Learning about Debt CrisesRadoslaw Paluszynski∗

Department of EconomicsUniversity of Houston

First version: October 2015This version: May 2019

Abstract

The European debt crisis presents a challenge to our understanding of the relationship be-tween government bond yields and economic fundamentals. I argue that information frictionsare an important missing element, and support that claim with evidence on the evolution ofGDP forecast errors in 2008-2014. I build a quantitative model of sovereign default whereoutput features rare disasters and agents learn about their realizations. Debt crises coincidewith economic depressions and develop gradually while markets update their expectationsabout future income. Calibrated to Portuguese economy, the model replicates the comove-ment of bond spreads and output before and after 2008.

Keywords: Sovereign debt, disaster risk, Bayesian learning, long-term debt

JEL Classification Numbers: D83,F34,G15

∗E-mail: rpaluszynski@uh.edu. I am grateful to Manuel Amador, Tim Kehoe and Motohiro Yogo fortheir continuous encouragement and support. I also thank Joao Ayres, Anmol Bhandari, Hyunju Lee, EllenMcGrattan, George Stefanidis, Kei-Mu Yi, Pei Cheng Yu and all participants of the Trade and Developmentworkshop at the University of Minnesota for many helpful comments. I acknowledge the generous supportof the Hutcheson-Lilly dissertation fellowship. The quantitative part of this paper was conducted using theresources of the Minnesota Supercomputing Institute. Consensus Economics Inc. owns the copyright to theConsensus Forecasts - G7 & Western Europe dataset which I use under a license agreement.

1 Introduction

The recent debt crisis in Europe has reopened the discussion about what puts governmentsat risk of sovereign default. The weak correlation between interest rates on public debt andeconomic fundamentals of the southern European countries challenges the theoretical linksestablished by a large body of research prior to 2008. This new evidence has led some re-searchers to revisit the hypothesis of self-fulfilling debt crises (Aguiar et al. (2019)). Othereconomists, motivated by the same observations, have argued that the European episode wasdriven by external factors such as the intra-EU politics (Brunnermeier, James and Landau(2016)). In this paper I propose a quantitative model of the European debt crisis based onidiosyncratic income fluctuations which feature disaster risk and information frictions.

I start by reviewing the comovement between government bond yields and economic funda-mentals such as GDP growth and external debt. The bond spreads1 were negligible since theintroduction of the euro, regardless of current economic performance. At the outset of theGreat Recession, peripheral EU countries were hit by negative income shocks in the rangebetween two to three standard deviations below their mean. Yet, despite the widespreadexpansion of external debt levels in 2009, government bond spreads temporarily rose above1.5-2.5% in 2009:Q1, only to fall back below 1-2% by the second half of 2009. The real stressdid not appear in the European bond markets until mid-2010 when Greece, followed by othercountries over the next two years, experienced a sharp increase in borrowing costs. The eco-nomic fundamentals were not much different relative to 2009, however, following a short-livedrecovery and debt reductions in 2010. This observed dynamics cannot be easily explained bycanonical quantitative models of sovereign default.2 In these models, the bond price functiontypically exhibits a high elasticity with respect to income shocks, even when debt maturitiesare long (if output is persistent enough). When I calibrate the model in a standard way toEuropean economies, the predicted spreads are positive even before the crisis while the re-sponse to bad shocks in early 2009 mirrors what in reality did not occur until two years later.

To address this puzzle, I build on two observations regarding that episode. First, the expe-rience of southern European economies in years 2008-2014 may easily be interpreted as therealization of a rare disaster risk. The peak-to-trough decline in quarterly real GDP rangesfrom around 10% for Italy, Portugal and Spain to 27% for Greece. As shown by Barro (2006),the risk of a recession of such magnitude is sufficient to explain the traditional puzzles in

1The bond spread is defined as a difference between the annual interest rates paid by the given country’sgovernment bonds and a risk-free asset, in this case the German long-term government bonds.

2I am referring to the class of models based on the seminal framework of Eaton and Gersovitz (1981).

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asset pricing.3 The size and persistence of GDP declines relative to a trend among thosecountries is also consistent with a great depression episode defined by Kehoe and Prescott(2002). Second, expectations about future income shocks evolved gradually among financialinstitutions in years 2008-2014, which I document using the real-time GDP forecast data.In particular, prior to 2008 the forecast root mean square errors are at a similar level acrossall analyzed countries and forecasting agencies, equivalent to 1.5% of GDP in 2010. Then,errors increase to 2.5-6% between 2008 and 2011, in all cases overestimating the future GDPgrowth. This indicates that negative output shocks at the time were perceived as a fairlytypical recession, corresponding in size and persistence to Europe’s post-war business cycle.Finally, in years 2012-2014 the forecasts become much more precise, with individual errorschanging sign in many cases (i.e. underestimating GDP) and overall root mean square er-rors falling to 1.5-3.5%. This increase in pessimism about the countries’ economic outlookcoincided with the dramatic spikes in interest rates on European governments’ bonds.

Motivated by these two observations, I develop an otherwise standard model of sovereigndebt that captures the two elements described above in a simple way. Starting with a long-term debt model as in Chatterjee and Eyigungor (2012) or Hatchondo and Martinez (2009),I first introduce a regime-switching income process with implied rare disaster risk. A disasteris represented by a large and negative shift in the long-run mean income of the economy.Then, I assume agents have incomplete information about the underlying switches betweenthe regimes and learn about them over time in Bayesian fashion. As it is typical for such asetting, bond yields carry a default premium which varies with the amount of outstandingdebt and the expected fluctuations in future GDP. Crucially for my model, spikes in defaultrisk coincide with rare disaster episodes rather than the recurring business cycle downturnsas it is the case for emerging markets. In normal times, which can last for decades, there islittle concern about a sovereign debt crisis in the foreseeable future and, as a result, bondprices carry a negligible default premium. When a rare disaster occurs, income is set on adownward trajectory; however agents are not aware of this immediately. In other words, theycannot tell if the shocks they are observing are temporary or permanent in nature. In thepresence of long-term debt, this information friction relieves the upward pressure on interestrates, because investors remain optimistic about the economy’s long-run outlook. Over time,as income continues to decline, agents increasingly recognize the looming disaster and revisetheir forecasts. The result is a sudden, sharp spike in default risk that follows long periodsof relative calmness in the bond markets.

3In contrast to Barro (2006), I use rare disasters to show why bond prices are higher for long periods oftime than what a standard model would predict.

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I use Portugal as a quantitative case study and calibrate the parameters of the disasterregime by targeting the actual path of long-term GDP forecasts discussed above. As a re-sult, agents in my model exhibit time-varying expectations about the direction of futureincome that mimic those observed in the data. The calibrated disaster regime features along-run mean income of 24 percent below trend and a switching probability of 1.6%, closeto the aforementioned benchmarks. I pick the remaining structural parameters to replicatethe key features of the Portuguese economy such as the average bond spread and average ex-ternal debt level. The calibrated model exhibits interesting behavior in several ways. First,it features a highly volatile bond spread even though average spread is targeted at a low level.This result reflects the fact that the spread tends to be negligible for long periods of timewhile the good regime is in place, and then it shoots up when a disaster turns on. Second,the government in the model sells bonds at steep discounts, resulting in high equilibriuminterest rates, reaching up to 100%. This occurs whenever the belief about an upcomingdisaster increases suddenly, while the income is still high and default is costly. The entirebond price schedule shifts downwards and the government is left with no other choice than toaccept very low prices, until either it manages to deleverage or until income falls low enoughto justify a default. Third, unlike in a standard default model, my calibration requires a highvalue of the discount factor, reducing the typical high volatility of consumption or trade bal-ance. This is due to the fact that defaults are generated by the occurrence of rare disasters,while the government behaves counter-cyclically during regular business cycles.

In an event study of Portugal’s debt crisis, I feed in the sequence of GDP data for years2000-2014. Consistent with the data, the model predicts a negligible spread and slow debtaccumulation prior to 2009. Then, the initial adverse income realizations cause an increasein the bond spread to 3% in the first quarter of 2009, compared with over 6% for thebenchmark model based on a simple AR1. This is because markets are unsure if they areobserving temporary shocks or a permanent regime switch. Over the next two years, thebelief about the latter converges to certainty, and agents become convinced that the processhas switched to a disaster mode. As a result, we observe a delayed jump in the bond spreadin the first quarter of 2012 combined with a sharp reduction in government debt (and, infact, an eventual sovereign default4). Finally, I show that learning is crucial to generatethis prediction. In a counterfactual exercise where agents are fully aware of the upcomingdisaster at the end of 2008, the bond spreads shoot up to the similar level as predicted bythe standard model, while the government begins a drastic path of debt reduction.

4Recall that Portugal, together with other European countries, received official bailouts in excess of 40percent of their GDP, to prevent them from defaulting, an element not present in my model.

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More generally, this paper shows that canonical sovereign debt models should place moreemphasis on modeling the stochastic income. The detrended AR1 process assumed in moststudies, while having the advantage of computational simplicity, does not fit the recentexperience of the European economies well. There are two arguments behind this claim.First, AR1 assumes implicitly that the data is stationary around a long-run linear trend.This cannot be confirmed with unit root tests if the 2008-2014 data is included. Second,and more fundamentally, the AR1 specification is unable to account for the time-varyingexpectations of the direction of future income, because it always mean-reverts back to thetrend. The agents know about it, and as a result they generate a consistent, time-invariantpattern of forecasts during a recession. This does not appear to be a major problem formodeling emerging market economies, where the expectations about future are generallynoisy and do not vary much over the business cycle (this is shown in Section 5 for the caseof Argentina). However, the data on forecast errors among advanced countries during theGreat Recession indicates that a precise modeling of market expectations may be one of thecrucial elements in understanding the recent events, in particular the European debt crisis.

1.1 Literature review

This paper is closely related to the quantitative sovereign debt literature, in particular onebuilding on the seminal work of Eaton and Gersovitz (1981) and, more recently, Aguiar andGopinath (2006) or Arellano (2008). Chatterjee and Eyigungor (2012) and Hatchondo andMartinez (2009) introduce long-duration bonds to these models and show that it is impor-tant in accounting for the amounts of debt and average spreads observed in the data.

The latest branch of quantitative default literature investigates the ability of such models tomatch the volatility of sovereign spreads, with a focus on the European debt crisis. Aguiar etal. (2016) point out that calibrated long-term debt models often deliver a standard deviationof the spread an order of magnitude lower than what it is in the data. Chatterjee andEyigungor (2019) obtain volatile spreads in a model with political frictions, while Aguiar etal. (2019) and Ayres et al. (2019) revisit models with multiple equilibria to justify why bondsare often sold at large discounts. Bocola and Dovis (2019) use the observed maturity choicesto identify the rollover risk components of sovereign spreads. Bocola, Bornstein and Dovis(2019) emphasize the role of domestic debt in generating a high spread volatility relative toits mean. The model in this paper achieves a similar objective of generating volatile andhigh-peaking spreads through a mechanism of learning about rare disasters. An attractivefeature of the model is that it does not rely on the borrower’s low discount factor.

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On a more general level, this paper derives from two well-known themes in macroeconomicsand finance. The first one concerns the role of economic disasters in asset pricing models.Rietz (1988) was the first article to propose that infrequent market crashes have a potentialto explain the equity premium puzzle using an Arrow-Debreu pricing framework. Barro(2006) shows that this result holds under plausible calibrations of rare disasters inferred us-ing historical data. Contrary to the usual logic of equity premium puzzle, this paper showsthat rare disasters can also explain why prices of risky bonds are unexpectedly high. The sec-ond theme incorporates learning about unobserved economic conditions in macroeconomics.Boz and Mendoza (2014) present a model where households learn about the probability ofswitching between credit cycles to produce a boom-bust cycle like the one observed around2008 in the US. Boz, Daude and Durdu (2011) show that learning about the permanent vs.transitory nature of shocks can explain some of the observed differences in volatility betweendeveloped and emerging economies. Contributing to this line of work, my paper shows howthe data on real time forecasts can be used to estimate deep parameters of the model andimprove its quantitative properties.

The remainder of this paper is structured as follows. Section 2 describes empirical evidenceregarding the European debt crisis. Section 3 introduces the main model. Section 4 calibratesthe model and uses it to analyze the European debt crisis and contrast the results with thoseobtained using a benchmark version of the model. Section 5 discusses the role of bailoutsand offers a further validation of the theory. Section 6 concludes.

2 Empirical motivation

In this section, I document the dynamic pattern of the European debt crisis and explainhow it is at odds with the predictions of typical models of sovereign debt. I also present theempirical evidence that motivates the theory developed in this paper, namely the depth ofthe output declines and the evolution of forecast errors in years 2008-2014.

2.1 European debt crisis

Prior to 2008, the Eurozone’s peripheral economies enjoyed a decade of relative prosperityand stable growth, fueled by the European integration, rising trade and the benefits ofcommon currency. On the other hand, sovereign debt crises had predominantly been aproblem of highly volatile emerging market economies over the last few decades. As a result,the securities issued by governments of European countries seemed risk-free to most financial

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market participants, in spite of the increasing debt stocks and low trend growth in manyof these economies. The tendency to put excessive faith in Europe’s ability to repay theirdebts continued even after the severe recession began in the second half of 2008. Figure 1depicts the times series of real GDP for the four southern European economies5, togetherwith the spreads on long-term government bond yields.6 As can be noticed, markets did notexpress significant concern about the European governments’ ability to repay for a long timefollowing the initial slump in output. Instead, the bond spreads exhibited minor “wiggles”when the financial crisis first hit, and only began a gradual increase afterwards, leading tothe dramatic spikes observed around 2011-2012. Appendix B additionally shows that thesovereign ratings assigned to these countries by the leading credit agencies closely mirroredthis dynamics, with major downgrades only starting in 2010-2011.

Note: The GDP series are in constant 2010 prices, and their values are normalized such that the thirdquarter of 2008 equals 100 (beginning of the financial crisis - shaded area). The bond spreads areexpressed in percentage points and are acquired from the OECD dataset.

Figure 1: Real GDP and government bond spreads of the peripheral European economies

5A similar pattern also holds in the case of Ireland which I leave out for the sake of clarity of exposition.6The bond spread is defined as the difference between the annualized interest rate on the given ten-year

government bonds and the interest rate on the German long-term bonds, assumed to be a risk-free asset.

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Another interesting aspect of the European crisis involves the debt policy of national govern-ments in years 2000-2014. Figure 2 shows the evolution of government external debt-to-GDPratios of the four economies of interest, contrasted with the same path of real GDP over time.7

Notice that as the negative output shocks first hit in 2008, we observe all governments re-sponding by increasing their foreign debt on impact, and avoiding any major adjustmentsfor the first two years of the crisis. The sharp debt reductions did not occur until late 2010and 2011 and were mostly enforced by the international financial institutions (IMF and theEuropean Commission), or took the form of an outright default in the case of Greece. Whilesuch countercyclical response may be natural in the canonical models of fiscal policy, it alsoleads to an increased risk of non-repayment through the lens of a sovereign default model.

Note: The GDP series are in constant 2010 prices, and their values are normalized such that the thirdquarter of 2008 equals 100 (beginning of the financial crisis - shaded area). The debt series representthe external debt securities of the general government, and are expressed as fraction of annualized GDP.The data are acquired from the World Bank’s Quarterly External Debt Statistics and start at differentpoints in time for different countries (hence the missing observations in the four figures above).

Figure 2: Real GDP and external debt of the peripheral European economies

7By “debt” I refer to short- and long-term debt securities and thus exclude other types of liabilities, inparticular official and emergency loans.

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2.2 Depth of GDP drops

In order to highlight the magnitude of the decline in economic activity among the peripheralEuropean countries, Table 1 lists the largest peak-to-trough drops since 2007. The numbersprovided refer to the drops in real GDP both at face value, and in relation to a 2% trend.Notice that the former ranges from almost 10 percent for Spain, Italy and Portugal, up to27.5 percent in the case of Greece. Recall that the cutoff size of contraction that defines arare disaster in Barro (2006) is 15 percent. He emphasizes however that using an alterna-tive threshold of 10 percent delivers similar results, fundamentally affecting the traditionalasset pricing mechanisms. On the other hand, Kehoe and Prescott (2002) define the “greatdepression” episode as a sustained negative deviation of at least 20 percent in the GDPlevel net of the 2 percent annual trend growth. Table 1 indicates that three out of the fouranalyzed economies satisfy this definition, while Spain falls just short of it. It should alsobe mentioned that using the countries’ individual trends rather than the common 2 percentgrowth rate would make this conclusion even starker.

Country Largest decline (in %) Quartersface value detrended

Greece 27.5 40.1 23Spain 9.60 19.2 21Italy 9.50 22.1 21Portugal 9.60 20.2 19

Note: This table shows the largest recorded declines in real GDP in the period 2007-2016,measured at face value and relative to a 2% trend as in Kehoe and Prescott (2002).

Table 1: Peak-to-trough GDP drops among the peripheral European economies

2.3 Market expectations during the recession

As a second piece of outside evidence on the European debt crisis, I investigate the beliefsabout GDP growth in real time. The distribution of future income shocks is a crucial elementdriving interest rate fluctuations in those models, and thus deserves particular attention.

Figure 3 once again presents the plot of real GDP over time for the European countries,along with the GDP forecasts published every year by OECD.8 As can be noticed for the

8For illustration, in this figure I use the two-year-ahead forecasts of real GDP growth from the Fall issuesof OECD’s Economic Outlook (the Spring version only provides one-year-ahead forecasts). In the followingsubsection I also present the forecast errors from other institutions, public and private alike.

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Note: The GDP series are annual and expressed in constant 2010 prices; their values are normalized suchthat the observation for 2010 equals 100. The red dotted lines represent one- and two-year ahead forecastspublished by the OECD Economic Outlook (fall edition) and start in the year when each of them is made.

Figure 3: Forecast and actual real GDP for the peripheral European economies

period prior to 2008, while the European economies are still growing along a stable trendthe observed forecast errors are small (with some overshooting for Italy and Portugal, whoseeconomies experienced a slowdown in the early 2000s). When the financial crisis breaksout, the forecasts are still fairly optimistic, predicting a recovery in years 2008-2010. Overtime however, as the GDP continues to plunge we also observe that the forecasts becomeflatter, indicating that the markets have realized the recovery of output cannot be expectedin the short and medium term. From 2012 on, the forecasts essentially line up again withthe subsequently realized data for all of the depicted economies. This is also the time whenthe European bond markets undergo unprecedented turbulences, with surging interest ratesand drastic reductions in debt levels, as documented in Figures 1 and 2.

To analyze the forecast errors more systematically around the Great Recession, I acquirereal-time predictions of three organizations: OECD, IMF and the European Commission.The historical forecasts of these institutions are available publicly and released every year in

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two vintages, Spring and Fall (roughly corresponding to May and November, respectively,so they are based on the knowledge of the data for first and third quarter of that year). Inaddition, I obtain the data from Consensus Economics, a survey of forecasters from privatebanks, government agencies, think-tanks and research centers. The queries are collectedmonthly and ask about a number of macroeconomic indicators, in particular the real GDPgrowth. A consensus forecast is defined as the average prediction across all participants of thesurvey. Broad literature documents that consensus forecasts do not suffer from many biasesthat may affect private and public institutions alike, and have traditionally performed betterthan any individual forecaster over the long run (see e.g. Batchelor (2001) and Cimadomo,Claeys and Poplawski-Ribeiro (2016)). In Table 2 I test this conclusion for the southernEuropean economies during the debt crisis, and investigate the direction of forecast errors.

Root mean square error OECD IMF EC CE

(a) Pre-recession sample: 2000-2007Greece 2.18 2.25 2.18 2.13Spain 0.76 0.91 0.99 0.95Italy 1.47 1.57 1.49 1.46Portugal 1.56 1.48 1.45 1.52(b) Recession - first stage: 2008-2011Greece 6.47 6.78 6.88 6.67Spain 2.92 3.26 3.22 3.09Italy 3.44 3.52 3.63 3.50Portugal 2.60 2.81 2.59 2.80(c) Recession - second stage: 2012-2014Greece 1.47 1.88 1.88 1.34Spain 1.09 1.40 1.00 1.10Italy 0.71 0.78 1.05 0.87Portugal 0.43 0.58 0.58 0.58

Note: The table presents root mean square errors of one-year-ahead forecasts of real GDP level. Forecasts areacquired from four sources: OECD, IMF, European Commission, and Consensus Economics Inc. The erroris expressed as percent of the 2010 level of real GDP for each country. All forecasts come in two vintages,Spring and Fall, which I use jointly. The number of forecasters participating in Consensus Economicssurveys varies over time and across countries, with a minimum of four and a maximum of twenty.

Table 2: Root mean square errors in real-time historical forecasts for different time frames

I use one-year-ahead forecasts from each of the four institutions, combining both vintagesin every year.9 To ensure direct comparability, I consider the May and November issues of

9While OECD and the European Commission also publish up to two-year ahead, and IMF up to five-year-ahead forecasts, the Consensus Economics survey is limited to next-year predictions only.

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the Consensus Economics survey which roughly coincide in time with the Spring and Fallreports of the IMF, OECD and the European Commission. Table 2 documents the rootmean square errors (RMSE) observed for each of the four countries of interest during threeseparate time periods: a pre-recession sample of 2000-2007 and the two stages of the GreatRecession, namely 2008-2011 and 2012-2014. The bias is expressed as percent of each coun-try’s 2010 real GDP level and can take positive values only.10 As a non-linear measure,RMSE punishes fewer but large mistakes more heavily than frequent and small ones. Thereare several interesting observations to be made about the data in Table 2. First, prior to 2008the RMSE is of similar size across all countries and forecasting agencies, between 1% and2% of 2010 real GDP. Second, in years 2008-2011 the errors increase sharply, ranging from2.5− 3.5% for Spain, Italy and Portugal, up to 6.5% for Greece. Third, in years 2012-2014the RMSE drops significantly for all analyzed countries, in most cases below the average levelfrom before 2008. Finally fourth, the Consensus Economics forecasts do not outperform theinternational organizations, but instead tend to do worse than either IMF or OECD, espe-cially during the first stage of the recession in 2008-2011. Interestingly, consensus forecastsdo better prior to 2008, in line with the findings of earlier studies such as Batchelor (2001).This indicates that private-sector expectations were likely to exhibit excessive optimism es-pecially at the beginning of the Great Recession.

Large forecast errors at the height of the crisis became a subject of intense critique and haverecently led the OECD to publish a study to evaluate the source of mistakes. In a “PostMortem”, Pain et al. (2014) write:

GDP growth was overestimated on average across 2007-12, reflecting not only errorsat the height of the financial crisis but also errors in the subsequent recovery. (...)

The OECD was not alone in finding this period particularly challenging. The pro-file and magnitude of the errors in the GDP growth projections of other internationalorganisations and consensus forecasts are strikingly similar.

In their ex post reflection, the OECD points to the repeated expectation of a swift recovery asthe main source of forecast errors, which suggests that a learning process was taking place.To not appear as the only culprit, the OECD also emphasizes that the overly optimisticforecasts have been common among other influential forecasters associated with internationalorganizations and consensus measures. This claim is confirmed by Figure 3 and Table 2.

10Appendix A contains additional analysis of the average bias in the forecasts which reveals their sign. Itshows that the errors are caused by overestimating growth in all years between 2008 and 2011. Interestingly,in 2012-2014 the average bias falls much more than RMSE or the mean absolute bias, indicating that theerrors of opposite signs canceled each other out.

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3 Model

In this section I present a model of sovereign debt that features an augmented specificationof output process and incomplete information about its realizations.

3.1 Economic environment

Consider a representative-agent small open economy with a benevolent sovereign governmentthat borrows internationally from a large number of competitive lenders. Time is discreteand there is no production or labor. Instead, the economy faces a stochastic stream of en-dowment realizations. Markets are incomplete and the only asset available for trading is themulti-period non-contingent bond.

Endowment process Suppose the country’s endowment follows an autoregressive regime-switching process. I assume that there are two possible regimes, High and Low, and each ofthem is characterized by its own long-run mean. For simplicity, the persistence and varianceparameters are assumed to be constant across regimes. Specifically, the evolution of output,detrended with a deterministic long-run mean growth rate,11 is given by

yt = µj(1− ρ) + ρyt−1 + ηεt (1)

where εt ∼ N (0, 1) is an i.i.d. random shock and ρ, η, {µj}j=L,H are parameters of the tworegimes. Regimes change according to a Markov process with the transition probabilitymatrix given by

Π = πLL 1− πLL

1− πHH πHH

(2)

The specification of a bimodal stochastic process of endowment in formula (1) is non-standardin the sovereign debt literature.12 It is motivated by the income pattern of Europeaneconomies in the recent decade, illustrated in Figures 1-2. Throughout the paper, I willconsider the two regimes as highly asymmetric, with the low one having the interpretationof a rare disaster or a great depression. In the context of the European debt crisis, such adisaster regime can for example be associated with the failure of the monetary union.

11Appendix C discusses details of the detrending method used in this paper.12Simultaneously with the present paper, a bimodal income process was also used by Chatterjee and

Eyigungor (2019) and Ayres et al. (2019). In contrast to these papers, the regimes here are meant to behighly asymmetric which can be interpreted as “normal times” and “rare disaster”. In such a setting, learningabout the underlying regime has a naturally powerful effect, as I show in section 4, because agents tend tohave a strong prior belief against a potential regime switch. I also show a new method of estimating thisincome process by incorporating real-time forecast data.

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Preferences The representative household has preferences given by the expected utilityof the form:

E0

∞∑t=0

βtu(ct) (3)

where I assume the function u(·) is strictly increasing, concave and twice continuously dif-ferentiable. The discount factor is given by β ∈ (0, 1).

Government In each period, the government chooses a consumption rule and the level ofdebt holdings to maximize the household’s lifetime utility. The only asset available is thelong-duration bond. In the spirit of Chatterjee and Eyigungor (2012), I assume that eachunit of outstanding bonds matures probabilistically in every period, or pays a fixed coupon.The government may save at an international risk-free interest rate. If it decides to borrow,however, the government is not committed to repay the debt next period. Consequently, thebond is priced endogenously by risk-neutral lenders to account for the possibility of defaultas well as debt dilution in the future. As it is commonly assumed in the sovereign debt liter-ature, the government who refuses to honor its obligations faces an exogenous cost of defaultand is further excluded from borrowing in the financial markets, with a certain probabilityof being readmitted in every subsequent period.

Market clearing There is no storage technology and, under the aforementioned assump-tions on the utility function, implies that the endowment is fully divided between currentconsumption and net borrowing. This market clearing condition is given by

ct = yt − bt (δ + (1− δ)κ) + qt (bt+1 − (1− δ)bt) (4)

where qt is the price of the debt stock bt+1 (to be repaid next period), δ is the rate at whichbonds mature every period and κ is a fixed coupon.

Bond prices International lenders are perfectly competitive and have “deep pockets” inthe sense that potentially even large losses do not affect their decisions. In equilibrium thelenders make expected zero profit and as a result, the bond pricing formula compensatesthem only for the default risk implied in the government’s decisions.

3.2 Information structure

The two state variables mentioned so far, current bond holdings (bt) and income (yt), arestandard in sovereign debt literature. In addition, this model features another exogenous

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stochastic variable, zt ∈ {zL, zH} representing the regime (Low or High) in which the econ-omy is currently operating. While all agents know the latest income realization, they haveincomplete information about the current regime. Instead of observing it directly, agentsform a belief pt defined as their perceived probability of being in the High regime, formallypt ≡ Prob(zt = zH). Intuitively, this variable can be thought of as market sentiment aboutthe economy’s expected future income path. As I show in Section 4, the belief about regimeis quantitatively significant and appears to have fluctuated significantly in years 2008-2014.

3.3 Timeline

In every period, the timing of events is as follows:

1. The new regime z ∈ {zL, zH} is drawn, with the probability distribution given by (2).

2. The new realization of endowment y is drawn, according to the newly updated regimez and conditional on its level from last period.

3. Agents observe y and mechanically form a new belief p about the regime, conditionalon the previous and current endowment, as well as the last period’s belief.

4. Default and redemption decisions take place:

• The government that has recently defaulted on its debt draws a random numberto determine whether it can be readmitted to the financial markets.

• The government that has recently been current on its debt decides whether torepay or default this period.

5. Equilibrium allocations take place:

• If the government defaults, it is excluded from financial markets this period andsimply consumes its endowment, subject to a default penalty.

• If the government repays, it chooses the new allocation of bonds b′, while thelenders post the bond price q(b′, y, p).

3.4 Recursive formulation

In the following section I formalize the economic environment by stating the problems facedby market participants in recursive form. To begin, define the vector of aggregate statevariables that are common knowledge as s = (b, y, p).

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Government The government that is current on its debt obligations has the general valuefunction given by

v0(s) = maxd∈{0,1}

{(1− d)vr(s) + dvd(y, p)

}(5)

A sovereign who defaults (d = 1) is excluded from international credit markets and hasprobability θ of being readmitted every subsequent period with zero debt. The assumptionthat all debt is wiped out upon readmission is not necessary and can be relaxed at theexpense of complicating the analysis. The associated default value is

vd(y, p) =u(y − h(y)

)+ β

∑z∈{zL,zH}

∑z′∈{zL,zH}

Prob(z) π(z′|z)× (6)∫fz′(y′, y)

[θv0(0, y′, p′) + (1− θ)vd(y′, p′)

]dy′

subject to the law of motion for the belief

p′(y, p, y′) =

[p π(zH |zH) + (1− p) π(zH |zL)

]fzH (y′, y)∑

z′=zL,zH

[p π(z′|zH) + (1− p) π(z′|zL)

]fz′(y′, y)

(7)

In equation (6), h(·) is a reduced-form representation of the output cost of defaulting13;fz′(y′|y) denotes the probability density of transitioning from state y to state y′ given thattomorrow’s regime is z′. Prob(zH) is equal to p and Prob(zL) is 1−p, π(z′|z) is the probabil-ity of transitioning from regime z today to z′ tomorrow. The next period belief p′, describedin equation (7), depends on the current and future income realization, as well as the currentbelief p. It is a simple application of Bayes’ rule and takes into account a potential regimeswitch at the beginning of next period, according to the transition matrix given by (2).

The value of the government associated with repayment of debt is given by

vr(s) = maxc,b′

u(c) + β∑

z∈{zl,zH}

∑z′∈{zL,zH}

Prob(z)π(z′|z)∫fz′(y′, y)v0(s′) dy′

(8)

subject to the law of motion for the belief in formula (7) and

c = y − b(δ + (1− δ)κ

)+ q(b′, y, p)

(b′ − (1− δ)b

)(9)

where equation (9) is the budget constraint.

13Quantitative sovereign debt models typically assume an exogenous punishment in the case of default inorder to facilitate calibration of the model to the data. For the specific functional form, see Section 4.3.

16

Having characterized the two value functions of the government, it is straightforward toderive the optimal default policy as a function of today’s state variables

d(s) =

1, if vd(y, p) > vr(s)

0, if vd(y, p) ≤ vr(s)(10)

International Lenders Every period the lenders only observe (b, y) and share a commonmarket belief p. Although they do not see the current regime z, they know its distributionand independently update their belief about it, as described by the law of motion in formula(7). The denominator in those equations is always greater than zero and the resulting nextperiod belief p′ is strictly interior on the interval (0, 1).

As it is common in the quantitative models of sovereign debt, lenders are competitive andrisk-neutral by assumption. The resulting equilibrium bond price is such that they make zeroprofit in expectation (according to their imperfect information). The bond price function is

q (b′, y, p) = 11 + r∗

(∑z

∑z′Prob(z) π(z′|z)×∫

fz′(y′, y)(1− d(s′)

)[δ + (1− δ)

(κ+ q (g(s′), y′, p′)

)]dy′)

(11)

where s′ =(b′, y′, p′(y, p, y′)

), d(·) and g(·) are the government’s optimal decisions with re-

spect to default and new debt, respectively, and r∗ is the risk-free rate of interest.

Concluding this section, Definition 1 introduces the standard concept of a Markov PerfectBayesian Equilibrium. In this equilibrium the posterior beliefs of agents must be specifiedat all states and for all strategies of other players (including those involving off-equilibriumactions). The agents’ best responses must belong to the set of stationary Markov strategies.

Definition 1 A Markov Perfect Bayesian Equilibrium for this economy consists of the gov-ernment value functions vr(s), vd(y, p) and policy functions c(s), b′(s), d(s) and the bondprice schedule q(b′, y, p

)such that:

1. Policy function d solves the government’s default-repayment problem (5).

2. Policy functions {c, b′} solve the government’s consumption-saving problem in (8).

3. Bond price function q is such that the lenders make zero expected profit (subject to theirimperfect beliefs).

17

4 Quantitative analysis

In this section I calibrate the model to Portuguese data and discuss its mechanics. I thenpresent the simulated behavior of the model and use it to study the European debt crisis.

4.1 Data

I use the data for Portuguese economy as a case study for the theory developed in this pa-per. The model could also be calibrated to other European economies discussed in Section2. The Portuguese episode is the most clear-cut case however, because it does not coincidewith other major economic events such as a banking crisis (like the one that occurred inIreland) or Mario Draghi’s “whatever it takes” speech and the introduction of the OutrightMonetary Transactions (OMT) program in the summer of 2012, at the peak of the debtcrises in Italy and Spain. Portugal is also a particularly relevant laboratory for this typeof sovereign default model because it easily satisfies its core assumptions, i.e. it is arguablya small open economy and the vast majority of its debt securities were held externally.14

While in principle Greece is also a plausible candidate, the validity of its macroeconomicdata has been questioned. Nonetheless, Appendix D extends the main estimation technique,of the regime-switching income process with learning, to other countries mentioned in theintroduction and discusses the usefulness of this theory in explaining these cases.

Quarterly data for real GDP are taken from OECD and cover the period 1960:Q1-2016:Q4.Consumption, current account series and interest rates on long-term government bonds, alsofrom OECD, span the time frame 1995:Q1-2014:Q4. Government debt data is acquired fromWorld Bank’s Quarterly External Debt Statistics (I use the real debt securities only).

The identification strategy for the model’s parameters is in line with the general approachin the literature. In what follows, I first calibrate the parameters of the income process(1) separately from the core model, by targeting the path of long-run GDP forecasts fromthe data. Then, I pick the remaining structural parameters of the model partly from theliterature and partly to match certain very general characteristics of the Portuguese economy.

4.2 Calibration of the income process

I start by calibrating the endowment process introduced in (1). Pinning down the parametersof this equation is conceptually the most significant element of the present paper, as it deter-

14See Andritzky (2012).

18

mines the speed with which markets learn about an underlying regime switch. Recall thatthe interpretation of the low regime assumed in this paper is that of a rare disaster. However,because we do not have enough historical data to account for such episodes (the previous onebeing the Great Depression), it is not appropriate to estimate a Markov-switching AR(1)process based on the current national accounts data. The result of such estimation will notbe capable of matching the size and frequency of large depressions, such as the one we haveobserved for European economies since 2008. Instead, the obtained low regime will representone of the regular recessions Europe has experienced in the recent decades.

I proceed to calibrate equation (1) in three steps. First, I fix the probabilities of switch-ing between regimes based on the historical experience. By all accounts, the recession insouthern European countries has been the worst since the Great Depression.15 This gives usroughly 60 years or 240 quarters of high regime duration.16 Conversely, the Great Depressionlasted for about 10 years, or 40 quarters, which I use to pin down the expected low regimeduration. The resulting probabilities of staying in the high and low regimes are therefore0.996 and 0.975, respectively. The exact numbers behind of these probabilities are not crucialfor the results because, given their predetermined values, I use another data source below todiscipline the level of the depression regime. What matters however is to capture the rightorder of magnitude - intuitively, the low regime ought to be rare and severe enough so thatit clearly stands out from a regular economic contraction.17

As a second step, I use historical GDP data to estimate the persistence and variance param-eters of the AR(1) process. The unconditional mean of the high regime µH is normalizedto zero, like in most previous studies. Then, for any given choice of the low regime meanµL (think of it as a generic step in the estimation algorithm), I use Maximum Likelihood toestimate ρ and η in (1) on GDP data from 1960:Q1 to 2016:Q4, assuming that the regimeswitches from high to low in 2008:Q3.18 The data is in constant prices and the linear trend

15While Portugal was not affected that much by the Great Depression itself, it subsequently sufferedduring the 1934-1936 civil war in Spain, with the peak-to-trough decline in real GDP of 12.5%. Followingthe 1930s, the 2008-2014 episode is by far the most severe contraction in Portuguese economic history. Itcomes close to satisfying the defining criteria of a great depression established by Kehoe and Prescott (2002).

16I ignore the second world war in my calculations as the model is not designed to account for such events.17This type of approach to calibration is common for models with disaster risks. For example, pooling

60 episodes in 35 countries Barro (2006) sets the probability of entering a disaster event at 1.7% annually,almost the same as the number I use. More recently, Coibion, Gorodnichenko and Wieland (2012) takea similar approach to approximate the frequency of interest rates in the US hitting the zero lower bound,which occurred for the first time since the second world war.

18While the precise beginning of the crisis is not indisputable, it is commonly associated with thebankruptcy of Lehman Brothers in September 2008. Around that time we can also observe the beginning ofa slump in GDP series for most European countries.

19

is removed.19 The resulting estimates for the persistence and variance of output (at theoptimal choice of µL, to be discussed below) are 0.96 and 0.01, respectively.

Finally, it remains to pin down the unconditional mean of the low regime (i.e. the size ofthe disaster). While it is impossible to find the “true” value of this parameter, what mattersis that it represents the markets’ expectation about the depth of the ensuing depression inyears 2008-2014. To extract this information, I use the historical GDP forecast data pub-lished by the agencies listed in Table 2. As discussed in section 2.3, the forecasts are releasedtwice a year (spring and fall) and consist of the projected GDP growth rates for up to fiveyears ahead.20 I use the posted growth rates to calculate the projected 5-year-ahead GDPlevels, take logarithms and remove the previously estimated trend (so that the forecasts aredirectly comparable to the model’s units).

For every given set of parameters of the output equation I generate a corresponding pathof 5-year-ahead forecasts made by the agents in my model.21 I first feed in the actual GDPobservations for Portugal between 2000 and 2014 and create a sequence of beliefs that resultfrom Bayesian updating. Then, for every first and third quarter between 2008 and 2014(corresponding to the spring and fall releases of the actual projections which are typicallypublished in May and November, respectively) I generate a sequence of model-implied annualforecasts, for three years ahead. I finally pick the value of low regime mean µL to minimizethe sum of deviations between the forecasts in the data and those implied by the model.

Figure 4 depicts the result of matching model-generated forecasts to the historical ones for5 years ahead. As can be noticed, the match is very close initially for years 2008-2009, andthen for 2012-2014. For the middle two years of the recession, 2010 and 2011, the two setsof forecasts diverge in that agents in the model are too optimistic about the prospects ofrecovery. This divergence is reasonable however given the simple two-regime specificationof the output process. It can be implied from this forecast data that in 2010 and 2011

19Because of the growth trend shifts for European economies over this relatively long sample, I followBai and Perron (1998) to identify statistically significant structural breaks in the growth rate of Portugueseeconomy. Two breakpoints are detected, at 1974:Q2 and 1999:Q4, which is intuitive as they coincide withthe democratic revolution in Portugal and accession to the Euro zone, respectively. The estimated quarterlytrend growth rates for the three time windows are 1.6%, 0.8% and 0.4%. See Appendix C for more details.

20For forecasts up to two years ahead, I use the average projection across all agencies. For forecastsranging from two to five years ahead I use the IMF projection as this is the only agency that publishes them.

21The exercise aims to match the forecasts at longest-possible horizon. For a shorter horizon it is plausiblethat forecasters form their projections using outside information (which shows up as the ε term in my outputprocess specification (1)). As the forecast horizon extends, they must instead rely on long-run trend of theeconomy, described in my model by the parameters of interest, i.e. persistence and unconditional means.

20

Note: Each point on the graph represents an annual detrended log-GDP level for five years ahead(for example, 2008 : Q1 corresponds to the GDP level in 2013). The solid blue line representsthe actual published forecasts (Q1 and Q3 refer to the spring and fall issues, respectively), whilethe dashed red line denotes the ones generated by the calibrated model. The dashed-dot blackline shows the actual realized data that the corresponding forecasts refer to (only available until2013 : Q3, when the projection for year 2018 was made). Additionally, the dotted green lineplots the forecasts made using a simple AR1 specification with zero unconditional mean.

Figure 4: Historical GDP forecasts: data- and model-generated projections

the forecasters were very cautious about predicting a recovery, but also did not foresee thesubsequent slump. The model is not capable of replicating this because the only two regimesavailable are “normal times” or “rare disaster”, and in the longer horizon the economy mustconverge to one of them. This divergence is not a problem however, because what reallymatters for calibration of the model is to capture the magnitude of the depression perceivedby the markets. This goal is achieved by selecting µL = −0.28, which means that the marketsexpect the average income level in the depression regime to be roughly 24% below the trend.

21

While this is lower than where Portugal eventually landed, the exercise aims at inferringthe forecasters’ belief rather than estimating the “true” depression level of the Portuguesedata generating process. This is reflected by the fact that both data- and model-generatedprojections underpredict GDP starting from 2012, a common feature of forecasts followinga recession. Finally, Figure 4 also plots the forecasts generated with an estimated standardAR1 process. Because this process always converges to mean zero in the long run, such amodel is not capable of capturing the reversal of expectations during the European debtcrisis. Table 3 summarizes the calibrated parameters of the income process.

Transition Prob.

Regime Mean µ Persistence ρ St. dev. η Low HighLow −0.2774 0.9637 0.0103 0.975 0.025High 0.00 0.9637 0.0103 0.004 0.996

Table 3: Parameters of the regime-switching endowment process

It should also be emphasized that the estimated regime-switching model in Table 3 providesa much better fit to Portuguese data than a single-mean AR1 process. I will use two argu-ments to make this point. First and foremost, notice that AR1 implicitly assumes that theunderlying data is stationary around its long-run mean. This is the case for Portugal’s GDPonly until 2008:Q2, where the Dickey-Fuller (DF) test rejects a null hypothesis of unit rootwith p-value below 0.01. However, Table 4 shows that as we include the most recent data(until 2011 and 2014), this conclusion no longer holds and a unit root cannot be rejected atany level of significance. On the other hand, conducting a Markov-switching version of theDF test under the parametrization from Table 3 allows us to reject the unit root hypothesiswith 2.5% significance. The intuition behind this conclusion is straightforward in that thepost-2008 drift away from the zero mean is largely explained by the regime switch, ratherthan a random walk component. Second, the goodness-of-fit is significantly higher underregime switching. To show this, I estimate the detrended AR1 process for years 1960-2011to be 0.9467 and 0.0107, respectively.22 Then, the likelihood ratio test statistic is 22.5 andhas approximately a chi-squared distribution with three degrees of freedom (the number ofadditional parameters in the extended model), resulting in a p-value below 0.01. This suggestthat we can reject the null model (single-regime AR1) at virtually all levels of significance.

22Once again, I use a broken linear trend with two statistically significant breakpoints detected using theBai and Perron (1998) test at 1974:Q2 and 1999:Q4. Due to the non-stationarity issue described above, Ido not use the final three years of the sample, i.e. 2012-2014, because including them makes the estimate ofpersistence ρ indistinguishable from 1. See Appendix C for more details.

22

Model Period t-statistic p-value

Simple AR1 1960:Q1-2014:Q4 0.570 0.8381960:Q1-2011:Q4 −1.059 0.2631960:Q1-2008:Q2 −2.608 0.009

Markov switching 1960:Q1-2016:Q4 −2.717 0.025

Note: The Markov-switching Dickey-Fuller test follows Hall, Psaradakis and Sola (1999).The empirical distribution of the t-statistic is constructed by simulating equation (1) underthe null hypothesis of a unit root with a discrete switch in unconditional means in 2008:Q3.

Table 4: Dickey-Fuller test for different specifications and time periods

Concluding this section, it is natural to ask about the results of a more typical estimationapproach which uses a variant of the Expectation-Maximization algorithm, as described byHamilton (1990). This method identifies regimes in the data with Kalman filter and yieldsthe following estimates for the sample from 1960 to 2011: ρ = 0.96, η = 0.009, µH = 0.04,µL = −0.43, πHH = 0.958, πLL = 0.771. Notice that now the high regime mean is nolonger normalized to zero and takes a small positive value, while the low regime has theunconditional mean twice as large (in absolute value) as in Table 3. As switching is muchmore likely under this specification, and it does not last very long, we detect several instancesof the low regime over time, in particular around 1969, 1974, 1983, 1992, 2008-2009 and 2011-2014. However, because the bad regime is now so stark and relatively more frequent, thelearning process about an underlying switch turns out to be very fast and does not alignwith the evidence from forecast data presented in Table 2.

4.3 Functional forms and calibration

To select the remaining structural elements of the model I follow the general trends in theliterature in that I fix the value of some non-controversial parameters, and I use a moment-matching exercise to pin down the more problematic ones. A representative household’sutility is a CRRA function of the form u(c) = c1−σ

1−σ , with risk aversion parameter set at thestandard level of 2. The risk-free interest rate is set equal to 1% (quarterly value) and theprobability of re-entry after default is fixed at 0.049, following Cruces and Trebesch (2013)who find that the average time to re-enter the credit market was 5.1 years in 1970-2010.Using OECD data I find that the average maturity of Portugal’s debt in years 1996-2010was 4.73 years, which translates into an average quarterly maturity rate of 0.053. The couponpayment is set to 0.0125 following Salomao (2017), which implies an annual coupon of 5%.The output cost of default is parametrized as h(y) = min

{y, y

}following Arellano (2008).

23

The parameter y is calibrated jointly with the discount factor β using the simulated methodof moments. The economy’s income path in years 1998-2014 is simulated 10,000 times, start-ing from the actual GDP and debt levels observed in 1998:Q1 and under the assumption thatthe regime switches from High to Low in 2008:Q3.23 The idea behind identification strategyis to match certain very general characteristics of the Portuguese experience during that timeperiod. To this end, I use information from the following moments of Portugal’s economy inyears 1998-2014: average ratio of external debt securities to GDP of 37.9%;24 and average5-year bond spread of 1.88%.25 The amount of debt in the economy is naturally an importantpiece of information to identify relative impatience of the government and the punishmentfor default. On the other hand, average bond spread in a model with risk-neutral lenders andzero recovery rate simply reflects the government’s average default probability. The target of1.88% is thus reasonable given all independent evidence on Portuguese historical default.26.Table 5 presents a summary of the parameter values that provide the closest match to theempirical moments. The model achieves an exact match in terms of both targeted moments.The calibration procedure results in a discount factor β of 0.987 and the default penaltyparameter y of 0.81.

Finally, in order to make a meaningful comparison with a literature benchmark, I also cal-ibrate a “standard” sovereign debt model with long-term debt, similar to Chatterjee andEyigungor (2012) or Hatchondo and Martinez (2009), which uses a simple AR1 specificationof the income process. As mentioned before, the estimated persistence and variance param-eters are 0.9467 and 0.0107, respectively. Calibration of the structural parameters closelyfollows the strategy described above and is summarized in the AR1 column of Table 5. It isimportant to emphasize that this benchmark is not meant to claim that no model based onan AR1 process can produce relevant predictions for the European debt crisis. Indeed, sev-eral papers such as Bocola and Dovis (2019) or Salomao (2017) have managed to do so. Thebenchmark thus refers to an “off-the-shelf” model following a routine calibration approach.

23An alternative would be to simulate the economy over many years and calibrate to its ergodic (long-run)business cycle statistics outside of default, following Chatterjee and Eyigungor (2012). However, as it willbecome clear from Table 6 describing the simulation results, years 1998-2014 were a highly non-stationaryperiod for the European economies, including slow accumulation of debt towards a steady state level.

24Because the model does not include post-default renegotiation, I follow Chatterjee and Eyigungor (2012)to calibrate only the true “unsecured” portion of the debt. While Portugal in the end did not default and itis difficult to know how much of its debt was in fact unsecured, the best guess is 0.535, the haircut rate inthe case of Greek default of 2012.

25I use a 5-year spread, rather than 10-year as in the introduction, because Portugal’s average debtmaturity is 4.73 years. The data on 5-year spread is acquired from Bloomberg.

26Reinhart and Rogoff (2009) identify four sovereign defaults in Portugal’s history since 1800, whileStandard & Poor’s (2014) identify three, implying an annual long-run probability of 1.5− 2%

24

An important conclusion from the moment-matching exercise is that it produces significantlydifferent values of the discount factor in the model with disaster risk relative to literaturebenchmark. In particular, in the “standard” model with a simple AR1 process, a value ofβ below 0.98 is needed to simultaneously generate high debt and defaults occurring with adesired frequency. On the other hand, in the model with disasters the same statistics areachieved with a high discount rate of 0.987. This is because defaults here occur predom-inantly under the circumstances of a rare disaster, rather than due to a myopic behaviorof the borrower. As a result, the model features a government that mostly uses debt forconsumption-smoothing purposes, but may occasionally default should the output collapsein a great-depression-like fashion. The following two sections illustrate the behavior of thegovernment in this model in more detail.

Symbol Meaning Learning AR1 Source

σ Risk aversion 2 2 Literaturer∗ Risk-free rate 0.01 0.01 Literatureθ Re-entry probability 0.049 0.049 Literatureδ Probability of maturing 0.053 0.053 Dataκ Coupon payment 0.0125 0.0125 Datay Default cost par. 0.810 0.922 Calibrationβ Discount factor 0.987 0.979 CalibrationCalibration targets Learning AR1 Data

E (debt/GDP) 37.9 37.9 37.9E (spread) 1.88 1.88 1.88

Note: targeted moments are given in percentage points. Simulations are repeated 10,000times for a period of 1998-2014.

Table 5: Calibration of structural parameters of the model

4.4 Characterization of the equilibrium

In the following section I first characterize some of the key properties of the equilibrium,and then show how the model’s simulated behavior compares with actual data. The modelis solved numerically by value function iteration using a continuous choice of next perioddebt and cubic spline interpolation to evaluate off-grid points, similarly as described inHatchondo, Martinez and Sapriza (2010). Expectations are approximated using Gaussianquadrature with 31 nodes and off-grid points for income and beliefs are linearly interpolation.I use 31 points for the grids of debt and income, and 21 points for the grid of beliefs.

25

4.4.1 Model mechanics

To understand how the model works, it is instructive to examine how the government’s op-timal decisions change with respect to state variables. Figure 5 shows the default and debtpolicies for different levels of prior belief. On the left-hand side panel, any combination ofcurrent debt and income above the line corresponding to some belief p indicates repayment,and below the line indicates default. Not surprisingly, higher belief about being in the goodregime induces the government to default in a smaller number of states. This relationshipis strictly monotonic in the level of prior belief (but not necessarily linear). The right-handside panel of Figure 5 shows that higher prior beliefs induce the government to borrow more.In this model, agents are impatient and would rather consume today than tomorrow. Whenmaking their debt decisions though, they need to weigh their impatience against the ex-pected income level in the future. A higher chance of being in economic depression nextperiod implies that the government must restrict its consumption today and reduce foreigndebt in order to decrease the probability of defaulting tomorrow and to secure a high bondprice today. Consequently, higher market belief has a strictly monotonic, increasing effecton the optimal debt level.

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.5 1 1.5 2 2.5

log

inco

me

current debt

Default decision

p=0.0p=0.2p=0.4p=0.6p=0.8p=1.0

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

-0.25 -0.15 -0.05 0.05 0.15

log income

Next period debt

p=0.0p=0.2p=0.4p=0.6p=0.8p=1.0

Figure 5: Default sets and bond price policy functions for different beliefs

26

Notice furthermore that policy functions are generally decreasing in income which impliesa consumption smoothing behavior during normal times. It is only when income falls lowenough, to a great-depression-like level, that the policy functions bend over and take a morecommon for this class of models, increasing shape, resulting in a procyclical fiscal policy.

Figure 6 plots government bond prices as functions of the next period debt choice, at severaldifferent levels of belief.27 The information about current regime is important in determiningfuture default risk and leads to large differences in the offered bond prices. The highest (red)line represents the bond price schedule when markets are fully convinced the economy is inthe high regime. As a result, the government is able to secure an almost maximum pricefor its bonds, regardless of its choice of next period debt (within reasonable bounds). Bycontrast, the lowest (black) line represents the schedule if the markets believe the economyis currently in a depression. Because default risk is much higher in such circumstances, thegovernment is offered very low prices for its debt. Finally, the schedules in the mid-rangeare increasing monotonically as the belief of being in the high regime improves.

0

0.2

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2

next period debt

Bond price function

p=0.0p=0.2p=0.4p=0.6p=0.8p=1.0

Figure 6: Bond price as function of next period debt for different beliefs

27This graph does not say anything about optimality of different debt choices, it only depicts the possibleprice schedules.

27

4.4.2 Business cycle statistics

In the next step, I analyze the model’s behavior in simulations. As discussed in Section 4.3,the model exhibits widely different behavior in the long run versus the short sample thataims at mimicking the period of 1998-2014 for the European economies.

Table 6 presents the simulated business cycle moments in the long- and short-run samples(i.e. ergodic and conditional distribution), along with the empirical ones. It can be noticedthat the model simulated in the short run performs closer to actual data in terms of correla-tions between the main variables, and the moments of bond spread and debt. In particular,note that the average debt-to-GDP ratio for Portugal in the long run implied by the modelis close to 62%, much higher than in the data for 1998-2014. This is due to the fact thatthe government is gradually accumulating debt towards its steady-state level for most ofthe 2000s. Unlike the ergodic distribution, short-run simulations are able to capture thisnon-stationarity. Related to the policy functions discussed in Figure 5, notice that the gov-ernment’s ergodic behavior implies a counter-cyclical fiscal policy, with a positive correlationof income and trade balance and a low volatility of consumption relative to output. Thesefeatures do not show up in the short-run statistics, however, due to the disproportionatepresence of a rare disaster which makes government behave like a classic sovereign defaulter.

Statistic Data Learning AR1Ergodic Conditional Ergodic Conditional

E(s) 1.88 1.56 1.88 2.18 1.88std(s) 3.46 5.56 4.89 1.92 1.33std(c)/std(y) 0.94 0.96 1.38 1.27 1.60std(tb)/std(y) 0.53 0.26 0.57 0.41 1.06corr(y, c) 0.97 0.96 0.94 0.96 0.78corr(y, tb) −0.94 0.29 −0.47 −0.50 −0.19corr(y, s) −0.72 −0.29 −0.77 −0.68 −0.65corr(s, tb) 0.72 0.38 0.71 0.73 0.69E(debt/y) 37.90 61.70 37.90 44.18 37.90

Note: moments for the bond spread, debt-to-GDP ratio and the long-run default probability (annual)are given in percentage points. Ergodic (long-run) simulations extend to 10,000 quarters and arerepeated 10,000 times, following closely Chatterjee and Eyigungor (2012). Conditional (short-run)simulations mimic the period of 1998-2014 (68 quarters) and are repeated 10,000 times starting fromthe actual levels of debt and GDP observed in 1998:Q1. Each short-run sample is constructed suchthat: i) the series start from the actual 1998:Q1 debt and income levels, and ii) the regime switchesfrom good to bad in 2008:Q3. Consumption data is detrended using the common GDP trend.

Table 6: Simulated behavior of the model

28

4.4.3 Simulated behavior of spreads

Table 7 highlights the most notable difference between the two models. The first two mo-ments of the simulated bond spreads are presented for each model, and contrasted with thedata. The average spread is a targeted moment, so it is matched in both cases exactly.However, the standard deviation of the spread is not a calibration target, and for the AR1model it falls short of the level observed in the data, producing a coefficient of variation ofthe spread smaller than 1 (reported in the last column). This result confirms the finding ofAguiar et al. (2016) who show that models of this type typically fail to deliver a realisticvolatility of the bond spread. By contrast, the model with disaster risk and learning gener-ates a standard deviation which actually exceeds what we observe in the data, resulting in acoefficient of variation much above 1. Mirroring this result is the fact that the trade balanceis much less volatile in the disasters model than in the AR1 model, as evident in Table 6. Byadjusting its trade balance more aggressively, the borrower is able to target a desired levelof bond spread with smaller variance.

Bond spreads (in %)

Country µ σ cv

Model-AR1 1.88 1.33 0.71Model-learning 1.88 4.89 2.58Data-Portugal 1.88 3.46 1.84

Table 7: Bond spread moments in the simulations and the data

The intuition behind the result presented in Table 7 is the following. In the standard AR1model with long-term debt a sovereign default is possible essentially always, within the ex-pected lifetime of an outstanding bond. Consequently, the spread never falls to 0, althoughit may on average be much smaller than the ones obtained for emerging market economiesas shown in Chatterjee and Eyigungor (2012) and other studies. On the other hand, inthe model with disasters and learning, sovereign defaults occur almost exclusively when theeconomy has switched to the disaster regime.28 Hence, most of the time while the economyis doing well and the market-wide belief is close 1, lenders do not fear that default is apossibility in any predictable future. As a result, bonds spreads are very close to zero, andthe average spread is low even if a debt crisis eventually does occur. Once that happens, thespread shoots up and can attain very high values, which I explain in the following paragraphs.

28In the long-run simulations summarized in Table 6, over 91% of all defaults occur in the low regime,and the average value of the belief at default is 0.088.

29

Table 8 generalizes this point by documenting the difference in bonds spread moments ofother peripheral European countries discussed in Section 2 and emerging market defaulters(which are the examples originally discussed by Arellano (2008)). As can be noticed, theformer tend to have a coefficient of variation of the bond spread above 1, implying thataverage spread is low relative to its volatility. On the other hand, emerging market defaulterstend to have average spreads that exceed their standard deviations significantly, resulting ina coefficient of variation smaller than 1. Consequently, as Table 7 shows, the benchmark AR1model seems to be a better description of the debt crisis experienced by an emerging marketeconomy, while the model with disaster risk developed in this paper is a better descriptionof the recent episode experienced by developed European nations. A natural caveat relatedto the interpretation of Table 8 is that emerging market economies are additionally affectedby recurring high rates of inflation, affecting the average levels of nominal spreads. Whilethis factor has not been present for the southern European economies since 1999, they tooexperienced this problem which resulted in much higher average spreads. Nevertheless, recentalternative studies such as Lo Duca et al. (2017) show that even before 1999, a dominantcomponent of the European spreads was the inflation risk, while external default risk wasrather negligible.

Bond spreads (in %)

European µ σ cvGreece 3.90 5.94 1.52Spain 0.98 1.33 1.36Italy 1.05 1.21 1.15Emerging µ σ cvArgentina 10.25 5.58 0.54Ecuador 16.91 10.72 0.63Russia 19.41 17.60 0.91

Note: bond spread moments for European countries are computedwith OECD data covering 1999:Q1-2014:Q4, while the momentsfor emerging economies are taken from Arellano (2008).

Table 8: Bond spread statistics for European vs. emerging market economies

Another interesting feature of the model is that bond spreads on equilibrium path can takemuch higher values than in the benchmark, i.e. the government sometimes sells bonds atdeep discounts. To illustrate this point, I compute the average maximum realized spreadin each simulated path of 10,000 periods. In the model with disasters and learning, thisamounts to around 125%, while in the benchmark model based on a simple AR1 process it

30

is 22%. The intuition behind this result is straightforward. In the latter, the governmenttargets a certain level of spread and only tolerates limited upward deviations. If spreadbecomes too high, it must be the case that income is low enough and default becomes amore attractive option. In the former however, a sudden fall in the belief may cause adownward shift of the entire bond price schedule, as shown in Figure 6. Spreads may shootup then while income remains relatively high, making default unattractive because of thenon-linear punishment function. As a result, the government sells bonds at steep discountsuntil one of two outcomes happen: either income falls enough to make default attractive, ordebt is reduced enough and spreads return to a desirable level.

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Figure 7: Histogram of spreads in simulated models

Figure 7 reinforces this point by comparing distributions of the spreads realized on equilib-rium path in the simulations. In the model based on a simple AR1, spreads do not carryany mass for values above 0.2, which yields the skewness of 4.2 and kurtosis of 28.5. Bycontrast, the distribution of spreads in the model with disasters and learning features a longupper tail extending all the way to 0.9 and beyond. As a result, the distribution is muchmore right-skewed (11.4) and leptokurtic (173.1). The ability of the model to generate highvalues of realized bond spread is akin to the result of Aguiar et al. (2019) who propose anenhanced theory of rollover crises to rationalize the existence of “desperate deals”. Crucially,here I show that a similar behavior can be obtained with fundamental factors.

4.5 Event analysis of the European debt crisis

In this section, I use the calibrated model to conduct an event study of the debt crisis inPortugal. I start with the benchmark AR1 model, and then move on to the predictions ofthe main model. I conclude by showing a counterfactual where agents have full information.

31

4.5.1 Benchmark case - AR1

I start by feeding the actual detrended GDP observations for Portugal into the benchmarkAR1 version of the model. Figure 8 presents the predicted evolution of debt-to-GDP ratioand the bond spread in that model and the data (I use the same time series that are depictedin Figures 1 and 2). The economy is launched in the first quarter of 1998 with the actualdebt and GDP levels from the data. I use the model-implied decision rules and bond pricesin response to the realized path of income shocks.

The figure highlights all the problems with the benchmark AR1 model that have been an-alyzed in the preceding sections. The government accumulates debt too fast, at least untilyear 2003, due to the fact that the model requires a low value of the discount factor in orderto hit the targets of average debt and average spread. Throughout this time, the bond spreadis strictly positive at around 1% and actively responds to current income shocks, includingthe minor recession in 2002-2003. When the Great Recession starts, the spread jumps upto almost 7% in 2009:Q1 due to the extreme negative shock that hits Portuguese GDP. Thegovernment reduces its debt sharply, which causes the spread to fall back. These predictionsare clearly at odds with the data - the bond spread was virtually zero up until 2009, and debtaccumulation was slower. In 2009 the 5-year bond spread temporarily rose to 1.2%, whilethe government actually increased its external debt-GDP ratio. Finally, the model-predictedspread rises again and predicts a default in the last quarter of 2011, unable to replicate theobserved peak level of 16% (the highest it can get is 11.3% in 2011:Q3).

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Figure 8: Event analysis in the model based on a standard AR1 process

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4.5.2 Model with disaster risk and learning

Now I conduct an event analysis in the model with rare disasters and learning about theirrealizations. The upper panel of Figure 9 presents the predicted evolution of debt-to-GDPratio and bond spread for the actual path of Portuguese GDP realizations. The economy isonce again started in 1998 with the actual levels of debt and GDP. The lower panel tracksthe evolution of agents’ belief about the economy type. The model offers a sharp improve-ment on all fronts of the analysis relative to the benchmark in Figure 8. The pace of debtaccumulation prior to 2008 is slower and consistent with the data due to the fact that weare able to use a much higher discount factor. At the same time, the predicted bond spreadis essentially zero due to the fact that agents have a near-certainty that the economy isoperating in the high regime in which a default is almost impossible. When the negativeshock of 2009:Q1 hits, the belief drops on impact, but only partially, leading the spread toincrease to 3% while debt is reduced on impact but then immediately picks up and stayson par with the data until the end of 2010. Finally, in 2011-2012 a new wave of low GDPshocks hits the economy which causes the belief to plunge. This contributes to an explosionof the bond spread which is predicted to reach 26% in 2011:Q4 and 33% in 2012:Q1 followed

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33

by a default. While in reality the Portuguese spread did not go that high, its upward pathwas halted by the successful bailout provided by the IMF and the European Commission.By contrast, the Greek 10-year bond spread reached 29% in February 2012 which shows thatsuch high values of the spread are feasible in equilibrium, and this paper shows that theycan be rationalized by a model based on fundamental factors.

Naturally, the model still falls short of predicting the data in a few ways. The 2009:Q1 bondspread increase is mostly attenuated but still higher than what it actually was, 1.22%. Thegovernment is also predicted to reduce its debt, due to a higher likelihood of the underlyingregime switch, while in reality most European countries actively increased their debt duringthat time in an attempt to provide an economic stimulus. Finally, the model still predictsa sovereign default to occur in 2012 which is not unreasonable, given that Portugal receiveda bailout from the European Commission and the IMF covering over 40% of its GDP (seeSection 5.1). Some of these shortcomings could potentially be fixed by introducing a lender oflast resort to this framework. How to model the European Commission in this role howeveris still a subject of active debate, and thus I leave this extension for future research.

4.6 Counterfactual exercise with full information

In this section, I show that learning is crucial for the model to generate sensible predictionsabout the actual debt crisis in Europe. Figure 10 presents the event study with the variant ofthe model in which agents have complete information about the underlying regime switches.As in the previous case, debt increases gradually prior to the Great Recession, while spreadsremain at zero. In 2008:Q3, upon learning about the regime switch, spreads shoot up toover 6% while the government embarks on a drastic debt reduction path. Interestingly,even though the debt is much lower in the second stage of the European crisis relative to thepredictions in Figure 8 and Figure 9, the spread still explodes in 2012 resulting in a sovereigndefault. This is because agents are aware that the economy is on a downward trajectory andwill likely reach a default stage before the outstanding bonds have matured.

5 Discussion

In this section, I discuss a common alternative explanation to the story proposed in thispaper, namely the role of international bailouts during the debt crisis. I also provide anexplanation for why the model presented here goes a long way in explaining the Europeanepisode, but does not necessarily help with the emerging market debt crises.

34

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Figure 10: Event analysis in the model with full information about the disaster regime

5.1 Bailouts and the European debt crisis

The impact of emergency loans on sovereign debt crisis has long been recognized by theliterature. In years 2008-2014 this factor has been particularly pronounced, with a total ofeight European nations getting official bailouts sponsored by the European Commission andthe International Monetary Fund. The scale of these loans was historically unprecedented,amounting to roughly 40% of GDP for Ireland and Portugal, and almost 100% in the caseof Greece, and with maturities extending up to 30 years into the future.29 The effect of suchinterventions cannot be disregarded - in their analysis of Argentina Aguiar and Gopinath(2006) show that incorporating the IMF loan leads to over 60 percent lower standard devi-ation of the interest rate than in the benchmark case. The same channel may explain whythe model-predicted volatility of the bond spread exceeds the one in the data, as in Table 7.Apart from impacting the levels of country risk, it is interesting to consider whether the EUpolitics over bailouts could account for the dynamics of the European crisis. While quanti-

29Detailed information about the bailout programs can be found at ec.europa.eu/info/business-economy-euro/economic-and-fiscal-policy-coordination/eu-financial-assistance.

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tative work on this topic is scarce, a narrative along these lines is popular (Brunnermeier,James and Landau (2016)). In that view, the European crisis began with the Deauvillemeeting between German chancellor Angela Merkel and French President Nicolas Sarkozyin October 2010, where they agreed on the “private sector participation” (i.e. allowed forhaircuts on private bond holdings). While relevant, this hypothesis also raises three doubts,best visualized in Figure 11 which presents the plot of bond spreads at monthly frequencyfor Greece, Ireland and Portugal along with the announcements of their emergency packagesmarked with vertical lines. First, the movements in bond spreads in early 2009 indicates thatthe debts of EU countries were not perceived as fully guaranteed, despite the fact that twoother member states (Hungary and Latvia) had been successfully bailed out in 2008. Second,the bond spreads for all countries had already been on the rise prior to the Deauville meeting(which occurred just before the Irish bailout), surpassing their levels from early 2009, andtheir slope did not change significantly following it. Third, the bond spreads continue to risein the aftermath of the bailout announcements at individual rates for each country, reachinga peak at separate points in time in the course of a year (starting with Ireland in June 2011,followed by Portugal in January 2012, Greece in March 2012 and Italy in June 2012). If thesudden change in the expectations about the EU policy was the main driver of default risk,it would have been an aggregate shock, affecting all members simultaneously.

Note: Bailouts are marked for the quarter when the official announcement of an emergency loanwas first made. The Deauville meeting occurred just before the Irish bailout (green vertical line).

Figure 11: Bond spreads and announcement of bailouts during the European debt crisis

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5.2 Learning about debt crises in emerging market economies

Aguiar and Gopinath (2006) show that emerging market bond spreads are better capturedby a model with permanent growth shocks to income, rather than transitory ones. It isnatural to ask how the model developed in this paper can be applied to the previous debtcrises. Figure 12 plots the evolution of GDP forecasts for Argentina, a representative emerg-ing market economy, around the time of its crash in 2001.30 Two differences stand out incomparison with Figure 3 which plots the analogous data for European countries. First, eventhough Argentina’s contraction is equally steep and much deeper than the one of Portugalor Italy, a swift recovery follows. Unlike the European economies, Argentina returns to itspeak output level of 1998 within six years. Second, forecasts for Argentina have an almostinvariant slope over time, regardless of whether the economy is currently in a boom or abust. As a result, large forecast errors arise most of the time, overestimating future GDPduring a recession and underestimating it during a recovery. This suggests that forecastershave noisy information about Argentina’s economy and form their projections based on theaverage long-run trend growth. As a result, while adding a regime-switching process with

Note: The GDP series is annual and in constant prices; values are normalized so that itequals 100 in 1998. The red dotted lines represent one- and two-year ahead forecasts publishedin the fall of each year by IMF. The shaded area marks Argentina’s debt crisis of 1998-2002.

Figure 12: Forecast and actual real GDP for Argentina

30I use IMF projections as it is the only source of forecast data out of the four I discuss in Section 2.3that contains Argentina.

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learning about its realizations to a model of emerging market debt crises is a possibility,there is relatively little information to be inferred from historical forecast data.

6 Conclusion

In their seminal contribution, Lucas and Sargent (1979) make the following remark aboutgeneral equilibrium macroeconomic models:

It has been only a matter of analytical convenience and not of necessity that equilibriummodels have used the assumption of stochastically stationary shocks and the assumptionthat agents have already learned the probability distributions they face. Both of theseassumptions can be abandoned, albeit at a cost in terms of the simplicity of the model.

This paper shows that learning about the probability distributions of future income shockswas an important driver of the European debt crisis. It impacted not only the movements inasset prices, but also the real variables such as government debt. I show that an otherwisestandard quantitative model of sovereign debt can be augmented to incorporate this learningprocess and match the markets’ gradually evolving beliefs over time. As a result, we obtaina delayed pattern of bond spread increases during the Great Recession in Europe.

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Appendix (for online publication)

A Further evidence on forecast errors

Table 9 presents the average bias in the GDP forecasts for the European countries. Thenegative signs in years 2008-2011 confirm that the increase in RMSEs was caused by over-estimating GDP growth, uniformly across all countries and all forecasters.

Average bias OECD IMF EC CE

(a) Pre-recession sample: 2000-2007Greece −0.37 0.045 −0.35 0.00Spain 0.22 0.14 0.19 0.28Italy −0.94 −1.15 −1.01 −0.94Portugal −1.00 −0.96 −0.78 −0.99(b) Recession - first stage: 2008-2011Greece −6.34 −6.66 −6.72 −6.54Spain −2.21 −2.44 −2.34 −2.36Italy −2.24 −2.03 −2.27 −2.10Portugal −1.47 −1.70 −1.26 −1.68(c) Recession - second stage: 2012-2014Greece −0.39 −1.27 −1.24 −0.35Spain 0.61 0.53 0.36 0.59Italy −0.49 −0.65 −0.80 −0.59Portugal 0.32 −0.16 −0.12 0.50

Note: The table presents average errors of one-year-ahead forecasts of real GDP level. Forecasts are ac-quired from four sources: OECD, IMF, European Commission, and Consensus Economics Inc. The biasis expressed in percentage of the 2010 level of real GDP for each of the four countries. A positive valuefor the bias indicates that forecast underestimate the actual values, while a negative bias indicates that theforecasts overestimate them. All forecasts come in two vintages, Spring and Fall, which I use jointly. Thenumber of forecasters participating in Consensus Economics surveys varies over time and across countries,with a minimum of four and a maximum of twenty in the entire sample.

Table 9: Average bias in real-time historical forecasts for different time frames

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B Dynamics of sovereign ratings over time

Figure 13 presents plots of the countries’ real GDP, together with the “sovereign bond ratingindex”.31 Even though the economies entered a recession as early as in the second quarterof 2008, markets continued to perceive their bonds as relatively risk-free investment untilabout two years later. As a result, only around 2010-2011 do we observe a sequence ofsovereign rating downgrades among peripheral European countries, indicating that marketexpectations about the sustainability of governments’ debt had deteriorated significantly.

Note: The GDP series are in constant 2010 prices, and their values are normalized such that the thirdquarter of 2008 equals 100 (beginning of the financial crisis - shaded area). The bond rating index isconstructed by converting the sovereign ratings of the three leading agencies - S&P, Moody’s and Fitch- into a numerical scale from 0 to 25 and computing a simple average.

Figure 13: Real GDP and sovereign bond rating index of the European economies: 2000-2014

31The index is a simple weighted average of the three leading rating agencies (S&P, Moody’s and Fitch),converted into a numerical scale from 0 to 25.

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C Detrending method

Figure 14 illustrates the detrending using a broken linear trend. For the baseline model,the trend is calculated until 2008:Q2 (because subsequently a presumed regime shift occurs)and for calibration of the simple AR1, the trend includes data until 2011:Q4. In each case,the two statistically significant breakpoints are detected using the Bai-Perron test (Bai andPerron (1998)) and continuity of the trendline is imposed. In each case, the two breakpointsare detected at 1974:Q2 and 1999:Q4, coinciding with the democratic revolution in Portugaland accession to the Euro zone, respectively. The estimated quarterly trend growth ratesfor the three time windows are 1.6%, 0.8% and 0.4% for the baseline case, and 1.6%, 0.8%and 0.3% for the simple AR1 case.

Detrending the data using a broken linear trend is not necessary for the results, but it doesallow one to use a longer time series in the estimation, going back to 1960. This would notbe possible with a single linear trend as the resulting residual would not be stationary.

(a) Baseline case (b) Simple AR1 case

Figure 14: Detrending the GDP using a broken linear trend

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D Estimation for other countries

In this appendix I present the results of the forecast-matching calibration strategy as de-scribed in Section 4.2 for all four southern European countries. The purpose is to show howapplicable the model is to other cases which jointly motivate the paper. For each country, Imaintain the assumption that the expected duration for the high regime and low regime is60 years and 10 years, respectively. Then, I pick the low regime mean to match the data-and model-generated paths of 5-year-ahead forecasts, estimating the persistence and vari-ance parameters for each choice using Maximum Likelihood for the sample 1960:Q1-2016:Q4.Figure 15 presents the results of this matching for all four countries, along with the pathsof realized data 5 years later, while Table 10 summarizes the parameter values. All panels

Note: Each point on the graph represents an annual detrended log-GDP level for five yearsahead (for example, 2008 : Q1 corresponds to the GDP level in 2013). The solid blue linerepresents the actual published forecasts (Q1 and Q3 refer to the spring and fall issues,respectively), while the dashed red line denotes the ones generated by the calibrated model.The dashed-dot black line shows the actual realized data that the corresponding forecasts referto (only available until 2013 : Q3, when the projection for year 2018 was made).

Figure 15: IMF- and model-generated projections for all countries

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confirm that the forecasts can be matched well and that there was a clear learning processfor each country. The estimated disaster regime means range from 24% below trend forPortugal to 41% for Greece.

Regime Mean µ Persistence ρ St. dev. ηGreece −0.8930 0.9745 0.0187Spain −0.4637 0.9703 0.0092Italy −0.3085 0.9658 0.0089Portugal −0.2774 0.9637 0.0103

Note: parameters are estimated independently for each country following the procedure described inSection 4.2. In each case, the regime-switching probabilities are the same as in Table 3. The GDPdata for each country is detrended in the same fashion as it is described for Portugal in Appendix C.

Table 10: Calibrated parameters of the regime-switching endowment process for all countries

Figure 16 presents the inferred paths of the belief about being in the disaster regime for eachcountry. The lower panels show the entire time period from 1960 to 2016, while the upperpanels focus on the most recent episode of interest, 2000-2016. The figures confirm thatfluctuations in the belief are generally rare and revert instantly for any time period beforethe Great Recession. The analysis also indicates that the model is likely to be applicable toGreece, in addition to Portugal. For both of these countries the belief about being in the highregime drops part ways on impact in 2009:Q1, and subsequently recovers before collapsingall the way to zero. By contrast, for Italy and Spain the belief drops to zero instantly in2009:Q1 which leads the agents in the model to underpredict future GDP level in that timeperiod (Figure 15). In other words, to minimize the sum of squared distances between thetwo paths of forecasts, agents in the model become overly pessimistic at the outset of theGreat Recession. This does not mean however that the theory of gradual learning is not ap-plicable to Italy or Spain, but rather that the model and the proposed calibration techniqueis not able to capture the slow learning process. One way to overcome this problem wouldbe to augment the income process with a third regime. In this scenario, the first two regimeswould have a standard expansion/recession interpretation, with frequent transitions betweenthem, while the third regime would be a rare disaster. The “regular recession” regime wouldthen help matching the forecasts in 2009:Q1 without an instant switch to the depression one.

It is also worth reiterating, as Section 4.1 explains, that Spain and Italy are not necessarilythe best countries to explain using the model in this paper. The reason is that both hadlarge stocks of domestic, rather than external, debt and the height of their crisis coincided

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with the unprecedented actions of the European Central Banks in the summer of 2012. Thismakes explanations based on rollover crises, such as in Bocola and Dovis (2019) or Aguiar etal. (2019), or domestic default as in Bocola, Bornstein and Dovis (2019) a more promisingavenue for rationalizing the events in these countries. On the other hand, countries suchas Greece and Portugal are a better target for analysis using this model and the learningprocess naturally turns out to be more relevant for them.

(a) Greece (b) Spain

(c) Italy (d) Portugal

Figure 16: Inferred path of beliefs for each country

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E Model with disasters and full information

In this section, I present the calibration and business cycle statistics for the model with raredisasters and full information about their realizations, which is used for the counter-factualexercise in Section 4.6. Note that the parameters of the income process are kept at the samelevel as in the main model. Table 11 summarizes the calibrated structural parameter values.It can be noticed immediately that the moments-matching exercise results in the values ofβ and y that are not much different from the model with partial information.

Symbol Meaning Full info Source

σ Risk aversion 2 Literaturer∗ Risk-free rate 0.01 Literatureθ Re-entry probability 0.049 Literatureδ Probability of maturing 0.053 Dataκ Coupon payment 0.0125 Datay Default cost par. 0.821 Calibrationβ Discount factor 0.987 CalibrationCalibration targets Full info Data

E (debt/GDP) 37.9 37.9E (spread) 1.88 1.88

Note: targeted moments are given in percentage points. Simulations are repeated 10,000times for a period of 1998-2014.

Table 11: Calibration of structural parameters in the model with full information

Table 12 presents the business cycle moments for this variant of the model. Note that themain differences relative to a benchmark AR1 model still hold in this case. It is also clearthat learning about rare disasters has an important quantitative impact, contributing to ahigher variance of the bond spread and a lower variance of consumption and trade balancerelative to income. It is also worth noting that 96.5% of all defaults in this model occurin the disaster regime. At the same time, the distribution of spreads becomes even moreskewed (13.1) and leptokurtic (235.7) relative to the model with partial information. Theaverage maximum spread across is 98.6% here, while it was 125% in the main model.

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Statistic Data Full infoErgodic Conditional

E(s) 1.88 1.12 1.88std(s) 3.46 3.87 4.10std(c)/std(y) 0.94 0.92 1.59std(tb)/std(y) 0.53 0.28 0.89corr(y, c) 0.97 0.96 0.84corr(y, tb) −0.94 0.41 −0.38corr(y, s) −0.72 −0.20 −0.71corr(s, tb) 0.72 0.26 0.63E(debt/y) 37.90 66.15 37.90

Note: moments for the bond spread, debt-to-GDP ratio and the long-run default probability (annual)are given in percentage points. Ergodic (long-run) simulations extend to 10,000 quarters and arerepeated 10,000 times, following closely Chatterjee and Eyigungor (2012). Conditional (short-run)simulations mimic the period of 1998-2014 (68 quarters) and are repeated 10,000 times starting fromthe actual levels of debt and GDP observed in 1998:Q1. Each short-run sample is constructed suchthat: i) the series start from the actual 1998:Q1 debt and income levels, and ii) the regime switchesfrom good to bad in 2008:Q3. Consumption data is detrended using the common GDP trend.

Table 12: Simulated behavior of the model with full information

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