Fractional integration and the volatility of UK interest rates

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1 DEPARTMENT OF ECONOMICS Fracional inegraion and he volailiy of UK ineres raes Simeon Coleman, Noingham Tren Universiy, UK Kavia Sirichand, Universiy of Leiceser, UK Working Paper No. 11/29 May 2011

2 Fracional inegraion and he volailiy of UK ineres raes Simeon Coleman a and Kavia Sirichand b a Economics Division, Noingham Business School, Noingham Tren Universiy, Buron Sree, Noingham, NG1 4BU, UK. simeon.coleman@nu.ac.uk b Deparmen of Economics, Universiy of Leiceser, Universiy Road, Leiceser, LE17RH, UK. ks62@le.ac.uk. Absrac Using fracional inegraion and GARCH modeling echniques, his paper invesigaes he dynamic properies of UK ineres raes. We find evidence ha, conrary o previous sudies for he US and Canada, shor raes are more nonsaionary compared o longer raes. Furher, differences in condiional volailiy exis beween raes of differen mauriies. We posi ha he dynamics of ineres raes may be boh mauriy-specific and counry-specific and any a priori generalizing assumpions may be misleading. Keywords: fracional inegraion, ineres raes, condiional volailiy JEL Codes: C22, E43 1. Inroducion The imporance of ineres raes in finance and economics is well esablished, hey are fundamenal in moneary policy formulaion and o invesmen decision making. As such, i is essenial o have a horough undersanding of he dynamic behavior of ineres raes. Empirically, ineres raes are found o exhibi I(1) behavior, see Campbell and Shiller (1991), Cuhberson e al. (2000) and Mishkin (1992). However, Tkacz (2001) and Lopes and Moneiro (2007) highligh ha he heoreical implicaion of ineres raes following a uni-roo process wihou drif is ha here are no bounds on is movemens, suggesing he possibiliy of negaive nominal raes. A furher implicaion is ha shocks have a permanen effec. In more recen examinaions of real raes, Tsay (2000), Tsakz (2001), Candelon and Gil-Alana (2006), and Karanasos e al. (2006) conclude in favor of fracional inegraion mehods as opposed o he knife-edge I(0)/I(1) approach. Improved knowledge of ineres rae dynamics is crucial for modeling and forecasing. Firs, heir dynamics are cenral o he valuaion of financial asses and he sudy of well-known macroeconomic We would like o hank Barry Harrison, Sephen Hall and Vior Leone for he insrucive commens hey provided on an earlier version of he paper. Corresponding auhor Simeon Coleman, simeon.coleman@nu.ac.uk, el.: and fax:

3 models. 1 Second, he flexibiliy and recen developmens in he use of long memory ess wih good size and power can provide beer insighs. Third, much of he exising empirical evidence, concerning he order of inegraion, focuses on US ex-ane and ex-pos raes. Despie is imporance, and he subsanial lieraure devoed o he opic, he jury is sill ou on he order of inegraion of ineres raes. In his paper, our empirical analysis has wo aims: Firs, o examine he order of inegraion of he shor and long raes over wo periods, defined as Pre (5/03/ /03/2007) aken as he pre-crisis period and Pos (4/04/ /07/2010) which includes he crisis period, o deermine if idiosyncraic differences exis beween he mauriies and across periods. We address his aim by employing fracional inegraion (FI) ess, which are more flexible compared o sandard uni-roo ess, esing he validiy of he common pracice of imposing I(0)/I(1) condiions. 2 Second, o invesigae volailiy in hese raes over he saed periods, o see which, if any, rae exhibis higher volailiy and if his volailiy changed over he periods. This aim is addressed by employing GARCH modeling echniques, allowing us o commen on asymmeries in condiional volailiy around heir means. The remainder of he paper is organized as follows. Secion 2 summarizes he economeric echniques and he main resuls, and Secion 3 concludes. 2. Economeric mehodology and resuls 2.1. Daa We use weekly observaions of UK risk-free discoun bonds for mauriies of 6, 12 and 120 monhs (hereafer r6, r12 and r120) beween 5/03/ /07/ Visual inspecion of r6, r12 and r120, (Figure 1) suggess significan co-movemen in he raes unil mid-2008, afer which r6 and r12 dipped significanly. See Table 1 for summary saisics. 1 For example he Fisher hypohesis and he uncovered ineres rae pariy hypohesis. 2 For an I(0) process shocks decay a an exponenial rae; for an I(1) process shocks have permanen effec and for an I(d) process shocks dissipae a a slow hyperbolic rae. 3 We use official Bank of England (BoE) daa on he Governmen liabiliy curve. Specifically, Wednesday observaions of nominal governmen spo raes are employed, where he yields are coninuously compounded.

4 12/03/ /03/ /03/ /03/ /03/ /03/ /03/ /03/ /03/ /03/ /03/ /03/ /03/ /03/2010 Figure 1: Time-series plos of ineres rae series March 97-July r6 r12 r % Dae Table 1: Summary Saisics r6 r12 r120 Pre Pos Pre Pos Pre Pos Mean Maximum Minimum Sd. Dev Observaions Uni roo ess To ascerain which raes can be beer explained by sochasic processes or deerminisic facors, we conduc Augmened Dickey-Fuller (1979) [ADF], Kwiakowski e al. (1992) [KPSS], and Ng and Perron (2001) [NP] uni roo ess. 4 Under he ADF es he null of a uni roo for r6 and r12 canno be rejeced for boh Pre and Pos periods; however, i is rejeced for r120 in he Pre bu no he Pos period. Under he KPSS es he null of saionariy is unambiguously rejeced and under he NP es he null of a uni roo canno be rejeced, in each series over boh periods. The inconclusive resuls (see Table 2), paricularly for r120, underscore he need o go beyond he I(1)/I(0) framework, making FI ess insrucive. 4 The NP es combines a Modified Informaion Crierion for he lag lengh and a Generalized Leas Squares mehod for derending he daa, i proposes four es saisics: MZ a, MZ, MSB and he MPT. In addiion o he convenional ADF, KPSS and NP individual uni roo ess, we also conduc hree panel uni roo ess, namely he Levin e al. (2002), Im e al. (2003) and ADF-Fisher Chi Square ess. Similarly, hese resuls of hese ess are inconclusive and are no repored here, bu are available upon reques.

5 Table 2: Uni roo ess r6 r12 r120 Pre Pos Pre Pos Pre Pos ADF ** KPSS 1.638* 1.491* 1.706* 1.486* 1.659* 1.161* NP GLS MZ GLS MZ GLS MSB GLS MP T Noes: ***, ** and * indicaes null rejeced a 10%, 5% and 1% respecively. The criical values for he ADF es criical are , and , and for he KPSS es are 0.216, and a he 10%, 5% and 1% levels respecively. The Ng and Perron ess include an inercep and he lag order was chosen using he modified AIC (MAIC). Fracile 1% % % Fracional inegraion ess Long memory in macroeconomic variables is well esablished. 5 Granger and Joyeux (1980) and Hosking (1981) showed ha a long memory process for y can be modeled as a fracionally inegraed, I(d), process d ( 1 L ) ( y ) (1) where L denoes he lag operaor, d is fracional difference parameer, μ is he uncondiional mean of y, and ε is saionary wih zero mean and finie variance. A flexible parameric process of order (p,d,q) called he ARFIMA(p,d,q) model incorporaes boh long-erm and shor-erm memory. d ( L )(1 L ) ( y ) ( L ) (2) where Φ(L) and φ(l) are auoregressive and moving average polynomials, respecively, wih roos ha lie ouside he uni circle and ε is Gaussian whie noise. y is saionary provided dє(-0.5,0.5); however, is lagged auocovariance decreases very slowly exhibiing long memory, see Table See examples Baum e al. (1999a, 1999b) and references cied herein. 6 y is inverible when d >-0.5. For deailed discussions of long memory esing and esimaion mehods, we refer he ineresed reader o Baillie (1996) and Baum e al. (1999a, 1999b).

6 Table 3: Summary of fracional inegraion parameer values d Variance Shock duraion Saionariy d=0 Finie Shor-lived Saionary 0<d<0.5 Finie Long-lived Saionary 0.5 d<1 Infinie Long-lived Nonsaionary d =1 Infinie Infinie Nonsaionary d>1 Infinie Infinie Nonsaionary Source: Tkacz (2001) We repor in Table 4, he Modified Log-Periodogram Regression esimaor proposed by Phillips (1999a, 1999b), which requires a choice of he number of harmonic ordinaes o include in he specral regression. For robusness, we use a range of powers ( ). 7 Table 4: Modified Log-Periodogram Regression esimaor [Phillips (1999a, 1999b) procedure] Fracional Inegraion es saisic (Modlpr) power Pre Pos Pre Pos Pre Pos Pre Pos r6 1.2*,^^^ 1.4*,^^ 1.4*,^ 1.5*,^ 1.3*,^ 1.3*,^^ 1.2*,^^^ 1.3*,^^ r12 1.2* 1.6*,^ 1.3*^^ 1.5*,^ 1.2*,^ 1.4*,^ 1.1* 1.3*,^^ r * 0.8* 0.9* 1.0* 0.8* 1.1* 0.9* 1.1* Noes: *, ** implies rejecion of he null d=0 a he 1%, 5% levels respecively; ^,^^,^^^ implies rejecion of he null of d=1 a he 1%, 5% and 10% levels respecively. Here, he null of d=0 is consisenly rejeced a all power levels in boh periods. Some furher observaions are noeworhy. Firs, under he Phillips es, which also provides a z-saisic o deermine wheher d is significanly differen from 1, for boh r6 and r12 (unlike r120), here is a high endency o rejec he null of d=1 across powers, suggesing ha d>1 i.e. explosive behavior. 8 This conrass sharply o he findings of Tkacz (2001) for he USA and Canada, who finds shorer raes o be less nonsaionary han longer raes. We posi ha since UK shor raes end o remain a he same level for prolonged periods before changing, i is likely ha hese sepwise movemens may be misinerpreed as srucural breaks by he FI ess. Second, he d esimaes for he Pos period appear o be larger han ha for he Pre period, suggesing higher endency of non mean-reversion. This resul can be explained by he fac ha he Pos period includes he curren financial crisis, and his 7 A desirable propery of his procedure is ha he dependen variable is modified o reflec he disribuion of d under he null hypohesis ha d=1. The esimaor gives rise o a es saisic for d=1 which is a sandard normal variae under he null. The regression slope esimae is an esimae of he slope of he series power specrum in he viciniy of he zero frequency; if oo few ordinaes are included, he slope is calculaed from a small sample. If oo many are included, medium and high-frequency componens of he specrum will conaminae he esimae. 8 As a robusness check, FI was also esed using wo oher widely used procedures suggesed by Geweke and Porer-Hudak (1983) and Robinson (1995) in STATA11. The resuls unequivocally confirm he rejecion of he null of d=0 for each series. In addiion, auocorrelaion funcions for each series (no shown here) confirm ha he decay in r120 (approximaely 60weeks) is faser han in r6 and r12 (approximaely 160 weeks).

7 sample period is no long enough for mean reversion o be observed i.e. ineres raes have no sared adjusing ye and have no begun revering o heir respecive means. 2.4 Volailiy esing We firs experimened wih differen combinaions of model orders and found ha a GARCH(1,1) model provided he bes fi for our series. 9 In he conex of ineres raes, our aim is o capure he series variance dependence (if any) on a weighed average of he long erm average of he series, news abou volailiy from he previous period ( ) and las period s forecas variance (β) respecively. Table 5 summarizes our resuls. 10 Table 5: Tess for Volailiy Pre Pos GARCH(1,1) IGARCH GARCH(1,1) IGARCH r6 c 0.004* 0.040* GARCH(-1) * ** * Resid(-1) * 0.691* 1.326* 3.743* r12 c GARCH(-1) *** 0.395* 0.574* * Resid(-1) * 0.605* 0.386* 3.244* r120 c 0.008* 0.003*** GARCH(-1) *** 0.545* 0.345* 0.523* Resid(-1) * 0.455* 0.709* 0.477* Noes: *, **, *** implies null of no significance rejeced a 1%,5% and 10% respecively. Based on he FI ess, he preferred model s resul is in bold fon. Following he FI esimaes (Table 4), where he null of d=1 is rejeced, he GARCH(1,1) model is applied. In which case, +β<1 infers mean-reversion, and +β>1, explosive behavior. Where he null canno be rejeced for a given series, an IGARCH model, which resrics +β=1 is employed. Therefore, we model r6 and r12 using GARCH(1,1) and r120 using an IGARCH model. For he shor raes, here is higher dependence of curren volailiy on he previous period s volailiy and unsurprisingly i is higher in he Pos period, his suggess a higher endency for 9 Noably, he condiional variance ( ) is he one-period ahead forecas variance based on pas informaion a consan erm (ø) and he ARCH erm ( ) and GARCH erm ( ). 10 We allow for he possibiliy ha residuals are no condiionally normally disribued, by compuing he Heeroskedasiciy Consisen Covariance.

8 explosive behavior in shor raes. However, for he long rae here appears o be consisency in volailiy over boh periods. 3. Concluding remarks This paper conribues o he debae on he order of inegraion of nominal ineres raes by analyzing raes wih differing erms o mauriy. We show ha he dynamic properies of shor and long raes are inherenly differen. Firs, conrary o previous sudies for he US and Canada, our resuls sugges ha, in he UK, shorer raes are more nonsaionary han long raes. Second, using GARCH echniques o measure uncerainy, we find ha volailiy in he shor raes end o be more dependen on news abou volailiy from he previous period; whereas he long rae ends o be fairly equally dependen on he level of, and news abou volailiy of he previous period. The level of volailiy in he Pos period appears o be more relevan han in he Pre period. In conclusion, erm o mauriy and origin counry appear o be imporan facors for he order of inegraion of ineres raes, so a priori generalizing assumpions abou he order of inegraion of ineres raes may be misleading. References Baillie, R.T., Long memory processes and fracional inegraion in economerics. Journal of Economerics. 73, Baum, C.F., Barkoulas, J.T., Caglayan, M., 1999a. Persisence in inernaional inflaion raes. Souhern Economic Journal. 65, Baum C.F., Barkoulas, J.T., Caglayan, M., 1999b. Fracional moneary dynamics. Applied Economics. 31, Campbell, J., Shiller, R. J., Yield spreads and ineres rae movemens: a bird s eye view. Review of Economic Sudies. 58, Candelon, B., Gil-Alana, L. A., Mean reversion of shor run ineres raes in emerging counries. Review of Inernaional Economics. 14, Cuhberson, K., Hayes, S., Nizsche, D., Are German money marke raes well behaved? Journal of Economic Dynamics and Conrol. 24, Dickey, D.A., Fuller W.A., Disribuion of he esimaors for auoregressive ime series wih a uni roo. Journal of he American Saisical Associaion. 74,

9 Geweke, J., Porer-Hudak, S., The esimaion and applicaion of long memory models. Journal of Time Series Analysis. 4, Granger, C.W.J., Joyeux, R., An inroducion o long memory ime series models and fracional differencing. Journal of Time Series Analysis. 1, Hosking, J.R.M., Fracional differencing. Biomerika. 68, Im, K.S., Pesaran, M.H., Shin, Y., Tesing for uni roos in heerogeneous panels. Journal of Economerics. 115, Karanasos, M., Sekioua, S. H. and Zeng, N., On he order of Inegraion of monhly US ex-ane and ex-pos real ineres raes: New evidence from over a cenury of daa. Economics Leers. 90, Levin, A., Lin, C.F., Chu, C., Uni roo ess in panel daa: asympoic and finie sample properies. Journal of Economerics. 108, Lopes, A. C., Moneiro, O. S., The expecaions hypohesis of he erm srucure: some empirical evidence for Porugal. MPRA Paper No Mishkin, F. S., Is he Fisher effec for real? a reexaminaion of he relaionship beween inflaion and ineres raes. Journal of Moneary Economics. 30, Ng, S., Perron, P., Lag lengh selecion and he consrucion of uni roo ess wih good size and power. Economerica. 69, Phillips, P. C. B., 1999a. Discree fourier ransforms of fracional processes. Unpublished working paper No. 1243, Cowles Foundaion for Research in Economics, Yale Universiy. Phillips, P. C. B., 1999b. Uni roo log periodogram regression. Unpublished working paper No. 1244, Cowles Foundaion for Research in Economics, Yale Universiy. Robinson, P.M., Log-periodogram regression of ime series wih long range dependence. Annals of Saisics. 23, Tkacz, G., Esimaing he fracional order of inegraion of ineres raes using wavele OLS esimaor. Sudies in Nonlinear Dynamics and Economerics. 5, Tsay, W. J. (2000). The long memory sory of he real ineres rae. Economics Leers. 67,