Deficits, explicit debt, implicit debt, and interest rates: some empirical evidence.

Wang, Zijun ; Rettenmaier, Andrew J.

1. Introduction

The effect of government borrowing on interest rates has been a controversial issue for about three decades. The federal deficits of the 1980s and the early 1990s caused speculation as to whether higher interest rates would follow. The current deficits have renewed interest in the link to higher interest rates. Since the influential work of Barro (1974), much of the discussion has focused on whether the Ricardian equivalence hypothesis holds. Bernheim (1987), Barth et al. (1991), Seater (1993), Elmendorf and Mankiw (1999), and Engen and Hubbard (2005) provide excellent reviews of the literature on the topic. Recent contributions include Cebula (1997); Gale and Orszag (2003); Laubach (2003); and Ardagna, Caselli, and Lane (2004) among others. Although empirical evidence obtained in earlier studies is mixed, as argued by Gale and Orszag (2003), major macroeconometric models imply an economically significant connection between changes in deficits and long-term interest rates. Empirical evidence assembled by recent studies seems to lean toward the existence of the relationship, but there is still no consensus about the magnitude of the effect.

A missing piece in the debate is a formal investigation of the effect of the implicit debt, as embodied in unfunded obligations of Social Security and Medicare. Theoretically, it could be argued that because the Ricardian equivalence proposition assumes that households are rational and make decisions for long horizons, then like the often-discussed public debt (explicit debt), the implicit debt (computed based on 75-year, 100-year, or infinite horizons) might also have a role to play in the determination of interest rates. The point we raise here is not new. In his 1996 paper "Reflections on Ricardian Equivalence," Barro writes "... (T)he basic invariance proposition for intergenerational transfers ... (is) that the government's transfers implied by budget deficits or pay-as-you-go social security would be fully undone if family members were connected through voluntary transfers based on altruism" (p. 2). More recently, Gokhale and Smetters (2005) also note that (Social Security's) pay-as-you-go financing may "crowd out" private saving and hence increase interest rates (p. 12). (1) Therefore, it is problematic to test the Ricardian equivalence hypothesis without considering the effect of the implicit debt. Empirically, the explicit debt, although large in absolute amount (about $4300 billion by the end of 2004), is only a part of the federal government's total obligations. According to the Office of the Chief Actuary of the Social Security Administration, the Social Security program alone carries unfunded obligations ranging from about $3700 to $11,200 billion in present values at the beginning of 2004 under various assumptions. These numbers are even larger if the trust fund balances are subtracted. In particular, projected Old Age, Survivors, and Disability Insurance payroll tax income will begin to fall short of outlays in 2017, which means that Social Security will require transfers from general revenues in the next decade (Social Security Trustees 2005).

While previous studies have investigated the effect of Social Security wealth on saving and consumption (e.g., Feldstein 1974, 1996; Smetters 1999), and some have also emphasized the importance of the implicit debt in the discussion of the relationship between government borrowing and interest rates (e.g., Gale and Orszag 2003, p. 463), little has been done to quantify the effect of the implicit debt on interest rates, with the exception of Wang (2005). The purpose of this paper is to characterize the dynamic effects of government borrowing on longo term interest rates using vector autoregression (VAR) models. This paper contributes to the discussion in two ways. First, we consider both explicit and implicit debt. Such a comprehensive total debt measure provides a more accurate indication of the burden imposed on future generations by government borrowing behavior than does a measure of annual deficits or a measure of the explicit debt alone. (2)

Second, we also wish to contribute to the literature with respect to the econometric method employed in the analysis. It is well known that estimating the effects of government borrowing on interest rates is complicated by the need to isolate the effects of fiscal policy from other influences, for example, the impacts of monetary policy actions of the Fed. Empirically, the identification in the context of reduced form VAR models is often achieved by assuming a Wold-causal order for the elements of the multivariate process so as to organize the triangular factorization of the innovations covariance matrix (Cholesky factorization). The major problem of the Cholesky factorization is that it critically depends on the ordering of variables in the VAR. Different orderings may lead to quite different results depending on the degrees of correlation between different innovations. Empirical researchers thus often rely on economic theory or other prior knowledge to determine the variable ordering. As is evident in the debate on the relationship between deficits, debt, and interest rates, predictions of economic theory are often ambiguous.

In this study, we adopt a structural VAR approach, which is of the Sims-Bernanke type (Bernanke 1986; Sims 1986). To achieve identification, we introduce a data-driven method to search for the causal structure in the innovations. The suggested method of directed graphs is the graph-theoretic analysis of causality. As demonstrated later in the paper, the study and application of the directed graph method is in line with the growing interest among econometricians and applied researchers in automated model discovery. (3)

An approach similar to this study is used by Wang (2005), who studied the impact of the implicit debt on short-term interest rates using a generalized VAR approach. The basic finding of Wang (2005) is that the implicit debt appears to have some moderate influence on real interest rates only at long horizons. The current paper focuses on the effect on long-term rates because, as argued by Engen and Hubbard (2005), if federal government borrowing crowds out private capital formation, then one would expect to find a larger impact on long-term interest rates than on short-term interest rates. Both the generalized impulse responses used by Wang (2005) and the directed acyclic graphs-based impulse responses adopted in the current paper are invariant to the ordering of variables in the VAR model. However, one calculates generalized impulse responses by conditioning on the future shocks, which embody the empirical correlation between shocks. By doing this, the generalized method tries to "separate" the impact of a particular shock from others by integrating out the effects of other shocks to the system. In this sense, the generalized method probably does not have or build on a clear-cut "structural" assumption about shocks, while the directed graph method explicitly sorts out contemporaneous causal relationships among the variables (more about this in the next section).

The overall evidence from our study suggests that deficits, explicit debt, and implicit debt all have some effects on the 10-year government bond interest rate, although with different dynamics. Based on our preferred model specification, we estimate that a 1 percentage point increase in the primary deficits, and explicit and implicit debt (all normalized by gross domestic product [GDP]) may lead to a maximum of a 56-, 10-, and 2-basis-point rise in the long-term interest rate, respectively. Nevertheless, these effects appear to be temporary and tend to die out within a 10-year horizon.

The rest of the paper is organized as follows. Section 2 briefly introduces the directed graph method and discusses its use in the VAR identification. Section 3 presents the data. Section 4 examines the contemporaneous causal relationships between the set of variables under study and presents the dynamic effects of federal deficits, explicit debt, and implicit debt on long-term interest rates. Section 5 summarizes the major results and concludes.

2. Directed Graphs and Structural VAR Identification

Directed graphs have been studied for decades. The recent developments are motivated by the research of Pearl (2000), Spirtes, Glymour, and Scheines (2000), and their coauthors. Swanson and Granger's (1997) work adapted the method to uncovering the causal order within a VAR model. Demiralp and Hoover (2003) and Hoover (2005) provide an accessible introduction to the method for causal analysis. In this section, we briefly describe how to conduct the directed graphs analysis using the variance-covariance matrix of the VAR innovations (residuals).

The basic idea behind directed graphs is to represent causal relationships among a set of variables using an arrow diagram. Mathematically, directed graphs are designs for representing conditional independence as implied by the recursive product decomposition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where pr is the probability of variables [v.sub.1], [v.sub.2] ..., [v.sub.n]. The symbol [[OMEGA].sub.i] refers to the realization of some subset of the variables that precede (come before in a causal sense) [v.sub.i] in order (i = 1, 2, ..., n), and [PI] is the product (multiplication) operator.

As an important contribution to the literature, Pearl (1986, 1995) proposed "d-separation" as a graphical characterization of the conditional independence relationship just described. Two vertices (for example, variables X and Y) are said to be d-separated if the information flow between them is blocked. This occurs when: (i) one variable is a common cause, say W in the graph X [left arrow] W [right arrow] Y, or a mediator in a causal chain, say U in the graph X [right arrow] U [right arrow] Y, and we condition on W or U; or (ii) if a variable Z is the middle variable (a collider) in an inverted fork (X [right arrow] Z [left arrow] Y) and we do not condition on Z or any of its descendents (descendents are not shown here).

If we formulate a directed graph in which the variables corresponding to [OMEGA]i are represented as the parents (direct causes) of [v.sub.i], then the independencies implied by Equation 1 can be read off the graph using the criterion of d-separation. Geiger, Verma, and Pearl (1990) showed that there is a one-to-one correspondence between the set of conditional independencies, X [perpendicular to] Y | Z (X is orthogonal to Y conditional on Z) and the set of triples (X, Y, Z) that satisfy the d-separation criterion in a graph G. Specifically, if G is a directed acyclic graph with variable set V, and if X and Y are in V, and Z is also in V, then the implied linear correlation between X and Y in G, conditional on Z, is 0 if and only if X and Y are d-separated given Z. Here, "acyclic" means that one cannot return to any starting variable by following arrows that lead away from that starting variable. Thus, the chain relationship X [right arrow] Y [right arrow] X is not allowed in a final directed graph. For convenience, this type of graph is abbreviated as DAG later in the paper. From the preceding brief description, we can see that conditional independence plays a central role in the graph method. Of course, the idea of conditional independence is not new in econometric modeling. For example, in the long-standing debate of the money-income causality, Sims (1980) found that the causality exists in the bivariate case but virtually disappears in the trivariate case when an interest rate variable is also included.

There exist several alternative search algorithms in the literature that can be used to implement directed graphs. Spirtes, Glymour, and Scheines (2000) describe the PC algorithm. A Bayesian search algorithm, the greedy equivalent search (GES) algorithm, is given in Chickering (2002). The GES algorithm is a stepwise search over alternative DAGs using Bayesian posterior scores. The algorithm consists of two stages, beginning with a DAG representation with no edges (independence among all variables). Edges are added and/or edge directions reversed in a systematic search across classes of equivalent DAGs if the Bayesian posterior score is improved. The first stage ends when a local maximum of the Bayesian score is found such that no further edge additions or reversals improve the score. From this final first stage DAG, the second stage commences to delete edges and reverse directions if such actions result in improvement in the Bayesian posterior score. The algorithm terminates if no further deletions or reversals improve the score. Both the PC and the GES algorithms are embedded in the TETRAD IV software (available online at www.phil.cmu.edu/projects/tetrad/). Similar to parametric tests that can make type I and type II errors, these search algorithms may make errors of two types: edge inclusion or exclusion and edge direction (orientation); the latter appears to be more likely than the former, especially when the sample size is small. Therefore, the following results should be viewed with caution and should be interpreted in light of other relevant information.

Swanson and Granger (1997) and Wang, Yang, and Li (2007) are two studies that apply the directed graph method to economic time series. We follow these papers in the study of innovations from a first stage VAR. Specifically, let [Y.sub.t] denote a (m x 1) vector of stationary processes. The dynamic relationship among these processes can be modeled as a VAR of order k,

[Y.sub.t] = [k.summation over (i = 1)][[PHI].sub.i][Y.sub.t-i] + [[epsilon].sub.t](t = 1, ..., T) (2)

Following Bernanke (1986), we can write the innovation vector ([[epsilon].sub.t]) from the estimated VAR model as A[[epsilon].sub.t] = [v.sub.t], where A is an m x m matrix and [v.sub.t] is a new m x 1 vector of orthogonal shocks. As has been discussed, the key issue here is how to specify the A matrix. As a special case, the Cholesky factorization provides an identified causal structure. Doan (2000) gives conditions for identification of the elements of A. The literature that follows the lead of Sims (1986) typically uses nonsample information to specify A, which can result in overidentification. Here, we apply the directed graph method to find the causal order using the reduced form innovations of the covariance correlation structure as input. Starting with an identity matrix M, we then replace element M (i, j) with a 1 if shocks in variable j contemporaneously cause shocks in variable i, based on the identified causal structure (DAG). Given the structural pattern matrix M, the decomposition factor matrix A can be estimated following Doan (2000, pp. 292-3).

3. Data and Variables

Based on both theoretical and empirical work, particularly Miller and Russek (1996) and Ardagna, Caselli, and Lane (2004), we include the following annual observations in our VAR analysis: The long-term interest rate (represented by the 10-year constant maturity Treasury note yield, rlong), the short-term interest rate (represented by the three-month Treasury bill secondary market rate, rshort), the inflation rate (infl), the GDP growth rate (growth), the primary deficit (national income and product account measures, deficit), federal debt held by private investors (explicit debt, debt), and Social Security unfunded obligations (uf) (deficit, debt, and uf are all measured as a ratio of GDP). The first six time series are easily accessible from the online resources of the corresponding government agencies.

Note that we use the primary deficit, rather than the total deficit, because it "strips out" the direct effect of interest rates on government spending, thus better capturing autonomous changes in fiscal policy (Ardagna, Caselli, and Lane 2004). To estimate the effect of public debt on interest rates, we use public debt held by private investors rather than total public debt. This is because government debt that is purchased by the Federal Reserve to increase the money supply may not have the same effect as federal debt held by private investors (Engen and Hubbard 2005). Furthermore, the inclusion of the inflation rate variable in the system should help capture the effect of the federal debt owned by the Federal Reserve. Nevertheless, the two series are highly correlated with each other. Further note that both deficits and public debt variables are included in the regression. While this might add some complication in interpreting the effects of the two variables, it helps capture possible nonlinear effects of government borrowing on interest rates (Ardagna, Caselli, and Lane 2004). By construction, the two variables are not perfectly correlated. Finally, Calomiris et al. (2003) find from their event study that news suggesting more robust economic growth is associated with an increase in interest rates. To control for this effect, GDP growth rate is included in the VAR model.

The Office of the Chief Actuary of the Social Security Administration calculates two major types of annual estimates of the Social Security unfunded obligations. The types are distinguished by the group of participants included in the unfunded obligation measure. The open group obligation measures that are calculated over a 75-year and an infinite horizon include both current and future participants in Social Security. In contrast the closed group measure is limited to current participants and is equal to the present value of the benefits they expect to receive less the remaining taxes they will pay. The most often cited estimate, however, is based on a rolling 75-year open group assumption (denoted as uf75). It is basically the present value of the difference between future revenues (taxable payroll) and expenditures of the program for the next 75 years less the value of the Trust Fund. This estimate is used to determine the required tax increase that is necessary to attain actuarial solvency for the program over the 75-year horizon. At the writing of this paper the series is available for the period of 1979 through 2003. For our analysis, the value of the Trust Fund, consisting of special Treasury notes, is removed from the estimates given that we are trying to capture a more complete measure for debt. We expand the series back to 1976 with data from Goss (1999, Table 3) (see Wang 2005 for more details). (4)

Similar to Levine, Loayza, and Thompson's (2000) measure of public debt, here both explicit and implicit debt series at year t are constructed as the average of debt stocks at period (t - 1) and period t (in case of the implicit debt, the period t value is actually the period [t + 1] beginning value).

4. Empirical Results

VAR Estimation

With annual observations spanning the period 1976 through 2003, we estimate a VAR system of the following variables: rlong, rshort, infl, growth, deficit, debt, and uf75. The augmented-Dickey-Fuller (ADF) test indicates that all variables except uf75 appear to be stationary at the 10% significance level (a maximum of three lags are allowed in the test). This is consistent with Ardagna, Caselli, and Lane's (2004) finding based on a panel of 16 OECD countries for the similar period. (5) The apparent nonstationarity property of the implicit debt series is probably caused by two sharp drops of the estimates following legislative changes to Social Security in 1977 and 1983. Because we have only 28 effective sample observations, we do not pursue this further. For the same reason, in estimating the high-dimension system, we include one lag for each variable in the VAR. In addition to an intercept term, a linear trend term is also included in model 2 because it is significant (at the 0.05 significance level) in some equations of the VAR. The estimated innovations (residuals) correlation matrix is a key input for the previously discussed statistical-based structural identification of contemporaneous causal links. We report them in Table 1. (6)

[FIGURE 1 OMITTED]

Causal Relationships and Decomposition of the Innovations

The PC algorithm has been widely used by applied researchers in the past. However, Monte Carlo simulation evidence suggests that the GES algorithm performs better in identifying directed acyclic graphs when the sample size is small or moderate. Following Wang and Bessler (2006), here we rely on GES to determine contemporaneous causal flows between interest rates and the other macro variables. Figure 1 plots the final directed acyclic graph, which is based on the innovations estimated from the VAR model. Notice that eight directed edges have been added to the graph. (7) We offer a brief discussion on the edges of interest in the following paragraph.

The innovation in short-term interest rate (rshort) is caused by two other innovation processes. The first factor is the inflation rate, which is not surprising because both short- and long-term interest rates are measured in nominal terms. Consistent with the findings reported by many empirical studies (as reviewed in the preceding discussion), the graph shows that changes in deficits have a contemporaneous impact on the short-term interest rate. Of central interest, the long-term interest rate (rlong) is caused by the short-term interest rate, not the other way around. This causal direction is in agreement with the familiar term structure of interest rates. Recall from Table 1 that rlong is also correlated with infl and deficit. However, when conditional on information on rshort, these two variables no longer have direct causal impact on the long-term interest rate. Alternatively, they do not have further impact on rlong beyond the part that has been incorporated in rshort. Because most new issues of debt (deficits) are held by private investors, changes in deficits have a direct impact on debt. This is confirmed by the search results in Figure 1.

[FIGURE 2 OMITTED]

Based on the DAG in Figure 1, the factorization pattern matrix M to decompose the VAR innovations can be specified as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Recall that the order of the seven variables is rlong, rshort, infl, growth, deficit, debt, and uf75. Clearly, model 2 is now overidentified. The usual chi-square test for the imposed restrictions has a statistic of 8.66 with 13 d.f., which equals 21 (the number of free parameters if the model is just identified) minus 8 (the number of free parameters implied by the causal graph in Figure 1, or number of 1s in off-diagonal M). Because the p-value is 0.80, we do not reject the restrictions.

We note from Figure 1 that deficits cause the short-term interest rate, which in turn causes the long-term rate. There is no direct casual flow from deficits to the long-term rate. This result is not in line with some of the existing empirical literature, which finds that long-term interest rates are affected by fiscal variables above and beyond the effect of fiscal variables on short-term interest rates. The latter finding would imply that an edge from deficit to rlong should remain in the final graph. A more complicated model allowing for this possibility has a p-value of 0.81, only slightly higher than the p-value of our chosen model, 0.8. The more complicated model is thus rejected by the Bayesian information criterion, which balances the model fit and parsimony.

Impulse Response Analysis

Given the factorization pattern matrix M and the VAR parameter estimates, we now study the impulse responses of the long-term interest rate to fiscal shocks in deficits, and in explicit and implicit debts. The responses are calculated over an out-of-sample 10-year horizon. We call these types of responses the DAG-based impulse responses. As a comparison, we also calculate the impulse responses based on the widely used Cholesky decomposition by simply making the reduced form innovations orthogonal. Following Ardagna, Caselli, and Lane (2004), we consider two extreme cases in ordering the variables in the VARs. In the initial ordering, fiscal policy variables come first. Specifically, the variables enter into the VAR in this order: deficit, debt, uf75, infl, growth, rshort, and rlong. Second, the monetary policy variable comes first. That is, the variables are ordered as follows: rshort, infl, growth, deficit, debt, uf75, and rlong. We refer to these two types of responses as orthogonalized responses of order I and order II, respectively. Note that we place rlong last in both orderings. While it is true that the short-term rate "comes before" the long-term rate in the contemporaneous causal sense, there is no theoretical foundation that deficits and debt should always "come before" rlong. The orthogonalized impulse responses might show a different pattern if rlong is ordered otherwise. Nevertheless, we focus on the DAG-based impulse responses, which are invariant to the ordering of the variables in the VAR.

Figure 2 plots the time profiles of the long-term interest rate's responses to shocks in deficits and explicit and implicit debt. Also plotted are the corresponding error bands generated by the bootstrap method with 1000 replications. Note that because the impulse responses often have a highly asymmetric distribution, especially when the sample size is small (as evident here), we follow Sims and Zha's (1999) recommendation to use the error band formed by the 0.16 and 0.84 fractiles rather than the one standard deviation band (under the normal distribution, one standard error band also has a coverage of 0.68). We will say that a variable has an insignificant effect on interest rates if the point estimate falls outside the error bands.

Consider the orthogonalized responses in Panel A where the fiscal policy variables are assumed to come first. An unexpected 1 percentage point increase in the deficit-to-GDP ratio raises the long-term rate by 0.50% in the same year as the shock [Graph (a)]. This estimate falls in between Laubach's (2003) estimate of 25 basis points (based on the projected deficit) and that of Canzoneri, Cumby, and Diba (2002) (53 to 60 basis points). However, our estimate is based on the impulse response functions while the estimates reported by others are often the coefficients of fiscal variables in interest rate regressions. Thus, the comparison should be made with caution. As is clear from the figure, the contemporaneous effect appears to overshoot. It decreases in the next three years and is statistically insignificant from zero. Graph (b) in Panel A shows that the explicit debt has a slightly larger contemporaneous effect than does the deficit; although, it is not significant for the longer horizons. The implicit debt does not have an immediate impact on rlong until three years after the shock.

A somewhat different picture emerges from Panel B, where the short-term interest rate, inflation, and GDP growth rate come before deficits and debt. Both the magnitudes and patterns of rlong's responses to shocks in deficit and debt differ. Specifically, with this ordering, shocks in deficits have a negative effect on the long-term interest rate in the first two years (-0.44%). In contrast to Panel A, debt now does not have significant contemporaneous effects on rlong. The implicit debt, uf75, does not have a significant contemporaneous effect either. However, note that the change in variable ordering has a relatively small impact on the estimates of uf75's effects.

Now consider the DAG-based impulse responses of the long-term interest rate in Panel C. Contemporaneously, a 1 percentage point shock to the primary deficit-to-GDP ratio leads rlong to increase by 0.56%, which is slightly larger than the orthogonalized response estimate (order I). After the overshooting in the first period, rlong decreases over time and is 0.15% lower at period 4 than its level in the absence of the deficit shock. The effect becomes positive again at periods 6, 7, and 8 before it dies out. A shock to the explicit debt has no contemporaneous effect on rlong, but it leads to about a 0.10% increase in rlong for periods 4 and 5. One possible reason for the delayed effect of debt is that the contemporaneous, or short-term, impact of government borrowing is mostly picked up by the other variable of deficits. This may also explain why the impact of debt is relatively small compared with that of deficit. Graph (c) shows the dynamic effect of the implicit debt. Similar to explicit debt, uf75 does not have a contemporaneous effect on rlong. However, the long-term rate increases by 0.02% at period 3 following a shock of a 1 percentage point increase in the implicit debt-to-GDP ratio. Because innovations in deficits, explicit debt, and implicit debt are of different magnitudes, to make the interest rate effects more comparable across variables, we also compute the impulse responses of the long-term interest rate to one standard deviation of innovations in the three variables (one standard deviation of the innovations are 0.80, 1.05, and 8.83 for deficit, debt, and uf75, respectively). The maximum effects over the 10-year horizon are 0.45, 0.05, and 0.15% increases in rlong due to shocks in deficit, debt, and uf75, respectively.

[FIGURE 3 OMITTED]

Comparing Panels A, B, and C, we can see that the results based on recursive structures (through Cholesky decomposition) are sensitive to the ordering of variables, either in direction or magnitude or both. In this sense, the structural identification based on the data-driven directed graph method should be emphasized more in empirical studies as the appropriate model to evaluate the effect of government borrowings on interest rates.

The causality findings based on the DAG can also be useful information in determining the Cholesky ordering. (8) For example, deficit and infl should precede rshort; growth is preceded by debt, which in turn is preceded by deficit. One possible ordering meeting these restrictions is as follows: deficit, infl, rshort, debt, uf75, growth, and rlong. Panel D of Figure 2 plots the corresponding orthogonalized impulse responses of rlong to the innovations in deficit, debt, and uf75. The impact is similar to that observed in Panels A and C, while the patterns of explicit and implicit debt are close to those in Panel B.

Additivity of Explicit and Implicit Debt

Thus far, we have modeled the implicit debt as a separate endogenous variable relative to the explicit debt. This treatment allows the implicit debt to have different impacts on the interest rates than the explicit debt because changes in the implicit debt may affect the public's expectation on equilibrium interest rates differently than the explicit debt does. The empirical evidence obtained in the preceding discussion largely supports this method: The effects of the two types of debt have different dynamics in many cases. A more restrictive assumption is that the two types of debt are additive. For example, Liu, Rettenmaier, and Saving (2002) propose that Social Security and Medicare entitlement commitments made by the federal government should be added to its balance sheet as debts on par with the debt held by the public. To evaluate the implication of this possibility for our empirical results, we form a new debt variable by simply adding the explicit and the implicit debt series together. The combined debt measure is denoted as debt75.

[FIGURE 4 OMITTED]

We reestimate model 2 using the combined debt measure and report the innovations (residuals) correlation matrix in the lower half of Table 1. Figure 3 provides the corresponding directed graph describing the contemporaneous causal relationships between innovations. The causal links between short and long interest rates, and between inflation rate, deficit, and the short rate all remain the same. The most noticeable change is that the edge between deficits and explicit debt no longer remains in the graph. This makes sense because the new debt measure consists of two parts with potentially different natures; the relationship between deficits and debt is thus less direct. The DAG-based impulse responses of the long-term interest rate using the new combined debt measure are presented in Figure 4. The basic results we have derived earlier all seem to hold using the combined debt measure. A unit shock (of the size of 1 percentage point in the deficit-to-GDP ratio) leads to a contemporaneous increase of 0.35% in the long-term interest rate. The effect dies out over time. In contrast, there is no immediate effect from a shock in the combined debt.

5. Summary

In this paper, we revisited the long-standing issue of whether federal government borrowing can cause changes in long-term interest rates. Based on the data-driven method of directed acyclic graphs for structural identification of contemporaneous causal links, we examined the impulse responses of the 10-year government bond interest rate to fiscal shocks. We included in the analysis both the explicit debt and the implicit debt of Social Security unfunded obligations.

Our primary finding is that the explicit debt and the implicit debt both appear to have some positive effect on the long-term interest rate within a 10-year horizon. Under the preferred model specification, a 1 percentage point increase in the deficit-to-GDP ratio causes the interest rate to increase by 0.56% in the period of the shock. Neither the explicit debt nor the implicit debt has any contemporaneous effect on the interest rate. However, a 1 percentage point shock to the two types of debt, explicit and implicit, as a percentage of GDP, causes an increase in the interest rate by 0.10 and 0.02%, respectively, a few years later. The smaller effect of the implicit debt may be because legislative action can change this debt's magnitude.

Secondly, the effects of deficits on interest rates may persist for up to eight years, but they are not permanent and tend to die out after that. The effects of the explicit and implicit debt appear to be less persistent. This implies that households or other market participants gradually adjust their behavior to the new level of debt.

In sum, the exact magnitude of the effect of the implicit debt on interest rates is clearly subject to further research given that we have a very small sample. Nevertheless, the overall evidence herein suggests that future studies on the impact of fiscal policy on interest rates and other macro variables may benefit from considering the significant magnitude of unfunded obligations embodied in the generational transfer programs. Furthermore, the financial problems of these programs are not unique to the United States, meaning they could have important implications for both domestic and world financial markets.

We are grateful to Kent Kimbrough (editor) and two anonymous referees for their comments, which significantly improved the paper. Financial support from the Lynde and Harry Bradley Foundation and the National Center for Policy Analysis (NCPA) is acknowledged.

Received August 2006; accepted June 2007.

References

Ardagna, S., F. Caselli, and T. Lane. 2004. Fiscal discipline and the cost of public debt service: Some estimates for OECD countries. NBER Working Paper No. 10788.

Barro, Robert J. 1974. Are government bonds net wealth? Journal of Political Economy 82:1095-117.

Barro, Robert J. 1996. Reflections on Ricardian equivalence. NBER Working Paper No. 5502.

Barth, James R., George Iden, Frank S. Russek, and Mark Wohar. 1991. The effects of federal budget deficits on interest rates and the composition of domestic output. In The great fiscal experiment, edited by Rudolph G. Penner. Washington, DC: Urban Institute Press, pp. 69-142.

Bernanke, B. 1986. Alternative explanations of the money-income correlation. Carnegie-Rochester Conference Series on Public Policy 25:49-99.

Bernheim, B. Douglas. 1987. Ricardian equivalence: An evaluation of theory and evidence. NBER Working Paper No. 2330.

Calomiris, Charles, Eric M. Engen, Kevin Hassett, and R. Glenn Hubbard. 2003. Do budget deficit announcements move interest rates? Mimeographed, American Enterprise Institute and Columbia University.

Canzoneri, Matthew B., Robert E. Cumby, and Behzad T. Diba. 2002. Should the European Central Bank and the Federal Reserve be concerned about fiscal policy? Presented at the Federal Reserve Bank of Kansas City's symposium on "Rethinking Stabilization Policy," Jackson Hole, Wyoming.

Cebula, J. Richard. 1997. An empirical note on the impact of the federal budget deficit on ex ante real long-term interest rates, 1973-1995. Southern Economic Journal 63:1094-9.

Chickering, D. M. 2002. Optimal structure identification with greedy search. Journal of Machine Learning Research 3:507-54.

Demiralp, S., and K. D. Hoover. 2003. Searching for the causal structure of a vector autoregression. Oxford Bulletin of Economics and Statistics 65:745-67.

Doan, T. A. 2000. User's guide." RATS 5.0. Evanston, IL: Estima.

Elmendorf, Douglas W., and N. Gregory Mankiw. 1999. Government debt. In Handbook of macroeconomics volume 1C, edited by John B. Taylor and Michael Woodford. Amsterdam: Elsevier Science B, pp. 1615-69.

Engen, E., and R. G. Hubbard. 2005. Federal government debt and interest rates. In NBER macroeconomies annual 2004, edited by M. Gertler and K. Rogoff. Cambridge, MA: MIT Press, pp. 83-138.

Feldstein, M. 1974. Social Security, induced retirement and aggregate capital accumulation. Journal of Political Economy 82:905-26.

Feldstein, M. 1996. Social Security and saving: New time series evidence. National Tax Journal 49:151-64.

Gale, William G., and Peter R. Orszag. 2003. Economic effects of sustained budget deficits. National Tax Journal 56:463-85.

Geiger, D., T. S. Verma, and J. Pearl. 1990. Identifying independence in Bayesian networks. Networks 20:507-34.

Gokhale, J., and K. Smetters. 2005. Measuring Social Security's financial problems. NBER Working Paper No. 11060.

Goss, Stephen C. 1999. Measuring solvency in the Social Security system. In Prospects for Social Security reform, edited by O. S. Mitchell, R. J. Myers, and H. Young. Philadelphia: University of Pennsylvania Press, pp. 16-36. Granger, C. W. J. 2005. Modeling, evaluation, and methodology in the new century. Economic Inquiry 43:1-12.

Hoover, K. D. 2005. Automatic inference of the contemporaneous causal order of a system of equations. Econometric Theory 21:69-77.

Laubach, T. 2003. New evidence on the interest rate effects of budget deficits and debt. Board of Governors of the Federal Reserve System, Finance and Economics Discussion Series 2003-12.

Levine, R., N. Loayza, and B. Thompson. 2000. Financial intermediation and growth. Journal of Monetary Economics 46:31-77.

Liu, L., Andrew J. Rettenmaier, and Thomas R. Saving. 2002. Meaningful measures of fiscal deficits and debt: The case for incorporating entitlement debts. Private Enterprise Research Center (PERC) Working Paper No. 0211.

Longstaff, F. A. 2000. The term structure of very short-term rates: New evidence for the expectations hypothesis. Journal of Financial Economics 58:397-415.

Miller, S. M., and F. S. Russek. 1996. Do federal deficits affect interest rates? Evidence from three econometric methods. Journal of Macroeconomics 18:403-28.

Pearl, J. 1986. Fusion, propagation, and structuring in belief networks. Artificial Intelligence 29:241-88.

Pearl, J. 1995. Causal diagrams for empirical research. Biometrika 82:669-710.

Pearl, J. 2000. Causality: Models, reasoning, and inference. New York: Cambridge University Press.

Seater, John J. 1993. Ricardian equivalence. Journal of Economic Literature 31:142-90.

Sims, C. A. 1980. Comparison of interwar and postwar business cycles: Monetarism reconsidered. American Economic Review 70:250-7. Sims, C. A. 1986. Are forecasting models useable for policy analysis? Federal Reserve Bank of Minneapolis Quarterly Review 10:2-16.

Sims, C. A., and T. Zha. 1999. Error bands for impulse responses. Econometrica 67:1113-56.

Smetters, Kent. 1999. Ricardian equivalence: Long run leviathan. Journal of Public Economics 73:395-421.

Social Security Trustees. 2005. 2005 Social Security trustees report. Washington, DC: U.S. Government Printing Office.

Spirtes, P., C. Glymour, and R. Scheines. 2000. Causation, prediction and search. 2nd edition. Boston: MIT Press.

Swanson, N. R., and C. W. J. Granger. 1997. Impulse response functions based on a causal approach to residual orthogonalization in vector autoregressions. Journal of the American Statistical Association 92:357-67.

Wang, Zijun. 2005. A note on deficit, implicit debt and interest rates. Southern Economic Journal 71:186-96.

Wang, Zijun, and D. A. Bessler. 2006. Price and quantity endogeneity in demand analysis: Evidence from directed acylic graphs. Agricultural Economics 34:87-95.

Wang, Zijun, J. Yang, and Q. Li. 2007. Interest rate linkages in the eurocurrency market: Contemporaneous and out-of- sample Granger causality tests. Journal of International Money and Finance 26:86-103.

(1) The Congressional Budget Office estimated in 1998 that one dollar's worth of closed-group obligations could reduce private saving between 0 and 50 cents.

(2) Throughout the paper, we consider only the unfunded obligations in Social Security because there has not been a consistent series of Medicare implicit debt. Projections of Medicare outlays involve far more uncertainty because of continued dramatic innovations in medical technology and procedures. Another complicating factor is that Medicare Part B is partially financed through general revenue transfers. The trustees started to report comprehensive measures of Medicare's unfunded obligations in their 2004 report.

(3) See Granger (2005) for a brief discussion on the relationship between directed graphs and modeling methodology in the new century; also see contributions from the twentieth anniversary issue of Econometric Theory (2005, volume 21) for more on the topic.

(4) We also examined another type of estimate on Social Security unfunded obligations, which is based on the 100-year closed group assumption (denoted as uf 100). While the basic causal relationships are similar to those summarized in Figure 1, the model using uf100 provides unrealistic estimates for the effects of deficits, explicit debt, and implicit debt. A possible reason is that uf75 corresponds more closely to projections of financial market players on the magnitude of the debt implicit in Social Security. That is, they may care about the solvency of Social Security but much less about whether the program is fully advance funded. Thus. uf75 is probably more relevant than uf100 for the purpose of studying the effect of Social Security unfunded obligations on long-term interest rates.

(5) However, should there be no interest payment, the debt variable would be I(1) by construction if the deficit variable is stationary (I(0)). There are probably two reasons that the ADF test rejects the nonstationarity of the variable debt. First, the two variables are not perfectly correlated as we pointed out earlier. More importantly, unit root or cointegration tests based on this small size of sample are known to have lower power. Nevertheless, because all variables enter the VAR in levels form, the bias in coefficients (hence computed impulse responses) incurred by regressing variables in different orders of integration are likely to be small. The previous discussions are motivated by an important comment from an anonymous referee.

(6) To account for small sample bias in the estimation, we follow Longstaff (2000).

(7) The GES algorithm is not able to direct the edge between infl and uf75. This is because the model with infl [right arrow] uf75 and the model with uf75 [right arrow] infl have equal Bayesian posterior scores. Because the two competing models have the same number of free parameters, we score the models by the trace of covariance matrix (because the determinants of the matrices of the two models are equal here). This leads to the choice of infl causing uf75. We use [right arrow] to indicate that the direction is based on a different criterion than the rest of the model.

(8) This interesting application was suggested by one of the referees.

Zijun Wang * and Andrew J. Rettenmaier ([dagger])

* Private Enterprise Research Center, Allen Building, Room 3028, Texas A&M University, College Station, TX 77843-4231, USA; E-mail z-wang@neo.tamu.edu; corresponding author.

([dagger]) Private Enterprise Research Center, Allen Building, Room 3028, Texas A&M University, College Station, TX 77843-4231, USA; E-mail a-rettenmaier@tamu.edu.

Table 1. Residuals Correlation Matrices

 rlong rshort infl growth deficit debt uf

From the VAR with implicit debt uf75

rlong 1.000
rshort 0.754 1.000
infl 0.138 0.308 1.000
growth 0.689 0.823 0.199 1.000
deficit 0.543 0.815 0.045 0.799 1.000
debt 0.666 0.824 0.120 0.825 0.856 1.000
uf75 -0.007 -0.028 -0.449 0.071 0.099 0.052 1.000

From the VAR with combined debt measure debt75

Hong 1.000
rshort 0.734 1.000
infl 0.139 0.250 1.000
growth 0.636 0.703 0.244 1.000
deficit 0.470 0.797 -0.064 0.499 1.000
debt75 -0.006 -0.109 -0.264 0.232 -0.182 1.000

See Figure 1 for variable definitions.