Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] Modelling Volailiy of Shor-erm Ineres Raes in Kenya Tobias Olweny Deparmen of Commerce and Economics, JKUAT-Kenya Email: oolweny@yahoo.com Absrac There is an exensive heoreical and empirical lieraure ha documens he link beween shor-erm ineres rae volailiy and ineres rae levels. This sudy sough o esablish he link beween he level of ineres and he volailiy of ineres raes in Kenya using he Treasury bill raes from Augus 99 o December 007. The main variable for he sudy was he shor erm ineres rae series. In Kenya, his is he Cenral Bank hreemonh Treasury bill rae. The ineres rae volailiy was sudied using he general specificaion for he sochasic behavior of ineres raes which is esed in a Sochasic Differenial Equaion (SDE) for he insananeous risk free rae of ineres as earlier defined by Chan e al. (99). The sudy applied he monhly averages of he 9-day T-BILL rae for he period beween Augus 99 and December 007 which were obained from he Cenral Bank of Kenya. The resuls of he sudy were consisen wih he hypohesis ha he volailiy is posiively correlaed wih he level of he shor erm ineres rae as documened by previous empirical sudies. The key findings revealed ha here exiss a link beween he level of shor-erm ineres raes and volailiy of ineres raes in Kenya. Secondly, he sudy s key findings revealed ha he GARCH model is beer suied for modeling volailiy of shor raes in Kenya, as opposed o ARCH models. The sudy furher esablishes ha GARCH models are able o capure he very imporan volailiy clusering phenomena ha has been documened in many financial ime series, including shor-erm ineres raes. The sudy recommends fuure research o examine if oher forms of he GARCH process can produce similar resuls (i.e., EGARCH, PGARCH, GARCH, and FIGARCH). Keywords: Volailiy, ineres rae, Kenya.. Background o he Sudy Tradiional heories define ineres rae as he price of savings deermined by demand and supply of loanable funds. I is he rae a which savings are equal o invesmen assuming he exisence of a capial marke. The loanable fund heory argues ha ineres rae is deermined by non-moneary facors. I assigns no role o quaniy of money or level of income on savings, or o insiuional facors such as commercial banks and he governmen. The liquidiy heory, on he oher hand, looks a he ineres rae as he oken paid for absinence and inconveniences experienced for having o par wih an asse whose liquidiy is very high. I is a price ha equilibraes he desire o hold wealh in he form of cash wih he available quaniy of cash, and no a reward of savings. Ineres rae is a funcion of income. Is primary role is o help mobilize financial resources and ensure he efficien uilizaion of resources in he promoion of economic growh and developmen (Ngugi and Kabubo, 998). Shor-erm ineres raes are charges levied by he lenders o he borrowers on loans ha mus be paid wihin a year such as Treasury bills and credi card loans. The Shor Term Ineres Raes are imporan variables in many differen areas of he economic and financial heory. They are imporan in many financial economic models, such as models on he erm srucure of ineres raes, bond pricing models and derivaive securiy pricing models. They are also imporan in he developmen of ools for effecive risk managemen and in many empirical sudies analyzing erm premiums and yield curves where risk free shor-erm raes are aken as reference rae for oher ineres raes. Besides, hey are also a crucial feaure of he moneary ransmission mechanism. Duguay (994) describes he moneary ransmission mechanism as saring wih a moneary auhoriy s acions influencing shor-erm raes and he exchange rae, which hen go on o ulimaely affec aggregae demand of inflaion. In order o undersand he characerisics of he moneary ransmission mechanism, i is herefore imperaive o have a good model of he behaviour of shor-erm ineres raes. Empirical evidence documens a level effec in he volailiy of shor erm raes of ineres (Olan and Sandy, 005; Turan and Liuren, 005). Tha is, volailiy is posiively correlaed wih he level of he shor erm ineres rae. Using Mone-Carlo simulaions, Olan and Sandy (005) examined he performance of he Engle- Ng (993) ess which differeniae he effec of good and bad news on he predicabiliy of fuure shor rae volailiy. The shor-erm ineres raes being he US hree monh Treasury bills raes aken from he Federal Reserve Bank of S. Louis Economic daabase were sampled a a weekly frequency over he period of 5 h January 965 o 4 h November 003 yielding 07 observaions. 89
The Special Issue on Conemporary Issues in Social Science Cenre for Promoing Ideas, USA Their resuls esablished ha he ess exhibi serious size disorions and loss of power in he face of a negleced level effec. The endency for ineres raes o be more volaile as shor erm raes rise is wha is commonly referred o as level effecs. The dynamics of shor-erm reasury ineres raes are cenral o he pricing of all fixed income insrumens and heir derivaives. Chan, Karolyi, Longsaff and Sanders (99), hereafer CKLS compared a variey of single facor coninuous-ime models of he shor-erm risk-less rae over he period 964 hrough 989. They found ha models ha allow he volailiy of ineres changes o be sensiive o he level of he risk-free rae ouperform oher models. Longsaff and Schwarz (99) presened a wo-facor general equilibrium model, wih he level and condiional volailiy of shor-erm raes as facors. They showed ha a wo-facor model carries addiional informaion abou he erm srucure and leads o beer pricing and hedging performance compared o a single facor model, which only uses he level of he shor rae. The facors ha affec shor-erm ineres raes include: he moneary policy, he Governmen fiscal policy, axaion, inflaion, demand for capial, social values, and poliical rends. The moneary policy is used by he governmen o conrol he supply of money in he economy. When supply of money in he economy is low hen he ineres raes are expeced o be high and vice versa. The volailiy in money supply growh may lead o higher ineres raes. Under he fiscal policy, he Governmen is supposed o finance all expendiure for he economy. In cases where expendiure exceeds revenue (budge defici), he Governmen is forced o borrow from he local markes. This in urn affecs he supply of money in he economy which in urn affecs he rend of ineres raes. Inflaion on he oher hand causes long-erm ineres raes o rise where invesors sell-off heir bonds in fear of inflaion eroding heir capial gains. Demand for capial influences ineres raes when he demand/supply of funds is below or above he equilibrium levels. If here are fewer borrowers and he demand for funds is low hen he ineres raes will be low and vice versa. In Kenya, he ineres raes charged by banks are deermined by: ineres rae on deposis; cos of liquidiy; cos of holding cash; and operaional coss. The ineres on deposis depends on he bank s cash raio, is overall financial sabiliy and he ype of he bank for example wheher i is a corporae bank or a nework bank. The cos of liquidiy covers boh he cash, which is mainained by he banks wih Cenral Bank as required cash raio, and he cash mainained by he banks as he minimum amouns o mee unexpeced demand from he cusomers. Cos of holding cash is derived from he cash held by he banks in form of liquid form o mee day-o-day cusomer s needs. The banks have o compare he coss of cash ous and he opporuniy coss associaed wih he cash held in liquid form. Operaional coss are mainly mean o cover he coss of running he bank and i includes capial coss, saff coss, and echnology coss. The base rae charged by he banks akes ino accoun all hese facors. The bank can reduce he base rae by improving efficiency... Shor-erm ineres raes in Kenya Prior o he implemenaion of Srucural Adjusmen Programme (SAP) in 983, he financial secor in Kenya suffered from severe repression. Ineres raes were mainained below marke-clearing levels, and direc conrol of credi was he primary moneary conrol insrumen of he auhoriies. Accompanying he SAP, ineres rae deregulaion ook place. In Sepember 99 he maximum lending rae was increased from 0% o 4 %. The rediscouning rae for crop finance paper was raised o.5 %, while he minimum savings deposi rae was raised o.5 %. Beween 983 and 987, he differenials beween he ineres raes of banks and non-bank financial insiuions were narrowed. This improved he compeiiveness of commercial banks. One of he firs seps owards freeing ineres raes was aken in 989, when he governmen sared selling Treasury Bonds hrough an aucion. In July 99, ineres raes were compleely freed. Since hen, ineres raes have been following a seep upward ascen, wih he gap beween loan deposi raes shrinking (Naude, 995). Afer he liberalizaion period, ineres raes were liberalized and indirec moneary policy ools adoped.seps were aken o esablish financial markes, deconrol foreign exchange, liberalize rade and ighen prudenial regulaions. The role of he Cenral Bank was srenghened and moneary policy was ighened. All hese were accompanied by declining economic performance. From he financial repression heory, a major achievemen in he financial liberalizaion is he deconrol of ineres raes. This has a posiive impac on economic performance and also in indicaing he direcion he financial secor akes afer he liberalizaion process (Ngugi and Kabubo, 998). High real shor-erm ineres raes have reduced he demand for capial marke insrumens and crowded-ou subsanial domesic savings o shor-erm governmen securiies (Kibuhu, 005). This siuaion was paricularly eviden in 00 when he Treasury bill (T-bill) rae was.6% compared o an inflaion rae of 0.8%. However, he siuaion is being reversed as T-bill raes have fallen o abou 8% resuling in increased demand for boh equiy and deb insrumens (World Bank, 00). Ineres rae spreads are high and currenly sanding a abou 3%. 90
Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] Deposi raes are oo low and lending raes oo high hereby discouraging domesic savings and invesmen. The domesic savings are less han 0% of Gross Domesic Produc (GDP) and hereby insufficien o mee invesmen needs and generae demand for equiies and deb insrumens (World Bank, 00). Risk free ineres raes play a fundamenal role in finance. Theoreical models of ineres raes are of ineres boh for he pricing of ineres rae sensiive derivaive conracs and for he measuremen of ineres rae risk arising from holding porfolios of hese conracs. There is a vas lieraure focusing on modelling is dynamics. This sudy sough o specify a model for modelling volailiy of shor-erm ineres raes in Kenya..3. Term Srucure of Ineres Raes The erm srucure of ineres raes concerns he relaionship among he yields of defaul free zero coupon bonds ha differ only wih respec o mauriy. Hisorically four compeing heories of he erm srucure have araced aenion. These are expecaion, liquidiy preference, hedging pressure of preferred habia and segmenaion heories of he erm srucure of ineres. According o he expecaion heory, he shape of he yield can be explained by invesors expecaions abou fuure ineres raes. The liquidiy preference heory argues ha shor erm bonds are more desirable han long erm bonds because former are more liquid. The preferred habia heory explains he shape of he erm srucure by he assumpions ha if an invesor is risk averse, he can be emped ou of his preferred habias only wih he promise of a higher yield. Marke segmenaion heory assumes ha here are wo disinc markes for he shor and long erm bonds. The demand and supply in he long erm bond marke deermines he long erm yield and he demand and supply in he shor erm bond Marke deermines he shor rae. This means ha he expeced fuure raes have lile o do wih he shape of he yield curve. Over he pas few decades, heoreical developmen of modelling erm srucure dynamics has been mainly along he following wo direcions. One direcion is, while keeping a simple, racable, and parsimonious srucure, o exend he model hrough more flexible specificaion in order o beer describe he dynamics of sae variables and projec he erm srucure movemens. Developmen along his direcion is evidenced in various one-facor models (Meron, 973; Vasicek, 977; Dohan, 978; Brennan and Schwarz, 979; and Cox, 980). Cox e al. (985) defined he erm srucure of ineres raes as he measure of he relaionship among he yields on risk-free securiies ha differ only in heir erm o mauriy. The yield is a rae a which he presen value of all fuure paymens of ineres and principal is equaed o he marke price of he securiy. The yield curve is posiively sloped implying ha he yields of long-mauriy securiies are higher han he yields of shor-mauriy securiies. According o Liernman e al. (99), he volailiy of he shor-erm rae has wo couneracing effecs on he yield curve. Firsly, higher volailiy of he shor-erm ineres raes induces higher expeced raes for he longer mauriies (premium effecs). Secondly, higher volailiy of he shor-erm ineres raes increases he convexiy of he discoun facor funcion and, herefore, reduces he yields for he longer mauriies. According o Kimura (997), he erm srucure of ineres raes is he relaionship beween long-erm and shor-erm ineres raes. Tha is, i is he relaionship beween an ineres rae and he mauriy on securiy assuming ha economic fundamenals such as inflaion, unemploymen, poliical environmen remain unchanged. The erm srucure of ineres raes shows he relaionship beween ineres rae level and he erm o mauriy of a securiy..4 Dynamics of Shor-erm Ineres Raes One of he mos puzzling pieces of evidence on he erm srucure of ineres raes is he weak link beween he slope of he erm srucure and fuure changes in ineres raes (Campbell, 995). Mankiw and Miron (986) relaed his evidence o he acive argeing of ineres raes on he par of he Federal Reserve. They argued ha prior o he founding of he Federal Reserve Sysem; he slope of he erm srucure of ineres raes was a fairly accurae predicor of fuure changes in shor-erm raes. During his period, ineres raes were quickly mean-revering and highly seasonal, and herefore fairly easy o predic. In conras, since he Federal Reserve s incepion, he sabilizaion of ineres raes was so successful ha seasonal effecs and volailiy were grealy reduced (Mankiw, Miron and Well, 987), and ineres raes began behaving in a way similar o a random walk. An imporan implicaion of Mankiw and Miron s (986) analysis was ha, by argeing he overnigh-fed funds rae, he Federal Reserve effecively enjoyed a subsanial amoun of conrol over erm-fed fund raes and longer-erm yields. Goodfriend (99) suggesed ha he argeing of he overnigh-fed funds rae was implemened wih exacly his goal, since longer-erm raes were more srongly linked o macroeconomic goals such as unemploymen and inflaion. 9
The Special Issue on Conemporary Issues in Social Science Cenre for Promoing Ideas, USA The exising lieraure suggess ha a Federal Reserve policy enforcing smooh ineres raes was desirable o avoid whipsawing he bond marke (Goodfriend 99), o conain he variabiliy of he inflaion ax (Barro 989), and o sabilize he macro economy (Mankiw, Miron, and Weil 987). In heir sudy, Pierluigi, Giuseppe, Silverio and Leora (997) documened a new sylized fac concerning he relaion beween ineres rae argeing and he dynamics of shor-erm raes. They showed ha during he 989-996 period, he Federal Reserve was able o closely arge he overnigh-fed funds rae, and especially o reduce he persisence of is spreads from he arge: hese spreads averaged one basis poin, and exhibied an auocorrelaion of only 0.07, afer one day. Sill, erm-fed funds raes of mauriy up o hree monhs flucuaed widely and persisenly around he arge. For example, he volailiy of daily spreads of he hree-monh ermfed funds rae from he arge was 36 basis poin, and he auocorrelaion of hese spreads afer 60 days was sill 0.58. Boh he volailiy and he persisence of spreads of erm-fed funds raes from he arge were an increasing funcion of he mauriy of he loan. This new sylized fac can be inerpreed as evidence ha, while cenral bank inervenion is imporan in deermining he shape and posiion of he erm srucure, even a igh argeing of he overnigh-fed funds rae does no mechanically ranslae ino a igh conrol of longererm raes. Some of he early work on erm srucure models focused on radiional facor analysis. Lierman and Scheinkman (99) compued he principal componens of yield changes and found ha he firs hree principal componens explained abou 96% of he variaion in yields. They referred o he hree facors as level, slope, and curvaure. The level facor referred o a parallel shif in he yield curve, he slope facors referred o a seepening or flaening, and he curvaure facors referred o he wising beween inermediae erm and shor and long erm yields. The level-slope-curvaure facors were closely relaed o he laen facors ha had been used for affine erm srucure models. Raher han using observed sae variables, he sae variables were backed ou from he observable yields. This approach was laer used in coninuous ime by Dai and Philippon (005), and Dai and Singleon (00) among ohers. The laen facors used in affine erm srucure models behave essenially like he level, slope, and curvaure facors. The major drawback of his approach was ha he facors were no observable, and so hey did no lend hemselves o good forecasing mehods. They also did no provide any explanaion of how macroeconomic variables affec he erm srucure. To cover he anomalies idenified in he facor analysis approach, Taylor (993) and McCallum (994) focused on using moneary policy rules o describe he dynamics of he shor raes. These approaches have been very successful a describing moneary policy. However, hese models assume a simple relaionship beween he shor rae and he longer erm yields. As a resul, alhough he models describe shor raes very well, hey do no fi longer erm yields very well. Some more recen work in he macro lieraure has focused on incorporaing macroeconomic variables in he erm srucure model. Evans and Marshall (00) used a vecor auo regression (VAR) model of his form ha includes facors for GDP and inflaion. Their model, however, did no impose he resricions of no-arbirage. Even hough he model did a beer job of explaining he effecs of macroeconomic variables on he full erm srucure, he lack of no-arbirage resricions means ha he model was fundamenally missing ou on imporan aspecs of erm srucure dynamics. In heir sudy, Turan and Liuren (005) performed a comprehensive analysis of he shor-erm ineres-rae dynamics based on hree differen daa ses and wo flexible parameric specificaions. They applied generalized auoregressive condiional heeroskedasic (GARCH)-ype models wih non-normal innovaions o capure he poenial impac of ime-varying volailiy and disconinuous ineres rae movemens. Esimaes on boh ses of models based on he hree ineres-rae series were performed using he quasi-maximum likelihood esimaion mehod. They found ha non-lineariies were srong in he federal funds rae and he seven-day Eurodollar rae, bu were much weaker in he hree-monh Treasury yield. They obained similar findings when hey esimaed a wo-facor diffusion model wih sochasic volailiy. They concluded ha he conflicing evidence was parially due o he use of differen daa ses as a proxy for he shor rae and he use of differen parameric/ non-parameric specificaions under which empirical sudies perform he saisical ess..5 Dynamics of Ineres Raes in Kenya The Treasury bill raes were sable from January 983 o November 990 where he lowes rae recorded was.5 % and he highes was 5.79%. In December 990, he Treasury bill raes shoo up o 6.68% and o 7.9% in January 99 bu remained sable in 99 and 99. In March 993 he raes increased o 4.94% from 7.85% in February 993 bu shoo up drasically o 45.8% in April 993. In July 993 he rae was 84.60%, which was followed by a general decline reaching 3.37% in Sepember 994. The raes flucuaed in he range of 6.7% and 7.5 beween Ocober 994 and November 998. 9
Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] In 003 and 004, he raes drasically declined o a level of 0.83% in Sepember 003, bu in December he rae was 8.04%. Since January 005 o dae he raes have been flucuaing beween 6% and 9%. According o he Cenral Bank of Kenya (005), he sabiliy of shor erm ineres raes beween 8% and 9%, have been vial o he financial secor sabiliy and overall economic growh. The sabiliy of domesic ineres raes in Kenya has conribued o he predicable macroeconomic environmen for invesors and business people. This in urn has increased he level of confidence in he economy and has led o increased shor erm capial inflows. Willem (995) conduced a comparaive empirical sudy beween Ghana, Kenya, Zimbabwe and Nigeria. The sample comprised of four counries, wo of he counries wih he mos advanced financial sysems in Sub- Saharan Africa (Kenya and Zimbabwe), and wo counries where srucural adjusmen had been an ongoing process for more han a decade (Kenya and Ghana). Willem applied shor-erm (less han 3 monhs) deposi raes and long-erm deposi raes (longer han monhs) from each of he four counries. The empirical findings from he sampled counries esablished ha: (i) lending raes iniially adjused more slowly han deposi raes, creaing iniial periods during which he gap beween lending and deposi raes narrowed, and even became negaive in he case of Zimbabwe, and (ii) he level and volailiy of ineres raes increased afer liberalizaion. In he Kenyan case, he sudy esablished ha ineres raes in Kenya have been fairly sable and ha a relaively consan gap had been mainained beween lending and deposi raes for mos of he period. However, i mus be borne in mind ha, alhough Kenya was one of he firs African counries o implemen a SAP, i was only in 99 ha full ineres rae liberalizaion ook place. Since hen, ineres raes have been following a seep upward ascen, wih he gap beween loan deposi raes shrinking afer ineres rae liberalizaion. Willem (995) furher revealed ha for he Kenyan case, only changes in conemporaneous shor-erm ineres raes seemed o have any effec on long-erm ineres raes, bu he value of his parameer was smaller han (0.69) which suggesed a less han perfec correspondence beween shor and long raes. Furhermore, he accepance ha lags of shor-erm ineres raes were insignifican, suggesed ha long-run ineres raes do no adjus sluggishly o shor-erm raes..6. Modeling sensiiviy of volailiy o he level of shor-erm ineres raes This secion discusses he basic ypes of models ha have been used o explain shor-erm ineres rae dynamics. The firs ype of model is he diffusion model ha is predominanly used in building erm srucure models. The second ype of model is he Auoregressive condiional heeroskedasiciy (ARCH) model ha has proven useful in modeling he dynamics of he second momen of many financial ime series. The hird model is an exension of he basic diffusion model which allows for sochasic volailiy..6.. Diffusion Models Mos erm srucure models assume ha shor-erm ineres raes evolve over ime as some ype of diffusion process. The beauy of he diffusion model is ha he insananeous change in he shor rae can be characerized as a sochasic differenial equaion (SDE hereafer) and Iˆo calculus can hen be uilized o characerize he erm srucure. This basic approach is used in boh he arbirage pricing and general equilibrium approaches o pricing he erm srucure. Chan e al. (99) (CKLS hereafer) provided a general framework for modeling ineres rae processes. They described ineres rae volailiy using he general specificaion for he sochasic behavior of ineres raes. They assered ha he single-facor diffusion processes o be sudied can be nesed in he following Sochasic Differenial Equaion (SDE) for he insananeous risk free rae of ineres r represened by equaion (): dr = ( α + βr ) d + σr dz () Where: dz denoes he sandard Wiener process or Brownian moion σr is he insananeous sandard deviaion of ineres rae changes which is ofen referred o as volailiy The dependence of he insananeous sandard deviaion on r is known as he levels effec. The drif componen of shor erm ineres raes is capured by α + βr while he variance of unexpeced changes in ineres raes equalsσ r. While σ is a scale facor, he parameer conrols he degree o which he ineres rae level influences he volailiy of shor erm ineres raes. A of.0 indicaes ha he volailiy of he ineres rae is independen of is level and a above uniy indicae ha he volailiy rises wih he level of ineres raes. 93
The Special Issue on Conemporary Issues in Social Science Cenre for Promoing Ideas, USA In equaion (), dz is he single facor driving he evoluion of he enire erm srucure. CKLS were concerned wih calibraing his general SDE economerically o evaluae he appropriaeness of hese compeing models for he shor rae. The exac funcional form of he shor rae SDE is of criical imporance for models of he erm srucure. For example, Vasicek (977) used an arbirage argumen o derive a parial differenial equaion for bond prices. His derivaion was sufficienly general o allow for any diffusion ype of SDE for he shor rae and hen proceeded o derive closed form bond process for he special case of an Ornsein- Uhlenbeck process for he shor rae. To empirically calibrae he general SDE, CKLS employed a simple discreizaion of equaion () o come up wih a calibraed equaion presened by equaion (): r = α + βr + σr ε. () Where: r is ineres rae a ime, r = r r is he change in he ineres rae during he period, and ε is a sandard normal random variable. They esimaed he parameers of his model by using he Generalized Mehod of Momens (GMM hereafer) esimaion echnique of Hansen (98). They found ou ha he shor rae is mean revering, and ha he elasiciy of volailiy parameer was.4999 (he sandard error was 0.59). The elasiciy parameer indicaes ha he volailiy of shor-erm ineres raes is explosive. Oher sudies includes he work of Broze, Scaille, and Zakoian (995) who used maximum likelihood based procedures and he indirec inference echnique of Gourieroux, Monfor, and Renaul (993) o accoun for he discreizaion bias which hey found o be very small. Anoher approach due o Ai Sahalia (996) esimaed he implied densiy of discree changes in he spo rae implied by various coninuous ime models, and compared hese wih he empirical disribuion of he discree changes in he spo rae..6.. GARCH Models The ARCH model was inroduced by Engle (98) and laer exended by Bollerslev (986), who developed he generalized ARCH, or GARCH model. In a GARCH (, ) model (equaion 3), he condiional mean and condiional variance of a ime series process are modelled simulaneously. r = α + βr + ε.. (3) Where he condiional volailiy of ε is given by equaion (4). 94 [ φ ] h E = ε. (4) and h = ω + θε + ψh (5) α, β, ω, θ and ψ are regression consans. r represens he ineres rae series. GARCH models are able o capure he very imporan volailiy clusering phenomena ha has been documened in many financial ime series, including shor-erm ineres raes (Bollerslev, Chou, and Kroner, 99), as well as heir lepokurosis. Noe ha in GARCH models he volailiy is a deerminisic funcion of lagged volailiy esimaes and lagged squared forecas errors. One problem wih GARCH models of he shorrae is ha he parameer esimaes sugges ha he volailiy process is explosive. Bollerslev (986) demonsraed ha he variance process is covariance saionary when p q + = j= α β < i i i j, p q α i= i + β j= j. In his case, i is usually assumed ha α, β 0 i j o ensure ha he condiional volailiy is nonnegaive, so i is usually considered for he cases where <. If his inequaliy is violaed, hen shocks o he volailiy process are regarded as persisen or explosive. If he sum of he coefficiens equals one, hen he process is ermed IGARCH (or inegraed GARCH). If he sum of coefficiens is sricly greaer han one, hen a shock o volailiy is explosive, and E[ ] = Lim φ. Parameer j h + j esimaes of GARCH (, ) models fied o shor-erm ineres raes indicae an explosive process for he condiional volailiy, or α + β >. For example, Gray (996) repored using weekly 30-day T-bill daa ha α + β =.0303, and Engle, Ng, and Rohschild (990) found ha α + β =.0096 for a porfolio of T-bills..6.3. Sochasic Volailiy Models The sochasic volailiy model allows log-volailiy o iself evolve sochasically over ime (Smih, 000). This is in direc conras wih he GARCH ype models which model volailiy as a deerminisic funcion of lagged squared forecas errors and lagged condiional volailiy. j
Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] The sochasic volailiy model is parsimonious and ye flexible, and has been successfully applied o a range of financial ime series including shor-erm ineres raes (Ball and Torous, 999); exchange raes (Harvey, Ruiz, and Shephard, 994); and sock prices (Sandmann and Koopman, 998). Mos sochasic volailiy models are se in discree ime. Ball and Torous (999) presened heir sochasic volailiy model as a simple exension of he discree ime diffusion models of he ype presened in equaion (). Their exension of equaion () is as shown in equaion (6): r = α + βr + σ r ε. (6) As he ime subscrip on σ in equaion (6) indicaes, he generalizaion employed allows he volailiy o be ime varying. The model allows log-volailiy o evolve sochasically as a simple firs-order auoregressive process represened in equaion (7): Logσ = ξ + κ log σ + η. (7) Where ξ and κ are regression consans while η iidn (0, σ n ). The disurbance erm η in (7) makes he process sochasic - he variance iself is subjec o random shocks. This process is parsimonious and able o capure ineresing dynamics. I can also be noed ha GARCH models can be derived as he discree ime limi of a coninuous ime sochasic volailiy model, bu ha he discree ime sochasic volailiy model here considered are more direc. One of he procedures available for esimaing sochasic volailiy models of his ype is he quasi-maximum likelihood procedure of Harvey, Ruiz, and Shephard (994). This approach uses a simple ransformaion of he residual in equaion (6) o wrie he sysem in sae-space form and hen applies he Kalman filer o recursively build up he likelihood funcion. The ransformaion is employed on he residual e = r α βr. Since e = σ r ε if he log of he squared residual is aken, a represenaion shown by equaion (8) is obained: log e = logσ + log r + logε.. (8) Equaion (8) can furher be simplified by inroducing some new noaion y = log e which is observable given he observed reurns and he parameers α and β; and x = logσ is he sae variable - logvolailiy. Using his noaion, he sysem of equaions can be re-wrien in sae-space form as shown by equaions (9) and (0): y = x + log r + logε. (9) x = ξ + κx + η. (0) The Kalman filer is an ieraive procedure ha forecass he sae variable one period ino he fuure by a linear projecion and hen updaes his forecas when he observaion on he variable y becomes available. If he disurbance erms are boh Gaussian, hen he linear projecion is also he condiional expecaion; and he condiional expecaion and is mean squared error are all ha is required o describe he condiional densiy. In his case, he Kalman filer enables he consrucion of he exac likelihood funcion, and hen full maximum likelihood esimaion. However, in his case he disurbance erm for he observaion equaion (9) is non-gaussian. In fac i is disribued as log-chi squared random variable wih one degree of freedom. Harvey, Ruiz, and Shephard (994) noed ha E[ logε ] =. 704 and [ log ] π Var ε =. They approximaed he observaion equaion disurbance erm by a normal random variable wih he same mean and variance as log e. The Kalman filering equaions and likelihood funcion are buil in wo seps. Sep I involves he forecasing of log-volailiy while sep II involves updaing of he forecass. Since he Gaussian densiy is used o approximae he rue densiy, his approach resuls in quasi-maximum likelihood parameer esimaes. The cenral limi heorem of Dunsmuir (979) is hen used o esablish he consisency and asympoic normaliy of he resuling parameer esimaes. 95
The Special Issue on Conemporary Issues in Social Science Cenre for Promoing Ideas, USA.0 Research Mehodology.. Concepual Model In GARCH and Sochasic modelling, he volailiy is regarded as a deerminisic funcion of lagged volailiy esimaes and lagged squared forecas errors. This implies ha for a shor-erm ineres rae process: r = ( ) f r and h = f ( h ) where r is he shor rae, and h is he condiional variance of he shor rae... Analyical Model Chan e al. (99) (CKLS) provided a general framework for modeling ineres rae processes. They described ineres rae volailiy using he general specificaion for he sochasic behaviour of ineres raes. They assered ha he single-facor diffusion processes o be sudied can be nesed in he following Sochasic Differenial Equaion (SDE) for he insananeous risk free rae of ineres r represened by equaion (): dr = α + βr d + σr () ( ) dz σ Where dz denoes he sandard Wiener process or Brownian moion and r is he insananeous sandard deviaion of ineres rae changes which is ofen referred o as volailiy. The dependence of he insananeous sandard deviaion on r is known as he levels effec. The r represens he level of he shor erm ineres rae. The drif componen of shor erm ineres raes is capured by α + βr while he variance of unexpeced changes in ineres raes equalsσ r. While σ is a scale facor, he parameer conrols he degree o which he ineres rae level influences he volailiy of shor erm ineres raes. A of.0 indicaes ha he volailiy of he ineres rae is independen of is level and a above uniy indicae ha he volailiy rises wih he level of ineres raes. The esimae β<0 if significan sugges ha he shor-erm rae is mean revering. Equaions (4) and (5) provide he condiional volailiy of he error erms..4. Diagnosic Tess.4.. Lagrange Muliplier (LM) Tes for Level Effecs and Asymmery In developing a es for he join null of asymmery and levels effecs an asymmeric GARCH model wih a level effec provides a naural saring poin given by he se of equaions in (): r = ε ε Ω ~ N(0, h ). () h = α 0 + αε + βh + br + α η Where β + α <, and β, α i, b > 0 for i = 0, and. If η = Min (0, ε ) hen negaive innovaions have a greaer iniial impac of magniude α + α on he volailiy of he shor rae change han a posiive innovaion of equal magniude which has iniial impac of size α. The level effec is capured by he dependence of he condiional volailiy of he shor rae change on he lagged shor rae level. Is persisence is governed by he parameers b and. Implicily he condiional mean of equaions under () is equivalen o r = α + βr + ε under he resricion α = β = 0. This resricion is consisen wih he evidence provided by Chan, Karolyi, Longsaff and Sanders (99), Longsaff and Schwarz (99), and Brenner, Harjes and Kroner (996). The null hypohesis o consider is ha of a symmeric GARCH (, ) while he alernaive is an asymmeric GARCH (, ) wih a level effec. This may be formulaed as follows H α 0 : = b = H : Eiher 0 α and/or b 0 Where α, α, and b are regression coefficiens derived from Equaion () above. Sequenial subsiuion for * h and a firs order Taylor series expansion abou o linearize he level effec erm in () yields 96
Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] h + = i= i= β α i 0 β φr i i * + ln r i= i α β + i= i α ε i β i + β η i h + i= β i br * i * ( ln r ) The null hypohesis of no level effec and no asymmery may be reformulaed as H 0 : b = φ = α = 0 where φ = b. Under he assumpion ha he residual ε is condiionally normally disribued, he Lagrange Muliplier es saisic LM ( * ) under he null hypohesis is approximaely disribued as a Chi-square wih hree degrees of freedom..4.. Likelihood Raio (LR) Tess The likelihood raio es (LRT) is a saisical es of he goodness-of-fi beween wo nesed models (Hanfeng, Jiahua and Kalbfleisch, 000). The LR ess was used o es for linear drif dynamics of he shor-erm raes. The form of he es as suggesed by is name, is he raio of wo likelihood funcions; he simpler model (s) has fewer parameers han he general (g) model. Asympoically, he es saisic is disribued as a chisquared random variable, wih degrees of freedom equal o he number of maximum lags beween he wo models. The es procedure is algebraically represened as shown in equaion (4). Ls θ LRT = log e.. (4) Lg θ Where LRT denoes he Likelihood Raio Tes Saisic, Log e denoes he naural logarihm, while L s and L g denoe he likelihood funcions from he simpler and he general models respecively..4.3. T-Tess The -es was used o es he hypohesis ha he regression coefficiens are significan o he respecive models. The es was performed a boh % and 5% levels of significance..5. Daa sources and Sample Empirical sudies on he dynamics of shor-raes have applied hree differen ineres-rae daa series namely: he federal funds rae (Conley e al., 997), he seven-day Eurodollar deposi rae (Hong and Li, 005; Jones, 003), and he hree-monh Treasury bill rae (Sanon, 997; Jiang, 998; Chapman & Pearson, 000; and Durham, 003). The shor erm ineres rae series in Kenya is he Cenral Bank hree-monh Treasury bill rae aken from he Cenral Bank of Kenya Daabase. The sudy applied he monhly averages of he 9-day T- BILL rae for he period beween Augus 99 and December 007. Prior o 983, he ineres raes used o be conrolled by he Governmen unil he implemenaion of SAP in 983. In July, 99, he ineres raes were fully liberalized. During his period, he facors influencing he ineres raes were mainly he Marke facors hence ideal for sudying he volailiy of he shor-erm ineres raes in Kenya. 3.0 Daa Analysis and Presenaion of Findings Figures 4. and 4. respecively presen he level and he differenced series of he monhly averages of he sample shor-erm raes used in he sudy. Visual inspecion of Figures 4. and 4. sugges ha he shor rae (i) was mos volaile beween January 993 and December 00 which includes he period of changes in he Kenyan moneary policies, (ii) ha he volailiy of he differenced series increases wih he level of he shor rae and (iii) ha he differenced series of he shor rae displays volailiy clusering. Volailiy clusering means ha he volailiy of he series varies over ime. Before performing he volailiy ess, he original series were ransformed ino saionary series and modelling was performed based on ransformed-saionary series. A special class of non-saionary process is he I() process (i.e. he process possessing a uni roo). An I() process may be ransformed o a saionary one by aking firs order differencing. This was achieved by employing he Augmened Dickey-Fuller (ADF) uni roo ess (Dickey and Fuller, 979) o check for saionariy for he T-BILL raes daa series. The null hypohesis, H 0 is ha r has uni roos while he alernaive hypohesis is ha r is inegraed of order zero, I (0). The hypohesis was esed a a criical level of 5% and %. (See Table 4.) i (3) 97
The Special Issue on Conemporary Issues in Social Science Cenre for Promoing Ideas, USA 4.. Time Series Properies of he Sample Shor-Term Raes Figure 4.: Level Form of Shor-erm raes Monhly Averages (Jan 99 June 008) 80 Monhly Averages of Shor raes 70 60 50 40 30 0 0 0 0 39 58 77 96 5 34 53 7 9 0 TIME Figure 4.: Differenced Series of Monhly Averages of Sample Shor-erm Raes (Jan 99 June 008) 5 0 Changes in T-Bill Average Raes 5 0 5 0-5 -0 Jan-97 Jan-96 Jan-95 Jan-94 Jan-93 Jan-9 Jan-9 Jan-03 Jan-0 Jan-0 Jan-00 Jan-99 Jan-98 Jan-08 Jan-07 Jan-06 Jan-05 Jan-04-5 Year Table 3.: Uni Roo Tes for he Sample Shor-Term Rae Variable ADF Criical Values (5%) Criical Values (%) Decision r -.387-3.45-4.04 Accep H 0 r -5.305-3.45-4.04 Rejec H 0 H 0 : r has uni roos The resuls of Table 4. were obained by lagging he variables once. The resuls also indicae ha he shorrae series was non-saionary a level form. This indicaed ha he series is an I() process and herefore differenced series was applied for modelling volailiy. The decision rule was based on rejecing H 0 : he series is non-saionary, if he ADF saisics are less han he criical values (Dickey and Fuller, 979). 98
Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] 3.. Modelling Volailiy of Shor-Term Raes 3... Lagrange Muliplier (LM) Tes for Level Effecs and Asymmery The residuals of he regressions of he differenced series were esed for level effecs using he ARCH Lagrange Muliplier (LM) es and he resuls are presened in Table 4. below. Table 4.: ARCH LM es for Level Effecs Lags Chi-square Criical Values Criical Values (p) saisic (5%) (%) d.f. Decision 88.783** 3.48 6.635 Rejec H 0 0.449** 5.99 9.0 Rejec H 0 3 04.740** 7.85.345 3 Rejec H 0 H 0 : no Level effecs vs. H : level effecs disurbance presen * Denoes significance a 5% criical level (P-values < 0.05) ** Denoes significance a % criical level (P-values < 0.0) The LM es was based on he null hypohesis ha he differenced series had no level effecs. The decision rule was based on rejecing he null hypohesis if he compued Chi-square saisics were greaer han criical values of a known chi-square disribuion a 95% and 99% levels of confidence. The findings are presened in Table 4.. The resuls shows ha he residuals developed for he T-BILL differenced shor rae had level effecs. Since he variance of he errors is no a consan, heeroscedasiciy exiss for he residuals of he shor-erm ineres rae. Thus, hough he serial correlaion es, (Breusch-Godfrey LM es for auocorrelaion, Table 4.3) show ha ARCH model is a good fi for implici yield on 9 day Treasury bill rae, he level effecs are presen and hence he model is no a good fi. The ess were based on procedures and decisions rules similar o hose of LM es above. Hence, i is necessary o develop a beer model o capure he ARCH level effecs in he shor-erm ineres rae series. Table 3.3: Breusch-Godfrey LM ess for auocorrelaion Lags (p) F-Saisic Criical values 95% 99% d.f. Decision 9.30** 3.84 6.64 (, 07) Rejec H 0 49.776** 3.00 4.6 (, 06) Rejec H 0 3 33.94**.6 3.78 (3, 05) Rejec H 0 H 0 : no serial correlaion Vs. H: Serial correlaion presen * Denoes significance a 5% criical level (P-values < 0.05) ** Denoes significance a % criical level (P-values < 0.0) 3... Modelling Volailiy Using ARCH/GARCH Models The objecive of modelling he sochasic volailiy underlying 9-day T-BILL rae changes in Kenya is o allow for deerminaion of beer forecasing models by players in he Kenyan financial markes. Empirical evidence indicaes ha parameers for he models shif over ime (Johnson and Sco, 999), herefore i is more appropriae o calculae model parameers from ime o ime. Accurae descripions of he shor erm disribuions would allow for developmen of improved forecasing models. In his sudy, he parameers of he GARCH (, ) and ARCH (, ) models were calculaed over he sample period, using maximum likelihood esimaion. The findings derived of he maximum likelihood esimaion are presened in Table 4.4 below. Table 3.4: Modelling shor-erm ineres raes using ARCH/GARCH Model (The variance equaion) Model Coefficien Value Z-Saisic P-values Decision ARCH (,) Consan 0.64067.74 0.08 Accep H 0 Lag () 0.69357 4.77** 0.000 Rejec H 0 Lag () 0.9307.48 0.40 Accep H 0 Lag (3) -0.437876 -.63** 0.008 Rejec H 0 GARCH (,) Consan 0.64067.74 0.08 Accep H 0 Lag () -0.85768 -.56 0.0 Accep H 0 Lag () 0.5983886 3.5** 0.000 Rejec H 0 Lag (3) 0.0868504. 0.7 Accep H 0 LR Saisic = -386. 564** Wald Chi-square Saisic (d.f. = ) = 7.43E+ ** H 0 : Value of Consans =0 vs. H : Oherwise * Denoes significance a 5% criical level (P-values < 0.05) ** Denoes significance a % criical level (P-values < 0.0) 99
The Special Issue on Conemporary Issues in Social Science 300 Cenre for Promoing Ideas, USA The findings of Table 3.4 above indicae ha he residuals of he wo models are in nonlinear form, ha is, hey have he volailiy clusering effec and his is indicaed by he significan coefficiens of he ARCH() and GARCH() erms in he variance equaion of he differenced 9 day Treasury bill rae. The sum of he significan coefficiens on he lagged squared error and lagged condiional variance is less han one in all he cases. The sum equals 0.5546 for he ARCH (,) model (equivalen o lag + lag 3 since lag is no significan) and 0.5983886 for he GARCH (,) model (equivalen o lag only since lag & lag 3 are no significan). This sum is close o uniy in he case of GARCH model indicaing ha shocks o he condiional variance will be highly persisen. A large sum of hese coefficiens implies ha a large posiive or a large negaive reurn will lead fuure forecass of he variance o be high for a proraced period. The variance inercep erm consan is very small (<) as expeced. 3..3. Likelihood Raio Tes The likelihood raio es (LRT) saisic presened in Table 4.5 indicae he significance of he goodness-of-fi beween he wo models as earlier idenified by Hanfeng, Jiahua and Kalbfleisch, (000). The form es represens he raio of wo likelihood funcions for boh he ARCH and GARCH series. Asympoically, he es saisic is disribued as a chi-squared random variable, wih degrees of freedom equal o he number of maximum lags beween he wo models. The es was based on he null hypohesis ha here was no goodnessof-fi beween he wo models. The decision rule was o rejec he null hypohesis if he absolue value of he compued saisic is greaer han he criical values a he designaed levels of significance. The null was hus rejeced hence implying ha here was significance of he goodness-of-fi beween he wo models a boh 95% and 99% levels of significance. Table 3.5: Likelihood Raio Tes (LRT) Criical values Number of lags LR Chi-square Saisic d.f. Decision 95% 99% 3-386. 564** 7.85.345 3 Rejec H 0 H 0 : no Goodness-of-fi beween he wo models vs. H : Oherwise * Denoes significance a 5% criical level (P-values < 0.05) ** Denoes significance a % criical level (P-values < 0.0) 3..4. ARCH Lagrange Muliplier Tes for Level Effecs The residual series obained from he esimaed GARCH models of Table 4.4 above were esed for level effecs o see if level effecs are capured well in he esimaed model. The findings are presened in Table 4.6 below. Table 3.6: ARCH LM es Residuals of he GARCH model Lags (p) F-Saisic d.f. Decision 0.0340 (, 07) Accep H 0 0.07686 (, 06) Accep H 0 3 0.40983 (3, 05) Accep H 0 H 0 : no ARCH level effecs presen vs. H : ARCH level effecs disurbance presen * Denoes significance a 5% criical level (P-values < 0.05) ** Denoes significance a % criical level (P-values < 0.0) The findings of Table 4.6 above indicae ha he ARCH effecs are no presen in he model esimaed afer aking ino accoun he GARCH erms. Thus, he GARCH model is beer han he ARCH model for modelling volailiy of shor-erm ineres raes. However, he GARCH models esimaed do no ake ino accoun he leverage effec and hence he E-GARCH models would be developed o es wheher asymmeric effecs are presen. 3..5. Summary The sudy idenifies ha he GARCH model is beer suied for modelling volailiy of shor raes in Kenya, as opposed o ARCH models. The general specificaion is herefore of he form of a Sochasic Differenial Equaion (SDE) for he insananeous risk free rae of ineres r represened by Equaion (5) below dr = α + βr d + σr.. (5) ( ) dz σ Where dz denoes he sandard Wiener process or Brownian moion and r is he insananeous sandard deviaion of ineres rae changes which is ofen referred o as volailiy. The drif componen of shor erm ineres raes is capured by α + βr where he resricion applied was ha β + α <, and β i, α i, > 0 for i = 0,, and 3. This resricion was found o be consisen wih he evidence provided by Chan, Karolyi, Longsaff and Sanders (99), Longsaff and Schwarz (99), and Brenner, Harjes and Kroner (996).
Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] 4.0 Conclusion The aim of his sudy was o develop a general specificaion ha can be used o model he sensiiviy of volailiy o he level of shor-erm ineres raes in Kenya. The following research quesions guided he sudy: Is here a link beween he level of shor-erm ineres raes and he volailiy of ineres raes in Kenya using he Treasury bills from Augus 99 o December 007. In answering his quesion, he sudy applied hisorical daa for he monhly (average) 9-day T-BILL raes which were obained from he Cenral Bank of Kenya. The key findings revealed ha here exiss a link beween he level of shor-erm ineres raes and volailiy of ineres raes in Kenya. Secondly, he sudy s key findings revealed ha he GARCH model is beer suied for modelling volailiy of shor raes in Kenya, as opposed o ARCH models.the resuls of he sudy were consisen wih he hypohesis ha he volailiy is posiively correlaed wih he level of he shor erm ineres rae as documened by previous empirical sudies (Olan and Sandy, 005; Turan and Liuren, 005). The key findings revealed ha here exiss a link beween he level of shor-erm ineres raes and volailiy of ineres raes in Kenya. Secondly, he sudy s key findings revealed ha he GARCH model is beer suied for modelling volailiy of shor raes in Kenya, as opposed o ARCH models. The GARCH model is a more general case han he ARCH model. In heir original form, a normal disribuion is assumed, wih a condiional variance ha changes over ime. For he ARCH model, he condiional variance changes over ime as a funcion of pas squared deviaions from he mean. The GARCH processes variance changes over ime as a funcion of pas squared deviaions from he mean and pas variances. Overall resuls demonsrae ha, alhough previous research indicaes ha volailiy clusering plays a role in ineres rae changes, i is no he primary facor generaing hese changes. GARCH models wih normaliy assumpions provide a beer descripion of exchange rae dynamics. Frequency disribuions show independence sill exiss in he daa afer removing he ARCH effecs. Likelihood raio ess indicae he significance of he goodnessof-fi beween he wo models as earlier idenified by Hanfeng, Jiahua and Kalbfleisch, (000). The sudy furher esablishes ha GARCH models are able o capure he very imporan volailiy clusering phenomena ha has been documened in many financial ime series, including shor-erm ineres raes (Bollerslev, Chou, and Kroner, 99), as well as heir lepokurosis. Noe ha in GARCH models he volailiy is a deerminisic funcion of lagged volailiy esimaes and lagged squared forecas errors. One problem wih GARCH models of he shor-rae is ha he parameer esimaes sugges ha he volailiy process is explosive. 4.. Furher Research Fuure research can examine if oher forms of he GARCH process can accoun for he independence found (i.e., EGARCH, PGARCH, GARCH, and FIGARCH). They should also be esed o deermine if hey are superior o he ARCH/GARCH specificaion in regard o modelling volailiy of shor-erm raes. Since all forms of he GARCH process are similar in form, focusing on volailiy clusering, i would be ineresing o see if hey are an improvemen. The sudy applied monhly observaions, as opposed o daily or weekly observaions. Therefore, furher research can be done using weekly daa on he 9-day T-BILL rae o ascerain if here would be any significan deviaions from he findings of his sudy. References Anderson, T., Lund, J., (997). Esimaing coninuous ime sochasic volailiy models of he shor-erm ineres raes. Journal of Economerics 77, 343-77. A ı Sahalia, Yacine, (996), Tesing Coninuous-Time Models of he Spo Ineres Rae, Review of Financial Sudies 9, 385 46. Ball, Clifford A., and Waler N. Torous, (999), The Sochasic Volailiy of Shor-erm Ineres Raes: Some Inernaional Evidence, Journal of Finance. Bollerslev, Tim, (986), Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics 3, 307 37. Bollerslev, Tim, Ray Y. Chou, and Kenneh F. Kroner, (99), ARCH Modeling in Finance: A Review of he Theory and Empirical Evidence, Journal of Economerics 5, 5 59. Brenner, R. J., Harjes, R., Kroner, K., (996). Anoher look a models of shor-erm ineres raes. Journal of Financial and Quaniaive Analysis 3, 85-07. Broze, Laurence, Oliver Scaille, and Jean-Michel Zako ı an, (995), Tesing for Coninuous-Time Models of he Shor-Term Ineres Rae, Journal of Empirical Finance, 99 3. Campbell, John Y. (995) Some Lessons from he Yield Curve. Journal of Economic Perspecives 9 (Summer), 3, 9-5. 30
The Special Issue on Conemporary Issues in Social Science Cenre for Promoing Ideas, USA Cenral Bank of Kenya (005) Banking Supervision Annual Repor 005 Nairobi Chan, K.C., G.A. Karolyi, F.A. Longsaff, and A.B. Sanders, (99), An empirical comparison of alernaive models of he shor-erm ineres rae, Journal of Finance 47, 09-7. Chapman, D.A., Pearson, N.D., (000) Is he shor rae drif acually nonlinear? Journal of Finance 55 (), 355 388. Conley, T.G., Hansen, L.P., Lumer, E.G. J., Scheinkman, J.A., (997) Shor-erm ineres raes as subordinaed diffusions. Review of Financial Sudies 0 (3), 55 577. Cox, J.C., (975), Noes on opion pricing I: consan elasiciy of variance diffusion," Working Paper, Sanford Universiy. Cox, John C., Ingersoll Jonahan E., and Ross Sephen A., (985) A Theory of he erm srucure of ineres raes, Economerica 53, 385-407. Dai, Q., and Philippon, T. (005) Fiscal Policy and he Term Srucure of Ineres Raes. Working Paper 574. NBER Working Paper Series Dai, Q., and Singleon K., 00. Expecaion puzzles, ime-varying risk premia and affine models of he erm srucure. Journal of Financial Economics, 63( ): pp.45-44. Dunsmuir, W., (979), A Cenral Limi Theorem for Parameer Esimaion in Saionary Vecor Time Series and is Applicaion o Models for a Signal Observed wih Noise, Annals of Saisics 7, 490 506. Durham, G. B., (00). Likelihood-based specificaion analysis of coninuous models of he shor erm ineres rae. Working Paper, Universiy of Iowa. Durham, G.B., (003) Likelihood-based specificaion analysis of coninuous-ime models of he shor erm ineres raes. Journal of Financial Economics 70 (3), 463 487. Engle, Rober F., (98), Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of U.K. Inflaion, Economerica 50, 987 008. Evans, C., and Marshall, D., (00). Economic deerminans of he nominal reasury yield curve. Working Paper, Federal Reserve Bank of Chicago. Goodfriend, Marvin (99) Ineres Raes and he Conduc of Moneary Policy. Carnagie Rocheser Series on Public Policy 34 (Spring 99), 7-30 Gourieroux, C., A. Monfor, and E. Renaul, (993), Indirec Inference, Journal of Applied Economerics 8, S85 S8. Hanfeng Chen, Jiahua Chen and John D. Kalbfleisch (000) A Modified Likelihood Raio Tes for Homogeneiy in he Finie Mixure Models Working Paper 000-0; Deparmen of Saisics and Acuarial Science, Universiy of Waerloo. Hansen, Lars Peer, (98), Large Sample Properies of Generalized Mehod of Momens esimaors, Economerica 50, 09 054. Harvey, Andrew, Esher Ruiz, and Neil Shephard, (994), Mulivariae sochasic variance models, Review of Economic Sudies 6, 47 64 Hong, Y., Li, H., (005). Nonparameric specificaion esing for coninuous-ime models wih applicaions o spo ineres raes. Review of Financial Sudies,8 (), 37 84. Jiang, G.J., (998) Nonparameric modeling of US ineres rae erm srucure dynamics and implicaions on he prices of derivaive securiies. Journal of Financial and Quaniaive Analysis 33 (4), 465 497. Jones, C.S., (003). Nonlinear mean reversion in he shor-erm ineres rae. Review of Financial Sudies 6 (3), 793 843. Kibuhu W. W. (005) Capial markes in emerging economies: A case sudy of he Nairobi Sock Exchange. A hesis Presened o he faculy of Law: The Flecher School of Law and Diplomacy Kimura J.H., (997), Ineres raes in Kenya IPAR/ICPAK seminar repor ; Nairobi Koedijk, K. G., Nissen, F. G. J. A., Scochman, P. C., Wolff, C. C. P., (997). The dynamics of shor-erm ineres rae volailiy reconsidered. European Finance Review, 05-30. Lierman, R.; J. Scheinkman; and L. Weiss (99) Volailiy and he Yield Curve Journal of Fixed Income, (99), 49-53. 30
Inernaional Journal of Business and Social Science Vol. No. 7; [Special Issue April 0] Longsaff, F.A and E.S. Schwarz, (99), Ineres rae volailiy and he erm srucure: A wo-facor general equilibrium model, Journal of Finance 47, 59-8. Mankiw, Gregory N., and Jeffrey A. Miron (986) The Changing Behavior of he Term Srucure of Ineres Raes. Quarerly Journal of Economics 0 (May 986), -8 McCallum, B. T. (994) Moneary Policy and he Term Srucure of Ineres Raes. Working Paper 4938. NBER Working Paper Series. Meron, R.C., (973), Theory of Raional Opion Pricing, Bell Journal of Economics and Managemen Science 4, 4-83. Ngugi R.W., and Kabubo J.W., (998) Financial secor reforms and ineres rae liberalizaion: The Kenya experience AERC Research Paper 7; African Economic Research Consorium, Nairobi. Olan T.H., and Sandy S., (005) Tesing for Asymmery in Ineres Rae Volailiy in he Presence of a Negleced Level Effec The Universiy of Melbourne & The Universiy of Queensland. Pierluigi Balduzzi, Giuseppe Berola, Silverio Foresi, Leora Kiapper (997) Ineres rae argeing and he dynamics of shor-erm raes; Cambridge: Naional Bureau of Economic Research Sandmann, Gleb, and Siem Jan Koopman, (998), Esimaion of Sochasic Volailiy Models via Mone Carlo Maximum Likelihood, Journal of Economerics 87, 7 30. Smih Daniel R, (000) Markov-Swiching and Sochasic Volailiy Diffusion Models of Shor-Term Ineres Raes Finance Division, Faculy of Commerce; Universiy of Briish Columbia Sanon, R., (997). A nonparameric model of erm srucure dynamics and he Marke price of ineres rae risk. Journal of Finance 5 (5), 973 00. Taylor, J. B. (993) Discreion verses policy rules in pracice. Carnegie-Rocheser Conference Series on Public Policy, 39( ): pp.95-4. Tse, Y.K. (995), Some inernaional Evidence on he Sochasic Behaviour of Ineres Raes, Journal of Inernal Money and Finance, 4(5), pp.7-738. Turan G. Bali, Liuren Wu, (005) A Comprehensive Analysis of he Shor-Term Ineres Rae Dynamics; New York: Baruch College, Zicklin School of Business, One Bernard Baruch Way, New York. Vasicek, Oldrich, (977), An Equilibrium Characerizaion of he Term Srucure, Journal of Financial Economics 5, 77 88. Willem Naudé (995) Financial Liberalizaion and Ineres Rae Risk Managemen in Sub-Saharan Africa; Oxford: Cenre for he Sudy of African Economies, Insiue of Economics and Saisics, Universiy of Oxford. World Bank (00). Capial Marke Inegraion in he Eas African Communiy. Washingon, DC: World Bank. 303