Informaion in he erm srucure for he condiional volailiy of one year bond reurns Revansiddha Basavaraj Khanapure 1 This Draf: December, 2013 1 Conac: 42 Amsel Avenue, 318 Purnell Hall, Newark, Delaware, 19716. Email: khanapur@udel.edu. I hank Prof. John H. Cochrane and Prof. Lars P. Hansen for numerous discussions and commens. I also hank Prof. Timohy G. Conley, Prof. Monika Piazzesi, Prof. Jeffrey R. Russell, Prof. Ruey S. Tsay and Prof. Piero Veronesi for valuable commens. I hank Alexi Savov for numerous proof reads.
Absrac The erm srucure of ineres raes capures a significan amoun of he ime variaion in he condiional volailiy of 1-year excess reurns of bonds of 2- o 5-year mauriy. A single linear combinaion of he forward raes capures mos of he variaion in he condiional volailiies across mauriies. The erm srucure drivers of he expeced excess reurns and he condiional volailiies are disinc. The overall level of ineres raes capures a significan amoun of he variaion in he condiional volailiy of excess reurns. The 4-5 year yield spread, which is an imporan posiive predicor of expeced excess reurns, has an opposing effec on he condiional volailiy. Thus, he en-shaped linear combinaion of forward raes driving he expeced excess reurns capures eiher he marke price of risk, or he correlaion wih he marke, bu no he condiional volailiy.
1 Inroducion The lieraure is filled wih evidence in favor of ime variaion of he expeced excess reurns of bonds. Fama and Bliss (1987), Campbell (1987), Campbell and Shiller (1991), Cochrane and Piazzesi (2005) show ha linear combinaions of bond yields or forward raes predic excess bond reurns. Cochrane and Piazzesi (2005) provide srong evidence in favor of common componen in he ime variaion of expeced excess reurns across mauriies. They find ha a single en-shaped linear combinaion of forward raes capures mos of he variaion in he expeced 1-year excess reurns of bonds of 2 o 5-year mauriies. The en-shaped facor is able o predic he excess reurns wih more han 30% R 2. Also, Fama and French (1989) find a common componen in he ime variaion of expeced excess reurns on bonds and socks. These aricles sugges ha he ime variaion in he expeced excess reurns of bonds is explained, a leas parially, by a ime-varying aggregae price of risk. In fac Wacher (2006) shows ha exernal habi preferences can generae a posiive forecasing relaion beween he yield spread and bond excess reurns. Boyd e al. (2005) and Andersen e al. (2005) show ha sock reurns, ineres raes and exchange raes have differen responses o macroeconomic news over he business cycle, hus suggesing he exisence of a business cycle componen o he variaion in he second momens of reurns. In his aricle, I focus on he variables explaining he ime variaion in he condiional volailiy or variance of 1-year excess bond reurns. I exrac he informaion in he erm srucure for esimaing he condiional volailiies of 1-year excess reurns of bonds of 2- o 5-year mauriies. The informaion in he erm srucure explains a subsanial amoun of variaion in he condiional volailiies of excess reurns. I also find ha a single linear combinaion of forward raes capures mos of he ime variaion in he condiional volailiies of excess reurns across mauriies. This single facor is differen from he en-shaped linear combinaion of forward raes ha drives he condiional mean. I es he exen o which he variaion in he expeced excess reurns is due o he variaion in he condiional 1
volailiy. I es he hypohesis ha he same linear combinaion of forward raes drives boh he condiional mean and he condiional volailiy. The es reveals ha such a resricion severely impedes he abiliy o forecas he condiional volailiy (and produces a model ha is a significan misfi o he daa). I herefore conclude ha he erm srucure drivers of he condiional mean and he condiional volailiy are no he same. Two differen linear combinaions of forward raes summarize he informaion in he erm srucure for he Sharpe raios of 1-year excess reurns of bonds of 2- o 5-year mauriies. Equilibrium asse pricing models obain variaion in he expeced excess reurns as a resul of variaion in he marke price of risk or variaion in he amoun of risk (bea) or boh. In his aricle, I sudy he condiional volailiy of 1-year bond reurns. I do no sudy he erm srucure drivers of he correlaion of 1-year bond reurns wih he marke porfolio, he oher imporan componen of bea. This precludes a conclusive inerpreaion of he driver of he expeced excess reurns as he marke price of risk. The resuls show ha he en-shaped facor capures he marke price of risk or he correlaion wih he marke or boh, bu no he condiional volailiy. I use forward raes as he primiives represening he erm srucure informaion. The specificaions in erms of log condiional volailiies help avoid resricions on he coefficiens of he forward raes. The condiional volailiy specificaion in erms of he shor rae and he spreads of he forward raes (relaive o he shor rae) reveals ha he overall level of ineres raes capures mos of he variaion in condiional volailiies. The condiional volailiy specificaion wihou he shor rae is srongly rejeced. In addiion, spreads do also conribue a non-rivial amoun o he variaion of he condiional volailiy. I disill he informaion in he erm srucure for forecasing he condiional volailiy ino wo componens, (1) he shor rae and (2) a facor formed from he spreads. The spread facor is a linear combinaion of he spreads of 5-year and 4-year forward raes relaive o he shor rae. The loading is excessively iled oward a posiive conribuion by he 5-year forward rae spread. Cochrane and Piazzesi (2005) noe ha he 4-5 year yield spread is 2
imporan for forecasing he 1-year excess reurns of all mauriies. The same 4-5 year yield spread has an opposing effec on he condiional volailiy. However, he shor rae drives boh he expeced excess reurns and he condiional volailiy in he same direcion, hough he shor rae dominaes he condiional volailiy variaion. The opposing effecs of spreads and he dominance of shor rae for he condiional volailiy variaion leave he drivers of he condiional mean and he condiional volailiy ou of perfec sync. L. R. Glosen and Runkle (1993) sugges ha he shor rae reflecs aggregae economic uncerainy in addiion o expecaions of fuure inflaion and he real ineres rae. Kim and Nelson (1999), McConnell and Perez-Quiros (2000) show ha he pos-1981 period is characerized by significanly lower real and nominal macroeconomic volailiy. This shif in macroeconomic volailiy does no affec he abiliy of he shor rae o forecas he condiional volailiy The paper closes o his aricle is Viceira (2007). Viceira focuses only on he realized second momens of reurns a 3 o 12 monhs horizons on a bond wih a fixed 5 years o mauriy and he ime variaion in business cycle proxies. I focus on he 1-year excess reurns of bonds of 2- o 5-year mauriies, for which Cochrane and Piazzesi (2005) find compelling evidence in favor of a common componen in he expeced excess reurns. I use a maximum likelihood approach o arrive a esimaes for he condiional volailiy using informaion in he enire erm srucure o forecas he condiional volailiy. 2 Daa I use monhly bond price daa for bonds of 1 o 5-year mauriy from he CRSP Fama- Bliss Discoun Bond file. I follow Cochrane and Piazzesi (2005) and only use he daa from January, 1964 onwards. Also, I consruc he realized bond reurn volailiies for bonds of fixed mauriies (2-year and 5-year mauriies) from he daily reurn daa in CRSP Fixed Term Index file. Boh he samples span over he period from January, 1964 o December, 2005. 3
3 Noaion I use following noaion for he bond prices, reurns, and forward raes. p (n) = Log price of he n-year discoun bond a ime. The log yield of he n year discoun bond a ime is: y (n) 1 n p(n). The uni ime inerval hroughou he aricle is one year. Less han a year s ime inervals are represened in fracions of 12. The log forward rae a ime for borrowing beween ime + n 1and + n is: f n 1 n = f (n) p (n 1) p (n). The log holding period reurn for buying a n-year bond a ime and selling i as a n 1-year bond a ime + 1 is: The 1-year excess log reurn is: hpr (j) +1 p(n 1) +1 p (n). 1 rx (n) +1 hpr(n) +1 y(1). I denoe he condiional volailiy or he condiional variance of he 1-year excess reurn of an n-year mauriy bond wih respec o he ime informaion se as +1, i.e. var (rx (n) +1 ) h(n) +1. Noe ha he ime index on he condiional volailiy is +1 hough he condiional volailiy is measurable wih respec o ime informaion. I use variaions of he following wo shor-hand noaions o succincly denoe cerain sums hroughou he aricle. 5 γ (n) 1,...,5 f (1,...,5) = f (j), β (n) 2,...,5 (f (2,...,5) f (1) )= j=1 5 j=2 γ (n) j β (n) j γ (n) and β (n) are he coefficiens in differen specificaions. 4 (f (j) f (1) ).
4 Informaion in he Term Srucure In his secion I es he hypohesis ha he erm srucure capures a significan amoun of he variaion in he condiional volailiy. The condiional volailiies of bonds of 2- o 5-year mauriies are esimaed using he maximum likelihood mehod. I find ha a single linear combinaion of he forward raes capures mos of he variaion in he condiional volailiies due o variaion in he forward raes. Furher I find ha he erm srucure drivers of he condiional mean and he condiional volailiy are no he same. rx (n) +1 = κ (n) + γ (n) 1,...,5 f (1,...,5) ɛ (n) +1 = +1 u (n) +1 wih u (n) +1 N(0, 1) + ɛ (n) +1 n =2,...,5 (S) +1 =exp(α(n) + δ (n) ) 2 + λ)) (S-0) +1 =exp(α (n) + δ (n) ) 2 + λ)+β (n) 1,...,5 f (1,...,5) ) (S-1) Specificaion (S) relaes he erm srucure informaion in he form of a linear combinaion of he forward raes o 1-year excess reurns of bonds of 2- o 5-year mauriies. Specificaion (S-1) relaes he forward raes o he log of he condiional volailiy. The log ransformaion avoids resricions on he coefficiens of he forward raes. I use a maximum likelihood approach o esimae he parameers in he condiional mean and he condiional volailiy specificaions. In all of he following specificaions λ is no an esimaed parameer bu a small consan o ensure he gradien of he likelihood funcion exiss 1. I use overlapping monhly daa of 1-year excess reurns of bonds of 2- o 5-year mauriies. The deails of he esimaion procedure and he mehod for correcing he errors are in Appendix A. I use maximum 1 The value of λ is se o 10 8 in all he specificaions (The unexpeced excess reurns in (S-1) are raw reurns wihou he conversion o percenage poins). Seing λ o lower values up o 10 12 resuls in a higher number of ieraions unil convergence. The parameer esimaes and he sandard errors are no [ ] much differen. 2ɛ(n) δ 1,f (1,...,5) (ɛ (n) ) 2 +λ 1 is a par of he gradien of he likelihood funcion. For λ =0,and 6 1 exremely small values of unexpeced reurns (relaive o machine precision) he muliple is no defined. 5
likelihood approach o obain he parameer esimaes. The covariance marix is based on he correced informaion marix. I also verify ha he inference and he conclusions are he same using he esimaes and he covariance marix based on he non-overlapping samples of he daa. One simple way o gauge he imporance of forward raes for he condiional volailiy esimaion is o compare he wo specificaions (S-0) and (S-1). (S-0) conains jus he lagged residual, whereas (S-1) also includes all he forward raes. The likelihood raio es saisics in Table 1 imply ha he erm srucure of ineres raes makes a significan conribuion oward predicing he condiional volailiy of bond excess reurns. The log likelihood values for specificaion (S-1) far exceed he log likelihood values for specificaion (S-0) which is devoid of forward raes. H 0 :S-0,H 1 :S-1 χ 2 P-Value 1% Cri-χ 2 n =2 50.19 0.00 15.09 n =3 52.88 0.00 15.09 n =4 53.38 0.00 15.09 n =5 62.12 0.00 15.09 Table 1: The χ 2 saisic for he likelihood raio es of he specificaion (S-0) wih jus he lagged residuals agains he alernae specificaion (S-1) which also includes he forward raes. The likelihood raio es of he specificaion (S-1) wih and wihou he log of lagged condiional variance implies ha he zero coefficien resricion on he log of he lagged condiional variance canno be rejeced. I discuss he convenional ARCH/GARCH specificaions and he specificaion wih he lagged condiional variance in Secion 8. Including he lagged condiional variance does no significanly improve he model s success a capuring he condiional volailiy. For he res of he aricle I do no include he lagged condiional variances in he 6
specificaions. Specificaion: S-1, H 0 : β (n) 1,...,5 =0 χ 2 P-Value 1% Cri-χ 2 n =2 31.18 0.00 15.09 n =3 22.78 0.00 15.09 n =4 35.93 0.00 15.09 n =5 26.61 0.00 15.09 Table 2: The χ 2 saisics for he join es of he hypohesis ha he coefficiens of forward raes in specificaion (S-1) are equal o zero. Tables 3 and 4 presen he coefficien esimaes and he sandard errors for he parameers in specificaion (S-1). None of he coefficiens of he forward raes in he condiional volailiy specificaion are significan. Small -raios and large coefficiens are a sign of possible nearmulicollineariy. I ake up he ask of reducing he number of explanaory variables in he nex few secions. However, he join significance es rejecs he hypohesis ha he coefficiens of he forward raes in he condiional volailiy specificaion are zero. The χ 2 saisics in Table 2 rejec he null even a a 1% significance level. Thus, he informaion in he erm srucure capures a non-rivial componen of he variaion in condiional volailiy. Fig. 1 and 2 include he plos of he condiional volailiies implied by he specificaion pair (S-1), (S-0) agains he magniudes of he unexpeced reurns. (S-1) implies high condiional volailiies over he 80s, a period of high ex-pos volailiy. Overall (S-1) implies greaer variaion in he condiional volailiy. Specificaion (S-0) essenially forecass differen average levels of condiional volailiies for differen mauriies along wih a few high volailiy evens. The sandard deviaion of he condiional sandard deviaion, σ(( +1 )1/2 ), is one way o quanify he variaion in he condiional volailiy. The values for he sandard deviaion of he condiional sandard deviaion are [0.39, 0.73, 0.99, 1.34] for he specificaion (S-1) 7
and [0.06, 0.21, 0.25, 0.33] for he specificaion (S-0) for bonds of mauriies n =2, 3, 4and 5 respecively. The unis are percen reurns, e. g. for he 1-year excess reurns (unis in percen) of bonds of 2-year mauriy he variaion in he condiional sandard deviaion (h (2) +1 )1/2 implied by he specificaion (S-1) over he enire sample period is of he order of 39 basis poins. The condiional volailiy implied by specificaion (S-1) varies much more han ha implied by specificaion (S-0). Thus, he erm srucure informaion helps capure a greaer amoun of variaion in he condiional volailiy. 8
rx (n) + γ (n) + ɛ (n) +1 = κ(n) 1,...,5 f (1,...,5) +1 κ (n) (κ (n) ) γ (n) 1 (γ (n) 1 ) γ (n) 2 (γ (n) 2 ) γ (n) 3 (γ (n) 3 ) γ (n) 4 (γ (n) 4 ) γ (n) 5 (γ (n) 5 ) n =2-0.02-5.02-0.83-6.12 0.15 0.58 1.19 5.03 0.46 2.41-0.76-5.05 n =3-0.03-4.90-1.72-7.67-0.02-0.05 3.09 7.72 0.59 1.78-1.64-6.77 n =4-0.04-5.26-2.20-6.90-0.15-0.24 3.59 6.09 1.54 3.15-2.35-6.46 n =5-0.05-5.51-2.88-7.47 0.26 0.35 3.97 5.47 1.53 2.57-2.35-5.39 Table 3: The coefficien esimaes and he -raios for he parameers in he condiional mean specificaion (S). The specificaion comprises of he pair, (S), (S-1), he condiional mean and condiional volailiy specificaions respecively. 9
ln( +1 )=α(n) + δ (n) ) 2 + λ)+β (n) 1,...,5 f (1,...,5) α (n) (α (n) ) δ (n) 1 (δ (n) 1 ) β (n) 1 (β (n) 1 ) β (n) 2 (β (n) 2 ) β (n) 3 (β (n) 3 ) β (n) 4 (β (n) 4 ) β (n) 5 (β (n) 5 ) n =2-10.03-17.77-0.04-1.06 5.78 0.42 11.68 0.47 4.76 0.21-13.35-0.69 9.14 0.71 n =3-9.17-20.20-0.06-2.03-9.36-0.68 23.94 0.92 5.16 0.22-12.79-0.67 12.33 0.94 n =4-8.19-16.69-0.02-0.52-15.53-1.16 28.74 1.10 3.29 0.14-10.45-0.57 12.77 0.93 n =5-7.89-16.75-0.02-0.54-17.89-1.35 30.22 1.13 7.08 0.29-11.06-0.61 11.83 0.86 Table 4: The coefficien esimaes and he -raios for he parameers in he condiional volailiy specificaion (S-1). The specificaion comprises of he pair, (S), (S-1), he condiional mean and condiional volailiy specificaions respecively. 10
8 6 ε (2) (S 0), n = 2 (S 1), n = 2 4 2 0 Oc65 Apr71 Oc76 Mar82 Sep87 Mar93 Sep98 Feb04 Aug09 12 10 8 ε (3) (S 0), n = 3 (S 1), n = 3 6 4 2 0 Oc65 Apr71 Oc76 Mar82 Sep87 Mar93 Sep98 Feb04 Aug09 Figure 1: Comparison: Magniude of unexpeced reurns, ɛ ( ), condiional volailiy implied by specificaions (S-0) and (S-1) for he excess reurns of bonds of mauriies n =2, 3. All values in % and condiional volailiies in sandard deviaion. 11
15 10 ε (4) (S 0), n = 4 (S 1), n = 4 5 0 Oc65 Apr71 Oc76 Mar82 Sep87 Mar93 Sep98 Feb04 Aug09 20 15 ε (5) (S 0), n = 5 (S 1), n = 5 10 5 0 Oc65 Apr71 Oc76 Mar82 Sep87 Mar93 Sep98 Feb04 Aug09 Figure 2: Comparison: Magniude of unexpeced reurns, ɛ ( ), condiional volailiy implied by specificaions (S-0) and (S-1) for he excess reurns of bonds of mauriies n =4, 5. All values in % and condiional volailiies in sandard deviaion. 12
γ (n) 1,,5 Coeff 4 3 2 1 0 1 (S), (S-1): Cond. Mean Coeff 2 3 4 5 β (n) 1,,5 Coeff 40 30 20 10 0 (S), (S-1): Cond. Volailiy Coeff 2 10 3 1 2 3 4 5 20 1 2 3 4 5 Coefficiens (S), (S-1): Dominan facor in Cond. Mean 8 6 4 2 0 2 4 Dominan Facor, 99.5% 6 1 2 3 4 5 Mauriy of Forward Raes Coefficiens (S), (S-1): Dominan facor in Cond. Volailiy 60 40 20 0 20 Dominan Facor, 97.4% 40 1 2 3 4 5 Mauriy of Forward Raes Figure 3: Top Row: Coefficiens of he forward raes in condiional mean (S) and condiional volailiy specificaions (S-1). Boom Row: Coefficiens of he forward raes in he linear combinaion ha defines he dominan facors ha capure mos of he variaion in he condiional means and he condiional volailiies across mauriies. Fig. 3 plos he coefficien esimaes for he specificaion pair (S), (S-1). A srong enshaped paern is eviden in he coefficiens of he forward raes in he condiional mean specificaion (S). In fac, as noed by Cochrane and Piazzesi (2005), a single en-shaped linear combinaion of he forward raes capures mos of he variaion (99.5%) in he condiional means across mauriies. I follow Cochrane and Piazzesi (2006) o obain he coefficiens of he forward raes in he linear combinaion ha defines his dominan facor. An eigenvalue decomposiion of he covariance marix of he condiional means yields he loadings of he 13
dominan facor on he condiional means of each of he mauriies 2. Similarly, a srong single facor srucure is eviden in he coefficiens of he forward raes in he condiional volailiy specificaion (S-1). A single linear combinaion of he forward raes capures 97.4% of he variaion in he log of condiional volailiies 2. I use he covariance marix of he log condiional volailiies o exrac he dominan facor. The paerns of he forward rae coefficiens in he linear combinaion ha define he wo dominan facors are quie differen. In fac he wo facors have a correlaion of only 0.33. Thus, he dominan facor explaining he variaion in he condiional mean has lile success a explaining he variaion in he condiional volailiy. The erm srucure drivers of he condiional mean and he condiional volailiy are no he same. Iseγ (n) 1,...,5 = β 1,...,5, (n) foreach mauriy n separaely, in specificaion (S-1 ) forcing he same coefficiens for he forward raes in he condiional mean and condiional volailiy specificaions, i.e. he same erm srucure drivers for he condiional mean and he condiional volailiy for he 1-year excess reurns of a bond. The drop in he likelihoods or he incremenal misfi of he specificaion is one measure of he differences in he erm srucure drivers of he condiional mean and he condiional volailiy. 2 I sor he eigenvecors from he eigenvalue decomposiion of he covariance marix by heir corresponding eigenvalues. The elemens of he eigenvecor, q 1, corresponding o he larges eigenvalue are he loadings of he mos dominan facor. The elemens of q 1 measure he movemen in he explanaory variable per uni movemen in he dominan facor (e.g. he firs elemen of (q1 rx) 4 1 is a measure of he change in he condiional mean for n =2,rx (2) +1, per uni movemen in he dominan facor for he condiional mean). The coefficiens of he forward raes, ξ, in he linear combinaion ha defines he dominan facor are he wighed sum of he esimaed coefficiens. The weighs are deermined by he elemens of he eigenvecor q 1. (e.g. for condiional mean: ξ rx =[(ˆγ (2) 1,...,5 ),...,(ˆγ (5) 1,...,5 ) ] 5 4 (q1 rx) 4 1, for condiional volailiy: ξln(h) = ( ) (2) [( ˆβ 1,...,5 ) (5),...,( ˆβ 1,...,5 ) ] 5 4 q ln(h) 1 4 1 ) 14
rx (n) +1 = κ (n) + γ (n) 1,...,5 f (1,...,5) + ɛ (n) +1 n =2,...,5 (S) ln( +1 )=α(n) + δ (n) ln( +1) =α (n) + δ (n) ) 2 + λ)+γ (n) 1,...,5 f (1,...,5) (S-1 ) ) 2 + λ)+β (n) 1,...,5 f (1,...,5) (S-1) H 0 :S-1 Vs. H 1 :S-1 χ 2 P-Value 1% Cri-χ 2 n =2 49.73 0.00 15.09 n =3 52.03 0.00 15.09 n =4 52.14 0.00 15.09 n =5 59.14 0.00 15.09 Table 5: The χ 2 saisics for he likelihood raio es of he resriced specificaion pair (S),(S-1) agains he alernae unresriced specificaion pair (S),(S-1 ). The log likelihoods for he specificaion pair (S), (S-1 ) wih he resricions are considerably lower han ha for he original specificaion wihou he resricions (S), (S-1). The log likelihood raio ess (Table 5) rejec he resricions even a a 1% significance level. Fig. 5 compares he condiional means implied by he resriced and unresriced specificaion pairs. The expeced excess reurns implied by he wo specificaions are almos he same. In fac he paerns of he coefficiens of he forward raes in he resriced specificaion pair (S),(S-1 ) resembles he en-shaped paern of he coefficiens of he forward raes in he condiional mean par of he unresriced specificaion pair (S),(S-1) (Fig. C-1). Fig. 4 includes he plos of he condiional volailiies for he resriced and unresriced specificaion pairs (for clariy only he plos for n = 2, 5 included). The wide flucuaions in he condiional volailiies implied by he unresriced specificaion over he sample period are no replicaed by he resriced specificaion. The sandard deviaions of he condiional sandard deviaions, 15
Condiional Volailiy in Percen Sandard Deviaion 12 10 8 6 4 2 Specificaion pairs S,S 1 and S,S 1 n = 2, S,S 1 n = 5, S,S 1 n = 2, S,S 1 n = 5, S,S 1 0 Oc65 Apr71 Oc76 Mar82 Sep87 Mar93 Sep98 Feb04 Aug09 Time (Monh of Year) Figure 4: Comparison of he condiional volailiies implied by specificaion pairs (S),(S-1) and (S),(S-1 ). For clariy only he plos for n =2, 5 included. The coefficiens of he forward raes in he condiional volailiy specificaion (S-1 ) are resriced o be he same as he coefficiens of he forward raes in he condiional mean specificaion (S). Volailiy is in % sandard deviaion. 16
20 15 Specificaion pairs S,S 1 and S,S 1 n = 2, S,S 1 n = 5, S,S 1 n = 2, S,S 1 n = 5, S,S 1 10 Condiional Reurn in Percen 5 0 5 10 15 Oc65 Apr71 Oc76 Mar82 Sep87 Mar93 Sep98 Feb04 Aug09 Time (Monh of Year) Figure 5: Comparison of he condiional means implied by specificaion pairs (S),(S-1) and (S),(S-1 ). For clariy only he plos for n =2, 5 are ploed. The wo plos for each n are on op of each oher and hard o disinguish. The coefficiens of he forward raes in he condiional volailiy specificaion (S-1 ) are resriced o be he same as he coefficiens of forward raes in he condiional mean specificaion (S). Reurns are in %. 17
σ(( +1) 1/2 ) are [0.06, 0.22, 0.25, 0.36] for he resriced specificaion pair and [0.39, 0.73, 0.99, 1.34] for he unresriced specificaion pair for mauriies n = [2, 3, 4, 5] respecively. The unresriced specificaion pair capures much more variaion in he condiional volailiy han he resriced specificaion pair, whereas he condiional means implied by he wo specificaions are almos he same. Thus, forcing he same erm srucure drivers for he condiional mean and he condiional volailiy limis he abiliy o forecas he condiional volailiy. The maximum likelihoods are subsanially lower for he resriced specificaion pair compared o he unresriced specificaion pair. Thus, he erm srucure drivers of he condiional mean and he condiional volailiy are differen. Thus, he erm srucure capures a significan amoun of variaion in he condiional volailiy of excess reurns. A single linear combinaion of he forward raes capures mos of he variaion in he log condiional volailiies due o he variaion in he forward raes. Hence, he informaion in he erm srucure for he variaion in he condiional volailiy is summarized by a single linear combinaion of forward raes. However, he erm srucure drivers of he condiional mean and he condiional volailiy are differen. A resricion o force he same erm srucure drivers is rejeced in he daa. Thus, wo differen linear combinaions of he forward raes summarize he variaion in he Sharpe raios of 1-year excess reurns of bonds of 2- o 5-year mauriies. The en-shaped facor ha capures he expeced excess reurns capures eiher he marke price of risk or he correlaion of he bond reurns wih he marke or boh, bu no he condiional volailiy. 5 Informaion in he Level In Secion 4, Fig. 1 and 2 plo he condiional volailiies implied by specificaion (S-1). Noe ha he high condiional volailiies over he eighies mach wih he high ex-pos volailiy over he same period. This mach could very well be jus due o he level effec. And he informaion in he erm srucure for condiional volailiy esimaion could jus be due o 18
he level of ineres raes. In his secion I find ha, in fac, mos of he variaion in he condiional volailiies is capured by he overall level of ineres raes. Specificaion: S-1, H 0 : 5 i=1 β(n) i =0 χ 2 P-Value 1% Cri-χ 2 n =2 20.44 0.00 6.63 n =3 25.30 0.00 6.63 n =4 25.04 0.00 6.63 n =5 31.45 0.00 6.63 Table 6: The join significance es for he coefficiens of forward raes in he condiional volailiy specificaion (S-1) (Condiional mean specificaion: (S)). One way o check wheher he level of ineres raes has any conribuion oward condiional volailiy esimaion is o check wheher he sum of he coefficiens of forward raes in (S-1) is zero. The χ 2 saisics in Table 6 rejec he hypohesis ha he sum of he forward rae coefficiens is zero. The null is rejeced even a a 1% significance level. Thus a rise in he overall level of ineres raes increases he condiional volailiy of one year excess reurns of bonds. +1 =exp(α(n) + δ (n) +1 =exp(α (n) + δ (n) +1 =exp(α(n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + β (n) (2,...,5) 2,...,5 (f f (1) )) (S-2) ) 2 + λ)+β (n) 2,...,5 (f (2,...,5) f (1) )) (S-3) ) 2 + λ)+β (n) 1 f (1) ) (S-4) I es he hypohesis ha he informaion in he level of ineres raes capures a significan amoun of variaion in he condiional volailiy of excess reurns. I use specificaion (S-2) ha separaes he overall level from he spreads in a parsimonious way. The shor erm rae proxies for he overall level of ineres raes in (S-2). 19
rx (4) +1 = κ(4) + γ (4) 1,...,5 f (1,...,5) + ɛ (4) +1 κ (4) γ (4) 1 γ (4) 2 γ (4) 3 γ (4) 4 γ (4) 5 Coeff -0.04-2.20-0.15 3.59 1.54-2.35 -Raio -5.26-6.90-0.24 6.09 3.15-6.45 Table 7: Specificaion Pair: (S), (S-2). Coefficiens and he -raios for he condiional mean par of he specificaion, (S), for he 1-year excess reurn of 4-year mauriy bond (n =4). ln(h (4) +1 )=α(4) + δ (4) 1 ln((ɛ (4) ) 2 + λ)+β (4) 1 f (1) + β (4) (2,...,5) 2,...,5 (f f (1) ) α (4) δ (4) 1 β (4) 1 β (4) 2 β (4) 3 β (4) 4 β (4) 5 Coeff -8.19-0.02 18.82 28.77 3.27-10.45 12.77 -Raio -16.69-0.52 5.00 1.10 0.14-0.57 0.93 Table 8: Specificaion Pair: (S), (S-2). Coefficiens and he -raios for he condiional volailiy par of he specificaion, (S-2), for he 1-year excess reurn of 4-year mauriy bond (n =4). The coefficien esimaes and he sandard errors for he level and spread specificaion (S-2) are in appendix Tables C-3 and C-4. Represenaive exracs for he bonds of 4-year mauriy (n = 4) are in Tables 7 and 8. The coefficiens for he shor erm rae are posiive and overwhelmingly significan. Thus, a higher overall level of ineres raes bodes higher condiional volailiy for he bond excess reurns. The significance of he coefficiens of he shor rae imply ha he informaion in he level of ineres raes capures a considerable amoun of he variaion in condiional volailiies. In fac, going by he individual -raios, level seems o be he dominan conribuion of he erm srucure of ineres raes for predicing he condiional volailiies. Fig. 6 depics he conribuion of he shor erm rae oward predicing he condiional volailiy. The level-and-spread specificaion (S-2) for he log of condiional volailiy is linear in he shor rae, he spreads and he log of lagged residual plus a consan. Fig. 6 shows ha he variaion in he shor erm rae is a significan componen of he variaion of log 20
5 5.5 Cons + Lagged Residual Level Spreads ln(h (4) ), Log condiional volailiy 6 6.5 7 7.5 8 8.5 1966 1970 1974 1978 1982 1986 1991 1995 1999 2003 Time (Leas Coun: Monh) Figure 6: Decomposiion of log condiional volailiy for n = 4 (Specificaion (S-2) ) ino hree componens: Consan and lagged residual: α (4) + δ (4) 1 ln((ɛ (4) ) 2 + λ), Level: β (4) 1 f (1) and Spreads: β (4) 2,...,5 (f (2,...,5) f (1) ). The baseline is shifed o -8.5 insead of 0, for convenience. 21
condiional volailiy. In fac, he shor rae dominaes he variaion and he spreads capure he second mos significan componen of he variaion. Fig. 6 srongly suggess ha he shor rae is barely saionary. The sandard errors and he maximum likelihood saisics, however, are asympoic and based on he saionariy assumpion of he daa generaing process. The asympoic maximum likelihood saisic is a poor choice for inference in he uni roo case. I inend o accoun for his issue in fuure by using ess based on he simulaed finie sample disribuion wih a roo close o uniy (e.g. boosrap procedure). Log likelihood values for differen specificaions. S-1 S-2 S-3 S-4 n =2 1335.44 1335.44 1319.51 1333.97 n =3 1055.74 1055.74 1037.22 1049.85 n =4 911.86 911.86 892.92 904.15 n =5 809.69 809.69 787.39 799.98 Table 9: The maximum log likelihood values for differen specificaions. The erm srucure variables in each condiional volailiy specificaion, (S-1): All forward raes, (S-2): Shor rae and all he spreads, (S-3): All he spreads, (S-4): Shor rae. Anoher way o check he imporance of he shor rae is o quanify he impac of he zero coefficien resricion on he shor rae for he maximized value of he log likelihood. Specificaion (S-3) employs only he spreads for he condiional volailiy esimaion and ses he coefficien on he shor rae o zero. Table 9 compares he maximized log likelihood values for differen specificaions. A move from he simplified condiional volailiy specificaion (S-1) o he level-and-spread specificaion (S-2) does no aler he log likelihoods. However, dropping he shor rae and keeping only he spreads (specificaion: (S-3)) significanly reduces he maximized log likelihood values. The likelihood raio ess rejec he zero coefficien 22
resricion on he shor raes even a a 1% significance level. Thus, he overall level of ineres raes capure a subsanial componen of he variaion of he condiional volailiies of one year excess reurns. σ(( +1) 1/2 ) for differen specificaions. n =2 n =3 n =4 n =5 S-1 0.39 0.73 0.99 1.34 S-2 0.39 0.73 0.99 1.34 S-3 0.24 0.40 0.53 0.78 S-4 0.39 0.67 0.86 1.14 Table 10: The sandard deviaion of he condiional sandard deviaion implied by differen specificaions. The erm srucure drivers of he condiional volailiy specificaion are (S-1): All forward raes, (S-2): Shor rae and all he spreads, (S-3): All he spreads, (S-4): Shor rae. Unis: percenage reurns. Table 10 presens he sandard deviaions of he condiional sandard deviaions, σ(( +1 )1/2 ), implied by differen specificaions. No noiceable differences exis beween he σ(( +1) 1/2 )values implied by specificaion (S-1) and hose implied by specificaion (S-2). Thus a move o level-and-spread represenaion of he erm srucure informaion does no aler he variaion capured by he condiional volailiy specificaion. However he drop in he sandard deviaion of he condiional sandard deviaion is subsanial once he shor rae is removed from he specificaion ((S-2) vs. (S-3)). The drop in, σ(( +1) 1/2 ), is anoher measure of he imporance of he overall level for predicing he condiional volailiy. Also of ineres is he drop in he variaion once all he spreads are removed. The drop in he variaion due o he absence of he spreads ((S-2) vs. (S-4) ) is minimal compared o he drop in he absence of he shor rae ((S-2) vs. (S-3)). In fac he absence of he spreads hardly affecs he variaion in he condiional volailiy of he 1-year excess reurns of bonds of 2-year mauriy. Thus, he shor capures a significanly greaer amoun of variaion of he condiional volailiy han 23
he spreads. Campbell and Ammer (1993) decompose he excess reurn (relaive o shor rae) on any asse ino funcions of news abou real cash flows, fuure real ineres raes and news abou expeced fuure excess reurns. Since he nominal coupons are fixed for bonds, news abou inflaion subsiues for he real cash flow componen of he decomposiion. Thus he inflaion uncerainy represening he uncerainy abou real cash flows feeds ino he condiional volailiy of he excess reurns. Fischer (1981) shows ha he variance of inflaion increases wih he level. To he exen ha he shor rae capures he expecaions abou inflaion, a higher shor rae predics higher condiional volailiy for excess bond reurns as implied by he condiional volailiy specificaion (S-2). 6 Informaion in Spreads In his secion, I check wheher he spreads capure any significan amoun of variaion in he condiional volailiy. I find ha a model wihou he spreads is inferior relaive o a model wih he spreads. In oher words, spreads do carry a non-rivial amoun of informaion for forecasing he condiional volailiy. I reduce he number of spreads in he model o jus wo spreads. Furher I find ha he 4-5 year yield spread has opposing effecs on he condiional mean and he condiional volailiy. +1 =exp(α (n) + δ (n) +1 =exp(α (n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + β (n) 2,...,5 (f (2,...,5) f (1) )) (S-2) ) 2 + λ)+β (n) 1 f (1) ) (S-4) In Secion 5, Fig. 6 shows ha he overall level of ineres raes capures a significan amoun of he variaion in he condiional volailiy of excess reurns. I compare he log likelihoods for he specificaion (S-4) ha conains only he shor rae represening he informaion in he erm srucure agains he log likelihoods for he specificaion (S-2) ha conains boh 24
he shor rae and he spreads. Specificaion (S-4) ses zero coefficiens resricion on he spreads. Table 11 presens he likelihood raio saisic for he zero coefficien resricion on he spreads. The drop in he maximum log likelihoods is significan, excep for he bonds of 2-year mauriy. However, he coefficiens of he spreads are no joinly significan (level of significance is beween 12% for n = 5 o 36% for n = 2). The implicaions of he χ 2 Wald es and he likelihood raio es differ. This is bacause he χ 2 Wald es and he likelihood raio ess are only asympoically equivalen. The sandard deviaion of he condiional sandard deviaion, σ(( +1) 1/2 ), is however lower once he spreads are dropped from he specificaion, excep for n = 2. As noed in Table 10, values for σ(( +1) 1/2 ) for he specificaion (S-4), are [0.39, 0.67, 0.86, 1.14] compared o he values [0.39, 0.73, 0.99, 1.34] for he specificaion (S-2). Thus he absence of spreads does drop he capured amoun of variaion in he condiional volailiy. Thus, he spreads do carry a non-rivial amoun of informaion useful for predicing he condiional volailiy. H 0 :S-4Vs.H 1 :S-2 χ 2 P-Value 1% Cri-χ 2 n =2 2.95 0.57 13.28 n =3 11.77 0.02 13.28 n =4 15.42 0.00 13.28 n =5 19.41 0.00 13.28 Table 11: Likelihood raio es of zero coefficien resricion on he spreads. (S-4) ses coefficiens of spreads o zero, whereas (S-2) employs boh he level and he spreads o predic he condiional volailiy. I drop he he spreads ha do no significanly conribue oward predicing he condiional volailiy. The iny -raios for he coefficiens of he spreads (Tables 8, C-4) in he condiional volailiy specificaion (S-2) are suggesive of near-mulicollineariy. The middle row of Fig. 7 25
plos he coefficiens of he forward raes, shor rae and he spreads for he specificaion pair (S), (S-2). The sharp changes in he signs of he coefficiens of he spreads in he condiional volailiy specificaion sugges near-mulicollineariy. (S), (S-6): Coeff. for condiional mean spec. Coefficiens γ Coefficiens γ Coefficiens γ 4 2 0 2 4 1 2 3 4 5 (S), (S-2): Coeff. for condiional mean spec. 4 2 0 2 4 1 2 3 4 5 (S ), (S-6): Coeff. for condiional mean spec. 4 2 0 2 4 1 2 1 3 1 4 1 5 1 Mauriy of Fwd Rae or Spreads 2 3 4 5 Coefficiens β Coefficiens β (S), (S-6): Coeff. for condiional volailiy spec. 30 20 10 0 10 1 4 1 5 1 (S), (S-2): Coeff. for condiional volailiy spec. 30 20 10 0 10 1 2 1 3 1 4 1 5 1 (S ), (S-6): Coeff. for condiional volailiy spec. Coefficiens β 30 20 10 0 10 1 4 1 5 1 Mauriy of Fwd Rae or Spread Figure 7: Condiional mean specificaion (S) (Top-Lef) and condiional volailiy specificaion (S-6) (Top-Righ). Condiional mean specificaion (S) (Middle-Lef) and condiional volailiy specificaion (S-2) (Middle-Righ). Condiional mean specificaion (S )(Boom-Lef)and condiional volailiy specificaion (S-6) (Boom-Righ). I use he log-likelihood values, he sandard deviaion of condiional sandard deviaion and Wald ess o guide hrough he specificaion search and choose he appropriae subse of spreads ha conribue he mos oward predicing he condiional volailiy. I refer o he 26
spreads beween he forward rae wih mauriy n and he shor rae ( n = 1 ) as he n-1 spread. Dropping he 5-1 spread resuls in a significan drop in he log-likelihood values. Whereas dropping he 3-1 spread resuls in no noiceable differences. The final reduced specificaion is (S-6) wih he shor rae and he 4-1 and 5-1 spreads. Deails of he specificaion search are in Appendix B. +1 =exp(α (n) + δ (n) +1 =exp(α(n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + β (n) 2,...,5 (f (2,...,5) f (1) )) (S-2) ) 2 + λ)+β (n) 1 f (1) + β (n) (4,5) 4,5 (f f (1) )) (S-6) Top row of Fig. 7 plos he coefficiens for he specificaion pair, (S), (S-6). The coefficien esimaes and he -raios for he condiional volailiy par of he specificaion pair are in Table 12. The shor rae is highly significan for all he mauriies and he -raios of he 5-1 spreads are beween a low of 1.28 o a high of 1.98. The coefficiens of he 4-1 spread are no significan. However, he coefficiens of he 4-1 and 5-1 spreads are joinly significan excep for n = 2 (significance levels for n =2, 3, 4, 5 are [34%, 8%, 2%, 0%], respecively). coefficiens for he condiional mean par of he specificaion pair (S), (S-6) are in appendix Table C-5. ln( +1) =α (n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + β (n) 4,5 (f (4,5) f (1) ) α (n) (α (n) ) δ (n) 1 (δ (n) 1 ) β (n) 1 (β (n) 1 ) β (n) 4 (β (n) 4 ) β (n) 5 (β (n) 5 ) n =2-10.05-19.22-0.04-1.03 18.50 5.01-7.56-0.56 13.59 1.28 n =3-9.23-20.94-0.06-1.94 20.40 5.48-3.10-0.23 20.01 1.77 n =4-8.21-17.10-0.01-0.29 19.92 5.62-0.07-0.01 20.75 1.88 n =5-7.95-17.80-0.02-0.43 21.48 6.34 1.87 0.14 21.36 1.93 Table 12: The specificaion pair: (S), (S-6). The coefficien esimaes and he -raios for he condiional volailiy par of he specificaion (S-6). The rx (n) +1 = κ (n) + γ (n) 1 f (1) + γ (n) 2,...,5 (f (2,...,5) 27 f (1) )+ɛ (n) +1 n =2,...,5 (S )
A higher level of overall ineres raes implies higher condiional volailiy. To exrac he effec of he level of ineres raes on he condiional mean I use (S ) for he condiional mean specificaion along wih (S-6) for he condiional volailiy specificaion. Specificaion (S ) capures he erm srucure informaion in he level and spreads. The shor rae proxies for he level of ineres raes in (S ). The boom row of plos in Fig. 7 corresponds o his combinaion. The coefficien on he shor rae is posiive and significan for all mauriies in (S ). The -raios (no included) indicae ha he spreads also capure significan amoun of he variaion in he condiional mean and he shor rae does no overwhelm he significance of spreads. Thus, he condiional volailiy and he condiional mean go in he same direcion wih respec o he overall level of ineres raes. Cochrane and Piazzesi (2005) find ha a single en-shaped facor of forward raes capures he risk premium (or he expeced excess reurns) associaed wih he one year reurns on bonds of 2 o 5-year mauriy. A similar en-shaped facor is eviden in he coefficiens of he forward raes in he condiional mean specificaion (S) (Fig. 7). They also noe ha he iny 4-5 year yield spread is imporan for forecasing all mauriy bond reurns. The high posiive coefficiens on he 5-1 spread in Fig. 7 indicae ha he 5-4 year yield spread (or he negaive of he 4-5 year yield spread) forecass higher condiional volailiy for he one year excess reurns of all mauriies. Thus he 4-5 year yield spread has opposing effecs on he condiional volailiy and condiional mean. However, he opposing effec of he 4-5 year yield spread does no ranslae ino a pronounced negaive correlaion beween he condiional volailiy and he condiional mean. This is because he variaion in he condiional volailiy is dominaed by he variaion in he level or he shor rae and no he spreads. Also he coefficiens on he 2-1 and 3-1 spreads are posiive in he condiional mean specificaion. The srong posiive correlaion beween he 5-1 spread and he 2-1, 3-1 spreads (0.85, 0.90 respecively) subdues a srong poenial negaive correlaion beween he drivers of he condiional mean and he condiional volailiy. The opposing effec of he 4-5 year yield spread on he condiional mean and he condiional 28
volailiy is in line wih he conclusion in Secion 4 ha he erm srucure drivers of he condiional mean and he condiional volailiy are differen. Mos of he variaion in he condiional volailiy is due o variaion in he overall level of ineres raes. However, he shor rae does no overwhelm he spreads conribuion o he variaion in he condiional mean. Thus, he drivers of he condiional mean and he condiional volailiy are differen. In fac, forcing he same coefficiens for he shor rae and he 4-1 and 5-1 spreads in he condiional mean and he condiional volailiy specificaions for each mauriy, n, resuls in a severe misfi o he daa. Similar o he resuls in Secion 4, he likelihood raio ess resoundingly rejec his resricion 3. Thus, he informaion in he erm srucure for he variaion in he Sharpe raios of 1-year reurns of bonds of 2- o 5-year mauriies is summarized by wo differen drivers. The variaion in he expeced excess reurns capures variaion in he marke price of risk or he correlaion wih he marke or boh, bu no he variaion in he condiional volailiy. 7 Effecs of change in macroeconomic volailiy The ineres rae daa sreches over 1964 o 2005. The US economy experienced a generally increasing shor-erm ineres rae and increasing inflaion unil mid-1981 and an equally declining shor erm rae and inflaion over he res of he sample period. L. R. Glosen and Runkle (1993) sugges ha he shor rae also reflecs aggregae economic uncerainy in addiion o he fuure expeced inflaion. Kim and Nelson (1999), McConnell and Perez-Quiros (2000) show ha he pos-1981 period is characerized by significanly lower real and nominal macroeconomic volailiy. I check wheher hese differences beween he pre- and pos-1981 periods affec he abiliy of he shor rae and he spreads o forecas he condiional volailiy. 3 Though he variaion in he condiional volailiy implied by he resriced specificaion pair drops, he condiional means implied by he resriced and unresriced specificaion pairs are he same. Resuls are similar o hose obained for he resriced specificaion pair (S), (S-1 ). 29
The common paern of he coefficiens of he shor rae and he 4-1, 5-1 spreads in Fig. 7 sugges ha a single linear combinaion of he shor rae and 4-1, 5-1 spreads capures mos of he variaion in he condiional volailiies due o he variaion in he erm srucure. However he overall level of he ineres raes dominaes and capures mos of he variaion in he condiional volailiies. Hence, I combine he 4-1 and 5-1 forward rae spreads ino a single facor and leave he level alone. I exrac he linear combinaion of he 4-1 and 5-1 spreads ha capures mos of he variaion in he condiional volailiies across mauriies due o he spreads. I summarize he informaion in he erm srucure for forecasing he condiional volailiy ino he shor rae (f (1) )andafacorx from he 4-1 and 5-1 spreads. This helps separaely quanify he effecs of lower macroeconomic volailiy in he pos-81 period on he abiliies of he shor rae and he spreads o forecas he condiional volailiy. +1 =exp(α(n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + β (n) (4,5) 4,5 (f f (1) )) (S-6) ln( +1) capures he par of he log condiional volailiies (ln( +1)) ha is due o variaion in he 4-1 and 5-1 spreads. ˆβ(n) 4,5 (n) (n) [ ˆβ 4, ˆβ 5 ] are he coefficiens of he 4-1 and 5-1 spreads from he maximum likelihood esimaion of he specificaion pair (S), (S-6). ln( (n) (4,5) +1 )= ˆβ 4,5 (f f (1) ) (n) = ˆβ 4 (f (4,5) f (1) )+ ˆβ (n) 5 (f (4,5) f (1) ) The linear combinaion x Γ 1 (f (4,5) f (1) )+Γ 2 (f (4,5) f (1) ) capures 99% of he variaion in he log condiional volailiies due o spreads, (ln( +1 )), across mauriies. Eigenvalue decomposiion of he covariance marix of he log condiional volailiies due o spreads [ln( h (2) +1),...,ln( h (5) +1)] T yields he dominan facor x 4. The coefficiens of he 4-1 and 5-1 spreads ha defines he linear combinaion x are [Γ 1, Γ 2 ] [ 1.92, 37.80]. 4 I sor he eigenvecors from he eigenvalue decomposiion of he covariance marix by heir corresponding eigenvalues. The elemens of he eigenvecor, q ln( h) 1, corresponding o he larges eigenvalue are he loadings of 30
ln( +1) =α (n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + φ (n) x (S-8) Wih he informaion abou spreads summarized in a single facor I fi he condiional volailiy specificaion (S-8) along wih he condiional mean specificaion (S). The parameer esimaes of he condiional volailiy specificaion are in Table 13. The coefficiens of he spread-facor x and heir significance increase wih mauriy. This reflecs an increasing imporance of he spreads for forecasing he condiional volailiy of 1-year excess reurns for higher mauriies. ln( +1 )=α(n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + φ (n) x α (n) (α (n) ) δ (n) 1 (δ (n) 1 ) β (n) 1 (β (n) 1 ) φ (n) (φ (n) ) n =2-10.12-20.35-0.04-1.08 19.05 5.77 0.22 1.58 n =3-9.25-22.68-0.06-2.00 20.55 5.95 0.49 3.09 n =4-8.20-17.87-0.01-0.28 19.85 5.97 0.57 3.65 n =5-7.93-18.17-0.02-0.42 21.32 6.52 0.63 4.01 Table 13: Specificaion pair: (S), (S-8). MLE esimaes of he parameers in he condiional volailiy specificaion (S-8) and he corresponding -raios. I modify specificaion (S-8) by inroducing an indicaor variable I. I spli he sample around he in-sample high of he shor rae. I is se o zero over he period January, 1964 o July, 1981 and se o one for he res of he period from Augus, 1981 o December, 2005. The condiional volailiy specificaion (S-9) includes he ineracion erms of he indicaor variable wih he shor rae and he spread-facor x. he mos dominan facor. The i h elemen of q ln( h) 1 is a measure of he change in ln( h (i) +1 ) per uni movemen in he dominan facor x. The coefficiens of he spreads, [Γ 1, Γ 2 ], in he linear combinaion ha defines x ˆβ (n) are he wighed sum of he esimaed coefficiens 4,5. The weighs are deermined by he elemens of he ( ) eigenvecor q ln( h) 1.[Γ 1, Γ 2 ] =[(ˆβ (2) 4,5 ) (5),...,( ˆβ 4,5 ) ] 2 4 q ln( h) 1 4 1 31
ln( +1) =α (n) + δ (n) ) 2 + λ)+β (n) 1 f (1) + β (n) I1 f (1) I + φ (n) x + φ (n) I x I, (S-9) where I =1 1981.08 2005.12 =0 1964.01 1981.07 ln( +1 )=α(n) + δ (n) β (n) 1 (β (n) 1 ) β (n) I1 ) 2 + λ)+β (n) 1 f (1) + β (n) I1 f (1) I + φ (n) x + φ (n) I x I (β (n) I1 ) φ(n) (φ (n) ) φ (n) I (φ (n) I ) n =2 18.51 1.84-1.36-0.06 0.77 1.94-0.81-0.60 n =3 18.47 3.49 1.83 0.61 0.88 2.79-0.63-1.98 n =4 17.45 4.08 2.57 1.04 0.87 3.24-0.53-1.74 n =5 18.25 4.16 3.42 1.34 0.92 3.42-0.53-1.73 Table 14: Specificaion pair: (S), (S-9). MLE esimaes of he parameers in he condiional volailiy specificaion (S-9) wih he indicaor variable I se o one over 1981:08-2005:12 and se o zero oherwise. Appendix Table C-9 presens all he coefficien esimaes. I only include he esimaes of β (n) 1,β (n) I1,φ(n) and φ (n) I in Table 14. The coefficiens of he shor rae ineracion erms are posiive (excep for n = 2),bu no saisically significan. Thus he coefficien esimaes do no sugges any significan change in he he shor rae s abiliy o forecas he condiional volailiy. The coefficiens of he ineracion erms of he spread-facor (φ (n) I ) are negaive and of he same magniude as he coefficiens of he spread-facor (φ (n) ). However, hese coefficiens are no significan and hus he spread-facor does no significanly lose is abiliy o forecas he condiional volailiy in he pos-1981 period. Thus, he abiliy of he shor rae and he spread-facor o forecas he condiional volailiy 32
is robus o he srucural shif in he macroeconomic volailiy beween he pre- and pos-1981 period. 8 Robusness Checks and ARCH/GARCH models In his secion I consider he condiional volailiy specificaion ha includes he lagged condiional volailiy as one of he explanaory variables. I also consider he ARCH/GARCH models of condiional volailiy. I find ha he lagged condiional volailiy does no significanly add o he abiliy o predic he fuure condiional volailiy. I also consider wheher he realized volailiy of daily bond reurns of fixed mauriy coupon bonds adds o he abiliy o predic he condiional volailiy. The realized volailiy drives ou he spread facor bu no he shor rae. +1 =exp(α(n) + δ (n) +1 =exp(α (n) + δ (n) ) 2 + λ)+η (n) 1 ln( ) 2 + λ)+β (n) 1,...,5 f (1,...,5) ) (S-1) )+β (n) 1,...,5 f (1,...,5) ) (S 1) +1 =exp(α(n) + δ (n) ) 2 + λ)) (S-0) +1 =exp(α (n) + δ (n) ) 2 + λ)+η (n) 1 ln( )) (S 0) The condiional volailiy specificaion (S 1) ness specificaion (S-1) and also uses he lagged condiional volailiy o forecas he condiional volailiy in he fuure. Similarly, (S 0) adds lagged condiional volailiy o he condiional volailiy specificaion (S-0). A comparison of (S 0) and (S 1) is similar o he comparison of (S-0) and (S-1) in Secion 4, overwhelmingly rejecing he zero coefficien resricions on he forward raes. The erm srucure informaion does improve he abiliy o forecas he condiional volailiy of 1-year excess reurns. Fig. 8 plos he condiional volailiies implied by (S 1) and (S 0) agains he absolue value unexpeced excess reurns ɛ (n) +1 for n =2and5. The lagged condiional volailiy, however, does no significanly add o he abiliy o predic he condiional volailiy in he fuure. The coefficiens of he lagged condiional volail- 33