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Liquidity constraints and aggregate consumption behavior.

LIQUIDITY CONSTRAINTS AND AGGREGATE CONSUMPTION BEHAVIOR

This paper presents time series evidence on the importance of liquidity constraints

in aggregate consumption expenditures. In contrast to previous studies, I find the

proportion of consumption attributable to liquidity constrained behavior to be large

and highly statistically significant. The estimation pays careful attention to the

problems of stochastic consumption and temporal aggregation, and the estimates are shown

to be robust to alternative specifications involving costly adjustment of consumption,

public spending, and to stochastically varying rates of return.

I. INTRODUCTION

In his seminal contribution to econometric testing of the permanent income hypothesis (PIH) under rational expectations, Hall [1978] showed that optimal saving behavior would make consumption close to a random walk. Hall found only minor departures from a random walk and concluded that there is little reason to doubt the hypothesis. Hall's work stimulated a vast and growing field of research on the econometric implications of optimal consumption behavior. (See Hall [1987] for references and a recent survey.) Econometric testing has taken two distinct approaches. The "Euler equation" approach employed by Hansen and Singleton [1982; 1983], and Nelson [1987] follows Hall by testing the unpredictability of changes in aggregate consumption. The second approach, followed by Hall and Mishkin [1982], West [1988], Deaton [1987], and Quah [1990], focuses on innovations in income and examines the sensitivity or volatility of consumption implied by the permanent income hypothesis. Flavin's [1981] work falls between these two approaches. Her formal work tests the restrictions of the "Euler equation" approach, but she interprets the magnitude of the rejections in terms of the sensitivity of consumption implied by the time-series process on income.

The problem with the first approach is that tests of the orthogonality conditions do not yield a meaningful metric with which to assess the size of the departure from the permanent income hypothesis. The results of the second approach are more directly interpretable, but the estimates are sensitive to assumptions on the stochastic structure of labor income and agents' information sets. This paper adopts the Euler equation approach of Hall [1978] but tests the permanent income hypothesis against a tightly parameterized alternative hypothesis. Following Hayashi [1982], I assume a fixed, but unknown, portion of consumption is accounted for by liquidity constrained agents. Estimating the percentage of liquidity constrained consumption provides a statistically powerful and economically meaningful test of the permanent income hypothesis.

The basic model in this paper is quite similar to that in the influential papers by Hayashi [1982] and Flavin [1981]. In contrast to those studies, however, I obtain plausible and highly statistically significant estimates of the percentage of liquidity constrained consumption. The results suggest that from 30 percent to 40 percent of U.S. consumption is accounted for by liquidity constrained consumers. Careful attention is given to the problems of stochastic consumption, temporal aggregation and coefficient instability. Further, the estimates are shown to be robust to alternative specifications involving costly adjustment, Aschauer's [1985] public spending hypothesis, and to stochastically varying rates of return. Flavin's basic point is confirmed -- small departures from the random walk predicted by the permanent income hypothesis could be the result of large structural departures from the permanent income hypothesis.

The paper is organized as follows. Section II presents Hall's model of the permanent income hypothesis under rational expectations and updates his basic results. Section III develops the model of liquidity constraints and the estimation strategy employed in this paper. Section IV presents the main empirical results. Section V compares my work to Flavin's and Hayashi's and explains why our results differ. Section VI demonstrates that the evidence is robust to a number of alternative specifications.

II. HALL'S MODEL OF CONSUMPTION

In a line of reasoning that is now standard, Hall [1978] pointed out that, under certain conditions, optimal savings behavior will make the marginal utility of consumption follow a first-order Markov process. The random walk restriction follows in the special case of constant real rates of interest and non-stochastic quadratic utility that is additively separable in the consumption good. Assume that the representative consumer maximizes [Mathematical Expression Omitted] subject to the series of budget constraints, (1) [Mathematical Expression Omitted] where [A.sub.t] is the level of real assets at the beginning of period t, [Mathematical Expression Omitted] is (disposable) labor income and R is the (constant) gross real rate of return.

An exact solution for consumption can be obtained in the special case of quadratic utility. If the instantaneous utility function is [Mathematical Expression Omitted] with [a.sub.t] a mean zero, serially uncorrelated shock to preferences. Sargent [1987, ch. XIII] shows the solution for consumption is (2) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] The error term, [[Epsilon].sub.t], is the sum of a white-poise term reflecting the innovation in labor income and an MA(1) term reflecting shocks to tastes.

Hall pointed out that equation (2) can be regarded as a local approximation, valid for small changes in consumption. I take equation (2), with [[Lambda].sub.1] close to one, as the restriction imposed by the permanent income hypothesis. The strong testable implication of equation (2) is that the error term, [[Epsilon].sub.t], is uncorrelated with any information known at time t-2. The exact restrictions on [[Lambda].sub.0] and [[Lambda].sub.1] are not tested because they are particularly sensitive to the assumption of quadratic utility.(1)

Hall [1978] conducted his original tests under the stronger assumption that shocks to utility are absent (and temporal aggregation problems can be ignored), in which case the error term in (2) is uncorrelated with all information known at time t-1. Hall regressed consumption on consumption lagged one period and tested whether lagged values of other variables (particularly disposable income) had significant predictive power. Table I updates Hall's findings, using quarterly, per capita data on consumption of nondurables and services for the period 1948:I through 1987:II. (See section IV for a full description of the data used in this study.)

Just as in Hall's original tests, the coefficient on lagged consumption is close to one and lagged values of disposable income are individually and jointly statistically insignificant. (The F-test on all lagged disposable income coefficients, F(4,152) = 1.79, has a marginal significance level of .134.) The inclusion of lagged disposable income reduces the standard error by a mere 34 cents and the sum of the coefficients is negative. The reduced-form evidence that led Hall to conclude "there is little reason to doubt the permanent income hypothesis" is apparent in Table I.

III. THE LIQUIDITY CONSTRAINED ALTERNATIVE

Flavin [1981] first pointed out the problem with Hall's reduced-form tests of the permanent income hypothesis. In a structural model, it is entirely possible for consumption to be close to a random walk even if the true model generating consumption is very far from the permanent income hypothesis. As the third line of Table I indicates, the univariate disposable income process is close to a pure random walk. Under the alternative hypothesis that consumers are liquidity constrained and simply consume their current income, consumption will also be close to a random walk. Although in this special case, the permanent income hypothesis and the liquidity constraints hypothesis both predict consumption will follow a random walk, the policy implications of the two models are very different. If consumption is governed by the permanent income hypothesis, the random walk process is invariant to changes in government tax or transfer policies. If instead, consumers are liquidity constrained, changes in taxes or transfers will alter the intertemporal behavior of consumption.

In this section I describe a simple generalization of the permanent income hypothesis suggested in Hall [1978] and Hayashi [1982]. A constant fraction, [Mu], of total consumption is assumed to be generated by liquidity constrained consumers. This tightly parameterized alternative hypothesis provides a meaningful and powerful test of the permanent income hypothesis.

The Model

Liquidity constrained individuals are assumed to simply consume all of their current disposable income. Per capita (liquidity constrained) consumption expenditure on any particular consumption category is (3) [Mathematical Expression Omitted] where [y.sub.t] is per capita disposable income and [Pi] (assumed constant) is the fraction of total consumption accounted for by this category. Unconstrained individuals consume according to the permanent income hypothesis, (4) [Mathematical Expression Omitted] Letting [Mu] denote the fraction of consumption accounted for by liquidity constrained individuals, per capita consumption is the weighted average of the two groups, (5) [Mathematical Expression Omitted]

Using the standard approximation [[Lambda].sub.1] = 1, equation (5) simplifies to(2) (6) [Mathematical Expression Omitted] Equation (6) may be viewed as a restricted version of the models considered in Flavin [1981] and Hayashi [1982]. Flavin's model of liquidity constraints allows for the effects of additional lagged values of income. Hayashi distinguished between the discount rate applied to labor income and non-human wealth.

The orthogonality condition, E([[Epsilon].sub.t] / [Z.sub.t-2]) = 0, imposes cross-equation restrictions on the system of equations generating changes in consumption and income. Following Flavin, I estimate equation (6) jointly with an auxiliary equation for changes in income, imposing the cross-equation restrictions. These full information-maximum likelihood (FIML) estimates are efficient, but can be sensitive to departures from normality. Therefore, I also estimate equation (6) by an instrumental variables approach (GMM). (See Hansen and Singleton [1982] for a discussion of the relative merits of the two approaches.) Because the two sets of estimates are similar, I present only the FIML estimates in the text and include the GMM estimates in the appendix.

The cross-equation restrictions generated by equation (6) provide a simple illustration of Flavin's criticism of reduced-form tests. Denote the least squares projection of [[Delta] c.sub.t] on a set of observable variables dated time t-2 or before, {[Z.sub.t-2]}, (7a) [Mathematical Expression Omitted] and the projection of [Delta] y on the same set as (7b) [Mathematical Expression Omitted] Equation (6) implies the cross-equation restrictions (7c) [a.sub.0] = (1 - [Mu]) [[Lambda].sub.0] + [[Phi] [[Mu] b.sub.0] (7d) [a.sub.i] = [[Phi] [Mu] b.sub.i] i = 1,2,.... The reduced-form coefficients, [a.sub.i], depend on the structural parameters [Phi] and [Mu] and also on the coefficients on the auxiliary equation, the [b.sub.i]. If the auxiliary equation contains only past income and income is close to a pure random walk (as suggested by the estimates in Table I) the [b.sub.i]'s will be small. Therefore the [a.sub.i]'s will be small even if, in a structural sense, the divergence from the null hypothesis is large.

The importance of examining the structural parameters can be illustrated by reevaluating Hall's analysis of the predictive power of lagged stock prices. Hall found lagged stock prices statistically significant in predicting consumption changes, but he argued that the divergence from the permanent income hypothesis is slight because the inclusion of lagged stock prices reduced the per capita prediction error by a mere 20 cents (in 1972 dollars). When his estimates are updated to the period 1948:IV-1987:II, similar results are obtained. Including twice-lagged real stock price reduces the standard error of prediction by a minuscule .04 cents (in 1982 dollars). The structural model, however, suggests a large divergence from the permanent income hypothesis. The reduced-form coefficients, [a.sub.1] = 4.52, [b.sub.1] = 14.43, [Pi] = .88527, imply that the percentage of liquidity constrained consumption is 35 percent!

Econometric Issues

Because the basic model is formulated in first differences, it is convenient to assume that consumption and income are difference stationary. Considerable controversy surrounds the issue of whether to model macroeconomic time series as difference stationary or stationary around a deterministic trend. Standard Dickey-Fuller tests cannot reject the hypothesis of difference stationarity, but doubts have been expressed on the appropriateness of these tests (See Sims [1988]). Fortunately, difference stationarity is not crucial to the results of this paper. Unlike in work on "excess smoothness" (Deaton [1987]), the time-series properties of income are exploited only to obtain one- or two-step-ahead predictions. These short-term prediction errors are relatively insensitive to the stochastic specification of the trend.

The potential problem with estimating consumption models using time-averaged data has recently been recognized by Hall [1988], Nelson [1987], and Christiano, Eichenbaum, and Marshall [1987]. A careful study of temporal aggregation, however, shows that the restrictions in (7) apply equally to time-averaged data. Consider forming a temporal aggregate, [X.sub.t], by taking overlapping averages of length m of the underlying series [x.sub.t]. The mth difference of [X.sub.t] is (8) [Mathematical Expression Omitted] where L is the lag operator and [a.sub.t] = ([x.sub.t] - [x.sub.t-1]). Applying the filter (8) to equation (6) yields (9) [Mathematical Expression Omitted] Because [[Epsilon].sub.t] is at most an MA(1), the error term in (9), [Mathematical Expression Omitted] is uncorrelated with all information known at time t-2m. Point sampling (9) at intervals T = t, t [+ or -] m, t [+ or -] 2m, yields the relationship among the temporal aggregates (10) [Mathematical Expression Omitted] where the error term is uncorrelated with information known at time T-2. Comparing (6) and (10) it is clear that, apart from the constant term, the restrictions in (7) apply to both point-sampled and time-averaged data.

Although the use of temporally aggregated data does not affect the estimation procedure, it does affect the interpretation of the error term. With point-sampled data, the transitory component in consumption induces negative autocorrelation in the error term. The error term using time-averaged data, however, will tend to be positively autocorrelated. As the averaging interval increases, the first-order autocorrelation coefficient approaches .25 (Working's [1960] result). Flavin's finding of little residual autocorrelation using quarterly data could be explained by a fortuitous canceling of the effects of stochastic consumption and temporal aggregation. The positive autocorrelation found by Hayashi (using annual data) is suggestive of temporal aggregation. In this study, the first-order autocorrelation coefficients tend to be close to .25, which suggests that temporal aggregation is empirically more important than transitory consumption.

IV. EMPIRICAL RESULTS

The data used in this study are seasonally adjusted, quarterly observations on real (1982 dollars) per capita consumption of non-durables and services and real per capita disposable income. All nominal variables are deflated by the consumption deflator. These measures are standard in empirical tests of the permanent income hypothesis, but they are, of course, not perfect. Consumption is measured by expenditures rather than the theoretically preferable flow of consumption services. Durable goods are excluded because the equality between expenditures and consumption is less likely to be true for this group. Blinder and Deaton [1985] suggest a number of modifications to the official measure of nondurable and services consumption (e.g., omitting clothing and shoes from nondurables), but they report that the modifications have minor effects on the series. Experimentation with various measures of consumption (excluding housing, clothing, and shoes, and including either the purchase of durables or the flow of services from durables) suggests that the results are insensitive to the particular measure.

I include in my auxiliary equation for income changes lags two through six of changes in consumption and disposable income plus an error correction term.(3) Although the entire parameter space was not searched, a number of other variables thought to predict disposable income were tried. I experimented with various combinations of lagged changes in consumption, disposable income, stock prices, deficits, government spending, money, and the relative price of nondurables and services, and I tried a variety of lag lengths. I found the estimates relatively robust to the choice of instruments.

I correct for MA(1) errors using a two-step, Hayashi-Sims procedure. In the first round the MA(1) error process is ignored and consistent parameter estimates are obtained from non-linear estimation of the restricted model, (11) [Mathematical Expression Omitted] (12) [Mathematical Expression Omitted] Next, using the residuals from the first-round estimates of equation (11), the MA(1) error process is approximated by a third-order AR with coefficients, [[Rho].sub.1], [[Rho].sub.2] and [[Rho].sub.3]. The AR coefficients are used to forward filter the dependent variables in (11) and (12) and the system is re-estimated. This procedure approximately whitens the error term in (11) while preserving the orthogonality conditions. Table II contains estimates of the parameters in (11) and (12) and the parameters of the AR(3) error process.(4)

Table II provides strong support for the hypothesis of liquidity constraints. The critical parameter [Mu] is large in an economic sense, .355, and the t-statistic of 3.86 is statistically significant at any reasonable level. The GMM estimate (Table I-A in the appendix) is a slightly lower .301 but is again highly statistically significant. The restriction implied by the generalized permanent income hypothesis model cannot be rejected at any reasonable significance level. The likelihood ratio test of the cross-equation restrictions is very near its expected level, and a visual inspection of the constrained and unconstrained parameter estimates (Table V-A in the appendix) confirms the fit of the restrictions.

Table III presents estimates broken down into various sub-periods. The first row summarizes the estimates from Table II for comparison. The second row reports the estimates for the sub-interval that drops pre-1955 data. This removes the disruptions associated with the Korean War. The third row drops both pre-1955 and post-1975 data. This sub-interval removes the disruptions associated with the supply disturbances of the 1970s and the temporary 1975 tax rebate. The final row reports the estimates for the period since 1975.

The estimate of [Mu] is large and statistically significant in each sub-period. The proportion of liquidity constrained consumption in the period prior to 1975, 46 percent, falls to 33 percent in the period after 1975. This decline is consistent with improvements in consumer credit services which would tend to relax constraints on borrowing. Similar, but slightly smaller estimates of [Mu] are obtained using the GMM approach.

V. COMPARISON WITH FLAVIN AND HAYASHI

Although the basic model in section III is very close to those used in Flavin [1981] and Hayashi [1982], the estimates are dramatically different. Hayashi's estimates are much lower, .17 and -.05, and are statistically insignificant. Flavin's implied estimate of [Mu] is large, but not statistically significant. In this section I carefully examine the differences in our specification and estimation. The major difference between my work and Flavin's is that I restrict liquidity constraints to operate only through current income. I show that this restriction is not rejected and that imposing the restriction greatly increases the precision of my estimates. I find that Hayashi's failure to find significant liquidity effects is explained by his use of an inappropriate measure of aggregate consumption.

Consider first Flavin's model. Flavin's model consisted of the two equation system, (13) [Mathematical Expression Omitted] and (14) [Mathematical Expression Omitted] The bracketed term in equation (14) is, apart from a constant term, simply expected changes in disposable income. As such, with the constraint [[Beta].sub.1] = [[Beta].sub.2] = , ..., = [[Beta].sub.7] = 0 imposed, Flavin's excess-sensitivity coefficient, [[Beta].sub.0], is equivalent to my measure of the percentage of liquidity constrained [Mu] divided by [Pi]. Her estimate of [[Beta].sub.0], .355, divided by her estimate of [Pi], .4488 (Flavin's measure of consumption is expenditures on nondurables only), gives a much higher point estimate of [Mu], .7767. Her estimate, however, with a t-statistic of 1.29, is statistically insignificant.

Table IV gives the results of an investigation into the factors responsible for our differing results. I begin with Flavin's specification (but using my measure of consumption, nondurables plus services, and the sample period 1955:I-1987:II) and sequentially change the underlying assumptions until my estimate is obtained. The first line of Table IV gives the estimate of [Mu] obtained using Flavin's exact specification, equations (13) and (14). The estimate and standard error are very close to those implied by Flavin's original work. Now, imposing the restriction that liquidity constraints operate only through current changes in income, the point estimate of [Mu] drops to .529 but the precision of the estimate increases dramatically. (The restriction cannot be rejected at any reasonable level of significance. The [[Chi].sup.2] (7) test statistic, -2 In([L.sub.u] - [L.sub.r]), is 5.874.) Subsequent changes in the specification, given in lines 2 to 6, result in smaller but more precise estimates of [Mu]. Note that the choice to model the trend as a deterministic exponential trend or as a stochastic trend has little effect on inference. The choice of a tightly parameterized alternative appears to be largely responsible for my finding a statistically significant estimate of [Mu].

Hayashi considered a generalization of the permanent income hypothesis model in which the rate of return to labor income differs from the rate of return to non-human wealth. Hayashi's solution for consumption is (15) [Mathematical Expression Omitted] where [r.sub.y] is the rate of return applied to human wealth and [Alpha] is the marginal propensity to consume out of wealth. The stochastic term, [[Upsilon].sub.t], is a composite error term containing the innovation to labor income at time t and an MA(1) component reflecting transitory shocks to consumption.

Assume now that some fraction of agents simply consume their current income. Let equation (15) describe consumption by unconstrained agents. Substituting the asset accumulation equation, (1), into (15), and denoting unconstrained consumption by a prime yields. (16) [Mathematical Expression Omitted] Letting total consumption be a weighted average of consumption by the two groups, we obtain (17) [Mathematical Expression Omitted] Equation (17) reduces to my equation (6) in the special case where the rate of return to human wealth is equal to the rate of return to non-human wealth, [r.sub.y] = r, and the marginal propensity to consume out of wealth, [Alpha], is equal to [r/(1 + r)]. Because our methods of nesting the liquidity constrained alternative differ, equation (17) differs slightly from the solution given in Hayashi (his equation (12)). Estimates of using exactly his model are similar to those presented below.

Table V updates Hayashi's estimates. As in Hayashi, equation (17) and the asset accumulation equation (1) are jointly estimated using TSP's non linear-instrumental variables procedure. I use as instruments a constant term and lags two through four of consumption and disposable income. The data definitions closely follow those used by Hayashi. Labor income is employee compensation plus government transfer payments plus labor's share of proprietor's income minus labor's share of income taxes minus contributions to social security. Quarterly data on household wealth are from the Federal Reserve flow-of-funds balance sheets. All variables are expressed in per capita terms and deflated by the consumption deflator. As in Hayashi, the estimates and standard errors are not corrected for residual serial correlation.

I use three different measures of consumption. For the first measure, I add the flow of durable services to nondurables and services expenditures to obtain total consumption. I constructed a series of durable services by multiplying the average stock of durables in each quarter by a constant quarterly rental rate of .06221 (a quarterly depreciation rate of .05221 plus a quarterly real interest rate of .01).(5) For comparison, I also used my standard measure and total consumption expenditures.

The estimate of from the first line of Table V, .366, is large, statistically significant and virtually identical to my estimate in Table III. Hayashi's corresponding estimate, (line 1 of his Table 3) is .0998 with a standard error of .261. The estimates in line 2, using nondurables and services, are similar. Modeling the flow of services as proportional to expenditures on nondurables plus services appears to work well. In other respects the results are similar to Hayashi's. The interest rate applied to labor income tends to be much larger than the interest rate applied to non-human wealth. Line 3 indicates that the purchase of new durable goods is more sensitive to current income than the consumption of durable goods.

The question remains--what accounts for our differing conclusions? The answer lies in differences in the measurement of the flow of durable services. Hayashi takes his consumption series from Christiansen and Jorgenson [1973]. They constructed the durable service flow by multiplying the stock of durables by an implicit rental rate, the depreciation rate plus the ex post real interest rate. This measure of the flow turns out to be more volatile than the National Income and Product Accounts (NIPA) series on purchases of new durable goods. Further, the flow of services and purchases of new durables are negatively correlated prior to 1968 but become strongly positively correlated in the 1970s. The anomalous behavior of their flow of services arises because changes in ex post rental rates are dominated by unexpected changes in the rate of durable goods inflation. During recessions in the earlier period, unexpected declines in inflation translated into increases in ex post rental rates and hence (given the smooth evolution of stocks of durables) increases in measured consumption flows. The variations in ex post rental rates imparted a spurious, countercyclical pattern to the flow of durable services and masked the influence of current income on actual consumption. I conclude that when the flow of services from durables is properly measured, Hayashi's formulation implies a large and statistically significant proportion of liquidity constrained consumption.

VI. MORE GENERAL ALTERNATIVES

In this section, I consider three popular generalizations of the permanent income hypothesis. First, I relax the assumption of time-separable utility and consider the case in which there are costs to rapidly adjusting consumption expenditures. Second, I allow some part of government expenditures to substitute directly for private consumption. Finally, I relax the assumption of constant real rates of return and test the liquidity constrained alternative in the popular constant relative risk-aversion/log-normal specification. In each case, I find strong support for the liquidity constrained alternative and little support for the generalization.

Costly Adjustment

The assumption of time-separable utility has come under increasing criticism. (See Browning [1986] for a survey and references.) The intuitively plausible alternative, that consumers face costs of rapidly adjusting consumption, has been incorporated into a number of studies. West [1988] suggested that costs of rapidly adjusting consumption could provide a rationale for Deaton's [1987] result that consumption is excessively smooth. In this section we test whether costly adjustment might also explain the rejections of the permanent income hypothesis reported in the previous sections.

Adding first-order adjustment costs to the basic model is the simplest way to introduce non-time-separability. Following Blundell [1987], assume that the momentary utility function is (18) [Mathematical Expression Omitted] Agents pay an increasing price of lost utility for abruptly changing consumption.

Using the utility function (18) and the approximation ([Beta] R) = 1, we obtain a simple characterization of consumption changes. The Euler equations for this problem area a set of second-order difference equations in consumption changes. Letting [Gamma] denote the stable root of the difference equation, the solution for consumption changes is (19) [[Delta] c.sub.t] = [[Lambda].sub.0] + [Gamma] [[Delta] c.sub.t-1] + [s.sub.t]. The parameter [Gamma] reflects the penalty for adjusting consumption. [Gamma] is bounded between zero and one and is monotonically increasing in [u.sub.3], the coefficient on changes in consumption in the utility function, (18). The error term, [[Epsilon].sub.t], as in the standard case, has a white noise component which reflects innovations in income and an MA(1) component which reflects shocks to the utility function. It is clear from (19) that the strong orthogonality condition tested in the previous sections is violated here. Changes in consumption are correlated with all past shocks to taste and all past innovations in labor income.

Proceeding in the manner of section III, I assume that a fraction of consumption, [Mu], is accounted for by liquidity constrained agents and the remainder consumed according to (19). The generalized model is (20) [Mathematical Expression Omitted]

The presence of lagged consumption in (20) complicates the analysis of temporal aggregation. To obtain the implied relationship among aggregates time-averaged over m periods, filter (20) with (21) [Mathematical Expression Omitted] and point sample at T = t, t [+ or -] m, t [+ or -] 2m,... This yields (22) [Mathematical Expression Omitted] where [[Eta]*.sub.T] is obtained by point sampling [Mathematical Expression Omitted] By inspection, [[Eta].sub.t] is uncorrelated with all information prior to period t - 3m + 2 (the largest power of L is 3m - 3 and [[Epsilon].sub.t] is an MA(1)). Therefore, temporal aggregates lagged three periods are uncorrelated with [[Eta].sub.T] and consistent estimation of the parameters of (20) requires restricting the auxiliary variables to those lagged three or more periods.

Formally, estimates of [Mu] and [[Gamma].sup.m] are obtained from estimating the projections (23a) [Mathematical Expression Omitted] (23b) [Mathematical Expression Omitted] (23c) [Mathematical Expression Omitted] (23d) [Mathematical Expression Omitted] subject to the cross-equation restrictions (23e) [Mathematical Expression Omitted] and (23f) [Mathematical Expression Omitted] The FIML estimates presented in Table VI were computed using the Hayashi-Sims procedure. The information set consisted of lags three to six of changes in consumption and disposable income.

Allowing costly adjustment does not alter the basic conclusion--estimates of [Mu] remain large and statistically significant. Estimates of the costly adjustment parameter, [[Gamma].sup.m], are either implausibly high (above one) or implausibly low (negative). In either case, the estimates are not statistically significant. The GMM estimates (Table II-A in the appendix) of [[Gamma].sup.m] are more plausible, but, again, [Mu] is large and statistically significant. The standard errors on [[Gamma].sup.m] are too large to confidently reject the hypothesis of costly adjustment, but the large estimates of [Mu] cannot be attributed to the omission of costly adjustment.

Utility from Government Consumption

Barro [1981] argues that a general model of consumption should include the direct effect of government purchases of goods and services on private utility. Aschauer [1985], using methods similar to those used here, found Hall's model of consumption could not be rejected once government purchases were allowed to partially substitute for private consumption. Aschauer modeled effective consumption as (24) [Mathematical Expression Omitted] where [Theta] is a parameter reflecting the substitutability of government consumption for private consumption. Aschauer's basic result follows directly from substituting (24) into the model of section II, (25) [Mathematical Expression Omitted] Information lagged two periods will be correlated with changes in consumption to the extent that the information is useful in predicting changes in government purchases.

Aschauer's hypothesis is easily nested in the model of section III. The general model is characterized by the following orthogonality condition, (2) [Mathematical Expression Omitted] Equation (26) imposes a set of cross-equation restrictions on the projections of changes in consumption, income, and government spending on instruments lagged two or more periods. Table VII presents the FIML estimates of [Mu] and [Theta] from jointly estimating these projections with the restrictions imposed. As instruments, I use lags two to six of changes in consumption, income, and government spending, and I correct for MA(1) errors using the Hayashi-Sims procedure.

Allowing for direct effects of public spending does little to alter the basic conclusion. Estimates of [Mu] remain statistically significant and even rise slightly. Estimates of [Theta] (the degree of substitutability between private and government consumption) are not statistically significant. Of course, given the size of the standard errors on [Theta], the hypothesis that [Theta] is large and positive cannot be rejected; however, the large estimates of [Mu] cannot be attributed to incorrectly constraining [Theta] to be zero.

Stochastic Returns.

Perhaps the most restrictive assumption of the models considered thus far is the constancy of real rates of return. Hansen and Singleton [1982; 1983] considered generalizations of the permanent income hypothesis that allow stochastically varying real rates of return. Hall's [1988] treatment suggests that to a first approximation, consumption can be modeled as (27) [Mathematical Expression Omitted] where [r.sub.t-1] is the real, after-tax, rate of return in period t-1, [e.sub.t] is the unpredictable innovation in wealth, k depends on the covariances of consumption and real returns and is assumed constant, and [Sigma] is the intertemporal elasticity of substitution. Changes in the log of consumption are predictable to the extent that changes in expected real rates of return are predictable. Hall [1988] estimates (30) taking proper account of temporal aggregation and finds [Sigma] close to zero; however, he provides no test of the model's restrictions. Hansen and Singleton [1982; 1983] test and reject the restrictions implicit in (27), but they provide no metric with which to assess the economic significance of their findings.

A simple test of equation (27) is obtained by nesting the model in a more general framework which allows some liquidity constrained consumers. The general model is (28) [Mathematical Expression Omitted] The parameter [Mu] provides a measure of the extent to which predicted changes in consumption are explained by predicted changes in disposable income rather than by predicted movements in asset returns.

Hall [1988] discusses at length the problem of estimating (27) (and thus (28)) with time-averaged data. Hall shows that [r.sub.t-1] must be replaced with r*, an average (using the filter in equation (8)) of returns observed over the true decision intervals. Values of r* lagged three or more periods and temporal aggregates lagged two or more periods are valid instruments for inclusion in the auxiliary regressions. Hall approximates r* from monthly data on real, after-tax returns on treasury bills. Rather than attempt to update his series, I use the exact data constructed in Hall [1988].(6) Because Hall's data run only through 1983:IV, I am unable to use the entire sample period.

With [r*.sub.t-2] omitted from {[Z.sub.t-2]}, the three projection equations (29a) [Mathematical Expression Omitted] (29b) [Mathematical Expression Omitted] (29c) [Mathematical Expression Omitted] are subject to the cross-equation restrictions, (30a) [Mathematical Expression Omitted] and (30b) [Mathematical Expression Omitted]

The results in Table VIII show large and statistically significant estimates of [Mu] and small, statistically insignificant estimates of [Sigma]. Close inspection of Table VIII, however, reveals two problems. First, estimates for the period 55:I to 74:I are not reported because the algorithm failed to converge. (The estimates of [Sigma] drifted toward large negative values and the estimates of tended towards one.) Nonetheless, the GMM estimates for this time period (Table III-A in the appendix) confirm the basic conclusion. Second, the large value of the [X.sup.2] statistic for the period 48:IV-83:IV, although not statistically significant at the 5 percent level (the 5 percent critical value for a [X.sup.2] (12) is 21.026), casts some doubt on the specification. Inspection of the data and some experimentation reveal that the lack of fit is due almost entirely to the tremendous (7 percent) increase in disposable income the first quarter of 1950. The poor fit of the restrictions appears to be attributable to the large, one-time payment of veterans benefits.(7) Because this was an unusual but perfectly anticipated event, its inclusion tends to distort the time-series behavior of disposable income.

VII. CONCLUSION

This paper found that between 30 and 40 percent of aggregate consumption expenditures is accounted for by liquidity constrained behavior rather than the permanent income hypothesis. In contrast to the work of Flavin [1981] and Hayashi [1982], the critical estimate of the proportion of liquidity constrained consumption is large and highly statistically significant. The estimation paid careful attention to the time-series properties of the consumption and disposable income processes and to the problems of temporal aggregation. Further, the estimates were shown to be robust to the hypothesis of costly adjustment, Barro's [1981] public spending hypothesis, and to stochastically varying rates of return.

Although the estimates indicate a large proportion of liquidity constrained households, there remains an even larger proportion that follow the standard permanent income hypothesis. As such, some caution must be exercised when considering the policy implications of this research. For example, if temporary tax cuts affect only unconstrained, high-income families, they may have only minor effects on aggregate spending. I am currently pursuing these policy implications.

APPENDIX

Tables I-A through IV-A contain GMM estimates of the models considered in the text. The GMM estimates are obtained from a two-step procedure. The first-round estimates minimize [[u [prime] Z(Z [prime] Z).sup.-1] Zu], where Z is the matrix of instruments and u is the vector of residuals. The second-round estimates minimize [u [prime] Z([W.sup.-1])Z [prime] u] where u is the vector of residuals from the first round and the matrix W is [Mathematical Expression Omitted] and the weighting kernel is a Newey-West-Bartlett window, [Mathematical Expression Omitted] I chose a window width of three (L = 3) to allow for a low-order MA process on the residuals. Hansen [1982] shows that these estimates are asymptotically normal with covariance matrix [[(X [prime] Z) [W.sup.-1] (Z [prime] X)].sup.01]. Under the null hypothesis, [u [prime] Z([W.sup.-1])Z [prime] u] is asymptotically distributed as a [[Chi].sup.2] with degrees of freedom equal to the number of overidentifying restrictions. All calculation were performed on Regression Analysis for Time Series (RATS).

TABLE I-A

PIH with Liquidity Constraints GMM Estimation of Equation 6 Instruments = ([Delta] [c.sub.t-2,..., [Delta] [y.sub.t-2,..., [Delta] [y.sub.t-6])
 48:IV-87:II 55:I-87:II 55:I-74:IV 75:I-87:II
 Mu .301** .365** .334** .290**
 (.059) (.065) (.075) (.074)
 [X.sup.2] (9) 7.254 6.723 7.38 3.883


Standard errors in parentheses. Statistically significant at the 5%, and 1% levels (two-tailed) are denoted by * and ** respectively. [Tabular Data I to VIII Omitted] [Tabular Data II-A to V-A Omitted]

(1)Lewbel [1987] shows that the exact constraints on [Gamma][sub.0] and [Gamma][sub.1] imposed by quadratic utility imply implausible estimates of the "bliss" level of consumption. (2)The emperical results are not sensitive to this approximation. Estimating the model with consumption and disposable income quasi-differenced by (1 - [Gamma][sub.1] L)for a number of reasonable values of [Gamma][sub.1] yielded essentially the same results. (3)If, as work by Engle and Granger [1987] and Campbell [1987] suggests, the levels of consumption and disposable income cointegrate, vector autoregressions in differences will be misspecified. I include an error correction term in the income equation, but it turns out to have little effect on inference. (4)The estimate of is the average proportion of non-durable and services in total consumption and is treated as as constant . All calculations were performed using Shazam's non-linear full information-maximum likelihood (FIML) routine. Standard errors are obtained from an approximation to the information matrix at the maximum. (5)I constructed a quarterly series on the stock of durables from the series on annual stocks, quarterly data on purchases and a constant depreciation rate of .05221. (6)I wish to thank Robert Hall for making his data available. (7)I wish to thank Tom Campbell for pointing this out to me. Campbell and Mankiw [1989] independently obtained results similar to those reported in this section.

REFERENCES

Aschauer, David Alan. "Fiscal Policy and Aggregate Demand." The American Economic Review, March 1985, 117-27. Barro, Robert, "Output Effects of Government Purchases." Journal of Political Economy, December 1981, 1086-121. Blinder, Alan S., And Angus S. Deaton. "The Time Series Consumption Function Revisited." Brookings Papers on Economic Activity 2, 1985, 465-511. Blundell, R. "Econometric Approaches to the Specification of Life-Cycle Labour Supply and Commodity Demand Behavior." Econometric Reviews 6(1), 1987, 103-65. Browning, Martin. "Testing Inter-Temporal Separability in Models of Household Behavior." Unpublished manuscript. McMaster University, 1986. Campbell, John Y. "Does Saving Anticipate Declining Labor Income? An Alternative Test of the Permanent Income Hypothesis." Econometrica, November 1987, 1249-74. Campbell, John Y., and N. Gregory Mankiw. "Consumption, Incme and Interest Rates: The Euler Equation Approach Ten Years Later." Paper presented at the NBER Macroeconomics Conferene, Cambridge, MA, March 1989. Christiano, Lawrence J., Martin Eichenbaum, and David Marshall. "The Permanent Income Hypothesis Revisited." National Bureau of Economic Research Working No. 2209, 1987. Christiansen, Laurits R., and Dale W. Jorgenson. "Measurin Economic Performance in the Private Sector," in The Measurement of Economic and Soial Performance, edited by Milton Moss. New York: Columbia University Press, 1973, Deaton, Angus. "Life-cycle Models of Consumption: Is the Evidence Consistent with the Theory?" in Advances in Econometrics: Fifth Congress, vol. 2, edited by Truman F. Bewley. New York: Cambridge University Press, 1987. Engle, Robert F., and Clive W.J. Granger. "Cointegration and Error-Correction: Representation, Estimation and Testing." Econometrica, March 1987, 251-76. Flavin, Marjorie A. "The Adjustment of Consumption to Changing Expectations aabout Future Income." Journal of Political Economy, October 1981, 974-1009. Hall, Robert. "Stochastic Implications of the Life-Cycle Permanent Income Hypothesis." Journal of Political Economy, October 1978, 971-88. __. "Consumption." National Bureau of Economic Research Working Paper No. 2265, May 1987. __. "Intertemporal Substitution in Consumption." Journal of Political Ecomomy, April 1988, 339-57. Hall, Robert, and Frederick S. Mishkin. "The Sensitivity of Consumption to Transitory Income: Estimates from Panel Data on Households." Econometrica, March 1982, 461-81. Hansenl, L.P. "Large Sample Properties of Generalized Method of Moments Estimators." Econometrica, September 1982, 1029-54. Hansen, L.P., and K.J. Singleton," Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models." Econometrica, September 1982, 1269-86. __. "Stochasti Consumption, Risk Aversion and the Temporal Behavior of Asset Returns." Journal of Political Economy, April 1983, 249-65. Hayashi, Fumio. "The Permanent Income Hypothesis: Estimation and Testing by Instrumental Variables." Journal of Political Economy, October 1982, 895-916. Hayashi, Fumio and Christopher A. Sims. "Nearly Efficient Estimation of Time Series Models with Predetermined, but Not Exogenous, Instruments." Econometrica, May 1983, 783-98. Lewbel, Arthur. "Bliss Levels That Aren't." Journal of Political Economy, February 1987, 211-15. Nelson, Charles R. "A Reappraisal of Recent Tests of the Permanent Income Hypothesis." Journal of Political Economy, June 1987, 641-46. Quah, Danny. "Permanent and Transitory Movements in Labor Income: An Explanation for |Excess Smootheness' in Consumption." Journal of Political Economy, June 1990, 449-75. Sargent, Thomas J. Macroeconomic Theory, 2nd ed. New York: Academic Press, 1987. Sims, Christopher A. "Bayesian Skeptcism on Unit Root Econometrics." Journal of Economic Dynamics Control, 12, 1988, 463-74. West, Kenneth D. "The Insensitivity of Consumption to News About Income." Journal of Monetary Economics, January 1988, 17-33. Working, Holbrook. "A Note on the Correlation of First Differences of Averages in a Random Chain." Econometrica, October 1960, 916-18.

MATTHEW J. CUSHING, Assistant Professor, Emory University. I wish to thank members of the Emory/Georgia State/Atlanta Fed Seminar Series, an anonymous referee and especially Mary McGarvey and Lucy Ackert for helpful comments.
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Date:Jan 1, 1992
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