |is greater than~ 0, (1)

   where |Beta~ is the discount rate, |c.sub.t~ is consumption at t and

   |s.sub.t~ = ||Theta~.sub.1~|c.sub.t-1~. (2)(1)

   In (2), if ||Theta~.sub.1~ = 0, the utility function becomes time-separable; if ||Theta~.sub.1~ |is greater than~ 0, the utility function shows local durability, which means consumption in the previous period and the present period are substitutes; and finally if ||Theta~.sub.1~ |is less than~ 0, the utility function shows habit formation, where a high level of consumption in the previous period changes the agent's habits, and the satisfaction the agent gains from the consumption of the current period depends on the difference between the present consumption and the habit. The marginal utility of consumption |c.sub.t~ divided by ||Beta~.sup.t~ is given by

   mr|u.sub.t~ = |(|c.sub.t~ + |s.sub.t~).sup.-|Gamma~~ + |E.sub.t~|Beta~||Theta~.sub.1~|(|c.sub.t+1~ + |s.sub.t+1~).sup.-|Gamma~~. (3)

   Equilibrium conditions in the market for assets imply that

   |Mathematical Expression Omitted~

   and

   |Mathematical Expression Omitted~

   where |Mathematical Expression Omitted~ is the realized return of the risky asset from period t to t + 1, and |Mathematical Expression Omitted~ is the return of the risk-free bond from period t to t + 1. The latter rate is assumed known with certainty at time t, whereas the former one is not. The parameters we wish to estimate are |b.sub.0~ = {|Beta~, |Gamma~, ||Theta~.sub.1~}.

3. GMM Estimation

   Define |Mathematical Expression Omitted~ as mr|u.sub.t~ without the conditional expectation operator on the second component of (3), i.e., |Mathematical Expression Omitted~ is the realized marginal utility. Denote the vector of parameters {|Beta~, |Gamma~, ||Theta~.sub.1~} by b. Given a scaling factor S|F.sub.t~(b), expressions similar to (4) and (5) can be rewritten as

   |Mathematical Expression Omitted~,

   where |Mathematical Expression Omitted~; and

   |Mathematical Expression Omitted~,

   where |Mathematical Expression Omitted~.

   Equations (4) and (5) also imply that in (4|prime~) and...