Ensaios em econometria de finanças

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(1)M´ arcio Poletti Laurini. Ensaios em Econometria de Finan¸ cas Tese apresentada ao Instituto de Matemática, Estat´ıstica e Ciência da Computa¸cão da Universidade Estadual de Campinas como requisito parcial para a obten¸cão do t´ıtulo de Doutor em Estat´ıstica. Orientador - Prof. Hotta. CAMPINAS 2009. Dr.. Luiz Koodi.

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(94) A` Louis Bachelier.. v.

(95) Agradecimentos Agradeço principalmente ao meu orientador Luiz Koodi Hotta pelo apoio fundamental dado durante todo o programa de doutorado, e principalmente pelo exemplo como acadêmico, pesquisador e ser humano. Esta convivência e´ um presente que irei valorizar por toda a vida. Aos demais professores do Programa de Pós-Graduaça˜ o em Estat´ıstica do Imecc, agradeço pela dedicaça˜ o e por tudo que pude aprender neste per´ıodo. Aos membros da banca, professores Benjamin Tabak e Pedro Morettin pelos comentários e sugestões dados durante a qualificaça˜ o da tese, e também pelos inúmeros artigos e livros que foram fundamentais até agora. Ao professor Flávio Ziegellman pelos comentários na defesa da tese, e pelos vários comentários em congressos. Ao professor Caio Ibsen de Almeida agradeço pela inspiraça˜ o para vários dos artigos que compõem essa tese. Ao professor Mauricio Zevallos por toda a convivência durante o programa de doutorado, e por todos os comentários dados no seminários que eu apresentei durante esse per´ıodo. A todos os meus colegas de turma, pelo apoio, amizade e por todas as discussões sobre estat´ıstica. Aos meus colegas de trabalho no Ibmec, pelo suporte e amizade. A todos meus alunos, por permitirem que eu trabalhe nos meus temas de pesquisa. A Lucinéia, pela minha alegria e todo o apoio e paciência. Aos meus pais, por tudo.. vii.

(96) “I’m not interested in doing research and I never been. I’m interested in understanding, which is quite a different thing.“ David Blackwell “Rather than love, than money, than faith, than fame, than fairness... give me truth.“ Christopher McCandless. ix.

(97) Resumo A tese compreende sete artigos sobre Econometria aplicada a problemas em Finanças. Dois artigos abordam a estimaça˜ o de modelos de fatores latentes para o ajuste e previsão da Estrutura a Termo de Taxas de Juros, utilizando métodos de Estimaça˜ o Bayesiana utilizando Markov Chain Monte Carlo, com o primeiro introduzindo uma estrutura de parâmetros variantes no tempo e o segundo uma generalizaça˜ o para Estruturas a Termo de Taxas de Juros em múltiplos mercados com a imposiça˜ o de condiço˜ es de não-arbitragem. Dois artigos discutem a aplicaça˜ o de Máxima Verossimilhança Emp´ırica e M´ınimo Contraste Generalizado na estimaça˜ o de equaço˜ es diferenciais estocásticas e modelos de volatilidade estocástica. O próximo artigo aborda a estimaça˜ o de equaço˜ es diferenciais estocásticas com uma estrutura de inovaço˜ es dirigida por um Movimento Browniano Fracionário, através de uma metodologia de Inferência Indireta. O artigo seguinte discute o uso de métodos não-paramétricos para a interpolaça˜ o de curvas de juros com a imposiça˜ o de condiço˜ es de não-arbitragem, utilizando splines suavizantes sujeitos a restriço˜ es de formato. A modelagem de microestruturas no mercado de câmbio e´ abordada no próximo artigo da tese, através do uso de metodologias paramétricas e semi paramétricas e testes para a presença de informaça˜ o assimétrica.. xi.

(98) Abstract The thesis consists of seven articles on Financial Econometrics. Two articles focus on the estimation of latent factor models to fit and forecast the term structure of interest rates using Bayesian estimation methods through Markov Chain Monte Carlo, with the first article introducing a structure of time-varying parameters and the second article a generalization for the term structure of interest rates in multiple markets with the imposition of no-arbitrage conditions. Two articles discuss the use of Empirical Likelihood and Generalized Minimum Contrast in the estimation of stochastic differential equations and stochastic volatility models. The next article discusses the estimation of stochastic differential equations with a structure of innovations driven by a Fractional Brownian motion, through a method of Indirect Inference. The following article discusses the use of nonparametric methods for interpolation of yield curves with the imposition of no-arbitrage conditions, using smoothing splines with shape restrictions. The modeling of microstructures in the exchange market is discussed in the next article of the thesis, through the use of semi-parametric and non-parametric methodologies and tests for the presence of asymmetric information.. xiii.

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(102) Introdu¸ ca õ Geral Esta tese consiste em uma cole¸cão de artigos sobre aplica¸cões de métodos estat´ısticos à problemas relacionados a modelagem de dados em mercados financeiros. Os dois artigos iniciais tratam da modelagem da Estrutura a Termo de Taxas de Juros utilizando modelos de fatores latentes. O primeiro artigo, “Bayesian Extensions to Diebold-Li Term Structure Model“, realizado em co-autoria com Luiz Koodi Hotta, apresenta um modelo de fatores latentes para a modelagem da estrutura a termo de taxas de juros, que generaliza o modelo de estrutura a termo proposto por Diebold and Li (2006). Propomos uma estrutura de fatores latentes, estimada de forma Bayesiana através de procedimentos de Markov Chain Monte Carlo, com uma forma funcional mais geral que permite um melhor ajuste ao formato da curva de juros. Nesta estrutura proposta tornamos variantes no tempo os parâmetros de descaimento utilizados no ajuste da curva de juros, e também adicionamos uma estrutura de volatilidade estocástica como um fator latente adicional. Esta estrutura permite capturar o padrão de heterocedasticidade observado nas taxas que compõem a estrutura a termo de taxas de juros. A metodologia proposta endere¸ca alguns dos problemas de inferência existentes no modelo original proposto por Diebold and Li (2006), permitindo uma inferência exata em amostras finitas para parâmetros, fatores latentes e previsões do modelo, não necessitando de um procedimento de inferência em dois estágios como no modelo original. Esta estrutura proposta também é vantajosa já que não necessita de um procedimento de pré-interpola¸cão da estrutura a termo de taxas de juros, como normalmente é realizado na estima¸cão de modelos para a curva de juros e que pode introduzir distor¸cões. Uma aplica¸cão emp´ırica da metodologia proposta é aplicada para a modelagem da estrutura ´ negociados na Bolsa de Mercadoria e Futuros a termo dos contratos de Swaps DI-PRE (BM&F), e um procedimento de análise de previsão é realizado, mostrando que o modelo proposto tem potencial preditivo e de ajuste dentro da amostra superior ao do modelo original de Diebold and Li (2006). O segundo artigo “Generalized Latent Factor Models For Yield Curves In Multiple Markets“, em co-autoria com Luiz Koodi Hotta, também aborda a estima¸cão de modelos. 1.

(103) de fatores latentes para a Estrutura a Termo de Taxas de Juros. Nesse artigo propomos uma estrutura geral de fatores latentes para a modelagem conjunta de m´ ultiplas curvas de juros. Partindo da metodologia Bayesiana de estima¸cão utilizando Markov Chain Monte Carlo discutida no artigo anterior, propomos uma estrutura geral que generaliza diversos modelos existentes da literatura de estrutura a termo de taxas de juros. Esta generaliza¸cão permite o uso de formas funcionais mais gerais que as utilizadas na literatura, com parâmetros de descaimento e volatilidades variantes no tempo, bem como a incorpora¸cão direta da possibilidade de intera¸co˜es entre movimentos na curva de juros entre mercados. O artigo também apresenta uma forma de incorporar restri¸cões de Não-Arbitragem na modelagem em m´ ultiplos mercados de juros. Nesse artigo também são discutidos problemas de identifica¸cão e do uso de métodos de Bayesian Shrinkage para reduzir o elevado n´ umero de parâmetros envolvido na estima¸cão de modelos para m´ ultiplos mercados, e a metodologia de inferência proposta permite obter as distribui¸cões exatas de parâmetros, fatores latentes e previsões do modelo. As metodologias propostas são aplicadas para a modelagem conjunta da curva de Cupom Cambial e da curva de Eurodólares, e no artigo temos uma discussão detalhada sobre especifica¸cão, compara¸cão de modelos e previsões, bem como uma discussão sobre a validade de condi¸cões de nãoarbitragem nestes mercados. Os dois próximos artigos, em co-autoria com Luiz Koodi Hotta, sobre aplica¸cões de métodos semi-paramétricos baseados em Verossimilhan¸ca Emp´ırica e M´ınimo Contraste Generalizados aplicados a problemas em finan¸cas. O primeiro artigo “Generalized Empirical Likelihood/Minimum Contrast Estimation of Stochastic Differential Equations“ trata da estima¸cão de equa¸cões diferenciais estocásticas, e o segundo artigo “Estimation of Stochastic Volatility Models Using Methods of Generalized Empirical Likelihood/Minimum Contrast“ da estima¸cão de modelos de volatilidade estoc´ astica. O ponto comum entre estes dois artigos está na dificuldade da avalia¸cão da fun¸cão de verossimilhan¸ca nestes dois problemas. Na estima¸cão de equa¸cões diferenciais estocásticas a avalia¸cão da fun¸cão de verossimilhan¸ca é dificultada pela não-existência de solu¸cões gerais para as equa¸cões diferenciais estocásticas, relacionadas à densidade de transi¸cão do processo que representa a fun¸cão de verossimilhan¸ca. Neste contexto mostramos que a estima¸cão de equa¸cões diferenciais estocásticas através do uso de métodos de Verossimilhan¸ca Emp´ırica e M´ınimo Contraste Generalizado nas discretiza¸cões dos processos de interesse rendem estimadores com boas propriedades em amostras finitas, como mostrado pelos procedimentos de Monte Carlo. 2.

(104) realizados neste estudo. Também discutimos como os métodos propostos tratam dos problemas de especifica¸cão incorreta representados pelo uso de discretiza¸cões. O artigo também contém uma aplica¸cão emp´ırica das metodologias propostas para uma série de taxas de juros de curto prazo de maturidade de um mês (T-Bills). No artigo que trata da estima¸cão de modelos de volatilidade estocástica a dificuldade da avalia¸cão da fun¸cão de verossimilhan¸ca está relacionada à presen¸ca do fator latente representado pela volatilidade não-observada do processo. Neste artigo propomos o uso de condi¸cões de momentos geradas pelo modelo log-normal de volatilidade para a constru¸cão de estimadores baseados em verossimilhan¸ca emp´ırica/m´ınimo contraste. Estas formas de estima¸cão podem ser pensadas como a generaliza¸cão dos métodos de momentos utilizados na estima¸cão de modelos de volatilidade estocástica através do uso de probabilidades impl´ıcitas dadas pela estima¸cão não-paramétrica da verossimilhan¸ca do processo. Mostramos através de uma série de estudos de Monte Carlo que as metodologias de estima¸cão propostas têm boas propriedades em amostras finitas, com desempenho superior aos métodos normalmente utilizados na estima¸cão de modelos de volatilidade estocástica. Também mostramos por meio de estudos de Monte Carlo que estes estimadores tem propriedades de robustez em rela¸cão a inova¸cões com caudas pesadas e outliers. O próximo artigo da tese “Indirect Inference in Fractional Short-Term Interest Rate Diffusions“, também em co-autoria com Luiz Koodi Hotta, discute o uso de metodologias de estima¸cão baseadas em simula¸cão para a estima¸cão de equa¸cões diferenciais estocásticas utilizando dados observados discretamente em um contexto não-trivial, que é o de processos cujas inova¸cões são dadas por um movimento Browniano Fracionário, que são processos não Markovianos e que não podem ser caracterizados como Semi-Martingales. Neste contexto não é poss´ıvel avaliar a fun¸cão de verossimilhan¸ca de forma exata, e propomos o uso do princ´ıpio de Inferência Indireta na inferência dos parâmetros deste processo. Este procedimento consiste no uso de um modelo auxiliar mais simples e do uso de corre¸cões de viés baseadas em simula¸cões do processo gerador dos dados. Mostramos que o procedimento proposto tem um desempenho satisfatório através de simula¸cões de Monte Carlo, e realizamos uma aplica¸cão emp´ırica da metodologia para a estima¸cão de modelos fracionários de taxas de juros de curto prazo através da estima¸cão do modelo Cox-Ingersoll-Ross para taxas de juros de T-Bills, Eurodólares e taxas de juros do Canadá. O artigo “Constrained Smoothing B-Splines For The Term Structure Of Interest Rates“, realizado em co-autoria com Marcelo Moura, discute o uso de métodos nãoparamétricos baseados em smoothing splines com restri¸co˜es de formato para procedi-. 3.

(105) mentos de interpola¸cão e suaviza¸cão de curvas de juros com a imposi¸cão de condi¸cões necessárias de não-arbitragem. Neste artigo mostramos que a metodologia proposta tem vantagens sobre algumas formas usuais de interpola¸cão de curva de juros através de estu´ dos de simula¸cão e aplica¸cões emp´ıricas para o mercado de instrumentos de Swap DIxPRE negociados na BM&F e para instrumentos de STRIPS (Separated Trading of Interest and Principals) de t´ıtulos do Tesouro Americano. Ou ´ ltimo artigo da tese, “Empirical market microstructure: An analysis of the BRL/US$ exchange rate market“, foi realizado em co-autoria com Luiz Gustavo Cassilati Furlani e Marcelo Savino Portugal. O artigo apresenta um estudo sobre a presen¸ca de microestruturas de mercado e processos de descoberta de pre¸cos para o mercado de câmbio R$/US$ utilizando dados de alta-frequência. Este artigo discute modelagens paramétricas e semi-paramétricas para estes efeitos de microestrutura, propondo modelos baseados em co-integra¸cão e regressão quant´ılica dinâmica para as cota¸cões de compra e venda neste mercado de câmbio. Os artigos estão formatados no padrão das revistas para o qual estão foram submetidos ou publicados, e desta forma estão em na l´ıngua inglesa de acordo com o padrão destas revistas. O artigo “Constrained Smoothing B-Splines For The Term Structure Of Interest Rates“ foi aceito para publica¸cão no Insurance: Mathematics and Economics, e o artigo “Empirical market microstructure: An analysis of the R$/US$ exchange rate market“ foi publicado no Emerging Markets Review, v. 9, p. 247-265, 2008.. 4.

(106) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL MÁRCIO POLETTI LAURINI LUIZ KOODI HOTTA Abstract. This paper proposes a statistical model to adjust, interpolate, and forecast the term structure of interest rates. This model is based on the extensions for the term structure model of interest rates proposed by Diebold and Li (2006), through a Bayesian estimation using Markov Chain Monte Carlo (MCMC). The proposed extensions involve the use of a more flexible parametric form for the yield curve, allowing all the parameters to vary in time using a structure of latent factors, and the addition of a stochastic volatility structure to control the presence of conditional heteroskedasticity observed in the interest rates. The Bayesian estimation enables the exact distribution of the estimators in finite samples, and as a by-product, the estimation enables obtaining the distribution of forecasts of the term structure of interest rates. Unlike some econometric models of term structure, the methodology developed does not require a pre-interpolation of the yield curve. The model is fitted to the daily data of the term structure of interest rates implicit in SWAP DI-PRÉ contracts traded in the Mercantile and Futures Exchange (BM&F) in Brazil. The results are compared with the other models in terms of fitting and forecasts. Keywords: Term Structure, Bayesian Inference, Markov Chain Monte Carlo. JEL Codes: G1,C22.. 1. 5.

(107) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 2. 1. Introduction The term structure of interest rates may be defined as a collection of interest rates, indexed in two dimensions: maturity and time. The first index shows the relation between the rates with different maturities for contracts of the same nature in a determined period. The second index shows the time evolution of the rates of contracts with the same maturities. The term structure of interest rates shows the dynamics of the yield curve, linking a functional structure of observations in cross-section (evolution of the rates over maturity) and the evolution of the yield curve over time. As such, the term structure may be represented by a multivariate stochastic process. There is a wealth of literature regarding the models of the term structure. To simplify, we can classify this literature into three classes of models. The first class encompasses the equilibrium models, such as Brennan and Schwartz (1979), Cox et al. (1985) and Duffie and Kan (1996). The second classification is based on the arbitrage-free models, of which Heath et al. (1992) is the representative framework. The third relevant literature is the use of statistical models without a structural interpretation, that is, models that synthesize data patterns and allow for the forecasting of the curve without necessarily representing the theoretical models that fit under equilibrium and free-arbitrage conditions. Examples of this model include the methodology of principal components (Litterman and Scheinkman (1991)), curve interpolation models such as splines (McCulloch (1971)), smoothing splines (Shea (1984)), kernel regression (Linton. et al. (2001)), and parametric models for curve fitting such as Nelson and Siegel (1987) and Svensson (1994). The dynamic extension of the Nelson-Siegel model, presented in Diebold and Li (2006) and the basis of the procedure studied in this article, are examples of a statistical model that can successfully forecast the term structure of interest rates. Despite its basis in the theoretical models for interest rates, the structural models based on equilibrium conditions have low forecasting power for the term structure. The calibration models based on no-arbitrage do not permit a direct forecasting of the yield curve. Statistical models are generally used in the fit and forecasting of the term structure of interest rates, because of their superior adjustment to equilibrium-based econometric models and for their greater simplicity.. 2. Diebold-Li Model Among the statistical models for interest rate, the influential model designed by Diebold-Li (Diebold and Li (2006)) is widely used in market applications. This model is a dynamic extension. 6.

(108) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 3. of the Nelson-Siegel model (Nelson and Siegel (1987)) for the cross-section fit for the yield curve. The Nelson-Siegel model corresponds to fitting the following equation for the yield curve observed in the market on a specific date:. (2.1). 1 − e−mit /τt 1 − e−mit /τt −mit /τt + ǫit + β3t −e yit (mit ) = β1t + β2t mit /τt mit /τt. where yit (mit ) is the observed rate on a given date, β1t , β2t , β3t are the time-varying parameters. The Nelson-Siegel model is a parsimonious way of fitting the yield curve while managing to capture a part of the stylized facts in interest rate process, such as the exponential formats present in the yield curves. The parameters βit have economic interpretations, where β1t presents a long-term level interpretation; β2t presents short-term components; and β3t indicates medium-term components. It may also be interpreted as the decompositions of level, slope, and curvature of the yield curve, respectively, according to the terminology developed by Litterman and Scheinkman (1991). An extension of this model is to use the formulation proposed by Svensson (1994) to fit the interest cross-sections. This formulation considers the inclusion of an additional term to the formulation proposed by Nelson and Siegel (1987), thus corresponding to:. (2.2) yit (mit ) = β1t +β2t. 1 − e−mit /τ1t 1 − e−mit /τ1t 1 − e−mit /τ2t +β3t − e−mit /τ1t +β4t − e−mit /τ2t +ǫit mit /τ2t mit /τ1t mit /τ2t. allowing a more flexible fit for the yield curve and enabling the capture of multiple changes in the yield-curve slope. The purpose of these models is to allow fitting and the subsequent interpolations and extrapolations of the yield curve based on a parametric structure, which concurs with other nonparametric fitting models such as smoothing splines. Besides the parsimonious estimation, the Nelson and Siegel (1987) model has two additional advantages over the nonparametric models. The first advantage is that the extrapolation of the curve has a better performance because of the exponential nature of this model. The second advantage is that this formulation avoids the problems in the construction of the forward curve, related to the absence of convexity adjustments, which occur in non-parametric methods. The extension formulated by Diebold and Li (2006) renders the Nelson and Siegel (1987) model dynamic (adjusting the several days observed for the yield curve) by means of a procedure in 3 stages:. 7.

(109) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 4. (1) The Nelson-Siegel model (with τ fixed, thus, making the model linear in the parameters) is fitted by ordinary least squares for each date, estimating the parameters β1t , β2t , β3t . (2) The dynamics of the system is modeled by a vector autoregressive (VAR) model for the parameters β1t , β2t and β3t estimated at the first stage. (3) Forecasts for these parameters are made through the VAR model estimated for vectors β1t , β2t and β3t . By substituting the forecasted parameters in Nelson-Siegel model given by Eq. 2.2, it is possible to forecast the future interest rate curves. According to Diebold and Li (2006), this dynamic formulation has the purpose of capturing the set of the existing stylized facts in the term structure of interest rates, such as the fact that while the yield curve is crescent and concave, it may also assume inverted shapes like decreasing curves and slope changes. Other stylized facts captured by Diebold and Li (2006) models are the high persistence in the time dynamics (rates with same maturity are highly dependent on the past) and the fact that persistence in the long-term rates is higher than that in the short-term rates. Though the Diebold-Li model is simple to implement and has a superior predictive potential when compared with other related models in the literature, some problems still arise when it is used. The three main limitations to this model are as follows: (1) To consider τ as fixed (linearization imposed in the model) may be troublesome for the more unstable yield curves, such as those of the emerging countries. (2) The functional form adapted from the Nelson and Siegel (1987) model does not allow for capturing more complicated yield curves, such as when there are multiple changes in the slope and curvature. (3) No econometric property of the estimation method has been presented. Consider that it is a two-step estimation, where the VAR is estimated on the basis of an estimated vector of beta parameters. The main problem is the construction of the confidence intervals in the finite samples for the forecasts obtained from this model. These intervals should take into account the uncertainty in the estimation of the vectors of the hyperparameters β1t , β2t and β3t . (4) As the forecasted curve may be contaminated by arbitrage situations, there is significant resistance to the use of models that are not based on no-arbitrage conditions. There are some proposed solutions to these problems. Problem 1 may be addressed by estimating the full Nelson and Siegel (1987) models without fixing the parameter τ , by generally using the nonlinear least squares. Yet, considering the limited number of observations in the yield curve,. 8.

(110) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 5. the problem of minimizing the nonlinear least squares may be complicated, presenting more than one local minimum, a possibility that may lead to an inappropriate fit of the yield curve. This is one of the justifications to keep the parameter τ fixed, to avoid the numeric optimization problems involved in the estimation of nonlinear models with a restricted number of observations. The simultaneous estimation of betas may be performed through the state-space formulation using the Kalman filter, but τ is kept fixed in the sample because of the need for linearity in the use of the linear Kalman filter. Some statistical properties of a model obtained from the Diebold and Li (2006) formulation were derived in Huse (2007), in which a form similar to the Nelson-Siegel model is used with the incorporation of spatial dependence and macroeconomic variables. The estimation is performed in two steps, but certain properties of the estimation method in the finite samples are studied using the Monte Carlo simulation. There are several works generalizing the Nelson-Siegel model in which the no-arbitrage condition is imposed (Christensen et al. (2008)), but they will not be considered here. 3. Proposed Extensions To overcome these problems, we have proposed an extended version of the Diebold and Li (2006) model using Bayesian methods. The Bayesian methods based on Markov Chain Monte Carlo (MCMC) are proposed as alternatives to the maximum likelihood estimation in large cases where the maximum likelihood methods are complicated or unfeasible to apply. Examples of estimation procedures using MCMC include: estimation of continuous-time diffusion processes for term structure of interest rates, option pricing, stochastic volatility, and regime switching models, as summarized in Johannes and Polson (2007). The advantages of the Bayesian formulation are that it enables us to treat both the parameters and state vectors as latent variables. This is carried out through the dynamic linear model formulation for the time evolution of those parameters. In the Bayesian formulation, it is not necessary to assume linearity, and hence, it is not necessary to fix the parameter τ , as it is done in the Diebold-Li method. It must be noted that the construction of the posterior distribution of the parameters is performed by simulation; hence, the various local minima that affect the estimation based on nonlinear least squares of Eqs 2.1 and 2.2 do not constitute a problem. The first Bayesian formulation of the model of Diebold and Li (2006), proposed by Migon and Abanto-Valle (2007), corresponds to an analogous specification of the original model, using Nelson-Siegel Eq. 2.1, with parameter τ kept fixed but estimated simultaneously with the other parameters of the model.. 9.

(111) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 6. We have proposed some extensions to the Bayesian formulation of the Diebold and Li (2006) model proposed by Migon and Abanto-Valle (2007). The first is to use the Svensson model (Eq. 2.2), rather than the original Nelson-Siegel formula (Eq. 2.1), which makes the curve format more flexible. The second extension is to make the parameters τ1 and τ2 time-varying, adding two latent factors to these components. The third extension is that the formulation of our model allows for different number of observations for each day, which avoids the first stage of curve interpolation to obtain a set of observations for the same maturities as it was originally performed in the article by Diebold and Li (2006), and may introduce distortions in the yield curves used in the estimation. The last extension introduced is to add a stochastic volatility structure to the model. This addition is of fundamental importance because one of the stylized facts in the interest rates is the presence of conditional heteroskedasticity, generally captured in no-arbitrage and equilibrium models by the addition of factors that specifically control the stochastic evolution of the variance. Examples of this kind of formulation include the Hull and White (1990) and Scott (1996) models, and a detailed discussion may be found in Fouque et al. (2000). The advantages of the Bayesian formulation are that the properties of the estimators are obtained in the exact form for finite samples, which allows calculating the confidence intervals for the hyperparameters and forecasting the term structure for interest rates, considering the uncertainty in the parameter estimation.. 4. Model Description We can describe the extensions proposed in this article by the following set of equations:. (4.1) yit (mit ) = β1t +β2t. 1 − e−mit /τ1t 1 − e−mit /τ1t 1 − e−mit /τ2t +β3 − e−mit /τ1t +β4t − e−mit /τ2t +eσt ηt mit /τ1t mit /τ1t mit /τ2t     µβ β 1 1t             β   µβ  2    2t               β3t   µβ3   + Φ  =         β4t   µβ4              τ   µτ  1    1t        µτ2 τ2t . (4.2).  β1t−1   β2t−1     β3t−1   + ǫt  β4t−1    τ1t−1    τ2t−1. 10.

(112) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. (4.3). 2 lnσt2 = φ0 + φ1 lnσt−1 + υt. (4.4). ηt ∼ IID(0, 1) e ηt ⊥ ηs ∀ t 6= s .  ση X  =  0  η,ǫ,υ 0. 0 Ωǫ 0. 7.  0   0    σv. In this specification, which may be considered as a nonlinear state-space model, Eq. 4.1 corresponds to a measurement equation, connecting the observed rates yit that describe the interest rate as the functions of maturities i at time t. In this specification, which may be seen as a nonlinear state-space model, Equation 4.1 corresponds to a measurement equation, connecting the observed rates yit that describe the interest rate as functions of maturities i at time t. The formulation of this equation follows the specification of Svensson model, but with the addition of latent factors βjt and τht , j = 1, 2, 3, 4 and h = 1, 2 are time-varying rather than fixed P in time. Matrix ε,ǫ,υ denotes the expanded variance-covariance matrix, where ση2 is a scalar. variance in the measurement equation, Ωǫ is the variance-covariance matrix between the latent. factors, and σv2 is a scalar variance in the stochastic volatility equation. We assume that the matrix is diagonal, except for the submatrix of components Ωǫ , which may be correlated. The evolution of the latent factors is given by Eq. 4.2, which describes a first-order autoregressive model for these components with a parameter matrix given by Φ, containing the coefficients of autoregressive estimation. We adopted a specification of first order for the autoregressive model, though noting that there is no theoretical limitation to a superior order. A possibility is to implement a restricted vector autoregressive structure, by working with only one autoregressive structure for each parameter. Although this may be imposed a priori, a possible alternative is the use of informative priors in the estimation of vector autoregressive models, as advocated by Doan et al. (1984). Finally, Eq. 4.3 describes the stochastic volatility components for the errors in the measurement equation. The formulation used is that of an autoregressive model for the unobserved stochastic. 11.

(113) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 8. volatility component, according to the original specification of the stochastic volatility model introduced by Taylor (1986). The addition of the stochastic volatility model represents a relevant extension, because the presence of conditional heteroskedasticity is a stylized fact in modeling the series of interest rates. We noted that the addition of stochastic volatility components is especially important at the moments of changes in the shape of the yield curve, especially because these moments are linked to greater uncertainties about future interest rates and the expectations about the ways assumed by the monetary and fiscal policy. A relevant stylized fact is that the volatility of the interest rates is greater in the emerging economies; thus, the component of stochastic volatility is especially relevant to the set of data used in this study. 5. Markov Chain Monte Carlo Estimation It must be noted that the model specification given by Eqs 4.1,4.2 and 4.3 corresponds to a nonlinear state-space model, and thus, cannot be treated by methods, such as the Linear Kalman Filter. A way to perform the simultaneous estimation is through methods of Bayesian Inference using the MCMC. The idea of the MCMC method is to simulate a Markov chain whose stationary distribution converges to the distribution p(Θ|y). The MCMC methodology simplifies the calculus, by factoring this distribution in a set of conditional distributions of inferior dimensions that can make the simulation easier. The Hammersley-Clifford theorem (see Robert and Casella (2004) for a derivation of this result) ascertains that under certain conditions, this set of conditional distributions will uniquely characterize the posterior distribution p(Θ|y), and the MCMC methodology is based on obtaining random samples of the conditional distributions, where a Markov Chain structure is used. An evident advantage of this method is that it does not involve any methodology of numerical maximization, thus avoiding the numerical problems involved in the nonlinear maximization of the functions such as those found in our problem. The validity of the methodology can be verified through methods that check the convergence of the Markov chains for its stationary distribution. The methodology of Bayes Hierarchical estimators is a convenient way to address the problem when the model to be estimated can be placed in a state-space formulation. Following the example given in Lehmann and Casella (1998), a form to represent these models is: X|θ ∼ f (x|θ) Θ|γ ∼ π(θ|γ) Γ ∼ ψ(γ). 12.

(114) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 9. Thus, we place a hierarchy structure among the prior distributions. This formulation is especially useful in state-space models, because the hierarchical specification allows for the estimation of the hyperparameters related to the latent factors using the avaliable data, specifying the dynamics for the latent factors. For example, the local level model is formulated as follows:. yt = µt + εt. (5.1). ,. µt = µt−1 + νt where we use as prior distribution of the latent factor µt , the value of µt−1 , and then µt ∼ p(µt−1 ), which corresponds to the idea of state equation in the state-space formulation. The specification of the latent factors uses a generalized formulation ξt ∼ p(ξt−1 ), where ξ denotes the set of latent factors in our model given by βit , τit and σi2 . This methodology is also known as empirical Bayes estimators (Lehmann and Casella (1998)). In our problem, we cannot directly sample all the conditional distributions, owing to the nonlinear forms involved. Thus, a Hybrid MCMC is used, where we simultaneously use the Gibbs algorithm and the Metropolis-Hastings algorithm, a methodology initially proposed in Tierney (1994). A hybrid MCMC algorithm (Robert and Casella (2004)) may be considered as iterations in the following stages: (t+1). For i=1,...,p ,and given (θ1. (t+1). (t). (t). , ..., θi−1 , θi , ...θp ). 1 - Simulate. 2 - Accept. (t+1) (t) (t) (t+1) , ..., θi−1 , θi , ...θp ) θei ∼ qi (θ|θ1.    θ(t) with probability 1 − ρ i (t+1) θi ,=   θei with probability ρ. where  (t+1) (t+1) (t) gii (θei |θ1 ,...,θi−1 ,,θi ,...θp(t) )    ^ e (t+1) ,...,θ (t+1) ,θ (t) ,θ (t) ,...θp(t) ) 1 i−1 i i qii (θi |θ ρ=1 (t+1) (t) (t) (t+1) ,...,θi−1 ,,θi ,...θp(t) ) gii (θi |θ1    (t) (t+1) (t+1) (t) (t) (t) qii (θi |θ1. ,...,θi−1 ,θi ,θi ,...θp ). where q is the so-called tentative distribution (we assume a multivariate gaussian distribution as tentative distribution) and g is the conditional distribution. To completely characterize our model, the prior distributions are the normal-gamma pair inverse for βit and τit , using the hierarchical characterization with the mean given by the vector. 13.

(115) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 10. Figure 6.1. Swap Di-PRÉ term structure(12/01/2004 - 05/12/2006). 0.22. 0.20. Intere. s t Ra. 0.18. te. 0.16 2500 0.14. Ma tur ity. 2000 1500 600. 1000 400 Tim e. 500 200. autoregressive structure. For the parameters Φ of the autoregressive vector, we assume a normal multivariate structure with the variance matrix given by Wishart distribution; for the latent factor 2 , τσ2 ), with a gamma distribution of stochastic volatility, we assume σt2 ∼ LogN ormal(φ0 + φ1t σt−1. for τσ2 , normal for φ0 , and finally φ1 ∼ Beta. For the parameters βit and φ0 , the parameters of Wishart distribution, and those of the gamma distributions, we used a Gibbs sampling step; for τit , we used Metropolis-Hastings; and for parameter φ1 , we used the algorithm known as Slice Sampler Neal (2003)).. 6. Application In this section, we have presented an application of this model for the fitting of term structure implicit in the SWAP DI-PRÉ curves provided by the BM&F (Mercantile and Futures Exchange) in Brazil. These instruments are swap contracts between floating and fixed interest rates, and are considered the more liquid fixed income market in Brazil. This yield curve is notoriously difficult to adjust by conventional methods. We have used the BM&F data on yield curves implicit in the SWAP operations for time interval from January 12, 2004 to December 12, 2006, a sample of 722 yield-curve days. Figure 6.1 shows the evolution of the yield curves over time. The interesting fact is that the curves in our study present several slope and curvature changes, going from the usual crescent shape to inverted curve, several times throughout the period. In the same interval, it also presents several days where the yield curves have two slope changes.. 14.

(116) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 11. This fact cannot be adequately captured by the model of Diebold and Li (2006), because the Nelson and Siegel (1987) formulation does not allow more than one slope and curvature change. Another point of importance is that the yield curve in Brazil has intense oscillation, both in terms of curve level and format, which reinforces the necessity to make the parameters time-varying and challenges the maintenance of parameter τ as fixed, as assumed by the Diebold and Li (2006) model. Another important point is that the yield curve lengthens and retracts in the mentioned period, that is, the maximal maturities observed in the SWAP contracts change in the analyzed sample, varying between 1800 and 2400 days. It must be noted that our study does not carry out a pre-interpolation and extrapolation on the data; the methodology permits to work with distinct maturities in each day. The average number of distinct maturities is 24, with a minimum of 20 and maximum of 29. This fact must be highlighted because the interpolation stage may distort the data, and the estimated model may be used to interpolate and extrapolate the curve if necessary. To estimate the model, we used 10,000 iterations of the MCMC algorithm described in Section 5, discarding the first 5,000 iterations (burn-in period) and using the other 5,000 in the construction of posterior distributions. The Gelman-Rubin convergence diagnoses indicate that the Markov chains converge to the stationary distributions, thus validating the estimation methodology used. Figure 6.2 shows the model fit inside the sample and the residuals in relation to the observed curves. Figures 6.3, 6.4, 6.5 and 6.6 show the evolution of the latent factors β1t , β2t , β3t and β4t obtained as medians of posterior distributions. The evolution of β1t clearly shows the level of interpretation of this parameter, following the evolution of the mean yield curve over time. The evolution of the other hyperparameters also adequately captures the evolution of the slope and curvature components of the term structure observed in the interest rates. Figure 6.7 and 6.8 are of of special importance because it they shows that the prior fixing of parameter τ assumed in Diebold and Li (2006) model is not a valid restriction, as it becomes evident by the great Timeral variation observed in parameters τ1 and τ2 . This gives an indication regarding the necessity to incorporate variation in these parameters for the yield curves with great variation of format, as observed in the emerging countries. The estimated Stochastic Volatility component (Figure 6.9) shows the capacity of the model to capture the stylized fact of the presence of conditional heteroskedasticity in the interest rates. The structure of conditional volatility captures the uncertainty existing in the periods of change. 15.

(117) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 12. Figure 6.2. Adjusted curve and residuals. 0.24. 0.22. Intere. 0.20. st Rat. 0.18. e 0.16 10 0.14. Ma turi ty. 8 6. 600 4 400 Tim e. 2 200. 0.005. e. st Rat. Intere. 0.000. 10 −0.005. 600 4 400 Tim e. Ma. turi. 6. ty. 8. 2 200. Figure 6.3. β1. 0.14. 0.16. 0.18. β1. 0.20. 0.22. 0.24. 0.26. β1. 0. 200. 400. 600. Time. in the yield curves’ shapes, because we can notice the correlation between increase in volatility and periods of inversion in the curve format. Figure 6.10 shows another fact captured by the Stochastic Volatility structure-the high persistence of shocks in the volatility. This is noticeable because parameter φ1 is concentrated on values close to 1.. 16.

(118) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 13. Figure 6.4. β2. −0.10. −0.05. β2. 0.00. 0.05. β2. 0. 200. 400. 600. Time. Figure 6.5. β3. −0.4. −0.3. −0.2. −0.1. β3. 0.0. 0.1. 0.2. β3. 0. 200. 400. 600. Time. Figure 6.6. β4. −1.5. −1.0. β4. −0.5. 0.0. β4. 0. 200. 400. 600. Time. Table 1 shows the credibility intervals calculated for the matrix of coefficients Φ. To verify the stationarity of the process, we calculated the eigenvalues of matrix Φ for the upper and lower limits of this matrix. The highest eigenvalues for the higher limit was 1.0029, and that for the. 17.

(119) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 14. Figure 6.7. τ1. 0.28 0.22. 0.24. 0.26. τ1. 0.30. 0.32. τ1. 0. 200. 400. 600. Time. Figure 6.8. τ2. 0.30. 0.35. τ2. 0.40. 0.45. τ2. 0. 200. 400. 600. Time. Figure 6.9. Stochastic Volatility. 0.035 0.030 0.015. 0.020. 0.025. Stochastic Volatility. 0.040. 0.045. 0.050. Stochastic Volatility. 0. 200. 400. 600. Time. lower limit was 0.9783, indicating that the region of nonstationarity is included in the credibility intervals.. 18.

(120) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 15. Table 1. Credibility Intervals 95% - Φ µ. β1t−1. β2t−1. β3t−1. β4t−1. β5t−1. β6t−1. Φβ1 (.025). .2310. 1.262. .5203. -.1122. -.4625. -.0152. -.1183. Φβ1 (.50). .2407. 1.281. .5504. -.1077. -.4432. -.0136. -.1115 -.0999. Φβ1 (.975). .2532. 1.293. .5721. -.1033. -.4266. -.0139. Φβ2 (.025). -.2582. -.2877. .4215. .1051. .4337. .0135. .1005. Φβ2 (.50). -.2453. -.2755. .4437. .1102. .4490. .0141. .1126 .1189. Φβ2 (.975). -.2316. -.2572. .4746. .1149. .4702. .0154. Φβ3 (.025). -.4741. -.4378. -.9778. 1.141. .5951. .0183. .1253. Φβ3 (.50). -.4020. -.3385. -.8204. 1.171. .7172. .0224. .1619. Φβ3 (.975). -.3331. -.2330. -.6642. 1.201. .8445. .0270. .1998. Φβ4 (.025). .1570. .1605. .3217. -.0822. .1397. -.0115. -.0896. Φβ4 (.50). .1764. .1918. .3705. -.0728. .1782. -.0100. -.0783. Φβ4 (.975). .1984. .2205. .4234. -.0639. .2124. -.0087. -.0659. Φβ5 (.025). .3965. .4719. .9298. -.5165. -2.2090. .9181. -.6076. Φβ5 (.50). .7881. 1.1180. 1.8360. -.3394. -1.5045. .9456. -.3833. Φβ5 (.975). 1.1680. 1.8060. 2.787. -.1688. -.7929. .9707. -.1819. Φβ6 (.025). .2788. .3752. .6515. -.1911. -.8212. -.0282. .7723. Φβ6 (.50). .3651. .5022. .8621. -.1567. -.6838. -.0232. .8116. Φβ6 (.975). .4360. .6283. 1.0570. -.1191. -.5263. -.0173. .8538. Figure 6.10. Posterior Distribution - Φ0 and Φ1 Histogram of phi1. 200 0. 0. 50. 100. 100. 150. Frequency. 300 200. Frequency. 400. 250. 500. 300. 600. Histogram of phi0. 0.032. 0.034. 0.036. 0.038. 0.040. 0.042. 0.9975. 0.9980. phi0. 0.9985. 0.9990. 0.9995. 1.0000. phi1. To demonstrate the predictive potential of the model, we showed the forecasts for some specific days, characterized by distinct shapes of the yield curve. We also showed the forecasts and onestep-ahead forecast errors for all the days observed in the sample. Figure 6.11 shows the one-step-ahead forecasts obtained by the extended Diebold-Li model, with confidence intervals at the 2.5% and 97.5% limits, for 4 days observed in the yield curve. The first subfigure shows the prediction for July 20, 2004, with the format generally observed in the interest rates, with a positive trend in maturity. The second curve, predicted for February 01, 2005, shows a curve with slope change, normally associated with expected changes in the long-term interest rates. The curve predicted for June 27, 2006 shows an opposite situation, with a decreasing. 19.

(121) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 16. 0.15. 0.16. 0.16. 0.17. 0.18. Interest Rate Forecast. 0.18 0.17. Interest Rate Forecast. 0.19. 0.19. 0.20. 0.20. Figure 6.11. One-step ahead forecasts and forecast errors-specific days. 0. 1. 2. 3. 4. 5. 0. 1. 2. Maturity. 3. 4. 5. Maturity. (b) 01-02-2005. 0.135 0.130. Interest Rate Forecast. 0.125. 0.160 0.155 0.150 0.140. 0.120. 0.145. Interest Rate Forecast. 0.165. 0.140. 0.170. (a) 20/07/2004. 0. 2. 4. 6. 8. 10. 0. Maturity. (c) 27/06/2006. 2. 4. 6. 8. 10. Maturity. (d) 06/12/2006. curve at the medium-term maturities and an increasing curve at the long-term maturities. The subfigure d shows a one-step-ahead forecast for the last observation in the sample, the observation referring to December 6, 2006. The one-step-ahead forecasts for all the samples, along with the inclusion of extrapolations for the unobserved maturities and associated prediction errors, are shown in Figure 6.12. The forecast errors have relatively low magnitude. It can be noticed that the bigger errors are concentrated at the moments of change in the shape of the yield curve. We also carried out a comparative forecast analysis between the extended Diebold-Li model and the original formulation of Diebold-Li with fixed τ , the time-varying specification of the Diebold-Li model, and a modification in the Diebold-Li model using Svensson Eq. 2.2 to replace the original formulation based on the Nelson-Siegel specification, with parameters τ1 and τ2 being fixed and time-varying, respectively. The estimation of these reference models for the time-varying τ was based on nonlinear least squares, whereas the linearized forms were based on the estimation by. 20.

(122) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 17. Figure 6.12. One step Ahead Forecasts and Forecast Errors. 0.24 0.005 0.22. Intere. Intere. 0.20. 0.000. st Rat. st Rat. 0.18. e. e 0.16 10. 10 −0.005. 0.14. 8. 600 4 400 Tim e. 2 200. (a) One step Ahead Forecasts. Ma turi ty. Ma turi ty. 8 6. 6. 600 4 400 Tim e. 2 200. (b) Forecast Errors. ordinary least squares. Table 2 presents the root mean square error using the one-step-ahead forecast errors for the five compared models. Parameters τ, τ1 and τ 2 were fixed by the mean value of the corresponding time-varying parameters. The results of this comparative analysis showed that the Diebold-Li model with the proposed extensions has a superior forecast performance when compared with the other models, as shown in Table 2. The original Diebold-Li model with parameter τ being fixed, using the Nelson-Siegel specification, is not a valid specification because it substantially reduces the predictive power of the model when compared with the varying-parameter version of the same model. In the case of the DieboldLi model using Svensson specification, the fixed parameters result in better predictive power than the estimation with free parameters. This result may be explained by the difficulty in estimating Svensson specification, because on many days, the estimation does not converge because of nonlinearity, which makes the model fitting inadequate, thus raising the mean quadratic error value with the presence of high forecasting errors for all the maturities observed on those days. This problem also contaminates the autoregressive vector estimation, which compromises on the curve forecasting for the following day. However, the use of Bayesian estimation with informative priors allows us to employ the more flexible Svensson specification, but without being affected by the instability problems in the nonlinear estimation, which occur in the classical estimation that use nonlinear least squares.. 21.

(123) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 18. Table 2. Root mean square forecast error Model Root Mean Squared Error Extended Diebold-Li 1.1987 Original Diebold-Li τ fixed 36.61 Original Diebold-Li τ varying 23.92 Diebold-Li-Svensson τ1 and τ2 fixed 17.05 Diebold-Li-Svensson τ1 and τ2 varying 204.6. 7. Conclusions In this article, we implemented some extensions for the Diebold and Li (2006) model, including the use of Bayesian estimation methods using MCMC for the parameters and latent factors of this model. The proposed extensions for the model were the changes in the functional form, making it more flexible with the use of the Svensson (1994) specification, including latent factors, to make the model parameters time-varying, and enabling the use of different observations on each day and the inclusion of a stochastic volatility structure. The more flexible form adopted allows capturing changes in the shapes associated with the yield curve in the emerging countries. This flexibility is reflected in the low forecasting and fitting errors observed in this model. The use of the Bayesian estimation associated with informative priors in the latent factors specification avoids the procedure of linearized estimation in the two stages employed in the Diebold-Li model, which results in more precise fits and forecasting. The parameter specification, as modeled latent factors through a Bayesian hierarchical structure, allows obtaining the distribution in finite samples of parameters and model predictions, thus enabling the quantification of the uncertainty present in the estimation of the term structure of interest rates We demonstrated that it is important to make the τ parameters time-varying to fit the term structure, in particular, to yield curves in the emerging countries with constant modifications in the shape of the yield curve, as observed by the behavior of the latent factors τ1t and τ2t . The latent factors, owing to their informative prior structure, allow us to overcome the common problem of numerical instability associated with the nonlinear estimation of the Nelson-Siegel and Svensson models in the presence of a restricted number of observations. On the other hand, the Bayesian estimation methodology, through the MCMC algorithms, carries out the estimation simultaneously and allows us to avoid the linearization of the model and the estimation in the two stages used in Diebold and Li (2006).. The specification of the model is based on a standard set of priors, and the estimation algorithm, based on a mixture of Gibbs and Metropolis-Hastings, is. 22.

(124) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 19. widely used and its properties are extensively studied, thus making the estimation of the model simple and trustworthy. References Brennan, M. J. and Schwartz, E. J.: 1979, A continuos time approach to the pricing of bonds, Journal of Banking and Finance 3, 133–155. Christensen, J. H., Diebold, F. X. and Rudebusch, G. D.: 2008, An arbitrage-free generalized nelson-siegel term structure model, Econometrics Journal forthcoming. Cox, J. C., Ingersoll, J. . E. and Ross, S. A.: 1985, A theory of the term structure of interest rates, Econometrica 53, 385–408. Diebold, F. and Li, C.: 2006, Forecasting the term structure of government bond yields, Journal of Econometrics 130, 337–364. Doan, T., Litterman, R. and Sims, C.: 1984, Forecasting and conditional projection using realistic prior distributions., Econometric Reviews 3, 1–100. Duffie, D. and Kan, R.: 1996, A yield-factor model of interest rates, Mathematical Finance pp. 379– 406. Fouque, J.-P., Papanicolaou, G. and Sircar, K. R.: 2000, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press. Heath, D., Jarrow, R. and Morton, A.: 1992, Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation, Econometrica 60(1). Hull, J. and White, A.: 1990, Pricing interest rate derivative securities, Review of Financial Studies 3(4), 573–94. Huse, C.: 2007, Term structure modelling with observable state variables. Unpublished Working Paper - FMG - LSE. Johannes, M. and Polson, N.: 2007, Handbook of Financial Econometrics, chapter MCMC Methods for Continuos Time Financial Econometrics. Lehmann, E. and Casella, G.: 1998, Theory of Point Estimation (2nd Edition), Springer. Linton., O., Mammen, E., Nielsen, J. and Tanggard, C.: 2001, Estimating yield curves by kernel smoothing methods, Journal Of Econometrics 105:1, 185–223. Litterman, R. and Scheinkman, J.: 1991, Common factors affecting bond returns, Journal of Fixed Income 1, 54–61. McCulloch, J.: 1971, Measuring the term structure of interest rates, Journal of Business (44), 19– 31.. 23.

(125) BAYESIAN EXTENSIONS TO DIEBOLD-LI TERM STRUCTURE MODEL. 20. Migon, H. and Abanto-Valle, C.: 2007, A Bayesian term structure modelling, in C. Fernandes, H. Schimidli and N. Kolev (eds), Proceedings of the Third Brazilian Conference on Statistical Modelling in Insurance and Finance, IME-USP, pp. 200–203. Neal, R.: 2003, Slice sampling (with discussions), Annals of Statistics 31, 705–767. Nelson, C. R. and Siegel, A. F.: 1987, Parsimonous modelling of yield curves, Journal of Business 60(4), 473–489. Robert, C. and Casella, G.: 2004, Monte Carlo Statistical Methods, Springer. Scott, L. O.: 1996, Simulating a multi-factor term structure model over relatively long discrete time periods, Procedings of the IAFE First Annual Computacional Finance Conference. Shea, G.: 1984, Pitfalls in smoothing interest rate structure data: Equilibrium models and spline approximation, Journal of Financial and Quantitative Analysis 19, 253–269. Svensson, L. E. O.: 1994, Estimating and interpreting forward interest rates: Sweden 1992-1994., NBER Working Paper (4871). Taylor, S. J.: 1986, Modelling Financial Time Series, John Wiley& Sons. Tierney, L.: 1994, Markov chains for exploring posterior distributions (with discussion), Annals of Statistics 22, 1701–1786.. 24.

(126) GENERALIZED LATENT FACTOR MODELS FOR YIELD CURVES IN MULTIPLE MARKETS MÁRCIO POLETTI LAURINI INSPER INSTITUTE AND IMECC-UNICAMP LUIZ KOODI HOTTA IMECC-UNICAMP Abstract. In this article we propose latent factors models to model simultaneously yield curves in multiple markets, generalizing several models found in the literature on the estimation of term structure of interest rates. The proposed models do not use some of usual restrictions adopted for estimation and identification, thus enabling us to use more flexible structures incorporating additional latent factors, stochastic volatility and the imposition of no-arbitrage consistency. The elimination of these restrictions is made possible through the Bayesian estimation methodology using the Markov Chain Monte Carlo (MCMC). This methodology makes it possible to obtain exact confidence intervals for the parameters, latent factors and forecasts, and also to address identification and dimensionality problems in the estimation of multimarket models. The models are applied to model jointly Cupom Cambial (USD interest rate in Brazil) and Eurodollar curves, carrying out an extensive procedure of model comparison and demonstrating the forecast and practical potential of the proposed models. Keywords: Term Structure, Latent Factors,No-arbitrage, Forecasting. JEL Codes: C11, G12, G17.. Adress- Insper Institute - Rua Quatá 300, 04546-042, São Paulo, SP. Brasil. email - Márcio Laurini - [email protected] Luiz Koodi Hotta - [email protected]. 1. 25.

(127) GENERALIZED LATENT FACTOR MODELS FOR YIELD CURVES IN MULTIPLE MARKETS. 2. 1. Introduction Modeling the term structure of interest rates is a fundamental point in the management of capital assets. A considerably large literature has been developed to obtain more precise forms for the modelling, forecasting and pricing of financial instruments with basis on the yield curve. Among these approaches an important part of the literature is based on the idea that the dynamic evolution of the yield curve may be described using a set of dynamic factors that determine the evolution of risk premiums for the various maturities observed. The most common way of considering these factors is through a representation using latent state variables, that is, as variables not directly observed. 1. The purpose of these latent factors is to summarize the whole set of relevant variables determining the yield curves’ movements. The methodologies for the extraction of these latent factors may arise from purely statistical mechanisms, such as the decomposition of principal components introduced in Litterman and Scheinkman (1991), where the latent factors are interpreted as components of level, slope and curvature. These latent factors may also be identified by methodologies of pricing by equilibrium, such as the shortrate models of Vasicek (1977) and Cox et al. (1985), which belong to the class of affine models (Affine Diffusions, e. g. Cox et al. (1985)). These equilibrium models may also be placed in a general framework based on no-arbitrage conditions by the Heath-Jarrow-Morton ( Heath et al. (1992)) formulation, which determines the evolution of forward rates as a stochastic process of infinite dimension. All these approaches, however, show partial success in the empirical modeling of the dynamic evolution of the term structure of interest rates. The equilibrium models and the affine models, though having important analytical properties such as the existence of closed formulas for asset pricing, are both characterized by a rather unsatisfactory fit of the rates observed and of the forecasts derived from these models as well. An additional difficulty is that, in general, the econometric estimation of these models suffers from problems of local maxima and identification, as pointed out by Duffe (2002). The models of no-arbitrage 1. For references about modeling the term structure of interest rates see, for example, Brigo and Mercurio (2006) for aspects related to the pricing of financial instruments, and Singleton (2006) about the estimation of models of the term structure of interest rates.. 26.

(128) GENERALIZED LATENT FACTOR MODELS FOR YIELD CURVES IN MULTIPLE MARKETS. 3. are calibrated so that they replicate perfectly the yield curve observed in the market by way of matching, using bond prices, but this calibration is cross-sectional and does not allow for forecasting future curves. It only allows pricing of derivative instruments. Moreover, these models are recalibrated daily using instruments observed in the yield curve. Diebold and Li (2006), whose main objective is to forecast the term structure of interest rates, propose a dynamic model using the parametric form for the yield curve proposed by Nelson and Siegel (1987), and interprete this model as a latent factor model. In this generalization each parameter of the cross-section fit of the Nelson-Siegel’s model is considered as a latent factor, and through the modeling and forecasting of these latent factors it is possible to obtain forecasts for the whole term structure of interest rates. The results obtained by Diebold and Li (2006) indicate that this formulation presents a fit and a forecast power superior to other methodologies of yield curve modeling, making this model the standard reference for term structure forecasting. The model proposed in Diebold and Li (2006) was also attractive because of its ease of implementation. With some restrictions about the parametric space, this model could be estimated by using only estimation by Ordinary Least Squares while the other models would require more complex estimation tools such as the Kalman filter (e.g. Duffe (2002)) or estimation methods such as the Simulated Method of Moments, employed in the estimation of affine models in Dai and Singleton (2000). Apart from simplifying its implementation, the restrictions imposed in the Diebold and Li (2006) model were necessary to avoid the usual problems in the estimation of term structure of interest rates models, such as the above mentioned problems of local maxima and non-identification. Based on the success they obtained in the dynamic extension of the Nelson-Siegel curve, Diebold et al. (2008) proposed a generalization of this model to fit multiple yield curves simultaneously, employing a methodology that consisted in building latent factors connected to a not directly observed global yield curve. In the Diebold et al. (2008) model, the yield curve of each market would be obtained as a linear displacement of the global yield curve plus an idiosyncratic factor, by means of these latent factors. It is important to note that the [Diebold et al., 2008] formulation is the first attempt at creating a model that. 27.

(129) GENERALIZED LATENT FACTOR MODELS FOR YIELD CURVES IN MULTIPLE MARKETS. 4. makes it possible to capture simultaneously the dynamics of several term structures. This formulation has also been adopted to model the yield curves of emerging countries in Morita and Bueno (2008), thus demonstrating the general applicability of this model. However, the model proposed by Diebold et al. (2008) employs a series of restrictions in its formulation. Given the high number of parameters involved in the estimation of the global model, Diebold et al. (2008) employ a rather limited specification for the general shape of the yield curve in each market. Instead of using a complete formulation of the Nelson-Siegel model with level, slope and curvature factors, Diebold et al. (2008) use only the components of level and slope, which makes the fit for the observed yield curves rather limited, although it is important to note that the primary purpose of this model was not its fit or forecast: its purpose was rather to verify the existence of a global factor influencing the movements of the term structure in most important markets. An additional restriction is also used in this model, in that the parameter defining the slope of the yield curve should be kept constant, which significantly impairs the model’s fit. Some other problems found in this formulation refer to the estimation procedures: the use of a procedure in two stages does not make it possible to obtain measures such as exact confidence intervals for the model’s parameters and for the yield curve forecasts. Further problems relate to the model’s identification, that is, to obtain conditions for a single vector of parameters able to define the maximum of the likelihood function employed in the model’s estimation. And finally, other problems found in this formulation relate to the presupposition of a constant conditional volatility, which contradicts one of the stylized facts in the modeling of yield curves. Furthermore, the formulation proposed in Diebold et al. (2008) does not overcome one of the fundamental criticism to the original model of [Diebold et al., 2006], that is, the model’s inconsistency with no-arbitrage conditions. This original limitation of the Diebold et al. (2008) model was resolved in Christensen et al. (2007, 2008), who demonstrate that, although the original formulation of the Diebold et al. (2008) model is incompatible with no-arbitrage conditions, it is nevertheless possible to work with an approximate form of this model which is arbitrage-free, reparametering the Diebold et al. (2008) model as an affine model of term structure and obtaining a term of correction that enables the incorporation of the no-arbitrage. 28.