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Government debt and the demand for money: an extreme bound analysis.


The article provides evidence that there is a relationship between government debt and interest rates via the demand for money. This relationship is examined through the wealth effect of government debt on money demand, and the robustness of the results is tested by the use of extreme bound analysis in addition to standard econometric techniques. We find that OLS regression shows government debt affecting the demand for money positively, implying that Federal government debt is net wealth. In addition, the extreme bound analysis shows that the estimates of the government debt coefficient are robust under alternative specifications of the Goldfeld model.


This paper studies the relationship between government debt and interest rates. The approach used in this paper differs from previous studies in two ways. First, this relationship is examined through the wealth effect of government debt on the demand for money. Second, the robustness of the results is tested by use of extreme bound analysis as an alternative to standard econometric techniques.

Theoretically, there is no clear cut consensus as to whether or not government debt should be net wealth. The argument used most often is that government debt will be net wealth if future tax liabilities to service the debt are, for the aggregate economy, only partially discounted. Two explanations for this partial discounting are offered. First, in dealing with finite-lived individuals, Thompson [1967] assumes that the average remaining lifetime of current taxpayers is shorter than for future interest payments on the federal debt. In this situation, the net present value of a stream of equal interest payments and taxes will be positive. As a counter argument, Barro [1974] has shown that current generations can act effectively as if they are infinite-lived when they are linked to future generations through a series of pairwise transfers of wealth. Here, the realization by current generations that tax liabilities will be left to future generations extends the lifetime of the relevant stream of tax liabilities for debt service. Government debt is then wealth neutral.

Mundell [1971] provides a second reason for partial discounting by assuming that capital markets are imperfect and that the relevant discount rate for tax liabilities is higher than the rate for interest payments. These discount rates are, respectively, thought of as applying to high-risk-of-loan default and low-risk-of-loan-default individuals. This difference in discount rates creates a positive wealth effect, since under Mundell's assumptions government debt held by taxpayers involves a loan from low-discount-rate to high-discount-rate individuals. As Barro [1974] has again countered, this wealth effect will exist only to the extent that the government is more efficient at providing such a loan than the private sector.

Finally, Barro [1974], Kormendi [1983] and later Aschauer [1985] advance an argument as to why government debt might be negative net wealth. Suppose there is considerable uncertainty about the intertemporal and cross-sectional burden of the future tax liability necessary to service the debt. In this case, the certainty-equivalent value of these future taxes might exceed the certainty-equivalent value of the future income stream of the debt, so that government debt would be negative net wealth.

The majority of the empirical studies of the net wealth-induced linkage between government debt and interest rates have attempted to measure this relationship using consumption functions. If privately-owned government debt is net wealth, the wealth effect of this debt working through a Keynesian consumption function will be positive, so that increased government borrowing will result in increased current consumption and interest rates. Several studies have assessed the relationship between government borrowing and consumption or interest rates via this mechanism. In general, the wealth-consumption effect of government debt in these studies has been inconclusive. Buiter and Tobin [1979] find that privately-held government debt is net wealth, while Tanner's [1979] results show government debt to be wealth neutral. Kormendi's [1983] and Evans's [1985] results lead one to conclude that federally-issued government debt may reduce the wealth of private asset holders, while Barth, Iden and Russek [1986a] show that tests of whether government debt is net wealth are sensitive to the sample period chosen. Finally, through the use of simulation, Barsky, Mankiw and Zeldes [1986] conclude that a debt-financed tax cut may result in a positive wealth effect on consumption by increasing current income, if there is a perceived sharing of risk of future tax liabilities for debt service among taxpayers.

Failure to find a clear cut consensus link between government debt and interest rates in the above studies is due to possible biases in the coefficients of government debt (or taxes) when these variables are introduced into consumption functions. Feldstein [1976; 1982] and later, Aschauer [1985] show that if increased government debt (or changes in taxes) provides a signal to private wealth holders that there will be future increases in government-provided services, which could substitute for private consumption, present consumption patterns may be altered. This in turn, biases the coefficients of government debt (or taxes) in these consumption studies, such as in Kormendi [1983], Kormendi and Meguire [1986], Barth, Iden and Russek [1986b] and Barsky, Mankiw and Zeldes [1986].

Use of a money demand equation is aimed at providing a remedy for the varying results of such studies. This alternative approach determines whether government debt is net wealth through its effect on the demand for money. The approach precludes the problem of private agents altering consumption patterns to perceived future-provided government services. Government debt (bonds) is viewed as a part of one's portfolio. Presumably, portfolio decisions are not directly influenced by government-provided consumption as are private sector consumption decisions. This is because portfolio demand is not a function of income but total nonhuman wealth. Therefore, the biases of government debt (or tax) coefficients in consumption studies are not present. If the coefficient of government debt is positive in the money demand equation, government debt is net wealth, at least in the short run. In this situation, even if there are no long-run effects of government debt on interest rates working through altered real consumption patterns, there may be short-run wealth effects of government debt on interest rates operating through the demand for money.

A potential problem exists for the measurement of this wealth effect using the demand for money. Cooley and LeRoy [1981] argue that most money demand studies are highly sensitive to specification search. Along the same lines, Plosser [1982] argues that the relationship between government financing decisions and interest rates is sensitive to the specification of the estimating equation. This paper handles the problem of possible specification bias in the money demand equation through use of extreme bound analysis [Leamer 1983; 1985] on the estimated coefficients. Performing this sensitivity analysis on the government debt coefficient across alternative money demand specifications will allow the authors to test the robustness of the conclusions about whether government debt is net wealth. This will also provide a test of robustness of any conclusion that can be drawn about the link between government debt and interest rates through a wealth effect on the demand for money.


Extreme bound analysis is a procedure suggested by Leamer and Chamberlain and Leamer [19781 and further developed by Leamer [1982; 1985], for dealing with uncertainty involved in constructing econometric models. Formally, the procedure allows the researcher to investigate the effect the inclusion or exclusion of variables he considers of "doubtful" importance has on the regression coefficient of a "focus" variable over alternative probability likelihoods generated by the sample data.

In economic modeling it is commonplace to have a model with a few explanatory variables that are known to belong in the equation and a longer list of "doubtful" variables. The former variables are usually of focus in the analysis, while the latter variables are often used to control for other factors. A problem arises when the list of doubtful variables is long, since the attempt to include all doubtful variables in the estimating equation leads to large standard errors for the coefficients of the "focus" variables.

In this situation, the standard practice is to experiment with different sets of doubtful variables, with the hope that the coefficients of the focus variables will not change greatly as the list of doubtful variables is altered. The flaw in this standard approach is that it is usually conducted haphazardly and does not allow the averaging of the many specifications implied by the examination of the list of doubtful variables into a single number. Extreme bound analysis is a formalization of the search for an econometric specification in this type of situation, aimed at correcting for the above two flaws.

Consider, then, a linear regression model in which the dependent variable [Y.sub.t] is thought to be a function of the three independent variables [X.sub.t], Wit and [W.sub.2t]. Such a model can be written:

[Y.sub.t] = [X.sub.t][beta] + [W.sub.1t][[alpha].sub.1] + [W.sub.2t][[alpha].sub.2] + [[mu].sub.t] (1)

where [beta], [[alpha].sub.1], and [[alpha].sub.2] are the regression coefficients to be estimated, and [[mu].sub.t] is a disturbance term. Suppose that the researcher's primary interest is in estimating the coefficient [beta] on the focus variable [X.sub.t], but that there is some uncertainty as to the correct specification for the above model. This uncertainty is handled by considering the variables [W.sub.1t] and [W.sub.2t] to be doubtful. In other words, the researcher is not certain of the effects of [W.sub.1t] and [W.sub.2t] on the dependent variable, but is unwilling to exclude these variables from the equation.

This equation can be rewritten by defining a composite variable,

[Z.sub.t]([theta]) = [W.sub.1t] + [theta] [W.sub.2t],

to obtain

[Y.sub.t] = [X.sub.t][beta] + [Z.sub.t]([theta])[alpha] + [[mu].sub.t]. (2) (2)

For properly defined values of [theta], this regression coincides with any of the four regressions defined by the inclusion or exclusion of the doubtful variables, [W.sub.1t] and [W.sub.2t]. However, as Cooley and LeRoy [1981] argue, there is no reason to concentrate on these four special regressions, because these particular regressions are a subset of a more general class in which [theta] can take any arbitrary value.

A different value for [theta] implies a new set of values (or location) for [[alpha].sub.1] and [[alpha].sub.2] and thus a different corresponding estimate for the focus coefficient. The set of all possible values of [beta], generated by alternative values of [[alpha].sub.1] and [[alpha].sub.2] which in turn is generated by varying [theta]) will define an ellipsoid of possible estimates constrained by the inclusion or exclusion of the doubtful variables.

The extreme bound analysis then entails generating the maximum and minimum point estimates of the focus coefficient on the locus of constrained estimates, over a sequence of likelihood ellipsoids, which are standard confidence intervals around the estimated coefficient of the focus variable in an ordinary least squares (OLS) model. If the gap between the upper and lower bound at a chosen likelihood ellipsoid is larger than the sampling standard error of the OLS coefficient for the focus variable, researchers are cautioned that no reliable inference can be drawn about the estimate of the focus variable.


The relationship between the demand for money and the government debt is investigated through specifying a money demand function having real income, total wealth of the household, and rate of interest as explanatory variables. Wealth is defined as consolidated net wealth of the private sector, including that sector's ownership of government debt. Next, the wealth variable is decomposed into the assets generated by the private sector (WP) and government debt (WG). The sign and significance level of the estimated coefficient for the government debt component of total wealth is used as the primary tool for the analysis. A positive coefficient for WG points towards a positive wealth effect for government debt, and a negative estimate indicates a negative wealth effect associated with an increase in government debt.

The outcome of the estimation depends on the specification of the demand for money that one uses. Therefore, before the test can be performed, a robust demand for money must be specified. This paper uses a version of Goldfeld's [1973] equation. The Goldfeld equation specifies the demand for money as a function of two rates of interest and real GNP as a scale variable. In this paper, this equation is expanded by entering the total net wealth of the household as an additional scale variable. This expansion is consistent with a portfolio demand for money and is necessary since the primary focus variable in this study, government debt, is a stock variable.

One shortcoming of the above procedure is that in general wealth effects are usually not found to be very important in money equations which focus on income as the primary scale variable. Thus a statistically significant government wealth coefficient should provide strong evidence in favor of a wealth effect in the demand for money. On the other hand, a statistically insignificant government wealth coefficient is necessary, but not sufficient, to reject the existence of a wealth effect in the money demand equation. This is particularly true because, as stated, a demand for money equation in terms of real income is shown to be inherently biased against wealth effects.

One possible solution to the above bias in the money demand equation is to drop real income from the estimating equation. This solution, although resolving the bias issue, can result in a possible specification problem.

Goldfeld [1973) and Friedman [1978] show that when income is replaced by real wealth in a money demand equation, speed of adjustment becomes implausibly low. This in turn leads to unrealistically high and possibly biased long-term scale and interest rate elasticities.

Given the above considerations, the Goldfeld equation is revised to give:

lnM1=C+[[beta].sub.1] lnWP+ [[beta].sub.2]InWG+ [[beta].sub.2]InY+ [[beta].sub.4]lnRCP+ [[beta].sub.5] lnRTD+ [[beta].sub.6]InM1(-1), (3)

where M1 = real money balances, WP = real net worth of the private sector, excluding that sector's ownership of government debt, WG = real government outstanding debt measured at market value, Y = real GNP, RCP = rate on three months' prime commercial paper, RTD = rate on passbook savings accounts and M1(-1) = M1 lagged one quarter.

After equation (3) is fitted to the data and the coefficients are estimated, this paper attempts to compute the extreme bounds of the [[beta].sub.2] coefficient. The null hypothesis to be tested here is that of the relative fragility of the estimate Of [[beta].sub.2]. If the bounds on [[beta].sub.2] are found to be wide, i.e., to contain both positive and negative values, then the empirical estimate of a wealth effect of government debt is argued to be a fragile one. In addition, if the difference between the extreme values of [[beta].sub.2] is large relative to the sampling uncertainty, then the uncertainty in the model specification can be argued to be a major contributor to the uncertainty of the value of [[beta].sub.2].

There is a potential problem with this approach. As the discussion between Leamer [1983; 1985] and McAleer et al. [1985] suggests, a hypothesis can always be made more fragile by indefinitely extending the list of doubtful variables. This lessens the usefulness of extreme bound analysis for drawing implications about the fragility of point estimates and points to the necessity of limiting the list of doubtful variables.

To deal with this problem the concept of a "free" variable is introduced by Leamer [1978]. A free variable is treated neither as a focus variable nor as a doubtful variable but nevertheless must be included in an estimating equation as dictated by economic theory, e.g., price in a demand equation.

Including such a variable in the estimating equation naturally limits the class of economic models under study. In addition, Leamer suggests that the list of doubtful variables should only include variables that should plausibly enter the estimating equation, again dictated by theory.

This approach is adopted here. Since this paper uses a portfolio demand for money, money demand will in all instances be taken to be a function of private wealth. Similarly, since all quarterly money demand estimates find some adjustment lag, in all cases the demand for money will be a function of MI lagged once. These two variables will therefore be treated as free variables, and not considered in the sensitivity analysis for the coefficient on government debt. And, following our extended version of the Goldfeld specification, our list of doubtful variables will be limited to interest rates, and in some instances, real income.


Equation (3) is estimated over the 1954:1-1980:4 time period. The length of the sample period was determined by the availability of a consistent series for the market value of real government debt (WG). The sample is then divided into two sub-sample periods covering 1954:1-1972:4 and 1973:1-1980:4, and a Chow test is employed. The reason for dividing the sample into two sub-samples results from the missing money episode. The results of the Chow test indicate that the hypothesis of the equality of the estimated coefficients of the two sub-sample structures, i.e. no structural change, is rejected. Given this, in order to minimize the influence of the structural break on the sensitivity analysis of the [[beta].sub.2] coefficient, equation (3) is fitted to the empirical observations of the data covering the separate periods. Least squares applied to the equation (3) yields the following estimates (standard errors in parentheses).

[Mathematical Expression Omited] (4)

R2 = .996 SEE = .004 Durbin h = .863 p=.433

Sample Period = 54.1 - 72:4

ln M1 = .808 -.031 ln WG + .029 ln WP (5)

(688) (.055) (.058)

+ 096 In Y -.026 In RCP -.295 ln RTD

(.119) (.015) (.068)

+ 806 ln M1(-1)


[R.sup.2] = .984 SEE = .007 Durbin h = .418

Sample Period = 73.1 - 80.4

The estimates of the coefficients over the earlier period are in line with those reported by others, i.e., Goldfeld [19731 and Friedman 1978]. With regard to the coefficient of interest, the coefficient on WG, the result over the earlier period indicates that government debt is positively related to the demand for money. In other words, government debt appears to be net wealth over the first sub-sample. There is, however, no evidence of any wealth effect of government debt for the second sub-sample. This finding is in line with discussion of Barth, Iden and Russek [1986a] and Darby [1984]. As reported by Barth, Iden and Russek, the results of the government debt neutrality proposition is sensitive to changes in the sample period. Given that the debt variable is found to have no relationship with the demand for money during 1973:1-1980:4 time period, this paper concentrates on the estimated equation over the earlier sub-period, that is equation (4).

In order to test for the robustness of the estimate Of [[beta].sub.2], sensitivity analysis is performed on this coefficient. In this experiment, private wealth and lagged M1 are assumed to be free variables and government debt and real income are taken as the "focus variables." The remaining variables are assumed to be doubtful variables. The extreme bound analysis is then applied to the coefficient of focus variables for various values of the data likelihood. The results for our primary focus variable, government debt.

Two important conclusions can be drawn from the results. First, both the lower and upper bounds of the debt coefficient include only positive values over the entire range of probability ellipsoids. This suggests that the hypothesis that government debt has positive net wealth effects on the demand for money function is not a fragile one using Leamer's [1983] definition of fragility.

The second conclusion that can be drawn is that the difference between the extreme bounds Of [[beta].sub.2] as a measure of the uncertainty in the estimated parameter is large relative to the sampling uncertainty of the data. This is notable given the fact that the uncertainty measure for the former, with 95 percent probability, is .065 relative to .040 for the latter. This implies that uncertainty in model specification can be a contributing factor to the conclusion about the value of the focus coefficient, although not its sign in the present case.

Although not reported here, the same conclusion can be drawn for the parameter on real income, the other focus variable. That is, the results show that there is not a sign change implicit in the bounds of the coefficient of this variable. Together, these results seem to suggest that there is no systematic bias against government debt induced wealth effects and also no bias against transactions effects in the money demand function estimated in this study.

For completeness, the list of doubtful variables is expanded to include real income, and extreme bound analysis is repeated. The issue of interest here is to test the robustness of the results reported in Table I to an expansion in the list of the doubtful variables. If the government debt is correlated with real income or some combination of real income and other doubtful variables, entering real income as a doubtful variable in the equation (4) should make the government debt coefficient fragile.

The null hypothesis that government debt is net wealth cannot be rejected, given the alternative money demand formulations specified by the extreme bound analysis and the expanded list of the doubtful variables.

The results suggest that government debt is net wealth. But since the bounds on the government debt coefficient are relatively wide, this conclusion appears to be sensitive to the money demand model used in this paper. To pursue this possibility a procedure initiated by Cooley and LeRoy [1981] is followed.

First note that equation (3) can be considered to be a static specification since it only contains contemporaneous values of the doubtful variables. In order to see if using this static specification has any impact on the results, equation (3) is expanded to include as explanatory variables all the doubtful variables tagged once. A more dynamic version of equation (3) is therefore specified, and extreme bound analysis is repeated for this expanded equation.

As can be seen, although the OLS estimate provides a positive government debt elasticity for the expanded model, the lower bound of the debt coefficient includes both positive and negative numbers over a wide range of probability ellipsoids. Therefore, no significance can be attached to the sign of any particular point estimates for [[beta].sub.2], as the additional doubtful variables are included in equation (3).

The primary conclusion to which this points is that the hypothesis that government debt is net wealth is robust within the context of the Goldfeld money demand equation. But, this conclusion appears to be relatively model specific, since the net wealth hypothesis for government debt is fragile within the context of this dynamic version of Goldfeld's equation.


This paper has studied the issue of whether government debt affects interest rates via a real wealth effect on the demand for money. This procedure has allowed a clearer assessment of the real wealth content of the federal debt than studies based on consumption functions, because the coefficients of the demand for money are not biased by debt-induced expectations of government provided services. The use of extreme bound analysis has allowed for accounting of specification bias inherent in money demand models.

The OLS regression shows government debt affecting the demand for money positively, implying that federal government debt is net wealth. In addition, the extreme bound analysis shows that the estimates of the government debt coefficient are robust in the sense of retaining positive values under alternative specifications of the model's doubtful variables. However, this conclusion appears to be specific to Goldfeld's money demand equation, in which the doubtful variables were limited to current values of income and interest rates.

How far Goldfeld's equation can be extended and still retain robust government debt coefficient estimates is a question for further research. Resolution of this question using extreme bound analysis, though, will first require a more complete theoretical model of the role of government debt in asset holders' portfolios than is usually assumed in the money demand literature.


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Author:Deravi, M. Keivan; Hegji, Charles E.; Moberly, H. Dean
Publication:Economic Inquiry
Date:Apr 1, 1990
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