How Forward Looking Are Consumers? Further Evidence for the United States.
This paper extends a standard model of consumption to test for the existence of "myopic" consumers. The extended model includes "rule-of-thumb" consumers as well as consumers who are assumed to solve a dynamic programming problem. The model allows this second set of consumers, however, to be "boundedly rational." For a variety of reasons, they may be unable to fully account for their future uncertain labor income. The model predicts that this inability to correctly value future resources leads to the breakdown of the simple permanent income hypothesis and that consumption responds to predictable changes in income. Using data for the United States for the period 1951u1990, the paper finds evidence of such myopic behavior.
A considerable amount of research has found that the strict form of the LCuPIH (lifecycle, permanent-income hypothesis) is not supported by the data. Consumption appears to be "too sensitive" to predictable changes in income. There are several explanations in the literature for this failure, but a Keynesian "rule-of-thumb" behavior is the explanation that has received most of the attention.
Campbell and Mankiw (1990) construct a model in which a fraction of income [lambda] accrues to individuals who, following "rules of thumb," consume their current income, while the remaining (1 - [lambda]) accrues to individuals who, being extremely farsighted, consume their permanent income. The rule-of-thumb behavior is subject to two interpretations (Shea 1995). One interpretation is that consumers are extremely "myopic." This nonoptimizing or myopic behavior implies that these consumers completely ignore their total wealth (or permanent income) when making consumption decisions. Another interpretation of the behavior of the first group of individuals is that they are liquidity constrained. These individuals have few or no real assets and have no access to capital markets. The empirical evidence from aggregate data is rather inconclusive, but evidence from more disaggregated data lends support to the liquidity constraints interpretation (Zeldes 1989a).
While the CampbelluMankiw specification, under either interpretation, generates excess sensitivity and results in an empirically more realistic consumption function, its treatment of the second group of individuals, who are assumed to consume their permanent income, is unnecessarily restrictive. As Kotlikoff, Samuelson, and Johnson (1988, p. 408) point out, "A second problem that is also routinely swept under the rug involves the implicit assumption that consumers optimize perfectly given their preferences and resources, and that they correctly value their resources." Their results, based on experimental evidence, indicate that "subjects made significant and systematic errors in their consumption choice, reflecting, in part, an overdiscounting of future income" (p. 408). This "quasi-myopic behavior" has a variety of sources, such as high computational costs, habit, self-control, lack of extreme rationality or perfect farsightedness, short multiperiod planning horizons, and so on (Mariger 1986; Boskin 1988; S hefrin and Thaler 1988; Blanchard 1997, p. 151; Lettau and Uhlig 1999, among others), which can lead to failure by consumers to fully account for their future after-tax labor income. I will show that this behavior, which is consistent with a limited degree of myopia that might exist among those who are assumed to be extremely farsighted, can lead to excess sensitivity and the failure of the PIH. Lettau and Uhlig (1999) call these consumers "boundedly rational" (p. 152). Boskin (1988) and Poterba (1988) discuss the important implications for fiscal policy resulting from such myopic behavior.
The evidence in favor of rule-of-thumb behavior in explaining consumption is too strong to ignore for empirical purposes (Dornbusch, Fischer, and Startz 1998, p. 308; Himarios 1995). Mariger (1986) argues that the failure to identify constrained and unconstrained consumers leads to models that are inherently misspecified. I will, therefore, use a specification that nests rules of thumb as a maintained hypothesis. The focus of the paper will be on the second set of consumers who are assumed to consume their permanent income, as in Campbell and Mankiw (1990). Based on the earlier discussion, I will relax this assumption and allow this set of maximizing consumers to behave in a "boundedly rational" manner.
The paper proceeds as follows. In section 2, I briefly develop an empirically tractable consumption function and discuss the empirical implications of the model. In section 3, I discuss the data and the econometric methodology, and in section 4, I present the estimation results. The evidence is consistent with the existence of rule-of-thumb consumers, a result that has been documented in previous studies, and also consumers who exhibit myopia of the type suggested previously.
2. Theoretical Background
Following Campbell and Mankiw (1990), I assume two types of households. Household type 1 behaves according to the LC-PIH. These consumers base their current consumption decisions on their lifetime resources, which consist of their financial wealth and the expected discounted value of their future after-tax labor income. A closed-form solution of this optimization problem is possible under certainty equivalence or under a quadratic utility function if income is assumed to be stochastic. Although quadratic utility can be justified on the grounds that it is a local approximation to the consumer's true utility function, its simplicity for computational problems is offset by serious shortcomings, and the resulting consumption function is likely to be severely misspecified (Hayashi 1982; Zeldes 1989b; Weil 1993). A more plausible utility function, assumed both by Hayashi and Zeldes, is the constant relative risk aversion function. Yet under such preferences and stochastic labor income, no closed-form solution is p ossible.
One way, though an imperfect one, to take into account this labor income uncertainty and derive an approximate solution to the optimization problem is to allow households to discount their uncertain future labor income at a rate higher than the rate of interest (Hayashi 1982; Zeldes 1989b). The assumption that the discount rate exceeds the rate of interest is amply supported by previous evidence (Hayashi 1982; Graham and Himarios 1996, among others). The approximate solution to the type 1 consumer's optimization problem is then
[C.sub.1,t] = [alpha][(1 + r)[W.sub.t-1] + [H.sub.1,t]], (1)
where [W.sub.t-1] is the end-of-period nonhuman wealth held by these consumers (I assume that type 2 consumers hold no assets), [alpha] is the propensity to consume out of total wealth, and r is the nonstochastic real interest rate. The variable [H.sub.1,t] is the present discounted value of future after-tax labor income ([Y.sub.1,t]). If we assume that the share of after-tax labor income that accrues to these consumers is (1 - [lambda]), where [lambda] is the share of income accruing to type 2 consumers, then [H.sub.1,t] is defined as
[H.sub.1,t] = [[[sum].sup.[infty]].sub.j=0] [(1 + [mu]).sup.-j][E.sub.t][Y.sub.1,t+j] = (1 - [lambda]) [[[sum].sup.[infty]].sub.j=0] [(1 + [mu]).sup.-j][E.sub.t][Y.sub.t+j], (2)
where [E.sub.t] is the expectations operator conditional on information available in period t and [mu] is the subjective discount rate that consumers apply to their uncertain future labor income.
This specification of the consumption function assumes that individuals do not face any of the constraints mentioned previously. In order to account for this possibility, I modify the consumption function for type 1 consumers as follows:
[C.sub.1,t] = [alpha][(1 + r)[W.sub.t-1] + [gamma][H.sub.1,t]]. (3)
The parameter [gamma] [epsilon] (0, 1) measures the degree to which consumers account for their future stochastic labor income. Admittedly, this is an ad hoc specification, but it is similar in spirit to others in this line of literature on consumption and captures one plausible form of myopic behavior: Type 1 consumers respond fully to wealth that they currently see and hold but fail to fully project their future income and future tax liabilities as implied by the government's budget constraint (Poterba and Summers 1987, p. 378).  This specification is also consistent with Zeldes's (1989b) formulation, although the interpretation is somewhat different. In Zeldes's formulation, the weight in front of the human wealth, ([gamma]), reflects precautionary saving behavior. His simulation results "show that current assets (which include income just received) and nonstochastic future income receipts are optimally given much more weight than future random labor income in making the current consumption decision" (p . 295).
I assume that type 2 households do not hold any assets, they have labor income [Y.sub.2,t] and they follow rules of thumb. Their consumption function is derived trivially from their budget constraint as
[C.sub.2,t] = [Y.sub.2,t] = [lambda][Y.sub.t], (4)
where [lambda] denotes the fraction of income accruing to the type 2 households.
Aggregate consumption is the sum of consumption over the two types of households:
[C.sub.t] = [C.sub.2,t] + [C.sub.1,t] = [lambda][Y.sub.t] + [alpha][(1 + r)[W.sub.t-1] + [gamma][H.sub.1,t]]. (5)
The general form of this consumption function is similar to the one postulated by Blanchard (1997, p. 149), who calls it a more realistic description of consumption behavior. The budget constraint for type 1 households in period t is
[W.sub.t] + [C.sub.1,t] = (1 + r)[W.sub.t-1] + [Y.sub.1,t] = (1 + r)[W.sub.t-1] + (1 - [lambda])[Y.sub.t], (6)
and hence the aggregate budget constraint is given by
[W.sub.t] + [C.sub.t] = ([W.sub.t] + [C.sub.1,t]) + [C.sub.2,t] = (1 + r)[W.sub.t-1] + (1 - [lambda])[Y.sub.t] + [[lambda]Y.sub.t] = (1 + r)[W.sub.t-1] + [Y.sub.t]. (7)
In order to eliminate the unobservable human wealth variable, H, from Equation (5), I make use of the following stochastic difference equation that describes the evolution of human wealth (Hayashi 1982):
[H.sub.1,t] = (1 + [mu])[[H.sub.1,t-1] - (1 - [lambda])[Y.sub.t-1]] + [u.sub.t]. (8)
The error term [u.sub.t] is defined as
[u.sub.t] = [[[sum].sup.[infty]].sub.j=0] [(1 + [mu]).sup.-j]([E.sub.t] - [E.sub.t-1])[Y.sub.1,t+j]. (9)
This error (or surprise) term represents the revisions of expectations between t - 1 and t and is orthogonal to the information set available to households in period t - 1. It is this surprise term that leads LC-PIH consumers to revise their consumption plans.
Lagging Equation (5) once, multiplying by (1 + [mu]), subtracting from (5), and using the private sector's budget constraint (7) along with the difference equation for human wealth (8), I derive the following consumption function in terms of observable
[Delta][C.sub.t] = ([mu] - [alpha][mu] - [alpha])[C.sub.t-1] + [lambda][Y.sub.t] - [alpha]([mu] - r)[W.sub.t-1] - (1 + [mu]([[lambda](1 - [alpha][gamma]) - [alpha](1 - [gamma])[Y.sub.t-1]] + [[epsilon].sub.t], (10)
[[epsilon].sub.t] = [alpha][gamma](1 - [lambda])[u.sub.t]. (11)
Consider now the implications of assuming that [lambda] = 0, [mu] = r, but that consumers cannot fully discount their future after-tax labor income, that is, ([gamma] [less than] 1). Equation (10) becomes
[Delta][C.sub.t] = ([mu] - [alpha][mu] - [alpha])[C.sub.t-1] + [alpha](1 + [mu])(1 - [gamma])[Y.sub.t-1] + [[epsilon].sub.t]. (12)
It is obvious that excess sensitivity now arises because consumers are optimizing under a number of constraints that prevent them from fully discounting their future after-tax labor income or because they exhibit quasi-myopic behavior due to habit, short planning horizons, and so on.
The consumption behavior captured by Equation (10) nests some of the independent explanations that have been offered in the literature for the failure of the LC-PIH. If we assume that LC-PIH consumers fully discount their future after-tax labor income ([gamma] = 1) and that they have infinite horizons ([mu] = r) (Blanchard 1985), then Equation (10) becomes
[Delta][C.sub.t] = ([mu] - [alpha][mu] - [alpha])[C.sub.t-1] + [lambda][Y.sub.t] - [lambda](1 + [mu])(1 - [alpha])[Y.sub.t-1] + [[epsilon].sub.t]. (13)
This is the case studied by Campbell and Mankiw (1990), and in this case excess sensitivity arises from rule-of-thumb behavior.  If we assume that all consumers are LC-PIH consumers but allow for the possibility of finite horizons ([mu] [neq] r), then Equation (10) becomes
[Delta][C.sub.t] = ([mu] - [alpha][mu] - [alpha])[C.sub.t-1] - [alpha]([mu] - r)[W.sub.t-1] + [[epsilon].sub.t] (14)
This is the equation estimated by Evans (1988) and Graham and Himarios (1996), among others. Here excess sensitivity arises from finite horizons or precautionary saving behavior.  If we assume that [mu] = r, [lambda] = 0, and [gamma] = 1, then Equation (10) reduces to the familiar martingale hypothesis:
[Delta][C.sub.t] = ([mu] - [alpha][mu] - [alpha])[C.sub.t-1] = [epsilon].sub.1]. (15)
It is clear from this analysis that "excess sensitivity" can arise from a combination of rule-of-thumb behavior, whatever its causes, finite horizons, and myopic behavior of the form proposed previously and that the single separate explanations proposed in the literature might not be adequate in describing consumption behavior. Indeed, as Himarios (1995) has shown, ignoring some of these explanations results in misspecification that biases the results of testing for the other hypotheses. Equation (10) provides a useful framework for evaluating these different hypotheses controlling for data, sample period, and estimating technique.
3. The Data and Econometric Methodology
Estimation Method and Econometric Issues
Equation (10) will form the basis for testing the hypotheses discussed previously. By assumption, the error term in this equation is white noise, and it is orthogonal to the information set [I.sub.t-1]. Because current income, however, appears in the equation and [E.sub.t]([[epsilon].sub.t]/[I.sub.t] [neq] 0, one can obtain consistent estimates only by using instrumental variables. There are additional reasons that require the use of instrumental variables. For example, I have assumed that transitory consumption or preference shocks are zero. However, the existence of such shocks will introduce an MA(1) error in the transformed equation and make the error term correlated with variables dated t - 1 as well. Consistent estimates can be obtained by using the generalized method of moments estimator (GMM) with instruments dated t - 2 and earlier to avoid misspecification arising from time-averaging and durability. 
Following Campbell and Mankiw (1990), I use lagged income to divide all the variables in Equation (10) before estimation to account for the fact that both consumption and income do not follow homoscedastic linear processes in levels. An added benefit of this transformation is the induced stationarity of the right-hand-side variables. Under the LC-PIH and the alternative described by Equation (10), consumption and income and income and wealth are cointegrated (i.e., they share common stochastic trends), and therefore their ratios should be stationary (Gali 1990; Graham and Himarios l996). 
The real rate of return r cannot be identified from Equation (l0).  Rather than fixing its value, however, as some previous studies have done, one can estimate Equation (10) along with the following equation for the real after-tax rate of return:
[varphi](L)([[rho].sub.t] - r) = [[eta].sub.t], (16)
where [varphi](L) is a polynomial in the lag operator and [[eta].sub.t] is a zero-mean, finite-variance white noise error term.  If the real after-tax rate of return for assets ([[rho].sub.t]) is stationary, then the expected real return can be estimated as its sample unconditional mean.
Following most previous studies, I measure consumption as the sum of expenditures on services and nondurables in constant 1987 dollars. I follow Campbell and Mankiw (1990) and divide this measure by 0.879, which is the mean ratio of services and nondurables consumption to total consumption over the sample period. Labor income is constructed as in Blinder and Deaton (1985) from NIPA data. The nonhuman wealth variable is an adjusted measure of the household sector's net worth constructed from Balance Sheets for the U.S. Economy. This measure is the result of adding household tangible assets and household financial assets and subtracting household liabilities. Where feasible, market rather than book values are used. All variables are divided by population to yield per capita measures. The measure for the real after-tax rate of return on assets is the ratio of after-tax corporate profits from current production to net reproducible assets valued at replacement cost. This measure is from the April 1991 Survey of C urrent Business. The data are annual observations from 1947 to 1990.  The Appendix provides more information on the variables and their sources.
4. Estimation Results
I estimate the system of Equations (10) and (16) using services and nondurables as a measure of consumption for the period 1950-1990. The first three observations are used up in transformations and lagged instruments. A series of diagnostic tests indicated that the real after-tax rate of return can be adequately represented by an AR(1) process. I use the GMM technique to estimate the system of Equations (10) and variants thereof and (16). Standard errors are calculated using the Newey-West estimator. The instrument set consists of the second and third lags of all the transformed variables in the regression and a constant. The choice of the instrument set was based on evidence that weak or irrelevant instruments can have deleterious effects on the GMM estimator and that the lagged right-hand-side variables are likely to have good explanatory power, thus alleviating the problems associated with poor instrument relevance (Fuhrer, Moore, and Schuh 1995). Given that the number of instruments is greater than the n umber of parameters to be estimated, the system of equations is overidentified. The overidentification restrictions imposed by the extra instruments, and thus the validity of the specification of the model, are tested by using the Hansen--Singleton J-test. The extra instruments also serve the purpose of improving the efficiency of the parameter estimates (Davidson and MacKinnon 1993, P. 595).
Table 1, line 1, presents the nonlinear ordinary least squares (OLS) estimates of [gamma], [alpha], [mu] [lambda], and r as a benchmark. The estimates of [alpha], [mu], [lambda], and r have the expected sign, and their size is consistent with previous estimates. The estimate of [gamma] is considerably less than one, indicating the possibility of myopic behavior. Significance tests are not appropriate with the OLS estimator, but if the GMM estimator is successful, the OLS and GMM estimates should be of comparable magnitudes. Rows 2 through 7 present the estimates with autocorrelation and heteroscedasticity consistent standard errors. All the estimates in row 2 have the correct sign and are statistically significant at conventional levels. The estimates are similar to the OLS estimates, indicating that the choice of the particular set of instruments is not critical to the results. The [X.sup.2] test (J-test) for the validity of the overidentifying restrictions is small and insignificant and does not provide any e vidence against the model. The estimate of [lambda]] (0.406) falls within the range estimated by Campbell and Mankiw (1990). The estimate of the discount rate ([mu]) of 8.6% exceeds the estimate of the interest rate (r) of 4.3% and this difference is statistically significant. A t-test for the difference between the two has a value of 3.17. The crucial parameter of interest ([gamma]) measuring the degree of myopic behavior is around 0.5, and it is statistically significant at the 1% level. This estimate indicates a significant degree of shortsighted behavior but certainly not extreme myopia, as it is significantly greater than zero. The t-statistic for the restriction that [gamma] = 1 is 3.618 and rejects the restriction at the 1% significance level.  I should note that there is a complication in testing the hypotheses concerning the value of [gamma]. This parameter cannot be negative or greater than one on a priori grounds. The complication arises because the null hypotheses of zero or one lie on the boundary of the parameter space.  There is no clear solution to this problem, but Pudney and Thomas (1995, p. 375) point out that "in practice, the usual procedure for a test of this type is to examine both the conventional [X.sup.2] statistic and also separate one-sided asymptotic t-ratios for each test." One-sided t-statistics for the hypotheses [H.sub.0]: [gamma] = 0, [H.sub.a]: [gamma] [greater then] 0 and [H.sub.0]: [gamma] = 1, [H.sub.a] [gamma] [less then] 1 reject the null in both cases at the 1% significance level. The restrictions are also tested with a likelihood ratio type test constructed as C = T[(J([[beta].sup.r]) - J([[beta].sup.u])]. This test follows a [X.sup.2] distribution with degrees of freedom equal to the number of restrictions imposed. The variable T is the number of observations, while J([beta]) is the minimized value of the objective function. The superscripts r and u show this value for the restricted version and the unrestricted version, respectively. The same estimator of the covariance matrix is used in both the restricted and the unrestricted version to ensure that the test statistic is properly signed. The numbers in the last column indicate the value of this C-statistic for the various restrictions with the significance level at which the restriction is rejected shown in parentheses underneath. Rows 3 and 4 impose the restrictions [gamma] = 1 and [gamma] = 0, respectively. The restriction that households fully discount their future after-tax labor income is rejected at the 1.5% significance level, while the restriction that they completely ignore it is much more strongly rejected. The next three lines impose the restrictions that result in the different hypotheses for the failure of the PIH that I discussed previously. In all cases the restrictions are rejected. Of particular interest is the restriction in row 6, which results in the Campbell-Mankiw specification. The restriction is rejected at the 1.9% significance level when tested against a more encompassing model. The Campbell-Mankiw specification thus appears overly restrictive compared to a model that allows households to discount their future income at a rate higher than the rate of interest and to have less than perfect foresight.
Although the sample is rather small, it would still be instructive to check the sensitivity of the results to different periods. I estimated the equations for four subperiods: 1951-1981, 1956-4990, 1960-1990, and 1951-1985. The estimates for the first three periods range in value from 0.36 to 0.44, and the restriction that [gamma] = 1 can be rejected in all cases. The last period is the exception, with [gamma] taking on a value of 0.984. The restriction cannot be rejected in this case.
The evidence from the modified model that I proposed previously supports the existence of both rule-of-thumb consumers and consumers who suffer, to some degree, from myopia with regard to their future income and taxes. This quasi-myopic behavior has many sources and does not necessarily mean that consumers are not optimizing. Faced with a multitude of constraints and lack of perfect information, problems that the PIH assumes away, consumers are unable to fully account for future uncertain income and tax streams. While useful as a benchmark for analyzing a multitude of problems, the strict version of the PHI fails as an accurate descriptor of consumer behavior. This paper has used a rather ad hoc modification of a well-specified model to capture this quasi-myopic behavior. The challenge for future research is to develop models that incorporate the constraints mentioned previously and generate such behavior in a rigorous way. Recent models of the type investigated by Lettau and Uhlig (1999), where consumers ar e assumed to behave "boundedly rationally," may prove useful in explaining observed consumption behavior more accurately.
(*.) Department of Economics, University of Texas at Arlington, Arlington, TX 76019, USA; E-mail email@example.com.
I wish to thank two anonymous referees for helpful comments and suggestions. Any remaining errors or omissions are my own.
Received April 1998; accepted August 1999.
(1.) Lee (1991) uses the same approach to measure the extent to which consumers perceive government bonds as net wealth. His parameter [theta] [epsilon] (0, 1) is imposed ad hoc in an otherwise optimizing model.
(2.) Campbell and Mankiw follow Flavin (1981) and essentially impose the restriction that [mu] = [alpha]. In this case, (1 + [mu])(1 - [alpha]) is roughly equal to one, and (13) reduces to that estimated by Campbell and Mankiw.
(3.) It is easy to substitute [W.sub.t-1] out of the equation and show that consumption responds to lagged income. Kimball and Mankiw (1989) show that [mu] [neq] [rho] is consistent with a precautionary saving motive. Haug (1996) tests Blanchard's finite horizon model with Canadian data and strongly rejects it.
(4.) The potential problems that may arise from time aggregation were investigated in Graham and Himarios (1996), who, using essentially the same data, found no evidence of biases.
(5.) Graham and Himarios (1996) eatablish cointegration between consumption and income and income and wealth for the period 1952-1991. Though the relatively small size limits the usefulness of unit root teats, I performed adjusted Dickey-Fuller and Phillips-Perron unit root tests on the ratios in the regression. These tests reject a unit root for the dependent variable and the ratio of current to lagged income at the 1% level but fail to reject it for the consumption-to-income and wealth-to-income ratios in the full sample. Even minor adjustments in the sample, however, reverse this result, and a unit root is rejected at the 5% level or better. The fragility of these results is well known, and it is common practice in such cases to use theory as a guide.
(6.) Since the equation is nonlinear only in the parameters, it can be estimated as an unrestricted linear model (Evans 1988; Graham and Himarios 1996). One can then test whether the estimated composite coefficients have the probability limits implied by the different assumptions. However, given that the model is overidentified, the underlying parameters cannot be recovered. By using a nonlinear estimator, one can get direct estimates of the parameters in question that will give a more meaningful measure of any rejection that might occur.
(7.) I wish to thank Paul Evans for suggesting this particular approach for identifying the after-tax rate of return on assets.
(8.) The end of the sample is dictated by the availability of some series.
(9.) Though not directly comparable, this result is consistent with the findings of Shapiro and Slemrod (1995), who conclude that 43% of the households surveyed behaved as if they were subject to myopia.
Blanchard, Olivier. 1997. Macroeconomics. Englewood Cliffs, NJ: Prentice Hall.
(10.) I wish to thank an anonymous referee for pointing out this problem.
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Generalized Method of Moments (GMM) Estimates of Equations (10) and (16): 1950-1990 [alpha] [gamma] [mu] r [lambda] J-test C-test 1 0.076 0.347 0.082 0.044 0.469 (0.028) (0.262) (0.028) (0.006) (0.048) 2 0.071 0.471 0.086 0.043 0.406 9.221 (0.012) (0.146) (0.013) (0.003) (0.051) (0.904) 3 0.043 1 0.079 0.042 0.357 15.164 5.943 (0.009) (0.018) (0.003) (0.047) (0.584) (0.015) 4 0.106 0 0.095 0.044 0.571 26.771 17.550 (0.007) (0.010) (0.005) (0.032) (0.061) (0.000) 5 0.020 1 [mu] = r 0.042 0.379 17.155 7.935 (0.003) (0.003) (0.046) (0.512) (0.019) 6 0.058 1 0.102 0.040 0 74.338 65.117 (0.007) (0.014) (0.002) (0.000) (0.000) 7 0.382 0.491 [mu] = r 0.043 0.408 15.049 5.829 (0.013) (0.188) (0.003) (0.051) (0.592) (0.016)
The J-test is a test of the validity of the overidentifying restrictions with its significance level shown in parentheses. The C-test is a likelihood ratio test for the validity of the imposed restrictions. The number in parentheses underneath indicates the significance level at which the restriction is rejected. Row 1 presents the nonlinear ordering least squares (OLS) estimates. The rest of the rows are GMM estimates with autocorrelation and heteroscedasticity consistent standard errors.
1. The consumption and income series come from the National Income and Product Accounts of the U.S. (vols. 1 and 2). Consumption is measured as expenditures on nondurable goods and services in 1987 dollars. The after-tax labor income (Y) was constructed as follows: Let L be the sum of wages, salaries, and other labor income and I be the sum of personal interest earnings, personal dividend earnings, and rental income to persons. Then Y = L+ government transfers to persons + (L/(I + L)) times proprietors' income minus social security contributions (by employees) minus (L/(I + L)) times personal income taxes.
2. Nonhuman wealth was constructed from end-of-year data from Balance Sheets for the U.S. Economy, 1945-1992. The variable W is defined as private sector net worth minus pension fund reserves minus par value of household holdings of tax-exempt obligations and U.S. government securities plus [P.sup.*] (defined shortly) times par value of household holdings of tax-exempt obligations plus the ratio of par value of household holdings of U.S. government securities to the par value of total outstanding U.S. government securities times the market value of privately held outstanding U.S. government debt (constructed by Michael Cox). The variable [P.sup.*] is a approximate market price for state and local obligations from Survey of Current Business, Current Business Statistics, Section 6: Finance, Bonds (Standard and Poors Corporation), domestic municipal bond prices per dollar of par value.
3. Per capita variables are constructed by dividing by the total population of the United States taken from Statistical Abstract of the United States, 1993, p. 8, Table 2.
4. Real variables, when not directly available, were constructed by dividing by the implicit price index constructed as the ratio of nominal to real expenditures on nondurable goods and services.