Forecasting the Term Structure of Philippine Interest Rates Using the Dynamic Nelson Siegel Model

May 3, 2018 | Author: Lester Umali | Category: Yield Curve, Forecasting, Time Series, Derivative (Finance), Bonds (Finance)


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190 E.P.de Lara-Tuprio, et al DLSU Business & Economics Review (2017) 27(1): 190-200 RESEARCH ARTICLES Forecasting the Term Structure of Philippine Interest Rates Using the Dynamic Nelson-Siegel Model Elvira P. de Lara-Tuprio, Ramil T. Bataller, Allen Dominique D. Torres, Emmanuel A. Cabral and Proceso L. Fernandez Jr. Ateneo de Manila University, Quezon City, Philippines [email protected] Abstract: The three-factor Nelson-Siegel model is a widely used model for forecasting the term structure of interest rates. Several extensions have recently been proposed. Even for the original model, different methods of treating the parameters have been shown. Ultimately, what works best depends on the data used to estimate the parameters. In this paper, the original three-factor model with fixed shape parameter was applied to forecast the term structure using market data from the Philippines. Instead of giving a pre-determined model for the latent factors, the best time series model for them was searched using standard statistical tools. Based on the historical data, the best model for each latent factor is of the form ARMA(p,q)+eGARCH(1,1). The dependence structure of these parameters was considered in generating their future values. This was carried out by finding the joint distribution of the residuals via appropriate copula. Results show that forecast of interest rates for different tenors is reliable up to the near future. For an active market, this is good enough since the models for the parameters can be adjusted as new information comes in. Keywords: Yield curve, Nelson-Siegel, forecasting, copula, time series JEL Classification: G17, E47, C53 In recent years, the influx of new financial products mathematical models. Loss estimates and pricing are in the Philippines, including financial derivatives, based on current and forecast of underlying economic has become inevitable. Some local banks have been variables such as interest rates and currency exchange granted limited authority by the Bangko Sentral ng rates. Currently, several financial institutions in the Pilipinas (BSP) to engage in specified derivatives Philippines are either using vendor-developed systems, transactions, while some others are still in the stage of which are essentially β€œblack box,” or have developed preparing to apply for a license for such transactions. their internal models which are mostly based on The bond market is also on the rise with an increasing popular or classical models. There is indeed a challenge participation of corporate bond issuers. Amidst these for financial institutions, particularly in the banking developments is the need for sound risk management industry, to develop their own models which are more structure and reliable models for the pricing and reliable, relevant, and suited to the market data. valuation of financial products. Public debt management also requires sound Financial risk management and valuation of mathematical models for forecasting the evolution of financial securities require sound and reliable macroeconomic factors such as interest rates, exchange Copyright Β© 2017 by De La Salle University TERM STRUCTURE FORECASTING TERM STRUCTURE TERM STRUCTURE FORECASTING FORECASTING 4 TERM STRUCTURE FORECASTING TERM STRUCTURE FORECASTING TERM STRUCTURE FORECASTING time series model will betools. 4 using standard statistica searched time series model will be searched using time standard series statistical The model that will be used model will be searched using standard 191 statistical tools. Philippine Interest Rates Using the Dynamic Nelson-Siegel Model time series model will be searched using standard statistical tools. The model that will be u time series model will of time be the form searched series model will beofsearched ARMA(p,q)+eGARCH(1,1) using standard statistical the form of using the tools. form or The standard ARMA ARMA(p,q)+eGARCH(1,1) model forstatistical ARMA(p,q)+eGARCH(1,1) thatthe will mean tools. be used equationor The model is ARMA and forthat or Exponential ARMA thewill forbe mean the u eq rates, and primary budget of the form ARMA(p,q)+eGARCH(1,1) deficit/surplus. Finding an tools. The model or ARMA that will for the be usedmeanisequation of the form and Exponential optimal debt strategy of the form ARMA(p,q)+eGARCH(1,1) means of the form determining GARCH for the variance ARMA(p,q)+eGARCH(1,1) a mix or ARMAof debtequation. for the mean GARCH GARCH Moreover, ARMA(p,q)+eGARCH(1,1) for the equationor for variance ARMAthe forand for variance forecasting, Exponential equation. the or mean equation. ARMAthe equation dependence Moreover, for the for and Moreover, mean Exponential for forecasti structure forecasting, ofthe thed instruments that minimizesGARCH cost to thefor government, the variance in equation. equation and Exponential Moreover, for forecasting, GARCH the for dependence the variance structure of terms of interest GARCH payment, for the variance beta GARCH subject parameters equation. for to a the Moreover,willvariance prudent beforrisk equation. considered forecasting, beta beta equation. by the parameters Moreover, usingparameters Moreover, befor an appropriate dependence will for forecasting, will be considered forecasting, structure considered copulaof bythe the to dependence the using byanusing dependence obtain theiranjoint appropriate structure appropriate copula of level. It may then be used beta to guide the policy makers structure of the parameters will be considered by using an appropriate copula to obtain their joint beta parameters will be considered in creating beta a short-term parameters orbeta will bedistribution. considered parameters medium-term by using will borrowing be considered an appropriate distribution. by an distribution. by copula using using to appropriate obtain an appropriate theircopula joint to copula obtaintotheir obtain joint their joint plan, which will hopefully distribution. improve the domestic debt distribution. market due to enhanced predictability distribution. distribution. and transparency The remainderThe of this paper asremainder of this proceedspaper as follows: proceeds as follows: The remainder of this paperThe proceeds remainder follows: of this paper The nextproceeds section as discusses follows: the TheNelsonextT related to public debt. The next section discusses the Nelson-Siegel equation The remainder of this paper proceeds as follows: The next section discusses the Nel Forecasting the termofstructure of interest The remainder this paper Siegel The equation proceedsfor rates remainder has theasyield. follows: ofSiegelWefor this The paperthe Siegel then equation yield. next describe We then section proceedsequation for the as the describe discusses follows: for data yield. the andWe the theThe yield. the data Nelson- next thengeneralWe and describe the section then general thediscusses describe procedure datathat the the anddatatheNel will an be ge created a huge literature spanning several decades. procedure that will be implemented, to be followed Siegel equation for the yield. We then describe the data and the general procedure that will Some of Siegel these make equation useyield. for the ofSiegel mean-reverting Weequation then describe stochastic for theyield. the data and with We thegeneral the then implemented,results describe of to the procedure the beimplementation data that followed and will the including begeneral with procedure theofresults theof thethat implemwill implemented, models, such as the models of Vasicek (1977) and Cox, to be followed with implemented, the results to be of the followed implementation with the resultsincluding the the time series implementation time series found for the beta parameters, the copula Ingersoll, and to Ross implemented, towhich be followed with to theobtainresults ofjoint the implementation including the time serie implemented, be (1985). followed Another with the implemented, classresults toparameters,has implementation beoffollowed the used with the beta found including results for their the of beta thethe time seriesthe parameters, implementation parameters, and copula the forecast including used the tothetime obtain serieth found gained popularity among government policy makers for the beta found the copula used to obtain their performance of the model. Finally, we provide the their joint for the parameters, the joint copula parameters, used to obtain and forecas due to for found its good fit toparameters, the beta observed found term forcopula the the beta structure parameters, includes used to obtain conclusion.the joint their copula used to obtain parameters, and the their forecastjoint parameters, and the fore found performance for the beta parameters, the copula performance used ofto obtain the Finally, wejoint model. their Finally, parameters, we provide and the theconclusfore Nelson and Siegel’s (1987) model andofitsthe model. Finally, variations. performance we provide of the model. the conclusion. provide the conclusion. This paperof performance aims thetomodel. presentperformance a method Finally, wefor offorecasting provide the model. Finally, we provide the conclusion. the conclusion. performance of the model. Finally, weNelson-Siegel provide the conclusion. The Nelson-Siegel Model Mo the term structure of interest rates that is applicable to TheThe Nelson-Siegel Model Model The Nelson-Siegel the Philippine market data. It is based on The theNelson-Siegel framework Model The Nelson-Siegel Model The aLetNelson-Siegel (𝜏𝜏)pricebe the atModel price atof time of a zero-coupon 1𝑑𝑑 peso 𝜏𝜏,bo developed by Diebold and LiLet 𝑝𝑝𝑑𝑑 (𝜏𝜏)which (2006), be the isprice a at time Let𝑑𝑑 𝑝𝑝of Let 𝑑𝑑 (𝜏𝜏) bebe𝑝𝑝the zero-coupon 𝑑𝑑the price bond at time time thatt 𝑑𝑑of pays a azero-coupon zero-coupon at time bond𝑑𝑑 +that t reinterpretation Let 𝑝𝑝𝑑𝑑 (𝜏𝜏)ofbethe themodel price at Let 𝑑𝑑𝑝𝑝𝑑𝑑of(𝜏𝜏) introduced time by be the pricebond Nelson a zero-coupon at time bond that 𝑑𝑑 of pays a that pays 1 peso at timezero-coupon 1 peso at time bond 𝑑𝑑 + that 𝜏𝜏, that pays 1 is, t is the peso at time 𝑑𝑑 + 𝜏𝜏 and Siegel (1987). Although is, 𝜏𝜏 is several Let 𝑝𝑝 the timeextensions (𝜏𝜏) be to maturity the price of 𝜏𝜏the areis, is at time is, bond 𝜏𝜏 thetotime 𝑑𝑑 is of the in years. a zero-coupon time to maturity ofLet to the𝑦𝑦bondmaturity (𝜏𝜏)the bond beyears.of that the the corresponding pays bond 1 in pesoyears. at Letthe𝑦𝑦continuously time Let 𝑦𝑦 𝑑𝑑 c𝜏𝜏 𝑑𝑑 + (𝜏𝜏) 𝑑𝑑 time maturity 𝑑𝑑of in bond in years. Let yt (t) be 𝑑𝑑 (𝜏𝜏) be the available, is, 𝜏𝜏 is thethe original time three-factor to maturity is,of𝜏𝜏 the model is thebond time will in tobematurity years. used. Let 𝑦𝑦𝑑𝑑of (𝜏𝜏) thebe bond the corresponding corresponding in years. continuously Let continuously be the corresponding 𝑦𝑦𝑑𝑑 (𝜏𝜏)compounded nominal continuous The procedure, however, is, will 𝜏𝜏 isbethe time modified compounded nominal yield.compounded to maturity and the of the Thisyield. bond compounded yieldThis in isnominal also years. yieldreferred is also Let nominal yield. 𝑦𝑦 toThis referred 𝑑𝑑 (𝜏𝜏) yield. be as ayield tozero the This as ais corresponding rate. zeroyield Then alsorate. isThen referred also referred tocontinuous as a zero to a results of the implementation compounded nominal yield. will be highly This yield isnominal compounded dependent also referred yield.toThis as ayield zero rate. is also Then referred to as a zero rate. Then on the data used. compounded nominal yield. This yield𝑝𝑝 is(𝜏𝜏) also = referred 𝑒𝑒 βˆ’πœπœπœπœπ‘‘π‘‘(𝜏𝜏) to as a zero rate. Then = 𝑒𝑒 βˆ’πœπœπœπœπ‘‘π‘‘(𝜏𝜏) ( (𝜏𝜏)𝑑𝑑(𝜏𝜏) 𝑝𝑝𝑒𝑒𝑑𝑑(1) βˆ’πœπœπœπœ 𝑑𝑑 𝑝𝑝 𝑑𝑑 (𝜏𝜏) = The Nelson-Siegel model is a popular𝑝𝑝𝑑𝑑model (𝜏𝜏) = for (𝜏𝜏) 𝑒𝑒 βˆ’πœπœπœπœπ‘‘π‘‘ 𝑑𝑑 (𝜏𝜏) (1) 𝑝𝑝𝑑𝑑 (𝜏𝜏) = 𝑒𝑒 βˆ’πœπœπœπœ the term structure of interest rates. It was introduced Ifisrate 𝑝𝑝is𝑑𝑑is(𝜏𝜏) the= the βˆ’πœπœπœπœ 𝑒𝑒 at instantaneous (𝜏𝜏) forward rate withatmaturity 𝑓𝑓𝑑𝑑 (𝜏𝜏) 𝑑𝑑at+ 𝑑𝑑 by 𝑓𝑓Nelson and Siegel If 𝑓𝑓𝑑𝑑 (𝜏𝜏) is the instantaneousIf forward (1987) as a class of parametric 𝑓𝑓𝑑𝑑 (𝜏𝜏)If the with instantaneous maturity instantaneousforwardtime 𝑑𝑑forward + 𝜏𝜏 with rate rate contracted maturitywith time at time 𝑑𝑑, then If 𝑑𝑑 (𝜏𝜏) is the instantaneous If forward 𝑓𝑓𝑑𝑑 (𝜏𝜏) is rate with the instantaneous maturity at time maturity rate forward 𝑑𝑑 + 𝜏𝜏 at time contracted with t +maturity at t contracted timeat time𝑑𝑑, then at time 𝑑𝑑 +t, 𝜏𝜏then contracted at time 𝑑𝑑, t functions to capture the If range of isshapes typically the instantaneous forward rate withπœ•πœ•maturity at time contracted atπœ•πœ•time 𝑓𝑓𝑑𝑑 (𝜏𝜏) associated with yield curves. Diebold and Li (2006) ln 𝑝𝑝 𝑑𝑑 (𝜏𝜏) 𝑑𝑑 + 𝜏𝜏 πœ•πœ• ln=π‘π‘π‘‘π‘‘βˆ’(𝜏𝜏) ln 𝑝𝑝𝑑𝑑𝑑𝑑,(𝜏𝜏)(t πœ•πœ• ln 𝑝𝑝𝑑𝑑 (𝜏𝜏) 𝑓𝑓𝑑𝑑 (𝜏𝜏) = βˆ’ πœ•πœ• ln 𝑝𝑝 (𝜏𝜏) 𝑓𝑓𝑑𝑑(2) (𝜏𝜏) = βˆ’ 𝑓𝑓𝑑𝑑 (𝜏𝜏) reformulated it as a dynamic factor model 𝑓𝑓𝑑𝑑 (𝜏𝜏)and=βˆ’ used πœ•πœ•πœ•πœ• 𝑑𝑑 πœ•πœ•πœ•πœ• πœ•πœ•πœ•πœ• πœ•πœ•πœ•πœ• 𝑓𝑓𝑑𝑑 (𝜏𝜏) = βˆ’ πœ•πœ• ln 𝑝𝑝𝑑𝑑 (𝜏𝜏) (2) it to forecast the yield curve. Several extensions and 𝑓𝑓𝑑𝑑 (𝜏𝜏) = βˆ’ πœ•πœ•πœ•πœ• It follows that It It follows follows that that πœ•πœ•πœ•πœ• variations It follows that of the model have been introduced with the goal of improving the out-of-sample It followsforecasts that (see for It follows example Christensen, Diebold, & Rudebuscha, 2011; that It follows that 1 𝜏𝜏 1 𝜏𝜏 1 𝜏𝜏 ( 1 𝜏𝜏 𝑦𝑦𝑑𝑑 (𝜏𝜏) = ∫1 𝑓𝑓𝑑𝑑 𝜏𝜏(𝑒𝑒)𝑑𝑑𝑑𝑑 (3) 𝑦𝑦𝑑𝑑 (𝜏𝜏) =(𝑒𝑒)𝑑𝑑𝑑𝑑 ∫ 𝑓𝑓𝑑𝑑 (𝑒𝑒)𝑑𝑑𝑑𝑑 𝑦𝑦 (𝜏𝜏) = ∫ 𝑓𝑓 (𝑒𝑒)𝑑𝑑𝑑𝑑 𝑦𝑦 (𝜏𝜏) = ∫ 𝑓𝑓 Exterkate, Dijk, Heij, & Groenen, 2013; 𝑑𝑑 Koopman, 𝑑𝑑 𝜏𝜏 0 𝑦𝑦𝑑𝑑 (𝜏𝜏) = 1 𝜏𝜏 0 𝜏𝜏 (𝑒𝑒)𝑑𝑑𝑑𝑑 𝑑𝑑 ∫ 𝑓𝑓𝑑𝑑 𝜏𝜏 0 𝑑𝑑 𝜏𝜏 0 Mallee, & Van der Wel, 2010). 𝑦𝑦𝑑𝑑 (𝜏𝜏) = 𝜏𝜏 ∫0 𝑓𝑓𝑑𝑑 (𝑒𝑒)𝑑𝑑𝑑𝑑 This paper will focus on theYield original three-factor Yield 𝜏𝜏 a0 curve curve (3) Yield curve at a particular point curve in timeat a 𝑑𝑑particular is a curve point Yield incurve that describes time at𝑑𝑑theis spotatinterest a particular a particular thatpoint describes rates in point timethe 𝑑𝑑inisspot time a curve 𝑑𝑑 is that interest a curve rate des Nelson-Siegel model. Following the method in Yield curve at a particular point in time 𝑑𝑑 is a curve that describes the spot interest r Diebold and Li (2006), the latent factors, which Yieldcurves curve will at abe particular point in different time 𝑑𝑑 is or a curve that describes theorspot interest 𝑦𝑦𝑑𝑑 (𝜏𝜏) for different maturities 𝑦𝑦𝑑𝑑 (𝜏𝜏) for different 𝜏𝜏. These referred to throughout this paper as beta parameters,𝑦𝑦𝑑𝑑 maturities are 𝜏𝜏. These typically (𝜏𝜏) (𝜏𝜏) 𝑦𝑦𝑑𝑑different curves monotonic, forYield for curve are a typically humped maturities at particular maturities 𝜏𝜏. monotonic, S-shaped These point These in𝜏𝜏.curves time humpedcurves t isarea curvetypically are monotor typically S-shaped 𝑦𝑦𝑑𝑑 (𝜏𝜏) for different maturities that 𝜏𝜏. These describes curves the spot are typically interest rates monotonic, yt (t) for different humped or S-shaped will be estimated from historical 𝑦𝑦𝑑𝑑 (𝜏𝜏) for data by fitting different the maturities 𝜏𝜏. These curves are typically monotonic, humped or S-shaped Nelson-Siegel equation to the yield curves over time maturities t. These curves are typically monotonic, and assumed to follow a time series model. However, humped or S-shaped (Nelson & Siegel, 1987). To instead of assuming independent univariate AR(1) generate this range of shapes, a parsimonious model processes for all the beta parameters, the best time introduced by Nelson and Siegel assumes that the series model will be searched using standard statistical forward rate follows the equation URE FORECASTING 0𝑑𝑑 0𝑑𝑑 TERM STRUCTURE FORECASTING 5 5 0𝑑𝑑 πœπœβ†’+∞ 𝑑𝑑 &STRUCTURE RE M Siegel, 1987). FORECASTING To generate this range FORECASTING of shapes, Nelson TERM and a5parsimonious Siegel STRUCTURE assumes 5 FORECASTING model that the introduced forward rate by 5 follows the equation πœπœβ†’+∞ TERM TERM STRUCTURE STRUCTURE FORECASTING FORECASTING yield curve shifts. The factor 𝛽𝛽1𝑑𝑑 is related to 𝛽𝛽the yield 0𝑑𝑑 and an curve increaseslope. 0𝑑𝑑5If in 𝛽𝛽0𝑑𝑑 the slope 5increases of theequally. all yields yield curve Thus, t and an 𝛽𝛽0𝑑𝑑1987). el, increase in 𝛽𝛽0𝑑𝑑 this To generate increases rangeallofyields equally. shapes, Thus, 𝛽𝛽 this andfactor a parsimonious yield in 𝛽𝛽 curve is responsible model an increase introduced forshifts. by allThe 0𝑑𝑑 parallel increases factor yields 𝛽𝛽1𝑑𝑑 isThus, equally. related thistofactor the yield curve slope. is responsible for If part and Siegel assumes (Nelson that&the Siegel, forward 1987). rate To follows generate 0𝑑𝑑 the equation this rangemodel of shapes, 0𝑑𝑑 a parsimonious model (Nelson introduced & βˆ’πœπœ/πœ†πœ† Siegel, by 1987). 𝑓𝑓𝑑𝑑 (𝜏𝜏)generate = 𝛽𝛽0𝑑𝑑 + egel, syield range 1987). of shapes, To generate a parsimonious this range model of shapes, is introduced defined a parsimonious as by lim 𝑦𝑦 (𝜏𝜏) βˆ’ lim introduced 𝑦𝑦 (𝜏𝜏), then itbyyield is equalcurve to shifts.βˆ’π›½π›½ The . 𝜏𝜏factor Lastly, 𝛽𝛽 𝛽𝛽 1𝑑𝑑 isisrelated related to To thethe to yield curve s el, lson rate 1987). & curve this Siegel, range To shifts. generate 1987). The of shapes, factor To this generate 𝛽𝛽a range is parsimonious related of this shapes, to (Nelson range the yield model a ofparsimonious & shapes, curve introduced Siegel, yield slope. curve a 5 1987). 𝑑𝑑 model parsimonious If the by shifts. slope To is The𝑓𝑓 introduced (𝜏𝜏) of defined generate factorthe 𝑑𝑑 model = 𝛽𝛽 yield as this by +introduced curve lim is range𝛽𝛽 related 𝑦𝑦 𝑒𝑒 (𝜏𝜏) βˆ’πœπœ/πœ†πœ† of to βˆ’ 𝑑𝑑by shapes, the lim + yield𝛽𝛽 1𝑑𝑑 𝑦𝑦 (𝜏𝜏), acurve 𝑒𝑒then parsimoniousslope. 1𝑑𝑑 it𝑑𝑑 2𝑑𝑑 is If equal the model slopeto βˆ’π›½π›½ of introduce the . Lastly, yield c el assumes TERM that TERM the STRUCTURE forward STRUCTURE rateFORECASTING FORECASTING follows the equation 𝑑𝑑 + 𝛽𝛽 0𝑑𝑑 1𝑑𝑑 2𝑑𝑑 5 bylim++5𝑦𝑦𝑑𝑑𝑑𝑑 (𝜏𝜏), then 1𝑑𝑑 πœπœβ†’+∞ πœπœβ†’0 𝑑𝑑 + E.P.πœ†πœ†de Lara-Tuprio, 𝑑𝑑 1𝑑𝑑 (Nelson & & Siegel, 1987). ToTo generate this range of shapes, aa parsimonious et al is model introduced by 1π‘‘π‘‘πœπœβ†’+∞ (Nelson 192 Nelson Siegel, and 1987). Siegel assumes generate that this the range forward 𝜏𝜏5 βˆ’πœπœ/πœ†πœ† of rate shapes, follows parsimonious the equation model defined (4) as introduced πœπœβ†’0+ lim 𝑦𝑦𝑑𝑑𝑑𝑑 (𝜏𝜏) 𝑑𝑑 βˆ’ it is equalthe to βˆ’π›½π›½ iegel assumes that the forward rate follows 5 βˆ’πœπœ/πœ†πœ† the equation where Nelson 𝛽𝛽 , and 𝛽𝛽 Siegel , 𝛽𝛽 , assumes and πœ†πœ†π‘‘π‘‘ areisthat constants for 11 ard rate follows the equation 𝑓𝑓 𝑑𝑑 (𝜏𝜏) = 𝛽𝛽 + 𝛽𝛽 𝑒𝑒 + 𝛽𝛽 𝑒𝑒 πœπœβ†’+∞ πœπœβ†’+∞ πœπœβ†’0 πœπœβ†’0 gel son s defined he forward rate assumes and Siegel as lim that 𝑦𝑦the assumes follows 𝑑𝑑 (𝜏𝜏) forward βˆ’ that lim the+equation 𝑦𝑦 the 𝑑𝑑rate (𝜏𝜏), 0𝑑𝑑 forwardfollows then 1𝑑𝑑 it is Nelsonπœπœπ›½π›½andβˆ’πœπœ/πœ†πœ† rate the equal curvature, 𝑑𝑑 equation followsto βˆ’π›½π›½defined 2𝑑𝑑 is Siegel defined the . Lastly, πœ†πœ†π‘‘π‘‘ 𝑑𝑑, 𝛽𝛽assumes 1𝑑𝑑 equation by as 𝑑𝑑 Diebold 𝛽𝛽 lim 2𝑑𝑑 is 𝑦𝑦 that related and (𝜏𝜏) curvature, the βˆ’ Li to as lim the forward defined 2𝑦𝑦𝑦𝑦 (24) (𝜏𝜏), rate then βˆ’ byπœ†πœ†FORECASTING 𝑦𝑦it follows Diebold (3) is equalβˆ’ and 𝑦𝑦0𝑑𝑑(120), theLithatto βˆ’π›½π›½ equation 1𝑑𝑑 where as 1𝑑𝑑 . 2𝑑𝑑 maturity Lastly, 𝛽𝛽 is related given to in the 𝑑𝑑 (24) 𝑦𝑦𝑑𝑑 (3)isβˆ’a 𝑦𝑦solution 𝑑𝑑 (120), 𝑑𝑑 𝑑𝑑 𝑑𝑑 2𝑦𝑦 TERM STRUCTURE FORECASTING 𝑑𝑑 where , and TERM 𝑑𝑑 are 𝑑𝑑TERM STRUCTURE constants 𝜏𝜏 (4) STRUCTURE + 𝑑𝑑 FORECASTING and 0. Note 5= βˆ’π‘“π‘“π‘‘π‘‘2𝑑𝑑 Li as (𝜏𝜏) πœπœβ†’+∞ π‘’π‘’πœπœ, 𝛽𝛽this πœ†πœ†the β‰  defined π‘₯π‘₯(𝜏𝜏) πœπœβ†’0 e this range Nelson Nelson of𝑓𝑓𝑑𝑑 (𝜏𝜏) (Nelson (Nelson shapes, and and=Siegel Siegel & 𝛽𝛽 & aSiegel, 0𝑑𝑑 parsimonious +assumes Siegel, assumes 𝛽𝛽1𝑑𝑑1987). βˆ’πœπœ/πœ†πœ† 𝑒𝑒1987). that that +To To model the the 𝛽𝛽generate 2𝑑𝑑 forward generateforward introduced 0𝑑𝑑 1𝑑𝑑 thisrate rate range range2𝑑𝑑 by follows follows πœπœβ†’+∞ of of βˆ’πœπœ/πœ†πœ† the shapes, shapes, – b equation equation . a a Lastly, πœπœβ†’0 parsimonious parsimonious b βˆ’πœπœ/πœ†πœ† curvature, is related 𝑑𝑑 model model to the introduced by introduced Diebold curvature, andby by (4) defined 2𝑦𝑦by 𝑑𝑑𝑑𝑑(24) βˆ’ 𝑦𝑦𝑑𝑑𝑑𝑑(3) βˆ’ 𝑦𝑦 this range of shapes, 𝜏𝜏 a=parsimonious model βˆ’πœπœ/πœ†πœ† 𝑑𝑑 months. 𝑓𝑓 π›½π›½πœ†πœ†π‘‘π‘‘0. introduced (𝜏𝜏) π‘¦π‘¦πœπœ =π‘’π‘’βˆ’βˆ’πœπœ/πœ†πœ† 𝛽𝛽(4) by 𝑑𝑑+πœπœπ›½π›½1𝑑𝑑where 𝑒𝑒 maturity 𝑑𝑑 +1t 𝛽𝛽 𝑒𝑒(4)2t(4) 𝑑𝑑 second-order (4)βˆ’ 𝑦𝑦𝑑𝑑 (3) linear ordinary differential 𝑓𝑓𝑑𝑑 (𝜏𝜏)is= eq 𝛽𝛽 𝑑𝑑 (𝜏𝜏) 𝑦𝑦𝑑𝑑0𝑑𝑑 curvature, defined 𝑓𝑓by Diebold 𝛽𝛽constants and + Li 𝛽𝛽= as 𝑒𝑒and + 𝑑𝑑 (3)βˆ’πœπœ/πœ†πœ† 2𝑑𝑑 is given in pes, a parsimonious βˆ’πœπœ/πœ†πœ† model βˆ’πœπœ/πœ†πœ† πœπœπ‘‘π‘‘(𝜏𝜏) introduced 2𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑(24) by βˆ’ (120), (4) βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ months. πœ†πœ† 𝜏𝜏 𝛽𝛽)+ 0𝑑𝑑 = 𝛽𝛽 , 1𝑑𝑑 𝛽𝛽 𝛽𝛽 𝑒𝑒 1𝑑𝑑 , + 𝛽𝛽 𝛽𝛽 𝑑𝑑 + 2𝑑𝑑 , 𝑓𝑓 𝑒𝑒and 𝛽𝛽 (𝜏𝜏) βˆ’πœπœ/πœ†πœ† 𝑑𝑑 2𝑑𝑑 πœ†πœ† = 𝑑𝑑𝑑𝑑+ 𝛽𝛽 𝑒𝑒 are 𝛽𝛽 0𝑑𝑑 𝑓𝑓+ 0𝑑𝑑 𝛽𝛽 𝑒𝑒1𝑑𝑑 𝑒𝑒 1𝑑𝑑 βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† 𝛽𝛽 + 𝑑𝑑0𝑑𝑑 second-order + πœ†πœ† 𝛽𝛽 𝑑𝑑 𝛽𝛽 β‰ 2𝑑𝑑 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 2𝑑𝑑 Note 𝑑𝑑 π‘‘π‘‘βˆ’πœπœ/πœ†πœ† 𝑒𝑒 πœ†πœ† curvature, + linear that 𝛽𝛽𝑑𝑑𝑑𝑑 π‘₯π‘₯(𝜏𝜏) defined 𝑒𝑒 ordinary (4) = 𝑓𝑓 by 𝑑𝑑 𝑓𝑓 Diebold (𝜏𝜏)𝜏𝜏 Diebold differential 𝜏𝜏 (𝜏𝜏) is = a and 𝛽𝛽 𝑑𝑑and solution Li Li equation + asas months. 𝛽𝛽 to 2𝑦𝑦 𝑒𝑒 the𝑑𝑑 βˆ’πœπœ/πœ†πœ† (24) + 𝛽𝛽 βˆ’ 𝑒𝑒 𝑦𝑦 𝑑𝑑 βˆ’πœπœ/πœ†πœ† (120),, where where maturity give orward 0𝑑𝑑 rate1𝑑𝑑 follows 𝑑𝑑 𝑑𝑑 πœ†πœ†and the&Siegel 0𝑑𝑑equation 𝑑𝑑 1𝑑𝑑 𝑑𝑑 1𝑑𝑑 2𝑑𝑑 2𝑑𝑑 (4) πœ†πœ†π‘‘π‘‘ (4) 𝑑𝑑 𝑑𝑑 Nelson Nelson (Nelson 𝑑𝑑and 2𝑑𝑑 Siegel, Siegel assumes 1987). that To the =generate πœ†πœ†the 𝑑𝑑 forward 𝑑𝑑 this πœ†πœ†rate 𝑑𝑑 𝑑𝑑𝑑𝑑follows range (Nelson of 𝑑𝑑(Nelson shapes, the & equation aSiegel, & 𝑑𝑑parsimonious Siegel, 1987). 1987). To model generate To2𝑑𝑑generate introduced this this range byrange of shapes, of shapes, a parsi ap 𝑑𝑑 πœ†πœ†assumes that forward π‘’π‘’βˆ’πœπœ/πœ†πœ† rate follows the π‘’π‘’π‘’π‘’βˆ’πœπœ/πœ†πœ† equation 𝑓𝑓𝑓𝑓𝑑𝑑𝑑𝑑(𝜏𝜏) = 𝛽𝛽 maturity is𝑑𝑑0𝑑𝑑given 1𝑑𝑑 in months. 0𝑑𝑑 + 𝛽𝛽of 𝑒𝑒= + 𝛽𝛽𝛽𝛽a2𝑑𝑑 πœ†πœ†and (𝜏𝜏) βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† months. rward 𝛽𝛽2𝑑𝑑 , and rate 𝑑𝑑 are constants πœ†πœ†follows the equation 𝑑𝑑 β‰  0. TERM Note EstimationSTRUCTURE 𝛽𝛽𝑑𝑑0𝑑𝑑 that + π‘₯π‘₯(𝜏𝜏) 𝛽𝛽1𝑑𝑑 1𝑑𝑑 FORECASTING Parameters 𝑓𝑓 𝑑𝑑 (𝜏𝜏) is 2𝑑𝑑 + solution to the ws the equation where 𝛽𝛽 , 𝛽𝛽 , 𝛽𝛽 , and πœ†πœ† are constants months. and πœ†πœ† β‰  0.πœ†πœ† πœ†πœ† Estimation 𝑑𝑑 Note that of Parameters π‘₯π‘₯(𝜏𝜏) Estimation = 𝑓𝑓 (𝜏𝜏) of is Parameters a solution to the π‘₯π‘₯ β€²β€² + order 𝛽𝛽 𝛽𝛽linear ,β‰  and ordinary πœ†πœ†where TERM 𝑑𝑑 are differential constants STRUCTURE 0π‘‘π‘‘πœπœ FORECASTING 1𝑑𝑑andequation 2𝑑𝑑 , and l 𝑑𝑑Note arethat constants and 𝑑𝑑 is a solution to the (𝜏𝜏) 𝑑𝑑 2 β€² 5to 1𝑑𝑑 where 𝛽𝛽0𝑑𝑑𝛽𝛽0𝑑𝑑 , 𝛽𝛽1𝑑𝑑 , 𝛽𝛽2𝑑𝑑 , and πœ†πœ†π‘‘π‘‘ are constaπœ†πœ† s,re constants 1𝑑𝑑 𝛽𝛽 and onstants𝛽𝛽, 2𝑑𝑑 𝛽𝛽 + πœ†πœ†and ,2𝑑𝑑 ,𝛽𝛽 𝑑𝑑 𝛽𝛽 and 𝑒𝑒 πœ†πœ†0. βˆ’πœπœ/πœ†πœ† Note are ,𝑑𝑑𝑑𝑑Parameters 𝛽𝛽 Nelson𝑑𝑑 ,+ constants and that 𝛽𝛽 . πœ†πœ†π‘‘π‘‘π‘₯π‘₯(𝜏𝜏) and Note areSiegel 𝑒𝑒 that and βˆ’πœπœ/πœ†πœ† that = π‘‘π‘‘πœ†πœ†π‘“π‘“π‘‘π‘‘π‘‘π‘‘assumes constants πœ†πœ†β‰  (𝜏𝜏) 𝑑𝑑 β‰  0. isand where 0. a solution Note t that (𝜏𝜏) πœ†πœ†π‘‘π‘‘π›½π›½is that is β‰ the aa 0. toNote π‘₯π‘₯(𝜏𝜏) forward solution solution π‘₯π‘₯(𝜏𝜏) the=(4) that to =𝑓𝑓(7), to , ratethe and 𝑓𝑓𝑑𝑑π‘₯π‘₯(𝜏𝜏) (𝜏𝜏) the is Nelson follows aare= solution Estimation Nelson (𝜏𝜏) 𝑓𝑓constants and the 𝜏𝜏𝜏𝜏 β€²β€²and π‘₯π‘₯ Siegel to isβˆ’πœπœ/πœ†πœ† equation + aofthe solution Parameters Siegel and assumes π‘₯π‘₯ + assumes the that π‘₯π‘₯ = Note the that forward the forward (4)(4)rate = π‘“π‘“πœ†πœ†π‘‘π‘‘rate follows isthea the follows equa the e πœ†πœ† β‰  0. π‘₯π‘₯(𝜏𝜏) = 𝑓𝑓 2that (𝜏𝜏) solution Estimation 2𝑑𝑑 0𝑑𝑑 1𝑑𝑑 of 2𝑑𝑑 𝑑𝑑 In , 𝛽𝛽 Equation , 𝛽𝛽 𝑑𝑑 𝑑𝑑 βˆ’πœπœ/πœ†πœ† the πœ†πœ† parameter 𝑑𝑑 πœ†πœ† is referred πœ†πœ† toβ‰  as 0. the shape π‘₯π‘₯(𝜏𝜏) parameter. For each t,towhen ear0𝑑𝑑ordinary βˆ’πœπœ/πœ†πœ†differential 𝜏𝜏 ,πœ†πœ†π›½π›½ equation 𝑑𝑑𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑(𝜏𝜏) ==𝛽𝛽Estimation ++πœ†πœ†π›½π›½π›½π›½ 𝑒𝑒𝑒𝑒0. 𝑑𝑑𝑑𝑑+ 𝛽𝛽𝛽𝛽2𝑑𝑑2𝑑𝑑In 𝑒𝑒𝑒𝑒𝑑𝑑 =πœ†πœ†π‘“π‘“π‘‘π‘‘ (𝜏𝜏) (𝜏𝜏) βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† 𝜏𝜏 1𝑑𝑑 where where 𝑑𝑑 𝛽𝛽𝛽𝛽 ,, 𝛽𝛽 𝛽𝛽 2𝑑𝑑 , 𝛽𝛽 , , and and πœ†πœ† πœ†πœ† are (Nelsonare constants constants & 0𝑑𝑑 Siegel, 𝛽𝛽 and 1𝑑𝑑 and 0𝑑𝑑0𝑑𝑑 (4) πœ†πœ† 1987). 2𝑑𝑑1𝑑𝑑 β‰  β‰  1𝑑𝑑 of0. To Note Note 𝑑𝑑 Parameters + generate that that π‘₯π‘₯(𝜏𝜏) π‘₯π‘₯(𝜏𝜏) Equation this = range 𝑑𝑑 𝑓𝑓 (7), (𝜏𝜏) of 𝑑𝑑 In is the isπœ†πœ† a shapes,Equation a2parameter solution solution a πœ†πœ† (7), to parsimonious the πœ†πœ† to theparameter is the referred 𝑑𝑑𝑑𝑑to model is as referred shape introduced as the pas 𝛽𝛽0𝑑𝑑 + 𝛽𝛽1𝑑𝑑 π‘’π‘’βˆ’πœπœ/πœ†πœ†second-order 𝑑𝑑 second-order + 0𝑑𝑑 0𝑑𝑑 𝛽𝛽 1𝑑𝑑 1𝑑𝑑 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 linear 2𝑑𝑑 2𝑑𝑑 linear𝑑𝑑 ordinary 2ordinary 𝑑𝑑𝑑𝑑 (4) differential1 differential 𝛽𝛽 equation equation 𝑑𝑑𝑑𝑑 In πœ†πœ† Equation πœ†πœ† 𝑑𝑑 (7), 𝑑𝑑 𝑑𝑑𝑑𝑑 the (5) parameter 𝑑𝑑 second-order l 𝑑𝑑 is referred 𝑑𝑑 linear to as ordinary the differenti + linear equation 𝛽𝛽2𝑑𝑑ordinary ordinary 𝑒𝑒 linear (Nelson 𝑑𝑑 differential 2𝑑𝑑& Siegel, 1987). To πœ†πœ†π‘‘π‘‘ differential equation generate this range 0𝑑𝑑of shapes, a parsimonious model 𝑑𝑑 introduced by From t near ond-order rential πœ†πœ†In𝑑𝑑equation Equation differential ordinary (7), the equation parameter π‘₯π‘₯ β€²β€² +πœ†πœ†π‘‘π‘‘ isπ‘₯π‘₯equation β€² referred second-order+πœ†πœ†π‘‘π‘‘ is2to π‘₯π‘₯ as=thelinear specified, shape theInmodel parameter. becomes shape For each linear 𝜏𝜏 5t,inβˆ’πœπœ/πœ†πœ† parameter. when the isFor πœ†πœ†π‘‘π‘‘π‘‘π‘‘π‘‘π‘‘πœ†πœ†parameters specified,each Equation t,the𝛽𝛽when model ,= 𝛽𝛽the (3), lt,becomes is and specified,𝛽𝛽𝛽𝛽(4) .𝑒𝑒Let linear the inπ‘Œπ‘Œ 𝑑𝑑 parameters 𝑑𝑑the =For 𝑦𝑦 (𝜏𝜏), 𝜏𝜏 βˆ’πœπœ 𝜏𝜏0 t,𝛽𝛽w +ordinary 𝛽𝛽Equation πœ†πœ†differential (7), the parameter 𝑒𝑒equation 𝑑𝑑 is referred to0𝑑𝑑𝑓𝑓 as shape parameter. ,each TERM STRUCTURE FORECASTING stants and second-order second-order πœ†πœ†where β‰  0. Note linear linear that 2 ordinary π‘₯π‘₯(𝜏𝜏) ordinary πœ†πœ† = 1 𝑑𝑑 differential 𝑓𝑓 From differential(𝜏𝜏) πœ†πœ†π›½π›½is𝑓𝑓Equation 0𝑑𝑑 a(𝜏𝜏) solution =πœ†πœ†π‘‘π‘‘π›½π›½20𝑑𝑑 equation equation (3), to the 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 is 𝑑𝑑model +specified, 𝛽𝛽2𝑑𝑑 becomes (5) thelinear model in becomes the 𝑑𝑑 (𝜏𝜏) 𝑓𝑓parameters 𝑑𝑑 (𝜏𝜏) linear 𝛽𝛽0𝑑𝑑 1𝑑𝑑 = in b + 𝛽𝛽the,0𝑑𝑑b 2𝑑𝑑+ 1𝑑𝑑 and π›½π›½βˆ’πœπœ/πœ†πœ† parameters 1𝑑𝑑b π‘’π‘’πœπœ.βˆ’πœπœ/πœ†πœ† + 𝛽𝛽 𝛽𝛽𝑑𝑑 𝑑𝑑+ 2𝑑𝑑 𝛽𝛽 𝛽𝛽2𝑑𝑑1𝑑𝑑𝑒𝑒, a 𝑑𝑑where 𝛽𝛽and β€²β€² ,Siegel ,+𝛽𝛽𝛽𝛽1𝑑𝑑 1𝑑𝑑, ,π‘₯π‘₯2 𝛽𝛽𝛽𝛽2𝑑𝑑 β€² 2𝑑𝑑, ,𝑓𝑓 and and Nelson πœ†πœ†πœ†πœ†the 𝑑𝑑 are 𝑑𝑑and 𝑑𝑑 constants Siegel assumes and 2 (5) 1𝑑𝑑 πœ†πœ†πœ†πœ†β€²π‘‘π‘‘π‘‘π‘‘equation β‰ β‰ that 0.10.Note the forward that 𝛽𝛽 πœ†πœ†π‘‘π‘‘ π‘₯π‘₯(𝜏𝜏) rate = follows𝑓𝑓 𝑑𝑑(𝜏𝜏) isisthe aasolution equation to t the (5) 0𝑑𝑑 πœ†πœ† πœ†πœ†π‘‘π‘‘ aare constants and Note that solution to the Nelson π‘₯π‘₯ 𝛽𝛽0𝑑𝑑0𝑑𝑑 assumes + that 2 π‘₯π‘₯ 𝑑𝑑= forward rate follows the π‘₯π‘₯(𝜏𝜏) (𝜏𝜏) (5)𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑑𝑑1βˆ’π‘’π‘’ 𝑑𝑑 0𝑑𝑑 πœ†πœ†ants 2is specified,and πœ†πœ†1𝑑𝑑 β‰ the 0. model Note 𝛽𝛽 that β€²β€² πœ†πœ†π‘₯π‘₯(𝜏𝜏) 2 = β€² πœ†πœ†π‘‘π‘‘π‘‘π‘‘1 (𝜏𝜏) 1the is πœ†πœ† 𝛽𝛽 2𝛽𝛽 solution π‘₯π‘₯ to β€²β€² the (5) 𝑑𝑑 + π‘₯π‘₯ + π‘₯π‘₯ = 𝑑𝑑 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘π‘‘π‘‘ 0βˆ’πœπœ/πœ†πœ† 1t 2t 1 𝜏𝜏 𝑑𝑑 𝑑𝑑Note β€² that 2 2π‘₯π‘₯(𝜏𝜏) (Nelson & =1 𝑓𝑓𝑑𝑑 π‘₯π‘₯ β€²β€²β€²β€²π‘₯π‘₯+ 0𝑑𝑑 Siegel, (𝜏𝜏) becomes 1987). is 𝛽𝛽0𝑑𝑑 + a solution 𝑑𝑑 To linear β€²β€²π‘₯π‘₯ β€²β€²+ to2the generate in this 2 range parameters π‘₯π‘₯π‘₯π‘₯1πœπœβ€²=+ 𝑑𝑑 𝑑𝑑 of 0𝑑𝑑 0𝑑𝑑 1 0𝑑𝑑 1βˆ’π‘’π‘’ shapes, βˆ’πœπœ/πœ†πœ† πœ†πœ† 𝛽𝛽 is a 0𝑑𝑑 , 𝛽𝛽 parsimonious 𝛽𝛽 specified, 0𝑑𝑑 πœ†πœ†1𝑑𝑑 , and (5) 𝑑𝑑 𝛽𝛽 the 1βˆ’π‘’π‘’ model 2𝑑𝑑 . 2 Let model βˆ’πœπœ/πœ†πœ† 𝑑𝑑 𝜏𝜏 =2𝑦𝑦𝑑𝑑 (𝜏𝜏), π‘Œπ‘Œ introduced becomes βˆ’πœπœ/πœ†πœ†by (5) ,linear 𝑑𝑑𝑑𝑑 𝑑𝑑 .2Then𝑋𝑋 = in 1βˆ’π‘’π‘’βˆ’πœπœ/πœ†πœ† the 1 (5) parameters , and and 𝛽𝛽 𝑋𝑋 𝑑𝑑𝑑𝑑 = 𝛽𝛽 1βˆ’π‘’π‘’βˆ’πœπœ/πœ†πœ† 1βˆ’π‘’π‘’ , 𝛽𝛽 𝑑𝑑𝑑𝑑 ,βˆ’ and 𝑒𝑒 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘π‘‘π‘‘ βˆ’πœπœ/πœ†πœ† 𝛽𝛽 . . ThenLet π‘Œπ‘Œ 𝑑𝑑 Equation = 𝑦𝑦𝑑𝑑π‘₯π‘₯(πœπœβ€²β€²( quation ntial π‘₯π‘₯ + (3), π‘₯π‘₯ = πœ†πœ†π‘₯π‘₯ + π‘₯π‘₯ + π‘₯π‘₯ 𝑋𝑋 = = 2 π‘₯π‘₯ 𝑑𝑑 , and = 𝑑𝑑 𝑋𝑋 𝜏𝜏 = πœ†πœ† βˆ’ 1βˆ’π‘’π‘’πœ†πœ† 𝑒𝑒 βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† 1𝜏𝜏 1𝜏𝜏 Equation (4)𝜏𝜏/πœ†πœ† 1βˆ’π‘’π‘’ βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† (7) 𝑑𝑑𝑑𝑑 2𝜏𝜏 𝜏𝜏0𝑑𝑑 becomes 2𝜏𝜏 𝜏𝜏/πœ†πœ†π‘‘π‘‘π‘‘π‘‘1𝑑𝑑 𝑦𝑦 (𝜏𝜏) 0π‘‘π‘‘πœπœ/πœ†πœ† 2𝑑𝑑 = ∫ 𝜏𝜏 [𝛽𝛽 0𝑑𝑑 + πœ†πœ†π‘‘π‘‘π‘₯π‘₯ equation 𝑑𝑑 𝑑𝑑 𝜏𝜏/πœ†πœ† βˆ’πœπœ/πœ†πœ† β€²β€² + βˆ’πœπœ/πœ†πœ† π‘₯π‘₯ β€² πœ†πœ†π‘‘π‘‘ 𝑑𝑑second-order + From 2 πœ†πœ†π‘‘π‘‘2Equation π‘₯π‘₯ = ,linear 2(3), ordinary πœ†πœ† 2 (𝜏𝜏) differential 2πœ†πœ† 𝜏𝜏/πœ†πœ†2 2 βˆ’πœπœ/πœ†πœ† equation 1 1 2𝜏𝜏 2𝑑𝑑 πœ†πœ† where βˆ’πœπœ/πœ†πœ† 𝛽𝛽 𝑑𝑑 𝑋𝑋 𝛽𝛽 𝜏𝜏/πœ†πœ†π‘‘π‘‘ 𝑑𝑑= Then 1 𝜏𝜏 𝑑𝑑 Equation π‘₯π‘₯ β€²β€² + , and (7) π‘₯π‘₯ 𝑋𝑋 β€² becomes + = 𝑑𝑑𝑑𝑑 π‘₯π‘₯ = βˆ’ 𝑒𝑒 𝑒𝑒 (5) (5) 𝑑𝑑 𝑑𝑑 . Then Equation 𝑑𝑑 (7) becom 2𝑑𝑑, ,βˆ’πœπœ/πœ†πœ† and are 𝑑𝑑 aconstants 𝑑𝑑 and πœ†πœ†π‘‘π‘‘ β‰ πœ†πœ†π‘‘π‘‘0.β‰ Note that tπ‘₯π‘₯ 2 2 where where second-order 𝛽𝛽assumes πœ†πœ†π›½π›½ linear 𝑑𝑑 , 𝛽𝛽 ordinary 2π‘‘π‘‘πœ†πœ†forward 𝑓𝑓 ,𝑑𝑑𝑑𝑑and2𝑑𝑑 π‘‘π‘‘πœ†πœ† πœ†πœ† = 𝛽𝛽 are differential 𝑑𝑑 πœ†πœ†π‘₯π‘₯ 2 + 𝑑𝑑 β€²β€²πœ†πœ†+ 0𝑑𝑑′′ 𝛽𝛽 𝑑𝑑constants1𝑑𝑑 𝑒𝑒 𝑑𝑑 𝑑𝑑 + β€²equation β€²πœ†πœ† 𝑑𝑑 and 𝛽𝛽 𝑒𝑒 =β‰  𝑓𝑓𝑑𝑑 (𝜏𝜏) 0.0𝑑𝑑 Note 0𝑑𝑑 1𝜏𝜏 𝛽𝛽 =0π‘‘π‘‘π›½π›½πœπœ/πœ†πœ† , 𝛽𝛽that𝛽𝛽𝑑𝑑0𝑑𝑑 +, ,+ 𝛽𝛽𝛽𝛽1𝑑𝑑𝛽𝛽 π‘₯π‘₯(𝜏𝜏) 𝛽𝛽 𝛽𝛽= 2𝜏𝜏 𝑓𝑓,𝑑𝑑𝑑𝑑and πœ†πœ†(𝜏𝜏) is πœ†πœ†π‘‘π‘‘π‘‘π‘‘π›½π›½ are solution constants βˆ’πœπœ/πœ†πœ† to the and 𝜏𝜏 0.0 Note 𝑑𝑑 theπ‘₯π‘₯π‘₯π‘₯ + π‘₯π‘₯πœ†πœ†π‘‘π‘‘π‘₯π‘₯𝑑𝑑𝑑𝑑(𝜏𝜏) 𝑑𝑑1𝑑𝑑 π‘’π‘’βˆ’πœπœ/πœ†πœ† +π‘‘π‘‘π‘‘π‘‘πœπœ/πœ†πœ† 𝛽𝛽 πœ†πœ†πœ†πœ†2𝑑𝑑𝑑𝑑 2𝑒𝑒 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ†π‘‘π‘‘ ]π‘Œπ‘Œπœπœπœπœπ‘‘π‘‘π‘‘π‘‘π‘‘π‘‘ al𝑑𝑑 equation 1βˆ’π‘’π‘’πœ†πœ†Nelson andπœ†πœ†π‘‘π‘‘Siegel 0π‘‘π‘‘πœ†πœ† 𝑑𝑑𝑑𝑑1𝑑𝑑 π‘₯π‘₯𝑑𝑑follows + + 𝑑𝑑= 𝑑𝑑 1𝑑𝑑 βˆ’π‘’π‘’/πœ†πœ† 2𝑑𝑑 2 0𝑑𝑑𝑑𝑑 πœ†πœ† 𝑑𝑑 rate 𝑑𝑑 that the equation 22𝑦𝑦 = 2𝑑𝑑2 ∫1βˆ’π‘’π‘’ [𝛽𝛽 1𝑑𝑑 𝑒𝑒 π‘‘π‘‘πœ†πœ†π‘‘π‘‘ +2𝑑𝑑 βˆ’πœπœ/πœ†πœ† 𝑋𝑋3), 𝑑𝑑 , and 𝑑𝑑 1βˆ’π‘’π‘’ 𝑑𝑑 βˆ’πœπœ/πœ†πœ† 1βˆ’π‘’π‘’ βˆ’πœπœ/πœ†πœ† 0𝑑𝑑 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 . Then Equation𝑑𝑑 (7) becomes 1𝜏𝜏 = 𝑋𝑋 = βˆ’ 𝑒𝑒 πœ†πœ† πœ†πœ† πœ†πœ† πœ†πœ†π‘‘π‘‘π‘‘π‘‘π‘‘π‘‘ , andπœ†πœ†π‘‹π‘‹π‘‘π‘‘π‘‘π‘‘2𝜏𝜏 =π‘Œπ‘Œ0𝑑𝑑 = πœ†πœ† 𝜏𝜏 βˆ’ 𝑒𝑒 𝑋𝑋 𝑑𝑑 . Then πœ†πœ† = 𝛽𝛽 + 𝛽𝛽 𝑋𝑋 + 𝛽𝛽2𝑑𝑑 2𝑑𝑑𝑋𝑋2 From Equation 1𝜏𝜏/πœ†πœ†, π‘‘π‘‘π›½π›½πœπœ (3), 𝑋𝑋𝑑𝑑1𝜏𝜏 𝑑𝑑 = 𝛽𝛽2𝑑𝑑Equation 𝑑𝑑 𝑑𝑑 𝑑𝑑 (7) becomes 2𝜏𝜏 0𝑑𝑑 0𝑑𝑑 1𝑑𝑑 1𝑑𝑑 1𝜏𝜏 1𝜏𝜏 on (3), 2 𝜏𝜏/πœ†πœ†π‘‘π‘‘ 1 𝛽𝛽 𝜏𝜏 𝑒𝑒 (5)𝜏𝜏/πœ†πœ† 𝜏𝜏 (𝜏𝜏) 𝜏𝜏/πœ†πœ†π›½π›½ 𝑑𝑑 0𝑑𝑑 + 𝛽𝛽1𝑑𝑑(6) 1𝜏𝜏 the + From 𝑋𝑋2𝜏𝜏 Equation 𝑑𝑑 (3), 𝑑𝑑(8) 𝑑𝑑 (8) where , and are constants and 𝑑𝑑 𝑒𝑒22 𝑒𝑒π‘₯π‘₯ ′𝑑𝑑 β€²+𝑑𝑑 ]1second-order Note that 𝑑𝑑𝑑𝑑π‘₯π‘₯ second-order 𝛽𝛽𝛽𝛽0𝑑𝑑𝑑𝑑0𝑑𝑑 linear is a linear solution ordinary to Evaluating ordinary differential π‘Œπ‘Œ differential = 𝜏𝜏the integral 𝛽𝛽 + (5) 0𝑑𝑑equation 𝛽𝛽 equation𝑋𝑋 1𝑑𝑑at1𝜏𝜏the right, + 𝛽𝛽 𝑋𝑋 𝑑𝑑𝑑𝑑 2𝑑𝑑 2𝜏𝜏we obt (3), 1 second-order 𝛽𝛽 = 1π‘‘π‘‘βˆ« linear , 𝛽𝛽 =ordinary πœ†πœ† differential πœ†πœ† β‰  0.equation 1 π‘₯π‘₯(𝜏𝜏) = 𝑓𝑓 (4) (5) m β€²β€²Equation 0𝑑𝑑 βˆ’π‘’π‘’/πœ†πœ† βˆ’π‘’π‘’/πœ†πœ† π‘₯π‘₯ β€² + 2 β€²π‘₯π‘₯ β€² +(3), 𝑦𝑦𝑑𝑑 (𝜏𝜏)π‘₯π‘₯ = 𝛽𝛽 0𝑑𝑑 𝜏𝜏 0𝑑𝑑 [𝛽𝛽 2𝑑𝑑 0𝑑𝑑 + 𝑓𝑓𝑑𝑑 (𝜏𝜏) 𝑑𝑑𝛽𝛽𝛽𝛽0𝑑𝑑From 1𝑑𝑑+𝑒𝑒𝛽𝛽1𝑑𝑑 𝑒𝑒Equation βˆ’πœπœ/πœ†πœ† 𝑑𝑑 + 𝑑𝑑 +𝛽𝛽 𝑑𝑑 𝑒𝑒 π‘₯π‘₯π‘₯π‘₯ β€²β€²+ 2𝑑𝑑 (5) 𝛽𝛽2𝑑𝑑 β€²β€² (3), βˆ’πœπœ/πœ†πœ† == Specific value of 𝜏𝜏 gives specific values of 𝑋𝑋1𝜏𝜏 1𝜏𝜏 and 𝑋𝑋2𝜏𝜏 𝑑𝑑𝑑𝑑 2𝜏𝜏. + 𝛽𝛽0𝑑𝑑 πœ†πœ†π‘₯π‘₯𝑑𝑑From + Equation πœ†πœ† π‘₯π‘₯21 = πœ†πœ† 𝜏𝜏 (3),2 π‘Œπ‘Œ 𝑑𝑑 = 𝛽𝛽 where + (5) Evaluating 𝛽𝛽 𝑋𝑋𝛽𝛽 + , 𝛽𝛽 𝛽𝛽 𝑋𝑋, the integral + πœ†πœ†π‘‘π‘‘πœ†πœ† 𝛽𝛽 , π‘₯π‘₯and(6) + πœ†πœ† at 𝑑𝑑the right, are π‘₯π‘₯ constants (6) and(8) πœ†πœ† β‰  0. Note that π‘₯π‘₯(𝜏𝜏) = 𝑓𝑓 (𝜏𝜏) is yield curve𝜏𝜏 to a solution From 2Equation (3),+ 𝛽𝛽ordinary πœ†πœ†π‘‘π‘‘π‘‘π‘‘we 𝑒𝑒obtain the Nelson-Siegel +2πœπœπ›½π›½. 1𝑑𝑑 𝑋𝑋1𝜏𝜏 + 𝛽𝛽2𝑑𝑑 𝑋𝑋model for the 𝑑𝑑 2 2 0 𝜏𝜏 𝑑𝑑 πœ†πœ†]πœ†πœ†π‘‘π‘‘π‘‘π‘‘π‘‘π‘‘ 2 2 𝑑𝑑 𝑑𝑑 𝑑𝑑 1𝑒𝑒value of π‘Œπ‘Œβˆ’π‘’π‘’/πœ†πœ†π‘‹π‘‹πœπœ1𝜏𝜏=and 𝛽𝛽0𝑑𝑑𝑋𝑋 𝑑𝑑 (6) βˆ’π‘’π‘’/πœ†πœ† βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 𝑑𝑑 π‘‘π‘‘πœπœ givesπœ†πœ†πœ†πœ†specific πœ†πœ†values (𝜏𝜏)πœ†πœ†π‘‘π‘‘= 𝑑𝑑 +Specific 0𝑑𝑑 1𝑑𝑑 𝑑𝑑of 2𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑑𝑑 = πœ†πœ†π‘‘π‘‘ 2𝑦𝑦𝑑𝑑where 𝑑𝑑 ∫ [𝛽𝛽 1𝑑𝑑 π‘’π‘’πœπœ 𝛽𝛽2𝑑𝑑 equation 𝜏𝜏 0𝑑𝑑 1𝑑𝑑 1𝜏𝜏 2𝑑𝑑 2𝜏𝜏 πœ†πœ†π‘’π‘’πœπœπœπœπ‘‘π‘‘ 2πœπœπ‘‘π‘‘0𝑑𝑑linear 𝛽𝛽10π‘‘π‘‘βˆ« second-order 1 differential 𝑒𝑒 𝑑𝑑 Specific βˆ’π‘’π‘’/πœ†πœ† Specific value value of (6) of 𝜏𝜏 t gives gives specific specific values 2𝜏𝜏 of of 𝑋𝑋 and and 𝑋𝑋 𝑑𝑑 . . (𝜏𝜏) 𝛽𝛽0𝑑𝑑1 𝜏𝜏, 𝛽𝛽01𝑑𝑑 , 𝛽𝛽2π‘‘π‘‘βˆ’π‘’π‘’/πœ†πœ† , and πœ†πœ†1𝑑𝑑 are constants βˆ’π‘’π‘’/πœ†πœ† 𝑦𝑦 (𝜏𝜏) 𝑑𝑑𝑑𝑑 and 𝑑𝑑 +πœ†πœ†π‘‘π‘‘π›½π›½β‰ π‘‘π‘‘π‘’π‘’ = πœ†πœ† 𝑑𝑑0. Note∫ [𝛽𝛽 (6) that βˆ’π‘’π‘’/πœ†πœ† 0𝑑𝑑 π‘₯π‘₯(𝜏𝜏) +𝑑𝑑2 𝛽𝛽 𝑒𝑒𝑑𝑑𝑑𝑑 = 1𝑑𝑑 𝑒𝑒 𝑓𝑓𝑑𝑑 (𝜏𝜏)1is a solution 𝑑𝑑 + 𝛽𝛽 π›½π›½πœπœ 0𝑑𝑑to the 2𝑑𝑑 (6)𝑒𝑒 𝑑𝑑 ] 𝑑𝑑𝑑𝑑In practice, (6) a record of 2 (5)β€²2 ′𝑦𝑦1𝑑𝑑 (𝜏𝜏) daily 1𝜏𝜏 datasets 2𝜏𝜏 is 1 collected, = 1𝛽𝛽each βˆ’ [𝛽𝛽 + 𝜏𝜏 πœ†πœ† βˆ’π‘’π‘’/πœ†πœ† 𝛽𝛽 𝑒𝑒 𝑦𝑦 (𝜏𝜏) + = 𝛽𝛽 ∫ 𝑒𝑒 (𝜏𝜏) [𝛽𝛽 𝑒𝑒 +] 𝑑𝑑𝑑𝑑𝛽𝛽 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† βˆ’π‘’π‘’/πœ†πœ† 𝜏𝜏 βˆ’π‘’π‘’/πœ†πœ† 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 π‘₯π‘₯+ 𝛽𝛽 βˆ’π‘’π‘’/πœ†πœ† β€²β€² + ] π‘₯π‘₯ (6) β€² βˆ’π‘’π‘’/πœ†πœ† + π‘₯π‘₯ 1 = In practice, πœ†πœ† a record of daily datasets 𝑒𝑒 π‘₯π‘₯ β€²β€² + π‘₯π‘₯ β€²β€² is + π‘₯π‘₯ collected, + π‘₯π‘₯ + π‘₯π‘₯ = π‘₯π‘₯ = 0𝑑𝑑 ng Specific ∫ [𝛽𝛽0𝑑𝑑 + the 𝑑𝑑 1𝑑𝑑 integral 𝑦𝑦 value 𝑑𝑑𝑑𝑑 𝛽𝛽 𝑒𝑒 𝑑𝑑 of 𝑑𝑑 = at 𝜏𝜏 gives βˆ’π‘’π‘’/πœ†πœ† the 2𝑑𝑑 ∫ πœπœπ‘¦π‘¦ 𝑑𝑑𝑑𝑑𝑑𝑑+ right, [𝛽𝛽 specific 𝛽𝛽 0𝑑𝑑 = + 0𝑑𝑑 we 𝑑𝑑 𝛽𝛽 valuesβˆ«π‘’π‘’ 1𝑑𝑑obtain 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ†1𝑑𝑑[𝛽𝛽 of 𝑋𝑋 second-order 𝑑𝑑 ] 𝑑𝑑𝑑𝑑 the 𝑑𝑑+ 𝑑𝑑 1𝜏𝜏 1+1 and 𝛽𝛽 Nelson-Siegel 𝛽𝛽 𝜏𝜏𝜏𝜏 2𝑑𝑑𝑒𝑒 𝑋𝑋 2𝑑𝑑 2𝜏𝜏In . πœ†πœ†π‘’π‘’0 linear Specific practice, 𝑑𝑑𝑑𝑑] 𝑑𝑑𝑑𝑑 model ordinary value a 𝑒𝑒 record 0𝑑𝑑 (𝜏𝜏) = of for 𝜏𝜏 𝑑𝑑 ]the differential of gives 𝑑𝑑𝑑𝑑𝑒𝑒 daily𝑒𝑒 yieldspecific datasets curve equation 𝑑𝑑 values isβˆ’πœπœ/πœ†πœ† of collected, 𝑋𝑋 βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 and each 𝑋𝑋 𝑑𝑑 . dataset (6) 𝑒𝑒(6) βˆ’π‘’π‘’/πœ†πœ† βˆ’πœπœ/πœ†πœ† 𝑦𝑦 indexed (𝜏𝜏) = by 𝛽𝛽 a 2 + particular 𝛽𝛽 𝜏𝜏 ( 2 𝑑𝑑 𝑑𝑑 𝑦𝑦 ∫In [𝛽𝛽 βˆ’+ 𝛽𝛽a1𝑑𝑑number 𝑒𝑒 +daily 𝛽𝛽indicating 𝑒𝑒 theπœ†πœ†isinterest ]collected, 𝑑𝑑𝑑𝑑 2 second-order From From 1𝑑𝑑 𝜏𝜏 Equation πœ†πœ† 0 2𝑑𝑑 πœ†πœ†(3), Equation 00 𝑑𝑑 ordinary linear 0𝑑𝑑 𝜏𝜏(3), 1𝑑𝑑 0𝑑𝑑 (𝜏𝜏) differential 𝑦𝑦𝑦𝑦 (𝜏𝜏) 0𝑑𝑑 𝑑𝑑 = = equation ∫ ∫ 1𝑑𝑑 2𝑑𝑑 π‘₯π‘₯ β€²β€² [𝛽𝛽 + [𝛽𝛽 2 πœ†πœ†π‘‘π‘‘π‘‘π‘‘ π‘₯π‘₯+ 𝑑𝑑 + β€² +𝛽𝛽 𝛽𝛽 1 2𝑑𝑑 𝑒𝑒 𝑒𝑒π‘₯π‘₯πœ†πœ† βˆ’π‘’π‘’/πœ†πœ†πœ†πœ†= βˆ’π‘’π‘’/πœ†πœ† 𝛽𝛽 𝑑𝑑𝑑𝑑𝑑𝑑+ πœ†πœ† + 𝛽𝛽𝛽𝛽 2 𝑑𝑑2𝑑𝑑 each 𝜏𝜏 𝑒𝑒𝑒𝑒 πœ†πœ†βˆ’π‘’π‘’/πœ†πœ† βˆ’π‘’π‘’/πœ†πœ† dataset 2 1𝑑𝑑0𝑑𝑑 𝑑𝑑practice, 𝑑𝑑]]𝑑𝑑𝑑𝑑 𝑒𝑒 day indexed 𝑑𝑑𝑑𝑑 𝑑𝑑 record (5) by 1𝜏𝜏 𝑑𝑑 of taand particular 12π‘‘π‘‘βˆ’ 2𝜏𝜏 datasets πœ†πœ† day 𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑑𝑑number πœ†πœ†π‘‘π‘‘βˆ’πœπœ/πœ†πœ† rates 0𝑑𝑑 πœ†πœ†π‘‘π‘‘π‘‘π‘‘t π‘¦π‘¦πœ†πœ†each (𝜏𝜏) πœ†πœ† 𝑑𝑑𝑑𝑑𝑑𝑑 1𝑑𝑑 dataset for πœ†πœ†π‘‘π‘‘πœπœ2 0 differ 𝑑𝑑 differe ntegralInatpractice, 0 the right, we obtain 𝑑𝑑 the 𝑑𝑑 Nelson-Siegel πœ†πœ†π‘‘π‘‘ model 0𝑑𝑑 0𝑑𝑑 1𝑑𝑑 1𝑑𝑑 𝑦𝑦 𝑑𝑑 for 2 𝑑𝑑 the(𝜏𝜏) = 𝑑𝑑 yield 2 𝛽𝛽 0𝑑𝑑 curve 2𝑑𝑑 + 𝛽𝛽 0 ( ) + 𝛽𝛽 ( 𝑑𝑑 βˆ’ 𝑒𝑒 ) isπœπœπ‘‘π‘‘πœπœthe πœ†πœ†dataset πœ†πœ†indexed πœ†πœ†πœ†πœ†π‘‘π‘‘π‘‘π‘‘aindicating 1𝑑𝑑 2𝑑𝑑 a record ofthe Evaluating Evaluating daily theintegral datasets integral at at collected, the 00 right, each 1 𝜏𝜏𝜏𝜏 we we obtain obtain In the practice,𝑑𝑑the Nelson-Siegel aand by record particular of (5) daily 𝜏𝜏/πœ†πœ† model the datasets for interest isthe collected, yield rates 𝜏𝜏/πœ†πœ† curve ymaturities each (t) for 𝑑𝑑 dataset different e tain integral the at Nelson-Siegel the right, we model 𝑒𝑒 obtain 1 for βˆ’ the 𝑒𝑒 the βˆ’πœπœ/πœ†πœ† Nelson-Siegel yield 2day number curve 1 1 βˆ’ 𝛽𝛽tmodel and 𝑒𝑒 βˆ’πœπœ/πœ†πœ† (6) indicating for the yield the interest curve 𝑒𝑒 rates 2 𝑦𝑦 𝑑𝑑 (𝜏𝜏) parameters (7) 1for Evaluating different 𝛽𝛽1𝑑𝑑1𝑑𝑑0𝑑𝑑, and t the 2𝑑𝑑 in(6) integral Equation 𝜏𝜏. indexed For(8) at each the are byt, right, athe estimated particula we us t, ntegral luating integral [𝛽𝛽 we +obtain at the 𝛽𝛽 𝑦𝑦 theintegral 𝑒𝑒(𝜏𝜏)βˆ’π‘’π‘’/πœ†πœ† the From right, = Nelson-Siegel 𝛽𝛽 Nelson-Siegel + at Equation we𝛽𝛽 the + 𝑒𝑒 obtain 𝛽𝛽 right,model 𝑒𝑒 ( βˆ’π‘’π‘’/πœ†πœ† (3), the we model for ] Nelson-Siegel obtain π‘₯π‘₯ 𝑑𝑑𝑑𝑑 β€²β€² the + for ) yieldπ‘₯π‘₯ β€² thethe + + 1 𝛽𝛽 Nelson-Siegel curve yield π‘₯π‘₯ ( model = 0𝑑𝑑 curve (6) for the model βˆ’ From yield 𝑒𝑒 day βˆ’πœπœ/πœ†πœ† From for maturitiesEquation curve number 𝑑𝑑 ) the π‘₯π‘₯ Equation ′′𝑒𝑒 yield + t t. and (3), For π‘₯π‘₯curve each (3), indicating β€² 𝑑𝑑 + The t, the π‘₯π‘₯ 𝛽𝛽 0𝑑𝑑, 𝛽𝛽interest the parameters =parameters 0𝑑𝑑 𝛽𝛽rates 𝛽𝛽 2𝑑𝑑 (6) , 𝑦𝑦𝛽𝛽 𝑑𝑑 (𝜏𝜏) , for and different 𝛽𝛽 are matur calle 𝛽𝛽 + 0𝑑𝑑 𝛽𝛽 𝑒𝑒 1𝑑𝑑 βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 From 𝑑𝑑 + 𝛽𝛽 𝑑𝑑 Equation0𝑑𝑑 2𝑑𝑑 2𝑑𝑑 1integral βˆ’ 𝑒𝑒 (3), βˆ’π‘’π‘’/πœ†πœ† 1𝑑𝑑 πœ†πœ†π‘‘π‘‘π‘’π‘’ interest βˆ’πœπœ/πœ†πœ† 𝑑𝑑 ]𝑑𝑑𝑑𝑑𝑑𝑑𝑦𝑦 𝑑𝑑 𝑦𝑦 Evaluating (𝜏𝜏)(𝜏𝜏) πœ†πœ†π‘‘π‘‘ = = 1we πœ†πœ† ∫ 2 2𝑑𝑑 ∫ the [𝛽𝛽 βˆ’πœπœ/πœ†πœ†[𝛽𝛽 πœ†πœ† integral 2 + + 𝛽𝛽𝛽𝛽 𝑒𝑒 𝑒𝑒at βˆ’π‘’π‘’/πœ†πœ†βˆ’π‘’π‘’/πœ†πœ† the 𝑑𝑑 right, 𝑑𝑑 + + 𝛽𝛽 𝛽𝛽 we 𝑒𝑒 obtain 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† (7) βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 𝑑𝑑]]the𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 Nelson-Siegel 2 2 model 0𝑑𝑑 for 1𝑑𝑑 the yield 2𝑑𝑑 curve day 0𝑑𝑑 number t and indicating the 𝜏𝜏/πœ†πœ† rates 𝑦𝑦𝑑𝑑 (𝜏𝜏) πœπœβˆ’ forobtain𝑒𝑒different 𝜏𝜏/πœ†πœ† maturities For each t, the πœ†πœ†βˆ’πœπœ/πœ†πœ† πœ†πœ†yield πœ†πœ†for 𝑑𝑑 𝑒𝑒Evaluating Evaluating the the πœ†πœ†integral atat) the the 𝑑𝑑𝑑𝑑right, 𝑑𝑑right, (6) we 𝜏𝜏 0obtain 0𝑑𝑑 1the the 𝑒𝑒,𝑑𝑑Nelson-Siegel 1𝑑𝑑 Nelson-Siegel 𝜏𝜏. 2𝑑𝑑 model model for the curve (8)for 𝑑𝑑the yield curve 1𝑑𝑑 𝑑𝑑 𝑑𝑑 0𝑑𝑑number 1𝑑𝑑 t and 2𝑑𝑑 The (parameters day 1𝑑𝑑𝑑𝑑,𝑑𝑑)𝛽𝛽 and indicating are πœ†πœ†the 1πœ†πœ†Equation 𝑒𝑒interest 𝑑𝑑 factors rates 𝑑𝑑 (𝜏𝜏) 𝑑𝑑 different maturities For each factor t, the 𝑑𝑑called 𝛽𝛽 βˆ’0𝑑𝑑 ,𝛽𝛽and βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† in𝛽𝛽2𝑑𝑑 in π‘‘π‘‘βˆ’ (8) are 𝑑𝑑(OLS) 𝑦𝑦and estimated their coefficients using, for example, (7) areregression. 𝜏𝜏.called + 𝑑𝑑 (𝜏𝜏) π‘¦π‘¦βˆ’πœπœ/πœ†πœ† 𝛽𝛽 =𝑒𝑒𝛽𝛽 0𝑑𝑑 + βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 ]𝛽𝛽 𝑑𝑑𝑑𝑑1𝑑𝑑 ( βˆ’πœπœ/πœ†πœ† 𝑑𝑑 1 βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ† +𝑑𝑑 𝛽𝛽2𝑑𝑑parameters 1 βˆ’0 π‘’π‘’πœπœπ›½π›½βˆ’πœπœ/πœ†πœ† ,βˆ’ 𝑑𝑑 𝑒𝑒 𝛽𝛽 2𝑑𝑑parameters Equation (7) are squares , estimated and in using, method Equation formultiple 𝜏𝜏for example, 𝜏𝜏 are linear (8) the estimated ordinary using, least 1 (7) is𝛽𝛽for 1,𝑒𝑒(aeβˆ’π‘’π‘’ 0𝑑𝑑 1𝑑𝑑 𝛽𝛽 , 𝛽𝛽 𝛽𝛽 βˆ’πœπœ/πœ†πœ† βˆ’ 𝑒𝑒 𝑦𝑦 πœ†πœ†(𝜏𝜏) 2𝑑𝑑 𝑑𝑑 From Equation 1 βˆ’ (3), 𝑒𝑒 1(βˆ’ 𝜏𝜏/πœ†πœ† 𝑒𝑒Equation 𝑑𝑑 βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† 𝑦𝑦 (𝜏𝜏) 1𝑑𝑑𝑑𝑑(8) βˆ’ = 1 𝑒𝑒are𝛽𝛽 𝛽𝛽 βˆ’πœπœ/πœ†πœ† 𝜏𝜏 + 𝜏𝜏/πœ†πœ† 1(0π‘‘π‘‘βˆ’+𝑒𝑒1𝑑𝑑 1 𝛽𝛽 βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† ( 1𝜏𝜏/πœ†πœ† 𝑑𝑑 βˆ’ βˆ’πœπœ/πœ†πœ† 𝑒𝑒𝑒𝑒example, 𝑒𝑒 ) + 𝛽𝛽 the ( ordinary 𝑒𝑒 (7) 0𝑑𝑑 least 1𝑑𝑑 βˆ’ squares 𝑒𝑒 (6) loadings. (7) 2𝑑𝑑 (OLS) 𝑑𝑑 ) 1 Themethod 1 factor βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ loading for (6) multiple 𝑦𝑦 (𝜏𝜏) = of 𝛽𝛽0𝑑𝑑 𝑒𝑒 (7) 𝑑𝑑𝛽𝛽+ 𝑑𝑑++ βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† 𝑑𝑑 𝑑𝑑𝑑𝑑(𝜏𝜏) 0𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑑𝑑 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 = 𝛽𝛽 𝑑𝑑𝑑𝑑 ,+ 𝛽𝛽 𝑑𝑑 π‘¦π‘¦βˆ’πœπœ/πœ†πœ† ) + 𝑑𝑑 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† βˆ’ ) π‘‘π‘‘βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑2𝑑𝑑 βˆ’πœπœ/πœ†πœ† parameters 1 βˆ’ ) 𝑑𝑑 𝑒𝑒+ 𝛽𝛽 𝛽𝛽 , 𝛽𝛽 ( and 𝛽𝛽 1in βˆ’ βˆ’ 𝑒𝑒 𝑒𝑒 = )𝑦𝑦 ∫ (𝜏𝜏) [𝛽𝛽 estimated = ∫ 𝛽𝛽 using, [𝛽𝛽 + for + 𝛽𝛽 𝛽𝛽 βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† 𝑒𝑒 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑the ]+ 𝑑𝑑𝑑𝑑 βˆ’πœπœ/πœ†πœ† ordinary 𝛽𝛽 1 𝜏𝜏/πœ†πœ† βˆ’ 𝑒𝑒 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† least 𝑦𝑦 𝑑𝑑 ]𝑑𝑑 𝑑𝑑𝑑𝑑 (𝜏𝜏) 𝑦𝑦 =(𝜏𝜏) 1 =∫ βˆ’ 𝑒𝑒 [𝛽𝛽 ∫ [𝛽𝛽+ 𝛽𝛽 +𝑑𝑑 𝑒𝑒𝛽𝛽 βˆ’π‘’π‘’/πœ†πœ† 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† 0𝑑𝑑 𝛽𝛽 𝛽𝛽 1𝑑𝑑 (𝑑𝑑𝑑𝑑𝑑𝑑(𝜏𝜏)the 𝑦𝑦obtain 𝛽𝛽=0𝑑𝑑 𝛽𝛽 𝑦𝑦1𝑑𝑑+1𝑑𝑑(𝜏𝜏) 𝛽𝛽1𝑑𝑑 =(𝛽𝛽𝛽𝛽 + 𝛽𝛽called () for + 𝑑𝑑 𝛽𝛽 (Equation )11𝜏𝜏+ 𝛽𝛽 (method βˆ’π‘¦π‘¦π‘‘π‘‘π‘‘π‘‘ 𝑒𝑒(𝜏𝜏) πœ†πœ†, 𝑑𝑑)𝛽𝛽1𝑑𝑑1 βˆ’ 𝑒𝑒𝛽𝛽 is()πœ†πœ†π‘‘π‘‘1, 𝑑𝑑 0𝑑𝑑 1𝑑𝑑 From 2𝑑𝑑 (3), ameters we the 0𝑑𝑑 2𝑑𝑑 Evaluating 𝑑𝑑 , + 𝛽𝛽Nelson-Siegel ,Nelson-Siegel and 2𝑑𝑑 are the model 𝜏𝜏/πœ†πœ† integral 𝑑𝑑 1 𝜏𝜏at factors the 𝑒𝑒the π‘‘π‘‘πœπœ2𝑑𝑑 βˆ’πœπœ/πœ†πœ† 0right,yield and βˆ’ βˆ’ we their curveparameters 1𝑑𝑑 𝜏𝜏/πœ†πœ† 𝑑𝑑 𝑒𝑒coefficients βˆ’πœπœ/πœ†πœ† 𝑒𝑒𝑒𝑒2𝑑𝑑 βˆ’πœπœ/πœ†πœ† 0𝑑𝑑 2𝑑𝑑 𝑑𝑑𝛽𝛽1𝑑𝑑 𝑑𝑑 𝑑𝑑 are ,linear 1of and βˆ’ βˆ’ called 2𝑑𝑑in 𝑒𝑒𝛽𝛽𝑒𝑒0π‘‘π‘‘π›½π›½βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† 𝑑𝑑 regression. 𝑑𝑑factor Equation 𝑑𝑑 (8)Diebold 𝑑𝑑 are estimated 𝑑𝑑 and Li (2006) using, 𝜏𝜏(7)(7) 0𝑑𝑑 used for 0𝑑𝑑 1𝑑𝑑 example, the same 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 1𝑑𝑑 𝑑𝑑 value the ordinary ofas 2π‘‘π‘‘πœ†πœ†π‘‘π‘‘π‘‘π‘‘ for πœ†πœ†πœπœπ‘‘π‘‘2𝑑𝑑 β†’aπœ†πœ† l 0𝑑𝑑 obtain the 0𝑑𝑑Nelson-Siegel model for the yield e𝜏𝜏/πœ†πœ† 𝛽𝛽 1𝑑𝑑obtain 0𝑑𝑑0𝑑𝑑 Evaluating ) 1𝑑𝑑 π›½π›½πœπœ/πœ†πœ† 2𝑑𝑑 2𝑑𝑑 ( 0𝑑𝑑model 𝑑𝑑the 𝜏𝜏/πœ†πœ† integral 1𝑑𝑑𝑑𝑑for the βˆ’πœπœ/πœ†πœ† at loadings. the yield 2𝑑𝑑 𝑑𝑑 right, squares 𝑑𝑑𝑑𝑑curve ) 𝜏𝜏/πœ†πœ† Thewe (OLS) factor obtain 𝑑𝑑 𝜏𝜏/πœ†πœ† loading the for= 𝛽𝛽multiple 𝑑𝑑 Nelson-Siegel 0𝑑𝑑 squares + 1𝑑𝑑 linear 2𝑑𝑑 (OLS) (6) model π‘‘π‘‘βˆ’ amethod constant regression.βˆ’πœπœ/πœ†πœ† for )for +that the 𝛽𝛽 multiple 2𝑑𝑑does yield (𝜏𝜏 curve curve0not linear 0decay regression. βˆ’to zero )even 𝜏𝜏/πœ†πœ† 𝑦𝑦 𝑦𝑦 (𝜏𝜏) 𝜏𝜏/πœ†πœ† (𝜏𝜏) (𝜏𝜏) 𝑑𝑑𝑑𝑑𝑑𝑑 factors0𝑑𝑑 𝑑𝑑𝑑𝑑== = 𝛽𝛽𝛽𝛽 + + 𝛽𝛽𝛽𝛽 ( ( βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 +𝑑𝑑 ) ) βˆ’π‘’π‘’/πœ†πœ† + + 𝑑𝑑 ] 𝛽𝛽 𝛽𝛽 (( βˆ’ 𝑒𝑒 𝜏𝜏/πœ†πœ† 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝑑𝑑𝑑𝑑) ) 𝜏𝜏/πœ†πœ† 𝛽𝛽0𝑑𝑑 , 𝛽𝛽1𝑑𝑑 on-Siegel squares (OLS) , and 𝑑𝑑 model 𝛽𝛽2𝑑𝑑 method for arefor the called yield multiple 𝑦𝑦 curve 𝑑𝑑 linear ∫ 𝜏𝜏at the [𝛽𝛽 and 0𝑑𝑑 0 regression. 0𝑑𝑑 their + 𝛽𝛽 1𝑑𝑑 𝑒𝑒 1𝑑𝑑 coefficients 1𝑑𝑑 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† 𝛽𝛽 2𝑑𝑑 πœ†πœ†π‘‘π‘‘ (OLS) 𝑒𝑒 are (7) called 𝑑𝑑𝑑𝑑 2𝑑𝑑2𝑑𝑑 factorDiebold and 𝑑𝑑Li (2006) 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† Thus, was called itused obtained has the significantsame based on this contribution 𝑑𝑑 value of single value of πœ†πœ†to the Evaluating The parameters the integral 𝛽𝛽 𝑑𝑑a , constant 𝛽𝛽 right, , and we 𝛽𝛽 obtain are squares thecalled 𝑑𝑑𝑑𝑑Nelson-Siegel factors method model and 1 for for 𝜏𝜏 𝑑𝑑multiple their 𝑑𝑑the yield coefficients linear curve 1𝑑𝑑 , 𝛽𝛽regression. 𝛽𝛽1𝑑𝑑 2𝑑𝑑 }are 𝑒𝑒 factor 𝑑𝑑𝑑𝑑 . Other Othe s. ers 1 1 The βˆ’ βˆ’ 𝛽𝛽 𝑒𝑒 𝑒𝑒factor , βˆ’πœπœ/πœ†πœ†βˆ’πœπœ/πœ†πœ† 𝛽𝛽 𝑑𝑑 loading , and Evaluating𝛽𝛽 1 of areβˆ’ 1 𝛽𝛽 βˆ’ 𝑒𝑒 called𝑒𝑒 βˆ’πœπœ/πœ†πœ† the is βˆ’πœπœ/πœ†πœ† 1, integralfactors 0𝑑𝑑 1𝑑𝑑 Thus, at and the that their itright, 2𝑑𝑑 has does Diebold coefficients significant 1we βˆ’ not (7) obtain 𝑒𝑒and (7) decay βˆ’πœπœ/πœ†πœ† Li 𝑑𝑑the are (2006) contribution to called Evaluating zero Nelson-Siegell used Evaluating for1 evenfactor βˆ’ all the to the 𝑒𝑒 as t. same βˆ’πœπœ/πœ†πœ† the 𝜏𝜏 Thus, integral the β†’ 𝑑𝑑value yield model +∞. integrala forseries 2𝑑𝑑 of forat πœ†πœ† the any the The at for of right, the parameters all maturity; yield estimatest. right, weThus, curve obtain we hence, a{ (7)(7)𝛽𝛽 series obtain the,it 𝛽𝛽 of , Nelson-Siegel isthe and estimates referred Nelson-Sieg 𝛽𝛽 to{𝛽𝛽are as,m ct ed parameters 𝛽𝛽0𝑑𝑑 (are factors , 𝛽𝛽1𝑑𝑑 0𝑑𝑑 𝑑𝑑 , and 1𝑑𝑑 and 𝛽𝛽0𝑑𝑑+ ) 𝛽𝛽their ,2𝑑𝑑 𝛽𝛽 𝛽𝛽2𝑑𝑑are 2𝑑𝑑 coefficients (, and called 0𝑑𝑑 𝛽𝛽2𝑑𝑑 𝑑𝑑 factors are are βˆ’ called βˆ’πœπœ/πœ†πœ† 𝑒𝑒 called and βˆ’πœπœ/πœ†πœ† 𝑑𝑑 ) factors their 𝑑𝑑 ) factor coefficients and 1βˆ’ their 𝑒𝑒modelβˆ’πœπœ/πœ†πœ† are coefficients 𝑦𝑦 𝑑𝑑 (𝜏𝜏) called = t factor ∫ are 1Dieboldβˆ’called [𝛽𝛽 βˆ’πœπœ/πœ†πœ† 𝑒𝑒 + and 𝛽𝛽 factor 𝑑𝑑 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† Liβˆ’πœπœ/πœ†πœ† (2006) 𝑑𝑑 𝑑𝑑 + used 𝛽𝛽 βˆ’π‘’π‘’/πœ†πœ† the𝑒𝑒 same ]value 𝑑𝑑 0𝑑𝑑 𝑑𝑑𝑑𝑑 1𝑑𝑑 of πœ†πœ†π‘‘π‘‘ for all t. Thus 2𝑑𝑑 0𝑑𝑑 actor 0𝑑𝑑called loading 1𝑑𝑑Evaluating Diebold ) factors βˆ’πœπœ/πœ†πœ†of + and 𝛽𝛽 2𝑑𝑑 𝛽𝛽0𝑑𝑑the ( and 1𝑑𝑑integral Li (2006) is 1, their at a𝜏𝜏/πœ†πœ†constant used thecoefficients βˆ’ right, 𝑑𝑑𝑑𝑑,(𝜏𝜏) 𝑦𝑦𝑦𝑦the 𝑒𝑒 (𝜏𝜏) we same ==The that obtain 𝛽𝛽 𝛽𝛽does value are parameters the ++ called 1of𝛽𝛽 Nelson-Siegel 𝛽𝛽1𝑑𝑑 not 𝑒𝑒𝑑𝑑 ( πœ†πœ†1𝑑𝑑 factor all𝛽𝛽t.0𝑑𝑑 ( 𝑑𝑑factors decay for to , zero Thus, 𝛽𝛽11𝑑𝑑)βˆ’π‘‘π‘‘for )a, + and the 𝑒𝑒+ even series yield 𝛽𝛽𝛽𝛽2𝑑𝑑Li𝛽𝛽 π‘‘π‘‘πœπœas of curve ((2006) 𝜏𝜏are (0}estimates was β†’ called 0𝑑𝑑 +∞. obtained{𝛽𝛽 factors 1𝑑𝑑 βˆ’βˆ’,same 𝑒𝑒𝑒𝑒 De based βˆ’πœπœ/πœ†πœ† long-term and 𝑑𝑑 )) on their 2𝑑𝑑 πœ†πœ†this πœ†πœ†factor. coefficients single value aare ofcalled factor valt 𝑑𝑑 𝑑𝑑for all The factor 𝛽𝛽1𝑑𝑑aofis called 1𝜏𝜏/πœ†πœ† 𝑑𝑑 2𝑑𝑑 𝛽𝛽𝛽𝛽(7) βˆ’ βˆ’πœπœ/πœ†πœ† Diebold and βˆ’πœπœ/πœ†πœ† 2𝑑𝑑 used theClaes, (7) value 𝑑𝑑 Ceuster, factor of & Zhang, 2012) t. Thus, considered series rangeestimates of valu βˆ’ 𝑑𝑑The 𝑑𝑑𝑒𝑒 parameters ,=and are called (and their coefficients are are called 0𝑑𝑑 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘π›½π›½ 𝑑𝑑,,𝑦𝑦𝛽𝛽 𝜏𝜏/πœ†πœ† The parameters loadings. 𝜏𝜏/πœ†πœ† 𝛽𝛽The 0𝑑𝑑 0𝑑𝑑 𝑑𝑑𝛽𝛽 𝑑𝑑factor (𝜏𝜏) 1𝑑𝑑 1𝑑𝑑 and 𝛽𝛽loading 0𝑑𝑑 +2𝑑𝑑 2𝑑𝑑 0𝑑𝑑 𝛽𝛽for are (,of called 𝛽𝛽 is factors )1, 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† +aβˆ’πœπœ/πœ†πœ† constant 𝛽𝛽𝑑𝑑2𝑑𝑑 and their that 2𝑑𝑑 βˆ’ coefficients does 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† not 𝑑𝑑 ) decay 0𝑑𝑑 called to zero factor even as 𝜏𝜏 β†’ +∞. has ( significant contribution βˆ’ 𝑒𝑒 ) to the yield 𝛽𝛽 any 𝛽𝛽 }maturity; was obtained hence, 𝑑𝑑 based it is lon . referred this Others single 𝑑𝑑to (Nelson 𝑑𝑑 as value the &of Siegel, πœ†πœ† . Others loadings. 1987; (Nelson Annaert, The & factor Siegel, Claes, loading 1987; Annaert, of 𝛽𝛽 isβˆ’πœπœ a𝛽𝛽eactor factor loading of is 1, a𝛽𝛽constant βˆ’that does not 1decay 𝑑𝑑to zero even as 𝛽𝛽not 1𝑑𝑑 0𝑑𝑑 𝜏𝜏was β†’ +∞. 1long-term 𝜏𝜏𝜏𝜏/πœ†πœ† factor. The βˆ’πœπœ/πœ†πœ† factor 𝜏𝜏/πœ†πœ† 𝛽𝛽1𝑑𝑑 t, is called theas βˆ’πœπœ/πœ†πœ† 𝑒𝑒obtained short-term based factor because (7) βˆ’πœπœ/πœ†πœ† its 𝑑𝑑factor βˆ’πœπœ/πœ†πœ† loading, to𝑑𝑑on this grid single 1 βˆ’value 𝑒𝑒of πœ†πœ†π‘‘π‘‘ .𝑑𝑑the Others 1(Nelso 1𝑑𝑑 2𝑑𝑑 𝑑𝑑 dings.2𝑑𝑑constant loading The𝜏𝜏/πœ†πœ† that factor of does 𝛽𝛽 loading is 0𝑑𝑑 1, decay a of constant to+single is zero 1, that a even constant βˆ’πœπœ/πœ†πœ† 𝑒𝑒 does as𝑑𝑑 not β†’that𝑑𝑑 +∞. decay 1 βˆ’ π‘’π‘’βˆ’ does to 𝑒𝑒not zero decay 𝑑𝑑 𝑑𝑑𝛽𝛽 even 𝑑𝑑 )1𝑑𝑑 to as 𝛽𝛽zero 2π‘‘π‘‘πœπœ 1}β†’ βˆ’ (7) even +∞. was 𝑑𝑑 𝜏𝜏 β†’ referred +∞. as search. 1𝑒𝑒Inβˆ’ this case, value of βˆ’0𝑑𝑑1 πœ†πœ†π‘’π‘’ βˆ’ is 1, a wasconstant obtained that based does on notthis decay value to zero of even . Others as (Nelson & Siegel, De 1987; Ceuster, Annaert, & Zhang, 2012) considered a range of β†’an βˆ’πœπœ/πœ†πœ† 0𝑑𝑑 ecalledificant 1𝑑𝑑 , 𝛽𝛽 calledfactors } factors 𝑑𝑑 2𝑑𝑑 contribution to the yield 𝑦𝑦 and 0𝑑𝑑 0𝑑𝑑 their 𝑦𝑦 𝑑𝑑 (𝜏𝜏) = coefficients 𝛽𝛽 0𝑑𝑑 0𝑑𝑑 for 𝛽𝛽 𝑑𝑑 (𝜏𝜏) 1𝑑𝑑 ( Evaluating any loadings. are = 𝜏𝜏/πœ†πœ† 𝛽𝛽𝑑𝑑called maturity; ) πœ†πœ† + +1,𝑑𝑑 the 𝛽𝛽The 𝛽𝛽 2𝑑𝑑 ( (factor integral 𝛽𝛽hence, factor , 𝜏𝜏/πœ†πœ†πœπœ β†’ 𝛽𝛽2𝑑𝑑𝑑𝑑 }itwas loading +∞.is βˆ’ at 𝑒𝑒 the referred ) obtained + of𝛽𝛽2𝑑𝑑 right, 𝛽𝛽to based0𝑑𝑑( we as isobtain 1, this on the a constant 𝑑𝑑 (𝜏𝜏) 𝑦𝑦single the 𝑒𝑒=value that βˆ’πœπœ/πœ†πœ† (𝜏𝜏) βˆ’π‘¦π‘¦Nelson-Siegel 𝛽𝛽 βˆ’πœπœ/πœ†πœ† = )does of 𝑑𝑑+ π›½π›½πœπœπœ†πœ†0𝑑𝑑𝛽𝛽 + 𝑑𝑑 .1𝑑𝑑 not Others 1𝑑𝑑decay (𝛽𝛽model ((Nelson for to the zero &) Siegel, +yield even )𝛽𝛽 +( 𝛽𝛽as curve 1987; 2𝑑𝑑 ( 𝜏𝜏 Ann 𝑑𝑑 𝑑𝑑 loadings. loadings. The and The their The parameters factor it1𝑑𝑑factorcoefficients loading 𝛽𝛽loading are ofof called 𝛽𝛽 𝛽𝛽Claes, 0𝑑𝑑isfactor is 1, a 1𝑑𝑑 constant aCeuster, constant 1𝑑𝑑 factors that 𝑑𝑑that does does not not decay decay toto zero zero 𝑑𝑑even 1βˆ’π‘’π‘’ even 0𝑑𝑑asas ,𝜏𝜏has β†’ +∞. β†’ofsignificant factor +∞. 𝜏𝜏/πœ†πœ† π‘‘π‘‘πœπœ/πœ†πœ† 2𝑑𝑑 𝜏𝜏/πœ†πœ† The Thus, parameters has significant 0𝑑𝑑𝛽𝛽 , , ,the 𝛽𝛽𝛽𝛽 1𝑑𝑑,short-term and ,𝛽𝛽,contribution and 𝛽𝛽𝛽𝛽called are are De called to hence, called the 𝜏𝜏/πœ†πœ† yield factors &their Zhang, itsand for and any 2012) values their theirmaturity; coefficients 𝜏𝜏/πœ†πœ† coefficients considered of lt the for𝑑𝑑hence, eacha range are it aret.𝜏𝜏/πœ†πœ†iscalled The of referred called valuesmethod itfactor toπœ†πœ†was𝑑𝑑as for the each referred contribution t.𝑑𝑑 22Thetoof method to the𝑑𝑑 The The parameters parameters and and 0𝑑𝑑 are are called factors andfactors coefficients are called factor m ignificant eClaes, factor. yield for The contribution any factor maturity; 𝛽𝛽 to is the hence, called 𝛽𝛽 , 0𝑑𝑑 𝛽𝛽1𝑑𝑑 yield 0𝑑𝑑 it is 1𝑑𝑑 for referred 1βˆ’π‘’π‘’ any 2𝑑𝑑 0𝑑𝑑 βˆ’πœπœ/πœ†πœ† 2𝑑𝑑 maturity; 2𝑑𝑑 𝑑𝑑 to as factorthe because itforis referredfactor Claes, De loading, to as Ceuster, 𝛽𝛽2𝑑𝑑 &} Zhang, were Thus, chosen 𝑑𝑑2012) based has considered on significant the highest a range 𝑅𝑅 contribution . Thus, values different of πœ†πœ†to da onnificant s, d their itto the hasDe contribution significant Ceuster, coefficients The yield and parameters for & their Zhang, any 𝛽𝛽 to contribution are0𝑑𝑑coefficients , the called 𝛽𝛽 maturity; 2012), yield and considered factor 𝛽𝛽 for to the are called are hence, any called Thus, yield it amaturity; factors is range factor for ,has referred it andof any hasloadings. hence, values their Claes, significantmaturity; significant to of coefficients as Dethe itπœ†πœ† The is 𝑑𝑑 contribution Ceuster, referred hence, contributionare each called&asZhang, t. itTheto grid to factor is asreferred method to search. 2012)thethetheβˆ’πœπœ/πœ†πœ† value yield to 2𝑑𝑑 In𝑑𝑑 this considered as for the of case, 𝑦𝑦a𝑑𝑑 (𝜏𝜏) any range the maturity; atofvalue shortervalues βˆ’πœπœ/πœ†πœ†hence, of maturities of ltπœ†πœ†and 𝑑𝑑 itforisthe eachreferred(smaller t. The to metho valu as 𝑑𝑑 t .s1,The 1,a aconstant factor 𝛽𝛽 that is does called notnot the 1𝑑𝑑 decay short-term 2𝑑𝑑 toloading zero factor because its factor loading, 1 βˆ’ 𝑒𝑒 1 βˆ’ 𝑒𝑒 constant that does ofdecay to ofzero ,𝛽𝛽even asathe 𝑑𝑑even as 𝑑𝑑 Thus, Thus, it1𝑑𝑑 loadings. loadings. The has itlong-term has significant significant parameters The The factor factor.factor contribution contribution loading The 𝛽𝛽factor 𝜏𝜏/πœ†πœ† was and of is to 1,to referred 𝛽𝛽 is𝜏𝜏constant the is β†’ are 𝜏𝜏1,+∞. yield yield 1, called to β†’anot called that as for +∞. for constant the grid does anyany factors short-termsearch. not maturity; maturity; that The decay and doesIn parameters The totheir this factor zerohence, not parametershence, case, even decay coefficients because 𝛽𝛽 it as the {is itgrid 𝜏𝜏,is value to 𝛽𝛽 β†’ referred 𝛽𝛽referred its+∞. zero , , 𝛽𝛽 of and are factor πœ†πœ†even even to ,called and 𝛽𝛽 to and as as loading, 𝛽𝛽the are the the factor set called are 𝜏𝜏𝜏𝜏case, β†’ of +∞. called estimates factors factors {𝛽𝛽 and ,and their𝛽𝛽1𝑑𝑑 their , coef tor. The factor loadings. factor is loading The called factor 𝛽𝛽 , is loading 1, aconstant 0𝑑𝑑 of 𝛽𝛽 constant 𝛽𝛽𝛽𝛽0𝑑𝑑 is that a𝑦𝑦 doesconstant (𝜏𝜏) not= 𝛽𝛽 that was + set does 𝛽𝛽 of referred (estimates not todecay as 0𝑑𝑑 to )0𝑑𝑑 zero + search. 1𝑑𝑑 𝜏𝜏). 𝛽𝛽 long-term Moreover, 1𝑑𝑑 𝑑𝑑( In 2𝑑𝑑 }as this were 2𝑑𝑑 β†’ factor. it chosen +∞. decreases βˆ’ the 𝑒𝑒 value The based βˆ’πœπœ/πœ†πœ† to ) 𝑑𝑑of factor 0 πœ†πœ† 0𝑑𝑑and as 𝛽𝛽 𝜏𝜏 β†’the is +∞ set call was the referred loadings. to as 𝛽𝛽contribution The grid factor search. loading Inthe thisof short-term 0𝑑𝑑 𝛽𝛽 case, is 1, 1𝑑𝑑 a theloading, value factor 1𝑑𝑑 𝑦𝑦𝑑𝑑 of 2𝑑𝑑 0𝑑𝑑 that because does and 𝑑𝑑 the decay itsset its to offactor zero 0𝑑𝑑 estimates even loading, as 1𝑑𝑑 𝜏𝜏 β†’ +∞. 2𝑑𝑑 thas short-term significant factor because to its the 𝜏𝜏factor 0𝑑𝑑 value of (𝜏𝜏) 𝑑𝑑at πœ†πœ†was shorter maturities (smaller {𝛽𝛽0𝑑𝑑 , In 𝛽𝛽 values , 𝑑𝑑 2 case,of 𝑑𝑑 𝜏𝜏/πœ†πœ† value of𝜏𝜏/πœ†πœ† estimates1𝑑𝑑{𝛽𝛽0𝑑𝑑 , 𝛽𝛽1 sr. g-term The factor factor. 𝛽𝛽The isfactor called the 1𝑑𝑑 is 𝛽𝛽even short-term called the factor .short-term because factor factor because loading, grid its factor search. loading, 1𝑑𝑑 referred to as 1𝑑𝑑this thedifferent πœ†πœ†π‘‘π‘‘ and 𝑑𝑑 the set ofhave does called notthe decay decay short-term 1𝑑𝑑 to 1𝑑𝑑 to zero zerofactor even as because β†’ long-term +∞. its factor Thus, it loading, has factor. significantThe factor 𝛽𝛽 on the is highest called the Rto.short-term Thus, factor datasets because could its factor loading, tothe theyield yield forfor Thus, any any it maturity; has maturity; significant hence, hence, contribution it to is itthe 𝜏𝜏).referred isany Moreover, to the to yield as it for the decreases any maturity; to 0 hence, as 𝜏𝜏 it β†’ is +∞. referred Lastly, as the 𝛽𝛽 is referred to as the medium-term fa isreferred to as the 1𝑑𝑑 2 nificant long-term long-term contribution Thus, factor. factor. it has to 𝑑𝑑 the The The significant value factor factor of 𝛽𝛽 𝛽𝛽 𝑦𝑦 contribution (𝜏𝜏) is 𝛽𝛽 called at }maturity; called were shorter the to thechosen the short-term short-term maturities yield based itfor iton isfactor factor (smaller any the highest because maturity; because tovalues 𝑅𝑅 The .its Thus, its of hence, factor different it isisloading, 2𝑑𝑑 datasets could loading, referred have different shape 𝑑𝑑to 𝛽𝛽as loadings. different loadings. The shape factor parameters loading ltof . even Furthermore, is 1, 2the aThus, the constant aissue of that does notcou 𝛽𝛽 } were Thus, chosen contribution loadings. it1βˆ’π‘’π‘’ Thus, has based significant βˆ’πœπœ/πœ†πœ† it has on the to The the contribution significant highest factor yield 𝑅𝑅 2 for loading contribution . 1𝑑𝑑 𝑑𝑑1𝑑𝑑 Thus, yield 2𝑑𝑑 of for different any 𝛽𝛽 to is maturity;thehence, datasets 1, ayield constant hence, could is for have any 𝛽𝛽that referred does }maturity; differentwere as the not chosen shape decay hence, factor based to it on loading zero referred the 𝛽𝛽of highest βˆ’πœπœ/πœ†πœ† 0𝑑𝑑 as to as 𝑅𝑅 0π‘‘π‘‘πœπœ is β†’.the 1, +∞. constant different βˆ’πœπœ/πœ†πœ† that datasets does codn 2𝑑𝑑 𝑑𝑑 , has significant The contribution parameters 0𝑑𝑑 𝛽𝛽2𝑑𝑑 to𝛽𝛽} 0𝑑𝑑 the were 𝛽𝛽value chosen ofbased 𝑑𝑑 (𝜏𝜏) 𝑦𝑦multicollinearity 2𝑑𝑑 itsonfactor atcalled the shorter highest factors maturities 2 𝑅𝑅 .because Thus, 1βˆ’π‘’π‘’ (smaller different itsand , has factor values datasets significant loading, couldof are have 1βˆ’π‘’π‘’ contribution different𝑑𝑑 factor βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ† shap to ysignificant contribution to the value of 𝑦𝑦isvalue (𝜏𝜏) 𝛽𝛽𝑑𝑑2𝑑𝑑at isshorter ,maturities 1𝑑𝑑 ,its isand (smaller 𝛽𝛽 are values of was and their considered coefficients ridge regression called βˆ’πœπœ/πœ†πœ† over, e maturity; value nificant itshort-term decreases of hence, 𝑦𝑦 contribution referred long-term (𝜏𝜏) at 𝜏𝜏/πœ†πœ† it𝑑𝑑shorter to is 0to factor. to as as referred the 𝜏𝜏the The β†’ maturities value long-term +∞. toistoof factor as 𝛽𝛽 Lastly,the (smaller factor. called at The the values shorter factor short-term referred of maturities factor to as because the (smaller medium-term 𝑑𝑑 values loading, of factor gnificant ion ed the, has significant long-term 𝑑𝑑 factor. factor contribution The (𝜏𝜏) factor because 𝛽𝛽 its called the factor 𝑦𝑦 1𝑑𝑑 1βˆ’π‘’π‘’ the(𝜏𝜏) π‘‘π‘‘βˆ’πœπœ/πœ†πœ† short-term loading, of 𝑦𝑦 (𝜏𝜏) factor at because shorter factor maturities 1βˆ’π‘’π‘’ 2𝑑𝑑 βˆ’πœπœ/πœ†πœ† loading, (smaller values of 𝜏𝜏/πœ†πœ†π‘‘π‘‘ 𝜏𝜏/πœ†πœ†π‘‘π‘‘ alledπœ†πœ†on to the the value short-termβˆ’πœπœ/πœ†πœ† 𝑑𝑑𝑑𝑑 of 𝑦𝑦factor at because shorter maturities its factor 𝑑𝑑𝑑𝑑 ,(smaller loading, has 𝑑𝑑significant values of contribution to the βˆ’πœπœ/πœ†πœ† value of 𝑦𝑦 (𝜏𝜏) at shorter maturities (smaller valu 1𝑑𝑑 1βˆ’π‘’π‘’ βˆ’πœπœ/πœ†πœ† called long-term Thus, the it short-term factor. has significant The factorfactor becausebecause 𝛽𝛽𝛽𝛽 isisits its called factor factor loading, loading, short-term Thus, was Thus, factor implemented itmaturity; has βˆ’itbecause𝑒𝑒significant has , starts significant as its a itsfactor itatiscontribution remedy contribution 𝑑𝑑0, increases, loading, (Annaert totothe then et to thedecays al., yield 2012).for any yield for to 0. maturity; any More matur 𝛽𝛽2𝑑𝑑contribution to the yield for any hence, referred as the 1βˆ’π‘’π‘’ 𝑑𝑑 𝑑𝑑 decreases1βˆ’π‘’π‘’ toβˆ’πœπœ/πœ†πœ†long-term 01βˆ’π‘’π‘’ ,as has ,βˆ’πœπœ/πœ†πœ† has πœπœπ‘‘π‘‘ β†’ 𝑑𝑑 significant significant +∞.factor. Lastly, The contribution contribution factor is 𝜏𝜏/πœ†πœ† 𝑑𝑑1𝑑𝑑to referred 1𝑑𝑑 to called the the tovalue value as the the of short-term of 𝑦𝑦𝑦𝑦𝑑𝑑𝑑𝑑(𝜏𝜏) medium-term (𝜏𝜏) atat 𝜏𝜏/πœ†πœ† factor shorter shorter 𝑑𝑑 factor because maturities maturities factor (smaller (smaller loading, values values ofof m, itfactor decreases 𝜏𝜏/πœ†πœ† because 𝜏𝜏/πœ†πœ† 𝑑𝑑𝑑𝑑 π‘‘π‘‘πœπœ). Moreover, , 𝜏𝜏/πœ†πœ† has its , has has factor loading, significant βˆ’πœπœ/πœ†πœ† significant significant contribution it decreasescontribution loadings. contribution to the to value to 0 as theThe to 𝜏𝜏 value β†’ of 𝑦𝑦𝑑𝑑 (𝜏𝜏) at shorter the factor +∞. of value 𝑦𝑦 loading (𝜏𝜏) Lastly, at maturities of shorter of 𝛽𝛽 𝛽𝛽 maturities (smaller is 2𝑑𝑑 0𝑑𝑑 valuesfactor referred is 1, a (smaller of constantto as values the that of precisely, medium-term does 𝜏𝜏). Moreover, not it approaches decay factor to it decreases zero 0 as 𝜏𝜏 to even β†’ 00 asand as + 𝜏𝜏 πœπœβ†’β†’+ a 𝑑𝑑 to 0referred 𝑑𝑑 as 𝜏𝜏 β†’ +∞. βˆ’πœπœ/πœ†πœ† Lastly, is referred to as the medium-term 1βˆ’π‘’π‘’ 𝑑𝑑 𝛽𝛽2𝑑𝑑 𝑑𝑑 ∞. otits Moreover, the Lastly, factor decreases value loading, π›½π›½πœπœ/πœ†πœ† it to of2𝑑𝑑 0is decreases 𝑦𝑦 as (𝜏𝜏) 𝜏𝜏 at β†’ at to +∞. shorter shorter to0 βˆ’ as 𝑒𝑒 Lastly, the 𝜏𝜏 β†’ maturities maturities medium-term 𝑑𝑑 , starts +∞. 𝛽𝛽 is Lastly, (smaller(smaller at referred 0, 𝛽𝛽factor increases, values to values is as referred of the of then medium-term t). to decays as the to 0. medium-term factor More factor 𝜏𝜏toβ†’the +∞. value Lastly, long-term βˆ’πœπœ/πœ†πœ† of 𝑦𝑦𝑑𝑑 ,,has 𝛽𝛽 is referred factor. Theto as factor 2𝑑𝑑 𝜏𝜏).the medium-term Moreover, 𝛽𝛽 is called 2𝑑𝑑 it decreasesfactor the short-term long-term to 0 long-term as factor 𝜏𝜏 β†’ factor. +∞. factor. because The Lastly, factor The its factor factor 𝛽𝛽 𝛽𝛽 is is loading, 𝛽𝛽 referred called is called to the as short-term the short-term medium-term factor factor bec fa itat shorter maturities (smaller values of 1βˆ’π‘’π‘’ βˆ’πœπœ/πœ†πœ† (𝜏𝜏) 𝑑𝑑 2𝑑𝑑 + 0precisely, 1𝑑𝑑 itto approaches 0ofis as and as Its 1βˆ’π‘’π‘’ 1βˆ’π‘’π‘’ 𝑑𝑑 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 2𝑑𝑑 𝜏𝜏/πœ†πœ† 𝑑𝑑 significant contribution the value of 𝑦𝑦𝑦𝑦 𝜏𝜏(𝜏𝜏) β†’ at0atshorter 𝜏𝜏 medium-term maturities β†’ +∞.2𝑑𝑑(smaller 1𝑑𝑑significant 1𝑑𝑑 valuescontribution of to the yield 𝑒𝑒has significant 00contribution isto the𝛽𝛽2𝑑𝑑 value shorter maturities (smaller values ofit is referred 𝑑𝑑 r loading, 𝜏𝜏). 𝜏𝜏). 𝜏𝜏).Moreover, Moreover,Moreover, 𝜏𝜏). Moreover, 𝜏𝜏/πœ†πœ† Moreover, βˆ’it decreases itdecreases itβˆ’πœπœ/πœ†πœ† decreases 𝑑𝑑 ,tostarts itdecreases decreases 0 as 𝜏𝜏to to atThus, to to β†’ +∞. 0,as as πœπœπœπœβ†’ asincreases, 𝜏𝜏Lastly, 1βˆ’π‘’π‘’ itβ†’ β†’ has +∞. +∞. 𝛽𝛽+∞. βˆ’πœπœ/πœ†πœ† Lastly, .Lastly, then 2𝑑𝑑𝑑𝑑significant Lastly, Lastly, referred decays toisas 𝛽𝛽𝛽𝛽 2𝑑𝑑 is referred contribution the 2𝑑𝑑 to referred 0. 𝑑𝑑to medium-term (𝜏𝜏) referred 𝑑𝑑Data More as the and to to to medium-term the factor as as the Procedurethe yield medium-term factor for anymedium maturity; factor factor hence, to 𝑑𝑑as th (𝜏𝜏) at shorter 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† because maturities 1βˆ’π‘’π‘’ 𝑑𝑑 π‘‘π‘‘βˆ’πœπœ/πœ†πœ† 𝑑𝑑 𝑑𝑑 its factor (smaller βˆ’πœπœ/πœ†πœ† loading, values of βˆ’ 𝑒𝑒because βˆ’πœπœ/πœ†πœ† 𝑑𝑑 ,decaysstarts at 0, increases, then decays the because to 0. 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Lastly, company contribution contribution 𝛽𝛽2𝑑𝑑Philippine is𝜏𝜏/πœ†πœ†to to𝑑𝑑 the referred the𝑑𝑑yield 𝜏𝜏/πœ†πœ† Dealing yield to isin Systems, in medium-term fact & Li, 2006). The factor b0t governs the 0𝑑𝑑level of the curve curve (Diebold & Li, 2006). The factor 𝛽𝛽 governs the level of the http://www.pds.com.ph/ curve since lim 𝑦𝑦 (𝜏𝜏) = ). The rates for tenors at most ts of el significant the curve since contribution since lim 𝑦𝑦𝑑𝑑 (𝜏𝜏) to the = byield is in πœπœβ†’+∞ 𝑑𝑑 and an increase in b increases one year were zero rates so0𝜏𝜏these thethe medium precisely, medium 𝛽𝛽0𝑑𝑑 and an πœπœβ†’+∞ term itterm increaseapproaches to toinmaturity. maturity. 0t increases 0 as 𝜏𝜏allβ†’ 0+equally. and 0t asThus, 𝜏𝜏 β†’this precisely, +∞. 1βˆ’π‘’π‘’ precisely, βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ Itsis responsible itβˆ’πœπœ/πœ†πœ† significant approaches it𝑑𝑑approaches contribution 0 as asβ†’to 𝜏𝜏0were + β†’theand +used. The data 0yield as 𝜏𝜏isas and β†’in𝜏𝜏+∞. β†’ +∞. Its significa Its signi all yields equally. Thus, 𝛽𝛽0𝑑𝑑 because this factor itsyields isfactor responsible loading, for 𝜏𝜏/πœ†πœ† factor for tenors longer than one year, however,then βˆ’ 𝑒𝑒 ,for starts parallel at 0, increases, weredecays not zeroto 0. More 𝑑𝑑 s, this factor is parallel responsible yieldfor curve parallel shifts. The 𝛽𝛽1𝑑𝑑 isfactor b1ttheisyield related curvetoslope. rates so the corresponding zero rates were obtained the theyield yield medium curve curveterm shifts. slope. The tofactor Ifmaturity. the slope of the yield curve related to the Ifmedium the the medium slope of term the from+Bloomberg (2013). Note that the zero rates can yield term to maturity. curve to maturity. ve slope. If the slope of the as yield limcurve precisely, it approaches 0 as 𝜏𝜏 β†’ 0 and as 𝜏𝜏 β†’ +∞. Its significant contribution to the yield is defined is defined as πœπœβ†’+∞ 𝑦𝑦 (𝜏𝜏) βˆ’ lim 𝑦𝑦 (𝜏𝜏), then it is equal to βˆ’π›½π›½ . Lastly, 𝛽𝛽 is related to the 𝑑𝑑 πœπœβ†’0+ 𝑑𝑑 , then it is equal to 1𝑑𝑑 also2𝑑𝑑be computed using a method called bootstrapping. 𝛽𝛽1𝑑𝑑 . Lastly, 𝛽𝛽2𝑑𝑑 is curvature, related todefined the and the by Diebold medium Li as 2𝑦𝑦term to maturity. 𝑑𝑑 (24) βˆ’ 𝑦𝑦𝑑𝑑 (3) βˆ’ 𝑦𝑦𝑑𝑑 (120), where maturity is given in months. βˆ’ 𝑦𝑦𝑑𝑑 (120), where maturity is given in Estimation of Parameters Nelson Nelson From Equation andand (3), NelsonSiegel Siegel and assumes assumes Siegel that that assumes the the forward that forward the rate rate forward follows follows rate thethe followsequation equation the equatio 1 𝜏𝜏 𝑓𝑓 (𝜏𝜏) βˆ’πœπœ/πœ†πœ† 𝑒𝑒𝑑𝑑 + βˆ’πœπœ/πœ†πœ† 𝜏𝜏 πœπœβˆ’πœπœ/πœ†πœ† 𝜏𝜏 𝑑𝑑 βˆ’πœπœ/ 𝑦𝑦𝑑𝑑 (𝜏𝜏) = ∫ [𝛽𝛽𝑑𝑑0𝑑𝑑𝑓𝑓𝑑𝑑+(𝜏𝜏) 𝑓𝑓= = 𝛽𝛽𝑒𝑒0π‘‘π‘‘βˆ’π‘’π‘’/πœ†πœ† (𝜏𝜏) 𝛽𝛽𝑑𝑑1𝑑𝑑 𝛽𝛽=0𝑑𝑑 +𝛽𝛽𝛽𝛽 + +𝛽𝛽+ 1𝑑𝑑 𝑑𝑑0𝑑𝑑 𝑒𝑒1𝑑𝑑 𝑒𝑒1𝑑𝑑 𝛽𝛽 𝛽𝛽2𝑑𝑑 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝛽𝛽 𝑒𝑒193+2𝑑𝑑 𝑑𝑑𝛽𝛽 𝑑𝑑 βˆ’π‘’π‘’/πœ†πœ† + ] 𝑑𝑑2𝑑𝑑 𝑒𝑒2𝑑𝑑𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝛽𝛽𝑑𝑑𝑑𝑑 𝑒𝑒 𝑑𝑑 Philippine Interest Rates Using the Dynamic Nelson-Siegel Model πœ†πœ† πœ†πœ† πœ†πœ† 𝜏𝜏 0 πœ†πœ†π‘‘π‘‘ 𝑑𝑑 𝑑𝑑 𝑑𝑑 Evaluating where where where𝛽𝛽0𝑑𝑑𝛽𝛽,0𝑑𝑑𝛽𝛽𝛽𝛽 , 1𝑑𝑑𝛽𝛽,1𝑑𝑑 𝛽𝛽𝛽𝛽 ,the ,2𝑑𝑑 , 2𝑑𝑑1𝑑𝑑 𝛽𝛽 , and, and 𝛽𝛽 πœ†πœ†are πœ†πœ†, 𝑑𝑑and 𝑑𝑑 areπœ†πœ†constants 𝑑𝑑 areconstantsconstants andand πœ†πœ†π‘‘π‘‘and πœ†πœ†β‰ π‘‘π‘‘ β‰  0. πœ†πœ†π‘‘π‘‘ 0.Note β‰  Note 0. that Note thatπ‘₯π‘₯(𝜏𝜏) π‘₯π‘₯(𝜏𝜏) that =π‘₯π‘₯( PDEx is one of the two major exchanges in the the integral the other at 0𝑑𝑑 method right, 2𝑑𝑑we that obtain used a thefixed Nelson-Siegel shape parameter model for the yiel Philippines, the other being the Philippine Stock second-order across all second-order linear datasets, linear ordinary which ordinary will differential be referred differential to for the rest second-order linear ordinary 1 βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 equation differential equationequation βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ† To(1 𝑑𝑑 Exchange (PSE). TERMPDEx STRUCTURE is a venue for trading fixed- FORECASTING of𝑦𝑦the (𝜏𝜏) paper= 𝛽𝛽 as+fixed 𝛽𝛽 ( lambda method. ) + 𝛽𝛽 determine 5 βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ ) 𝑑𝑑 0𝑑𝑑 1𝑑𝑑 2𝑑𝑑 income and other securities, most of which are this fixed shape parameter, 𝜏𝜏/πœ†πœ†first, for2each lambda 𝜏𝜏/πœ†πœ† 2 β€² 1β€² 1 1 𝛽𝛽0𝑑𝑑𝛽𝛽0𝑑𝑑𝛽𝛽0𝑑𝑑 in β€²β€² β€²β€² β€² 2 𝑑𝑑 𝑑𝑑 government securities. From daily trading, PDEx the same range used in grid search, π‘₯π‘₯ β€²β€²π‘₯π‘₯+ π‘₯π‘₯+beta π‘₯π‘₯+π‘₯π‘₯+parameters+ π‘₯π‘₯ + π‘₯π‘₯ = π‘₯π‘₯ 2=π‘₯π‘₯2=2 2 (Nelson & Siegel, 1987). To generate this range of shapes, a parsimonious model πœ†πœ†π‘‘π‘‘ πœ†πœ†π‘‘π‘‘πœ†πœ†π‘‘π‘‘ 2πœ†πœ†π‘‘π‘‘ 2πœ†πœ†by πœ†πœ†π‘‘π‘‘ introduced 𝑑𝑑 πœ†πœ†π‘‘π‘‘ πœ†πœ†π‘‘π‘‘ πœ†πœ†are calculates and publishes Philippine Dealing TheSystem parameters 𝛽𝛽0𝑑𝑑 , 𝛽𝛽1𝑑𝑑 , and 𝛽𝛽2𝑑𝑑 are were estimated called factors andand corresponding their coefficients 𝑑𝑑 calle Treasury Reference Rates such as PDS Treasury R were taken for each t. The average over all t was 2 Nelson From From Equation From Equation Equation (3),(3), (3), Reference Rate AM and Siegel assumes (PDST-R1) and PDS that loadings.the forward Treasury The factor rate then follows loading calculated. the of 𝛽𝛽equation 0𝑑𝑑 is 1, a the Finally, constant shapethat does not that parameter decay to zero eve Reference Rate PM (PDST-R2). Both PDST-R1 and produced the highest 𝜏𝜏 βˆ’πœπœ/πœ†πœ† average 1 1was 𝜏𝜏 𝜏𝜏 chosen. Because the 𝑒𝑒 𝑒𝑒 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ 1 𝜏𝜏 (4) 𝑑𝑑 it βˆ’π‘’π‘’/πœ†πœ† 𝑒𝑒 𝑑𝑑 βˆ’π‘’π‘’ PDST-R2 benchmarks are intended to become Thus, 𝑓𝑓𝑑𝑑 (𝜏𝜏) it has =the𝛽𝛽0𝑑𝑑 significant + 𝛽𝛽results1𝑑𝑑 𝑒𝑒 ofcontribution +the𝛽𝛽2𝑑𝑑 fixed 𝑦𝑦𝑑𝑑to(𝜏𝜏) 𝑦𝑦𝑑𝑑𝑒𝑒lambda (𝜏𝜏) = 𝑦𝑦the 𝑑𝑑 = (𝜏𝜏) yield method ∫ = ∫ [𝛽𝛽for [𝛽𝛽 ∫ 0𝑑𝑑 +any were[𝛽𝛽 0𝑑𝑑 𝛽𝛽 + maturity; 1𝑑𝑑more 𝛽𝛽𝑒𝑒1π‘‘π‘‘βˆ’π‘’π‘’/πœ†πœ† + 𝑒𝑒1𝑑𝑑 𝛽𝛽 +𝑑𝑑hence, stable, βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑 βˆ’π‘’π‘’/πœ†πœ† 𝑒𝑒 𝛽𝛽 +2𝑑𝑑𝛽𝛽+ 2𝑑𝑑 𝛽𝛽𝑒𝑒2𝑑𝑑is𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† referr 𝑒𝑒] 𝑑𝑑𝑑𝑑 πœ†πœ†π‘‘π‘‘ 𝑑𝑑 𝜏𝜏 𝜏𝜏 𝜏𝜏 0𝑑𝑑 πœ†πœ† πœ†πœ† πœ†πœ† source of reference rates for the repricing of loans, the succeeding steps proceeded 0 0from 0 these. 𝑑𝑑 𝑑𝑑 𝑑𝑑 securities, derivative transactions, and other long-term interestfactor. The It isfactor important 𝛽𝛽1𝑑𝑑 is called to notethe that short-term the results factor because its factor lo in either where 𝛽𝛽0𝑑𝑑 , 𝛽𝛽1𝑑𝑑 , 𝛽𝛽2𝑑𝑑 , and πœ†πœ†π‘‘π‘‘ are constants Evaluating and Evaluating Evaluating πœ†πœ†π‘‘π‘‘ the β‰ the 0. Note integralintegral the that at the integral atorπ‘₯π‘₯(𝜏𝜏) theright, at = right, right, the 𝑑𝑑 (𝜏𝜏) 𝑓𝑓we we is obtain obtain we a obtain solution thethe toNelson-Siegel Nelson-Siegel the Nelson-Siegel the model mode mo fo rate sensitive instruments to be issued. They are also method, grid search fixed lambda, were dependent βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ intended to become the bases for market1βˆ’π‘’π‘’ valuation , has equation on the choice significant contribution of rangetoofthe values value ofofl𝑦𝑦t.βˆ’πœπœ/πœ†πœ† second-order linear ordinary differential (𝜏𝜏) 𝑑𝑑 at βˆ’πœπœ/πœ†πœ† 𝑑𝑑 shorter 1maturities βˆ’πœπœ/πœ†πœ† 𝑑𝑑 (sm βˆ’πœπœ/πœ†πœ† of Government Securities and other Philippine-peso- 𝜏𝜏/πœ†πœ† 𝑑𝑑 Having 𝑦𝑦chosen the fixed 1βˆ’ shape 1 π‘’π‘’βˆ’1 𝑑𝑑 𝑒𝑒 βˆ’ parameter 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 and βˆ’1 π‘’π‘’βˆ’1 π‘’π‘’βˆ’ 𝑒𝑒 βˆ’πœπœ/ 𝑑𝑑 (𝜏𝜏) 𝑦𝑦 (𝜏𝜏) = 𝑑𝑑 𝑑𝑑 𝑦𝑦𝑑𝑑 0𝑑𝑑= =𝛽𝛽 (𝜏𝜏) 𝛽𝛽 + 𝛽𝛽 + 0𝑑𝑑𝛽𝛽0𝑑𝑑 𝛽𝛽 1𝑑𝑑 + ( ( 1𝑑𝑑 𝛽𝛽1𝑑𝑑 ( ) + ) 𝛽𝛽 + ) 𝛽𝛽 2𝑑𝑑 + ( ( 2𝑑𝑑𝛽𝛽2𝑑𝑑 ( βˆ’ π‘’π‘’βˆ’ denominated fixed income securities. 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† 𝑑𝑑the𝑑𝑑 second 𝜏𝜏/πœ†πœ† 𝑑𝑑 t,𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† 𝑑𝑑 𝜏𝜏/πœ†πœ† 𝜏𝜏). Moreover, β€²β€² 2itthe decreases β€² resulting 1 to 0beta 𝛽𝛽as parameters for each 0𝑑𝑑 𝜏𝜏 β†’ +∞. Lastly, 𝛽𝛽 is referred (5) to as the 𝑑𝑑 𝑑𝑑 medium In this work, TERMthe STRUCTURE PDST-R2 rates from Januaryπ‘₯π‘₯2, + step FORECASTING π‘₯π‘₯ +was2 π‘₯π‘₯to=fit a2 univariate time series model for 2𝑑𝑑 5 The Theπœ†πœ†parameters parameters πœ†πœ†π‘‘π‘‘ { 𝛽𝛽 𝛽𝛽,},πœ†πœ†π›½π›½π‘‘π‘‘π›½π›½ 𝛽𝛽,1𝑑𝑑 factorsfactors 2008 to December 25, 2013 were chosen. The PDST-R2 𝑑𝑑The each parameters of 0𝑑𝑑 0𝑑𝑑 ,{1𝑑𝑑 0𝑑𝑑 ,and , and }𝛽𝛽 𝛽𝛽 , 2𝑑𝑑 1𝑑𝑑and and are 𝛽𝛽2𝑑𝑑{ 𝛽𝛽are called 2𝑑𝑑 }. called are called First, each and factors and series their andtheircoefficients their coefficiecoeffi 1βˆ’π‘’π‘’ βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ rate is the weighted average of the yields because from done its factor was loading, tested for stationarity βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 ,and starts was at 0, found increases, to have then decays to 0 transactions ofFrom (Nelson Equation of& (3), 1987). Siegel, To generate this range of shapes, 𝜏𝜏/πœ†πœ†π‘‘π‘‘ a parsimonious model introduced by the set benchmark securities for each loadings. loadings. loadings. a unit The The root.factor The factor loading factor Hence, loading of of loading each is 𝛽𝛽1, 𝛽𝛽0𝑑𝑑𝛽𝛽of0𝑑𝑑 series is 0𝑑𝑑1, was aisconstant atransformed 1,constant a constant that that does by does that not does notdecay decay not de to tenor up to 4:15 p.m. For simplicity, weeklyprecisely, data were differencing to remove 𝜏𝜏 it approaches 0 as 𝜏𝜏 β†’ 0 and as 𝜏𝜏 β†’ +∞. Its significant contribution to + the unit root. Each series also Nelson and Siegel assumes that1 the forward Thus, Thus, it rate has it has follows itsignificant significant 𝑒𝑒equation thecontribution (6) extracted by taking the rates Wednesday of each 𝑦𝑦𝑑𝑑 (𝜏𝜏) = week. ∫ [𝛽𝛽 Thus, showed 0𝑑𝑑 + 𝛽𝛽1𝑑𝑑 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ† has strong 𝑑𝑑 + significant 𝛽𝛽2𝑑𝑑presence 𝑒𝑒contribution contribution βˆ’π‘’π‘’/πœ†πœ† 𝑑𝑑of ] 𝑑𝑑𝑑𝑑 ARCHto to thethe toyield effect.yield the for yield for The any any for bestmaturity; maturity; any maturity; hence, henc h Index t is attached to each week. 𝜏𝜏 model obtained for πœ†πœ† 𝜏𝜏 each differenced series followed 0 𝑑𝑑 the medium term to maturity. (4)factor 𝑓𝑓𝑑𝑑 (𝜏𝜏) =long-term long-term 𝛽𝛽0𝑑𝑑 𝛽𝛽1𝑑𝑑 factor. +long-term factor. 𝑒𝑒 βˆ’πœπœ/πœ†πœ† ARMA(p,q)+eGARCH(1,1). The 𝑑𝑑factor. + The 𝛽𝛽 2𝑑𝑑factor Thefactor is 𝛽𝛽called 𝛽𝛽1𝑑𝑑𝛽𝛽𝑑𝑑1𝑑𝑑 𝑒𝑒 βˆ’πœπœ/πœ†πœ† factor isThen 1𝑑𝑑called thethe is called short-term error short-term the short-term terms factor of factor because because beca it Evaluating the integral at the right, we obtain the Nelson-Siegel πœ†πœ†π‘‘π‘‘ model for the yield curve Procedure the estimated models for the betas were considered as βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† In using the Nelson-Siegel model to forecast the 1βˆ’π‘’π‘’ 1βˆ’π‘’π‘’ a𝑑𝑑 ,random 1βˆ’π‘’π‘’ 𝑑𝑑 has , has 𝑑𝑑 significant vector whose significant contribution joint probability contribution distribution where 𝛽𝛽0𝑑𝑑 , 𝛽𝛽1𝑑𝑑 , 𝛽𝛽2𝑑𝑑 , and πœ†πœ†π‘‘π‘‘ are constants 1 βˆ’ 𝜏𝜏/πœ†πœ†π‘’π‘’ and βˆ’πœπœ/πœ†πœ† 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† π‘‘π‘‘πœ†πœ†π‘‘π‘‘ ,β‰  has 0. significant Note 1was βˆ’that 𝑒𝑒 βˆ’πœπœ/πœ†πœ† contribution π‘₯π‘₯(𝜏𝜏) 𝑑𝑑 = 𝑓𝑓to𝑑𝑑 (𝜏𝜏) to thethe to isvalue value athe solutionof of value 𝑦𝑦𝑑𝑑 (𝜏𝜏) 𝑦𝑦of to (7) at𝑑𝑑 (𝜏𝜏) 𝑑𝑑 (𝜏𝜏) 𝑦𝑦 the shorter at shorter at shorte matu m term structure, three major steps were implemented 𝑦𝑦𝑑𝑑 (𝜏𝜏) = 𝛽𝛽0𝑑𝑑 + 𝛽𝛽1𝑑𝑑 ( 𝑑𝑑 function 𝑑𝑑 𝑑𝑑 ) + 𝛽𝛽2𝑑𝑑 ( (pdf) to be determined βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ† 𝑑𝑑 ) by the copula in this work. The first step was to run multiple linear 𝜏𝜏/πœ†πœ†method. 𝑑𝑑 Finally, random 𝜏𝜏/πœ†πœ†π‘‘π‘‘ numbers from the joint pdf second-order linear ordinary differential 𝜏𝜏).𝜏𝜏). equation Moreover, Moreover, it decreases it decreases to to 0 as 0toas 𝜏𝜏0β†’ 𝜏𝜏asβ†’ +∞. 𝜏𝜏 +∞. Lastly,Lastly, 𝛽𝛽2𝑑𝑑𝛽𝛽2𝑑𝑑is 𝛽𝛽referred is2𝑑𝑑referred to to as as th regression in the form of Equation (8) for each dataset of Moreover, 𝜏𝜏). the error terms it decreases were generated. β†’ These +∞. Lastly, were then is referred to t to obtain the The beta parameters 𝛽𝛽0𝑑𝑑 , 𝛽𝛽1𝑑𝑑 , and 𝛽𝛽2𝑑𝑑 .are This called 2used factors in and the final 1 their 𝛽𝛽0𝑑𝑑step, coefficients forecasting βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† are called 𝑑𝑑 βˆ’πœπœ/πœ†πœ† the term factor structure (5) β€²β€² β€² 1βˆ’π‘’π‘’1βˆ’π‘’π‘’ 1βˆ’π‘’π‘’ 𝑑𝑑 𝑑𝑑 βˆ’πœπœ/πœ†πœ† βˆ’πœπœ/πœ†πœ† step required a prior choice of value or values of the π‘₯π‘₯because because ofπ‘₯π‘₯its + because interestits + factor factor π‘₯π‘₯loading, its2 rates. factor=loading, loading, 2 βˆ’ 𝑒𝑒 βˆ’ π‘’π‘’βˆ’ 𝑑𝑑 , βˆ’πœπœ/πœ†πœ† 𝑒𝑒 starts 𝑑𝑑 , starts at 0, 𝑑𝑑 , starts at 0, increases, atincreases, 0, increases, then then dt πœ†πœ†π‘‘π‘‘ πœ†πœ†that 𝑑𝑑 doesπœ†πœ†π‘‘π‘‘not decay to zero 𝜏𝜏/πœ†πœ† 𝜏𝜏/πœ†πœ† 𝑑𝑑 π‘‘π‘‘πœπœ/πœ†πœ†π‘‘π‘‘ shape parameter loadings.lt. TheThe nextfactor steploadingwas to of 𝛽𝛽0𝑑𝑑 is a1, a constant assume even as 𝜏𝜏 β†’ +∞. model for the time series of beta parameters obtained. precisely, precisely, precisely, it approaches it approaches it approaches 0 as 0 as 𝜏𝜏0β†’ 𝜏𝜏asβ†’0𝜏𝜏+0β†’ + + and 0and asand as 𝜏𝜏 β†’ 𝜏𝜏asβ†’ +∞. β†’ Its 𝜏𝜏 +∞. +∞. Its significant significant Its significant contrco From Thus, itEquation has to (3), significant contribution to the yield for any maturity; hence, it is referred to as the This model was then used forecast future values Implementation and Results of these parameters. The last step was to plug in the 1 𝜏𝜏thethe medium medium the medium term term to term to maturity. maturity. to𝑒𝑒maturity. (6) future valueslong-term of the beta factor. The factor parameters 𝑦𝑦and 𝑑𝑑 𝛽𝛽 (𝜏𝜏) the = 1𝑑𝑑 is called same ∫ [𝛽𝛽 the 0𝑑𝑑 + short-term Fitting𝛽𝛽 1𝑑𝑑 𝑒𝑒 the βˆ’π‘’π‘’/πœ†πœ† factor 𝑑𝑑Yield + 𝛽𝛽 2𝑑𝑑 because Curve 𝑒𝑒 βˆ’π‘’π‘’/πœ†πœ†π‘‘π‘‘its ] 𝑑𝑑𝑑𝑑 factor loading, shape parameter in Equation (7) to obtain forecast 𝜏𝜏 0of A total of 313 datasets πœ†πœ†π‘‘π‘‘ of zero rates from January 2, βˆ’πœπœ/πœ†πœ† interest rates y1βˆ’π‘’π‘’ (t). 𝑑𝑑 , has significant contribution to the 2008 of value to 𝑦𝑦 December 25, 2013 were considered. Each t 𝑑𝑑 (𝜏𝜏) at shorter maturities (smaller values of For the firstEvaluating step, 𝜏𝜏/πœ†πœ†π‘‘π‘‘ the integral regression usingata the fixed right, shape we obtain dataset the Nelson-Siegelgave the interest model for thein rates yield hundred curve basis parameter across all datasets and using a grid search points for maturities 1 month (Mo), 3 Mo, 6 Mo, 1 𝜏𝜏). Moreover, were both considered it decreases initially. For𝑦𝑦 the grid to search, 0 as 𝜏𝜏 β†’ the 1 βˆ’ Lastly, +∞. βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ 𝛽𝛽 is referred βˆ’πœπœ/πœ†πœ† 𝑒𝑒year 13Yr,βˆ’ 𝑒𝑒 4Yr, to as𝑑𝑑 the βˆ’πœπœ/πœ†πœ† medium-term factor(7) (Yr), ) + 𝛽𝛽2Yr, 2𝑑𝑑 5Yr, βˆ’ 𝑒𝑒 6Yr, 𝑑𝑑 ) 7Yr, 8Yr, 9Yr, 𝑑𝑑 (𝜏𝜏) = 𝛽𝛽0𝑑𝑑 + 𝛽𝛽1𝑑𝑑 ( 2𝑑𝑑 ( values of lt ranged from 2.000000 to 40.000000 with 𝜏𝜏/πœ†πœ† 10Yr, 𝑑𝑑 𝜏𝜏/πœ†πœ† 15Yr, 20Yr, and𝑑𝑑30Yr. Figure 1 shows sample 1βˆ’π‘’π‘’ βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ increments ofbecause0.000001 for each its factor loading,t. The lt together βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ , starts yield at 0, increases, then decays to 0. More curves. 𝜏𝜏/πœ†πœ†π‘‘π‘‘ with the set of betaThe parameters parameters 𝛽𝛽0𝑑𝑑 , 𝛽𝛽1𝑑𝑑 , and 𝛽𝛽 2𝑑𝑑 that are called factors Before and their coefficients performing linear regression, are called allfactorrates were highest R2itfor produced the precisely, each t were approaches 0 as 𝜏𝜏 β†’ 0+ and taken. It was as 𝜏𝜏first β†’ converted +∞. Its significant to continuously contribution compounding to the yield rates. is For in noted that quiteloadings. a number The factor loading of datasets had veryofhigh 𝛽𝛽0𝑑𝑑 Ris2 1, a constant the processing that does of data, not decay from estimation to zero even to as time 𝜏𝜏 β†’ series+∞. but the beta parameters the mediumobtained were not realistic, term to maturity. modelling of the beta parameters, only the rates from that is, the values Thus, were it has verysignificant far from the contribution trend in the to the Januaryyield for 2, any 2008 maturity; to December hence, it26, is referred 2012 were to asused. the majority of datasets. This problem did not appear in Thus, 261 datasets were considered. The remaining long-term factor. The factor 𝛽𝛽1𝑑𝑑 is called the short-term factor because its factor loading, 1βˆ’π‘’π‘’ βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ 𝜏𝜏/πœ†πœ†π‘‘π‘‘ , has significant contribution to the value of 𝑦𝑦𝑑𝑑 (𝜏𝜏) at shorter maturities (smaller values of justified justifiedbybyKPSS KPSStesttest(for (forthe theactual actualseries) series)and andARCH-LM ARCH-LMtest test(for (forthe thedifferenced difference All statistical procedures were implemented using the software R. First, univariate time giving all significant p-values. These are s giving giving all allsignificant significant p-values. p-values. These These 2 andare arestandard standard ofprocedures procedures inindetecting detectingthese thesefeat fea RM STRUCTUREseries models were fitted on the FORECASTING estimated betas. As seen in Figures 3, presence non- financialE.P. timede series. Lara-Tuprio, et al 10 194 stationarity and of conditionalfinancial heteroscedasticity financial time were apparent. These observations were timeseries. series. justified by KPSS test (for the actual series) and ARCH-LM test (for the differenced series), giving all significant p-values. These are standard procedures in detecting these features of most financial time series. Figure 2. Time serie Figure Figure2.2.Time Timeseries seriesplots plotsofofthe theestimated estimatedbetas. betas. Figure 2. Time series plots of the estimated betas. Figure 1.Figure Actual1. Actual yield curves for selected dates. yield curves for selected dates. Table 1. Minimum, Maximum, Mean Values, and Standard Deviations of R2 and Beta Parameters Min max mean standard deviation Before performing linear R regression, 80.26% all rates 99.72% were first 3.04% 96.22% converted to continuously 2 Figure 3. Time serie Ξ²0 0.02242 1.49527 0.17384 0.18941 Figure Figure3.3.Time Timeseries seriesplots plotsofofthe thedifferenced differencedbetas betas. . mpounding rates. For theΞ²processing -1.41810 of 0.01044 data, from estimation0.17833 -0.13947 to time series modelling of the 1 Figure 3. Time series plots of the differenced betas. Ξ²2 -1.80766 0.24361 -0.04411 0.25549 a parameters, only the rates from January 2, 2008 to December 26,The 2012 best were models used. Thus, for all three 2 differ TheThebest bestmodels modelsfor forall The allbest three three models differenced differenced for all three series seriesdifferenced 𝑑𝑑𝛽𝛽 𝑑𝑑𝛽𝛽𝑖𝑖 ,𝑖𝑖 ,𝑖𝑖 𝑖𝑖==0,1,2, serieswere 0,1,2, wereARMA ARMA historical data were to be used to check the reliability The best models for all three differenced series 𝑑𝑑𝛽𝛽𝑖𝑖 , 𝑖𝑖 = 0,1,2,, were wereARMA equation ARMA forfor and the the mean Exponential mean equation GARCH for the of the forecast. asets were considered. Theequation fixed and lambda The method remaining equation equationand described historical andExponential Exponential anddata GARCH GARCH Exponentialwerefor forto the GARCH the be variance used variance for the tovariance equation. equation. check In the reliability Inparticular, particular, equation. based basedon on Exponential GARCH for theearliervariance equation. In particular, In particular, based on the Bayesian Information basedBayesian on the Information Criterion (BIC), bot produced lt = 9.807527. To summarize the results of 2 Bayesian Bayesian Information Information Criterion Criterion Criterion TERM STRUCTURE (BIC), (BIC),FORECASTING (BIC), both bothboth{𝑑𝑑𝛽𝛽 {𝑑𝑑𝛽𝛽00}}and and{𝑑𝑑𝛽𝛽 and {𝑑𝑑𝛽𝛽11}}followed followed followed MA(1)+eGA MA(1)+eGA of R and(BIC), forecast. linear regression, of Equation (8)TERM the valuesCriterion Bayesian Information for the different datasets STRUCTURE TERMare FORECASTING beta parameters both {𝑑𝑑𝛽𝛽 STRUCTURE TERM0 } and describedFORECASTING {𝑑𝑑𝛽𝛽 STRUCTURE 1 } followed MA(1)+eGARCH(1,1) FORECASTING MA(1)+eGARCH(1,1) models, models, while while {𝑑𝑑𝛽𝛽 2 } followed followed 12 ARMA(1,1 in Table 1. models, while {𝑑𝑑𝛽𝛽2 } followedmodels, models, while while{𝑑𝑑𝛽𝛽 ARMA(1,1)+eGARCH(1,1). {𝑑𝑑𝛽𝛽22}}followed ARMA(1,1)+eGARCH(1,1). followed ARMA(1,1)+eGARCH(1,1). ARMA(1,1)+eGARCH(1,1). TERM STRUCTURE FORECASTING The Theestimated Theestimated estimated models models weremodels given were were by the given given by by following the the following 12 equations. equatio Let 𝑑𝑑𝛽𝛽 Forecasting of interest rates requires forecast of The fixed lambda method The estimated described the beta parameters. Thus, the next step was to find models The were in estimated givenSection models by the following III were following produced given equations. by the following equations. πœ†πœ† Let Let 𝑑𝑑 = 𝑑𝑑𝛽𝛽 9.807527. equations. 𝑖𝑖,𝑑𝑑 = 𝛽𝛽 𝑖𝑖,𝑑𝑑 βˆ’ Let 𝛽𝛽 𝑑𝑑𝛽𝛽 𝑖𝑖,π‘‘π‘‘βˆ’1 To , 𝑖𝑖,𝑑𝑑 = 𝛽𝛽 𝑖𝑖,𝑑𝑑 βˆ’ 𝛽𝛽𝑖𝑖,π‘‘π‘‘βˆ’ appropriate time series models for them. models were 𝑖𝑖 = 0,1,2, 𝑖𝑖 =𝑑𝑑 = 0,1,2,1, 2,𝑑𝑑…=, 𝑇𝑇, 1, where 2, … , 𝑇𝑇, T is where wherethe sample TT is thesize. sample Thensize. size. Then The estimated 𝑖𝑖 = 0,1,2, 𝑖𝑖 = 0,1,2, 𝑑𝑑 = 1, 2, … , 𝑇𝑇, where T is the sample 𝑑𝑑 = 1, 2, given … , 𝑇𝑇, by where Thensize. Thenthe Tfollowing is the equations. sample size. Let Then 𝑑𝑑𝛽𝛽 𝑖𝑖,𝑑𝑑 = 𝛽𝛽 𝑖𝑖,𝑑𝑑 βˆ’ 𝛽𝛽 𝑖𝑖,π‘‘π‘‘βˆ’1 , 2 mmarize the results of linear Time Series Models for0,1,2, the Beta regression, Parameters the values of 𝑅𝑅 and𝑑𝑑𝛽𝛽beta 𝑖𝑖,𝑑𝑑 = πœ‡πœ‡π‘‘π‘‘π›½π›½ parameters 𝑖𝑖 +𝑖𝑖,π‘‘π‘‘πœ‘πœ‘= 𝑖𝑖 π‘‘π‘‘π›½π›½πœ‡πœ‡π‘–π‘–,π‘‘π‘‘βˆ’1 𝑖𝑖 + πœ‘πœ‘ +𝑖𝑖 𝑑𝑑𝛽𝛽 of π‘Žπ‘Žπ‘–π‘–,𝑑𝑑 +Equation 𝑖𝑖,π‘‘π‘‘βˆ’1 πœƒπœƒ+ π‘Žπ‘Žπ‘–π‘–,𝑑𝑑 + πœƒπœƒπ‘–π‘– π‘Žπ‘Žπ‘–π‘–,𝑑𝑑 𝑖𝑖 π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 (8 𝑖𝑖 = 𝑑𝑑 = 1, 2, … , 𝑇𝑇, where T is the sample 𝑑𝑑𝛽𝛽 size. = Then πœ‡πœ‡ + πœ‘πœ‘ 𝑑𝑑𝛽𝛽 + π‘Žπ‘Ž + πœƒπœƒ π‘Žπ‘Ž (9) (9) All statistical procedures were implemented 𝑑𝑑𝛽𝛽𝑖𝑖,𝑑𝑑 using = πœ‡πœ‡π‘–π‘– + πœ‘πœ‘π‘–π‘– 𝑑𝑑𝛽𝛽𝑖𝑖,π‘‘π‘‘βˆ’1 + π‘Žπ‘Žπ‘–π‘–,𝑑𝑑 + πœƒπœƒπ‘–π‘– π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 𝑖𝑖,𝑑𝑑 𝑖𝑖 𝑖𝑖 𝑖𝑖,π‘‘π‘‘βˆ’1 𝑖𝑖,𝑑𝑑 𝑖𝑖 𝑖𝑖,π‘‘π‘‘βˆ’1 π‘Žπ‘Žπ‘–π‘–,𝑑𝑑 = πœŽπœŽπ‘–π‘–,𝑑𝑑 πœ€πœ€π‘–π‘–,𝑑𝑑 π‘Žπ‘Ž = 𝜎𝜎 πœ€πœ€ 𝑖𝑖,𝑑𝑑 𝑖𝑖,𝑑𝑑 π‘Žπ‘Ž = 𝜎𝜎 πœ€πœ€ π‘Žπ‘Žπ‘–π‘–,𝑑𝑑 = πœŽπœŽπ‘–π‘–,𝑑𝑑 πœ€πœ€π‘–π‘–,𝑑𝑑 (10) (10) the software R. First, univariate time series models the different datasets are described in Table 𝑑𝑑𝛽𝛽𝑖𝑖,𝑑𝑑 = 1. πœ‡πœ‡π‘–π‘– +𝑖𝑖,π‘‘π‘‘πœ‘πœ‘π‘–π‘– 𝑑𝑑𝛽𝛽𝑖𝑖,π‘‘π‘‘βˆ’1 𝑖𝑖,𝑑𝑑 𝑖𝑖,𝑑𝑑+ π‘Žπ‘Ž + πœƒπœƒ π‘Žπ‘Ž ln(𝜎𝜎 2 𝑖𝑖,𝑑𝑑 2 𝑖𝑖 𝑖𝑖,π‘‘π‘‘βˆ’1 ln(𝜎𝜎 πœ”πœ”π‘–π‘–π‘–π‘–,𝑑𝑑+)|π‘Žπ‘Ž 2 𝛼𝛼=𝑖𝑖𝑖𝑖,π‘‘π‘‘βˆ’1 π‘Žπ‘Žπœ”πœ” 𝑖𝑖 |++ 𝛼𝛼 2 |π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 | π‘–π‘–π›Ύπ›Ύπ‘Žπ‘Žπ‘–π‘– 𝑖𝑖,π‘‘π‘‘βˆ’1 (π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 + 𝛾𝛾𝑖𝑖 βˆ’ |π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 ( 𝐸𝐸 (| π‘Žπ‘Ž(9) (11) | 𝑖𝑖,π‘‘π‘‘βˆ’1 βˆ’2|)) π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 𝐸𝐸 (|+ πœ…πœ…π‘–π‘– ln | were fitted on the estimated betas. As seen in Figures2 π‘Žπ‘Ž |π‘Žπ‘Ž = 𝜎𝜎| 2 πœ€πœ€)𝑖𝑖,𝑑𝑑= 𝑖𝑖,𝑑𝑑 π‘Žπ‘Ž 𝑖𝑖,π‘‘π‘‘βˆ’1 (11) (10) 2 ln(𝜎𝜎 2 ) = πœ”πœ” + 𝑖𝑖,𝑑𝑑 𝛼𝛼𝑖𝑖 π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 + 𝛾𝛾𝑖𝑖𝑖𝑖,π‘‘π‘‘βˆ’1 𝑖𝑖,π‘‘π‘‘βˆ’1 𝑖𝑖,𝑑𝑑 ( βˆ’ 𝐸𝐸 (| 𝜎𝜎2𝑖𝑖,π‘‘π‘‘βˆ’1 |)) +πœŽπœŽπœ…πœ…π‘–π‘–,π‘‘π‘‘βˆ’1 πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1𝑖𝑖,𝑑𝑑 ) πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 𝑖𝑖 ln(𝜎𝜎 2 and 3, presence of non-stationarity ln(πœŽπœŽπ‘–π‘–,𝑑𝑑 )and= πœ”πœ”of𝑖𝑖 conditional + 𝛼𝛼𝑖𝑖 π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 + 𝛾𝛾𝑖𝑖 ( 𝑖𝑖 𝑖𝑖,𝑑𝑑 βˆ’ 𝐸𝐸 (| |)) πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 + πœ…πœ…π‘–π‘– ln(𝜎𝜎 πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 𝑖𝑖,𝑑𝑑 ) (11) πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 |π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 |π‘Žπ‘Ž πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 Tablewere1 justified by KPSS test (for ln(𝜎𝜎 heteroscedasticity were apparent. These2 observations 𝑖𝑖,𝑑𝑑 ) = πœ”πœ”π‘–π‘– + π‘Žπ‘Ž 𝛼𝛼𝑖𝑖 π‘Žπ‘Ž 2 𝑖𝑖,π‘‘π‘‘βˆ’1 + where 𝐸𝐸where π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 𝛾𝛾(|𝑖𝑖 ( 𝐸𝐸|)(|= βˆ’ 𝐸𝐸|) 𝐸𝐸(|πœ€πœ€ 𝑖𝑖,π‘‘π‘‘βˆ’1 (| 𝑖𝑖,𝑑𝑑=|) =0 𝜈𝜈 𝐸𝐸(|πœ€πœ€ |)) when |)+= πœ…πœ…βˆš 𝑖𝑖 0ln(𝜎𝜎 𝜈𝜈 when 2 πœ€πœ€π‘–π‘–,𝑑𝑑 )√2 . The 𝑖𝑖,𝑑𝑑 ~𝑑𝑑 𝜈𝜈 coefficients πœ€πœ€π‘–π‘–,𝑑𝑑 ~𝑑𝑑 2 . The coeffi and π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 thewhere actual𝐸𝐸series) (|πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1and |) = 𝐸𝐸(|πœ€πœ€ |)𝜎𝜎= πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 𝜈𝜈 𝑖𝑖,𝑑𝑑 𝑖𝑖,π‘‘π‘‘βˆ’10 whenπœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 √ πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 πœ€πœ€ 𝑖𝑖,𝑑𝑑 𝑖𝑖,𝑑𝑑 ~𝑑𝑑 2 . The πœˆπœˆβˆ’2 coefficients πœˆπœˆβˆ’2 and corresponding Minimum, ARCH-LM Maximum, where test 𝐸𝐸 (| (for the 𝜎𝜎Mean differenced |) = 𝐸𝐸(|πœ€πœ€ 𝑖𝑖,π‘‘π‘‘βˆ’1π‘Žπ‘Ž Values, series), 𝑖𝑖,𝑑𝑑 |)giving 0and =𝑖𝑖,π‘‘π‘‘βˆ’1 all Standard when βˆšπœˆπœˆβˆ’2 πœ€πœ€π‘–π‘–,𝑑𝑑 ~𝑑𝑑2 . Deviations 𝜈𝜈 The coefficients πœˆπœˆβˆ’2 ofandR2 correspondingand Beta Paramete significant p-values.where These𝐸𝐸are 𝑖𝑖,π‘‘π‘‘βˆ’1 (| standard |) = 𝐸𝐸(|πœ€πœ€ procedures |) = standard 0inwhenstandard √errorsπœ€πœ€π‘–π‘–,𝑑𝑑 are ~𝑑𝑑given errors 2 . The arein coefficients Table 2. given in Table and 2. corresponding πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 standard 𝑖𝑖,𝑑𝑑 errors are given in Table 2. detecting thesestandard Min features of most financial errors are given in Table 2.max time series. mean πœˆπœˆβˆ’2 standard deviation Table 2 Table 2 2 standard errors are givenTable in Table 2 96.22% 2. R 80.26% Table 2 99.72% Estimated Parameters 3.04% of ARMA +ofeGARCH Estimated Parameters ofEstimated ARMA + eGARCH Parameters Models and ARMA +Models Corresponding eGARCH andModels Corresponding Standardand Errors Corres Sta Table 2Parameters of ARMA + eGARCH Models and Corresponding Standard Errors Ξ²0 0.02242Estimated 1.49527 0.17384 i=0 0.18941 i=1 i=0 i=1 i=2 i=2 Estimated Parameters of ARMA + eGARCH i=0 Models and i=1Corresponding i=2 Standard Errors galalltime series. p-values. These are standard procedures in detecting these features of m significant ial time series. Philippine Interest Rates Using the Dynamic Nelson-Siegel Model 195 Figure 2. Time series plots of the estimated betas. Figure 2. Time series plots of the estimated betas. Figure 2. Time series plots of the estimated betas. TERM STRUCTURE FORECASTING i=2 0.00144197 0.02802909 TERM STRUCTURE TERM FORECASTING STRUCTURE FORECASTING 1212 The assumed distribution of the error πœ€πœ€π‘–π‘–,𝑑𝑑 was the t distribution with 2 degrees of The Theestimated models estimated were models given were byby given thethe following equations. following Let equations. 𝑑𝑑𝛽𝛽𝑑𝑑𝛽𝛽 Let == 𝑖𝑖,𝑑𝑑 𝑖𝑖,𝑑𝑑 𝛽𝛽𝑖𝑖,𝑑𝑑𝛽𝛽𝑖𝑖,𝑑𝑑 βˆ’βˆ’ 𝛽𝛽𝑖𝑖,π‘‘π‘‘βˆ’1 , , 𝛽𝛽𝑖𝑖,π‘‘π‘‘βˆ’1 or 𝑑𝑑2 . Such distribution captures the heavy-tailedness of the series since it has infinite va 𝑖𝑖 = 0,1,2, 𝑖𝑖 = 𝑑𝑑 = 0,1,2, 1, 1, 𝑑𝑑 = 2, 2, …… where , 𝑇𝑇,, 𝑇𝑇, TT where is is thethe sample size. sample Then size. Then In generating future values of the beta parameters using the formulas in Equation 𝑑𝑑𝛽𝛽𝑑𝑑𝛽𝛽 == 𝑖𝑖,𝑑𝑑 𝑖𝑖,𝑑𝑑 πœ‡πœ‡π‘–π‘–πœ‡πœ‡+𝑖𝑖 +πœ‘πœ‘π‘–π‘–πœ‘πœ‘π‘‘π‘‘π›½π›½ 𝑖𝑖 𝑑𝑑𝛽𝛽 𝑖𝑖,π‘‘π‘‘βˆ’1 𝑖𝑖,π‘‘π‘‘βˆ’1 ++ π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘π‘Žπ‘Žπ‘–π‘–,𝑑𝑑 ++ πœƒπœƒπ‘–π‘– πœƒπœƒ π‘Žπ‘Žπ‘–π‘–π‘–π‘–,π‘‘π‘‘βˆ’1 π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 (9)(9) (11), their underlying π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘π‘Žπ‘Ž = = πœŽπœŽπ‘–π‘–,π‘‘π‘‘πœŽπœŽπœ€πœ€π‘–π‘–,π‘‘π‘‘πœ€πœ€dependence structure based on their (10) observed values were consider (10) Figure 3. Time𝑖𝑖,𝑑𝑑series𝑖𝑖,𝑑𝑑plots 𝑖𝑖,𝑑𝑑 of the differenced betas. (11) |π‘Žπ‘Ž|π‘Žπ‘Ž | | π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 (11) 𝑖𝑖,π‘‘π‘‘βˆ’1 𝑖𝑖,π‘‘π‘‘βˆ’1 π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 Figure 3. Time series plots of the differenced betas 2 2 2 2 2 2 ln(𝜎𝜎 ln(𝜎𝜎 ) 𝑖𝑖,𝑑𝑑 𝑖𝑖,𝑑𝑑=) πœ”πœ” = πœ”πœ” 𝑖𝑖 +𝑖𝑖 required 𝛼𝛼 + 𝑖𝑖 π‘Žπ‘Ž 𝛼𝛼 π‘Žπ‘Ž 𝑖𝑖𝑖𝑖,π‘‘π‘‘βˆ’1 + 𝑖𝑖,π‘‘π‘‘βˆ’1 estimating +𝛾𝛾𝑖𝑖 ( 𝛾𝛾 ( 𝑖𝑖 𝜎𝜎 the βˆ’ βˆ’πΈπΈdistribution (| 𝐸𝐸 (| πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 |)) |))of + +the πœ…πœ…π‘–π‘– πœ…πœ…π‘–π‘– random ln(𝜎𝜎ln(𝜎𝜎 ) 𝑖𝑖,𝑑𝑑 𝑖𝑖,𝑑𝑑 ) vector (πœ€πœ€.0 , πœ€πœ€1 , πœ€πœ€2 ) consisting of the e πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 𝑖𝑖,π‘‘π‘‘βˆ’1 πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 where π‘Žπ‘Ž π‘Žπ‘Žπ‘–π‘–,π‘‘π‘‘βˆ’1 variables. 0 when As𝜈𝜈assumed 𝜈𝜈 in thein time the series time series models, models, the thedistribution ofeach distribution of eacherror error variable w where (|(|𝑖𝑖,π‘‘π‘‘βˆ’1 𝐸𝐸 𝐸𝐸 |)|)== 𝐸𝐸(|πœ€πœ€ 𝑖𝑖,𝑑𝑑 |) 𝐸𝐸(|πœ€πœ€ == 𝑖𝑖,𝑑𝑑 |) 0 when √ βˆšπœˆπœˆβˆ’2 πœ€πœ€π‘–π‘–,π‘‘π‘‘πœ€πœ€~𝑑𝑑 2 . 2The 𝑖𝑖,𝑑𝑑 ~𝑑𝑑 . The coefficients coefficients andandcorresponding corresponding where when Figure 3. Time series plots of the differenced betas πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 πœŽπœŽπ‘–π‘–,π‘‘π‘‘βˆ’1 πœˆπœˆβˆ’2 variable was t2. The dependence structure . of (e0 , e1 , The coefficients and corresponding standard errors The dependence structure of (πœ€πœ€ , πœ€πœ€ , πœ€πœ€ ) was obtained using a copula function. e2 ) was 0 obtained 1 2 using a copula function. standard are givenerrors standard inerrors Table areare given 2. giveninin Table Table2. 2. With T as the sample size, the Copula estimatedfor initial The best models for all three differenced series 𝑑𝑑𝛽𝛽 , 𝑖𝑖 = 0,1,2, were ARMA for the me values to Table Table be used 2 2 in forecasting Using are given in Table 3. the Joint Density Using 𝑖𝑖 of the Copula Error for the JointVariables Density of the Error Variables TheEstimated assumed distribution Parameters ofof of the ARMA error ei,t was + eGARCH Models the and Corresponding Standard Estimated Parameters ARMAFirst, pseudo-observations + eGARCH Models and defined First, Corresponding by Errors pseudo-observations Standard Errors defined by t distribution with 2 degrees of freedom or t2. Such onThe andbest Exponential GARCH models for fordifferenced all three the variance distribution captures the i=0 equation. series 𝑑𝑑𝛽𝛽 , 𝑖𝑖 In = particular, 0,1,2, werebased heavy-tailedness i=0 ARMA on the i=1for the me ofi=1the series i=2i=2 𝑖𝑖 𝑅𝑅𝑖𝑖 since it has ΞΌi ΞΌinfinite variance. -0.000385 -0.000385 0.000386 0.000386 -0.000247 -0.000247 π‘ˆπ‘ˆπ‘–π‘– = i 𝑇𝑇 + 1 (12) In generating future(0.000480) values (0.000480) of the beta(0.000426) parameters (0.000426) (0.000409) (0.000409) an on Information Criterion and Exponential GARCH(BIC), forboth using theΟ†formulas {𝑑𝑑𝛽𝛽 } and the variance i Ο†i {𝑑𝑑𝛽𝛽 } followed equation. MA(1)+eGARCH(1,1) In particular, in Equations based on the 0 0 (9) to (11), their 0 00 -0.152738 1 -0.152738 underlying dependence structure were created, based whereon their T is the sample were (0.043782) size and 𝑅𝑅 is the rank of the 𝑖𝑖 π‘‘π‘‘β„Ž observation. These created, where T𝑖𝑖 is the sample size and Ri is the (0.043782) observedΞΈivalues were-0.215126 considered. This required rank of the ith observation. These pseudo-observations s,ian while {𝑑𝑑𝛽𝛽 } followed Information estimating 2 ARMA(1,1)+eGARCH(1,1). Criterion ΞΈi the distribution -0.215126 (0.053564) (0.053564) -0.199544 -0.199544 (BIC), both {𝑑𝑑𝛽𝛽 } and {𝑑𝑑𝛽𝛽 } followed MA(1)+eGARCH(1,1) ofobservations were used the random(0.050340) vector (0.050340) -0.099143 -0.099143 (e0 ,0 to (0.039429) check were the association used (0.039429) to1 check theamong the error association among variables the errorand in estimat e1 , e2 ) consisting of the error variables. As assumed variables and in estimating the parameters of the Ο‰iΟ‰ -0.095836 -0.095836 -0.105876 -0.105876 -0.079393 -0.079393 i (0.036828) parameters(0.049583) of the copula model. The pairwise scatterplots of the pseudo-observations are (0.051335) (0.036828) (0.049583) (0.051335) s, while {𝑑𝑑𝛽𝛽2 } Ξ±followed Ξ± ARMA(1,1)+eGARCH(1,1). 0.390405 0.390405 -0.276128 -0.276128 -0.316096 -0.316096 i i (0.207581) in Figure 4.(0.188705) (0.206423) (0.207581) (0.188705) (0.206423) Ξ³i Ξ³i 0.537282 0.537282 0.626099 0.626099 0.558905 0.558905 196 E.P. de Lara-Tuprio, et al Table 2. Estimated Parameters of ARMA + eGARCH Models and Corresponding Standard Errors i=0 i=1 i=2 -0.000385 0.000386 -0.000247 ΞΌi (0.000480) (0.000426) (0.000409) 0 0 -0.152738 Ο†iΒ  Β  Β  (0.043782) -0.215126 -0.199544 -0.099143 ΞΈiΒ  (0.053564) (0.050340) (0.039429) -0.095836 -0.105876 -0.079393 Ο‰iΒ  (0.036828) (0.049583) (0.051335) 0.390405 -0.276128 -0.316096 Ξ±iΒ  (0.207581) (0.188705) (0.206423) 0.537282 0.626099 0.558905 Ξ³i (0.277760) (0.343980) (0.360170) 0.984873 0.983536 0.983487 ΞΊiΒ  (0.006315) (0.008814) (0.008566) Table 3. Estimated Initial Values Based on the Fitted ARMA + eGARCH Models Β  ai,T Οƒi,T i=0 -0.00492075 0.01729929 i=1 0.00121345 0.01571196 i=2 0.00144197 0.02802909 copula model. The pairwise scatterplots of the pseudo- was based on maximum pseudo-likelihood estimation observations are shown in Figure 4. proposed by Genest, Ghoudi, and Rivest (1995) and Based on the scatterplots of the pseudo- the goodness of fit test was based on the empirical observations, the first error variable had strong copula proposed by Genest, RΓ©millard, and Beaudoin negative association to the second and third error (2009). The results are shown in Table 4. Note that ri,j variables, while the second and the third had strong is the Pearson’s correlation coefficient of the ith and jth pseudo-observations. positive correlation. Notice that the associations among the error variables showed a symmetric- Table 4. Results of the Goodness-of-Fit Test Under type dependence. That is, the dependence on the the Gaussian Copula Assumption tails was almost the same as the distributions on the middle part of the distributions. One model that Estimated parameter Sn p-value captures such symmetry in association between two uniform random variables, as depicted by the ρ0,1= -0.981 0.0047 0.9296 scatterplots, is the elliptical family. This family ρ0,2= -0.959 Β  Β  contains the Gaussian and t copulas. ρ1,2= 0.909 Β  Β  Estimation and goodness of fit test were performed for the Gaussian copulas. The estimation procedure s of the copula model. The pairwise scatterplots of the pseudo-observations positive correlation. Notice that the associations among the error variables showed a tric-type dependence. That is, the dependence on the tails was almost the same as the 4.tions on the middle Philippine Interest Rates Using the Dynamic Nelson-Siegel Model part of the distributions. One model that captures such symmetry in 197 tion between two uniform random variables, as depicted by the scatterplots, is the al family. This family contains the Gaussian and t copulas. Estimation and goodness of fit test were performed for the Gaussian copulas. The ion procedure was based on maximum pseudo-likelihood estimation proposed by Genest, , and Rivest (1995) and the goodness of fit test was based on the empirical copula ed by Genest, RΓ©millard, and Beaudoin (2009). The results are shown in Table 4. Note is the Pearson’s correlation coefficient of the 𝑖𝑖 π‘‘π‘‘β„Ž and 𝑗𝑗 π‘‘π‘‘β„Ž pseudo-observations. Table 4 Results of the Goodness-of-Fit Test Under the Gaussian Copula Assumption Estimated parameter Sn p-value ρ0,1= -0.981 0.0047 0.9296 ρ0,2= -0.959 Figure 4 . Pairwise scatterplots of the pseudo-observations. Figure 4 . Pairwise scatterplots of the pseudo-observations. ρ1,2= 0.909 With p-value of 0.9296, it seemed that the Gaussian Forecasting sed on the scatterplots copula was adequate to captureof the the pseudo-observations, dependence of the thea fraction To validate the model, With p-value of 0.9296, it seemed that the Gaussian copula was adequate to capture the first oferror variable ha the available pseudo-observations. The estimated correlation matrix data, from January 2 to December 25, 2013 (52 was given by ence of the pseudo-observations. The estimated correlation matrix was given by Wednesdays), was taken out. The goal was to compare ssociation to the second and third error variables, while the second and the 1 βˆ’0.981 βˆ’0.959 the forecasts and the actual data taken out. First, 10,000 triples of random numbers (e0,1 , e1,2 , (βˆ’0.981 1 0.909 ) e2,1 ) from the joint distribution of the random vector βˆ’0.959 0.909 1 (e0 , e1 , e2 ) of the residuals were produced. Each triple ERM STRUCTURE FORECASTING was substituted into the time 15series model to obtain the stimated parameters were then used to simulate trivariate data with 𝑑𝑑2 margin and normal These estimated parameters were then used to simulate values of b01 , b11 , b21 . Thus, 10,000 triples (b01, b11 , trivariate data with t2 margin and normal copula to to capture the dependence. b21 ) were obtained. Then the median of each beta was From Equations capture the(9) to (11) and the random numbers15generated dependence. taken from the distribution to obtain one triple (bof01 , b11 , b21 ). This median From Equations (9) to (11) and the random numbers triple was then used in Equation (7) to get the yield he random vector generated (πœ€πœ€0 , from 2 ), future πœ€πœ€1 , πœ€πœ€the distributionvalues of ofthethe beta parameters random vector wereofcomputed curve and used week 1 (January in 2, 2013), y1 (t), where t is ) and the random (e0 , enumbers 1 generated , e2 ), future values of from the distribution the beta parameters wereof the maturity. he Nelson-Siegel computed equation and used in the Nelson-Siegel equation For the second week (t = 2),10,000 triples of random uture values of the beta parameters were computed and used in numbers (e0,2 , e1,2 , e2,2 ) from the joint distribution of 1 βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ 1 βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ†the 𝑑𝑑 random vector (e0 , e1 , (13) e2 ) were again produced. βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ 𝑦𝑦𝑑𝑑 (𝜏𝜏) = 𝛽𝛽0𝑑𝑑 + 𝛽𝛽1𝑑𝑑 ( ) + 𝛽𝛽2𝑑𝑑 ( βˆ’ 𝑒𝑒 ) 𝜏𝜏/πœ†πœ†π‘‘π‘‘ 𝜏𝜏/πœ†πœ† These and the betas of previous weeks were plugged (13) 𝑑𝑑 in the time series model to obtain 10,000 triples (b , 02 1 βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ 1 βˆ’ 𝑒𝑒 βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ (13) b12 , b22 ). Again, the median of each beta was taken to with( πœ†πœ†π‘‘π‘‘ = 9.807527. ) + 𝛽𝛽2𝑑𝑑 The( initial values βˆ’ 𝑒𝑒were βˆ’πœπœ/πœ†πœ†π‘‘π‘‘ those obtained for the December 26, 2012 dataset: ) get one triple (b02 , b12 , b22 ). This median triple was 1𝑑𝑑 𝜏𝜏/πœ†πœ†π‘‘π‘‘ 𝜏𝜏/πœ†πœ†π‘‘π‘‘ used in the Nelson-Siegel equation to get the yield 𝛽𝛽0,0 = 0.042670232 , 𝛽𝛽0,1 = 0.035735575, and 𝛽𝛽0,2 = 0.107586694. curve yThe (t)randomness of the 9, 2013). This process for week 2 (January 2 values were those obtained for the December with lt = 9.807527. The initial values were those 26, 2012 dataset: was continued to find the yield curve for week t, t > eta parameters were due to the residuals πœ€πœ€π‘–π‘–,𝑑𝑑 , 𝑖𝑖 = 0,1,2. obtained for the December 26, 2012 dataset: b 0,0 2. Notice that medians of beta parameters were taken 35735575, and 𝛽𝛽0,2 = 0.107586694. = 0.042670232, The randomness b 0,1 = 0.035735575, and ofb 0,2the = first before computing the yield yt (t). An alternative Forecasting 0.0107586694. The randomness of the beta parameters approach is to compute for each t the yield yt (t) using were due esiduals πœ€πœ€π‘–π‘–,𝑑𝑑 , 𝑖𝑖 = 0,1,2. to the residuals e i,t ., i = 0,1,2. each of the 10,000 triples (e0,t , e1,t , e2,t ) then take To validate the model, a fraction of the available data, from January 2 to December 25, 013 (52 Wednesdays), was taken out. The goal was to compare the forecasts and the actual data action of the available data, from January 2 to December 25, he actual data 5fortocomparison. Figures 8 present sample forecast of interest rates per tenor and per week along with E.P. de Lara-Tuprio, et al he actual data 198 for comparison. Figure 5. Forecast for 3-month tenor. Figure 5. Forecast for 3-month tenor. Figure 5. Forecast for 3-month tenor. Figure 6. Forecast for 5-year tenor. ng Head: TERM STRUCTURE FORECASTING Figure 6. Forecast for 5-year tenor. Figure 6. Forecast for 5-year tenor. Figure 7. Yield curve forecast for week 1. Figure 7. Yield curve forecast for week 1. ing Head:Philippine TERMInterest STRUCTURE FORECASTING Rates Using the Dynamic Nelson-Siegel Model 199 FigureFigure 7. Yield curve 8. Yield forecast curve forecast for for week 9. week 1. the median yt (t) for each t. This, however, will still Forecast of interest rates was based on the resort to taking a single triple of beta parameters assumption that yield curve would follow the Nelson- from previous weeks; otherwise, the process will be Siegel equation with the same shape parameter as computationally expensive. produced from historical data and with beta parameters Figures 5 to 8 present sample forecast of interest generated from the time series model. The dependence rates per tenor and per week along with the actual data structure of the beta parameters was considered in for comparison. generating their future values. This was carried out by It was observed that the forecasting accuracy is finding the joint distribution of the error variables via weakened as one goes farther into the future. This is appropriate copula. a limitation of any time series models. Thus, instead Results showed that forecast of interest rates for of 1-year forecast, only 12 weeks were considered. different tenors, short, medium or long, was relatively The root mean square errors (RMSE) per tenor and good up to the next three months. From then on, the per week are shown in Tables 5 and 6. In can be seen accuracy weakened. This showed that the model, that the best forecast is for 5-year yield, with only 6 wherein parameters were based on historical data, Figure 8. Yield curve forecast for week 9. bps RMSE, and the least accurate is for 10-year yield could be reliable only for the near future. For an active with 27 bps RMSE. market, this is good enough since the models for the parameters can be adjusted every trading day. As mentioned earlier, a model for forecasting the It wasConclusion observed and Recommendation that the forecasting accuracy is weakened as one goes farther into term structure of interest rates is essential in designing In this paper, the three-factor Nelson-Siegel model an optimal debt strategy for the government. This paper was applied to forecast the term structure of interest presents one such model and how it can be applied to e. This is a limitation of any time series models. Thus, instead of 1-year forecast, only rates using Philippine market data. Using a fixed shape the Philippine market data. Forecast of interest rates will help the policy makers determine the appropriate parameter, the equation for the yield became linear in the beta parameters. Such equation was fitted to each mix of debt instruments that will minimize cost subject to a prudent risk level. ks were considered. historical dataset of zero rates thereby producing a series of beta parameters. An appropriate time series model was then obtained for each beta parameter. The root mean square errors (RMSE) per Acknowledgement tenor and per week are shown in Tables 5 Based on the historical data, the best model for each beta was of the form ARMA(p,q)+eGARCH(1,1). It E. de Lara-Tuprio and E. Cabral are grateful for the is important to note that a different set of historical financial support provided by the National Research Council can be seen that the best forecast is for 5-year yield, with only 6 bps RMSE, and the l data may produce a different time series model for the beta parameters. of the Philippines. rate is for 10-year yield with 27 bps RMSE. 200 E.P. de Lara-Tuprio, et al Table 5. RMSE Per Tenor Β  1 Mo 3 Mo 6 Mo 1 Yr 2 Yr 3 Yr 4 Yr 5 Yr RMSE 0.1106% 0.1556% 0.1489% 0.1071% 0.1821% 0.1460% 0.1310% 0.0635% Β  6 Yr 7 Yr 8 Yr 9 Yr 10 Yr 15 Yr 20 Yr 30 Yr RMSE 0.0830% 0.1399% 0.1921% 0.2378% 0.2743% 0.2636% 0.1507% 0.1655% Table 6. RMSE Per Week Week 1 2 3 4 5 6 RMSE 0.1236% 0.1280% 0.1163% 0.1100% 0.1265% 0.1389% Week 7 8 9 10 11 12 RMSE 0.1407% 0.1491% 0.1321% 0.1640% 0.1719% 0.2246% References Genest, C., Ghoudi, K., & Rivest, L. P. (1995). A semiparametric estimation procedure of dependence Annaert, J., Claes, A., De Ceuster, M., & Zhang, H. (2012). parameters in multivariate families of distributions. Estimating the yield curve using the Nelson-Siegel Biometrika, 82, 543–552. model: A ridge regression approach. International Genest, C., RΓ©millard, B., & Beaudoin, D. (2009). Goodness- Review of Economics & Finance, 27, 482–496. of-fit tests for copulas: A review and a power study. Bloomberg (2013). Bloomberg Professional. [Online]. Insurance: Mathematics and Economics, 44, 199–213. Available at: Subscription Service (Accessed: May 7, Koopman, S. J., Mallee, M. I. P., & Van der Wel, M. (2010). 2013). Analyzing the term structure of interest rates using Christensen, J. H. E., Diebold, F. X., & Rudebuscha, G. 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