A new estimation technique of sovereign default risk

Soytas, Mehmet Ali; Volkan, Engin Article A new estimation technique of sovereign default risk Central Bank Review (CBR) Provided in Cooperation with: Central Bank of The Republic of Turkey, Ankara Suggested Citation: Soytas, Mehmet Ali; Volkan, Engin (2016) : A new estimation technique of sovereign default risk, Central Bank Review (CBR), ISSN 1303-0701, Elsevier, Amsterdam, Vol. 16, Iss. 4, pp. 119-125, https://doi.org/10.1016/j.cbrev.2016.11.002 This Version is available at: https://hdl.handle.net/10419/217297 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ A new estimation technique of sovereign default risk * Mehmet Ali Soytas¸ * , Engin Volkan Department of Economics, € Ozye gin University, Nisantepe Mah. Orman Sok. No:34-36, 34794, Cekmekoy, Istanbul, Turkey article info Article history: Received 11 September 2016 Received in revised form 29 November 2016 Accepted 29 November 2016 Available online 27 December 2016 Keywords: Sovereign default risk Hotz-Miller estimation Endogenous default risk Conditional choice probabilities PML GMM abstract Using the fixed-point theorem, sovereign default models are solved by numerical value function iteration and calibration methods, which due to their computational constraints, greatly limits the models' quantitative performance and foregoes its country-specific quantitative projection ability. By applying the Hotz-Miller estimation technique (Hotz and Miller, 1993)- often used in applied microeconometrics literature- to dynamic general equilibrium models of sovereign default, one can estimate the ex-ante default probability of economies, given the structural parameter values obtained from country-specific business-cycle statistics and relevant literature. Thus, with this technique we offer an alternative solution method to dynamic general equilibrium models of sovereign default to improve upon their quantitative inference ability. ©2016 Central Bank of The Republic of Turkey. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction In developing countries, economic crises are often followed by sovereign debt crises, which results in sudden drop or halt in foreign credit to real sector, a surge in cost of borrowing in the international financial markets, a sharp decline in foreign trade, a slowdown in economic growth, and as a consequence a decline in consumption and investment. 1 To avoid the aforementioned large negative impacts of sovereign debt crises, it is crucial for policymakers to identify the optimal external borrowing need, assess debt sustainability, and understand the (speculative) pricing strategies of foreign investors. In addition, considering the high stakes, it also becomes crucial for foreign investors to identify default tendencies of countries during high global/local economic volatilities. With this motivation, in order to make the aforementioned quantitative inferences and to estimate the ex-ante default probabilities of countries, we develop the framework to apply the Hotz- Miller estimation technique on dynamic general equilibrium models of sovereign default. The Hotz-Miller estimation technique (Hotz and Miller, 1993)is popularly used in microeconomics literature as an alternative to Nested Fixed Point (NFXP) estimation technique. This technique estimates the structural parameter values of a recursive competitive partial equilibrium (general equilibrium under some conditions) model with discrete choice, given the actual probability of the (discrete) endogenous choice extracted from real-time data. By applying this technique to the dynamic general equilibrium models of sovereign default, one can estimate the ex-ante default probability of economies for a given set of structural parameter values that are obtained from real business-cycle statistics and relevant literature. Thus, with this technique we offer an alternative solution method to dynamic general equilibrium models of sovereign default to improve upon their quantitative inference ability. Macroeconomic studies on sovereign default literature became popular in the 1980s, with the seminal work of Eaton and Gersovitz, (1981) which studies sovereign default through an exchange economy general equilibrium model. However, during the period of 1990e2000 during which the financial problems of developing countries seemed to be predominantly concerned with privately issued debt and liquidity crises, this area of research lost attention. However, after Argentina's default in 2001, the macroeconomic literature on sovereign default revived with (Arellano, 2008) that reused the general equilibrium model of sovereign default introduced by the seminal paper of Eaton and Gersovitz, (1981). * This paper (Project Number:113K754) has been funded with support from The Scientific and Technological Research Council of Turkey (TUBITAK). *Corresponding author. E-mail addresses: [email protected] (M.A. Soytas¸), engin.volkan@ gmail.com (E. Volkan). Peer review under responsibility of the Central Bank of the Republic of Turkey. 1 Arteta and Hale (2008), Borensztein and Panizza (2008), Dias and Richmond (2009), Fuentes and Saravia (2010), Furceri and Zdzienicka (2011), Gelos et al. (2011), Rose (2005) and Yeyati and Panizza (2011) study the impacts of debt crises on economic activity. Contents lists available at ScienceDirect Central Bank Review journal homepage: http://www.journals.elsevier.com/central-bank-review/ http://dx.doi.org/10.1016/j.cbrev.2016.11.002 1303-0701/©2016 Central Bank of The Republic of Turkey. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/). Central Bank Review 16 (2016) 119e125 Following Arellano (2008), there is a still growing literature that focuses on identifying the factors that induce emerging economies to default despite the sanctions and economic costs 2 that followed 3 . The most common feature of the this literature is that they are all based on Eaton-Gersovitz model which endogenizes default. Almost all of these studies incorporate new features into the Eaton- Gersovitz model to refine its ability in matching the two most important empirical stylized facts of default: high debt-to-income ratio observed in developing countries, accompanied with high spread. Even though these studies achieve some improvements, their models' ability to predict default probabilities and optimal debt strategies is highly limited given their solution technique that is numerical value function iteration with calibration. For example, we often observe that the numerical value function iteration generates discontinuous policy functions. Moreover, calibratio 4 does not allow us to use the model to quantitatively study those countries that did not default in their past and binds the model to a fixed set of parameter values that sometimes are not economically intuitive. By applying Hotz-Miller estimation technique to general equilibrium sovereign default models, this paper contributes to the literature in two aspects. First, by using this technique and avoiding calibration, we can quantify the debt-default strategies and most importantly estimate the ex-ante default probabilities of all countries, even those that have never defaulted. Second, through this technique, we avoid using the fixed-point theorem which together with calibration bind the model to a fixed set of parameter values that sometimes are not economically intuitive. As opposed to the numerical iteration function and calibration, this will give us room to study countries' debt-default strategies at any given set of parameter values that are economically intuitive and representative of their real business cycle facts. The paper is organized as follows. In section 2,wedefine the sovereign default problemandcharacterize itsrecursive formulation. In section 3, we show the application of the Hotz-Miller estimation technique to our sovereign default problem. Section 4will outline the econometric estimation. Finally, in section 5we conclude. 2. The sovereign default model The sovereign default model we introduce is adopted from Hatchondo and Martinez (2009). It is a dynamic general equilibrium (endowment) model, specifically with long-term debt and endogenous default 5 . The economy we characterize below is an emerging economy inhabited by a representative agent, in other words a sovereign. 2.1. The setup The sovereign derives utility from consumption c t . The sovereign's expected life-time utility is given by UðcÞ¼EX ∞ t¼0 b tc1 g t 1 g (1) where the discount factor b 2ð0;1Þ, the relative risk aversion parameter g 1. 6 At period t0, the representative agent receives a stochastic endowment stream of a single tradable good, y t , drawn from a compact set Y¼½ y;y3R þþ with probability p ðy t jy t1 Þconditional on previous period realization of y t1 . In this model, we can assume the income to follow an AR (1) process given as log yt¼ð1 r Þmþ r log yt1þεtεtN0; s 2(2) After the income realization, the sovereign is then required to repay its debt obligations. Given that the sovereign's debt stock is l t , its debt obligations is ½ d þ k ð1 d Þb t where the first term is the portion of the debt that matures and the second term is the coupon payments of those that are still outstanding. However, the sovereign may opt to default, that is repudiate on its debt obligations, which in turn would incur an output loss of fðy t Þ 7 and face autarky at the period of default. After, the sovereign does not incur a contraction 8 and with probability ð1 m Þregains access to the capital markets. These assumptions are supported by empirical evidence. 9 On the other hand, if the sovereign opts to repay its debt, it maintains its access to the international capital markets to issue new debt in the form of long-term bonds, denoted by l tþ1 . With the new bond issuance the economy's outstanding debt stock becomes btþ1¼ð1 d Þbtþ l tþ1(3) We assume d 2ð0;1Þwhich implies that on average the bonds mature in 1= d periods. Note that if d ¼1 the debt would become a one-period discount bond or if d ¼0 and k >0 the debt would become a consol bond promising to pay k units infinitely. In the above equation, b tþ1 <ð1 d Þb t means that the sovereign is borrowing from the international financial markets by selling bonds, i.e. l tþ1 <0 at a price of qðb tþ1 ;y t Þset by risk-neutral investors. Additionally, b tþ1 >ð1 d Þb t means that the sovereign is saving by purchasing bonds from the international financial markets amounting l tþ1 >0 at the risk-free interest rate r. Finally, 2 Default costs are identified as financial sanctions imposed by international creditors, decline in international trade, temporary output loss, and decline in output growth rate. Some of the papers that study default costs are Arteta and Hale (2008), Borensztein and Panizza (2008), Fuentes and Saravia (2010), Rose (2005) and Yeyati and Panizza (2011). 3 For example, Aguiar and Gopinath (2006) proposes that it is more the permanent income shocks that induces emerging economies to accumulate debt. Additionally, Yue (2010) highlights the enforcing role of debt renegotiation and its recovery in an emerging economy's ex-ante incentive to default. More differently, Cuadra and Sapriza (2008) shows the direct effect of political uncertainty on the frequency of default by emerging economies (Lizarazo, 2013). characterizes foreign investors as risk averse agents (Chatterjee and Eyigungor, 2012; Hatchondo and Martinez, 2009; Volkan, 2008). contribute mainly by increasing the termstructure of sovereign bonds. Finally, Bai and Zhang (2012), Mendoza and Yue, (2012), Park (2012) study impacts of sovereign default on production and investment. 4 This is an estimation technique often used by macroeconomists to calibrate the values of the structural parameters of a general equilibrium model so that the model's simulated data statistics match to the target statistics obtained from the actual data. 5 For another dynamic general equilibrium (endowment) model, specifically with long-term debt and endogenous default please see (Chatterjee and Eyigungor, 2012). 6 When g ¼1 the period utility becomes logarithmic, i.e. lnðc t Þ. 7 Empirical evidence shows that there are several mechanisms through which default affects output. Firstly, sovereign defaults are associated with direct output loss. It is evident that the direct loss increases when the economy suffers a banking crisis in addition to default. Second, sovereign defaults cause output loss through its restrains on trade, foreign direct investments and foreign as well as domestic credit to the private sector. 8 Using quarterly data on defaults that occurred during 1970e2005 period (Yeyati and Panizza, 2011), finds that growth rates in post-default period are never significantly lower than in normal times. On the contrary, output reaches its minimum at the time of default and starts recovering after. Using yearly data on defaults that occurred during 1980e2000 period (Sandleris, 2012), finds similar results. 9 Gelos et al. (2011) and Alessandro (2011) show that the duration of the exclusion is short-lived. Accordingly, in the aftermath of defaults, average period of autarky was 4.5 years during 1980e2000 and 2.9 years during 1990e2000. M.A. Soytas¸, E. Volkan / Central Bank Review 16 (2016) 119e125120 b tþ1 ¼ð1 d Þb t means that the sovereign is neither borrowing nor saving. The risk-neutral investors price the bond, i.e. qðb tþ1 ;y t Þ, so that they make zero-profit in expectations. Note that the investors can borrow at the risk-free interest rate and have perfect information regarding the emerging economy's endowment. Let dðb tþ1 ;y t Þbe an indicator function that represents the default decision of the sovereign at its outstanding debt stock and income. Accordingly, the function takes on value 1 for default and 0 for no default. where bðdðb tþ1 ;y tþ1 Þ;b tþ1 ;y tþ1 Þis the optimal debt policy of the sovereign at its outstanding debt stock and income given its default decision. As can be seen in the above equation, the price internalizes the information that in the event of default, they will not receive any repayment, otherwise, they will receive redemption of the debt that matures and the coupon payment due from the rest of the outstanding debt. Finally, this price is both the bid price of the new issues as well as the repurchase price of the outstanding bonds. 2.2. Recursive formulation Let Vðb;yÞbe the value function of the sovereign at the beginning of period tafter the default or no default decision is made. Vðb;yÞ¼ max dðb;yÞ2f0;1gfdV0ðb;yÞþð1dðb;yÞÞV1ðyÞg (5) where drepresents the optimal default policy of the sovereign for all pairs of outstanding debt and income. Thus, the state space ðb;yÞ2BYf0;1gconsists of the outstanding debt stock, income and default decision at period t. We assume the debt stock level is from a compact set of B¼½ b;03ℝ  . In the above equation, V 0 ðb;yÞis the value of no default defined as V0ðb;yÞ¼max b 0 8 < : c1 g 1 g þ b X y 0 2Y Vðb0;y0Þ p ðy0jyÞdy09 = ;;(6) with c¼y½ d þð1 d Þ k bqðb0;yÞ½b0ð1 d Þb0 and l 0¼b0ð1 d Þb0. On the other hand, V 1 ðyÞis the value of default defined as V1ðyÞ¼8 < : ðyfyÞ1 g 1 g þ b X y02Y ½ m V1ðy0Þ þð1 m ÞVð0;y0Þ p ðy0jyÞdy09 = ;:(7) As introduced before with probability m , the economy will stay in the default state, while with probability ð1 m Þit regains access to the capital markets. 2.3. Definition of equilibrium The equilibrium of the above recursive problem is a Markov Perfect Equilibrium which is characterized by a set of value functions fV 0 ðb;yÞ;V 1 ðyÞ;Vðb;yÞg, a set of decision rules fdðb;yÞ;bðd;b;yÞg, and a bond price qðb0;yÞ¼ X y 0 2Y½ d þð1 d Þ½ k þqðbðdðb0;y0Þ;b0;y0Þ;y0Þ ½1dðb0;y0Þ p ðy0jyÞ1 1þr; such that given the bond price qðb0;yÞ, the set of decision rules fdðb;yÞ;bðd;b;yÞg solve the recursive problem defined by Equations (5)e(7). 3. Applying Hotz-Miller estimation technique The application of the Hotz-Miller estimation technique will be illustrated using a stationary environment. Suppose the income in any period can take one of the values from Y¼fy 1 ;y 2 ;y 3 ;…:; y n ;…y N g. Then transition of income can be given by a transition matrix P ðy0jyÞ. Note that AR (1) income process mentioned in Section 2can be transformed into a Markov Process using the methodology introduced by Hussey and Tauchen (1991). Additionally, assume that in a given period, the country can choose among the following debt alternatives b02ðb 1 ;b 0 ;b 1 ;b 2 ;b 3 ;…;b j ;…b J Þwhere b 1 or j¼1, represents the default decision and the following choices are the ascending debt level choices of the sovereign, respectively. Note also that b 0 or j¼0 represents the country's 0 debt level choice in the current period. Let the state variable relevant for period utility be s¼ðy;b;εÞ, such that s2S¼YBF. First two state variables are already introduced. Then, following the functional form given in Section 2 the period utility can be written as qðbtþ1;ytÞ¼ X y tþ1 2Y½ d þð1 d Þ½ k þqðbðdðbtþ1;ytþ1Þ;btþ1;ytþ1Þ;ytþ1Þ ½1dðbtþ1;ytþ1Þ p ðytþ1jytÞ1 1þr;(4) uðs;jÞ¼ 8 > > > > > > < > > > > > > : ðyfyÞ1 g 1 g þεjj¼1 yþ d þð1 d Þ k bqy;bjbjð1 d Þb1 g 1 g þεjj¼0;1;2;3;…J ; M.A. Soytas¸, E. Volkan / Central Bank Review 16 (2016) 119e125 121 where the additive unobserved components in the utility function (ε j ) have a continuous joint distribution function GðεÞwhere ε¼ðε 1 ;ε 0 ; ::; ε J Þ. Given the characterization above, the recursive formulation of the sovereign default model then becomes VðsÞ¼max j2J"uðs;jÞþ b X s02s p ðs0js;jÞVðs0Þ;(8) where p ðs0js;jÞ¼ p ðy0;b0jy;b;jÞgðε0jy0;b0Þ:(9) In the state transition in Equation (9), we assume that conditional on the state ðy;bÞ, 10 the unobserved component is conditionally independent over periods. This is a founding assumption in most microeconometric applications using the Hotz-Miller or Nested Fixed point environment. Violation of this assumption would bring more computational complexity, since then one should include all the relevant past states in the state transition equation for ε. Moreover presumably some of those past states will include the unobserved ε0s, we will need to integrate out those components from the value functions. Therefore conditional independence assumption will be assumed throughout the representation of the estimation. However it should be clear that the technique can still be applied in a more general dependency structure in the transition of εin the expense of computational flexibility. For now, we will assume no autarky period, in other words, an instant access to the international capital markets after default. This assumption can be removed. However, for now, assuming no autarky provides us an applicative ease and removing this assumption presumably can only improve the quantitative performance of the estimation technique. Given the setup presented above, the value equation (8) can be rewritten as the integrated valuation function Vðy;bÞ¼Z0 @max j2J2 4ujðy;bÞþεðjÞþ b X y02Y p ðy0jy;jÞVðy0;b0Þ3 5 1 Agðdεjy;bÞ: (10) Now let the conditional choice probability (CCP) be defined as Pðb0jy;bÞ¼ZIj¼arg max j2JVjðy;bÞþεðjÞgðdεjy;bÞ;(11) where V j ðy;bÞ¼u j ðy;bÞþ b P y02Y p ðy0jy;jÞVðy0;b0Þ. The right hand side of Equations (6), (7), and the bond price equation can be expressed in closed form given the distributional assumption about GðεÞ.Ifεare distributed extreme value type I, we have the following form for the conditional valuation functions. V1ðy;bÞ¼8 > > < > > : yfyþ bm P y02Y p ðy0jyÞV1ðy0;0Þ þ b ð1 m ÞP y02Y p ðy0jyÞ l þP J k¼1 eV k ðy0;0Þ!9 > > = > > ; j ¼1 (12) Vjðy;bÞ¼8 > < > : yþ½ d þð1 d Þ k bqy;bjbjð1 d Þb þ b P y 0 2Y p ðy0jyÞ l þP J k¼1 eV k ðy0;b j Þ!9 > = > ;j ¼0;1; ::; J (13) where l ¼0:577215665 is the Euler's constant. In Hotz and Miller (1993) representation, we know there is a one to one mapping between the conditional choice probabilities (CCPs) and the normalized valuation functions. lnpjðy;bÞp1ðy;bÞ¼Vjðy;bÞV1ðy;bÞwhere j ¼0;1; ::; J It remains how one can obtain the value of qðy;b0Þ as a function of model parameters and fundamentals. It is given in the bond price equation that the value of qðy;b0Þ depends on the solution to a fixed point. The natural way to accommodate this object in the Hotz- Miller framework is to make use of the CCPs. One can write the value of qðy;b0Þ at a particular choice of debt level as: qðy;b0Þ¼Eððd 00 ;b 00 Þ;y 0 jy;b 0 Þ½1d00½ d þð1 d Þ½ k þqðy0;b00Þ 1þr; where d 00 ¼dðy0;b0Þ and b 00 ¼bðy0;b0Þ denotes the default and the debt level choices in the next period. The choice on the next period depends on the valuation function comparisons, therefore we can characterize the above equation in terms of CCPs and the finite number of qðy;bÞ’s given the discrete nature of the state space. Obviously existence of the unobserved shock in the utility function requires integrating this component. qðy;b0Þ ¼ X y 0 2Y8 > < > :X J j2Jf0g2 6 4 PVjðy0;b0Þ þ εjVkðy0;b0Þ þ εk  d þð1 d Þ k þqy0;bj 1þr 3 7 5 9 > = > ; p ðy0jyÞ The first term in the square brackets is the CCP of choosing action j, therefore we can write the simplified version of the above equation for qas qðy;b0Þ¼ X y02Y8 < :X J j¼0 pjðy0;b0Þ  d þð1 d Þ k þqy0;bj 1þr9 = ; p ðy0jyÞ; and qðy;b0Þ ¼ X y02Y8 < :X J j¼0 pjðy0;b0Þ d þð1 d Þ k þqy0;bj 1þr9 = ; p ðy0jyÞ 10 Transition from bto b0is determined entirely by the choice j, therefore, in the following representations we will denote the function p ðy0;b0jy;b;jÞby p ðy0jy;jÞ. M.A. Soytas¸, E. Volkan / Central Bank Review 16 (2016) 119e125122 for the possible values of y02ðy 1 ;y 2 ;…;y n ;…y N Þand b 00 2fb 0 ;b 1 ;…;b j ;…;b J g,qðy;b0Þ ¼ H½ðqðy 1 ;b 0 Þ;qðy 1 ;b 1 Þ;qðy 1 ;b 2 Þ ;qðy 1 ;b 3 Þ;…:; qðy 3 ;b 3 Þ;…;qðy N ;b J ÞÞ 00 . Given the solutions to these functions, we may obtain the following representation of the problem using the stationarity of the problem. Let u j ¼E½ε j y;b0¼b j . In the case of Extreme Value Type I, u j¼ l lnpjðy;bÞ Using the conditional valuation functions, the valuation function can be expressed as Vðy;bÞ¼ X J j¼1 pjðy;bÞnujðy;bÞþEhεjy;b0 ¼bjiþ b X y02Y p ðy0jyÞVy0;bj9 = ;:(14) Let us substitute the conditional expectation of the errors and stack the Mequations for each possible value of the state vector ðy;bÞ2M≡ðY  BÞ. In compact matrix notation we get V¼X j2J Pjujþ u jþ b FjV;(15) where is the Hadaramad (element by element) product, Vis the M1 vector of value functions; P j ,u j , and u j are M1 vectors that stack the corresponding elements at all states for alternative j; and F j is the MMmatrix of conditional transition probabilities F j ¼Fðs0jsÞ¼ p ðy0jyÞPðb0b j Þ. This system of fixed point equations can be solved for the value function Vas a function of Pwhere Pis the MðJþ1Þ1 11 vector of CCPs V¼IM b FUðPÞ18 < :X j2J Pjujþ u j9 = ;;(16) where F U ðPÞis the MMmatrix of unconditional transition probabilities induced by P. Now, we can define and calculate the vector of expected utilities. 4. Estimation strategy Standard estimates of dynamic discrete choice models involve forming the likelihood functions from the CCPs derived in Equation (11). This involves solving the value function for each iteration of the likelihood function. The method used to solve the value function depends on the nature of the optimization problems and falls into one of two cases; finite-horizon problems: in that case the solution will involve a backward induction starting from the last period of the model Tðt¼0;1;…;TÞ; stationary infinite-horizon problem: the valuation is obtained by a contraction mapping as described in the model section. We will describe the estimation relying on the stationary infinite horizon environment, however model can be generalized to a finite life-cycle setting without loss of generality. The estimation can be done in two ways, the first is a PML (as used in Aguirregabiria &Mira (2002) and the second is a GMM (as used in the original Hotz and Miller (1993). With Mpossible state variables, the PML needs to estimate the fP j g J j¼1 probabilities which can be constructed as parametric or non-parametric functions of state variables. Assuming a parametric functional form for the P j and denoting its dependence on the parameter vector q as p j ðy m ;b m ; q Þ, the PML function can be obtained via maximizing b q PML ¼arg max q 0 @X M m¼1X J j¼1 lnpjðym;bm; q Þ1 A;(17) Or a GMM estimator can be constructed using the inversion found in Hotz &Miller (1993). As already introduced, under the assumption that εis distributed independently and identically as type I extreme values, then the Hotz and Miller inversion implies that lnpjðym;bm; q Þ=p1ðym;bm; q Þ¼Vjðym;bm; q ÞV1ðym;bm; q Þ (18) for any normalized choice, but we set this choice to the default alternative conveniently. We can use the set of structural parameter values obtained from real business-cycle statistics and relevant literature to construct the valuation functions (specifically let G ¼ð b ; g ; r ;m; s 2 ; l ; d ; k ;fÞdenote those parameters used in the model introduced) up to the parameters q of the CCPs. Then we can proceed to form empirical counterparts of equation (18) and estimate the parameters of the model. The moment conditions can be obtained from the difference in the conditional valuation functions calculated for choice jand the base choice 1. The following moment conditions are produced for a particular state variable ðy m ;b m Þ: x jmð q Þ≡Vjðym;bm; q ÞV1ðym;bm; q Þlnpjðym;bm; q Þ =p1ðym;bm; q Þ:(19) Therefore, there are Jþ1 orthogonality conditions and thus j¼0;…;J. Letting x m ð q Þbe the vector of moment conditions at state m, these vectors are defined as x m ð q Þ¼ð x 0m ð q Þ; x 1m ð q Þ;… x Jm ð q ÞÞ 0 . Therefore, E½ x m ð q o Þy m ;b m ; G converges to 0 for every consistent estimator of true CCPs, p j ðy m ;b m ; q Þ, for m2f1;…;Mg;and where q o is the true parameter of the model. Then the GMM estimate of q is obtained via b q GMM ¼arg min q "1=MX M m¼1 x mð q Þ#0"1=MX M m¼1 x mð q Þ#:(20) 4.1. Model moment functions and GMM estimator Returning back to the model introduced. in Section 2, the moment conditions can be obtained by the differences in the choices considering the default and debt level choice. We have the following value function differences and moment conditions obtained from a particular state ðy m ;b m Þ: 11 Jþ2 is the number of choice alternative including the default ðb 1 ;b 0 ;b 1 ;b 2 ; ::; b J Þ. Knowing Jþ1 probabilities automatically gives the last one. M.A. Soytas¸, E. Volkan / Central Bank Review 16 (2016) 119e125 123 For j¼0;1;2; ::; Jand m¼1;::M. x m ð q Þis the vector of moment conditions at state mas introduced before. This vector is defined as before: x mð q Þ¼ 0 B B B B B B B B B B B B B B @ V0ðym;bmÞV1ðym;bmÞlnp0ðym;bm; q Þ p1ðym;bm; q Þ « Vjðym;bmÞV1ðym;bmÞlnpjðym;bm; q Þ p1ðym;bm; q Þ « VJðym;bmÞV1ðym;bmÞlnp0ðym;bm; q Þ p1ðym;bm; q Þ 1 C C C C C C C C C C C C C C A(22) The optimal GMM estimator for, q satisfies equation (20). 4.2. Estimation of the conditional choice probabilities Let's denote the probability at the mth row of Pcorresponding to the jth choice as: pmj ¼fjðym;bmÞ; and also let p m collect the choice alternatives for the state ðy m ;b m Þ. pm¼pm;1;pm0;…;pmJ ¼ðf1ðym;bmÞ;f0ðym;bmÞ; ::; f3ðym;bmÞÞ The estimation methodology proposes a functional form for f j ðy m ;b m Þand then the PML or the GMM estimates the function fð:Þ. For instance if we use logistic function for the probabilities and linear dependency to ðy;bÞ 12 and using the fact that: f1ðym;bmÞþf0ðym;bmÞþ:: þfJðym;bmÞ¼1 we obtain the following equations: f1ðym;bmÞ¼ 1 1þXJ j¼0ea j þb j y m þc j b m f0ðym;bmÞ¼ ea 0 þb 0 y m þc 0 b m 1þXJ j¼0ea j þb j y m þc j b m « fkðym;bmÞ¼ ea k þb k y m þc k b m 1þXJ j¼0ea j þb j y m þc j b m « fJðym;bmÞ¼ ea J þb J y m þc J b m 1þXJ j¼0ea j þb j y m þc j b m Replacing the above functions to form the F U ðPÞand P k terms in the value function differences in equation (21), and similarly obtaining p j ðy m ;b m ; q Þterms in equation (22) in terms of the functions defined for the CCPs, we obtain a natural framework for estimating the coefficients (a 0 b 0 ,c 0 a 1 b 1 c 1 ,…,a j ,b j ,c j ,…,a J ,b J ,c J ). The estimation can be done either by GMM or ML as introduced in Section 4. Once the coefficients are estimated, we can construct the CCPs associated with each state pair ðy;bÞusing the logistic functions. 13 4.3. Beyond the solution of the original model The estimation technique we introduced for the sovereign default model in fact provides a solution to the model valuation functions by optimally estimating the CCPs consistent with those valuation functions. Therefore any policy simulation including generation of future paths from the equilibrium follows as usual as when the model is estimated traditionally. However, one should observe that other than finding the optimal solution to the model (which is essential and important for further analysis), the estimated CCP functions can serve for other purposes outside the model. For instance assuming a logistic form for the choice probabilities, once the coefficients are estimated, the flexible logit functional form gives us a tool for predicting default probabilities for a variety of possible states. Of those states, we are not restricted to the ones that are actually visited in the solution of the model in ðy m ;b m Þ. This actually let the researcher to predict the default Vjðym;bmÞV1ðym;bmÞ¼ 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ymþ d þð1 d Þ k bmqym;bjbjð1 d Þbm1 g 1 g ðymfymÞ1 g 1 g þfy0;b0ym;b0¼bj0ðy0;b0jym;b0¼b1Þ0 IM b FUðPÞ18 < :X k2J Pk½ukþ u kg 9 > > > > > > > > > > > > > = > > > > > > > > > > > > > ; (21) 12 The p mj functions can be constructed parametrically as the logit example introduced or non-parametrically. Only requirement for consistently estimating those functions would be that P J j¼1 p mj ¼1, and each probability p mj 0 for j¼1;0; ::; Jand m¼1; ::; M. 13 Logistic functions can be generalized to include interactions and higher order terms of the state variables ðy m ;b m Þ. In this case the approximation will be uniformly better to the true CCPs. M.A. Soytas¸, E. Volkan / Central Bank Review 16 (2016) 119e125124 probability associated with a particular scenario. We can further generalize this idea to the estimation of the sovereign default model itself. To estimate the probabilities consistent with a particular model (with the other model parameters are either calibrated, observed, estimated elsewhere as the current sovereign default models generally do), theoretically we are not required to have the country defaulted. Most of the literature use Argentina or countries with at least one default in their histories. However given the parametrization of the model with the relevant parameters from the literature, there exists a set of CCPs consistent with those parameters that solve the model. 14 In this association between the model and the CCPs, there is no reference to the default probability other than its influence on the calibrated, estimated, observed parameters used from the literature. This particular property makes the CCP framework a potential tool for further exploring the sovereign default models. 5. Conclusion In this paper we present a new estimation technique, namely Hotz-Miller estimation technique (Hotz and Miller, 1993), to solve dynamic general equilibrium models of sovereign default. By applying Hotz-Miller estimation technique to general equilibrium sovereign default models, this paper contributes to the literature in two aspects. First, by using this technique and avoiding calibration, we can quantify the debt-default strategies and most importantly estimate the ex-ante default probabilities, even those that have never defaulted. Second, through this technique, we avoid using the fixed-point theorem which together with calibration bind the model to a fixed set of parameter values that sometimes are not economically intuitive. As opposed to the numerical iteration function and calibration, this gives us room to study countries' debt-default strategies at any given set of parameter values that are economically intuitive and representative of their real business cycle facts. Therefore the methodology is promising a substantial improvement in quantitative performance of dynamic general equilibrium models of sovereign default. Using the proposed technique, how well the results can replicate the main behaviors of the emerging economies is an empirical question, and is a topic of another paper currently in progress. References Aguiar, M., Gopinath, G., June 2006. Defaultable debt, interest rates and the current account. J. Int. Econ. 69 (1), 64e83. Aguirregabiria, V., Mira, P., 2002. Swapping the nested fixed-point algorithm: a class of estimators for discrete markov decision models. Econometrica 70 (4), 1519e1543. Alessandro, M., September 2011. Three essays on sovereign debt and financial markets. 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Yeyati, E.L., Panizza, U., January 2011. The elusive costs of sovereign defaults. J. Dev. Econ. 94 (1), 95e105. URL. http://ideas.repec.org/a/eee/deveco/v94y2011i1p95- 105.html. Yue, V.Z., March 2010. Sovereign default and debt renegotiation. J. Int. Econ. 80 (2), 176e187. URL. http://ideas.repec.org/a/eee/inecon/v80y2010i2p176-187.html. 14 Certainly, one potential problem becomes the uniqueness of the solution to the CCP functions. Since the estimation framework is new, such topics are let for future research. However if one uses the information of past defaults to set a target (this is another way of saying defining a moment condition as traditionally the literature does), the uniqueness can be guaranteed with many functional forms. M.A. Soytas¸, E. Volkan / Central Bank Review 16 (2016) 119e125 125