ARCH IN SHORT-TERM INTEREST RATES: CASE STUDY USA

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1 Arch in Shor-Term Ineres Raes: Case Sudy USA ARCH IN SHORT-TERM INTEREST RATES: CASE STUDY USA Adrian Ausin, Universiy of Wes Georgia Swarna (Bashu) Du, Universiy of Wes Georgia ABSTRACT We invesigae ARCH effecs in shor erm ineres raes. Many of he models used o sudy ineres rae dynamics posi a linear funcion for he condiional mean. Recen work has shown ha here are significan non-lineariies in he srucure of ineres raes. We use a neural nework model o capure he non-lineariies. We find ha he evidence for ARCH in shor-erm ineres raes is somewha oversaed. There is some evidence of ARCH effecs, bu he persisence is no as long as prior esimaes have indicaed. JEL Classificaion: F40 INTRODUCTION The volailiy of he shor rae is a key elemen in he valuaion of ineres sensiive coningen claims. Thus considerable aenion has been paid o characerizing he daa generaing process of his rae. The fac ha large changes in a ime series end o be followed by large changes and small changes end o be followed by small changes (volailiy persisence) has been noed in he lieraure since Mandelbro (1963). There has been an explosion in volailiy modeling in he las 5 years since he seminal papers by Engle (198) and Bollerslev (1986) who inroduced he AuoRegressive Condiional Heeroskedasiciy (ARCH, henceforh), and Generalized AuoRegressive Condiional Heeroskedasiciy (GARCH, henceforh) models respecively. In he volumes of work since hen, here exiss significan evidence ha volailiy also appears o conain long-memory characerisics, he paern of which seems o be a rapid decrease a firs, and hen a hyperbolic decrease for many periods. This has led o he developmen of fracionally inegraed GARCH model (FIGARCH) of Baillie, Bollerslev and Mikkelsen (1996) and long memory sochasic volailiy models (Breid, Crao and DeLima 1998). There has also been a lo of recen ineres in non-linear modeling of he condiional mean and variance (Ai-Sahalia 1996, Tauchen 1997, ec). Non-lineariies in he underlying ime series may lead o over rejecion of he no ARCH hypohesis and may be he cause of some of he evidence for long-memory. We use he mehod described by Blake and Kapeanios (007) o apply nonparameric modeling using neural neworks o capure possible non-lineariies in he daa generaing process. Having modeled he condiional mean, we can hen apply any convenional ess for ARCH o invesigae he remaining condiional heeroskedasiciy. These models are inended o capure he persisence in volailiy, which seems o be a salien feaure in financial markes. 15

2 Souhwesern Economic Review The paper proceeds as follows. Secion II shows he naure of he problem. Secion III inroduces he Neural Nework mehod of modeling flexible funcional forms, followed by he esimaion and resuls in Secion IV. Secion V concludes he sudy. ARCH AND NON-LINEARITIES Volailiy persisence has long been known o be a feaure of financial ime series. Since Engle s seminal paper (Engle198) here have been a proliferaion of models in he ARCH/GARCH line, and oday i is a sandard feaure of any ime series and financial sofware. The esimaes from hese models end o indicae ha volailiy is very persisen. Indeed he GARCH model and long-memory models were proposed o explicily ake ino accoun he very long lag srucure in volailiy modeling. Mos of he models used assume a linear funcion for he condiional mean of he ime series. Recen work indicaes ha non-linear models may be more appropriae. Chan, e al (199), Brenner, Harjes, and Kroner (1996), Ai-Sahalia (1996), Conley e al (1997), Tauchen (1997), provide empirical evidence casing doub on he lineariy of single facor coninuous models for he ineres rae. The Markov regime-swiching models proposed by Hamilon (1988, 1989, 1990) and applied o ineres raes by Ausin (00), Bansal and Zhou (00), and Gray (1996) were creaed o model non-lineariy in discree ime. The basic srucure of he ARCH model is given by: y = μ ( y, y, y, ; β) + ε (1) 1 3 k 0 i i i= 1 h ε = + () where h is he condiional volailiy. These models ypically use a linear specificaion for he condiional mean. However as is well undersood, if he condiional mean is acually non-linear his may lead o over-rejecion of he null hypohesis of no ARCH. Blake and Kapeanios (007) show ha non-lineariies in he condiional mean can significanly affec ess for ARCH since he mis-specified model can lead o squared residuals ha are correlaed even in cases where here is no ARCH. Since he volailiy is a key componen of many pricing models, his may have serious consequences for he accuracy of hese resuls. Following Blake and Kapeanios (007), we use neural neworks o esimae he non-lineariies in he condiional mean. This produces residuals which converge in probabiliy o he rue error erms and can hus be used o es for he presence of ARCH. Our resuls are wofold. Firs, we find ha alhough here is srong evidence for ARCH effecs in he series, he lag lenghs are much shorer han previously esimaed. (1) Second, we also show he ype of mis-specificaion inheren in linear-models, which suggess he manner in which hese models should be modified. 16

3 Arch in Shor-Term Ineres Raes: Case Sudy USA NEURAL NETWORKS AND THE RADIAL BASED FUNCTIONS (RBF) An Arificial Neural Nework (ANN, henceforh) is a non-linear saisical compuaional ool based on adapive biological neural neworks. Radial Based Funcion (RBF, henceforh) neworks are ANNs ha can approximae any funcion N N R R defined on a compac subse of R wih any degree of precision (Campbell, Lo, and MacKinlay 1997). We use he RBF nework o approximae he condiional mean funcion of he ransiional densiy of he shor rae. We concenrae on he following univariae model for he shor rae: r = f( r 1; β ) + ε (3) where r is he shor erm ineres rae a ime, f ( ) is an unknown, coninuous, funcion and ε is a random variable wih mean 0 and condiional variance given by: k 0 i i i= 1 h ε = + (4) Given he universal approximaion propery, we know ha we can use an RBF nework o wrie f as: m f ( r; β ) = b0 + br 1 + aig(( d0, i + dir)) (5) i= 1 where g is given by: r d j ( ) τ grd ( ;, γ ) = e (6) j and d j is he h j cener. The funcion has a maximum of 1 when r coincides wih he cener, and goes o 0 as r goes furher away from he cener. The rae a which he funcion decreases is deermined byτ. The nework is defined by he choice of m ceners and radiusτ. If τ and he ceners{d j } j = 1 are known hen he RBF is easy o esimae using leas squares. m We follow Blake and Kapeanios (007) by choosingτ, and {d j } j = 1 by a daa dependen mehod independenly of he RBF. τ is chosen o be he variance of he daa (or alernaively 1 for normalized daa), and we allow here o be T poenial ceners. Each of he ceners is an observaion drawn from he daa. The ceners are hen ranked by heir abiliy o reduce he unexplained variance when enered individually. Then we successively add he ranked ceners unil we minimize an informaion crierion. Having chosen he ceners and he radius, he neural nework becomes linear in a i. Afer fiing he nework, he residuals can be used o es for he presence of ARCH, and we esimae he model given above. 17

4 Souhwesern Economic Review ESTIMATION We perform he ess on weekly 3-monh ineres raes for he U.S. The raes are from he 90 day reasury bill from o , colleced on Thursdays. If Thursday daa is no available, we use Wednesday s numbers. The daa are provided by he Federal Reserve Bank of S. Louis, and is commonly used as a proxy for he shor erm risk-free rae. Figure 1 3-monh T-bill Rae (Unied Saes) Ineres Rae (%) Dae Figure Condiional mean funcion E[r()] = f(r(-1)) E[r()] r(-1) The iniial esimaes from he linear model are given in able 1. Using he AIC informaion crierion we find ha he US series show ha a shock has consequences for volailiy for a very long ime. We hen re-esimae he model using he Radial based neural nework (which we refer o as he non-linear model). The resuls are also given in able 1. 18

5 Arch in Shor-Term Ineres Raes: Case Sudy USA The series shows evidence of ARCH effecs, bu when we esimae is srucure, we find ha he lag lengh o be much shorer, and fewer overall number of parameers han he linear model. TABLE 1 AKAIKE INFORMATION CRITERIA (AIC) ARCH lag lengh US Linear US non-linear Table 1: The AIC informaion crieria for he models wih linear condiional mean and non-linear condiional mean a varying ARCH lag lenghs. i represens a model wih i lags in he condiional variance. Noes: 1) This noe is par of a more comprehensive projec where we apply his esimaion procedure across a larger se of counries. Our resuls hold consisenly over all he ime series examined. Resuls available upon reques. ) We are currenly examining ineres raes for he Unied Kingdom, Canada, Japan, and Europe. Preliminary findings show similar resuls o he US case. Looking a he condiional mean funcion (figure ) also gives us some insigh ino he srucure of he ineres rae process. We find ha over a cerain range of he daa (up o abou 14) he funcion is close o linear and here is almos no evidence of mean reversion. However a he high end of he range, he funcion 19

6 Souhwesern Economic Review becomes non-linear and he evidence for mean reversion becomes very srong indeed. This may help o explain some of he conroversy over he exisence of mean reversion in ineres raes, and i also fis in wih he findings of Ai-Sahalia (1996) and Gray (1996). CONCLUSION We find ha for shor erm US ineres raes, here is evidence of boh ARCH effecs and non-lineariy in he condiional mean. We find ha wih he condiional mean, he volailiy persisence is much shorer han is ypically found in esimaion of long-erm volailiy effecs. Furher sudy is needed o deermine wheher hese characerisics are unique o he US, or are consisen wih oher counry daa. This could also have consequences for he pricing of bonds and opions when non-lineariies are presen. REFERENCES Ai-Sahalia, Y Tesing Coninuous-Time Models of he Spo Ineres Rae. Review of Financial Sudies 9 (): Ausin, A. 00. Non-nesed Models of he Shor Term Ineres Rae: A Cross Counry Examinaion. working paper, Universiy of Miami. Bansal, R., and H. Zhou. 00. Term Srucure of Ineres Raes wih Regime Shifs. Journal of Finance 57: Baillie, R. T., T. Bollerslev, and Hans Ole Mikkelsen Fracionally Inegraed Generalized Auoregressive Condiional Heeroskedasiciy. 74 (1): Blake, A. P., and G. Kapeanios Tesing for ARCH in he presence of nonlineariy of unknown form in he condiional mean. Journal of Economerics 137 (): Bollerslev, T Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics 31 (3): Breid, F., Crao, J., and P. de Lima The Deecion and Esimaion of Long Memory in Sochasic Volailiy, Journal of Economerics 83 (1): Brenner, R. J., R. H. Harjes, and K. F. Kroner Anoher Look a Models of he Shor-Term Ineres Rae. Journal of Financial and Quaniaive Analysis 31 (1): Campbell, J. Y., A. W. Lo, and A. C. MacKinlay The Economerics of Financial Markes Princeon Universiy Press. Chan, K. C., G. A. Karolyi, F. A. Longsaff, and A. B. Sanders An Empirical Comparison of Alernaive Models of he Shor-erm Ineres Rae. Journal of Finance 47: Conley, T.G., L.P.Hansen, E.J. Lumer, and J.A. Scheinkman Shor-erm Ineres Raes as Subordinaed Diffusions. Review of Financial Sudies 10: Engle, R. F Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion. Economerica, 50 (1): Gray, S. F Modeling he Condiional Disribuion of Ineres Raes as a Regime-swiching Process. Journal of Financial Economics 4:

7 Arch in Shor-Term Ineres Raes: Case Sudy USA Hamilon, J. D Raional Expecaions Economeric Analysis of Changes in Regime: an Invesigaion of he Term Srucure of Ineres Raes. Journal of Economic Dynamics and Conrol 1: Hamilon, J. D A New Approach o he Economic Analysis of Non-saionary Time Series and he Business Cycle. Economerica 57: Hamilon, J. D Analysis of Time Series Subjec o Changes in Regime. Journal of Economerics 45: Mandelbro B The Variaion of Cerain Speculaive Prices. Journal of Business 36 (4): Tauchen, G New minimum chi-square mehods in empirical finance. in Advances in Economerics, Sevenh World Congress, eds. D. Kreps, and K. Wallis, Cambridge Universiy Press, Cambridge. Acknowledgemen: Du would like o hank he Universiy of Wes Georgia for heir Faculy Research Gran , and Mr. Ravi Nandan Sahay and Mr. Saurabh Das for heir suppor. 131

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