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A test of steady-state government debt neutrality.

A TEST OF STEADY-STATE GOVERNMENT-DEBT NEUTRALITY

This paper investigates whether in the steady state the real interest rate is an increasing function of both the government debt and government spending. Using data from the period between January 1981 and March 1986, the paper finds no evidence of such a relationship. These data afford an especially powerful test because the ratio of federal debt to trend output, which had fallen from 93 to 33 percent between January 1948 and December 1980, reversed course after the enactment of the Economic Recovery Tax Act, reaching 47 percent in March 1986.

I. INTRODUCTION

In conventional macroeconomic analysis, government debt is not neutral because households view it as contributing to their net wealth. As a result, the larger the government debt is, the wealthier households feel and the more they consume. More consumption in turn spells less investment over time and thus a lower steady-state capital stock. As a result, output is ultimately lower and the real interest rate higher.

In principle, however, households need not view government debt as net wealth.(1) If households accurately foresee the future taxes that will service the government debt, if they face perfect capital markets, if taxes do not distort their decisions, and if they internalize future generations, then they treat the future taxes servicing the government debt as an exact offset. Consequently, government debt is neutral since it does not make households feel wealthier and hence affects nothing.(2)

Nevertheless, because most macroeconomists doubt that households accurately foresee future servicing taxes, have access to perfect capital markets, and internalize future generations, they do not model government debt as neutral. Like perfect competition, however, government-debt neutrality may be a good approximation in many applications even if the assumptions underlying it are unrealistic. Therefore, empirical analysis rather than introspection should determine how one models government debt. This paper provides such an empirical analysis.

The empirical analysis focuses on the steady-state relationship between government debt and the real interest rate.(3) In the steady state, a larger government debt leads to a smaller capital stock and hence a higher marginal product of capital if households view government debt as net wealth. Consequently, the real interest rate is also higher. If instead government debt is neutral, nothing should happen to the real interest rate.

U.S. data from the period between January 1981 and March 1986 have been used here to investigate whether a larger government debt leads to a higher real interest rate in the steady state and hence whether government debt is neutral. These data afford an especially powerful test because the ratio of federal debt to trend output, which had fallen from 94 to 33 percent between January 1948 and December 1980, reversed course after the enactment of the Economic Recovery Tax Act, reaching 47 percent by March 1986. Moreover, this upward trend is likely to continue well beyond 1986. Such an enormous increase in federal debt should have raised by steady-state real interest rate appreciably if households view government debt as net wealth to any important extent. In the absence of evidence that the steady-state real interest rate did rise, government debt is judged to be neutral in the steady state.

Section II formulates the model, which is estimated in sections III and IV. Section V summarizes the paper and then briefly reviews an extensive literature consistent with government-debt neutrality.

II. FORMULATION OF THE ECONOMETRIC MODEL

The appendix demonstrates that if households view government debt as net wealth, the real interest rate should be an increasing function of both the government debt and government spending in the steady state (see also Evans [1988] for this analysis). If instead government debt is not net wealth, the real interest rate should depend on neither of these variables.(4) This section formulates an econometric model with which one can estimate how the government debt and government spending affect the real interest rate in the steady state; the null hypothesis that government debt is neutral can therefore be tested against the alternative hypothesis that government debt is net wealth.

A linear version of the model under consideration is

r(*)t = Beta o + Beta dd(*)t + Beta gg(*)t + ut, (1) where r(*)t, d(*)t, and g(*)t are, respectively, the expected steady-state interest rate and the expected steady-state ratios of the government debt and government spending to output; the expectations are those of bond traders during period t; Beta d and Beta g are parameters; and ut is an error term whose statistical properties are specified below. If government debt is neutral, then Beta d = O and Beta g = O; if government debt is net wealth, then Beta d > O and Beta g > O.

Because direct measures of r(*)t, d(*)t, and g(*)t are unavailable, one must use indirect measures. To measure r(*)t, it is assumed that

r(*)t = i(*)t - Pi (*)t (2) and

ft = i(*)t + e(*)t, (3) where t is an index of time; i(*)t is the expected steady-state nominal interest rate: Pi (*)t is the expected steady-state inflation rate; ft is the forward interest rate during period t on a coupon bond that will be issued in period Tau and will mature in period T; T >> Tau >> t; and et is a term premium whose statistical properties are specified below. Equation (2) is simply Fisher's equation.(5) Equation (3) is a variant of the expectation theory of the term structure. In this variant, bond traders expect that the U.S. economy will behave over long periods of time far in the future as if it were in a neoclassical steady state. Hence, between periods Tau and T, the one-period nominal interest rate is expected to average not only ft but also i(*)t.

According to Shiller, Campbell and Schoenholtz [1983], one can calculate ft to a close approximation using the formula

ft = (XTt ITt - X Tau t I Tau t)/(XTt - X Tau t), (4) where ITt and I Tau t are the yields to maturity in period t on coupon bonds maturing in periods T and Tau; where XTt and X Tau t, the durations of the two bonds, are given by

XTt = [CT j=1 to T Summation j(1+ITt) raise to -j + T(1+ITt) raise to -T]/[CT j=1 to T Summation (1+ITt) raise to -j + (1+ITt) raise to -T]

(5) and

X Tau t = [C Tau j=1 to T Summation j(1+I Tau t) raise to -j + Tau(1+ITt) raise to -Tau]/[CT j=1 to T (1+I Tau t) raise to -j + (1+I Tau t) raise to -Tau]; (6) and where CT and C Tau are the coupon rates paid by the two bonds.

It is assumed that bond traders have rational expectations and that they expect the economy to converge toward its steady state. Therefore, Pi(*)t, d(*)t, and g(*)t are their optimal forecasts infinitely far in the future of the inflation rate and of the ratios of government debt and government spending to output. Formally,
 d(*)t = j to infinity lim Etdt+j, (7)
 g(*)t = j to infinity lim Etgt+j, (8)


and

Pi(*)i = j to infinity lim Et Pi t+j, (9) where dt and gt are the ratios of government debt and government spending to output in period t, Pi, is the inflation rate in period t, and Et is the expectations operator conditional on all information known to bond traders in period t. It is assumed further that bond traders forecast using the vector autoregressions
 [dt ] [vdt ]
 H(L)Delta [gt ] = [vgt ] , (10)
 [Pi t] [v Pi t]
 [zt ] [vzt ]


where zt is a vector of variables useful in forecasting dt, gt, and Pi t; vdt, vgt, v Pi t, and all entries of the vector vzt are serially uncorrelated error terms with zero means and a constant positive definite covariance matrix; Delta is the difference operator; H(L) = I + j=1 to infinity HjL raise to j; H1, H2, H3,...are matrices; L is the lag operator; and the roots of det[H(L)] lie outside the unit circle so that dt, gt, Pi t, and each entry of zt are difference-stationary series. The assumption that these series are difference-stationary is in accord with the finding of Nelson and Plosser [1982] that U.S. macroeconomic time series are typically well characterized in this way.

Equation (10) implies that(6)
 [d(*)t ] [vdt ]
 Delta [g(*)t ] = H raise to -1 (1) [vgt ] , (11)
 [Pi(*)t] [v Pi t]
 [z(*)t ] [vzt ]


where z(*)t = j to infinity lim Etzt+j. Differencing equation (1) and substituting from equations (2), (3), and (11) yields
 [vdt ]
 [vgt ]
 Delta(ft - Pi(*)t - et) = (Beta d,Beta g,O,O)H raise to -1 (1) [v Pi t] + De
lta ut. (12)
 [vzt ]


Let
 [ Kdd Kdg Kd Pi Kdz ]
 H raise to -1 (1) = [ Kdd Kgg Kg Pi Kgz ] (13)
 [ K Pi d K Pi g K Pi Pi K Pi z ]
 [ Kzd Kzg Kz Pi Kzz ]


From equations (11) and (13),

Delta Pi(*)i = K Pi dvdt + K Pi gvgt + K Pi Pi v Pi t + K Pi zvzt.

(14)

Substituting equations (13) and (14) into equation (12)and rearranging, one obtains

Delta ft - K Pi dvdt - K Pi gvgt - K Pi Pi v Pi t - K Pi zvzt - Delta et = (Beta dKdd (15) + (Beta dKdg + Beta gKgg)vgt + (Beta dKd Pi + Beta gKg Pi)v Pi t + (Beta d Kdz + Beta gKgz)vzt + Delta ut or

Rho t = fo + fdvdt + fgvgt + Eta t, (16) where
 Rho t = Delta ft - K Pi dvdt - K Pi gvgt - K Pi Pi v Pi t - K Pi zvzt,
 (17)
 (fd,fg) = (Beta d,Beta g) [ Kdd Kdg ] , (18)
 [ Kgd Kgg ]


and Eta t = Delta(ut + et) - fo + (Beta dKd Pi + Beta gKg Pi)v Pi t + (Beta dKdz + Beta gKgz)vzt. (19) In this and the next section, it is assumed that ut + et is a random walk(7) with the drift rate fo and that Delta(ut + et) is uncorrelated with vdt and vgt(8) and has a constant variance. Under the null hypothesis, equation (19) then implies that eta t is a constant-variance, serially uncorrelated error term,(9) which is uncorrelated with vdt and vgt. In this case, equation (16) is a regression model that ordinary least squares can estimate consistently and efficiently. One can therefore estimate fd and fg using ordinary least squares, recover estimates of the parameters Beta d and Beta g using equation (18) and estimates of the Ks, and then test the null hypothesis Beta d=0 and Beta g=0 against the alternative hypothesis Beta d>0 and Beta g>0.(10)

III. ESTIMATES OF THE MODEL

One should choose the maturity date Tau as far in the future as possible and the maturity date T as much after Tau as possible. Furthermore, Tau and T should coincide with the maturity dates of Treasury bonds frequently traded during the sample period, which extends from January 1981 to March 1986. Finally, these bonds should have similar coupon rates.(11) Trading off these competing desiderata, the interest rates on Treasury 9s of February 1994 and Treasury 9 1/8s of May 2004-2009 are used as measures of I Tau t and ITt. With this choice, Tau exceeds t by as much as 158 months.(12) If 95 months suffices for the U.S. economy to approach its steady state fairly closely and if over periods as long as 123 months the U.S. economy can be well characterized as neoclassical, the calculated ft should proxy well for i(*)t.

dt was calculated by dividing the par value(13) of the gross public debt of the federal government by the consumer price index and by trend output.(14) The variable gt is real federal purchases (as estimated by the Federal Reserve Bank of Minneapolis) divided by trend output. The variable Pi t, is the annualized growth rate of the consumer price index between months t-1 and t. The gross public debt and the consumer price index come from the Federal Reserve System's [1976] Banking and Monetary Statistics, a printout provided by the Bureau of Labor Statistics (U.S. Department of Labor), and various issues of the Survey of Current Business (U.S. Department of Commerce).

In addition to dt, gt, and Pi t, the vector autoregressions include yt, the logarithm of the industrial production index; Mu t, the growth rate of the M1 money supply; and Alpha t, Moody's Aaa bond rate. Industrial production is included because output affects dt (federal tax collections rise and transfer payments fall as output increases) and Pi t (there may be a short-run Phillips curve).(15) The growth rate of the money supply is included because of the causal link between it and the inflation rate. The bond rate is included because traders in the long-term bond markets may anticipate dt, gt and Pi t.(16)

No doubt the vector zt includes many more entries than just yt, Mu t, and at. Consequently, the ordinary least squares are inconsistent unless H(L) is block triangular, permitting the exclusion of the other variables from the vector autoregressions for dt,gt and Pi t. One must hope that any such inconsistencies are empirically inconsequential.

Using ordinary least squares, the vector autoregression (10) was estimated over the sample period between March 1949 and March 1986.(17) Pretesting suggests that twelve lags suffice. Estimating the vector autoregressions yields
 [ Kdd Kdg ] raise to -1 = [ .270 -.369 ]
 Estimated [ Kgd Kgg ] = [ .022 .076]


Using the residual vector [estimates (vdt, vgt, v pi t, vyt, v mu t, v at) from the estimated vector autoregression in lieu of the error vector (vdt,vgt,v pi t,v upsilon t,v mu t, v alpha t), Rho t was calculated with formula (17) and then equation (16) was estimated with ordinary least squares over the sample period between January 1981 and March 1986. Pagan [1984] has shown that this procedure is asymptotically equivalent to joint estimation of the system (10) and (16) using full-information maximum likelihood. In particular, the resulting estimators of the coefficients fd and fg and of their standard errors are consistent and efficient.(18)

The following regression was obtained:(19,20)

Estimated Rho t = .102 - .842 estimated vdt - 2.50 estimated vgt,
 (.095) (.279) (3.05)
 R square = -132, S.E.E. = .664, D-W = 2.20, Q(12) = 20.3. (21)


The coefficient on estimated vdt is significantly negative. The Durbin-Watson statistic and the Box-Pierce Q statistic provide no evidence that the error team Eta t is serially correlated. Therefore, the maintained assumption that Eta t is white noise seems to be in reasonable accord with the data.

Substituting estimated fd,estimated fg, and equation (20) into the formula (18) yields(21)

Estimated Beta d = (-.842)(.270) + (-2.50)(.022) = -.282 (22) and

Estimated Beta g = (-.842)(-.369) + (-2.50)(.076) = .121. (23)

Because Beta d is negative, the alternative hypothesis that Beta d and Beta g are positive is not supported.(22) Indeed, if anything, the estimates support the hypothesis that increasing the federal debt lowers the steady-state real interest rate.

One of the assumptions used in deriving equation (16) is that the tax system is neutral with respect to expected inflation. In fact, nominal interest is taxable income to lenders and deductible expense to borrowers, realized nominal capital gains income to the recipient, and depreciation allowances are not indexed. As a result, Fisher's equation actually takes the form

r(*)t = Lambda ti(*)t - Pi(*)t, (24) where Lambda t depends on parameters of the tax code and may differ from one.

In keeping with conventional macroeconomic analysis, equation (1) implicitly assumes that fiscal policy affects the steady-state real interest rate primarily by altering the steady-state ratios of government debt and government spending to output. If instead government debt is neutral, a better assumption would be that fiscal policy affects the steady-state before-tax real interest rate primarily by altering how the income from capital is taxed in the steady state. With this interpretation, the point estimates reported in equation (21) suggest that bond traders expect a lower steady-state marginal tax are on capital when they observe an unanticipated increase in the government debt (vdt > 0) and a higher tax rate when they observe an unanticipated increase in government spending (vgt > 0). See the appendix for a demonstration that the steady-state before-tax real interest rate rises as the steady-state marginal tax rate on the income from capital rises.

Another assumption made above is that the error Eta t is uncorrelated with vdt abd vgt. One might object to this assumption, however, arguing that a larger Eta t is associated with higher output, higher tax collections, lower transfer payments and hence a smaller vdt. This objection, which would have considerable weight if quarterly or annual data had been used, has little weight in this empirical analysis. Because the lags in the collection of federal taxes and in the administration of federal entitlement programs are least a month, including yt in the vector autoregressions for dt should adequately account for this causal connection.

IV. FURTHER ESTIMATES OF THE MODEL

In order for the estimates reported in the previous section to be efficient, the error term ut + et must be random walk. Suppose instead that it is covariance-stationary. Suppose further that dt, gt, Pi t, and the one-period nominal interest rate it are integrated processes of order one. Then d(*)t, g(*)t, Pi(*)t, and i(*)t must be within covariance-stationary error terms of dt, gt, Pi t, and it. Therefore, equations (1) and (2) imply that

it =Beta ddt + Beta ggt + Beta Pi Pi t + Gamma t, (25) where Beta Pi = 1 and Gamma t, which equals Beta o + ut + et + (it - i(*)t) + Beta d(d(*)t - dt) Beta g(g(*)t - gt) + Beta Pi(Pi(*)t - Pi t), is covariance-stationary.(23)

By assumption, dt, gt, Pi t, and hence it are integrated processes, and Gamma t is convariance-stationary. It allows that these variables are cointegrated. Consequently, as Engle and Granger [1987] have shown, applying ordinary least squares to equation (25) yields consistent estimates of Beta d, Beta g, and Beta Pi whether dt, gt, or Pi t is correlated with Gamma t or not. Intuitively, this is true because ordinary least squares selects the estimators estimated Beta d,estimated Beta g, and estimated Beta Pi in order to minimize the sample variance of it - estimated Beta ddt - estimated Beta Pi Pi t, a quantity whose population variance can be finite if, and only if, estimated Beta d = Beta d, estimated Beta g = Beta g, and estimated Beta Pi = Beta Pi.

Table II reports ordinary least squares estimates of equation (25). The three-month Treasury bill rate and the series described in the previous section have been used as measures of it, dt, gt, and Pi t. The estimates are for the sample periods extending from February 1948 to December 1986, from February 1948 to December 1966, from January 1967 to December 1986, from February 1948 to December 1960, from January 1961 to December 1970, from January 1971 to December 1980, and from January 1981 to December 1986. All estimates of Beta d are negative, and only one estimate of Beta g is positive.

V. CONCLUSIONS

Between January 1981 and March 1986, the federal debt increased from 33 to 47 percent of trend output. If households view government debt as net wealth, the resulting increase in the steady-state federal debt should have appreciably raised the steady-state real interest rate. In fact, the steady-state real interest rate did not rise. The experience since 1981 has therefore been consistent with government-debt neutrality.

Many papers besides this one have also provided empirical support for government-debt neutrality.(25) Kormendi [1983], Aschauer [1985], Seater and Mariano [1985], and Evans [1988] have found no evidence that federal debt raises consumption. Dwyer [1982] has found no evidence that federal debt raises the price level or nominal interest rates. Plosser [1982; 1987], Evans [1985; 1987a], and Tanzi [1985] have found no evidence that the federal budget deficit raises nominal interest rates in the United States.(26) Evans [1987b] has found no evidence that budget deficits raise nominal interest rates in the United States and five other industrial countries. Evans [1986] has found no evidence that the federal budget deficit raises the exchange rate of the dollar. Evans [1988] has found no evidence that the sharp increase in federal debt during World War II raised forward interest rates.

APPENDIX

This appendix derives the steady-state relationship between the government debt and the real interest rate in a simple neoclassical model. It is hoped that the relationship thus derived adequately describes the average behavior of the U.S. economy over long periods of time.

Consider the model

C(t) = ( Theta + Gamma)[K(t) + D(t) + integral t infinity W(s)e raise to -integral t s r(x)dx
-Gamma(s-t) ds], (A1)
 W + r(K+D) = C + K + D, (A2)
 Y = C + K + G, (A3)
 Y = K raise to alpha (A4)


and

r = Alpha Y/K, (A5) where C is consumption; t is an index of time; K is the capital stock; D is the real stock of government debt; W is disposable wage income; r is the real interest rate; Y is output; G is real government spending; and Theta, Gamma and Alpha are parameters with Theta > 0, Gamma > or equal to 0, and 0 < Alpha < 1. Where no confusion can result, the argument t has been suppressed.

Equation (A1) describes how households choose consumption over time. Blanchard [1985] has derived it from an intertemporal maximization problem in which households discount utility at the subjective discount rate Theta, property income at the real interest rate r, and wage income and the taxes that service government debt at the rate r + Gamma.(27) They consume at the rate Theta + Gamma from total wealth, which consists of the capital stock, the government debt and the present discounted value of disposable wage income. The parameter Gamma measures how short household horizons are(28) and is a metaphor for how illiquid human capital is and how imperfectly households foresee future taxes. The larger Gamma is, the more net wealth household perceive to be embodied in government debt. If government debt is not net wealth, Gamma is zero and disposable wage income and property income are discounted at the same rate.

Equation (A2) states that total disposable income is either consumed or invested or used to purchase government debt. Differentiating equation (A1) with respect to time and substituting from equations (A1) and (A2), one obtains

C = (r-Theta)C - Gamma(Gamma + Theta)(K+D). (A6)

Equation (A3) is the aggregate budget constraint. Output must be consumed, invested, or purchased by the government. The production function in equation (A4) takes the Cobb-Douglas form. For simplicity, it is assumed in equations (A3) and (A4) that the economy is closed and that employment and population are fixed. Equation (A5) equates the real interest rate to the marginal product of capital, a condition for profit maximization in neoclassical models.

Equations (A1), (A4) and (A6) implicitly assume that taxes do not distort consumption, labor supply, and investment. This formulation was chosen not because taxes are brought to have no important effects on these decisions, but because the purpose of this paper is to test the hypothesis that government purchases and government debt affect the economy primarily through their effects on aggregate demand.

One can reduce equations (A2) to (A6) to the three equations
 c = (Alpha/k - Theta)c - Gamma(Theta + Gamma)(k + d) - [Alpha/(1-Alpha)]c(k/k)
, (A7)
 k = 1 - g - c, (A8)


and

r = alpha/k, (A9) where c = C/Y,k = K/Y,d = D/Y, and g = G/Y. Suppose that d and g converge to the constant steady-state values d(*) and g(*). This equation system then implies that c,k and r converge along a unique path to their steady-state values c(*), k(*) and r(*).(29) In the steady state, c = k = 0. With these restrictions imposed, equations (A7) and (A8) imply that

(Alpha/k(*) - Theta)c(*) = Gamma(Theta + Gamma)(k(*) + d(*)) (A10) and

c(*) = 1 - g(*). (A11)

Consider the properties of the steady state if households do not view government debt as net wealth; i.e., if Gamma = 0. In this case, equations (A10) and (A9) imply that

r(*) = Theta. (A12)

Consequently, r(*) depends on neither d(*) nor g(*).

Eliminating c(*) between equations (A10) and (A11) yields

(Alpha/k(*) - Theta)(1 - g(*)) = Gamma(Theta + Gamma)(k(*) + d(*)) (A13)

The feasible solution to this quadratic equation is

k(*) = (1/2)[-[(Theta/Gamma)(1-g(*))/(Theta + Gamma) +d(*)]

+ {[(Theta/Gamma)(1-g(*))/(Theta + Gamma) + d(*)] square + 4(Alpha/Gamma)(1-g(*))/(Theta+Gamma)} raise to 1/2].

(A14)

Implicit differentiation of (A13) results in

ak(*)/ad(*) = - Gamma(Theta + Gamma)/[Alpha(1-g(*))/k(*)square + Gamma(Theta + Gamma)] (A15) and

ak(*)/ag(*) = - (Alpha/k(*) - Theta)/[Alpha(1-g(*))/k(*) square + Gamma(Theta + Gamma)]. (A16) Since Gamma > 0, k(*) + d(*) > 0 and g(*) < 1, equations (A10) and (A11) imply that Alpha/k(*) - Theta > 0 and hence that ak(*)/ad(*) < 0 and ak(*)/ag(*) < 0. Equations (A9) and (A14) then imply that r(*) > Theta, that r(*) is a function of d(*) and g(*), and that ar(*)/ad(*) > 0 and ar(*)/ag(*) > 0.

Increasing disposable wage income by one unit in the steady state raises the term integral t infinity W(s)e raise -(r(*)+Gamma)(s-t) ds in equation (A1) by 1/(r(*) + Gamma) units and hence raises consumption by (Theta + Gamma)/(r(*) + Gamma) units. Because r(*) > 0, the steady-state marginal propensity to consume from income is less than one as was asserted in footnote 4.

Consider the properties of the steady state if Gamma = 0 and if the government levies a marginal tax rate K on the income from capital. The only modification that must be made in the model above is that r should be replaced by (1-K)r in equations (A1) and (A2). Therefore, in the steady state,

(1-K(*))r(*) = Theta (A17) so that

ar(*)/K(*) = Theta/(1 - K(*))square > 0. (A18)

(*) Associate Professor, Ohio State University. I am grateful to Dick Sweenney and two anonymous referees for helpful comments.

(1). This was originally pointed out by Ricardo (as discussed by Sraffa [1951]. More recent references are Bailey [1971] and Barro [1974].

(2). See Barro [1984] for further discussion.

(3). Poterba and Summers [1987] have shown that the short-run effects of budget deficits on consumption and hence other variables may be trivial even if households do not internalize future generations. Therefore, directly testing for these short-run effects may not be revealing. In contrast, the test performed in this paper should be revealing because the steady-state effects of government debt are nontrivial unless households largely internalize future generations.

I also focus on the steady state because it is by now fairly well established that the short-and medium-run relationships one can find between budget deficits and nominal or real interest rates are not consistent with conventional macroeconomic theory. (See section V.)

(4). If government debt is neutral, government spending does not affect the steady-state real interest rate because the steady-state marginal propensity to consume from income is one. Consequently, increasing steady-state government spending and taxes by a given amount reduces consumption by the same amount, thereby leaving investment, the steady-state capital stock, and the steady-state real interest rate unchanged. In contrast, if government debt is not neutral, the steady-state real interest rate must exceed the subjective rate of discount. It then follows that the steady-state marginal propensity to consume from income is less than one. (See the appendix for proof.) Therefore, increasing government spending and taxes by a given amount reduces consumption by a smaller amount, crowds out investment, reduces the steady-state capital stock, and raises the steady-state real interest rate.

(5). In writing it in this way, I am implicitly assuming that the tax code neutral to expected inflation. In the empirical work below, this assumption is relaxed.

(6). One can rewrite equation (10) as

Delta Xt = n=0 to infinity Summation Knvt-n, where x't, = (dt,gt,pi,z't), n=0 to infinity Summation KnL raise to n = F raise to -1 (L) and

v't = (vdt,vgt, v pi t, v'zt). Then

x(*)t = j to infinity lim EtXt+j = j to infinity lim (Xt-1 + m=0 to j Summation Et Delta Xt+n)

= Xt-1 + m=0 to infinity Summation (n=0 to infinity Summation Kn+m vt-n) = Xt-1

+ n=0 to infinity Summation (m=n to infinity Summation Km)vt-n. Because vt, is optimally forecasted as 0 in period t-1,

x(*)t-1 = Xt-1 + n=1 to infinity Summation (m=n to infinity Summation Km)vt-n. Therefore,

Delta X(*)t = (m=0 to infinity Summation Km)vt. The right-hand member of this equation is simply what one obtains by setting L = 1 in H raise to -1 (L).

(7). Section IV entertains the hypothesis that ut+et is covariance-stationary.

(8). Although this condition is restrictive, it is no doubt less restrictive than the conditions that are routinely imposed in empirical work when variables are classified as exogenous. Variables must often be assumed to be strictly exogenous while here one need only assume no contemporaneous correlation.

(9). The regression results reported in the next section are consistent with Eta t's being white noise.

(10). Plosser [1982] and Huang [1986] have estimated models of the form

hmt = Tau mdvdt + Tau mgvgt + Tau m mu v mu t + upsilon mt, where hmt is the excess holding-period yield on bonds with m periods to maturity, the Taus are parameters, and the upsilons are error terms. Expanding h Tau t to first-order terms in a Taylor series as in Shiller, Campbell, and Schoenholtz [1983] yields

h Tau t = Omega Delta ft + f Delta I Tau t, where Omega and f are negative parameters. Therefore, their estimates of Tau Td and Tau Tg and my estimates of fd and fg are related. Hence it is not surprising that their estimates Tau md are positive while my estimates of fd are negative. However, my fs can be used to estimate Beta d and Beta g, which have a structural interpretation, while their estimates are not related in any obvious way to structural parameters.

(11). It is desirable to choose similar coupons because the approximation (4) is more accurate, the closer are the coupon rates C Tau and CT in equations (5) and (6).

(12). It is assumed here that the public expected the Treasury 9 1/8s to be called in May 2004. Alternative assumptions do not affect any conclusions.

(13). Using the market value of the privately held federal debt from Cox [1986] alters none of the conclusions reached in this paper. See footnote 22 for estimates based on Cox's data. Results based on par values are reported in the text because changes in the level of interest rates can immediately affect market values, thereby inducing appreciable simultaneity.

(14). I have divided by trend output rather than output itself in order to avoid the simultaneity that would result if output is not predetermined with respect to changes in Eta t. Trend output is simply the exponential trend fitted to quarterly real GNP over the sample period 1947I-1986I and interpolated to months; i.e., 1115.5 exp(.0026343t), where t is an index of time that equals 1 in January 1947 and that increases by 1 every month.

(15). Industrial production comes from U.S. Department of Commerce [1984] Business Statistics and various issues of the Survey of Current Business. The data and programs used in this paper are available upon request.

(16). It is hoped that including at, in the vector autoregressions is a close substitute for including all the other variables that bond traders use in forming expectations since at should incorporate that information.

(17). Data availability prevents the sample from beginning earlier or ending later. The sample periods beginning in January 1953, January 1961, January 1969 and January 1981 have also been considered. None of the conclusions depends on the sample period chosen.

(18). The Monte Carlo simulations reported in Hoffman, Low and Schlagenhauf[1984] suggest that the two-step estimator used here actually has better small-sample properties than full information maximum likelihood (FIML). This result provides another reason besides ease of calculation for preferring the two-step estimator.

(19). Because the variables ft, dt, gt and Pi t are expressed in units of percent, estimated Rho t, estimated vdt and estimated vgt are also expressed in units of percent. The figures in parentheses are standard errors. The estimated coefficients are not much affected by reducing the sample period to 1982:1-1986:3, 1983:1-1986:3, 1984:1-1986:3, 1981:1-1984:12, 1981:1-1983:12, or 1982:1-1984:12.

(20). One of the referees wonders whether the inclusion of at in the vector autoregressions appreciably affects these estimates. Since the estimates are estimated fd = -.610 and fg = -3.00

( .257) (3.11) when at is excluded, the answer is no.

(21). Standard errors are not reported because they are difficult to calculate and because the inference made below is independent of what they are.

(22). Using the market value of the privately held federal debt in lieu of the par value of the gross federal debt results in

estimated Beta d = (-1.064)(.442) + (.740)(-.573) = -.894 and

estimated Beta g = (-1.064)(.015) + (.740)(.071) = .037. Clearly, these estimates yield the same inference.

(23). If the tax system is not neutral to expected inflation, then B Pi need not be one.

(24). I have tested whether dt, gt, Pi t, and it are integrated series using augmented Dickey-Fuller tests. Each Dickey-Fuller regression was augmented with twelve lags and was fitted over the sample period extending from March 1949 to December 1986, obtaining the t-ratios 2.20, 1.96, 3.10, and 1.67. Since the critical value of these t-ratios is 2.87 at the .05 significance level, dt, gt, and it easily pass master as integrated series, but Pi t does not. Applying the augmented Dickey-Fuller test to the residuals from equations (25) yields the t-ratio 2.86. According to Engle and Yoo [1987], the critical value for this t-ratio is 3.9. Therefore, one cannot reject the null hypothesis that it, dt, gt, and Pi t are not cointegrated.

(25). See Seater [1985] for a more complete review of this literature.

(26). After having had the point raised by Spiro [1987], Tanzi [1987] has admitted that in his 1985 study he inadvertently used the cyclically adjusted federal surplus as his measure of the deficit. Thus, correctly interpreted, Tanzi has presented evidence that, if anything, federal budget deficits lower nominal interest rates. Spiro also shows that Hoelscher [1986], a study often cited as evidence that the budget deficit raises interest rates, is actually consistent with a negligible or negative effect.

(27). Equation (A1) corresponds to Blanchard's equation (5). Gamma and W are what Blanchard denotes by p and Y.

(28). In Blanchard's model, Gamma is the rate at which households die and are replaced by households with which they are entirely unconnected. Households therefore wish to consume a fraction Gamma of wealth every period in addition to the fraction Theta that they would wish to consume, if they had infinite horizons. They also discount disposable wage income at a higher rate than they discount property income because after they die their disposable wage income accrues to the households that replace them.

(29). Existence requires that g not be too large. Uniqueness reflects the saddlepoint property of the system (A7) and (A8).

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Date:Jan 1, 1989
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