Federal Funds Rate Prediction *

Transcription

1 Federal Funds Rae Predicion * Lucio Sarno abc,,, Daniel L. Thornon c and Giorgio Valene a a: Universiy of Warwick b: Cenre for Economic Policy Research (CEPR) c: Federal Reserve Bank of S. Louis Firs version: Augus 00 - Second revised version: May 004 Absrac We examine he forecasing performance of a range of ime-series models of he daily US effecive federal funds (FF) rae recenly proposed in he lieraure. We find ha: (i) mos of he models and predicor variables considered produce saisfacory oneday-ahead forecass of he FF rae; (ii) he bes forecasing model is a simple univariae model where he fuure FF rae is forecas using he curren difference beween he FF rae and is arge; (iii) combining he forecass from various models generally yields modes improvemens on he bes performing model. These resuls have a naural inerpreaion and clear policy implicaions. JEL classificaion: E43; E47. Keywords: federal funds rae; forecasing; erm srucure; nonlineariy.

2 . Inroducion The imporance of he effecive federal funds (FF) rae in US financial markes is unquesionable. The Federal Reserve (Fed) implemens moneary policy by argeing he effecive FF rae. The abiliy of marke paricipans o predic he FF rae is imporan o modern analyses of moneary policy in ha oher ineres raes are believed o be linked o he FF rae by he marke expecaion of moneary policy acions ha direcly affec he FF rae. I is herefore no surprising ha a vas body of research has sudied he behavior of he FF rae and proposed empirical models designed o explain i. One srand of his lieraure focuses on he FF rae using daa a monhly or quarerly frequency o esablish he exen o which argumens of ineres o he Fed - such as inflaion and he oupu gap - are sufficien o explain he variaion in he FF rae (e.g. Taylor, 993, 999; Clarida, Gali and Gerler, 998, 000, and he references herein). A relaed lieraure invesigaes he impac of moneary policy shocks on key macroeconomic aggregaes, again using low frequency daa, idenifying shocks o moneary policy using he FF rae in srucural vecor auoregressions (e.g. Chrisiano, Eichenbaum and Evans, 999, and he references herein). Oher sudies focus on he high-frequency behavior of he FF rae, using daa a he daily frequency. This frequency is appealing because each day he Trading Desk of he Federal Reserve Bank of New York (hereafer Desk) conducs open marke operaions designed o move he FF rae in he desired direcion (e.g. Hamilon, 996; Roberds, Runkle and Whieman, 996; Balduzzi, Berola and Foresi, 997; Taylor, 00). The lieraure has suggesed ha several variables have predicive power for explaining FF rae movemens: FF fuures raes (Krueger and Kuner, 996), he FF rae arge (Rudebusch, 995; Taylor, 00), and oher ineres raes linked o he FF rae via no-arbirage condiions or he erm srucure of ineres raes (e.g. Enders and Granger, 998; Hansen and Seo, 00; Sarno and Thornon, 003; Clarida, Sarno, Taylor and Valene, 004). A number of models, boh univariae and mulivariae, linear and nonlinear, have been proposed o capure he unknown process ha drives FF rae movemens. To dae, however, here appears o be no consensus on wha variables and models bes characerize he behavior of he FF rae a he daily frequency. This paper aemps o fill his gap in he lieraure by examining a variey of univariae and mulivariae, linear and nonlinear empirical

3 models of he FF rae, largely aken direcly from or inspired by previous research in his conex. We esimae hese models using daily daa for he period from January 990 hrough December and generae forecass over he remaining four years of daa. We also examine he poenial o improve on he individual or primiive models by using combinaions of forecass (see, iner alia, Diebold, 00; Sock and Wason, 999b, 003; Swanson and Zeng, 00). To anicipae our resuls, we find ha, in general, mos of he models and predicor variables considered produce saisfacory one-day-ahead forecass of he FF rae. However, he bes forecasing model is a very parsimonious univariae model where he one-day-ahead funds rae is forecas using he curren difference beween he funds rae and is arge. This model can be hough of as a simple varian of he Desk s reacion funcion proposed by Taylor (00). Combining he forecass from various models provides generally modes improvemens on his Desk s reacion funcion. We argue ha hese resuls have a naural inerpreaion and ha hey are in line wih he growing empirical evidence suggesing ha he Fed s policy is well described as a forward-looking ineres rae rule. 3 The remainder of he paper is se ou as follows. In Secion we describe he empirical models of he daily FF rae considered in he paper. We also briefly discuss some economeric issues relevan o esimaion of hese models. In Secion 3 we describe he daa. Some preliminary daa analysis and he in-sample esimaion resuls are given in Secion 4, while in Secion 5 we presen our forecasing resuls using boh he primiive models and he combinaions of forecass, including evidence on poin forecas accuracy and marke iming abiliy. A final secion concludes.. Empirical models of he federal funds rae This secion describes he empirical models of he daily US effecive FF rae considered in his paper. Our aim is o model he daily behavior of he FF rae. However, we canno rely on sandard macroeconomic variables (such as inflaion and oupu) ha one may expec o drive moneary policy decisions and ineres rae movemens since hese variables are no available a he daily frequency. Thus, he informaion se considered includes he FF rae arge, oher US ineres raes o which he FF rae is likely o be linked via no-arbirage condiions, and FF fuures raes, in addiion o lagged values of he FF rae. 4

4 . Univariae models The firs specificaion considered is a simple drifless random walk (RW) model: i = u, () FF where FF i is he daily change in he effecive FF rae. Alhough several researchers have concluded ha he ineres rae fails o follow a random walk (e.g. Shiller, Campbell and Schoenholz, 983; Campbell, 987; Barre, Slovin and Sushka, 988; Lasser, 99; Hamilon, 996; Roberds, Runkle and Whieman, 996; Balduzzi, Berola and Foresi, 997; Lanne, 999, 000), mos sudies canno rejec he uni roo hypohesis for ineres raes (e.g. Sock and Wason, 988, 999a). Given he large empirical work suggesing ha very persisen series wih a roo very close (if no equal) o uniy are beer approximaed by I () processes han by saionary ones (e.g. see Sock, 997), i seems reasonable o consider a RW model for he FF rae as one of our specificaions. 5 A more general model of FF rae movemens is a linear ARMA ( p, q) model: p q FF FF γ j i j ε φsε s j= s= () i = + +. This model generalizes he RW model o accoun for higher-order auoregressive dependence and for moving-average serial correlaion in he residual error. The hird specificaion considered is a varian of he model recenly proposed by Taylor (00): i = i i + error erm (3) FF FF T ξ FF T FF T where i i is he lagged difference beween he FF rae, i and he FF rae arge, i. Under his model, fuure daily changes in he FF rae are driven by curren deviaions of he FF rae from is arge. Taylor (00) suggess ha he Desk srives o keep he FF rae close o is arge level, which, since 994, is publicly announced by he Federal Open Marke Commiee (FOMC). According o he lieraure on forward-looking ineres rae rules (Clarida, Gali and Gerler, 998, 000), he FF rae arge is se on he basis of consideraions ha may be parsimoniously summarized 3

5 in expeced inflaion and oupu gap. Hence, equaion (3) may be seen as he Desk s reacion funcion designed o minimize deviaions of he FF rae from he arge, which is se by he FOMC a a level believed o deliver he desired inflaion and oupu objecives. Taylor argues ha he adjusmen o deparures of he FF rae from is arge is parial a daily frequency, implying ha < ξ < Anoher univariae model of he daily FF rae, firs examined by Hamilon (996), is he exponenial generalized auoregressive condiional heeroskedasiciy or EGARCH model. An EGARCH ( p, q) for he daily change in he FF rae may be wrien as follows: i = η i i + σ v FF FF FF 3 log v v v p q s s s σ = ω + ϑjσ j + ρs E ς s j= s= σ s σ + s σ s (4) FF FF where i i 3 is he cumulaive change of he FF rae over he preceding wo days, and v denoes independenly and idenically disribued (i.i.d.) innovaions wih zero mean and uni variance. The condiional variance σ is modeled in he spiri of he analysis of Nelson (990), where in order o ake ino accoun he asymmeric effec beween fuure condiional variances and curren FF rae changes, boh sign and magniude of he innovaions are aken ino accoun. Anoher univariae nonlinear model we consider is a Markov-swiching p h-order auoregressive model wih M regimes, MS-AR ( M, p) (Hamilon, 988, 989; Gray, 996): p FF j j j= FF ϕ ( ) σ ( ) ω (5) i = z i + z z =,,, M. p Model (5) allows for he auoregressive srucure, ϕ ( z ) erm, σ ( ) z j= i FF j j, and he variance of he error, o be shifing over ime across regimes. In his model, he variance of he innovaions ω is ime-varying bu, unlike for he EGARCH, he dynamics is governed by an unobservable variable z which is assumed o follow a firs-order Markov chain. Markov-swiching models have ofen been employed o model ineres raes wih some degree of success. Such models are a 4

6 plausible alernaive o models of he ARCH family designed o capure fa-ailed disurbances (e.g. Hamilon, 988; Hamilon and Susmel, 994; Ai-Sahalia, 996; Gray, 996; Bansal and Zhou, 00; Dai, Singleon and Yang, 003; Clarida, Sarno, Taylor and Valene, 004).. Mulivariae models Among he mulivariae models we consider, we include he Momenum-Threshold Auoregressive (M-TAR) model proposed by Enders and Granger (998): p αβ ( ) αβ (6) y = Λ y + I y + I y + error j j j= I ifβy 0 = 0 if β y < 0 where y = i, i ; FF TB TB i is he 3-monh Treasury Bill (TB) rae. This model explicily akes ino accoun he exisence of asymmeries ha may occur in he adjusmen process along he shor-end of he yield curve. Essenially, he adjusmen parameers, α, α, differ depending on wheher he slope of he shor-end of he yield curve is posiive or negaive. Anoher mulivariae hreshold model, applied o he FF rae by Hansen and Seo (00), is he bivariae TAR or BTAR model: p p j j j j j= j= αβ ( ) αβ (7) y = I Λ y + y + I Λ y + y + error I ifβy = 0 if β y k < k where y = i, i. Alhough boh he M-TAR model (6) and he BTAR model (7) belong o he FF TB family of hreshold auoregressive models, hey differ in several respecs. Firs, in he BTAR model (7) he Heaviside indicaor funcion I is equal o zero or uniy according o wheher he value of he coinegraing residual is smaller (or larger) han a hreshold k, which mus be esimaed. In conras, in he M-TAR model (6) he hreshold is assumed o be equal o zero. Second, in he BTAR model (7) he whole se of parameers, Λ i, α i, is shifing over ime, while in he M-TAR model (6) only he 5

7 speed of adjusmen parameer, namely α (for i =, ), is shifing over ime. i The final model considered is a Markov-Swiching Vecor Error Correcion Model or MS- VECM of he erm srucure of FF fuures raes. This model is parly inspired by he recen work of Krueger and Kuner (996) and Kuner (00), where i is shown ha FF fuures raes are useful in predicing fuure changes in he FF rae. If he FF rae and FF fuures raes are I () variables, i is sraighforward o demonsrae ha he FF rae and he FF fuures raes should coinegrae wih a coinegraing vecor [, ]. 8 In urn, via he Granger represenaion heorem (Granger, 986), he join dynamics of he FF rae and he FF fuures raes can be described by a VECM: p / ( ) ( ) ( ) (8) y = Γ z y +Π z y +Σ z ε i j j= where y i f f =,,, and he long-run impac marix ( z ) ( z ) FF Π = α β. The VECM in equaion (8) has been generalized o a Markov-swiching framework where he parameers can shif o ake ino accoun he evidence ha ineres rae changes are heeroskedasic and ha heir disribuion is well approximaed by a mixure of normal disribuions (e.g. see Hamilon, 988, 996; Gray 996; Bansal and Zhou, 00; Dai, Singleon and Yang, 003; Clarida, Sarno, Taylor and Valene, 004) Daa issues The daa se consiss of daily observaions on he effecive FF rae FF, nc i, he 3-monh T-bill i TB, he FF rae arge i T, and he FF fuures rae f for m =,. The FF rae is a weighed average of he m raes on federal funds ransacions of a group of federal funds brokers who repor heir ransacions daily o he Federal Reserve Bank of New York. Federal funds are deposi balances a Federal Reserve banks ha insiuions (primarily deposiories, e.g. banks and hrifs) lend overnigh o each oher. These deposi balances are used o saisfy reserve requiremens of he Federal Reserve Sysem. Because reserve requiremens are binding a he end of he reserve mainenance period, called selemen Wednesday, he FF rae ends o be more volaile on selemen Wednesdays. 0 The federal funds rae ime series was adjused in order o eliminae he effec of he increased volailiy on selemen days, as done, for example, in Sarno and Thornon (003). This yielded he ime series 6

8 for he correced FF rae, FF i, which is he ime series we employ in our empirical work. m f is he rae on a FF fuures conrac wih mauriy m, raded on he Chicago Board of Trade (CBT). Fuures conracs are designed o hedge agains or speculae on he FF rae. The CBT offers FF fuures conracs a several mauriies; however, he mos acive conracs are for he curren monh and a few monhs ino he fuure. The conracs are marked o marke on each rading day, and final cash selemen occurs on he firs business day following he las day of he conrac monh. The FF rae i FF and he 3-monh T-Bill rae i TB were obained from he Federal Reserve Bank of S. Louis daabase, Federal Reserve Economic Daa (FRED). The FF rae arge, i T was aken from Thornon and Wheelock (000). The FF fuures raes m f were obained from he CBT. The sample period spans from January 990 hrough December 3 000, yielding a oal of, 869 observaions. This sample period was chosen for wo reasons. Firs, while he Fed has never explicily saed when i began argeing he FF rae in implemening moneary policy, an emerging consensus view is ha he Fed has been explicily argeing he FF rae since a leas he lae 980s (e.g. see Meulendyke, 998; Hamilon and Jordá, 00; Poole, Rasche and Thornon, 00). Second, while he FF fuures marke has exised since Ocober 988, rading aciviy in his marke was iniially small (Krueger and Kuner, 996, p. 867). To insure agains he possibiliy ha he empirical analysis would be affeced by he hinness of he FF fuures marke during he early years of is operaion, we decided o begin he sample in January 990. Since we are ineresed in he predicive power of alernaive ime series models, we iniially esimae each model over he period January 990 hrough December Forecass over he remaining four years of daa are generaed using a recursive forecasing procedure, described in Secion Empirical resuls 4. Preliminary daa analysis and uni roo ess Table presens summary saisics for he series of ineres, boh in levels (Panel a) and firs difference (Panel b). These summary saisics show ha all raes - he (non-correced and correced) 7

9 FF rae, he TB rae, he FF fuures raes, and he FF rae arge - display similar values for he mean, variance, skewness and kurosis. I is, however, clear ha he correcion for selemen Wednesdays discussed in he previous secion make he mean of he FF rae closer o he mean of he FF rae arge; indeed, afer his correcion he mean difference beween he FF rae and he FF rae arge, i FF i T, is only 0.05 and saisically insignificanly differen from zero when one akes ino consideraion is sandard deviaion. Also, an examinaion of he hird and fourh momens indicaes he exisence of boh excess skewness and kurosis, suggesing ha he underlying disribuion of each of hese ime series may be non-normal. This is confirmed by he rejecions of he Jarque-Bera es for normaliy repored in he las row of Panels a-b in Table. 4. Esimaion resuls and in-sample performance We esimae each model described in Secion over he sample period January 990 and December Alhough he core of our empirical work relaes o he ou-of-sample performance of hese models, his secion provides deails on he abiliy of each model o explain he FF rae in sample. The full esimaion resuls for each model are given in Tables A o A3 in he Appendix. Table A repors he esimaion resuls for he univariae models of he FF rae. For model () we found an ARMA(,) specificaion o be saisfacory in ha no serial correlaion was lef in he residuals (Table A, column ). Esimaion of model (3), namely he Desk s reacion funcion, suggess very fas, albei parial, adjusmen of he FF rae oward he FF rae arge (Table A, column 3). The speed of reversion parameer ξ is consisen wih over 70 percen of he daily difference beween he FF rae and he arge being dissipaed he following day, which is in line wih he evidence presened in Taylor (00). The EGARCH model (4) was esimaed following he specificaion procedure adoped in Hamilon (996); see Table A, column 4. 3 The univariae Markov-swiching model (5) was specified and esimaed by employing he boom-up procedure suggesed by Krolzig (997, Ch. 6). This procedure is designed o deec Markovian shifs in order o selec he mos adequae characerizaion of an M-regime p-h order MS- AR for FF i. The boom-up procedure suggesed in each case ha wo regimes were sufficien o characerize he dynamics of FF rae changes, confirming previous resuls repored in he lieraure on 8

10 modeling shor-erm ineres raes (e.g. Gray, 996; Ang and Bekaer, 00). Furher, a specificaion which allows he auoregressive srucure and he variance o shif over ime (Markov-Swiching- Auoregressive-Heeroskedasic-AR, or MSAH-AR) was seleced since a regime-shifing inercep was found o be insignifican a convenional saisical levels (Table A, column 5). Turning o he mulivariae models, he M-TAR (6) and he BTAR (7) were esimaed following he procedures described in Enders and Granger (998) and Hansen and Seo (00) respecively (see Table A in he Appendix). In paricular, boh models (6) and (7) were esimaed by imposing he coinegraing vecor β = [, ]. Wih respec o he BTAR model (7), he hreshold parameer was esimaed, as in Hansen and Seo (00), using 300 gridpoins, and he rimming parameer was se o Finally, he MS-VECM (8) was esimaed according o he convenional procedure suggesed by Krolzig (997) and employed, for example, by Clarida, Sarno, Taylor and Valene (003, 004), designed o joinly selec he appropriae lag lengh and he number of regimes characerizing he dynamics of he FF rae and he FF fuures raes. The VARMA represenaions of he series suggesed in each case ha here are beween wo and hree regimes. We adoped a specificaion which allows he whole se of parameers - i.e. inercep, auoregressive srucure, coinegraing marix and variance-covariance marix - o shif over ime (Markov-Swiching- Inercep-Auoregressive-Heeroskedasic-VECM or MSIAH-VECM) wih he number of regimes M = 3, which was found o adequaely characerize he join dynamics of he FF rae and FF fuures raes - see Table A3 in he Appendix. The esimaion yields fairly plausible esimaes of he coefficiens for all he specificaions considered. Furher, for any of he regime-shifing models (5)-(8) he null hypohesis of lineariy is rejeced in all cases wih very low p-values, suggesing ha nonlineariies and asymmeries of he kind modeled here may be imporan ingrediens for characerizing in sample he dynamics of he effecive FF rae. Saisics measuring he in-sample performance of he esimaed models ()-(8) are given in Table, where he R and convenional informaion crieria (namely, AIC, BIC and HQ) are repored. The goodness of fi of he models is saisfacory and all he R are larger han 0., which 9

11 is a saisfacory R if one considers ha we are modeling a daily ineres rae ime series in firs difference. Five models ou of eigh exhibi an R > 00. and wo of hem, namely Taylor s (00) Desk reacion funcion and he MSIAH-VECM, display an 030 R >.. Inspecion of he informaion crieria ells us ha, alhough he MSIAH-VECM has he highes R recorded, his model may be overparameerized. In fac, wihin he group of mulivariae models, he MSIAH- VECM is ouperformed by he wo compeing TAR models and i is also ouperformed by he Taylor s (00) Desk reacion funcion, which displays he second highes R and he lowes informaion crieria wihin he se of compeing models. 5. Ou-of-sample forecasing resuls 5. Mehodological issues In order o evaluae he forecasing performance of he empirical models of he FF rae considered, dynamic ou-of-sample forecass of he FF rae were consruced using each of he models esimaed in he previous secion. In paricular, we calculaed one-sep-ahead (one-day-ahead) forecass over he period January 997 and December The ou-of-sample forecass are consruced according o a recursive procedure ha is condiional only upon informaion available up o he dae of he forecass and wih successive re-esimaion as he dae on which forecass are condiioned moves hrough he daa se. 4 We assess he forecasing performance of each of he eigh individual models examined and hen consider combinaions of forecass (models) using he forecas pooling approach proposed by Sock and Wason (999b, 003). For each ime series we choose wo separae periods: (i) a sar-up period over which forecass are produced using he eigh individual compeing models bu no he pooling procedures; (ii) he simulaed real-ime forecas period over which recursive forecass are produced using all individual models as well as he pooling procedures. Le T 0 be he dae of he firs observaion used in his sudy (namely January 990) and T be he firs observaion for he forecas period (namely January 997). Then he sar-up period ends a T = T + 6 (unil he end of 0

12 997) and he forecas period goes from T o T 3, where T 3 is he dae of he final observaion in our daa se (December 3 000). All he forecasing resuls repored in he following sub-secions refer o he simulaed real-ime forecas period T o T 3 (inclusive) Forecasing resuls: primiive models In Table 3 we repor he forecasing resuls obained using he eigh models esimaed in Secion 4, which we erm primiive models. In Panel a) of Table 3 we repor he mean absolue error (MAE) and he roo mean square error (RMSE) for each of he esimaed models. Using he random walk (RW) model as a benchmark in assessing he relaive forecasing performance of he primiive models, we hen repor he p-values of ess for he null hypohesis of equal poin forecas accuracy based on boh he MAE and he RMSE. 6 We use he RW model as a benchmark since i is he benchmark used in he analysis of much research on he properies of he FF rae, cied in Secion., and in paricular he heoreical benchmark considered by Hamilon (996). Moreover, he RW model is ofen used as a benchmark in forecasing sudies on financial variables, such as, for example, exchange raes (Meese and Rogoff, 983). An alernaive benchmark we migh have chosen is he ARMA model in equaion () (e.g. Nelson, 97). However, he exercise carried ou in his paper is he firs comprehensive ou-ofsample forecas comparison of FF rae models. Ou-of-sample forecas comparisons are popular in applied economic and finance largely because some landmark papers (e.g. Nelson, 97; Meese and Rogoff, 983) found ha simple benchmarks do as well as heory-based models. Our calculaions appear o sugges ha he Taylor (00) Desk reacion funcion (3) exhibis he bes ou-of-sample performance: he MAE and he RMSE obained for he Desk reacion funcion are he lowes obained across all models. However, he p-values for he null of equal predicive accuracy, calculaed by boosrap 7, indicae ha he null hypohesis is no rejeced in each case. Hence, he differences in erms of MAEs and RMSEs repored in Panel a) of Table 3 are no saisically significan and do no enable us o discriminae among he models examined. Neverheless, his resul should be aken wih cauion as he non-rejecion of he null of equal poin forecas accuracy may be due o he low power of he relevan es saisic (e.g. see Kilian and Taylor,

13 003). 8 Clark and Wes (004) recenly invesigaed ou-of-sample mean squared predicion errors (MSPEs) o evaluae he null ha a given series follows a zero-mean maringale difference agains he alernaive ha i is linearly predicable. Despie he fac ha under he null of no predicabiliy he populaion MSPE of he null model is equal o he MSPE of he linear alernaive, Clark and Wes show ha he alernaive model s sample MSPE is greaer han he null s, which is wha we find in our resuls in Panel a) of Table 3. Clark and Wes s simulaions sugges ha he es power is abou 50 percen in heir seup. This led us o consider addiional ess. Formal comparisons of he prediced and acual FF rae changes can be obained in a variey of ways. We consider a se of ess for marke iming abiliy of he compeing models, including he hi raio (HR), calculaed as he proporion of imes he sign of he fuure FF rae change is correcly prediced over he whole forecas period, as well as he ess proposed by Henriksson and Meron (98) and McCracken and Wes (998) - hereafer HM and WM ess. The idea behind he HM es is ha here is evidence of marke iming if he sum of he esimaed condiional probabiliies of correc forecass (ha is he probabiliy of correc forecas sign eiher when he FF rae is rising or falling) exceeds uniy. The HM es saisic is given by: HM n n0n0 = n N, n0n0n0n0 n ( n ) ( 0) (9) where n is he number of correc forecass when he FF rae is rising; n 0, n 0 are he number of posiive FF rae changes and forecass of posiive FF rae changes respecively, while n 0 and n 0 denoe he number of negaive FF rae changes and forecass of negaive FF rae changes respecively. The oal number of evaluaion periods is denoed by n. The HM es is asympoically equivalen o a one-ailed es on he significance of he slope coefficien in he following regression: I HM HM FF = 0 + { i 0 FF } I + error erm (0) > i > 0 { } + + where FF FF + i+ i, denoe he realized and forecas firs difference of he FF rae respecively, and I is he indicaor funcion equal o uniy when is argumen is rue and zero oherwise. The oher es employed is he efficiency es inroduced by Mincer and Zarnowiz (969)

14 and sudied by McCracken and Wes (998). This es exends he HM es o ake ino accoun no only he sign of he realized reurns, bu also heir magniude. This involves esimaing he auxiliary regression: MW MW + + φ0 φ + i i = + i + error erm () or alernaively MW MW i = + i + error erm () MW MW MW where 0 = φ0 and = + φ MW. As for he HM es, he null hypohesis of no marke HM iming abiliy is ha he slope coefficien is equal o zero agains he one-sided alernaive ha i is sricly greaer han zero. Differenly, for he WM es he null hypohesis of marke iming abiliy MW MW is ha he slope coefficien is equal o uniy (or alernaively he slope coefficien φ is equal o zero). The resuls from calculaing he hi raio and execuing he HM and WM ess are repored in Panel b) of Table 3. A fairly clear-cu resul emerges from hese ess. The analysis of he hi raio saisics shows ha mos of he models (wih he excepion of he BTAR) exhibi evidence of marke iming - i.e. he hi raio is above 50 percen. This finding is corroboraed by he resuls of he regression-based ess of marke iming, which provide general evidence ha he primiive models have marke iming abiliy. However, he evidence of marke iming abiliy is clearly sronger for he more parsimonious univariae models han for he more sophisicaed nonlinear mulivariae models, as evidenced by he fac ha he p-values for he null of no marke iming are drasically lower for he univariae models. This resul is in line wih he general finding ha nonlinear models do no subsanively ouperform linear models in ou-of-sample forecasing (e.g. see Clemens and Krolzig, 998; Sock and Wason, 999b). I seems reasonable o conclude ha, while all models examined display evidence of marke iming abiliy, univariae models perform beer han mulivariae models. Moreover, he Taylor (00) Desk reacion funcion exhibis he bes performance in erms of hi raios and marke iming, displaying p-values much lower han he ones recorded by he alernaive models. Several caveas are in order. We have seleced he bes performing model on he basis of 3

15 comparisons of he hi raios and he p-values from carrying ou marke iming ess, essenially selecing he bes performing model as he one for which he rejecion of he null hypohesis of no marke iming is sronges. We did no direcly es a model agains anoher in erms of marke iming, however, since a es for equal marke iming abiliy beween compeing models is no available o dae. 9 Also, one may be concerned abou he imporance of he selemen days in our forecasing exercise even if we correced he FF rae ime series o eliminae he effec of selemen Wednesdays prior o beginning he empirical work. Hence, we checked he robusness of a fracion of he resuls in Table 3 by comparing he forecasing performance of he simples models - he RW model, he ARMA model, and Taylor Desk reacion funcion - when he final day of he selemen period is aken ou of he calculaions in consrucing he forecass errors used in he forecasing comparison of he models. The resuls, repored in Table C in Appendix C, sugges ha excluding selemen days does no affec our resuls repored in Table 3 in ha here is no qualiaive difference and only small quaniaive differences beween he resuls in Table 3 and in Appendix C. 5.3 Combinaions of forecass In his secion we invesigae wheher here may be gains from combining forecass from he primiive models. Following Sock and Wason (999b, 003), we employ five combinaions of forecass: simple combinaion forecass; regression-based combinaion forecass; median combinaion forecass; discouned MSFE forecass; and shrinkage forecass. These mehods differ in he way hey use hisorical informaion o compue he combinaion forecas and in he exen o which he weigh given o a primiive model s forecas is allowed o change over ime. 0 Le FF i +, j be he one-sep-ahead forecas of he FF rae change a ime implied by he primiive model j =,, N. Mos of he combinaion forecass are weighed averages of he primiive models forecass, i.e. FF N FF i+, c = w, j i+, j where w, j is he weigh associaed a ime j= wih model j. In general, he weighs w, j depend on he hisorical performance of he individual forecas from model j. As discussed in Secion 5., in order o obain he firs esimaes 4

16 of he weighs w, j, we rain he individual models during he sar-up period (i.e. 997) and hen we apply he following combinaion schemes. The simple combinaion forecass scheme compues he weighs based on he relaive forecasing performance, measured by he MSFE, of he primiive models. This relaive performance is conrolled by a parameer ω, which is se o zero in he simples scheme ha would place equal weigh on all he forecass. As ω increases, more imporance is given o he model ha has been performing beer, in erms of MSFEs, in he pas. In our empirical exercise we use values of ω = 05,,. The regression-based combinaion forecass scheme compues he weighs applied o he FF combinaion forecas as he resul of esimaing a regression of i + on he one-sep-ahead forecas of he FF rae change a ime implied by each primiive model and an inercep erm (Granger and Ramanahan, 984; Diebold, 00). If forecas errors are non-normal, hen linear combinaions are no longer opimal. The median combinaion forecass scheme akes his ino accoun and compues he combinaion forecass as he median from a group of models. This scheme avoids placing oo big a weigh on forecass ha are srongly biased upwards or downwards for reasons such as parameer breaks or parameers which have been esimaed by achieving local (raher han global) opima. The discouned MSFE forecass scheme (Diebold and Pauly, 987; Diebold, 00) compues he combinaion forecas as a weighed average of he primiive forecass, where he weighs depend inversely on he hisorical performance of each individual forecas according o a discoun facor, d which we se equal o 095. and 090. in our calculaions. The shrinkage forecass scheme compues he weighs as an average of he recursive Granger- Ramanahan regression-based esimaes of he weighs and equal weighing (see Diebold and Pauly, 990; Giacomini, 00). The resuls of he combinaion forecas exercises are repored in Table 4. Panel a) of Table 4 shows he mean absolue error (MAE) and he roo mean square error (RMSE) for each combinaion forecas and he bes performing primiive model - i.e. he Taylor (00) Desk reacion funcion. The 5

17 resuls sugges ha he performance of he bes primiive model is difficul o mach even by using sophisicaed pooling forecas echniques. The resuls of he es for equal poin forecas accuracy indicae ha we are no able o rejec he null of equal predicive accuracy in each case. Hence, he differences in MAEs and RMSEs repored in Table 4 are no saisically significan. This conclusion is of course subjec o he cavea ha he es saisic for he null of equal poin forecas accuracy may have low power in his conex. Tess for marke iming abiliy are repored in Panel b) of Table 4. Using his meric, we find clear evidence of marke iming for all of he forecass examined. This finding is corroboraed by he values of he hi raios and he resuls of he ess of marke iming (HM and WM ess). On he basis of he size of he p-values (smaller p-values indicae sronger marke iming abiliy), he bes performing primiive model displays sronger marke iming abiliy han mos of he combinaion forecass. The excepions, which appear o have sronger marke iming abiliy han Taylor s (00) Desk reacion funcion, are he discouned MSFE forecas scheme and, very marginally, he shrinkage forecass scheme. 5.4 Inerpreing he forecasing resuls The forecasing resuls repored in his secion provide several insighs. Firs, we confirm ha, in general, mos of he models and predicor variables considered produce saisfacory one-day-ahead forecass of he FF rae. Second, he bes forecasing model is a simple and very parsimonious univariae model where he fuure FF rae is forecas using he curren difference beween he funds rae and is arge. Third, combining he forecass from various models may improve on he bes performing model, bu he improvemens are generally modes. These resuls may be seen as consisen wih he growing empirical evidence uncovering ha he Fed s policy is well described as argeing he FF rae according o a forward-looking version of Taylor s (993) ineres rae rule. For example, if he Fed implemens moneary policy on he basis of expecaions of fuure inflaion and oupu gap, hen he FF rae arge se by he Fed will presumably conain informaion abou fuure inflaion and oupu gaps, which a priori one would expec o be imporan in predicing fuure ineres raes (Clarida, Gali and Gerler, 998, 000). 6

18 Furher, our resuls are also consisen wih he Fed s descripion of is moneary policy operaing procedure and is undersanding by he economics profession. Indeed, here seems o be general agreemen ha he Fed has explicily argeed he funds rae a leas since he lae 980s and, herefore, hroughou he sample period under invesigaion in his paper (see Meulendyke, 998; Hamilon and Jordá, 00). Also, since hence hroughou he forecas period examined - he Fed has announced arge changes immediaely upon making hem. Before 994, arge changes were no announced: he marke had o infer he Fed s acions by observing open marke operaions and he FF rae (e.g., Cook and Hahn, 989; Rudebusch, 995; Taylor, 00; Thornon, 004). If one believes ha he FF rae does in fac display reversion oward he FF rae arge, hen clearly his procedure would make i easier for he marke o forecas he nex-day FF rae by publicly announcing wha he Fed s desired FF rae is. Our resuls suppor he view ha reversion o he arge is a prominen feaure of FF rae behavior during he sample examined and i is a crucial feaure in forecasing ou of sample he FF rae a he daily frequency. Alhough he mechanism linking he FF rae o he FF rae arge is one of parial, no full, adjusmen a he daily frequency, we found i very hard o improve on he simple Desk reacion funcion linking he FF rae and he arge by using much more sophisicaed mulivariae or nonlinear models and alernaive predicor variables. A he very leas, our resuls sugges ha he simple univariae Desk reacion funcion model suggesed by Taylor (00) is a very good firs approximaion o he FF rae behavior and represens a difficul benchmark o bea in oneday-ahead forecasing of he FF rae. 6. Conclusion In his paper we repored wha we believe o be he firs broad-based analysis of a variey of empirical models of he daily FF rae and examined heir performance in forecasing ou-of-sample he one-dayahead FF rae. Our research was inspired by encouraging resuls previously repored in he lieraure on he predicabiliy of he FF rae using linear and nonlinear models and on he explanaory power of variables such as he FF fuures raes, he FF rae arge and oher US ineres raes o which he FF rae is likely o be linked via no-arbirage condiions. 7

19 Using daily daa over he period from January 990 hrough December 3 996, we confirmed ha he predicor variables suggesed by he lieraure have subsanial explanaory power on he FF rae in sample and ha accouning for nonlineariy in he unknown rue daa generaing process governing he FF rae may yield saisfacory characerizaions of he ime-series properies of he FF rae. We hen used a wide range of univariae and mulivariae, linear and nonlinear models o forecas ou of sample over he period January 997 hrough o December 3 000, using boh convenional measures of poin forecas accuracy based on mean absolue errors and roo mean squared errors as well as hi raios and marke iming ess designed o evaluae he abiliy of he models o forecas boh he direcion and he magniude of fuure FF rae changes. The forecasing resuls were ineresing. Using convenional measures of poin forecas accuracy we found ha he reacion funcion proposed by Taylor (00), where he gap beween he FF rae and is arge is used o predic he nex-day FF rae, produces he lowes mean absolue errors and roo mean square errors. However, general ess for equal poin forecas accuracy did no enable us o disinguish among he compeing models, possibly because of he low power of hese ess in his conex. Using hi raios and marke iming ess, we found ha he simple univariae reacion funcion emerges as he bes performing model, forecasing correcly over 66 percen of he imes he direcion of he nex-day FF rae and showing saisfacory marke iming abiliy. Combining he forecass from various models may improve on he bes performing model, bu he improvemens are generally modes in size. In urn, hese resuls have a naural inerpreaion and may be seen as consisen wih he growing empirical evidence suggesing ha he Federal Reserve s policy may be characerized as a forward-looking ineres rae rule. Our resuls suppor he view ha reversion o he arge is a key ingredien in models designed for characerizing in sample and forecasing ou of sample he FF rae a he daily frequency. The simple univariae Desk reacion funcion suggesed by Taylor (00) is a very good firs approximaion o he FF rae behavior and represens a difficul benchmark o bea in one-day-ahead forecasing of he FF rae. 8

20 Table. Summary saisics Panel a): Levels FF nc i, FF i T FF T i i i Mean Variance Skewness Kurosis JB x0 TB i f f Panel b): Firs differences FF, nc i FF i T FF T i i i Mean Variance Skewness Kurosis TB i JB f f Noes: i FF, nc, i, FF i, T i, TB f, f, denoe he non-correced effecive federal funds rae, he correced effecive federal funds rae (adjused o accoun for selemen days, as discussed in Secion 3), he federal funds rae arge, he 3-monh T-Bill rae, he one-monh and wo-monh federal funds fuures raes respecively. The sample period spans from January 990 hrough December JB denoes he Jarque-Bera es for he null hypohesis of normaliy, for which only p-values 4 are repored; 0 denoes p-values lower han 0. is he firs difference operaor. 9

21 Table. In-sample performance R AIC SIC HQ RW ARMA(,) Taylor (00) EGARCH(,) MSAH-AR(,) M-TAR BTAR MSIAH-VECM(3,) Noes: The definiions of he models are given in Secion. All models are esimaed over he sample period from January 990 o 3 December 996. AIC, BIC and HQ denoe he Akaike, Schwarz and Hannan-Quinn informaion crieria respecively. 0

22 Table 3. Ou-of-sample performance: primiive models a) Poin forecas evaluaion MAE p value RMSE p value RW ARMA(,) Taylor (00) M-TAR EGARCH(,) MSAH-AR(,) BTAR MSIH-VECM(3,) b) Regression-based es for marke iming HR HM WM RW ARMA(,) ( 009. ) Taylor (00) (0. 07) M-TAR (0. ) EGARCH(,) (0. 3) MSAH-AR(,) (0. 7) BTAR (0. ) MSIH-VECM(3,) (0. 9) Noes: The definiions of he models are given in Secion. The forecas period goes from January 998 o December Panel a): MAE and RMSE denoe he mean absolue error and he roo mean square error respecively. p value is he p-value from execuing es saisics for he null hypohesis ha he model considered has equal poin forecas accuracy as he random walk (RW), calculaed by boosrap using he procedure described in Appendix B. Panel b): HR is he hi raio calculaed as he proporion of correcly prediced signs. HM and WM are he Henriksson and Meron (98) and he efficiency es calculaed as in Wes and McCracken (998), using he auxiliary regressions (0) and () respecively (see Secion 5.). HM and WM is calculaed using he Newey-Wes (987) auocorrelaion- and heeroskedasiciy-consisen covariance marix. For he 4 HM es saisics only p-values are repored; 0 denoes p-values lower han 0. Values in parenheses are esimaed sandard errors.

23 Table 4. Ou-of-sample performance: combinaions of forecass a) Poin forecas evaluaion MAE p value RMSE p value Taylor (00) LCF, ω = LCF, ω = LCF, ω = RCF Median DCF, d = DCF, d = SCF, κ = SCF, κ = SCF, κ = b) Regression-based es for marke iming HR HM WM Taylor (00) (0.06) LCF, ω = (0.09) LCF, ω = (0.09) LCF, ω = (0.09) RCF (0.07) Median (0.0) DCF, d = (0.06) DCF, d = (0.05) SCF, κ = (0.06) SCF, κ = (0.06) SCF, κ = (0.06) Noes: See Noes o Table 3. The full forecas period goes January from 997 o December 3 000, while he sar-up period spans from January 997 and December LCF denoes he combinaion of forecass where he weighs assigned o each primiive model s forecass are calculaed by using he simple combinaion forecass scheme described in Secion 5.3; and ω = 05,, is a parameer conrolling he relaive weigh of he bes performing model wihin he panel. RCF denoes he combinaion of forecass where he weighs assigned o each primiive model s forecass are calculaed using he regression-based mehod of Granger and Ramanahan (984). Median is he combinaion of forecass calculaed as he median of he forecass from he primiive models. DCF denoes he combinaion of forecass calculaed according o he discouned forecass scheme, and he discoun facor d = ,.. SCF is he combinaion of forecass where he weighs assigned o each primiive model s forecass are calculaed as an average of he recursive ordinary-leas-squares esimaor and equal weighing, wih a degree of shrinkage κ=0.5, 0.5,.0.

24 Appendix A. Furher esimaion resuls Table A. Univariae models ARMA(,) Taylor (00) EGARCH(,) MSAH-AR(,) γ 0.98 (0.03) γ (0.03) φ (0.0) ξ (0.0) η (0.0) ω -.07 (0.05) ϑ 0.44 (0.0) ρ (0.0) ς (0.0) ( z ) ( z ) ϕ = 0. (0.03) ϕ = (0.05) σ σ = 0.0 ( z ) ( ) σ = 0.6 z ARCH LR lineariy Noes: The definiions of he models are given in Secion. This able repors esimaes of he parameers of he models defined by equaions ()-(5) in Secion. The sample period used for he esimaion goes from January 990 o 3 December 996. σ denoes he residual sandard deviaion. Values in parenhesis are sandard errors calculaed using he Newey-Wes auocorrelaion and heeroskedasiciy consisen covariance marix. ARCH is he Lagrange muliplier (LM) es for auoregressive condiional heeroskedasiciy (ARCH) in he residuals (Engle 98). The LR lineariy es is a Davies (977, 987) es for he null hypohesis ha he rue model is a linear AR ( p ) agains he alernaive of a MS-AR ( M, p). Is p-value is calculaed as in Davies (977, 987). For each of ARCH and LR lineariy ess, we only repor p-values; 0 denoes p-values lower han

25 Table A. TAR models Panel a) M-TAR (0. 0) (0. 4) (0. 0) (0. 4) Λ = ;Λ = ; (0. 004) (0. 0) (0. 003) (0. 0) (0. 0) (0. 03) α = ; α = (0. 005) (0. 004) LR : {0} Panel b) BTAR (0. 0) (0. 5) (0. 0) Λ = ; α = ; (0. 005) (0. 03) (0. 004) (0. 03) (0. 7) (0. 04) Λ = ; α = ; (0. 004) (0. 04) (0. 005) k = , 0 suplm : {0. 04} Noes: The definiions of he models are given in Secion. This able repors esimaes of he parameers of he models defined by equaions (6) and (7) in Secion. The sample period used for he esimaion is from January 990 o 3 December 996. Panel a): he M-TAR model (6) is esimaed imposing β = [, ], as in Enders and Granger (998, p. 30), and selecing he lag lengh according o he resuls of AIC, BIC and HQ crieria. LR is he likelihood raio es for he null hypohesis ha α = α, for which we repor he p-value; 0 denoes a p-value lower han 4 0. Panel b): he BTAR model is esimaed by imposing β = [, ]. The hreshold parameer is esimaed as in Hansen and Seo (00) and compued wih 300 gridpoins, whereas he rimming parameer is se o 005. (see, Andrews, 993). Figures in parenheses are asympoic sandard errors. 0 suplm is he p-value from execuing a es for hreshold coinegraion, compued by parameric boosrap (Hansen and Seo, p. 99). 4

26 Table A3. Markov-swiching VECM MSIAH-VECM wih hree regimes and one lag (0. 07) (0. 05) (0. 78) (0. 09) (0. 5) (0. 8) = = ; Γ = = ; (0. 004) (0. 05) (0. 064) ( 0. 00) (0. 03) (0. 06) (0. 006) (0. 064) (0. 066) (0. 00) (0. 08) (0. 0) ( z ) ( z ) Γ ( z 3) Γ (0 6) ( 89) ( 53) = = ; (0. 05) (0. 445) (0. 350) (0. 07) (0. 56) (0. 395) (0 054) (0 04) (0 007) (0 006) ( ) α z = = ; α( z = ) = ; (0. 05) (0. 0) (0. 005) (0. 005) (0. 09) (0. 05) (0. 008) (0. 006) (0 036) (0 030) α ( z = 3) = ; (0. 056) (0. 055) (0. 055) (0. 050) (coninued...) 5

27 (...Table A3 coninued) = = ; = =.. ; ( z ) ( z ) ( ) z = 3 = ; P = ρ ( A) = 0. 35; LR lineariy es: 0 Noes: The model esimaed is given by equaion (8) in Secion. Tildes denoe esimaed values obained using he expecaion maximizaion (EM) algorihm for maximum likelihood (Dempser, Laird and Rubin, 977). Figures in parenheses are asympoic sandard errors. Symbols are defined as in equaion (8). P denoes he M M ransiion marix. ρ ( A) is he specral radius of he marix A calculaed as in Karlsen (990). I can be hough as a measure of saionariy of he MS- VECM, and for saionariy ρ ( A) < 0 is required. The LR lineariy es is a Davies-ype es for he null hypohesis ha he rue model is a linear VECM ( p ) agains he alernaive of a MSIAH ( M, p ) -VECM (Davies, 977, 987). For he LR lineariy es, we only he repor p- 4 value; 0 indicaes a p value below 0. 6

28 Appendix B. Boosrap procedure for he p-value of he es of equal poin forecas accuracy The boosrap algorihm used o deermine he p-values of he es saisic of equal poin forecas accuracy consiss of he following seps:. Given he sequence of observaions { x } where x = i, z and z denoes he FF explanaory variables, esimae each of he models ()-(8) in Secion and consruc he es saisic of ineres, θ (i.e. es saisic for he null hypohesis, H 0, of equal poin forecas accuracy).. Posulae a daa generaing process (DGP) for each of he models ()-(8) given in Secion, where he FF rae is assumed o follow a drifless random walk under H 0 and he innovaions are assumed o be i.i.d. x 3. Based on he model specified in sep ), generae a sequence of pseudo observaions of he same lengh as he original daa series { x } and discard he firs,000 ransien. The pseudo innovaion erms are random and drawn wih replacemen from he se of observed residuals. Repea his sep 5,000 imes. x 4. For each of he 5,000 boosrap replicaions consruc he es saisic of ineres, θ., esimae each of he models ()-(8) and 5. Use he empirical disribuion of he 5,000 replicaions of he boosrap es saisic, θ o deermine he p-value of he es saisic θ. 7

29 Appendix C. Robusness resuls Table C. Ou-of-sample performance of hree primiive models wihou using he forecass of selemen days a) Poin forecas evaluaion MAE p value RMSE p value RW ARMA(,) Taylor (00) b) Regression-based es for marke iming HR HM WM RW ARMA(,) ( 009. ) Taylor (00) (0. 08) Noes: The definiions of he models are given in Secion. The forecas period goes from January 998 o December Panel a): MAE and RMSE denoe he mean absolue error and he roo mean square error respecively. p value are p-values from execuing es saisics for he null hypohesis ha he model considered has equal poin forecas accuracy as he random walk (RW), calculaed by boosrap using he procedure described in Appendix B. Panel b): HR is he hi raio calculaed as he proporion of correcly prediced signs. HM and WM are he Henriksson and Meron (98) and he efficiency es as in Wes and McCracken (998) calculaed using he auxiliary regressions (0) and () respecively (see Secion 5.). HM and WM is calculaed using he Newey-Wes (987) auocorrelaion and heeroskedasiciy consisen covariance marix. For he 4 HM es saisics only p-values are repored; 0 denoes p-values lower han 0. Values in parenheses are esimaed sandard errors. 8