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Consumption and asset prices an analysis across income groups.

Abstract

Employing aggregate consumption data to test the consumption-based capital asset pricing model (CCAPM) is likely to lead to a specification error since a significant portion of consumers live from paycheck-to-paycheck and, therefore, are constrained in their ability to intertemporally allocate consumption. Furthermore, these consumers lack the savings needed to directly influence an equilibrium between consumption expenditures and asset returns. Using consumption expenditures grouped by consumer income, this paper examines the issue of whether the CCAPM is more consistent with the consumption of unconstrained (high-income) consumers as compared to constrained (low-income) consumers. Several traditional methods of analyzing the CCAPM are explored utilizing five time series of consumption expenditures delineated by consumer income. This approach allows us to indirectly test whether liquidity constraints affect the CCAPM without imposing additional specification on the model. Overall, the tests fail to find any discernible patterns across income groups that are consistent with the idea that liquidity constraints bind lower income consumers. [C] 2002 Elsevier Science Inc. All rights reserved.

1. Introduction

The theoretical models that bridge microeconomics and finance have been the subject of numerous empirical studies which, for the most part, find an inconsistent relationship between US consumption data and asset returns. (1) The inconsistency stems from the finding that equity risk, as measured by the covariance between stock returns and consumption growth within the context of the consumption-based capital asset pricing model (CCAPM), is too small to warrant the large average risk premium on stocks. In other words, the CCAPM shows that aggregate investors, or the representative single agent, must be extremely averse to consumption risk (corresponding to a large coefficient of relative risk aversion) to demand such a large equity premium. The disparity between the predictions of these models and the data are so pronounced that the literature has continued to refer to this phenomenon as the "equity premium puzzle," based on the work of Mehra and Prescott (1985), who first brought the issue to light.

Attempts to rescue the CCAPM or solve the equity premium puzzle are bountiful. They seek to unravel the assumptions employed by Mehra and Prescott (1985) in hopes of explaining this puzzle. The assumptions, as well as the literature attacking these assumptions, are typically categorized into three (not necessarily mutual exclusive) areas: consumer preferences, the representative agent, and market frictions. In brief, papers focusing on the consumer preference assumption seek to generalize the very convenient time-separable power utility function employed by Mehra and Prescott. Papers focusing on the representative agent assumption examine the implications of using aggregate consumption data to test the CCAPM. And finally, papers studying market frictions examine transaction costs, the implication of incomplete markets, and economically constrained consumers.

The purpose of this paper is to examine the implications of economically constrained and unconstrained consumers on the standard CCAPM. It therefore spans the area of surrounding the representative agent assumption, as well as, market frictions (see Campbell, Lo, & MacKinlay, 1997, p. 317 for an in-depth review of these issues). More specifically, this paper seeks to determine whether real asset returns are more consistent with the consumption patterns of higher income consumers than lower income consumers. The hypothesis is that the consumption patterns of lower income consumers, who live from paycheck-to-paycheck, should bear a weaker relation to real asset returns than higher income consumers since they lack the ability to save and manipulate consumption. This hypothesis suggests that traditional tests of the CCAPM based on national per capita consumption are misspecified since a significant portion of consumers either do not save or have negligible savings rates. Furthermore, even if these consumers have some savings, their awkward economic position would indicate that they probably do not manipulate their consumption patterns based on expected real asset returns. An important element of the tests employed in this paper is that they do not impose any form of liquidity constraint on the CCAPM, thus eliminating the possibility of additional misspecification.

The paper is organized as follows. Section 2 reviews the CCAPM literature and its relation to the equity premium puzzle of Mehra and Prescott (1985). Section 3 covers the tests used to measure the differences in the relationship between consumption growth and asset returns for low-income and high-income consumers. Specifically, the generalized method of moments (GMM) is employed to exploit the orthogonality conditions implied by the model. In Section 4, the data are described and the coefficients of relative risk aversion across income groups are estimated using methods that rely on rather strong distribution assumptions for consumption growth and asset returns as well as distribution-free methods such as the GMM. The overall results indicate that there appears to be no clear pattern between consumption and asset returns across income quintiles. In other words, the consumption-based CAPM specification does not unambiguously become stronger as income rises. Consequently, categorizing consumption expenditures b y income quintiles appears to provide little, if any, information about the equity premium puzzle in that the relationship between consumption and asset returns shows no discernible systematic patterns. Section 5 concludes.

2. Background

2.1. Literature review

The lack of consumption data sorted by consumers' income level makes testing of the constrained/unconstrained hypothesis difficult, and consequently, there appear to be few papers that directly address this issue within the context of the CCAPM. For example, Mankiw and Zeldes (1991) examine this issue indirectly, without consumption sorted by income. They examine a unique data set that distinguishes between the consumption of stockholders and nonstockholders. (2) Their objective is to test the hypothesis of whether stockholders' consumption is more consistent with asset returns as compared to nonstockholders. Their data set, however, has severe limitations in that it only covers food consumption. The use of food consumption has both positive and negative implications for the CCAPM. On the positive side, food consumption probably comes closest to matching the assumption of time-separable utility since consumers are assumed not to store goods intertemporally under the standard power utility function. On the ne gative side, Attanasio and Weber (1995) indicate that food is considered a necessity and is not likely to change proportionally to income changes. Furthermore, the use of only food consumption data requires the assumption that the utility derived from food is separable from the utility derived from other goods and services.

Mankiw and Zeldes (1991) conclude that the food consumption of stockholders differs from that of nonstockholders. Food consumption data of stockholders are more volatile and consistent with the CCAPM. Specifically, they find that food consumption is more highly correlated with excess returns of the S&P 500 over 3-month T-bills. Their conclusions are tenuous, at best, however. Mankiw and Zeldes's work is plagued by severe data limitations not only because they use just food consumption data, but also because they have only 13 observations in which to draw their conclusions.

In this paper, we attempt to provide more conclusive evidence concerning the relationship between consumption and investment behavior. (3) Specifically, we examine the relationship between consumption spending and asset returns across income classes as designed by the Consumer Expenditures Survey: Quarterly Data from the Interview Survey (CES), published by the Bureau of Labor Statistics (BLS). The CES provides 51 quarterly observations across five different income groups.

The BLS publishes household consumption and income data quarterly. (4) Income is divided into quintiles where the first quintile corresponds to consumers with the lowest 20% of reported income and the fifth quintile denotes the top 20% of consumers in terms of reported income. The assumption made in this paper is that there are five independent representative economic agents in the economy, one corresponding to each income quintile. Each agent faces the same set of expected asset returns; however, the lowest quintile representative agent is hypothesized not to invest since he or she lives from paycheck-to-paycheck and hence has no savings. Representative agent 2, corresponding to the second income quintile, may have some savings, but is generally limited in his or her ability to consistently invest in assets. The other representative agents can be categorized accordingly, with the fifth agent consistently saving income depending on expected asset returns. The preferences of each agent are assumed to be chara cterized by the power utility function.

An examination of consumption across higher income quintiles should show stronger and more consistent results with the CCAPM, since higher income consumers are assumed to have the capacity to alter consumption patterns with the use of financial assets. The representative agent corresponding to the lowest income quintile, on the other hand, does not have the luxury of manipulating consumption patterns intertemporally, and therefore, has no direct influence on asset returns. Since there is a significant proportion of consumers in the United States living from paycheck-to-paycheck, such a finding may be relevant to the equity premium puzzle and the general failure of the consumption-based CAPM based on national per capital consumption.

2.2 Consumption-based capital asset pricing model

A brief review of the consumption-based CAPM and its assumptions will provide the background necessary to understand the issues involved. The equity premium puzzle was first brought to light my Mehra and Prescott (1985) and focuses on the tightly parameterized intertemporal equilibrium imposed by a standard CCAPM. The standard Euler equation is

u' ([C.sub.t]) = [beta][E.sub.t][(1 + [R.sub.i,t+1])u'([C.sub.t+1])] (1)

where [beta] is the rate of time preference. Higher values of [beta] imply that consumers place a higher value on future consumption, meaning that they would prefer to save today and to consume tomorrow. The term [R.sub.i,t + 1] denotes the expected real return on asset i for the period ending t + 1. Assuming that the model is tested with some measure of aggregate consumption, u'([C.sub.t]) denotes the marginal utility from aggregate consumption measured at time t. Eq. (1) states that the representative agent will forego a dollar of consumption today and invest it in asset i so long as the expected additional discounted marginal utility that it provides is greater than today's marginal utility. Eq. (1) is parameterized with a state-independent and time-separable power utility function (5)

u([C.sub.t]) = [C.sup.1-[gamma].sub.t]/1 - [gamma] (2)

where [gamma] > 0 is the constant relative risk aversion coefficient (CRRA). When [gamma] = 1, u([C.sub.t]) = log([C.sub.t]). Marginal power utility in this case is u'([C.sub.t]) = [C.sup.-[gamma].sub.t] so that when Eqa. (1) and (2) are combined, the first-order conditions are

1 = [beta][E.sub.t] [(1 + [R.sub.i,t+1]) [([C.sub.t+1]/[C.sub.t]).sup.-[gamma]] (3)

where the term ([C.sub.t + 1]/[C.sub.t]) denotes consumption growth. An economically plausible [beta] coefficient should be loss than 1.0 indicating that consumers discount future consumption. According to Mehra and Prescott (1985), economically plausible estimates for [gamma] are greater than zero and less than or equal to 10. (6) Coefficients of relative risk aversion that are around 10 or more suggest extremely risk-averse consumers who prefer low consumption growth (less volatile consumption over time). Coefficients of less than zero contradict Eq. (2) by indicating nonconvex preferences (increasing marginal utility) and the nonexistence of an intertemporal equilibrium for asset prices.

Nearly all of the empirical studies that examine Eq. (3) suffer from serious data limitations: the weight of these studies employ aggregate nondurable goods and/or services consumption data from the National Income and Product Accounts (NIPA) published by the Bureau of Economic Analysis (BEA). By assuming homogeneous preferences and identical investment portfolios, a representative agent is said to exist and allows Eq. (3) to be tested using per capita consumption data. However, consumption data in the aggregate are very smooth over time causing many to wonder whether the aggregation process eliminates the volatility needed to satisfy the restrictions imposed by the Euler equation (3). This may be one explanation for the equity premium puzzle.

Eqs. (1-3) assume that all consumers (the representative agent) invest in financial assets. However, there is a significant portion of lower income consumers who are constrained in their ability to invest since they tend to live paycheck-to-paycheck. These consumers have no direct impact on the return of financial assets, and suggests that a significant proportion of national per capita consumption may not bear a direct relation to real asset returns. Empirical tests of Eq. (3) using national per capita consumption, therefore, are likely to fail. In this context, the failure of the model may help explain the equity premium puzzle in that lower income groups may be distorting the economic relationship consumption and asset returns.

3. Generalized method of moments

Hansen and Singleton (1982) test Eq. (3) by employing Hansen's (1982) GMM

Q([theta]) = [g([[theta].sub.T])]'[W.sup.-1.sub.T] [g([[theta].sub.T])]. (4)

GMM selects the parameter vector [[theta].sub.T] = ([beta], [gamma])' so that the sample average of Eq. (3), stacked by asset and denoted by g([[theta].sub.T]), minimizes Eq. (4). (7) [W.sup.-1.sub.T] denotes an [N.sub.r] x [N.sub.r] optimal weighting matrix so that the GMM estimates are consistent and asymptotically normal. Q([theta]) is [chi square] distributed with ([N.sub.r] - [N.sub.[gamma],[beta]]) degrees of freedom, where [N.sub.r] represents the number of orthogonal conditions and [N.sub.[gamma],[beta]] the number of parameters estimated in Eq. (4), respectively. If the number of orthogonal conditions exceeds the number of parameters to be estimated, Q([theta]) represents a test statistic for the overidentifying restrictions imposed on the model. To start the process of minimizing Q([theta]), the initial weighting matrix is the identity matrix. An iterating process is employed which results in an optimal weighting matrix that also is used to estimate the standard errors of the GMM parameter estimates. The main advantage to using the GMM is that it makes no assumptions concerning the distribution of real asset returns and consumption.

While the parameters in the GMM do not rely to distribution assumptions, Hansen and Singleton (1983), nevertheless, discuss the empirical difficulties and possible measurement errors in attempting to specify the exact timing of each variable in Eq. (3). Consequently, they develop an alternative method to estimating Eq. (3) by assuming that asset returns and consumption data are jointly homoskedastic and lognormal. By transforming Eq. (3) in log form, they derive the following expression that relates the log real excess return on asset i to the coefficient of relative risk aversion times the covariance between the log real returns on asset i and log real consumption

[E.sub.t][[r.sub.i,t+1] - [r.sub.f,t+1]] + [[sigma].sup.2.sub.i]/2 = [gamma][[sigma].sub.i,c] (5)

The term [[sigma].sup.2.sub.i]/2 is a Jensen's inequality adjustment and lower case variables are in log form. All variables are readily determined by the data, except for [gamma] which is solved for algebraically.

A third method of testing the relationship between asset returns and consumption growth is instrumental variables (IV) regression estimation. Following Hall (1988) and Hansen and Singleton (1983), Campbell et al. (1997) examine the following IV (two-stage) regression

([C.sub.t+1]/[C.sub.t]) = [v.sub.i] + [pi][r.sub.i,t+1] + [[xi].sub.i,t+1] (6)

where [[xi].sub.i,t + 1] = [pi]{([E.sub.t][[r.sub.i,t + 1]] - [r.sub.i,t + 1]) + [gamma]([C.sub.t + 1]/[C.sub.t]) - [gamma][E.sub.t][[C.sub.t + 1]/[C.sub.t]]} and [pi] denotes the asymptotic reciprocal of the coefficient of relative risk aversion. The first stage regresses consumption growth onto a constant, real Treasury bill rates, real S&P 500 total returns, and real consumption growth. Each dependent variable is lag 1 or lag 1 and 2. The second stage regresses real Treasury bill rates onto a constant and the fitted consumption growth values from the first-stage regression. Since [pi] represents the reciprocal of the coefficient of relative risk aversion, economically plausible estimates should lie in the range .1 and higher.

4. Data and empirical results

4.1. Data

All data are sampled quarterly from first quarter 1984 to third quarter 1996. The consumption and income data are from two separate sources, the CES published by the BLS and NIPA published by the BEA. The CES is produced with the goal of periodically revising the Consumer Price Index (CPI) and is collected in independent samples that are representative of the United States population. The survey is composed of two separate parts that are later integrated: a diary survey and interview survey. The diary captures small and frequently purchased items, whereas the interview survey captures large and "regularly recurring" purchased items. With the interview survey, a rotating panel method is employed where each panel is interviewed every 3 months for five consecutive quarters then dropped so that about 20% of the consumer units are new each quarter. Approximately 5000 consumer units are sampled with about 95% of all expenditures covered. (8)

The CES reports both total income before incomes taxes and expenditure data by quintiles based on complete income reporters. Complete income reporters are ranked in ascending order by total income before taxes reported by the unit consumer. (9) Since the models employed in this paper depend upon time-separable utility functions, only nondurable goods and services (NDS) and food are examined, rather than longer lasting durable goods. Blinder and Deaton (1985) argue that NDS is a better time series than durable goods and the use of NDS and food also follows a convention generally employed by the literature.

The CES does not report NDS separately, and must be estimated. In keeping with the spirit of the NIPA, NDS are computed from the CES by subtracting an estimate of durable goods expenditures from total expenditures. Durable goods are considered to be the CES line items "House Furnishing and Operations" plus a fraction of "Other Transportation Expenses" which includes expenditures on motor vehicles and parts. (10) The derived NDS series includes food expenditures; however, for comparative purposes with Mankiw and Zeldes (1991), food expenditures also are examined separately. Food expenditures are a separate line item in both the CES and the NIPAs.

The CES data has several, possibly severe, limitations. First, the data used in this paper exclude consumer units that failed to report their income, but nevertheless responded to expenditure questions. In this paper, it is assumed that the impact from this under-reporting of income uniformly affects each quintile so that there are no systematic biases from one quintile to another. Second, the CES only surveys about 7000 consumer units per year which is relatively small compared to other national economic surveys. Finally, according to the BLS, the data are preliminary and may be subject to minor revisions.

Reliable consumer expenditure data, sorted by income, are notoriously difficult to obtain for extended sample periods. The only other conventional data set used in this type of analysis, besides the CES, is the Panel Study of Income Dynamics (PSID). However, the only consumer expenditure line item sorted by income is food making it substantially less appealing than the CES. CES data are used in many papers including Attanasio and Weber (1995), Caroll (1992), and Lusardi (1992).

The implicit price deflator for NDS from the BEA is used to adjust all nominal variables. The index is set equal to 100 in the first quarter of 1984. Per capita figures for the CES are computed by dividing average household NDS consumer expenditures by the number of unit individuals in the household. The BEA publishes per capita expenditures by dividing aggregate NDS expenditure by population estimates published by the Bureau of Census.

The real interest rate is computed from the nominal monthly auction-average for the 3-month US and inflation based on the NDS implicit price deflator. The T-bill data are published by the Board of Governors of the Federal Reserve in Statistical Release H. 15. Total stock returns are based on the S&P 500 index with dividends immediately reinvested. Following Campbell et al. (1997), the log dividend to price ratio is employed as an instrument in the IV regressions. This ratio is constructed using S&P 500 dividends divided by the beginning of the period S&P 500 index.

The timing of the data implicitly assumes that consumers make economic decisions that coincide with the sampling interval of the data. Consumers make their consumption decisions at the beginning of the current quarter, based on knowledge of their investment returns for that quarter (see Campbell et al., 1997, p. 308). The 3-month T-bill returns are from the first month of the current quarter (as consumers earn this rate during the quarter) and total stock returns are based on the current quarter ending S&P 500 index (including reinvested dividends). Consumers are assumed not to intertemporally store purchased items.

4.2. Results

The GMM results based on Eq. (4) are presented in Table 1 for NDS and Table 2 for food consumption. The instruments used for both tables are lagged and represent information that is easily accessible to consumers. They are real consumption growth, the real risk-free rate of return, the real total return on the S&P 500, and a constant. (11) The coefficients of constant relative risk aversion, which should lie within the range of 0 < [gamma] [less than or equal to] 10, are frequently negative, but statistically insignificant. Furthermore, the model appears to show larger (less negative) coefficients for wealthier consumers (consumers in Quintiles 4 and 5) in all three lag structures, but are statistically insignificant in most cases. The rates of time preference, which should be less than 1.0, are consistent with the model and are statistically significant. The J-statistics, generally fail to reject the overidentifying restrictions on the model with no real tendency for the model to perform better for higher inco me consumers. (12)

Consequently, the model fits the data reasonably well; however, it suggests that consumers are risk neutral and there is no strong pattern confirming the constrained/unconstrained hypothesis.

It is worth mentioning that these results are consistent with other studies. For example, our economically implausible estimates for the CRRA are consistent with other studies such as Campbell et al. (1997, p. 312) and Hansen and Singleton (1982, 1983 Errata). In addition, our plausible and statistically significant time preference estimates are consistent with Hansen and Singleton.

With the NIPA data in Table 1, the estimated CRRAs are positive and statistically significant at the 5% level for lags 2 and 6 and significant at the 10% level for lag 4. The time preference factors are also in line with expectations. The J-statistics generally fail to reject the overidentifying restrictions and vary considerably across income levels and lag lengths.

The GMM results for the food consumption data from the CES in Table 2 are mixed overall. The CRRAs are negative and insignificant at the lag 2 specification. For lags 4 and 6, the CRRAs are positive and significant in several cases; however, these results tend towards the lower income quintiles, thereby providing less support for the constrained/unconstrained consumer hypothesis. The [beta] coefficients, like the NDS results, all fall within the economically plausible range of less than 1.0. The J-statistics fail to reject the overidentifying restrictions on the model and show no real pattern supporting the constrained/unconstrained hypothesis. For the NIPA data in Table 2, the lags 4 and 6 show positive and statistically significant CRRA coefficients. The [beta] coefficient is consistent with the model, while the J-statistics fail to reject the overidentifying restrictions.

Overall, for both the NDS data and the food data, the GMM estimates provide mixed results for Eq. (3). The parameter estimates for the CRRA are questionable yet the time preference parameter appears economically reasonable. The overidentifying restrictions tend to show no real pattern towards confirming the constrained/unconstrained hypothesis. For food consumption, the results across quintiles are not consistent with the constrained/unconstrained consumer hypothesis especially at lag 6 -- the results contradict the hypothesis by showing positive statistically significant CRRAs in low-income quintiles. The J-statistics indicate only a slight pattern toward rejecting the model. The NIPA data are much more consistent with the model's predictions on a coefficient by coefficient basis and the overidentifying restrictions are not rejected.

The GMM estimates presented are very sensitive to the timing assumptions made when specifying consumer behavior relative to financial asset returns. In an effort to sidestep these potential measurement errors, Table 3 presents estimates of the implied CRRA based on Eq. (5) for NDS. The constrained/unconstrained hypothesis suggests that the CRRA should decrease as income increases. While the results tend to show that the implied CRRAs corresponding to the top two quintiles are smaller than the bottom two quintiles, the estimates do not decrease steadily in progressively higher income groups. The most notable exception to the hypothesis is Quintile 3 which produces a negative implied CRRA. While the overall results for this test are somewhat consistent with the constrained/unconstrained hypothesis, the inconsistencies across the income groups weaken any strong conclusions that can be made.

Table 4 presents the implied CRRA statistics for food consumption. Overall, the CRRAs for the CES data are somewhat more consistent with the lognormal/power utility model (5) than NDS data in Table 3. The exception is the fourth income quintile with a CRRA coefficient equal to -- 83.6. The results shown in Table 4 make it difficult to discern any patterns across income quintiles that are consistent with the constrained/unconstrained hypothesis. The two extreme income quintiles, however, with CRRAs of 7.8 and 7.92, are within the range (less than 10) that Mehra and Prescott (1985) consider economically plausible. The results for the NIPA food consumption data show an implausibly high CRRA of 334.6.

The instrumental variable results are presented in Table 5 for NDS and Table 6 for food consumption. (13) Overall, the IV estimates reject the CCAPM. The instruments used are the lagged real 3-month T-bill rate, the lagged real total return on the S&P 500, lagged real consumption growth, and the lagged log dividend to price ratio. (14) The NDS results in Table 5 for both the CES and NIPA data show that the estimated [pi] coefficients are generally positive with the exception of the S&P 500 instrument for lags 1 and 2. Since [pi] is the reciprocal of [gamma] (the CRRA), economically plausible estimates range above .1 since smaller estimates of [pi] imply larger [gamma]. All of the estimates for [pi] are statistically insignificant so that a meaningful interpretation is difficult to develop within the context of the constrained/unconstrained consumer hypothesis. The [R.sup.2] statistic measures the joint explanatory ability of the instruments about the residuals from the first stage. The [R.sup.2] statistics s hould be small since the instruments should be uncorrelated with the residuals from the IV regression. The significance levels for the test statistic (T x [R.sup.2]) are in brackets. The results generally fail to reject the overidentifying restrictions in many instances at the 10% significance level, but there is no discernable pattern among the CES data other than most of the rejections that occur for the T-bill instrument with 1 and 2 lags. For the NDS NIPA data, the results for [pi] indicate that they are all statistically insignificant; however, the low [R.sup.2]s and their corresponding high test statistics indicate that the overidentifying restrictions are not rejected.

Table 6 presents the results for food consumption. For the CES data, [pi] is frequently negative and in all cases is statistically insignificant. The overidentifying restrictions are generally not rejected at the 10 percent level for one lag, but are generally rejected for lags 1 and 2. For the NIPA data, again, the [pi] are small and positive, but statistically insignificant. The data fail to reject the overidentifying restrictions of the model.

5. Conclusion

Utilizing US aggregate consumption data poses problems for testing the CCAPM since lower income (constrained) consumers live paycheck-to-paycheck, and therefore do not influence asset returns directly through savings. Most CCAPM research, however, neglects to consider this notion. Consequently, this paper seeks to determine whether the CCAPM is more consistent with the consumption of higher income consumers than with the consumption of lower income consumers. An important aspect of this paper is that it eliminates the possibility of additional misspecification error by not modeling a liquidity constraint.

Overall, using consumption expenditure data split into quintiles based on consumer income, this paper shows that the CCAPM behaves erratically across income groups. Since we have not employed a specific model for the liquidity constraint, the rejections found in this paper are a function of the data rather than a misspecification of the constraint itself. The distribution-free GMM tests performed fails to show any discernible patterns across income groups for both NDS and food consumption, but in general, the data fail to reject the model. The implied coefficient of relative risk aversion exercise, which assumes that consumption and asset returns are jointly homoskedastic and lognormal, shows results that are slightly more consistent with the constrained/unconstrained consumer hypothesis. Any conclusions based on these results, however, would be untenable since the results are not consistent across income groups. Finally, the estimated coefficients under the instrumental variables tests are generally statist ically insignificant so that conclusions about consumption patterns across income groups are absent. Our constrained/unconstrained consumer analysis, therefore, fails to shed light on the equity premium puzzle.
Table 1

Generalized method of moments. Nondurable goods and services consumption
growth by income. First quarter 1984 to third quarter 1996

                                  Consumer expenditures survey


NLAG                   CG1           CG2              CG3       CG4

2         [gamma] (a)   -.1343       - .0930            .0018    -.0222
                        (.1017)       (.0725)          (.0280)   (.0162)
          [beta] (a)     .9833         .9831            .9831     .9839
                        (.0047)       (.0032)          (.0018)   (.0018)
          J(12)         5.379        14.119           14.256    11.343
                        [.9441]       [.2932]          [.2846]   [.4997]
4         [gamma] (a)   -.0094        -.0038           -.0061    -.0105
                        (.0073)       (.0130)          (.0085)   (.0088)
          [beta] (a)     .9777         .9826            .9840     .9823
                        (.0013)       (.0015)          (.0015)   (.0014)
          J(24)        27.512        21.437           24.836    20.777
                        [.2812]       [.6128]          [.4147]   [.6519]
6         [gamma] (a)   -.0058        -.0862            .0094    -.0177
                        (.0060)       (.0149)          (.0068)   (.0052)
          [beta] (a)     .9746         .9708            .9732     .9831
                        (.0008)       (.0011)          (.0011)   (.0005)
          J(36)        34.076        37.461           36.257    29.799
                        [.5604]       [.4020]          [.4566]   [.7573]

           Consumer
          expenditur
          es survey

                    NIPA
NLAG      CG5       NCG

2          -.0143     .6982
           (.0059)   (.2957)
            .98367    .9878
           (.0017)   (.0049)
          11.548    12.477
           [.4826]   [.4081]
4          -.0041     .1330
           (.0051)   (.0706)
            .9804     .9810
           (.0013)   (.0019)
          23.614    25.023
           [.4838]   [.4045]
6          -.0034     .4128
           (.0041)   (.0935)
            .9772     .9680
           (.0007)   (.0012)
          30.980    40.046
           [.7061]   [.2953]

[gamma] is the coefficient of relative risk aversion where economically
plausible estimates are suggested to be between 0 and 10, [beta]
is the rate of time preference. Smaller values of [beta] imply
that consumers place smaller weight on future events.
J{2(3NLAG + 1) - 2} is Hansen's test statistic for the number of
overidentifying restrictions implied by the model and is chi-square
distributed. The term NLAG represents the number of quarters in which
the instruments are lagged. The instruments are log consumption growth,
the log risk free rate of return, the log total return on the S&P 500,
and a constant.

(a) Asymptotic standard errors are in parentheses. P values for the [chi
square] statistic are in brackets; larger values suggest stronger
evidence against the model.
Table 2

Generalized method of moments. Food consumption growth by income. First
quarter 1984 to third quarter 1996

                                Consumer expenditures  survey   NIPA
NLAG                       FCG1        FCG2           FCG3      FCG4

2         [gamma] (a)   -.0160      -.0280         -.0370    -.0469
                        (.0098)     (.0303)        (.0224)   (.0202)
          [beta] (a)     .9834       .9826          .9827     .9826
                        (.0019)     (.0020)        (.0019)   (.0020)
          J(12)         9.501      12.140         10.431     9.473
                        [.6597]     [.4345]        [.5782]   [.6621]
4         [gamma] (a)   -.0038       .0063          .0414    -.0239
                        (.0045)     (.0112)        (.0152)   (.0098)
          [beta] (a)     .9832       .9848          .9856     .9799
                        (.0011)     (.0012)        (.0014)   (.0012)
          J(24)        21.257      24.435         21.258    21.210
                        [.6235]     [.4370]        [.6235]   [.6263]
6         [gamma] (a)    .2534       .0675         -.0260    -.0333
                        (.0186)     (.0150)        (.0084)   (.0059)
          [beta] (a)     .9452       .9663          .9793     .9796
                        (.0028)     (.0012)        (.0009)   (.0009)
          J(36)        37.252      39.000         28.109    28.478
                        [.4113]     [.3364]        [.8233]   [.8097]

           Consumer
          expenditur
          es  survey    NIPA
NLAG          FCG5      FNCG

2          -.0049    -.2188
           (.0072)   (.3791)
            .9833     .9822
           (.0014)   (.0035)
           7.368    13.137
           [.8324]   [.3592]
4          -.0049     .0781
           (.0072)   (.0955)
            .9833     .9796
           (.0014)   (.0015)
          22.681    20.725
           [.5387]   [.6549]
6          -.0015     .2517
           (.0037)   (.0236)
            .9795     .9821
           (.0009)   (.0004)
          32.029    33.539
           [.6580]   [.5862]

[gamma] is the coefficient of relative risk aversion where economically
plausible estimates are suggested to be between 0 and 10. [beta] is the
rate of time preference. Smaller values of [beta] imply that consumers
place smaller weight on future events. J{2(3NLAG + 1) - 2} is Hansen's
test statistic for the number of overidentifying restrictions implied by
the model and is chi-square distributed. The term NLAG represents the
number of quarters in which the instruments are lagged. The instruments
are log consumption growth, the log risk free rate of return, the log
total return on the S&P 500, and a constant.

(a) Asymptotic standard errors are in parentheses. P values for the [chi
square] statistic are in brackets; smaller values suggest stronger
evidence against the model.
Table 3

Summary sample statistics for nondurable goods and services consumption
growth by income, stock returns, real returns, and risk premium. First
quarter 1984 to third quarter 1996

                                Consumer expenditures survey

                             CG1           CG2         CG3      CG4

Mean                       .0006         .0035       .0085    .0092
Standard deviation         .2348         .1881       .1861    .1848
Correlation (CG, SR)       .0243         .0138      -.0223    .1921
Covariance (CG, SR)        .0016         .0007      -.0012    .0101
Implied CRRA             83.6         191.2      -111.5     13.2

                       Consumer
                      expenditur
                      es survey
                                    NIPA
                          CG5        NCG  SR        RR        RP

Mean                    .0102      .0162  .1137     .0221     .0916
Standard deviation      .2167      .0244  .2906     .0169     .2890
Correlation (CG, SR)    .0499      .0814
Covariance (CG, SR)     .0031      .0006
Implied CRRA          43.2      223.0

Consumption growth is the annualized change in log real per capita
consumption of nondurable goods and services. Data from the Consumer
Expenditure Survey are split into quintiles by income rank, where CG1 is
the consumption growth associated with the lowest income quintile
through CG5, the highest income quintile. SR is the annualized log of
the real total return to the S&P 500. RR is the real risk-free rate of
return estimated from the 3-month T-bill. RP is the market risk premium.
The implied coefficient of relative risk aversion, CRRA, assumes power
utility with joint conditional lognormality of asset returns and
consumption, and is computed as CRRA = (RP + .5[[sigma].sup.2.sub.SR]/
([[sigma].sub.CG,SR]). See Eq. (5) where CRRA = [gamma] and RP =
[r.sub.i] - [r.sub.f].
Table 4

Summary sample statistics for food consumption growth by income, stock
returns, real returns, and risk premium. First quarter 1984 to third
quarter 1996

                              Consumer expenditures survey

                      FCG1               FCG2           FCG3

Mean                  - .0065            - .0027        - .0070
Standard deviation      .2549              .1756          .1561
Correlation (FCG,SR)    .2368              .0966          .1426
Covariance (FCG,SR)     .0172              .0048          .0063
Implied CRRA           7.8               27.9           21.2

                      Consumer expenditures
                             survey
                                           NIPA
                      FCG4       FCG5      FNCG      SR        RR

Mean                    - .0055  - .0034      .0085  .1137     .0221
Standard deviation        .1614    .2022      .0275  .2906     .0169
Correlation (FCG,SR)    - .0345    .2917      .0556
Covariance (FCG,SR)     - .0016    .0169      .0004
Implied CRRA          - 83.6      7.9      334.6



                      RP

Mean                  .0916
Standard deviation    .2890
Correlation (FCG,SR)
Covariance (FCG,SR)
Implied CRRA

Consumption growth is the annualized change in log real per capita
consumption of food. Data from the Consumer Expenditure Survey are split
into quintiles by income rank, where FCG1 is the food consumption growth
associated with the lowest income quintile. SR is the annualized log
change in the real total return the S&P 500. RR is the real risk-free
rate of return estimated from the 3-month T-bill. RP is the market risk
premium. The implied coefficient of relative risk aversion (CRRA)
assumes power utility with joint conditional lognormality of asset
returns and consumption, and is computed as CRRA=(RP +
.5[[sigma].sup.2.sub.SR])/[[sigma].sub.FCG.SR]. See Eq. (5) where CRRA =
[gamma] and RP = [r.sub.1] - [r.sub.f].
Table 5

Instrumental variables estimation. Nondurable goods and services
consumption growth by income. First quarter 1984 to third quarter 1996

                                      Consumer expenditures survey

Return (lag)                  CG1                CG2            CG3

T-bill (1)         [pi]          .2347            3.8182         1.4019
                   (S.E.)      (3.2664)          (2.4248)       (2.5603)
                   [R.sup.2]     .1611             .0918          .3190
                   df=4         [.1313]           [.4005]        [.0072]
S&P 500 (1)        [pi]          .0906             .4816          .5428
                   (S.E.)       (.5812)           (.4697)        (.5997)
                   [R.sup.2]     .1552             .0437          .1432
                   df=4         [.1453]           [.7501]        [.1777]
T-bill (1 and 2)   [pi]       - 1.4420            1.7641         2.7430
                   (S.E.)      (3.5378)          (2.7584)       (2.9701)
                   [R.sup.2]     .5078             .2676          .3410
                   df=8         [.0111]           [.2356]        [.1019]
S&P 500 (1 and 2)  [pi]        - .8963           - .1266          .2651
                   (S.E.)       (.6342)           (.2955)        (.3373)
                   [R.sup.2]     .2105             .2809          .2624
                   df=8         [.4134]           [.2043]        [.2490]

                   Consumer expenditures
                          survey
                                        NIPA
Return (lag)       CG4       CG5        NCG

T-bill (1)          2.4676      1.7472   .2300
                   (2.5883)   (3.0823)  (.3425)
                     .0927      .0072    .0114
                    [.3955]    [.9886]  [.9733]
S&P 500 (1)          .0831      .0697    .0039
                    (.4112)    (.4896)  (.0297)
                     .1020      .0128    .0200
                    [.3440]    [.9670]  [.9272]
T-bill (1 and 2)    4.0814      .2023    .2622
                   (3.0270)   (3.4757)  (.3974)
                     .3763      .6056    .0607
                    [.0658]    [.0027]  [.9676]
S&P 500 (1 and 2)  - .7341   - 1.1951    .0176
                    (.3703)    (.4262)  (.0254)
                     .1613      .0713    .0552
                    [.6145]    [.9473]  [.9760]

[pi] is the reciprocal of the estimated coefficient of relative risk
aversion and is from the second-stage regression with standard errors in
parentheses; see Eq. (6). Economically plausible estimates of [pi] are
suggested to range above.1.[R.sup.2] is from the regression of the
residuals (from the Stage 1 regression) onto the instruments as in
Hansen (1982). The significance levels for the test statistic (T x
[R.sup.2]) are in brackets; significant levels less that .10 reject the
overidentifying restrictions implied by the model at the 10% level based
on a chi-square distribution.
Table 6

Instrumental variables estimation. Food consumption growth by income.
First quarter 1984 to third quarter 1996

                                  Consumer expenditure survey


Return (lag)                        FCG1         FGG2      FCG3
T-biIl (1)         [pi]        - 1.8677      -1.3089   -1.4247
                   (S.E.)       (3.5515)     (2.4944)  (2.2080)
                   [R.sup.2]      .0700        .0506     .0389
                   df=4          [.5445]      [.6945]   [.7887]
S&P 500 (1)        [pi]          -.1286       -.1047    -.2230
                   (S.E.)        (.3025)      (.2439)   (.3105)
                   [R.sup.2]      .0749        .0544     .0387
                   df=4          [.5098]      [.6641]   [.7901]
T-bill (1 and 2)   [pi]         -1.0796      -1.4443    -.5504
                   (S.E.)       (4.1422)     (2.8766)  (2.5306)
                   [R.sup.2]      .4409        .1933     .4890
                   df=8          [.0281]      [.4799]   [.0145]
S&P 500 (1 and 2)  [pi]          -.0042        .1032     .0974
                   (S.E.)        (.3050)      (.2281)   (.2934)
                   [R.sup.2]      .4425        .1874     .4378
                   df=8          [.0275]      [.5037]   [.0293]

                   Consumer expenditure
                          survey
                                          NIPA

Return (lag)           FCG4      FCG5     FNCG
T-biIl (1)         -1.0239    -.0034    .2259
                   (2.2906)  (2.854)   (.3767)
                     .0260     .1439    .0071
                    [.8873]   [.1756]  [.9891]
S&P 500 (1)          .0590     .1790    .0271
                    (.3364)   (.3659)  (.0676)
                     .0292     .1232    .0092
                    [.8643]   [.2467]  [.9821]
T-bill (1 and 2)    -.6230    1.4912    .2070
                   (2.6312)  (3.2548)  (.4234)
                     .4659     .6323    .0399
                    [.0200]   [.0018]  [.9917]
S&P 500 (1 and 2)    .1212     .0093    .0237
                    (.3015)   (.3180)  (.0584)
                     .4271     .6291    .0363
                    [.3389]   [.0019]  [.9940]

[pi] is the asymptotic reciprocal of the estimated coefficient of
relative risk aversion and is from the second-stage regression with
standard errors in parentheses; see Eq. (6). Economically plausible
estimates of [pi] are suggested to range above .1. [R.sup.2] is from the
regression of the residuals (from the Stage 1 regression) onto the
instruments as in Hansen (1982). The significance levels for the test
statistic (T x [R.sup.2]) are in brackets; significant levels less that
.10 reject the overidentifying restrictions implied by the model at the
10% level based on a chi-square distribution.


Received 29 May 2000; received in revised from 5 November 2000; accepted 12 December 2001

(1.) For theoretical models, see Breeden (1979, 1986), Lucas (1978), and Rubenstein (1976). For the empirical studies, see Grossman and Shiller (1981), Kocherlakota (1996), and Mehra and Prescott (1985) for an in-depth literature review. Our discussion follows Campbell et al. (1997).

(2.) The data Mankiw and Zeldes employ is from the Panel Study of Income Dynamics (PSID). The survey, starting in 1984, questions consumers as to the market value of their holdings in publicly traded corporate shares, mutual funds, and stocks held in IRAs.

(3.) The CCAPM is often tested by employing a Markov chain to estimate, for example, consumption growth with the assumption that dividends from a major stock index equals consumption (see, e.g., Cecchetti, Lam, & Mark, 1990, 2000). We have decided not to follow this method for three reasons. First, since we are dealing with five different consumption streams (by income group) and only one dividend series, from the S&P 500, the assumption that dividends equals consumption is difficult to overcome. Second, estimating a Markov chain for five different consumption streams would be difficult under one set of assumptions because parameter estimates may or may not be significant from one stream to the next. Lastly, estimating a Markov chain for each consumption stream adds another layer of subjectivity to the analysis making comparisons across income groups much more difficult.

(4.) The CES also contains information such as the average number of members in the household and the average respondent's age.

(5.) State independence implies that the utility an individual receives from consumption is independent of the state of the world, that is, the utility function is the same regardless of whether times are good or times are bad. Time separable implies that utility today does not influence utility tomorrow, that is, habits are disallowed.

(6.) Mehra and Prescott (1985) base their range on estimates derived from microeconomic literature; for a listing of this literature, see their paper, p. 154.

(7.) In other words, GMM is used to estimate a nonlinear system of simultaneous equations, where the number of stacked equations is equal to the number of different assets employed in Eq. (3).

(8.) Roughly, a consumer unit represents all members of a particular household.

(9.) Complete income reporters are respondents who provide at least one of the major sources of income such as wages and salaries, self-employment income, and Social Security income. See the "Technical Notes" section of the CES for details on the definition of a consumer unit.

(10.) Both furniture/household equipment and motor vehicles are considered durable goods by the BEA. The CES, however, does not separate the portion of line "other transportation expenses" between durables and nondurables. The fraction of "other transportation expenses" considered durable goods expenditures is derived from the percentage of durables transportation expenses found in the NIPA.

(11.) Lags of 2, 4, and 6 quarters are used as in Hansen and Singleton (1982, 1983). NDS consumption growth is used in Table 1 and food consumption growth is used in Table 2.

(12.) Since there are more orthogonal conditions than parameters in this model, the J-statistic tests whether all the sample moments of g([[theta].sub.T]) are statistically equal to zero.

(13.) These tables are designed to be directly comparable to Table 8.2 (bottom), p. 312 of Campbell et al. (1997).

(14.) These instruments are frequently employed in the CCAPM literature. The logged dividend-to-price ratio is found in the literature to forecast excess returns (see Campbell et al., 1997).

References

Attanasio, O. P., & Weber, G. (1995). Is consumption growth consistent with intertemporal optimization? Evidence from the consumer expenditures survey. Journal of Political Economy, 103 (6), 1121-1157.

Blinder, A. S., & Deaton, A. S. (1985). The time-series consumption revisited. Brookings Papers on Economic Activity, 465-521.

Breeden, D. T. (1979). An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics, 7 (3), 265-296.

Breeden, D. T. (1986). Consumption, production, inflation, and interest rates: a synthesis. Journal of Financial Economics, 16, 3-39.

Campbell, J. Y, Lo, A. W., & MacKinlay, A. C. (1997). The econometrics of financial markets. Princeton, NJ: Princeton University Press.

Caroll, C. D. (1992). How does future income affect current consumption? Board of Governors of the Federal Reserve System, mimeo.

Cecchetti, S. G., Lam, P.-s., & Mark, N. C. (1990). Mean reversion in equilibrium asset prices. American Economic Review, 43 (3), 398-418.

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 (4), 787-805.

Grossman, S. J., & Shiller, R. J. (1981). The determinants of the variability of stock market prices. American Economic Review, 71(2), 222-227.

Hall, R. E. (1988). Intertemporal substitution in consumption. Journal of Political Economy, 96 (2), 339-357.

Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029-1054.

Hansen, L. P., & Singleton, K. J. (1982). Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica (Errata: Econometrica 52 (1), 267-268), 50 (5), 1269-1286.

Hansen, L. P., & Singleton, K. J. (1983). Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Journal of Political Economy, 91(2), 249-265.

Kocherlakota, N. R. (1996). The equity premium puzzle: it's still a puzzle. Journal of Economic Literature, 34 (1), 42-71.

Lucas Jr., R. E. (1978). Asset prices in an exchange economy. Econometrica, 46 (6), 1429-1445.

Lusardi, A. (1992). Permanent income, consumption, and precautionary saving: an empirical investigation. PhD Thesis, Princeton University, Princeton, NJ.

Mankiw, N. G., & Zeldes, S. P. (1991). The consumption of stockholders and nonstockholders. Journal of Financial Economics, 29 (1), 97-112.

Mehra, R., & Prescott, B. C. (1985). The equity premium: a puzzle. Journal of Monetary Economics, 15 (2), 145-161.

Rubenstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics, 7 (2), 407-425.

H.J. Smoluk *

* Corresponding author. Tel.: +1-207-780-4407; fax: +1-207-780-4662.

E-mail address: hsmoluk@usm.maine.edu (H.J. Smoluk).
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Author:Smoluk, H.J.; Neveu, Raymond P.
Publication:Review of Financial Economics
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Geographic Code:1USA
Date:Jan 1, 2002
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