A note on deficit, implicit debt, and interest rates.

Wang, Zijun

1. Introduction

In the past three decades, economists and policy makers have debated the following two questions: Does federal government borrowing cause changes in interest rates? If so, what is the timing of the impacts? Recently, people have also started to ask whether the debt implicit in the Social Security and Medicare programs plays a role in the discussions. Despite extensive research effort, the questions remain controversial. Feldstein (1982), Hoelscher (1986), Abell (1990), Miller and Russek (1991), Raynold (1994), Cebula (1993, 1997), and Vamvoukas (1997) provide evidence in favor of the existence of a positive relationship between government deficits and interest rates (often related to the Keynesian proposition). On the other hand, Barro (1987), Evans (1985, 1987, 1988), and Darrat (1990) argue that government borrowing does not crowd out private investment, hence, it does not lead to higher interest rates (the Ricardian equivalence outcome). Surveying both theoretical and empirical studies on the issue, Seater (1993) concludes that it can be almost certain that Ricardian equivalence "is not literally true; Nevertheless, equivalence appears to be a good approximation" (p. 184).

In this paper, we plan to contribute to the discussions in two dimensions. First, we use a more comprehensive measure of federal government debt. The often-used public debt (official debt), although large in absolute amount (about $6400 billion in gross measure by the end of 2002), is only a part of the total obligations of the federal government. According to the Office of Chief Actuary of the Social Security Administration, the Social Security program alone carried unfunded obligations of approximately $3350 to $12,162 billion in present value at the beginning of 2002 under various assumptions. Recently, Liu, Rettenmaier, and Saving (2002) proposed that Social Security and Medicare entitlement commitments made by the federal government (implicit debt) should be added to its balance sheet as debts on par with the debt held by the public. They argue that such a comprehensive total debt measure conveys more accurate information on the burden imposed on future generations by government borrowing than does the official debt. (1) Because this implicit debt is still mounting and likely to persist in the future, it is important to quantitatively investigate its impact on financial markets. While many researchers have investigated the effect of Social Security wealth on saving and consumption (e.g., Feldstein 1974, 1996), it appears that no one has examined whether the unfunded Social Security obligations have a role to play in the determination of interest rates. By including the implicit debt in the analysis, we wish to provide some new evidence on the discussion of the debt and interest rates. (2)

In this paper, we also hope to add to the literature with respect to the econometric method employed in the empirical analysis. Traditionally, univariate regressions have been widely used in the empirical studies. However, recognizing that most macro variables are probably determined simultaneously in the economy, researchers have increasingly relied on VAR models. Miller and Russek (1991, 1996) and Vamvoukas (1997) are a few researchers that use VAR models to investigate the dynamic relationships between interest rates and deficits and/or debt. Forecast error variance decomposition is an important tool, based on the VAR models, to summarize the dynamic interactions among economic series.

There is an unresolved issue in VAR analysis. To provide parameter estimates from reduced form VAR structural interpretations, researchers often use either the recursive Cholesky factorization pioneered by Sims (1980), or a nonrecursive strategy suggested by Bernanke (1986), Blanchard and Watson (1986), and Sims (1986). Both methods rely on economic theory or other prior knowledge to determine the ordering of variables in VAR models and to provide information about the linkages between innovations. Different orderings may lead to quite different decomposition results, depending on the degree of correlations between shocks. A particular ordering implies that we impose in priori economic structure on the multivariate processes (when the structure itself is often the subject of study). Unfortunately, as is evident in the debate on interest rates and deficit and/or debt, predictions of economic theory are often ambiguous.

Instead of relying on a particular form of matrix factorization, the generalized decompositions, as developed by Koop, Pesaran, and Potter (1996) and Pesaran and Shin (1998), measure the effect of a particular shock by integrating out the effects of other shocks to the system. Hence, these generalized decompositions are invariant to the ordering of variables in a VAR. For the question of whether there is a causal relationship between deficits and interest rates, Miller and Russek (1996) find that the answer is dependent on the sensitivity of the variance decompositions to the various structural specifications. Therefore, the use of the generalized variance decompositions in this paper may offer some help in solving the problem of ambiguity.

The rest of the paper is organized as follows: Section 2 explains the VAR modeling and the forecast error variance decompositions; Section 3 briefly discusses the data; Section 4 examines the impact of deficits and implicit debt on short-term interest rates represented by three-month and one-year Treasury bill rates; and Section 5 summarizes the major results and concludes.

2. VAR and Forecast Error Variance Decompositions

Let [Y.sub.t] denote an (m x 1) vector of stationary processes under investigation. The dynamic relationship among these processes can be modeled as a VAR of order k,

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

where [Y.sub.t] = ([Y.sub.1t], [Y.sub.2t], ..., [Y.sub.mt])', [[PHI].sub.i] and B are (m x m) and (m x n) coefficient matrices, t is an (m x 1) vector of innovations following a multivariate normal distribution with variance [??]. Furthermore, [[epsilon].sub.t] can be correlated only contemporaneously. Model 1 has an infinite moving average representation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

The error in forecasting [Y.sub.t] s periods into the future, conditional on information available at t - 1 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

with a variance of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The conventional orthogonalized variance decompositions are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [[theta].sub.ij,s] measures the contribution of the jth orthogonalized innovation to the total forecast error variance of variable [y.sub.it] at horizon s, P is the recursive-form Cholesky factor of [??], and [e.sub.i] is the selection vector that has all elements equal to 0 except for the ith element being 1. The variance decompositions based on Equation 5 critically depend on the ordering of [Y.sub.t], because P does.

Koop, Pesaran, and Potter (1996) and Pesaran and Shin (1998) developed an alternative to the above method. Their basic idea is to consider the proportion of the s-period forecast error in Equation 3, which is explained by conditioning on the nonorthogonalized shocks, [[epsilon].sub.jt], [[epsilon].sub.j,t+1], ..., [[epsilon].sub.j,t+s], while explicitly allowing for the contemporaneous correlations between these shocks (recall that all future shocks are assumed to be 0 in computing orthogonalized decompositions). The conditional forecast error variance is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Comparing Equation 6 with Equation 4, it can be seen that by conditioning on future shocks to the jth variable, the forecast error variance declines. Similar to Equation 5, normalizing the ith diagonal element in Equation 6 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Pesaran and Shin (1998) call Equation 7 the generalized forecast error variance decomposition. Note that the variance [??] in Equation 7 replaces its orthogonalized factor, P, in Equation 5. Therefore, the generalized decomposition is invariant to the ordering of variables in the VAR model. (3)

To end this section, it is important to note that the forecast error decompositions as defined either in Equation 5 or in Equation 7 measures the dynamic impact of a temporary shock to the jth variable on variable i. This impact is different than the impact that a permanent change in the jth variable may impose on variable i.

3. Data and Variables

We use annual data from 1959 through 2002 on the three-month (or one-year) Treasury bill rate, federal government spending, the national income and product (NIPA) measures of the federal government deficit, money supply (M2), and Consumer Price Index (CPI). All these time series are easily accessible from the online resources of corresponding government agencies. The last series we use is the Social Security unfunded obligations.

The Office of the Chief Actuary (OCAT) of the Social Security Administration calculates several alternative estimates of Social Security's unfunded obligations. Because of availability, we concentrate on two estimates. The first estimate is based on a rolling 75-year open group assumption, that is, including both current and future participants. It is basically the present value of the difference between future revenues and expenditures of the program for the next 75 years. Another estimate is based on the 100-year closed group assumption and is equal to the present values of future cost less future taxes over the next 100 years for all current participants. The two measures are defined on the basis of the intended financing of the program. Because the current system (more strictly, its Old-Age and Survivors Insurance and Disability Insurance [OASDI] component) is financed on essentially a pay-as-you-go basis, the open group obligation is often considered most applicable for assessing the actuarial status of the program. The second obligation measure would be more appropriate if the program is to be fully advance funded (Goss, Wade, and Shultz 2004).

The open-group series of unfunded obligations is available for the period of 1979 through 2003. The second series is available from 1980 to 2003. Both series are expanded back to 1976 with data from Table 3 of Goss (1999). (4) Figure 1 is a graphic illustration of the two implicit debt series. Note that these measures included the Social Security Trust Fund balances (currently running surpluses). However, the Trust Fund balances are nothing but specially issued bonds by the government, which are real debt against current and future generations. Therefore, total unfunded obligations are larger than the figure indicates. It can be seen from Figure 1 that the two measures are approximately equal at the start but tend to diverge over years. At the beginning of 2003, the 100-year closed group estimate is about 2.5 times that of the 75-year open group estimate. As is obvious in Figure 1, the new rules of the Social Security Amendments Act of 1977, enacted on December 20, 1977, resulted in the sharp drop in the 75-year open group series (and less dramatically in the alternative measure).

[FIGURE 1 OMITTED]

Based on the aforementioned previous theoretical and empirical studies, particularly Evans (1985, 1987) and Barro (1987), both four-variable and five-variable VAR systems are estimated. The four-variable VAR that examines the impact of deficits on the interest rates consists of the following endogenous variables: the Treasury bill rate, the ratio of real federal government spending to trend real GDP, the ratio of real deficit to trend real GDP, and the ratio of real M2 to trend real GDP. Nominal and two types of real interest rates are considered. The ex post real interest rate is calculated by subtracting the realized (actual) rate of inflation from the nominal interest rate. The ex ante real interest rate is obtained by subtracting the expected rate of inflation from the nominal interest rate. The expected rate of inflation is generated as forecasts from a univariate model with two lags of actual inflation and one lag of monetary growth as regressors (Barro, 1987).

In addition to the preceding four variables, the five-variable VAR also includes the ratio of change in real implicit debt to trend real GDP.5 In all regressions, a constant term is included.

4. Empirical Results

Estimation without Implicit Debt

We first study the relationship between the deficit and interest rates without considering implicit debt. Model 1 is first estimated using the observations from 1959 to 2002. In all combinations (real or nominal interest rates, one-year or three-month Treasury bills), two information criteria (AIC and HQ) and the log likelihood ratio test conclude with k = 2 in Model 1. The SC at lag order 1 is only slightly smaller than it is at lag order 2. Furthermore, diagnosis tests on residual series for serial correlation and normality also indicate that the specification of VAR(2) is appropriate for the data. Based upon this evidence, k = 2 is chosen. (6) Model 1 is also estimated with observations from 1975 to 2002 to be comparable to the later five-variable models that include the change in the implicit debt variable.

Table 1 reports the generalized forecast error variance decompositions (in percentages) of the interest rates (represented by the three-month Treasury bill rate) due to shocks in the federal government deficits for the sample periods 1959 to 2002 (Panel A) and 1975 to 2002 (Panel B), respectively. The decompositions are calculated over a 10-year horizon. The 95% confidence intervals are based on the bootstrap proposed by Runkle (1987). (7) The replication number is 1000. Note that the variance decomposition cannot be negative by definition. Hence, its confidence interval does not contain zero even when the true value is zero. Because our goal is to assess the relative importance of the deficit and implicit debt (among other endogenous variables) in the determination of interest rates, a variable is assumed to have a small effect on interest rates if the point estimate of the decomposition is less than 5%, or if the lower bound of the 95% confidence interval is less than (1 - 95%)/2 = 2.5%.

The first three columns of Panel A summarize the mean estimates (M) of the variance decompositions of the ex post interest rates, and their lower (L) and upper bounds (U). The impact of deficits on interest rates is negligible within one period (year) of the shocks. Deficits explain virtually no portion of the interest rate variance (0.05%). The estimate is smaller than 5% in the first five periods. However, the impact accumulates over time. At period 10, deficits account for 10.00% of the variability in interest rates. This estimate is significant, as its lower bound is also estimated to be 3.22%. The middle three columns of Panel A show that the impact of deficits on the ex ante real interest rates is larger than that on the ex post real interest rates. The contemporaneous impact is 12.46% although with a small value of lower bound (0.28%). The estimate becomes smaller in the next six periods, but it is always larger than 8%. After 10 periods, deficits can explain approximately 14% of the total variance in the ex ante real interest rates. It is clear from the last three columns that, compared to their impact on the real interest rates, deficits have a more significant influence on the nominal interest rates over all horizons under consideration. At period 1, 22.75% of the forecast error variance of the nominal interest rates is due to shocks in the variable of deficits. The estimate is quite stable over time. It decreases only to 20.71% at period 10.

In short, the impact of deficits on the nominal interest rates is larger than that on the ex ante real interest rates, which in turn is larger than that on the ex post real interest rates. This result is consistent with the traditional belief that deficits increase aggregate demand, and hence affect the formation of inflation expectation.

Panel B contains the forecast error variance decomposition of the ex post real, ex ante real, and nominal interest rates, respectively, estimated with the subsample period of 1975 through 2002. The results obtained above from the full sample still hold true. The impact of deficits on the ex post real interest rates is insignificant in the first three periods after shocks but appears to be significant afterward. The percentage of interest rate variance attributable to deficits is 11.63% at period 10. As in the full sample, deficits have more impact on nominal interest rates than on real interest rates.

Estimation with Implicit Debt

Panel C and D of Table 1 summarize the forecast error variance decompositions of interest rates due to deficits estimated from the five-variable VAR that includes implicit debt for the period of 1975 through 2002. In Panel C, the implicit debt measure is based on the 75-year open group assumption. The estimated impact of deficits on the real interest rates is generally larger than that estimated with the same subsample period but excluding the implicit debt (Panel B). Within one period of shocks, about 8.38% of the ex post real interest rate variance is due to deficits. Nevertheless, this point estimate has a lower bound of 0.01% only. At period 10, a significant part of the interest rate variance (18.85%) is attributable to deficits, which is higher than the estimate of 11.63% at the same horizon when the implicit debt variable is not included in the regression. In the case of ex ante real interest rates, the estimates are only slightly larger than the previous ones. But following similar patterns, the estimates in the first few periods have relatively small lower bounds. The comparison of the last three columns in Panel C with those in panel B shows that the inclusion of implicit debt in the regressions does not seem to affect the estimate of the deficit contribution to the nominal interest rate variations.

We report the decomposition results using the 100-year closed group implicit debt measure in Panel D. In the first five periods, the impact of deficits on ex post real rates is smaller and even less significant than that estimated with the 75-year open group debt measure. For example, at period 1, deficits explain only as little as 2.13% of variations in the real interest rates (compared to 8.38% in Panel C). In contrast, the estimate of the long-run deficit impact is larger using the second measure. It can also be seen from Panels B and C that using the second implicit debt measure does not make much difference in the regressions of the ex ante real interest rates and nominal interest rates.

Table 2 illustrates the impact of the implicit debt on the three different measures of interest rates for the two different implicit debt measures. These results are also derived from the previous five-variable VARs estimated with the observations from 1975 to 2002. With the first measure, the implicit debt does not appear to have much impact on real interest rates in the short term. It explains only 0.16 and 3.93% of the interest rate variance during the first two periods (the second column of Panel A). However, it does have some impact in the long run. The estimate increases to around 11% in the last five periods. Evidence presented in the middle three columns indicates that some of the variability in the ex ante real interest rates can also be explained by the implicit debt (that is, 9.31% at period 5). The impact of implicit debt on the nominal interest rates is marginal at all horizons. The highest estimate is 6.83% (reached at period 1), and the lower bound is never larger than 1.07%. It is clearer from Panel B that the impact of implicit debt (using the second measure) on the nominal interest rates is negligible in both the short and long runs.

Two points are clear. First, over the long horizons, the impact of implicit debt is larger on the ex post real interest rates than on the ex ante interest rates. The latter is, in turn, larger than the impact on nominal interest rates. This may imply that changes in the implicit debt do not affect the public's expectation on inflation as much as deficits do. Second, in all six combinations of interest rates and implicit debt measures in Table 2, the impact of implicit debt is generally smaller than the impact of deficits in Table 1 during the sample period of 1975 through 2002.

Impacts on One-Year Treasury Bill Rates

We repeat the previous analyses with the short-term interest rates represented by the one-year Treasury bill rates. To save space, the detailed estimates are not reported here (but are available upon request). The basic results obtained previously hold with the alternative measure of interest rates. The magnitudes of estimates and their dynamic patterns remain largely the same in most cases. The major changes are in the regressions that involve the nominal interest rates. In both sample periods, the percentage of variance in the one-year bill rates accounted for by deficits is 2 to 4%, lower than its counterpart in the three-month bill rates at each horizon. This is true no matter which type of implicit debt is used.

Recall from Table 2 that if the implicit debt is measured on the 100-year closed group basis, it has no impact on the three-month interest rates (Panel B). The evidence is marginal if the 75-year open group measure is used. The implicit debt does not have a significant impact on the nominal one-year bill rates for both debt measures (the maximum impact is estimated to be 2.18 and 0.9% for the two debt measures, respectively).

Orthogonalized Decompositions

As a comparison, Table 3 reports the orthogonalized forecast error decompositions of three-month Treasury bill rates due to the implicit debt. In Case I, the implicit debt is the first variable in the VAR. It is ordered as the last variable in Case II. Clearly, the ordering of variables now matters, in Case I, the implicit debt has a significant effect on the interest rates in all model combinations with one exception (on nominal interest rates when the debt is measured on the open group basis). In contrast, no impact is found if the debt is assumed to be the last variable in the VAR. Note that the corresponding generalized decomposition estimates in Table 2 fall in between the two sets of orthogonalized estimates.

5. Summary

In this note, we revisit the long-standing issue of whether federal government borrowing causes changes in interest rates. Based on a relatively new generalized method, we decompose the forecast error variance of the interest rates estimated from VAR models. Both official deficit and implicit debt (long-run unfunded obligations of the Social Security program) is used in the analysis. Our major findings include:

(i) Deficits do not have a significant impact on the ex post real interest rates in the short term; however, they do appear to have some moderate impact in the long term (five or more periods). They can also explain a sizable portion of variability in the ex ante real interest rates, especially over long horizons. Deficits have a significant impact on nominal interest rates in all periods.

(ii) The inclusion of implicit debt in the regressions generally increases the preceding estimates of deficits but does not change their dynamic patterns.

(iii) The implicit debt has some explanatory power in the ex ante real interest rate variations over both short and long horizons. It affects the ex post real rates only at long horizons. There is not strong evidence that the implicit debt has an impact on the nominal interest rates.

(iv) The implicit debt possesses a smaller effect than the regular federal deficits. One possible explanation may be that investors are vaguely aware of the implicit debt of the Social Security system, but put a discount factor on it. (8)

(v) The preceding results hold true regardless of whether three-month or one-year Treasury bill rates are used. Furthermore, they also hold when either the 75-year open group measure or the 100-year closed group measure of the implicit debt is used.

Admittedly, the number of observations on unfunded U.S. Social Security obligations used in this study is small, which partly explains why most decomposition estimates have wide confidence intervals. In this sense, a more comprehensive study on the relationship between total government obligations and interest rates would probably have to wait for more observations on the future solvency of Social Security and Medicare. The results obtained here should be interpreted with caution and better understood in connection to other relevant information. Nevertheless, our finding that the implicit debt seems to have some impact on the real interest rates in the long run suggests that it is necessary to include such a measure in empirical studies.

Table 1. Generalized Forecast Error Variance
Decompositions of Interest Rates Due to Deficits

 Ex Post Real Rates Ex Ante Real Rates

Horizons L M U L M U

Panel A. Without implicit debt, sample period 1959-2002

 1 0.01 0.05 11.42 0.28 12.46 25.32
 2 0.10 0.21 13.69 1.60 12.06 25.89
 5 0.99 3.88 24.73 2.85 8.77 24.81
 10 3.22 10.00 27.90 6.42 13.95 31.65

Panel B. Without implicit debt, sample period 1975-2002

 1 0.01 1.46 21.74 0.06 7.37 25.67
 2 0.33 3.91 27.46 0.64 8.15 25.34
 5 1.66 5.84 32.59 2.24 10.09 28.32
 10 2.60 11.63 36.79 3.89 17.10 34.40

Panel C. Implicit debt based on the 75-year open group assumption

 1 0.01 8.38 22.78 0.02 9.12 27.36
 2 0.66 16.24 29.47 0.38 9.38 26.07
 5 3.16 17.87 31.30 1.99 11.90 27.97
 10 3.46 18.85 32.42 3.51 18.42 30.27

Panel D. Implicit debt based on the 100-year closed group assumption

 1 0.01 2.13 20.51 0.02 8.34 26.52
 2 0.47 9.78 30.19 0.27 8.84 25.73
 5 2.45 14.78 36.94 1.88 11.47 28.26
 10 3.71 22.63 41.70 4.07 19.59 32.79

 Nominal Rates

Horizons L M U

Panel A. Without implicit debt, sample period 1959-2002

 1 10.88 22.75 31.71
 2 8.56 22.46 32.84
 5 9.16 21.09 33.29
 10 9.28 20.71 33.12

Panel B. Without implicit debt, sample period 1975-2002

 1 12.78 27.46 36.05
 2 10.49 24.26 34.58
 5 10.46 22.02 35.46
 10 11.14 24.90 38.15

Panel C. Implicit debt based on the 75-year open group assumption

 1 16.95 27.54 34.21
 2 11.73 24.12 32.02
 5 11.91 22.65 33.64
 10 12.21 23.89 34.27

Panel D. Implicit debt based on the 100-year closed group assumption

 1 14.37 27.12 34.35
 2 11.29 24.14 32.55
 5 10.20 21.93 33.39
 10 10.95 24.72 34.42

"M" stands for mean (point) estimates of the generalized
forecast error variance decompositions of interest rates
due to deficits. "L" and "U" are the corresponding lower
and upper bounds of the 95% confidence intervals based on
the empirical distribution of the bootstrapped
decomposition estimates.

Table 2. Generalized Forecast Error Variance
Decompositions of Interest Rates Due to Implicit Debt

 Ex Post Real Rates Ex Ante Real Rates

Horizons L M U L M U

Panel A. Implicit debt based on the 75-year open group assumption

 1 0.01 0.16 21.15 0.05 6.45 24.10
 2 0.29 3.93 22.85 0.45 6.38 23.63
 5 2.04 9.56 27.62 1.62 9.31 24.69
 10 2.26 11.17 28.08 2.02 8.77 23.90

Panel B. Implicit debt based on the 100-year closed group assumption

 1 0.02 7.26 30.25 0.03 7.26 28.52
 2 0.31 7.25 27.34 0.28 6.95 27.95
 5 1.36 10.50 29.60 1.01 6.98 27.74
 10 1.65 11.32 29.88 1.10 6.27 25.45

 Nominal Rates

Horizons L M U

Panel A. Implicit debt based on the 75-year open group assumption

 1 0.04 6.83 21.07
 2 0.20 6.32 20.86
 5 0.83 6.36 21.99
 10 1.07 6.34 22.35

Panel B. Implicit debt based on the 100-year closed group assumption

 1 0.00 1.03 15.23
 2 0.07 0.74 15.78
 5 0.58 2.21 20.25
 10 0.73 1.97 19.65

"M" stands for mean (point) estimates of the generalized
forecast error variance decompositions of interest rates
due to implicit debt. "L" and "U" are the corresponding
lower and upper bounds of the 95% confidence intervals
based on the empirical distribution of the bootstrapped
decomposition estimates.

Table 3. Orthogonalized Forecast Error Variance
Decompositions of Interest Rates Due to Implicit Debt

 Ex Post Real Ex Ante Nominal
 Rates Real Rates Rates

Horizons Case I Case II Case I Case II Case I Case II

Panel A. Implicit debt based on the 75-year open group assumption

 1 0.22 0.00 8.29 0.00 17.04 0.00
 2 5.87 0.55 8.27 0.07 14.74 0.13
 5 13.07 1.42 11.38 0.07 12.17 0.55
 10 15.14 1.43 11.29 0.05 12.49 0.73

Panel B. Implicit debt based on the 100-year closed group assumption

 1 9.08 0.00 9.10 0.00 2.39 0.00
 2 10.57 0.35 8.77 0.10 1.67 0.13
 5 15.08 2.45 8.18 0.58 4.34 1.34
 10 15.99 2.41 8.12 0.44 3.86 1.47

Each entry is the mean (point) estimate of the orthogonalized
forecast error variance decomposition of interest rates due to
implicit debt. To save space, the 95% confidence intervals are
omitted. In Case I, the variable of implicit debt is the first
variable in the VAR. In Case II, it is ordered as the last variable.

I would like to thank Orlo R. Nichols at the Office of Chief Actuary of the Social Security Administration for providing their estimates of Social Security unfunded obligations. I am grateful to Andrew J. Rettenmaier and Thomas Saving for their helpful comments and suggestions.

Received July 2003; accepted January 2005.

(1) Policy makers also start to recognize the importance of this. "... (As) longer-term commitments have come to dominate tax and spending decisions, (current) cash accounting has been rendered progressively less meaningful as the principal indicator of the state of our fiscal affairs" (Greenspan 2003).

(2) Clearly, a more complete measure of the national implicit debt should also include the unfunded Medicare obligations, which is of increasing importance. As of 2001, the accrued Medicare unfunded liability totaled $16.9 trillion according to Liu, Rettenmaier, and Saving (2002). However, a consistent series of this type of debt is not available.

(3) Because the generalized variance decompositions, as defined in Equation 7, explicitly allow for contemporaneous correlations among shocks, unlike the orthogonalized decompositions, they do not sum to 100 over j. However, of interest here is the relative importance of the component factors in the variables to be explained. Therefore, we further divide the decomposition values by their sum, so that they still sum to 100 (the original decomposition values would be higher depending on the magnitude of the correlations).

(4) The estimates provided by OCAT are as of January 1 of the valuation year. The annual estimates in Goss (1999) are as of October 1, covering the period of 1976 through 1997. We use the simple average ratio of the first three overlapping years to convert the October 1 estimates to the January 1 estimates for years 1976 through 1978 and years 1976 through 1979, respectively for 75-year open group estimates and 100-year closed group estimates. (The ratios of the October 1 estimates and the January 1 estimates are stable for both series during the overlapping years). These estimates are further adjusted by the Social Security Trust Fund and lagged one period in regressions.

(5) We test the null hypothesis that the VAR coefficients of deficit and implicit debt are equal. We fail to reject the null at the 0.05 significance level only in the regression of ex ante real interest rates. Later empirical results also suggest that investors respond differently to the explicit and implicit debt. Therefore, we decide not to combine the two types of debt.

(6) The Augmented Dickey-Fuller test results indicate that one-year and three-month Treasury bill rates (nominal), government spending, and money supply appear to be nonstationary, while most of the other variables are stationary. Thus, we also estimated both a four-variable VAR(1) in first differences and a cointegration model with rank 1. However, the variance decompositions based on these two specifications are similar to those from the chosen VAR(2) in levels. Because the sample size is small, both the stationarity and cointegration tests might be biased. In the remaining portions of the paper, we concentrate on the VAR(2) in levels.

(7) The OLS estimates for VAR parameters may be biased in small sample sizes. To adjust for this bias in calculating confidence intervals for the impulse response, Kilian (1998) and Kilian and Chang (2000) propose a bootstrap-after-bootstrap method. Following a similar method, we find that the adjusted intervals are generally tighter. However, the differences between the adjusted and unadjusted intervals are relatively small.

(8) We owe this point to an anonymous referee.

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Zijun Wang, Private Enterprise Research Center, Allen Building, Room 3028, Texas A&M University, College Station, TX 778434231. USA; E-mail: z-wang@tamu.edu.