Probabilistic Sustainability of Public Debt: A Vector Autoregression Approach for Brazil, Mexico, and Turkey

Evan C Tanner; Issouf Samaké

doi:10.5089/9781451865554.001.A001

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Probabilistic Sustainability of Public Debt

A Vector Autoregression Approach for Brazil, Mexico, and Turkey

Author: Mr. Evan C Tanner and Issouf Samaké

Contributor Notes

E-Mail Addresses: etanner@imf.org; isamake@imf.org

Publication Date:: 01 Dec 2006

eISBN:: 9781451865554

Language:: English

Keywords:: WP; time horizon; exchange rate; debt accumulation; interest rate shock; debt stabilization; debt reduction path; baseline debt need; stabilization benchmark; surplus ratio

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This paper examines the sustainability of fiscal policy under uncertainty in three emerging market countries, Brazil, Mexico, and Turkey. For each country, we estimate a vector autoregression (VAR) that includes fiscal and macroeconomic variables. Retrospectively, a historical decomposition shows by how much debt accumulation reflects unsustainable policy, adverse shocks, or both. Prospectively, Monte Carlo techniques reveal the primary surplus that is required to keep the debt/GDP ratio from rising in all but the worst 50 percent, 25 percent, and 10 percent of circumstances. Such a value-at-risk approach presents a clearer menu of policy options than currently used frameworks.

Abstract

I. Introduction

A sustainable fiscal policy is often defined as one that can be continued into the future without modification—no adjustment of the primary surplus and no default (by inflation or otherwise). At a minimum, the intertemporal solvency criteria must be satisfied. More often, sustainability has come to mean that the debt stock (or its ratio to output) does not rise.

In developing countries, it may be difficult to design fiscal policy in such a way that no further modification will ever be necessary. Despite the best intentions of policymakers to stabilize the debt, there is some chance that adverse movements of interest rates, exchange rates, output, and other key variables may cause persistent increases in government debt. In this sense, sustainability is probabilistic. Accordingly, fiscal authority may seek to reduce the probability that such adjustments will be necessary, through further increases of the primary surplus. A primary surplus that reduces the probability of future adjustments to less than 50 percent will, on average, reduce the debt.

This paper examines the sustainability of fiscal policy under uncertainty in three emerging market economies, namely Brazil, Mexico, and Turkey. Sustainability is assessed both retrospectively (“If historical policies were to be continued into the future, would fiscal policy be sustainable—or will a modification of policies be required?”), and prospectively (“What policies should be undertaken today in order to prevent the need for further adjustments in the future?”)

Other retrospective assessments of fiscal sustainability do not emphasize uncertainty. The fiscal gap calculations of Blanchard and others (1990) and Talvi and Végh (2000) (see also Croce and Juan-Ramón (2003)) tell us how high a primary surplus must be in order to ensure sustainability. Such accounting-based (not econometric) calculations typically assume full knowledge about certain key variables, namely long-run GDP growth and interest rates.

Econometric solvency tests introduced by Hamilton and Flavin (1986) (later extended by Wilcox (1989), Trehan and Walsh (1990, 1991), Hakkio and Rush (1991), and others) are also retrospective. They tell us whether the historical trajectory of fiscal data can be sustained into the future—but not how shocks to key variables will affect debt accumulation.

Prospectively, sustainability assessments have most often been implemented with accounting (not econometric) frameworks. Such approaches introduce uncertainty in a rudimentary way. For example, the International Monetary Fund (2003) advocates a stress test approach that examines outcomes of a fiscal program in the event of adverse shocks certain key variables in isolation (interest rates or growth economic growth).

However, for a prospective assessment of fiscal sustainability under uncertainty, an econometric approach may be better suited than an accounting one to model the interactions of key variables—and thus their joint impact on debt accumulation. For this reason, several authors (see, for example Hoffmaister and others (2001); Garcia and Rigobon (2004); Hostland and Karam (2005); Celasun, Debrun, and Ostry (2006); and Penalver and Thwaites (2006) have proposed the use of multivariate stochastic simulations of future debt behavior (potentially based on an econometric model)).

Our approach to sustainability, both retrospective and prospective, differs from previous work. Retrospectively, beginning in some base period, the evolution of debt is attributed to either a baseline policy or accumulated shocks—movements of certain key explanatory variables that were not anticipated at the base period. A simple (near) vector autoregression (VAR) model yields such historical decompositions. Hence, if under the baseline forecast, the debt/GDP ratio does not rise, fiscal policy is sustainable in the way discussed by Blanchard and others (1990) and Talvi and Végh (2000). To the extent that adverse (beneficial) shocks to nonpolicy variables (those not directly controlled by the fiscal authority) contribute to debt increases (decreases), the country is said to be “unlucky” (“lucky”). Likewise, shocks to fiscal policy itself (that is, the primary deficit) may be thought of as departures from an implicit fiscal policy rule (possibly including discretionary policy).

For the prospective analysis, we then simulate this model. This aspect of our work is similar to papers by Garcia and Rigobon (2004) and Celasun, Debrun, and Ostry (2006), Penalver and Thwaites (2006). However, their work is positive in nature. They present forecasts of debt—the mean value along with upper- and lower-confidence intervals that increase with time (“fan charts”).

However, our work takes a more normative tack. We link such “fan chart” forecasts to an objective function (similar to Tanner and Carey (2005)) that summarizes the maximum fiscal adjustment that a country is willing to target in order to avoid further increases in debt— probabilistically, over a given horizon. Thus, we thus calculate the average primary surplus required to stabilize debt with probability 90 percent (and, where applicable, 75 percent) for one- to five- year horizons.²

Of course, when policies change, so might the behavior of market participants—in ways not captured by the econometric model. This is the “Lucas critique.” For example, the level of the primary surplus may itself affect interest rates (both level and volatility) in a way not captured in the data. Such issues may be addressed on an ad-hoc basis; this defect is also inherent in other extant debt sustainability frameworks. While it would be better to address the Lucas critique in a full-fledged general equilibrium model of debt sustainability, such models are still in their infancy.

The paper is organized as follows. In Section II, we review previous work on fiscal sustainability. In Section III, we provide an overview of our methodology. In Sections IV, V, and VI, we analyze the cases of Brazil, Mexico, and Turkey. Section VII concludes.

II. Fiscal Sustainability: Some Previous Work

In this section, we briefly discuss some previous work on fiscal sustainability. In Section A, basic identities are presented. In Section B. previous work on retrospective sustainability is discussed. In Section C, previous and current work on prospective sustainability is discussed.

A. Basic Identities

Any notion of fiscal sustainability should begin with public sector’s budget constraint. In any period, this is:

\begin{matrix} (1) & b_{t - 1} (1 + r) + γ_{t} - τ_{t} = b_{t} \end{matrix}

where b is real government debt, γ is noninterest expenditures, and τ are tax revenues. Forward substitution of budget constraint over an infinite horizon (1) yields the intertemporal constraint:

\begin{matrix} (2) & b_{-1} (1 + r) - E {Σ_{t = 0}^{\infty} p s_{t} / {(1 + r)}^{t}} = E {\lim_{t \to \infty} (b_{t} / {(1 + r)}^{t - 1}} \end{matrix}

where ps_t = τ_t-γ_t is the expected primary surplus. The “no-Ponzi game” condition is:

\begin{matrix} (3) & E {\lim_{t \to \infty} b_{t} / {(1 + r)}^{t - 1}} = 0 \end{matrix}

Now, assume that interest expenditures (θ_t = rb_t-1) in any period have both a deterministic and a time-varying component:

\begin{matrix} (4) & θ_{t} = θ^{P} + z {(θ)}_{t} \end{matrix}

The primary surplus can be decomposed into three elements: one that tracks interest expenditures θ^P, a “tax gap” (Blanchard and others, 1990) and an “own” shock z(ps)_t.

\begin{matrix} (5) & {ps}_{t} \equiv τ_{t} - γ_{t} = θ^{P} - κ + z {(ps)}_{t} \end{matrix}

In any period, the deficit is thus:

\begin{matrix} (6) & b_{t} - b_{t - 1} \equiv γ_{t} + θ_{t} - τ_{t} \equiv κ + z {(θ)}_{t} - z {(ps)}_{t} \equiv κ + z_{t} \end{matrix}

B. Intertemporal Solvency and Tax Smoothing

Arguably, the least restrictive notion of fiscal sustainability is intertemporal solvency: satisfaction of conditions (2) and (3). Note that the government remains solvent even if κ > 0. This point was emphasized by McCallum (1984), who showed that a government can run a constant deficit (inclusive of interest payments) forever—and still remain solvent. Here, debt growth is less than the interest rate (r).

However, continued debt accumulation may not be desirable: to service an ever rising debt stock, the government must correspondingly increase the primary surplus over time. As one benchmark, a benevolent, far-sighted planner would wish to distribute the burden of the primary surplus evenly over time. Such a planner might set κ = 0—a policy of debt stabilization (Blanchard and others (1990), Talvi and Végh (1999), International Monetary Fund (2003a), Burnside (2005)).

Formally, debt stabilization closely resembles another well-known policy, namely tax smoothing (discussed by Barro, 1979). Sargent (1987, pp. 385–88) noted that, under a tax-smoothing regime, the government minimizes a quadratic loss function based on collection costs:

\begin{matrix} (7) & Φ (τ_{t}) = Φ τ_{t}^{2} \end{matrix}

subject to budget constraint (2) and no-Ponzi game condition (3). In this case, for ϕ = r, the Euler equation yields a well-known result, namely that taxes follow a random walk:

\begin{matrix} (8a) & τ_{t} = E_{t} (τ_{t + 1}), \forall t . \end{matrix}

However, substituting the Euler condition into budget constraint (2) yields another (and perhaps more important) implication of tax smoothing emphasized by Barro: taxes τ_t move one-to-one with total expenditures (γ^P + θ^P) over the long run:³

\begin{matrix} (8b) & τ^{P} = γ^{P} + θ^{P} \end{matrix}

where τ^P is the long-run tax rate. Note that (8b) is equivalent to the debt stabilization policy (κ=0). Thus, such a policy has an implicit objective function that is similar to Barro (1979) and Sargent (1987, pp. 385–88).

C. Retrospective Sustainability

A test for retrospective sustainability seeks to answer the following question: “If historical policies were to be continued into the future, would fiscal policy be sustainable—or will a modification of policies be required?” Table 1 presents a summary of past work on this topic.

Table 1.

Approaches to Retrospective Sustainability:

“If historical policies were to be continued into the future, would fiscal policy be sustainable—or will a modification of policies be required?”

Table 1.

Approaches to Retrospective Sustainability:

“If historical policies were to be continued into the future, would fiscal policy be sustainable—or will a modification of policies be required?”

Description of Approach	Citations	Remarks
1. Fiscal gap (noneconometric)	Blanchard and others (1990); Talvi-Végh (2000)	Evaluates historical primary surplus against debt stabilizing benchmark ps*={(r-λ)/(1+λ)}b, λ = GDP growth.
2. Stationarity of deficit (time- series econometric).	Hamilton and Flavin (1986); Wilcox (1989), Kremers (1989), Trehan and Walsh (1990,91), Corsetti and Roubini (1991), Hakkio and Rush (1991); Tanner (1995); extensions include Feve and Henin (2000), Martin (2000), Uctum and Wickens (2000), Arestis and others (2004).	Solvency guaranteed by stationarity of real deficit Δb, but debt may still rise (since deficit may fluctuate around non-zero mean).
3. Cointegration of revenues and expenditures (time- series econometric).	Hakkio and Rush (1991); Bohn (1991), Haug (1991), Tanner and Liu (1994), Ahmed and Rogers (1995), Quintos (1995), Telatar and others (2005), Leachman and others (2005)	Solvency guaranteed by cointegration of primary expenditures, revenues, and debt (b), but debt may still rise unless vector of coefficients is [1,-1,r] with no constant or trend.
4. Link between primary surplus and debt (econometric model).	Bohn (1998, 2005), International Monetary Fund (2003b), Tanner and Ramos (2003).	Solvency guaranteed by positive relationship between primary surplus and debt (ps_t=κ+αb_t-1, α >0).
5. Historical Decomposition used to distinguish “policy” from “luck” (shocks) (time- series econometric).	This paper only.	For details, see text; evolution of “baseline” debt ratio corresponds to approach (1) above.

Table 1.

Approaches to Retrospective Sustainability:

“If historical policies were to be continued into the future, would fiscal policy be sustainable—or will a modification of policies be required?”

Description of Approach	Citations	Remarks
1. Fiscal gap (noneconometric)	Blanchard and others (1990); Talvi-Végh (2000)	Evaluates historical primary surplus against debt stabilizing benchmark ps*={(r-λ)/(1+λ)}b, λ = GDP growth.
2. Stationarity of deficit (time- series econometric).	Hamilton and Flavin (1986); Wilcox (1989), Kremers (1989), Trehan and Walsh (1990,91), Corsetti and Roubini (1991), Hakkio and Rush (1991); Tanner (1995); extensions include Feve and Henin (2000), Martin (2000), Uctum and Wickens (2000), Arestis and others (2004).	Solvency guaranteed by stationarity of real deficit Δb, but debt may still rise (since deficit may fluctuate around non-zero mean).
3. Cointegration of revenues and expenditures (time- series econometric).	Hakkio and Rush (1991); Bohn (1991), Haug (1991), Tanner and Liu (1994), Ahmed and Rogers (1995), Quintos (1995), Telatar and others (2005), Leachman and others (2005)	Solvency guaranteed by cointegration of primary expenditures, revenues, and debt (b), but debt may still rise unless vector of coefficients is [1,-1,r] with no constant or trend.
4. Link between primary surplus and debt (econometric model).	Bohn (1998, 2005), International Monetary Fund (2003b), Tanner and Ramos (2003).	Solvency guaranteed by positive relationship between primary surplus and debt (ps_t=κ+αb_t-1, α >0).
5. Historical Decomposition used to distinguish “policy” from “luck” (shocks) (time- series econometric).	This paper only.	For details, see text; evolution of “baseline” debt ratio corresponds to approach (1) above.

Naïvely, one might simply ask whether debt growth (in real terms or relative to GDP) was positive between some initial period M and a future date M+j, j>0. If so, one might say that the fiscal policy was “unsustainable” between M and M+j.

However, such a (tautological) conclusion ignores some important features of fiscal policy. Typically, fiscal policy (as gauged by the structural primary balance) is adjusted at intervals that are measured in years—not weeks or months. Adverse shocks that were not anticipated at some initial date M and are not under the control of the policy maker (interest rates, GDP growth, oil prices, etc.) may render an otherwise well-intentioned fiscal policy unsustainable. In the absence of such shocks (given the forecasts for growth, interest rates and other variables based only on information available at period M) might policies initiated at period M have been sustainable? Essentially, Blanchard and others (1990) and Talvi and Végh (1999) attempt to answer such a question. They calculate a fiscal gap: the difference between a country’s primary surplus and that primary surplus required to stabilize the debt (real interest payments, adjusted for economic growth). If the gap at period M was zero—if a tax smoothing policy was then in place—we would have expected that the debt remain constant from M onward, in the absence of shocks, forecast errors, and policy changes.

Time-series econometrics methods have also been used to examine retrospective sustainability. Specifically, Hamilton and Flavin (1986), and later Trehan and Walsh (1990,1991), Hakkio and Rush (1991), and others tested for the stationarity of the deficit Δb_t = κ + z_t. Consistent with McCallum (1984), stationarity of the interest-inclusive deficit implies solvency.

In an alternative formulation, Bohn (1998, 2005), International Monetary Fund (2003b, Chapter 3), and Tanner and Ramos (2003), test for a positive relationship between the primary surplus and debt by running the regression ps_t = κ + ab_t-1; solvency is assured if α > 0.⁴

However, neither test constrains κ to be zero. Thus, while tests such as these may indicate that the government is solvent (either Δb is stationary or α > 0), the debt (or its ratio to GDP) will still be growing over time if κ exceeds zero; in this case, ever-increasing primary surpluses are required to offset ever-increasing interest payments. A more stringent test might be for a one-to-one cointegrating relationship between tax revenues τ and interest inclusive expenditures γ+rb (with no intercept). Several authors including Trehan and Walsh (1990,1991), Hakkio and Rush (1991), Bohn (1991) and Tanner and Liu (1994), also use such tests. Such a test is viewed as a more precise characterization of the government’s effective fiscal rule– the long-run linkage between expenditures and revenues.

D. Prospective Sustainability

For policy makers, the question of prospective sustainability is of critical interest: “What policies should be undertaken today in order to prevent the need for further adjustments in the future?” Previous approaches to this issue are summarized in Table 2.

Table 2.

Approaches to Prospective Sustainability:

“What policies should be undertaken today in order to prevent the need for further adjustments in the future?”

Table 2.

Approaches to Prospective Sustainability:

“What policies should be undertaken today in order to prevent the need for further adjustments in the future?”

Description of Approach	Citations	Remarks
1. Fiscal gap (noneconometric).	Blanchard and others (1990); Talvi-Végh (2000).	Primary surplus should be equal to debt stabilizing benchmark ps*={(r-λ)/(1+λ)}b, λ = GDP growth.
1a. Fiscal gap (noneconometric).	Croce and Juan-Ramon (2003).	Like approach (1) but permits gradual adjustment.
2. Stress test (noneconometric).	IMF (2003a) and subsequent country reports.	Targeted primary surplus typically aims at debt reduction. Alternative scenarios for two standard deviation shocks to interest rates, growth, etc.
3. Value-at-Risk (noneconometric).	Kopits and Barnhill (2003, applied to Ecuador).	Examines main sources of shocks to net worth.
4. Value-at-Risk (econometric).	Adrogué (2004, applied to Central American countries).	Forecasts deficit, debt.
5. Simulated debt projections, baseline policies (econometric or other stochastic model).	Celasun, Debrun, and Ostry (2006), and Garcia and Rigobon (2004); see also Hostland and Karam (2005).	Projects debt accumulation under uncertainty (means and confidence intervals, including “fan charts”).
6. Simulations debt projections, baseline and adjustment policies (econometric).	This paper; see also Hoffmaister and others (2001), Guerson (2004), Koeva (2005), Penalver and Thwaites (2006).	Uses projections similar to (5), includes alternative policies, consistent with objective to avoid further adjustment for all but the worst w-percent of cases.

Table 2.

Approaches to Prospective Sustainability:

“What policies should be undertaken today in order to prevent the need for further adjustments in the future?”

Description of Approach	Citations	Remarks
1. Fiscal gap (noneconometric).	Blanchard and others (1990); Talvi-Végh (2000).	Primary surplus should be equal to debt stabilizing benchmark ps*={(r-λ)/(1+λ)}b, λ = GDP growth.
1a. Fiscal gap (noneconometric).	Croce and Juan-Ramon (2003).	Like approach (1) but permits gradual adjustment.
2. Stress test (noneconometric).	IMF (2003a) and subsequent country reports.	Targeted primary surplus typically aims at debt reduction. Alternative scenarios for two standard deviation shocks to interest rates, growth, etc.
3. Value-at-Risk (noneconometric).	Kopits and Barnhill (2003, applied to Ecuador).	Examines main sources of shocks to net worth.
4. Value-at-Risk (econometric).	Adrogué (2004, applied to Central American countries).	Forecasts deficit, debt.
5. Simulated debt projections, baseline policies (econometric or other stochastic model).	Celasun, Debrun, and Ostry (2006), and Garcia and Rigobon (2004); see also Hostland and Karam (2005).	Projects debt accumulation under uncertainty (means and confidence intervals, including “fan charts”).
6. Simulations debt projections, baseline and adjustment policies (econometric).	This paper; see also Hoffmaister and others (2001), Guerson (2004), Koeva (2005), Penalver and Thwaites (2006).	Uses projections similar to (5), includes alternative policies, consistent with objective to avoid further adjustment for all but the worst w-percent of cases.

Here again, debt stabilization is frequently used as a benchmark. Following Blanchard and others (1990) and Talvi and Végh (2000), the government should target the primary surplus to a level that stabilizes the debt—conditional on its forecasts of GDP growth and interest rates.

Such an approach is well-suited to an economic environment with no uncertainty. However, the question of fiscal sustainability has increasingly emphasized the importance of uncertainty: exogenous shocks to nonpolicy variables that impact the evolution of public debt. For example, the IMF’s Sustainability Framework (2003) includes a (noneconometric) stress test that shows how high the debt would be (relative to a baseline scenario) if the economy suffered certain adverse shocks (two standard deviations in magnitude).

More recently, several authors have proposed stochastic simulations of debt accumulation: see, for example Hoffmaister and others (2001), Garcia and Rigobon (2004), Hostland and Karam (2005), and Celasun, Debrun, and Ostry (2006). An important feature of such simulations is that the forecast variance of debt increases with the forecast horizon; this gives rise to a “fan chart” similar to those frequently used in the forecast of monetary aggregates. At this juncture, the objective function of the authority becomes crucial: the normative implications a “fan chart” for policy decisions will reflect that objective function. If the authorities’ objectives are characterized by the quadratic objective function (7), such a “fan chart” would be of little interest.

However, if the objective function emphasizes the avoidance of undesirable outcomes—that is, if it is an objective function with a third moment that is, “prudence;” see Tanner and Carey (2005), the “fan chart” is of greater interest. In this case, the upper tails of the “fan chart” illustrate precisely the undesirable outcomes that the authority would want to avoid.⁵ Such an objective function is discussed in greater detail below.

III. Overview of Our Methodology

The recent emphasis of uncertainty in the fiscal sustainability literature highlights the role of both deliberate policies and exogenous, non-policy shocks in the process of debt accumulation. Accordingly, our methodology to assess sustainability—both retrospectively and prospectively—is based on a simple near-vector autoregression (VAR) model of fiscal policy that helps quantify the role of policy and luck. The vector of variables X will include both the key fiscal variables (the real debt, b, and the real primary deficit, pd) and nonpolicy variables (interest rates, exchange rates, industrial output, oil prices).⁶

As summarized in Figure 1 and detailed in an appendix, our time series model is developed through several familiar steps, including unit root tests of individual variables in the model, determination of optimal lag length, an analysis of issues related to model structure and identification, and other diagnostic procedures.

To isolate distinct economic regimes, we rely mainly on prior knowledge of the country; in some cases, certain statistical tests supplement such knowledge. In some cases it was possible to lengthen our dataset by accounting for regime changes with discrete intercept shifts (dummy variables chosen using prior knowledge.) Alternative permutations of sample and dummy may yield similar results; we feature our preferred choice in the text, leaving discussion of alternatives for an appendix.

A. Retrospective Sustainability

Our methodology features a well-known representation of a VAR model, namely the historical decomposition. We also present the (more familiar) variance decomposition. Both tell us which shocks were most important for debt accumulation.⁷ However, only the historical decomposition reveals when such shocks occurred and whether such shocks increased or decreased debt.

In a historical decomposition, each element of X is expressed as the sum of: (i) a baseline projection of that variable, conditional on all information available in the base period M; plus (ii) the (orthogonal) impacts of shocks from all variables thereon, accumulated from M+1 forward. Thus, in any period M+j (j=1,2,3,4…J) the change in the debt (that is, the deficit) Δb_M+j is:

\begin{matrix} (9a) & Δ b_{M + j} = Δ b {(base)}_{M + j} + {z^{*}}_{b 1 j} + {z^{*}}_{b 2 j} + \dots . . {z^{*}}_{b I j} \end{matrix}

where Δb(base)_tB+k incorporates all information about the evolution of deficit that is available before time M+1, and z*_bij represent the impacts of the i^th variable (i = 1,2,3,..I) on the deficit, accumulated from M+1 through M+j. The variables corresponding to z*_bij are both policy and nonpolicy; they are discussed below. Thus, a country’s debt level at the end of period M+j is:

\begin{matrix} (9b) & b_{M + j} = b_{M + j - 1} + Δ b {(base)}_{M + j} + {z^{*}}_{b 1 j} + {z^{*}}_{b 2 j} + \dots {z^{*}}_{b I j} \end{matrix}

Absent shocks, fiscal policy is sustainable over the period M+1 through M+j if:

\begin{matrix} (10) & b {(base)}_{M + j} / {GDP}_{M + j} \leq b_{M} / {GDP}_{M} . \end{matrix}

That is, a country’s policy is “unsustainable” if the debt stock rises under certainty (the baseline projection); otherwise, policy is “sustainable.”

Note that observed and baseline debt need not be equal at end of the sample. It would be tempting to conclude that, since all shocks will have impacts that have a zero mean, all shocks should eventually cancel out, thus implying that b(base)_M+J = b_M+J, where J is the end-of-sample period. If this were the case, sustainability criteria would only be of interest for periods prior to M+J.

However b(base)_M+J and b_M+J are not always identically equal. To see this, first note that the total deficit has two components: interest payments and the primary balance: Δb_t = pd_t + rb_t-1. In any period t, the baseline is: Δb(base)_t = pd(base)_t+r(base)_t*b(base)_t-1. The observed primary deficit pd_t fluctuates about a fixed mean, namely pd(base). Positive shocks to the interest rate – r_t > r(base)_t—will raise the debt in the absence of an offsetting movement in the primary balance. More importantly, cases are easily found for which interest rate shocks have impacts on the debt that do not cancel out—even though the interest rate itself fluctuates about mean value r(base).

Thus, consider a simple example in which bad interest rate shocks happen before good ones. This helps boost the debt. At the end of the horizon (period J) observed debt may exceed baseline debt. Two factors are at work: (i) the baseline (mean) primary surplus is may not be high “enough;” (ii) period-by-period responses of the primary balance to interest rate shocks may not be sufficient to neutralize debt increases.

The framework permits alternative definitions of baseline fiscal policy. Typically, pd(base) might be estimated as a fixed mean; around which fluctuations in pd reflect both exogenous shocks and responses to shocks in other variables.⁸ However, such a definition might be unnecessarily restrictive. We may have prior knowledge that the mean primary surplus changed during the sample—although not as a systematic response to shocks from other variables. Such prior information might be reflected in a time trend or dummy variable.

The framework permits analysis of the effects of individual shocks. We also analyze the effect of individual shocks on the debt. We calculate the counterfactual value of the deficit at time M+j if the shocks to variable i had not occurred:

\begin{matrix} (11a) & Δ b (omit i)_{M + j} = Δ b_{M + j} - {z^{*}}_{bij} \end{matrix}

For any period M+j, the debt purged of the impact of shocks to variable i thus evolves according to:

\begin{matrix} (11b) & b {(omit i)}_{M + j} = b {(omit i)}_{M + j - 1} + Δ b_{M + j} - {z^{*}}_{bij} \end{matrix}

With respect to the i^th shock to a non-policy variable a country may be said to be “lucky” if, in the absence of that shock the debt would have been higher (b(omit i)_M+j > b_M+j); or “unlucky” if, in the absence of that shock, the debt would have been lower (b(omit i)_M+j < b_M+j).

This taxonomy is informally summarized in Table 3. A country’s policy is “unsustainable” if the debt stock rises under certainty (the baseline projection); otherwise, policy is “sustainable.” Whether the country is “lucky” or “unlucky” depends on the impact of nonpolicy shocks on the debt level.

Table 3.

Taxonomy: Fiscal Policy and Shocks

Table 3.

Taxonomy: Fiscal Policy and Shocks

		Shocks
Policy		“Lucky”—Beneficial shocks help reduce debt	“Unlucky”—Adverse shocks help increase debt
	“Sustainable”—Debt constant or falling under baseline projection	Sustainable and lucky: debt does not rise	Sustainable but unlucky
	“Unsustainable”—Debt rising under baseline projection	Unsustainable but lucky	Unsustainable and unlucky: debt rises

Table 3.

Taxonomy: Fiscal Policy and Shocks

		Shocks
Policy		“Lucky”—Beneficial shocks help reduce debt	“Unlucky”—Adverse shocks help increase debt
	“Sustainable”—Debt constant or falling under baseline projection	Sustainable and lucky: debt does not rise	Sustainable but unlucky
	“Unsustainable”—Debt rising under baseline projection	Unsustainable but lucky	Unsustainable and unlucky: debt rises

Of course, there will also be shocks to policy variables. Insofar as such shocks are departures from an estimated policy reaction function, they may be thought of as containing both discretionary policy (similar to Fatás and Mihov (2003)) and other random shocks to the primary surplus process (spending, tax collection). Counterfactually, if an expansionary (contractionary) shock had not occurred, the debt would have been lower (higher).

B. Prospective Sustainability/Objective Function

To prospectively assess fiscal sustainability, simulations of the VAR system with randomly generated shocks are presented. Specifically, the simulated value of the debt b(sim) for any period t > J is:

\begin{matrix} (11c) & b {(sim)}_{t} = b {(sim)}_{t - 1^{*}} (1 + r {(sim)}_{t}) + pd {(sim)}_{t} \end{matrix}

where simulated values of the interest rate and primary deficit, r(sim) and pd(sim) respectively, are:

\begin{matrix} (11d) & r {(sim)}_{t} = ζ_{r 0} + {ζ^{*}}_{r 1 t} + {ζ^{*}}_{r 2 t} + \dots {ζ^{*}}_{rIt} \end{matrix}

\begin{matrix} (11e) & pd {(sim)}_{t} = ζ_{p 0} + {ζ^{*}}_{p 1 t} + {ζ^{*}}_{p 2 t} + \dots {ζ^{*}}_{pIt} \end{matrix}

where ζ_r0 and ζ_p0 are the assumed mean levels of the real interest rate and primary surplus and the terms ζ*_rit and ζ*_pit are simulated impacts of shocks to variable i on the real interest rate and primary deficit, respectively.

Papers that present simulations similar to these include Garcia and Rigobon (2004) and Celasun, Debrun, and Ostry (2006). However, these papers are primarily positive in nature. Their simulations yield forecasts for the mean debt level over some horizon, conditional on a preexisting fiscal rule. Their analysis features “fan charts” designed to display a forecast variance that increases with horizon length.

By contrast, our emphasis is more normative in nature. We consider an authority whose objective would be to cushion the country’s residents from future adjustments. As an illustrative example, Tanner and Carey (2005) discuss a constant elasticity of risk aversion (CARA) / exponential objective function, specifically Φ(τ_t) = −1/φ exp(−φτ_t), where φ ≤ 0 is a “prudence” parameter. They show that minimizing such a collection cost function typically yields a long-run (expected) relationship between tax rates, expenditures, and debt:

\begin{matrix} (8c) & τ_{t}^{*} = κ_{t}^{*} + γ_{t} + {rb}_{t - 1} \end{matrix}

where $κ_{t}^{*} \equiv - ϕ σ_{t}^{2} / 2 \geq 0$ . Thus, for φ < 0, equation (8c) describes a government with a precautionary motive: its primary surplus at any time is positively related to the variance of the tax burden (γ+rb). Of course, for φ < 0, the debt ratio itself should fall over time. (That is, $κ_{t}^{*}$ is time-varying: the variance of the tax burden (γ+rb) falls as the debt ratio falls.) And, note that, if φ = 0, (8c) collapses to (8b)—the debt stabilization benchmark proposed by Blanchard and others.

The prudence parameter -φ has an alternative, common-sense interpretation, consistent with the value-at-risk approach: it summarizes the willingness to pay for precautionary cushion. If -φ equals or exceeds some critical value -φ(z), 0<z <1, the government will be willing to levy taxes today sufficient to “cover itself” with probability (1–z).

Thus, for a baseline scenario, the simulations report the means, standard deviations, and fractiles (median, 75^th, and 90^th percentiles) of simulated debt/GDP ratios. But, we are also able to depart from the baseline scenario by modifying the mean primary surplus ζ_p0. Such simulations thus yield a menu of policy options: they reveal what primary surplus is required to keep the debt ratio constant for all but the worst w-percent of cases (w = 50 percent, 75 percent, 90 percent)—over a given horizon.⁹

Note that debt reduction is a by-product of this objective function (φ < 0). Also, the horizon itself is a choice variable. A longer horizon implies a less stringent adjustment than a shorter horizon. That is, trying to prevent a bad outcome after (say) five years of continual debt reduction is easier (lower primary surplus) than trying to prevent that same outcome after just one year—precisely because the benefits of debt reduction are cumulative.

To inform current policy making, data should be as recent as possible. However, it may be also appropriate to present a counterfactual prospective analysis whose data end (and whose simulations begin) at some previous date M+J*, J* < J. Such an exercise thus provides a “rear view mirror” for policy makers. It asks: “If we had used our model at some date in the past— using only information available at that date—might we have reached different conclusions about policy?”

IV. Brazil, 2000–05

In Brazil, public debt has grown from about 30 percent of GDP during the 1990s to about 51 percent in 2005.¹⁰ This happened in the context of several dramatic changes in economic policies, including the Real Plan of 1994 (when inflation fell dramatically), increased exposure to global market shocks (spillovers from the Mexican, Asian, and Russian crises of 1995, 1997, and 1998, respectively), and again in 1999, when a pegged exchange rate regime (within a narrow band) ended in crisis and devaluation.

The period 2000–05 is well suited for sustainability analyses like those proposed in this paper. In 2000, in the aftermath of the crisis and while both an IMF program and an inflation targeting regime were being inaugurated, the primary surplus was substantially increased, from about zero to just over 3 percent of GDP. At that time the debt/GDP ratio was about 49 percent, but it was envisaged (as reflected in IMF Staff Reports) that the fiscal adjustment would be large enough to gradually reduce the debt ratio to about 46 ½ percent of GDP by 2005.

However, this hope was dashed. Instead, between 2000 and 2004, the debt ratio rose. Notwithstanding additional primary adjustments during this period, electoral uncertainties helped boost the real interest rate, real depreciation, and hence debt (whether denominated in domestic or foreign currency). In 2002, the debt ratio peaked at about 60 percent of GDP. By 2004, however, both output and the Real recuperated, helping the debt ratio to fall—but not enough to meet the initial projections.

A. Econometric Preliminaries

The analysis of Brazil focuses on a vector X_t of endogenous variables:

\begin{matrix} (12) & X {(Brazil)}_{t} = [{ip}_{t}, {pd}_{t}, ε_{t}, r_{t}] \end{matrix}

where ip is the industrial production index, pd is the primary deficit, ε is real depreciation (bilaterally, against the U.S. dollar), and r is the (implicit) average real interest factor calculated directly from the budget constraint, namely r_t = ([b_t+pd_t]/b_t-1)-1, a measure of the real cost of government borrowing. Exogenous variables include the change in oil prices and discrete intercept dummy variables, discussed below.

The frequent, dramatic regime changes in Brazil make it difficult to choose the sample period for estimation. Our principal estimates (see Table A.2, Appendix) use the period from mid-1995 to mid-2005. To account for regime shifts, we include two dummy intercept variables: one to isolate the exchange rate crisis (D=1 for t =1999:1–1999:4 and D = 0 otherwise), and one to isolate the more recent floating rate period (D=0 for t =1995:5–1999:4, D = 1 thereafter). We also include a time trend in the estimates (thus capturing increases in the primary surplus during 2000–05). Standard tests (Aikaike, Schwarz) suggest that 4 lags should be included (see Appendix).

We also discuss alternative estimates in the Appendix. These include results using data only post-1999 data. While these options do not yield identical results, they do confirm that the upsurge in debt from 2001–03 was largely the result of innovations to exchange rates and interest rates, rather than baseline policy.

B. Retrospective Analysis, 2000–05

Table 4a presents results of a retrospective analysis whose baseline begins in mid-2000 (M = 2000:5) and ends in mid-2005. At the beginning of this period, Brazil’s debt ratio was about 50 percent of GDP. Over the period, the debt ratio rose and then fell dramatically; it returned to its initial GDP ratio by the end-period.

4a.

Brazil: Retrospective Analysis, 2000:5–2005:6

Source: Central Bank of Brazil and author’s estimates.

4a.

Brazil: Retrospective Analysis, 2000:5–2005:6

Historical Decomposition			Percent of GDP
Initial Debt (b_M)			50.5
End Period Debt (b_M+J)			50.8
	Baseline Projection (b(base)_M+J)		50.7
	Shock Component (b_M+J- (b(base) _M+J)		0.2
Shocks: Var. and Historical Decomposition		Percent of total variation in debt	b(omit)_M+J Percent of GDP
	Oil Price (poil)	0.75	51.2
	Exchange rate+int rat (ε+r)	97.2	50.2
	Industrial Production (ip)	1.9	51.0
	Primary Deficit (pd)	0.13	50.9
Initial primary surplus (ps(base))			3.5
Debt stabilizing constant primary surplus			4.0

Source: Central Bank of Brazil and author’s estimates.

4a.

Brazil: Retrospective Analysis, 2000:5–2005:6

Historical Decomposition			Percent of GDP
Initial Debt (b_M)			50.5
End Period Debt (b_M+J)			50.8
	Baseline Projection (b(base)_M+J)		50.7
	Shock Component (b_M+J- (b(base) _M+J)		0.2
Shocks: Var. and Historical Decomposition		Percent of total variation in debt	b(omit)_M+J Percent of GDP
	Oil Price (poil)	0.75	51.2
	Exchange rate+int rat (ε+r)	97.2	50.2
	Industrial Production (ip)	1.9	51.0
	Primary Deficit (pd)	0.13	50.9
Initial primary surplus (ps(base))			3.5
Debt stabilizing constant primary surplus			4.0

Source: Central Bank of Brazil and author’s estimates.

Movements in exchange rates and interest rates (reflecting shifts in investor sentiment) jointly explained over 97 percent of variation in the debt ratio over this period. Figure 2 illustrates this: for most of the period, the debt level purged of these two shocks (b(omit ε+r)) is close to the baseline level (b(base). When the debt peaks in September 2002, the cumulative (adverse) impact of these shocks (b_t – b(omit ε+r)_t > 0) was about 13 percent of GDP.

Thereafter, pressures subsided and the Real recuperated. By mid-2005 (end sample) the gap (b_t – b(omit ε+r)_t < 0) fell to about 0.6 percent of GDP.¹¹

As Figure 3 shows, innovations in output (industrial production) also affected the debt accumulation process—but to a much smaller extent than exchange rates and interest rates (under 2 percent of total variation). In 2003, debt levels purged of the output shocks are higher than observed levels (b(omit ip)_t – b_t > 0)—that is, during this period, industrial production shocks helped reduce the debt. In August 2003, the gap between b(omit ip)_t and b_t is about 1.3 percent of GDP.

As mentioned above, there were several fiscal adjustments between 2000 and 2005: the primary surplus ratio rose from about 3.5 percent of GDP at the end-2000 to over 6 percent by mid-2005 (4.8 percent end-year). From a tax-smoothing perspective, it might have been preferable if the initial adjustment in 2000 had been somewhat larger, thus avoiding the need for subsequent adjustments. As an illustrative calculation (Table 4a, bottom) if the primary surplus had initially been adjusted to about 4 percent of GDP (rather than 3.5 percent), the debt ratio would have been stabilized—without the need for further increases in the primary surplus (relative to GDP).¹²

C. Prospective Analysis from 2005 Onward

Prospective analyses are presented in Tables 4b(i) and 4b(ii). The upper portion of each table shows a baseline scenario: the primary surplus for the initial year of the simulation (2005:6) is designed to roughly correspond to the actual policy: a primary surplus of about 4 ½ percent of GDP. The second portion of both tables presents the menu alternatives for probabilistic debt sustainability.

Table 4b(i).

Brazil: Prospective Sustainability from 2005:6 Onward; Higher Interest Rate