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Finite horizons, infinite horizons, and the real interest rate.

FINITE HORIZONS, INFINITE HORIZONS, AND THE REAL INTEREST RATE

Using an overlapping-generations model in which households may have either finite

or infinite horizons, I derive the implications of each horizon for the steady-state real

interest rate. I then formulate an econometric model of the steady-state real interest

rate and devise tests that can distinguish between finite and infinite horizons. These

tests are applied to annual and quarterly U.S. data, which span the period 1875-1988.

The results are inconsistent with finite horizons, and broadly consistent with infinite

horizons.

I. INTRODUCTION

In conventional macroeconomic analysis, government budget deficits raise perceived wealth and stimulate aggregate demand because households are modeled as having finite horizons. If households are instead modeled as having infinite horizons, budget deficits need not stimulate aggregate demand and hence need not affect output, employment, interest rates, and the price level. The reason is that households with infinite horizons do not view the government debt as net wealth; thus they may treat budget deficits (i.e., future taxes) and current taxes as equivalent. Ricardian equivalence is said to hold if households treat budget deficits as equivalent to current taxes.(1)

Three reasons are typically advanced for assuming households have finite horizons. First, the individuals who comprise current households have finite lifetimes. Second, some households face binding liquidity constraints either now or in the future.(2) Third, boundedly rational households may choose to behave myopically, acting as if their current decisions do not affect allocations occurring beyond some horizon.(3)

These three reasons, however, do not necessitate modeling households as having finite horizons. First, Barro [1974] has shown that finite lifetimes need not imply finite horizons. Second, Hayashi [1987] and Yotsuzuka [1987] have shown that lenders may impose liquidity constraints that vary one-for-one with the government debt. In such an environment, the government debt relaxes no liquidity constraints. Third, because bounded rationality assumes fixed costs of decision making, it may be able to explain why aggregate demand would respond to small, short-term budget deficits but cannot explain why aggregate demand would respond to large, or long-term, budget deficits.

Because logically consistent reasons can be adduced for modeling households as having either finite or infinite horizons, the issue of which is the better modeling strategy should ultimately be decided by empirical analysis. This paper offers such an analysis. It investigates the implications of finite and infinite horizons for the steady-state behavior of the real interest rate. The evidence reported here is inconsistent with finite horizons and broadly consistent with infinite horizons.

The rest of the paper is organized as follows. Using Barro's [1974] overlapping-generations model, I derive its implications for the steady-state real interest rate in section II. These implications differ according to whether households have inoperative bequest motives and finite horizons or operative bequest motives and infinite horizons. In section III, I formulate an econometric model that encompasses the implications of the theoretical model of section II for both of these cases. In sections IV, V, VI, and VII, I test these implications using U.S. data. In section VIII, I investigate whether the real interest rate evidences any short-term departures from Ricardian equivalence. Finally, in section IX, I draw some tentative conclusions.

II. AN OVERLAPPING-GENERATIONS MODEL OF THE STEADY STATE

In this section, I lay out Barro's overlapping-generations model, which is the simplest model in which one can derive the implications of finite and infinite horizons for the steady-state real interest rate. The same basic results can be obtained in more complicated models but only at the expense of more elaborate analysis.

The economy is assumed to be closed and to experience no aggregate growth. Each period its population consists of a young household born that period and an old household born the previous period. Time and generations are indexed by the subscript t. The household born in period t chooses [C.sub.yt [prime]] [C.sub.ot + 1 [prime]] and [B.sub.t] in order to maximize the utility function (1) [Mathematical Expression Omitted] subject to the budget constraints (2) [Mathematical Expression Omitted] and (3) [Mathematical Expression Omitted] and to the nonnegativity constraint (4) [B.sub.t] [is greater than or equal to] 0.

The variables [C.sub.yt] and [C.sub.ot] are the household's consumption in the first and second period of its life, [B.sub.t] is its bequest at the end of period t to the household that will be born in period t + 1, [A.sub.t] is its stock of assets net of bequests at the end of period t, [r.sub.t] is the real interest rate on assets carried from period t - 1 to period t, [W.sub.t] is its wage income, [T.sub.t] is its tax payment, and [J.sub.t] is the value function for this problem. For simplicity, I have assumed that the household supplies one unit of labor inelastically in the first period of its life and that the only taxes are lump sums collected from young households. The utility function is increasing in each period's consumption and in the utility obtained by the next generation and has all of the properties usually assumed of utility functions.

The first-order conditions for the maximum are equations (2) and (3) together with(4) (5) [Mathematical Expression Omitted] and either [B.sub.t] > 0 and (6) [Mathematical Expression Omitted] or (7) [B.sub.t] = 0. Equations (5) and (6) have the straightforward interpretation that one unit of goods must contribute the same amount on the margin to utility whether consumed in the first period of life, invested for one period and then consumed in the second period of life, or left as a bequest when it is optimal to leave a positive bequest.

Suppose that the economy converges to a steady state. Two possible cases arise: (i) bequests converge to zero; and (ii) bequests converge to a positive value. In case (i), each household acts as if it is only concerned with events that take place in its own lifetime. Its horizon is thus finite. In case (ii) by contrast, each household acts as if its horizon is infinite even though its lifetime is finite. As a result, its current decisions completely internalize the effects of all future events. See Barro [1974] for further discussion of this point.

I consider case (i) first. Equations (2), (3), (5), and (7) imply that (8) [Mathematical Expression Omitted] where an asterisk attached to a symbol indicates a steady-state value. If consumption in both periods of life is a normal good, the function a(.) has the property 0 < [a.sub.1] <1.(5) To complete the model, I make the following assumptions. Assets consist of reproducible capital [K.sub.t] and one-period government debt [D.sub.t]. The government budget constraint is (9) [G.sub.t] + (1 + [r.sub.t]) [D.sub.t-1] = [T.sub.t] + [D.sub.t [prime]] where [G.sub.t] is government purchases. Production is F ([K.sub.t]), where the production function F(.) is increasing, strictly concave, and satisfies the Inada conditions. Competition in the rental market for capital, in the labor market, and in the product market equates the real interest rate to F [prime] [K.sub.t]], the marginal product of capital, and wage income to F ([K.sub.t]) - ([[K.sub.t] F [prime]) ([K.sub.t]). With these assumptions, equilibrium in the steady state is characterized by (10) [Mathematical Expression Omitted] where [Kappa] ([r.sup.*]), the steady-state demand for capital, is defined by the identity F [prime] [Kappa] ([r.sup.*]) = [r.sup.*]. Applying the implicit function theorem to equation (10) reveals that (11) [r.sup.*] = [Rho] ([D.sup.*], [G.sup.*]) with (12) [Mathematical Expression Omitted] and (13) [Mathematical Expression Omitted] In order for equilibrium to be stable, the expression in brackets in equations (12) and (13) must be positive. Therefore, if horizons are finite, it follows from equations (12) and (13) that [[Rho].sub.1] > 0 and [[Rho].sub.2] > 0.(6)

The intuition for [[Rho].sub.1] > 0 is straightforward. Each period the old household finances its consumption by selling its assets to the young household. At any given real interest rate, the assets sold by the old household rise one-for-one with an increase in the steady-state government debt, and the assets bought by the young household fall because increased taxes are required to service the increased government debt. If equilibrium is stable, the steady-state real interest rate must rise in order to reestablish equilibrium. The key ingredient in reestablishing equilibrium is the crowding out of capital induced by the increased real interest rate. With less capital existing, the old household has fewer assets to sell. If [a.sub.2] - [aa.sub.1] > 0, the increased real interest rate also facilitates the reestablishment of equilibrium by increasing the amount of assets that the young household is willing to buy. The crowding out of capital, however, suffices, or more than suffices, to reestablish equilibrium if [a.sub.2] - [aa.sub.1] [is less than or equal to] 0 so long as [a.sub.2] - [aa.sub.1] > 1/F [double prime].

I now turn to case (ii). In the steady state, equations (2), (3), and (9) imply that (14) [Mathematical Expression Omitted] Examining the budget constraint (14) reveals that [J.sup.*], the steady-state value function, must depend only on [W.sup.*] - [G.sup.*] + [r.sup.*] ([B.sup.*] - [D.sup.*]). Therefore, equations (5), (6), and (14) imply that (15) [Mathematical Expression Omitted] (16) [Mathematical Expression Omitted] and (17) [Mathematical Expression Omitted] where [c.sub.y] ([.]), [c.sub.o] ([.]), and b ([.]) are functions. According to equation (17), bequests move one-for-one with the government debt in the steady state; i.e., the government debt passes intact from generation to generation. Consequently, consumption in the two periods of life is independent of the government debt in the steady state, as is the amount of assets that each period's old household sells to the young household.(7)

The intuition for this result is straight-forward. Because households have infinite horizons, they view the government debt as being exactly offset by the future taxes that will service it and hence wish to make allocative decisions that are independent of the government debt. They can generate this outcome by always leaving the government debt to the next generation as a bequest. The household of each generation can then use the interest received from holding the government debt in order to pay the taxes required to service the government debt. The optimal allocations therefore remain feasible and are chosen.

Using the equations (15)-(17) and the assumptions made above, one can rewrite equation (6) as (18) [Mathematical Expression Omitted] Applying the implicit function theorem to equation (19) then yields (19) r* = [Eta] (G*). Consequently, in the steady state, the real interest rate is independent of the government debt but may depend on government purchases. The derivative [Eta]', however, cannot be signed in general.(8) In the special case in which the utility function is separable between the consumption of the current generation and the utility of the next generation, however, [U.sub.3] is constant, and hence the real interest rate is independent of government purchases as well as the government debt. In this case of intergenerational separability, the real interest rate is equated in the steady state to the constant rate at which each generation discounts the next generation's utility. Equation (18) can then be rewritten as (20) [Mathematical Expression Omitted] where [Rho] [tilde] is the constant intergenerational discount rate.

I derived this result assuming that all households face perfect capital markets and have either finite or infinite horizons. These assumptions can be relaxed somewhat. Suppose that only some households have infinite horizons and face perfect capital markets in the steady state. If the intergenerational discount rate is the constant [Rho] [tilde], the real interest rate must assume that value in the steady state even though all other households have finite horizons or face liquidity constraints.(9) Consequently, Ricardian equivalence holds in the steady state if any households at all have infinite horizons, face perfect capital markets, and discount the next generation's utility at a constant rate. Of course, unless the fraction of such households is appreciable, the economy can deviate substantially from Ricardian equivalence in the short run. Nevertheless, because the steady state is Ricardian, a short-run deviation in one direction must be followed by compensating deviations in the other direction. For example, if a surprise budget deficit induces one unit of disinvestment, extra investment totaling one unit must occur in subsequent periods.

The next section formulates an econometric model with which one can estimate how government debt and government purchases affect the real interest rate in the steady state. One can then distinguish whether households have finite or infinite horizons.

III. ECONOMETRIC SPECIFICATION OF THE MODEL

A linearized, stochastic analogue to equation (14) is (21) [Mathematical Expression Omitted] where t is an index of time; [r.sub.t] is the ex post real interest rate; [d.sub.t] and [g.sub.t] are logarithms of the per capita government debt and per capita government purchases; [z.sub.t] is the vector of all other stochastic variables that either affect the real interest rate or are useful in predicting [d.sub.t] and [g.sub.t]; [E.sub.t] is the expectation operator conditional on [d.sub.t], [d.sub.t-1] [d.sub.t-2], ..., [g.sub.t], [g.sub.t-1], [g.sub.t-2], ..., and [z.sub.t], [z.sub.t-1], [z.sub.t-2], ...; [[Rho].sub.d] and [[Rho].sub.g] are parameters; and [[Rho].sub.z] is a parameter vector. Equation (21) encompasses both equations (19) and (20) as well. Under the null hypothesis that horizons are infinite, [[Rho].sub.d]] = 0; under the alternative hypothesis that horizons are finite, [[Rho].sub.d] > 0. If the intergenerational discount rate is also constant, then [[Rho].sub.g] = 0 under the null hypothesis and [[Rho].sub.g] [is not equal to] 0 under the alternative hypothesis.(10) If the intergenerational discount rate is not constant, [[Rho].sub.g] may be either positive or negative.

In the next five sections, I use U.S. data to investigate the empirical implications of this model. It will become apparent that these implications depend in important ways on the stochastic properties of the government debt, government purchases, and the other variables affecting the real interest rate.

IV. ARE THE GOVERNMENT DEBT AND GOVERNMENT PURCHASES STATIONARY?

It is important to investigate whether [d.sub.t] and [g.sub.t] are stationary or difference-stationary because of the following two propositions.(11,12)

PROPOSITION 1. Whether horizons are finite or infinite has no implications for the relationship between [d.sub.t] and [r.sub.t] if [d.sub.t] is stationary.

PROPOSITION 2. Whether horizons are finite or infinite and whether the inter-generational discount rate is constant or not have no implications for the relationship between [g.sub.t] and [r.sub.t] if [g.sub.t] is stationary. Loosely speaking, these propositions state that if [d.sub.t] and [g.sub.t] never change permanently, it is impossible to determine how purely hypothetical permanent changes in [d.sub.t] and [g.sub.t] would affect [r.sub.t].

I use two variants of the augmented Dickey-Fuller test in order to investigate whether [d.sub.t] and [g.sub.t] are stationary or difference-stationary. One performs these tests by applying ordinary least squares to (22) [Mathematical Expression Omitted] and (23) [Mathematical Expression Omitted] where [x.sub.t] is the series under consideration; [Alpha], [Beta], [[Phi].sub.[Mu]], [[Phi].sub.[Tau]], and the [Theta] s are parameters; [Rho] is an appropriately chosen lag length; and [Xi] is a serially uncorrelated error term with a zero mean and a finite and constant variance. Next, one calculates the t-ratios [[Tau].sub.[Mu]] and [[Tau].sub.[Tau]] for testing the null hypotheses [[Phi].sub.[Mu]] = 1 and [[Phi].sub.[Tau]] = 1 against their alternatives [Phi] < 1 and [[Phi].sub.[Tau]] < 1 and the F-ratios [[Phi].sub.1] and [[Phi].sub.3] for testing the null hypotheses [Mathematical Expression Omitted] against their alternatives [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. One then compares these ratios with the critical values of their distributions, which are nonstandard but have been tabulated by Fuller [1976] and Dickey and Fuller [1981]. Under the null hypothesis, [x.sub.t] is difference-stationary either without or with drift; under the alternative hypothesis, [x.sub.t] is stationary around either a fixed mean or a deterministic trend.

My measures of [d.sub.t] and [g.sub.t] are the logarithm of the market value of the privately held federal debt at the end of period t divided by the consumer price index for the last month of the period and by mid-period population and the logarithm of real federal purchases of goods and services divided by mid--period population.(13,14) Table I reports [[Tau].sub.[Mu]], [[Tau].sub.[Tau]], [[Phi].sub.1], and [[Phi].sub.3] for these series, based on both annual and quarterly data. The statistics for [d.sub.t] do not enable one to reject the null hypothesis that [d.sub.t] is difference-stationary.(15) The statistics for [g.sub.t] give conflicting signals: one can reject the null hypothesis that [g.sub.t] is difference-stationary with no drift in favor of the alternative hypothesis that it is stationary around a fixed mean, but one cannot reject the null hypothesis that it is difference-stationary with drift in favor of the alternative that it is difference-stationary around a deterministic trend. It is, however, implausible that government purchases are stationary around a constant mean. Therefore, on the whole, [d.sub.t] and [g.sub.t] appears to be difference-stationary.

V. IS THE REAL INTEREST RATE STATIONARY?

The evidence presented in the previous section suggests that [d.sub.t] and [g.sub.t] are difference-stationary. It is therefore important to know whether the real interest rate is stationary or difference-stationary because of the following proposition.

PROPOSITION 3. Suppose that [d.sub.t] and [g.sub.t] are difference stationary. Then [r.sub.t] can be stationary only if horizons are infinite and the intergenerational discount rate is constant. If either horizons are finite or the intergenerational discount rate is variable, then [r.sub.t] must be difference stationary.(16)

The intuition for this proposition is straightforward. If [d.sub.t] and [g.sub.t] are difference-stationary, their levels change permanently over time. If [r.sub.t] is stationary, its level never changes permanently. Therefore, [r.sub.t] can be stationary and [d.sub.t] and [g.sub.t] can be difference-stationary only if the level of [r.sub.t] is not related in the long run to the level of [d.sub.t] and [g.sub.t].

My measure of the real interest rate is the continuously compounded quarterly average of the ex post real commercial paper rate.(17) I have adopted this definition here in order to have as long a sample as possible: 114 years of quarterly data. A long sample is important because unit-root tests have notoriously low power (Cochrane [1988]).

Table I reports [[Tau].sub.[Mu]], [[Tau].sub.[Tau]], [[Phi].sub.1], and [[Phi.sub.3] for this measure of [r.sub.t] for three sample periods: 1877:II-1988:IV, 1919:I-1988:IV, and 1947:I-1988:IV. One can easily reject the null hypothesis that [r.sub.t] is difference-stationary in favor of the alternative hypothesis that it is stationary for each sample period.

Schwert [1987] has shown that augmented Dickey-Fuller tests are subject to large size distortions if the series under consideration cannot be well represented as a low-order autoregression. When moving--average structure is important, the size distortions typically found result in too frequent rejection of the null hypothesis. Phillips [1987] and Phillips and Perron [1988] have suggested alternative tests that may have better properties in this case. Applying their tests, however, still leads to the conclusion that [r.sub.t] is stationary.(18) Therefore, horizons appear to be infinite and the intergenerational discount rate appears to be constant.(19)

VI. STEADY-STATE RELATIONSHIPS ON LEVELS

The evidence in section IV suggests that [d.sub.t] and [g.sub.t] are difference-stationary. Therefore, the following proposition appears to be germane.

PROPOSITION 4. Suppose that [d.sub.t] and [g.sub.t] are difference-stationary and that ordinary least squares is applied to the equation. (24) [Mathematical Expression Omitted] where [e.sub.t] is an error term. Superconsistent estimates result if [e.sub.t] is stationary; i.e., if only [d.sub.t] and [g.sub.t] have permanent effects on [r.sub.t]. The restriction that [e.sub.t] be stationary is crucial. If [e.sub.t] is instead difference-stationary, ordinary least squares yields inconsistent estimates and danger of what Granger and Newbold [1974] have termed "spurious regressions." It is therefore important to test whether [e.sub.t] really is stationary.(20) It is straightforward to do so if [e.sub.t] can be well approximated by the autoregression. (25) [Mathematical Expression Omitted] where [Phi] and the [Theta] s are parameters, [Rho] is an appropriately chosen lag length, and [[Epsilon].sub.t] is a serially uncorrelated error term with a zero mean and a finite and constant variance. Under the null hypothesis that [Phi] = 1 [e.sub.t] is differences-stationary; under the alternative hypothesis that [Phi] < 1, [e.sub.t] is stationary.

The superconsistency of the ordinary-least-squares estimates [[Rho] [caret].sub.d] and [[Rho] [caret].g] implies that they converge in probability to [[Rho].sub.d] and [[Rho].sub.g] even if [e.sub.t] is correlated with [Delta] [d.sub.t] and [Delta] [g.sub.t]; see Engle and Granger [1987]. Moreover, in large samples, the residuals can be treated as if they are the true error terms. Therefore, applying ordinary least squares to equation (25) with the residuals' replacing the error terms yields a consistent estimate of [Phi]. The null hypothesis can then be tested against the alternative hypothesis by comparing the t-ratio ([Phi] [caret] - 1)/s([Rho] [caret] with the critical values of its distribution, where s([Rho] [caret]) is the standard error of the ordinary least squares estimate [Rho] [caret].

I fitted equation (24) to two samples: annual averages of the ex post real commercial paper rate, the real federal debt, and real federal purchases for the sample period 1929-1988; and quarterly averages of the same series for the sample period 1947:I-1988:IV.(21) The resulting equations are (26) [Mathematical Expression Omitted] and (27) [Mathematical Expression Omitted] The standard errors from the computer output are not reported because they are inconsistent. Pretesting suggests that p = 1 for the annual data and p = 4 for the quarterly data. Fitting equation (25) to the residuals from each regression yields t-ratios of -4.92 for the annual data and -3.31 for the quarterly data. Monte Carlo simulations reveal that these t-ratios are statistically significant at the .01 and .11 levels, respectively.(22) Consequently, the former t-ratio provides strong evidence that [e.sub.t] is stationary, but the latter t-ratio provides only weak evidence.

Suppose that [e.sub.t] is in fact stationary so that ordinary least squares provides superconsistent estimates of [[Rho].sub.d] and [[Rho].sub.g]. Clearly, in that case, equations (26) and (27) provide no evidence of finite horizons since [[Rho] [caret].sub.d] is negative for both samples. This evidence is weak, however. If [e.sub.t] is stationary, then plim [R.sup.2] = 1. Because the [R.sup.2] s in equations (26) and (27) are actually quite small, [[Rho] [caret].sub.d] and [[Rho] [caret].sub.g] may be subject to much bias even if they are superconsistent.

VII. STEADY-STATE RELATIONSHIPS ON DIFFERENCED DATA

The evidence presented in section IV suggests that [d.sub.t] and [g.sub.t] are difference-stationary. Suppose that notwithstanding the evidence presented in sections V and VI, [e.sub.t] is also difference-stationary; i.e., variables other than [d.sub.t] and [g.sub.t] have permanent effects on [r.sub.t]. The following proposition would then be germane.

PROPOSITION 5:

If [d.sub.t], [g.sub.t], and [e.sub.t] are difference-stationary, then [r.sub.t] has a representation that can be approximated arbitrarily well as (28) [Mathematical Expression Omitted] where n is an appropriately chosen lag length, [u.sub.t] is a stationary error term, and the [[Delta].sub.i] and [[Gamma].sub.i] parameters satisfying (29) [Mathematical Expression Omitted] and (30) [Mathematical Expression Omitted] Suppose further that the error term [u.sub.t] in equation (28) is orthogonal to [[Delta] d.sub.t], [[Delta] d.sub.t-1], [[Delta] d.sub.t-2], ... and [[Delta] g.sub.t], [[Delta] g.sub.t-1], [[Delta] g.sub.t-2], ... and can be well approximated as an autoregression of order p. Then maximum-likelihood estimation of the coefficient sums [[Sigma] [Delta].sub.i] and [[Sigma] [Gamma].sub.i] yields consistent estimates of [[Rho].sub.d] and [[Rho].sub.g].

These orthogonality conditions, however, are unlikely to be satisfied if equation (28) is fitted to the same measures of [r.sub.t] and [d.sub.t] used in the previous section. Each quarterly average of the ex post real commercial paper rate is realized over the entire quarter and has a term to maturity that is twice the sampling interval. Because these attributes of this series can generate spurious correlation, I measure [r.sub.t] in this section as the continuously compounded three-month Treasury bill rate in the last month of quarter t less the annualized inflation in the consumer price index from the last month of quarter t to the last month of quarter t+1. Changes in real interest rates induce changes in the market value of the government debt. Furthermore, fluctuations in economic activity not only affect the demand for and supply of credit and hence the real interest rate but also induce changes in tax collections, transfer payments, and hence the government debt. Therefore, in this section, I measure [d.sub.t] as the cyclically adjusted par value of the federal debt at the end of quarter t divided by the consumer price index for the last month of quarter t and by mid-quarter population.(23,24)

Table II reports the results of estimating equation (28). Because theory provides no guidance in selecting an appropriate lag length, results are reported for n = 0, 1, 2, 3, 4, 6, 8, 12, and 16 quarters. The error term [u.sub.t] appears to be well approximated as a second-order autoregression. All of the estimates of [[Sigma] [Delta].sub.i] are negative, and all but one are significantly so. All of the estimates of [[Sigma] [Gamma.sub.i] are positive, but only two are significantly so. Consequently, the finite horizons hypothesis receive no support, and variability of the intergenerational discount rate receives only weak support.

VIII. IS THERE A SHORT-RUN RELATIONSHIP?

The evidence presented in section IV suggests that [d.sub.t] and [g.sub.t] are difference-stationary, and the evidence presented in sections V and VI suggests that [r.sub.t] is stationary. In that case [[Rho].sub.d] = [[Rho].sub.g] = 0 and [e.sub.t] is stationary. Therefore, the following proposition appears to be germane.

PROPOSITION 6: If [d.sub.t] and [g.sub.t] are difference-stationary but [r.sub.t] is stationary, then [r.sub.t] has a representation that can be approximated arbitrarily well as (31) [Mathematical Expression Omitted] where n is an appropriately chosen lag length, [[Upsilon].sub.t] is a stationary error term, and the [[Lambda].sub.i] and [[Omega].sub.i] are parameters that characterize the short-run responses of [r.sub.t] to [d.sub.t] and [g.sub.t]. If the error term [[Upsilon].sub.t] in equation (31) is also orthogonal to [[Delta] d.sub.t], [[Delta] d.sub.t-1], [[Delta] d.sub.t-2], ... and [[Delta] g.sub.t], [[Delta] g.sub.t-1], [[Delta] g.sub.t-2], ... and can be well approximated as an autoregression of order p, maximum-likelihood estimation yields consistent estimates of these parameters.

I fitted equation (31) to the data described in section VII for values of n as long as sixteen quarters over the sample period 1955:IV-1988:IV. After much pretesting, I obtained the following regression: [r.sub.t] = 1.42 - 7.5 [[Delta] d.sub.t] + 7.8 [[Delta] g.sub.t] + [[Upsilon] [caret].sub.t] (0.93)(16.0) (6.6) with [[Upsilon] [caret].sub.t] = [.293 [Upsilon].sub.t-1] + [.251 [Upsilon].sub.t-2] + [.280 [Upsilon].sub.t-3], (.091) (.088) (.087)

SEE = 2.03, [R.sup.2] = .519. Including lagged [Delta] ds and [Delta] gs yields estimated coefficients that are virtually all insignificant and that scatter around zero.

According to equation (32), an increase of 1 percent in the real cyclically adjusted federal debt lowers the real interest rate by a statistically insignificant 7.5 basis points for one quarter and has no effect thereafter, and an increase of 1 percent in federal purchases raises the real interest rate by an insignificant 7.8 basis points in the quarter in which the increase occurs but has no effect in subsequent quarters. These estimates provide no evidence of either statistically significant or empirically important deviations from Ricardian equivalence even in the short run.

IX. CONCLUSION

If households have finite horizons, the real interest rate should be an increasing function of the government debt in the steady state. In contrast, if households have infinite horizons, the steady-state real interest rate should not depend on the government debt. The paper has investigated which of these characterizations of the steady state accords better with U.S. data. I find no support for finite horizons, and infinite horizons are broadly consistent with the data. Therefore, for many purposes, assuming that households have infinite horizons may be a better modeling strategy than assuming that they have finite horizons.

APPENDIX

Since by definition the vector [[d.sub.t,] [g.sub.t], [z.sub.t] [prime]] contains all variables that affect [r.sub.t], it is reasonable to suppose that [r.sub.t] has the representation(25) (33) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] L is the lag operator; [[Delta].sub.0], [[Delta].sub.1], [[Delta].sub.2], ... and [[Gamma].sub.0], [[Gamma].sub.1], [[Gamma].sub.2], ... are parameters such that [Delta] (L) and [Gamma](L) are square-summable; and [[Zeta].sub.0], [[Zeta].sub.1], [[Zeta].sub.2], are ... vectors of parameters such that [Zeta] (L) is square-summable. Suppose further that [[d.sub.t], [g.sub.t], [z.sub.t [prime]] has the representation (34) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] [H.sub.0] = I; [H.sub.1], [H.sub.2], [H.sub.3], ... are matrices such that H(L) is square-summable; and [[Upsilon].sub.t] is a serially uncorrelated error vector with a zero mean vector and a constant positive definite symmetric covariance matrix. From equations (33) and (34), one then has (35) [Mathematical Expression Omitted] Equations (35) and (34) imply that(26) (36) [Mathematical Expression Omitted] and (37) [Mathematical Expression Omitted] From equations (36) and (37), it follows that (38) [Mathematical Expression Omitted] Comparing equations (21) and (38) results in (39) [Delta] (1) = [Rho.sub.d], (40) [Gamma] (1) = [[Rho].sub.g], and (41) [Zeta] (1) = [[Rho].sub.z]. First, consider the implications of assuming that [d.sub.t] and [g.sub.t] are stationary. Let H(L) be partitioned as [[h.sub.d] (L), [h.sub.g] (L), [H.sub.z] [prime] (L)] [prime], where [h.sub.d] [prime] (L) and [h.sub.g] [prime] (L) are the first two rows of H(L) and [H.sub.z] (L) consists of the rest of H(L). Because [[Delta] d.sub.t] = [h.sub.d] [prime] (L) [[Upsilon].sub.t] and [[Delta] g.sub.t] = [h.sub.g] [prime] (L) [[Upsilon].sub.t], the variables of [d.sub.t] and [g.sub.t] can be stationary only if 1-L is a factor of [h.sub.d] (L); i.e., only if [h.sub.d] (1) = [h.sub.g] (1) = 0 of [h.sub.d] (L) and [h.sub.g] (L); i.e., only if [h.sub.d] (1) = [h.sub.g] (1) = 0. Equation (36) then implies that [Mathematical Expression Omitted] and equations (37) and (41) imply that (42) [Mathematical Expression Omitted] Because [[Rho].sub.d] and [[Rho].sub.g] do not appear in equation (42), they are not identified. Clearly, [[Rho].sub.d] remains unidentified if [d.sub.t] is stationary and [g.sub.t] is difference-stationary, and [[Rho].sub.g] remains unidentified if [g.sub.t] is stationary and [d.sub.t] is difference-stationary. Therefore, propositions 1 and 2 follow immediately.

Next, consider the implications of assuming that only [d.sub.t] and [g.sub.t] permanently affect [r.sub.t], i.e., that [Mathematical Expression Omitted] I now show that this condition is equivalent to the stationarity of [[Rho].sub.z] [prime] [z.sub.t]]. According to equation (33), [[Delta] z.sub.t] = [[H.sub.z] (L) [[Upsilon].sub.t]]. Therefore, [[Rho].sub.z] [prime] is stationary if, and only if [[Rho].sub.z] [H.sub.z] (L) takes the form (1 - L) [Psi] (L), where [Psi] (L) is square-summable. Therefore, [[Rho].sub.z] [prime] [H.sub.z] (1)] = 0. It follows from equation (37) that [Mathematical Expression Omitted] and from equations (33), (34), and (30)-(41) that [Mathematical Expression Omitted] or (43) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] and (44) [Mathematical Expression Omitted] Clearly, the error term [e.sub.t] is stationary since the error vector [v.sub.t] is stationary and the term in braces is square-summable. Therefore, I have established proposition 4.

Under the null hypothesis that [[Rho].sub.d] = 0 and [[Rho].sub.g] = 0, equation (43) implies that [r.sub.t] must be stationary. If either the alternative hypothesis [[Rho].sub.d] > 0 or the alternative hypothesis [[Rho].sub.g] [is not equal to] 0 holds then [r.sub.t] must be difference-stationary. Therefore, I have established proposition 3.

Equations (33), (34), and (39)-(41) imply that (45) [Mathematical Expression Omitted] so that (46) [Mathematical Expression Omitted] Because [Lambda] (L), [Omega] (L), [Phi] (L), and H (L) are square-summable and [v.sub.t] is stationary, [e.sub.t] is difference-stationary if, and only if, [[Rho].sub.z]

[prime] [z.sub.t]] is nontrivially difference-stationary. This condition, however, can be met only if [z.sub.t] is difference-stationary. Because [Delta] (L) and [Gamma] (L) are square-summable, differencing equation (33) and substituting for [[Delta] z.sub.t] from equation (34) results in an equation that can be approximated arbitrarily well by equation (28) with [u.sub.t] = [Zeta] [prime] (L) [H.sub.z] (L) [[Upsilon].sub.t]. This error term is stationary since [Zeta] (L) and H(L) are square-summable and [v.sub.t] is stationary. Equations (29) and (30) follow from equations (39) and (40). Therefore, I have established proposition 5.

Suppose that [d.sub.t] and [g.sub.t] are difference-stationary but that [r.sub.t] is stationary; i.e., that [[Rho].sub.d] = [[Rho].sub.g] = 0. Equations (43), (44), and (34) then imply that (45) [r sub. t] = [Lambda] (L) [[Delta] [d.sub.t]] + [Omega] (L) [[Delta.sub.t]] + [v.sub.t], where [[Upsilon].sub.t] = [[[Phi] [prime] (L) H(L) + [Psi] (L)] [v.sub.t]]. Equation (45) can be approximated arbitrarily well by equation (31) because [Delta] (L) and [Gamma] (L) are square-summable, and the error term [v.sub.t] is stationary because [Zeta] (L), [Psi] (L), H(L) are square-summable and [v.sub.t] is stationary. Therefore, I have established proposition 6. [Tabular Data 1 and 2 Omitted]

(1)Ricardian equivalence is named after David Ricardo, who discovered it in the early 1800s. See Barro [1990] for further discussion of Ricardian equivalence. (2)Consider a household facing a binding constraint on its assets T periods from now. Even if the constraint is temporay and the household's lifetime greatly exceeds T periods, the household maximizes its utility over the next T periods subject to a terminal wealth constraint and, therefore, behaves as if it has a horizon of T periods. Of course, T is itself a choice variable unless the household has a lifetime of exactly T periods. (3)Cochrane [1989] has shown that near-rational intertemporal allocation of consumption can entail trivial costs. (4)For simplicity, I assume an internal equilibrium for [C.sub.yt] and [C.sub.ot+1. I indicative derivatives with primes if the function has only one argument and with numeric subscripts if it has more than one argument. Note that if [B.sub.t] > 0, the evelope theorem implies that [Mathematical Expression Omitted] (5)It is straightforward to show that [Mathematical Expression Omitted] and [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. Strict concavi ty requires that [Mathematical Expression Omitted]. Consumption in the first and second perio ds of life is a normal good [Mathematical Expression Omitted] and [Mathematical Expression Omitted ]. The derivative [a.sub.2] is composed of a substitution term [Mathematical Expression Omitte d], which must be positive, and an income term [Mathematical Expression Omitted] which must be negative. (6) If government purchases are inputs into the productive process and are Edgeworth substitutes for capital, then [Rho][.sub.2] can be negative. Similarly, if government purchases in the steady state affect the marginal rate of substitution between the consumption in the two periods of life, then the sign of [Rho][.sub.2] can also be negative. Furthermore, other variables that affect the marginal rate of substitution or the marginal product of capital also affect the steady-state real interest. (7)It is also straightforward to show that [B.sub.t] moves one-for-one with [D.sub.t] along the transition path to this steady state if [B.sub.t] > 0 prevails everywhere. See Barro [1974]. (8)The sign of [Eta]' is even more ambiguous if government purchases directly affect the steady-state discount rate on the utility of the next generation. Furthermore, if other environmental variables affect this discount rate, they would also affect the steady-state real interest rate. Note, however, that r* is entirely independent of technology and market structure as well as the government debt, depending only on the determinants of preferences. (9)See my 1991 paper for more discussion of this point. That paper also demonstrates that in a closed economy, infinite horizons may not be distinguishable in practice from long finite horizons (e.g., twenty-five years) and that as much as 25 percent of disposable wage income can be received by households facing permanent liquidity constraints without affecting the behavior of the economy in empirically important ways. (10)Section II also showed that in this case, [Rho][.sub.z] = 0 if the intergenerational discount rate is constant and [Rho][.sub.z] [is not equal] 0 if it is variable. (11)A variable is difference-stationary if it is not stationary in levels but becomes stationary if differenced once. (12)The appendix establishes the propositions stated in this and the next fousections. (13)The data on the market value of the privately held federal debt come from Seater [1981] for 1919-1941 and from Cox and Lown [1989] for 1942-1987. Michael Cox has also kindly provided updates for 1988. The consumer price index comes from a printout provided by the Bureau of Labor Statistics and from the Citibase data tape. Real federal purchases come from the National Income and Product Accounts. The data on population come from Citibase and Historical Statistics of the United States. All data and programs used in this paper are available upon request. Deflating the end-of-year federal debt by December's consumer pride index minimizes the time between when the two series are realized. (14)Considering the debt and purchase of only the federal government is appropriate if households and factors of production are perfectly mobile across state and local jurisdictions in the steady state. In that case, state and local governments must behave in the steady state like firms selling their services to households in competitive markets. Furthermore, site values subject to state and local taxes fall in value by the market value of any government debt not backed by productive capital. As a result, state and local governments should be treated as if they are part of the private sector. Ideally, the option value of federal credit-market guarantees such as those provided by the FDIC and FSLIC should be included in the federal debt, and the market value of federal holdings of financial assets and capital should be netted off against the federal debt. To do so with any degree of accuracy, however, would be difficult, if not impossible. (15)All tests are performed at the 0.5 level unless stated otherwise. (16)For convenience in stating propositions 3 and 3, I consider neither the case in which [[d.sub.t] , [g.sub.t]]' is cointegrated with the cointegrating vector [[Rho][.sub.d], [Rho][.sub.g]] nor the case in which [[d.sub.t],[g.sub.t],[z.sub.t']]' is cointegrated with the cointegrating vector [[Rho] [.sub.d], [Rho][.sub.g], [Rho][.sub.z']]. Note also that I do not take explicit account of tax distortions either in the models laid out in section II or in the empirical work. Abstracting from distorting taxes on labor is justified if the intergenerational discount rate is independent of leisure. Abstracting from distorting taxes on capital is justified if the marginal tax rate on capital is stationary since changes in it would then have no long-run implications. Theory suggests that the marginal tax rate on capital should indeed be stationary. Judd [1985] has shown that driving the marginal tax rate on capital to zero in the steady state is optimal if the government can commit to a preannounced path for the marginal tax rate on capital. This is true even if the government cannot very lump-sum taxes and has a redistributive motive. If the government cannot perfectly commit, the steady-state marginal tax rate on capital is determined by such factors as the nature and strength of the available commitment mechanisms and the nature and strength of its incentives to deviate from its commitments. These determinants are likely to be stationary. (17)In other words, [r.sub.t] = 200 [1n(1+.005CP[R.sub.t]) -In(PGNP[.sub.t+2]/ PGNP[.sub.t])], where CPR is the commercial paper rate and PGNP is the GNP deflator. The data come from Balke and Gordon [1986] and Citibase. I have used the six-month commercial paper rate and the GNP deflator here because the three-month Treasury bill rate is not available prior to 1941 and the consumer price index is not available prior to 1914. Their use maximizes the length of the sample period but also entails the costs described in footnote 24. Note that using the GNP deflator to calculate [d.sub.t] would have contaminated the measure with prices realized as many as three months before year's end. (18)The [Z.sub.[Tau] [Mu]] and [Z.sub.[double Tau]] statistics are -11.7 and -26.5 for 1875:II-1988: IV, -7.7 and -17.5 for 1919:I-1988:IV, and -5.9 and -15.1 for 1947:I-1988:IV Under the null hypothesis that [Delta] [x.sub.t] = [v.sub.t] - [Theta][v.sub.t-1]' the [Z.sub. [double Tau] statistics suffice for rejection at the .05 significance level if [Theta] [is less than or equal to] .8, and the [Z.sub. [Mu] [Tau]] statistics suffice if [Theta] [is less than or equal to] .5. See Schwert [1987]. (19)If d and g were cointegrated with a cointegrating vector [1,[Rho][.sub.g]/ [Rho][.sub.d]], then [r.sub.t] could be stationary even if [Rho][.sub.d] > 0 and [Rho][.sub.g] > 0. It is unlikely that d and g are cointegrated with this cointegrating vector since government debt and government purchases would then be negatively related in the steady state. The estimated cointegrating vector is [1, -.657] over the sample period 1929-1988 and [1, .160] over the sample period 1947:I-1988:IV. (20)Note that r can be stationary only if e is stationary as well as [Rho][.sub.d] = [Rho][.sub.g] = 0. Therefore, the evidence presented in the previous section is also evidence that [e.sub.t] is stationary. (21)The series d and g have trended upward over time while r has evidenced no trend. For this reason, I subtracted 1.912908 + .01621305t from the annual data for d and g 1.953047 + .004808583t from the quarterly data. I obtained these trends by fitting the logarithm of per capita real GNP to a constant term and a time trend over the sample periods 1929-1988 and 1947:I-1988:IV. (22)The annual simulations assume that [Delta] d and [Delta] g are generated by a first-order vector autoregression, that [Delta] e is a first-order autoregression, and that all parameters and covariance matrices take their fitted values. The quarterly simulations assume that [Delta] d and [Delta] g are generated by a fifth-order vector autoregression, that [Delta] e is a fourth-order autoregression, and that all parameters and covariance matrices take their fitted values. Each simulation was replicated one thousand times, and the .01 and .11 critical values for the t-ratio were equated to the tenth and one-hundred-tenth smallest t-ratios obtained. (23)The continuously compounded three-month Treasury bill rate is -400ln(-.002528TBR), where TBR is the three-month Treasury bill rate published in the Federal Reserve Bulletin. The cyclically adjusted federal debt comes from the Survey of Current Business. (24)In the previous two sections, the use of overlapping, averaged, endogenous series did not create problems because only unit-root tests were performed and because cointegrating vectors are estimated super-consistently. The use there of the measures used here would have appreciably shortened the samples but would not have produced materially different results. (25)For notational convenience, the equations below are written without constant terms and time trends. (26)See my 1988 paper for a derivation.

REFERENCES

Balke, Nathan S., and Robert J. Gordon. "Historical Data," in The American Business Cycle, edited by Robert J. Gordon. Chicago: University of Chicago Press, 1986, 781-850. Barro, Robert . "Are Government Bonds Net Wealth?" Journal of Political Economic, December 1974, 1161-76. __. Macroeconomics, 3rd ed. New York: Wiley, 1990. Board of Governors of the Federal Reserve System. Federal Reserve Bulletin. Washington: Board of Governors, 1955-1989. Bureau of the Census. Historical Statistics of the United States. Washington: Government Printing Office, 1975. Bureau of Economic Analysis. Survey of Current Business. Washington: Government Printing Office. Cochrane, John H. "How Big Is the Random Walk in GNP?" Journal of Political Economic, October 1988. 893-920. __. "The Sensitivity of Tests of the Intertemporal Allocation of Consumption to Near-Rational Alternatives." American Economic Review, June 1989, 319-37. Cox, W. Michael, and Cara Lown. "The Capital Gains and Losses on U.S. Government Debt: 1942-1987." Review of Economics and Statistics, February 1989, 1-14. Dickey, David A., and Wayne A. Fuller. "Likelihood Ratio Statistics for Autoregression Time Series with a Unit Root." Econometrica, July 1981, 1057-72. Engle, Robert F., and Clive W. J. Granger. "Cointegration and Error Correction: Representation, Estimation and Testing." Econometrica, March 1987, 251-76. Evans, Paul. "Are Government Bonds Net Wealth? Evidence for the United States." Economic Inquiry, October 1988, 551-66. __. "Is Ricardian Equivalence a Good Approximation?" Economic Inquiry, October 1991, 626-44. Fuller, Wayne A. Introduction to Statistical Time Series. New York: Wiley, 1976. Granger, Clive W. J., and Paul Newbold. "Spurious Regressions in Econometrics." Journal of Econometrics, July 1974, 111-20. Hayashi, Fumio. "Tests for Liquidity Constraints: A Critical Survey," in Advances in Econometrics Fifth World Congress, Vol. II, edited by Truman Bewley. New York: Cambridge University Press, 1987, 91-120. Judd, Kenneth. "Redistributive Taxation in a Simple Perfect Foresight Model." Journal of Public Economics, October 1985, 59-83. Phillips, Peter C. B. "Time Series Regression with a Unit Root." Econometrica, March 1987, 227-301. Phillips, Peter C. B., and Pierre Perron. "Testing for a Unit Root in Time Series Regression." Biometrika, June 1988, 335-46. Said, Said E., and Dickey, David A. "Testing for Unit Roots in Autoregressive--Moving Average Models of Unknown Order." Biometrika, December 1984, 599-607. Schwert, G. William. "Effects of Model Specification on Tests for Unit Roots in Macroeconomic Data." Journal of Monetary Economics, July 1987, 73-104. Seater, John J. "The Market Value of Outstanding Government Debt, 1919-1975." Journal of Monetary Economics, July 1981, 85-101. Yotsuzuka, Toshiki. "Ricardian Equivalence in the Presence of Capital Market Imperfections." Journal of Monetary Economics, September 1987, 411-36.

PAUL EVANS, Department of Economics, Ohio State University, Columbus, Ohio 43210-1172, 614-292-0072. I am grateful to Dick Sweeney and an anonymous referee for helpful comments and to the Osaka Institute of Economic and Social Research for their hospitality and financial support during Autumn 1990.
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