Journal Issue

Parameter Estimate Uncertainty in Probabilistic Debt Sustainability Analysis

International Monetary Fund. Research Dept.
Published Date:
March 2010
  • ShareShare
Show Summary Details

This paper extends the probabilistic approach to debt sustainability analysis (DSA) developed by Celasun, Debrun, and Ostry (2006, herein CDO). As noted by these authors, the fan charts generated by their approach underestimates the degree of uncertainty in the debt projections because it ignores the stochastic nature of the parameter estimates used in the forecasts. We build on their work by explicitly accounting for the probability distribution of the estimators in both the projected path of nonfiscal determinants of public debt, and in the fiscal policy reaction function (FRF). Hence, the resulting debt forecast reflects both the intrinsic volatility of debt determinants and the inaccuracy of the parameter estimates of econometric models employed in the projections.

This extension is of general applicability to other areas that make use of probabilistic approaches for macroeconomic forecasting, such as inflation targeting or budget forecasts. Currently, policymaking bodies such as the Bank of England, Norway’s Norges Bank, Sweden’s Riksbank, and the U.S. Congressional Budget Office communicate projections of key economic variables through similar fan charts. although the procedures used to develop these vary, like CDO’s approach, they forecast from estimated parameters that have a probability distribution describing their estimations. For example, the Bank of England describes its estimated probability distribution function at any period as a mixture of past forecast errors and subjective assessment, rather than mechanically extrapolating its results (Bank of England, 2002, pp. 48–49). The U.S. Congressional Budget Office adopts a similar strategy in basing the published probability distribution of its forecasts on the historical distribution of past forecast errors, after adjusting for the business cycle and structural changes such as new tax legislation (CBO, 2007, pp. 8–18). The Reserve Bank of New Zealand’s forecast algorithm for monetary policy is similar to CDO. They employ Monte Carlo forecasts based on estimated vector autoregression (VAR) and policy reaction function error correlations (Hunt and Drew, 1998). Like in CDO, these alternate methodologies characterize the intrinsic forecast uncertainty from past residuals or estimation errors. Nevertheless, they do not explicitly consider the additional uncertainty introduced by employing estimated rather than known models in the projections.

The standard International Monetary Fund and World Bank DSA generates a medium-term path for the debt-to-GDP ratio resulting from assumptions about future realizations of macroeconomic and fiscal policy variables. Sensitivity analysis of the baseline debt forecast is conducted by subjecting the main determinants of debt to shocks, typically proportional to historical volatility. This exercise has drawbacks. For example, the sensitivity analysis is deterministic, as it involves simulating specific shocks one variable at a time, or at most combining tailored shocks to more than one variable. In the latter case, the correlation of the shocks is ignored, as is the dynamic response of the other endogenous variables to these shocks. Also, the DSA assumes no policy response to changing economic conditions.

A growing literature addresses the shortcomings of the IMF/World Bank DSA. Work by Ferrucci and Penalver (2003), Garcia and Rigobon (2005), and Tanner and Samake (2008) among others, rely on VARs to simulate probabilistic scenarios that take into account the statistical properties of macroeconomic shocks, in particular their persistence, variance and covariance. CDO and Penalver and Thwaites (2006) incorporate in addition a fiscal policy reaction function that accounts for systematic policy responses to varying economic scenarios. These two studies build on the literature on endogenous fiscal policy and solvency, for example, Bohn (1998), Galí and Perotti (2003), and Abiad and Ostry (2005).

The probabilistic approach to fiscal sustainability analysis proposed by CDO relies on an algorithm consisting of three building blocks: (1) the estimation of a VAR model of the nonfiscal determinants of public debt, which is used to predict a set of possible future economic scenarios; (2) the estimation of a fiscal policy reaction function that endogenizes fiscal policy responses to potential future economic scenarios; and (3) the combination of the first two blocks to produce a probability distribution of future paths of the public debt-to-GDP ratios for the projection period. This distribution is graphically displayed as a fan chart that shows the probability distribution of the debt ratio at each future period.

This study augments the CDO algorithm by incorporating the statistical distribution of the estimated coefficients of the VAR of nonfiscal debt determinants and the fiscal policy reaction function at each step of the projection. Hence, this approach attempts to address the problem of the inherent accuracy limitations highlighted by Wyplosz (2007). In CDO’s algorithm, uncertainty is incorporated by allowing for random shocks to the nonfiscal debt determinants and to fiscal policy. However, by not considering the error in the econometric model, uncertainty is being underestimated. In this paper, the risk of substantial widening of the fan chart from incorporating parameter estimation errors is attenuated by developing a country-specific fiscal reaction function—robust to historical credit events—and based on long-dated series of economic aggregates rather than the country panel regression approach to fiscal reaction function estimation of CDO. In addition, parameter uncertainty is further restrained by estimating a restricted VAR.

This augmented algorithm is assessed on the public debt sustainability of Uruguay. For this purpose, a VAR is estimated using quarterly data for the period 1988–2008 and a fiscal policy reaction function is estimated based on annual data for the period 1965–2008. The algorithm is also used under alternative fiscal policy rules, to show its applicability in cases where a fiscal reaction function may not be available or may not provide a reasonable working hypothesis. The alternative rules consist of a constant primary surplus target and a countercyclical fiscal policy rule. In these cases, inaccuracy of the econometric model only feeds into the simulations through the VAR.

We find that for the case of Uruguay, the fan charts are not substantially modified by accounting for the probabilistic nature of the estimators. The fan charts do widen, but less than may have been anticipated a priori: the debt level on the last year goes up by about 25 percent of GDP, and widens the interquartile range by 15 percent of GDP. Nevertheless, the results also show that a better specification of the underlying econometric model may significantly reduce the variance of debt projections. The reduction to the maximum forecasted debt level and interquartile range from using a country-specific fiscal reaction function and a restricted VAR are 33 and 4 percent of GDP, respectively. Hence, these improvements help compensate for the additional uncertainty stemming from the probability distribution of the estimated parameters.

I. Methodology

This section extends the probabilistic DSA approach developed by CDO to incorporate the inaccuracy stemming from using estimated rather than known parameters in the debt projections. It also describes a fiscal reaction function specific to Uruguay and restricted VAR, which are used in the proposed methodology.

The first block of CDO’s algorithm estimates a standard VAR of the economic aggregates driving fiscal policy and public debt. The VAR includes the foreign interest rate (rF), the real domestic interest rate (rD), the real GDP growth rate (g), and the log change in the real exchange rate (e). As in CDO, we estimate an unrestricted VAR in these four variables and then improve the fit—and hence subsequent projections—by restricting the VAR to a more parsimonious model. This limits the additional uncertainty introduced into the projections by poorly estimated parameters. Specifically, we restricted the VAR to only include right-hand side variables that are both economically relevant and statistically significant in the unrestricted VAR.1 That is, the foreign interest rate is a univariate reggression reflecting the implausibility that Uruguayan factors impact U.S. interest rates. The domestic interest rate is explained by its lagged value and the lagged foreign interest rate, to account for international liquidity conditions. GDP growth is estimated as a function of its lagged value and the domestic interest rate. Finally, the level of the exchange rate is estimated as a function of its lag and the level of the output growth. Given the restrictions imposed to the VAR, the resulting system of equations is estimated by seemingly unrelated generalized least squares regression. The restricted system is:

where αnk denotes the coefficient for equation and variable k, and εnt is the error term of equation n. The vector εt = (ε1t, ε2t, ε3t, ε4t) is assumed to be distributed normally, N(0, Ω). The variance-covariance matrix of residuals, Ω, characterizes the joint statistical properties of the nonfiscal determinants of public debt dynamics.

The estimated system of equations is used to generate quarterly projections of the nonfiscal macroeconomic fundamentals driving debt dynamics. The quarterly outturn is annualized and also used to generate projections of the output gap (computed as the difference between the predicted GDP growth and the steady-state growth rate resulting from the VAR), to feed the fiscal reaction function and the debt projections (see below). The residuals from equation (1) are used to estimate the variance-covariance matrix, Ω, which is then used for the draws of jointly Normal random vectors εt+1,…, εt+T, such that for all τ ϵ [t + 1, T]; ετ = Wητ, where ητ~N(0,1) and W is the Choleski factorization of Ω (such that Ω = W’ W).

The second block of the algorithm is a fiscal policy rule. Under our baseline formulation, this rule takes the form of a fiscal reaction function that endogenizes fiscal policy in response to varying macroeconomic conditions. This study postulates a functional form relating the primary fiscal balance to the lagged level of public debt and the level of the output gap.2 In addition, to account for the fact that the Uruguayan economy has been historically highly dollarized, the rate of change of the real bilateral Uruguayan peso-U.S. dollar real exchange rate is also included as a determinant of fiscal policy.3 The estimated equation is therefore:

where pt is the primary balance of the central government as a percent of GDP in year t, dt−1 is the public sector debt-to-GDP ratio, Δet is the annual percentage change of the end-of-period bilateral Uruguayan peso-U.S. dollar exchange rate (a positive sign indicates a real peso depreciation), and ygt is the output gap. The contemporaneous shock to the primary balance is υtN(0,συ2), assumed to be independently distributed from the other macroeconomic shocks. Alternative fiscal rules can also be used in lieu of equation (2).

The final block of the algorithm combines simulated economic scenarios with the fiscal policy equation to recursively compute the path of public debt over a specified projection period. The evolution of the debt-to-GDP ratio is governed by the standard law of motion:

dt1Ddt1F and denote the domestic and foreign currency components of public debt at the end of the previous period, respectively. Each constellation of macroeconomic shocks drawn from the estimated probability distribution drives a path of debt over the forecast period. The frequency distribution of the debt paths generated by drawing a large number of these shock constellations (1,000 in our case) is summarized by “fan charts,” depicting confidence bands around a median debt projection.

This study attempts to account for the errors in the estimated parameters. In each of the 1,000 scenarios, the drawings of the macroeconomic debt determinants are combined with a set of VAR parameters drawn from their (assumed joint-normal) probability distributions. Similarly, the fiscal policy stance is the result of combining the predicted macroeconomic variables with a draw of the parameters of the FRF. That is, the estimators used at each step of the forecast are treated as random variables rather than known values. The probability distributions of the FRF and VAR parameters are:


where α^ and β^ are point estimates that center the sampling distribution of the estimated parameters, Z = (dt, Δet, ygt) and X=(rtf,rtd,gt,et) contain the right-hand side variables in equations (1) and (2), and Ω is the estimated weighting matrix.

II. Results

Vector Autoregression

The VAR in equation (1) is estimated using quarterly data for the period 1988–2008, with one lag for each variable.4 Descriptive statistics of the data underlying the VAR are presented in Table 1, which shows a median four-quarter growth rate exceeding 4 percent, as well as substantial volatility in the economic variables, driven largely from the crisis in 2002. The results of the restricted VAR estimation are presented in Table 2. The four equations fit the data well with highly significant estimated coefficients in all cases but for two of the leveling constants.

Table 1.Economic Fundamentals in Vector Autoregession(In percent)
U.S. Real

Uruguay Real



Standard deviation0.363.025.619.44
Annual percent change.Source: Authors’ estimates; Uruguayan authorities.Note: Quarterly observations, 1988:Q1–2008:Q4. REER = real effective exchange rate.
Annual percent change.Source: Authors’ estimates; Uruguayan authorities.Note: Quarterly observations, 1988:Q1–2008:Q4. REER = real effective exchange rate.
Table 2.Restricted Vector Autoregression Estimation
CoefficientStd. Errort-StatisticProb.
Adjusted R-squared0.500.440.880.69
SE of regression0.
Mean dependent variable0.0080.0260.0310.028
SD dependent variable0.0040.0300.0540.101
Sum squared resid.0.0010.0380.0270.239
Source: Authors’ estimates.
Source: Authors’ estimates.

The restricted VAR in equation (1) limits the additional uncertainty introduced into the projections by poorly estimated parameters. The excess volatility introduced by including statistically insignificant and economically irrational estimators into the projection can be either potentially very large if insignificant, or reflecting a spurious and unknown correlation if significant. The estimation is shown in Figure 1, where each panel corresponds to a parameter of the VAR (for example, the top-left chart corresponds to the constant in the real foreign interest rate equation). The horizontal line shows the point estimate of the coefficient, whereas the dark area is a scatter plot corresponding to random draws of the estimates. The figure illustrates the variability introduced at each draw from the coefficient distribution for the restricted VAR (left) and the unrestricted one (right). The large variation of those estimators, depicted in the right-hand panel but omitted from the restricted VAR on the left, highlights the magnitude of the imprecision that may feed through to the CDO algorithm from an inaccurately estimated econometric model, and motivates the restricted VAR.

Figure 1.Randomized Vector Autoregression Coefficients

Note: The left panel shows the simulated coefficients from the restricted vector autoregression in equation (1) used in the simulations, with the solid line showing the point estimate and the shaded area showing the simulated coefficient draws. The right panel shows the analogous coefficient draws for the unrestricted vector autoregression.

Figure 2 shows the distribution of the residuals for each of the equations of the restricted VAR, as well as the correlation of the residuals across equations. The right panel shows the histogram of estimated residuals, where the potential for nonnormality is evident. Jarque-Bera tests of joint normality on both the restricted and unrestricted system fail to reject the null of normality under some specifications but reject it under others. We take these results as potential but inconclusive evidence of nonnormality in the system residuals, and leave for future work addressing such issues via bootstrapping as done in Frank and Ley (forthcoming).

Figure 2.Estimated Vector Autoregression Residuals

Note: The left panel shows scatter plots of the residuals resulting from the vector autoregression estimation. The right panel shows histograms of the equations’ residuals.

Fiscal Policy Reaction Function

Equation (2) is estimated with annual data for the central government of Uruguay for the period 1965–2008. The properties of the data are summarized in Table 3, which shows a negative average primary balance over the 44-year period, and an average debt level of 38 percent of GDP, with a standard deviation of debt of 26 percent of GDP. Given the length of the period covered by the data, and the bouts of economic instability that Uruguay experienced in this period, we check for the possibility of structural breaks in the series by applying the Andrews (1993) and Bai and Perron (1998) tests. These tests fail to reject the null hypothesis of no breaks in the estimated equation throughout the entire period.5,6

Table 3.Economic Fundamentals in Fiscal Reation Function(In annual percent of GDP)
DebtPrimary BalanceOutput GapREER1
Standard deviation0.
Source: Authors’ estimates; Uruguayan authorities.Note: Annual observations, 1965–2008. REER = real effective exchange rate.

Annual percent change.

Source: Authors’ estimates; Uruguayan authorities.Note: Annual observations, 1965–2008. REER = real effective exchange rate.

Annual percent change.

Another potential problem is that of endogeneity bias stemming from the correlation of the output gap with contemporaneous fiscal policy shocks ut. To the extent that fiscal policy affects the output gap in the current year, this regressor would be endogenous and ordinary least square (OLS) estimates would be biased. Nevertheless, the Durbin-Wu-Hausman test failed to reject the null hypothesis of exogeneity of the output gap with p-value exceeding 0.18. Notwithstanding, the FRF is estimated by OLS, GMM and instrumental variables (IV) to account for the possibility of endogeneity.

The output gap is instrumented with its lagged value and the lagged real GDP growth of Argentina, given the history of regional spillover from that country. For the debt simulations discussed below, the OLS results are employed.

The results for the estimation of the FRF are shown in Table 4. The estimated specification includes also a dummy to try to capture the effects from the crises in 1982 and 2002. Under OLS, all macroeconomic variables are significant at conventional levels. The results suggest a sustainable fiscal policy insofar as higher levels of public debt historically led to increases of the primary balance. As shown by Bohn (1998), a positive estimated response of the primary balance to public debt is sufficient to ensure long-run solvency. The positive coefficient on the output gap indicates that on average fiscal policy has been countercyclical. The crisis dummy is not statistically significant. Finally, the coefficient on the contemporaneous real exchange rate depreciation is negative and robust to changes in the estimation approach and alternative specifications. It signals that a currency depreciation has a negative impact on the fiscal position. This could reflect a deterioration in public sector balances from higher import costs, as the public sector is net buyer of foreign currency denominated inputs. The main difference between the GMM and IV estimations (in columns 2 and 3, respectively) and the OLS is the loss of significance of the output gap coefficient. The estimated equation seems to reflect quite well the historical patterns in the data, including during the crisis periods of 1982 and 2002. To see this, Figure 3 plots the estimated fiscal reaction function estimated by OLS alongside the historical primary balance.

Table 4.Fiscal Policy Reaction Function Estimation
Source: Authors’ calculations.Note: *p <0.1; **p <0.05; ***p <0.01. OLS = ordinary least square; GMM = generalized method of moments.
Source: Authors’ calculations.Note: *p <0.1; **p <0.05; ***p <0.01. OLS = ordinary least square; GMM = generalized method of moments.

Figure 3.Fiscal Reaction Function and Debt Level

Having estimated the fiscal reaction function, Table 4 shows the simulated coefficients used in forecasting debt. The charts along the diagonal present histograms of the realizations of the drawn coefficients but the off-diagonal scatter plots reflect cross-correlations between the equation estimators. At each step of the projection, the simulated fundamentals are paired with a draw from these coefficients to produce a primary balance and move forward the debt projection. Each realization of the debt projection simultaneously reflects a constellation of economic shocks and a draw of randomized coefficients from their estimated distributions (Figure 4).

Figure 4.Simulated Fiscal Reaction Function Coefficients

Note: The left panel shows scatter plots of the simulated coefficients used in forecasting. The right panel shows histograms for the coefficient draws, which are centered on the estimation presented in Table 4. REER = real effective exchange rate.

Debt Projections

Endogenous Fiscal Policy

The quarterly path for the nonfiscal determinants of debt is forecasted by the VAR for a five-year period, with shocks to these variables drawn from the joint distribution summarized by the variance-covariance matrix of the estimated VAR. The quarterly projections are then annualized and fed into the fiscal reaction function to produce first a projection of the primary surplus, and then a debt-to-GDP ratio from the debt motion equation. Frequency distributions for public debt, as well as for all the other forecasted variables are obtained by simulating 1,000 stochastic paths.

Figure 5 shows three fan charts with five-year debt forecasts for Uruguay under endogenous fiscal policy. The median projection is represented by the bold line. The first four shaded surfaces at each side of the median represent an interval of 10 percent of the distribution of the debt ratio, and the last two, 5 percent. Hence, the lighter area on both sides of the median represents a 20 percent confidence interval around that value, and the total span of the fan corresponds to a 95 percent probability of public debt being within that range in any given year. The fan chart is thus a graphical representation of the probability distribution of debt forecasts and summarizes the risks to debt dynamics.

Figure 5.Debt Forecasts Based on Restricted Vector Autoregression and Fiscal Reaction Function

The three fan charts portray the two sources of uncertainty entering into the DSA. The upper left panel shows the fan chart most closely corresponding to the CDO algorithm, where uncertainty derives only from the shocks to the nonfiscal debt determinants and to fiscal policy, through the estimated joint residual distribution of the VAR and of the residuals in the fiscal reaction function, and the estimated coefficients of the VAR and the FRF are treated as known constants equal to their point estimates. As noted by CDO, failure to take into account the probability distribution of the estimated parameters implies that the fan chart presented conveys a lower bound of the risks to public debt. By contrast, the lower left panel assumes certainty regarding nonfiscal debt determinants and fiscal policy, and instead depicts the situation where uncertainty derives only from drawing from the joint distribution of the estimated parameters of the VAR and the FRF. Finally, the right panel shows the resulting fan chart when both sources of uncertainty in the forecast are incorporated. In the latter, the median projection shows a steady decline of public debt from an estimated 58 percent of GDP at end-2008 to below 50 percent in 2013. Although sustainable, the fan also reveals that given the responsiveness of fiscal policy embedded in the reaction function, and the possible configurations of macroeconomic scenarios, there is still a risk below 15 percent of reaching the end of 2013 with a higher level of debt than at the beginning of the period. In this sense, the overall shape of the cone points to downside risks to debt. The span of the cone gives an idea of the intrinsic volatility of the Uruguayan economy as captured by the VAR.

Figure 5 shows the analogous set of fan charts generated using the unrestricted VAR. The median forecast for the projection period remains roughly unchanged relative to the forecast generated with the restricted VAR, below 50 percent in all cases. The focus is, hence, on other moments of the resulting probability distributions. Table 5 contains summary statistics of the dispersion of debt forecasts using the restricted and unrestricted VARs, and both sources of forecast uncertainty (shocks and parameters), as with the fan charts, distinguishes the cases where the two sources of risks are considered separately and when they are combined. The upper panel shows the restricted VAR, the middle panel shows the unrestricted VAR, and the lower panel displays the difference between the two. The estimation analogous to CDO is represented by the middle column of results in the middle panel, but the full stochastic forecast is represented by the right column of results (and the left represents forecasting with stochastic parameter draws as the only source of uncertainty). The middle panel shows that accounting for parameter uncertainty in the unrestricted VAR forecast (analogous to CDO) does not affect the interquartile range, but increases the maximum projected debt level by 23 percent of GDP and widens the range of observed debt levels in 2013 by 25 percent of GDP. Moreover, the increases in the skewness and kurtosis suggest that the distribution is tilted toward the higher debt ranges, as suggested visually by Figure 6. The upper panel of results shows the forecasts based on the restricted VAR depicted in Figure 1, and the bottom panel shows the difference between the estimates. In most cases, the dispersion estimates in the forecasts can be reduced by half by excluding the irrelevant right-hand side variables in the unrestricted VAR when estimating the system using some form of generalized least squares. The reduction in the range of estimates for the last year of the forecast (when the uncertainty is the greatest) is 37 percent of GDP, offsetting the increase in uncertainty in the forecast from accounting for the stochastic distribution of the estimated parameters.

Table 5.Summary Statistics of the Last Year of Debt Forecasts(In percent of GDP)
Stoch. ParametersStoch. ErrorsBoth
Restricted VAR
Unrestricted VAR
Unrestricted—Restricted VAR
Source: Authors’ estimates.Note: VAR = vector autoregression.
Source: Authors’ estimates.Note: VAR = vector autoregression.

Figure 6.Debt Forecasts Using Unrestricted Vector Autoregression

The relatively tight debt forecasts produced, compared with those presented in CDO, despite incorporating an additional source of uncertainty are due, in part, to the stronger underlying econometric specification, namely the more parsimonious restricted VAR (as shown in Table 5) and the country-specific fiscal reaction function. By contrast, by relying on a panel of countries and a VAR with larger estimation errors, CDO’s estimates utilize a more imprecise econometric model which translates into more imprecise forecast. This point is also illustrated in Table 6, where debt forecasts are produced assuming arbitrarily larger (double) values for συ (as well as the standard errors in the FRF equation), resulting in a roughly doubling of the dispersion of the forecasts.

Table 6.Restricted Debt Forecasts while Doubling Standard Errors(In percent of GDP)
Stoch. ParametersStoch. ErrorsBoth
Restricted vector autoregression

Alternative Fiscal Policy Rules

The FRF captures the average long-term relationship between fiscal policy and the state of the economy. However, it is useful to study departures from the behavior postulated by the FRF as the future policy stance is itself an uncertain factor in forecasting. Two alternative policy rules are considered here—a fixed primary balance target, and a countercyclical fiscal policy. As above, the nonfiscal determinants of public debt are simulated with the restricted VAR, but instead of having the primary balance endogenously determined by the FRF, it is set exogenously according to each of the alternative rules.

The fixed primary balance target rule is intended to mimick the medium-term fiscal framework in place in Uruguay, which establishes a constant primary surplus of 2.75 percent of GDP for the forcasting period. This is broadly in line with the estimated outturn of 2008, and consistent with the government’s medium-term objective of keeping the primary surplus between 2½ and 3 percent of GDP. The countercyclical fiscal policy rule is operationalized by setting the cyclically adjusted primary balance as the fiscal policy objective, in line with the Uruguayan authorities’ expressions about the desirability of moving in the medium term to a framework that takes into account cyclical considerations. We conduct simulations for a target cyclically adjusted primary balance of 2.75 percent of GDP.7

The results of the debt forecast simulations under the two alternative policy rules are presented in Figure 7. Under both rules the expected debt-to-GDP ratio declines significantly faster than under the FRF. In the case of the fixed primary surplus rule, median debt declines to about 42 percent of GDP in 2013. It is evident that the estimated reaction function reflects historical policies. The current level of the primary balance, used as a reference for the fixed primary rule, implies a tighter stance consistent with the objective of lowering public debt to more conservative levels. Under the countercyclical rule median debt declines to 37 percent of GDP in 2013. This rapid decline results in part from the fact that the projection period starts at a high point of the cycle and the persistence of GDP growth identified by the VAR, implying the need for large cyclical balances.8 The implications for public debt forecasts of following the different rules can be easily understood from looking at Figure 8, which shows fan charts of the predicted levels of the primary surplus under the different policy rules, respectively.9 Either of these two rules yields a higher primary balance, which in turn drives debt levels down faster.

Figure 7.Debt Forecasts under Alternative Policy Rules

Figure 8.Primary Balance Forecasts under Alternative Policy Rules

An important caveat here is that in the absence of a fiscal rule, policy implications drawn from fiscal reaction function based forecasts about the primary balance should also reflect the constellation of macroeconomic shocks driving debt realizations in the fan chart. The strength of fan chart debt forecasts is precisely the endogenization of jointly distributed macroeconomic shocks and their impact on fiscal policy and the debt. Hence, an observed sequence of primary balances projected by the fiscal reaction function that yields debt within a desired range is more accurately described in the context of the sequence of macroeconomic scenarios influencing the realized path of the primary. For example, one of the most important factors influencing the debt forecast is the sequence of GDP growth forecasts. Hence, without a rule fixing the fiscal policy, the path of primary balances that yields a desired debt level by the last forecast period is more precisely characterized as a combination of observed fiscal balances and observed GDP growth paths.

Figure 9 attempts to describe such a path by showing a three-dimensional surface composed of the five-year average annual growth rate of GDP graphed against the five-year average primary balance, and against the debt level in the last year of the forecast. Each point in the surface represents debt in the last year of the forecast for a combination of average GDP growth and primary balance over the period. The figure shows this jointly in the top panel, and individual GDP growth and primary balance cut-aways are shown in the lower panels. One observes, for example, that the range of observed debt levels drops as the average GDP growth rate approaches 5 percent (lower left panel). For the primary balance, there is a slow secular decline in the range of debt levels observed in the last year of the forecast for average balances greater than 1 percent, and below this average level, the debt projected to stagnate or increase.

Figure 9.Policy Implications of Debt Forecasts with Fiscal Policy Reactive Function

III. Conclusions

This study extended the fan chart approach developed by Celasun, Debrun, and Ostry (2006) by taking into account explicitly the uncertainty embedded in the use of an estimated restricted VAR and policy reaction functions to make macroeconomic forecasts. The CDO methodology contrasts with the standard DSA in IMF staff reports, which are framed in a deterministic setup, relying on bound tests of the main assumptions to incorporate uncertainty. The analysis presented here extended the CDO algorithm and performed a stochastic DSA for Uruguay. The extended algorithm explicitly accounts for parameter estimates uncertainty in constructing the forecast fan chart, and uses a restricted VAR. The impact of parameter uncertainty in the debt fan chart forecast was limited, in part because of more precise estimates based on a country-specific policy reaction functions, long dated country-specific times series, and the exclusion of irrelevant variables in the VAR.

Disregarding parameter uncertainty could be potentially important in other applications. In this sense, the paper presented a simple way to operationalize incorporating parameter uncertainty in the computation of debt forecasts. This procedure could be adapted to other environments, for example in the use of Taylor rules for forecasting inflation. Moreover, important extensions, such as incorporating time consistency (as mendtioned by Woodford, 2007), addressing the robustness of assumed distributions, potential structural breaks, and other important sources of uncertainty could also be included in future work.

The paper assessed debt sustainability in Uruguay in a stochastic environment that takes into account the joint probability distribution of the macroeconomic determinants of public debt dynamics. Fiscal policy was endogenized through a fiscal reaction function specific to this country. For the estimation of this function a new database for Uruguay going back more than 40 years was constructed. The results indicate that on average fiscal policy in Uruguay has been conducted in a manner consistent with long-term debt sustainability although the implied return to lower debt levels is slow. The findings show that the primary surplus increases with rising level of public debt and that fiscal policy over the studied period has been moderately countercyclical. Alternative fiscal policy rules were also considered, namely a fixed primary surplus and a cyclically adjusted surplus of 2.75 percent of GDP, under which public debt was found to decline to approximately 40 percent in the medium term.


Data Sources and Descriptions

Primary balance: the annual time series for the primary balance of the central government was constructed from revenues and total expenditure published in the IMF’s International Financial Statistics (IFS) database for the period 1965–90 and from IMF’s staff reports for 1991–2008. Interest payments for the period 1965–72 were taken from IMF staff reports, from IFS for the period 1973–90, and from staff reports thereafter.

External debt of the combined public sector: Data for the period 1965–98 were compiled from the figures published in IMF staff reports. For the period 1999–2008 the data are from the Banco Central del Uruguay.

Real GDP: Obtained from the IFS.

Real domestic interest rate: this was computed using the time deposit rate as the nominal interest rate, and the consumer price index, both published by IFS, in the following way:

where it is the nominal interest rate and πt+1 is the CPI inflation one quarter ahead.

Interest rates on one-year U.S. government bonds: FRED database, the Federal Reserve Bank of St. Louis.

Real U.S. dollar exchange rate: computed as

where et is the U.S. dollar/Uruguayan peso nominal exchange rate, ptF is the U.S. CPI index, and ptD is the Uruguayan CPI index. These three variables were obtained from IFS.


Alejandro Hajdenberg is an economist with the IMF Fiscal Affairs Department. Rafael Romeu is an economist with the IMF Western Hemisphere Department. The authors thank Nigel Chalk, Robert Flood, Eduardo Ley, Pau Rabanal, Marcos Souto, Andy Wolfe, Michael Woodford, and an anonymous referee for helpful comments and discussions.

Alternatively, one could estimate a Bayesian VAR centered on a zero prior for the restricted coefficients and allow the estimation to confirm that assumption or not via the posterior distribution. Nevertheless, the resulting posterior standard errors of the estimators would depend to a significant degree on the assumed variance of the priors, and hence would make difficult comparisons with previous work. Comparing the performance of classical and Bayesian VARs in fan charts is outside the scope of this paper and left for future work.

As the vast majority of Uruguay’s debt has historically been denominated in foreign currency, the FRF is estimated using foreign debt. The projections allow for foreign and domestic debt as domestic currency issues have increased in recent years.

As suggested by Abiad and Ostry (2005), we explored other potentially important drivers of fiscal policy, including the real price of oil and the price of Uruguay’s major exports, but none of these variables were statistically significant. We discarded using the presence of an arrangement with the IMF as Uruguay had arrangements in place during most of the period under study.

Data sources are presented in the Appendix.

In addition, Durbin’s alternative test for autocorrelation, and the Breusch-Godfrey LM test for autocorrelation both failed to reject the null of autocorrelation, and the Dickey-Fuller (including augmented) as well as the Phillips Perron tests reject unit roots.

A shorter time series would have the advantage of basing forecasts on data only reflecting the recent institutional framework. This study opts for conducting the analysis with the full sample, given the good fit of the estimation and no evidence of structural breaks.

The cyclically adjusted primary balance (sp) is approximated by the expression spt= pt+ Gt × ygt, where Gt is the ratio of public spending to GDP in the year t and Gt is assumed to remain constant at the estimated 2008 level. This approximation assumes that the elasticity of public revenue and expenditure to the output gap are one and zero, respectively.

The calculation of the cyclically adjusted primary balance overestimates the required adjustment. The results are strongly influenced by the 2008 GDP growth rate on account of the one-off impact of the opening of a large pulp mill, which, incidentally, does not significantly affect the fiscal balance as it benefits from substantial tax exemptions.

All the variables simulated with the VAR and the primary balance shock have the same mean and standard deviation by construction given the large number of simulated scenarios.

Other Resources Citing This Publication