1 Purpose of the paper

Transcription

1 Moneary Economics 2 F.C. Bagliano - Sepember 2017 Noes on: F.X. Diebold and C. Li, Forecasing he erm srucure of governmen bond yields, Journal of Economerics, Purpose of he paper The paper presens a novel mehodology for forecasing he yield curve of governmen bonds using he Nelson-Siegel (1987) mehod for deriving a coninuous curve wih desirable properies from a nie se of observed bond prices a di eren mauriies. The Nelson-Siegel (N-S) echnique is inerpreed as a facor model and each N-S componen is given an inerpreaion in erms of level, slope and curvaure e ecs. A simple ime-series model for he evoluion over ime of such facors is esimaed and used o forecas he facors ou-of-sample. Using monhly US daa for he period, he forecasing performance of his mehodology is compared wih alernaive mehods for forecasing yields and found superior a leas over a one-year forecasing horizon. 2 Fiing he yield curve Given a limied se of observed bond prices a a poin in ime, several echniques can be used o exrac a coninuous yield curve, delivering he yield o mauriy on bonds wih any mauriy. In order for his derived curve o be useful for nancial analysis and forecasing, i mus possess a se of desirable properies, such as exibiliy (i.e. i mus be able o reproduce he various observed shapes of he yield curve, such as upward-sloping, downward-sloping, hump-shaped), parsimony (i.e. i mus usefully summarize he feaures of he erm srucure using a limied se of parameers) and economic inerpreabiliy. Perhaps he mos widely used mehod for ing he yield curve is he hree-componen exponenial mehod of Nelson and Siegel (1987), on which he Diebold-Li paper is based. 2.1 Preliminaries Focusing on he simpler case of pure discoun bonds (paying 1 uni a he mauriy dae o he holder, wih no coupon paymens in he inervening 1

2 periods), le s recall he heoreical relaionships among hree basic conceps: he discoun curve, he forward rae curve and he yield curve. In wha follows, ime is coninuous, ineress are coninuously compounded, and denoes he mauriy of he discoun bonds (measured in monhs in he empirical analysis of he paper). Denoing he yields o mauriy on -period discoun bonds a ime as y (), he discoun curve gives he prices of he bonds P () as a funcion of mauriy as P () = e y() discoun curve from which y () = 1 log P () The relaionship beween he yields o mauriy and he implici isananeous forward raes f () gives he forward rae curve 1 and he yield curve f () = dy () + y () d P 0 = () 1 y () + y () P () P 0 = () forward rae curve P () y () = 1 Z 0 f (u) du yield curve 2.2 The Nelson-Siegel mehod and is inerpreaion A a given dae, a which a se of yields on bonds wih di eren mauriies is available, he Nelson-Siegel mehod s a smooh coninuous curve of he following hree-componen exponenial form: y () = e e wih he following relaed isananeous forward rae curve: f () = e + 3 e e 1 Equivalenly, he isananeous forward rae may be seen as he rae of decay of he discoun funcion d() P () =e y() = e R 0 f(u) du. 2

3 The properies of he curves are de ned by he four parameers 1, 2, 3 and. The parameer governs he exponenial decay of he wo funcions in brackes as mauriy goes from 0 o 1, wih higher values of deermining a faser decay. 2 Given a value for, he hree erms in he curve for y () can be inerpreed as a parameer i (i = 1; 2; 3) muliplied by a funcion of mauriy as follows: 1. he rs componen is simply 1 and does no depend on. I is inerpreed as a long-erm componen capuring he level of he yield curve. In fac, leing! 1 y (1) = 1 2. in he second erm, a decay funcion 1 e (monoonically going o 0 as! 1) is applied o he parameer 2. Since he size of his componen declines as mauriy increases, i is inerpreed as a shorerm componen. In fac, as! 0 we have y (0) = ha, combined wih he previous propery, gives y (1) y (0) = 2 The parameer 2 has herefore he inerpreaion of (minus) he slope of he yield curve. 3. in he hird erm, he parameer 3 is muliplied by a decay funcion 1 e e which sars a 0 for = 0, increases for inermediae values of up o a maximum (whose posiion is deermined by he chosen value for ) and hen decreases o 0 for! 1. Therefore, his componen has a smaller size for boh shor and long mauriies and a larger size for medium-run mauriies: i is hen inerpreed as a medium-erm componen. Moreover, he parameer 3 is closely relaed o a (convenional) measure of he curvaure of he yield curve, given by (wih measured in monhs and for a chosen = 0:0609) 2 y (24) (y (3) + y (120)) = 0: :37 3 Hence, he parameer 3 capures he curvaure of he yield curve. 2 The parameer > 0 capures he speed of convergence of he spo and isananeous forward raes o heir values for! 1 (he consol rae). A lower shifs he hump of he curve owards shorer mauriies, hereby acceleraing convergence o he consol rae. 3

4 Overall, he Nelson-Siegel funcional form for he yield curve is inerpreed by Diebold and Li as a hree-facor (saisical) model for he erm srucure wih he parameers i capuring he relevan facors a ecing he shape of he yield curve a each dae : 1! level facor 2! slope facor 3! curvaure facor The e ecs of a (uni) change in each facor i on he yields a di eren mauriies is given by he values of he decay funcions, or loadings (displayed in Figure 1 of he paper) and are consisen wih he above inerpreaion of he facors as level, slope and curvaure. 3 Modelling and forecasing he yield curve Diebold and Li propose a muli-sep empirical sraegy for forecasing he yield curve ou-of-sample using a ime-series of cross-secions of yields. The daa se consiss, for each monh beween January 1985 and December 2000, of yields on discoun bonds of 17 di eren mauriies, ranging from 3 monhs o 10 years. The seps of he mehodology are he following: 1. Facor esimaion. For each monh, he N-S coninuous funcion for y () is ed o he 17 available yields (imposing = 0:0609, which implies a maximum value for he loading of he medium-erm facor a a mauriy of 30 monhs) by means of he following OLS regression: y () = e e e + u () (1) where he 17 yields are regressed on a consan and he wo regressors in brackes, wih u () represening pricing errors. From hose regressions, ime series for he esimaed ^ 1, ^ 2, ^ 3 and ^u () are obained. 2. Facor modelling and forecasing. To obain forecass of he yield curve over h-period horizons on he basis of informaion available a monh, ^y +hj (), he hree facors are modelled as auoregressions of he form (for i = 1; 2; 3) ^ i = c i + i ^i h + " i (2) 4

5 so ha he values of facors i forecas a for monh +h are obained as ^ i +hj = ^c i + ^ i ^i As an alernaive, forecass can be obained from a mulivariae model of he form ^ = c + ^ h + " where ^ = ( ) 0 and is a 3 3 marix, so ha ^ +hj = ^c + ^ ^ 3. Yield curve forecasing. Once forecass of he facors are obained, he yield curve is forecas ou-of-sample over a h-period horizon as ^y +hj () = ^ 1 +hj + ^ 1 e 2 +hj + ^ 1 e 3 +hj e (3) Forecass are consruced recursively. Saring from he sample 1985(1)- 1993(12), he auoregressive models for he facors (2) are esimaed and used o produce facor forecass from 1994(1) o 2000(12) a he 1- monh, 6-monh and 1-year horizons; hen, forecass of he yield curve are consruced from (3). The esimaion sample for he facor models (2) is hen updaed by adding one observaion a a ime (becoming 1985(1)-1994(1), 1985(1)-1994(2), and so on) and new forecass of facors and yields are obained for he same horizons over he remaining par of he period (being 1994(2)-2000(12), 1994(3)-2000(12), and so on). 4. Forecas evaluaion. Finally, forecass are evaluaed and compared wih hose obained by alernaive models (such as a simple random walk for he yield on each mauriy, V AR models for he levels or he changes in he yields, V ECM models wih one or wo common sochasic rends) using various saisics, including he roo mean squared error for each forecasing horizon h and each mauriy : RMSE(h; ) = s PT (+h) i=0 y+i+h () ^y +i+hj+i () 2 T ( + h) + 1 where T = 2000(12) and = 1993(12). 5

6 4 Conclusion Diebold and Li presen a reinerpreaion of he Nelson-Siegel mehod for yield curve ing as a hree-facor model, capuring movemens in he level, slope and curvaure of he curve, and assess is ou-of-sample forecasing performance. The resuls show ha such a parsimonious model ouperforms several compeiors in forecasing exercises over he one-year horizon. 6