Fuzzy Regression and the Term Structure of Interest Rates Revisited

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1 Fuzzy Regresson and the Term Structure of Interest Rates Revsted Arnold F. Shapro Penn State Unversty Smeal College of Busness, Unversty Park, PA 68, USA Phone: , Fax: , E-mal: Abstract Recent artcles of Sánchez and Gómez (3a, 3b, 4) addressed the subect of fuzzy regresson (FR) and the term structure of nterest rates (TSIR). Followng Tanaka et. al. (98), ther regresson models ncluded a fuzzy output, fuzzy coeffcents and an non-fuzzy nput vector. The fuzzy components were assumed to be trangular fuzzy numbers (TFNs). The basc dea was to mnmze the fuzzness of the model by mnmzng the total support of the fuzzy coeffcents, subect to ncludng all the gven data. Key components of the Sánchez and Gómez methodology ncluded constructng a dscount functon from a lnear combnaton of splnes, the coeffcents of whch were assumed to be TFNs, and usng the mnmum and maxmum negotated prce of fxed ncome assets to represent the fuzzness of the dependent varable observatons. Gven the fuzzy dscount functons, the authors provded TFN approxmatons for the correspondng spot rates and forward rates. The purpose of ths artcle s to revst the fuzzy regresson portons of the foregong studes, and to dscuss ssues related to the Tanaka approach, ncludng a consderaton of fuzzy least-squares regresson models. Keywords: fuzzy lnear regresson, fuzzy least-squares regresson, fuzzy coeffcents, possblstc regresson, term structure of nterest rates Acknowledgments: Ths work was supported n part by the Robert G. Schwartz Faculty Fellowshp and the Smeal Research Grants Program at the Penn State Unversty. The assstance of Mchelle L. Fultz s gratefully acknowledged. AFIR4_Shapro_5.doc

2 Introducton Recent artcles of Sánchez and Gómez (3a, 3b, 4) addressed the subect of fuzzy regresson (FR) and the term structure of nterest rates (TSIR). Followng Tanaka et. al. (98), ther models took the general form: Y = A A x A () + + L+ n x n where Y s the fuzzy output, Ã, =,,..., n, s a fuzzy coeffcent, and x = (x,..., x n ) s an n-dmensonal non-fuzzy nput vector. The fuzzy components were assumed to be trangular fuzzy numbers (TFNs). Consequently, the coeffcents, for example, can be characterzed by a membershp functon (MF), µ A (a), a representaton of whch s shown n Fgure. Fgure : Fuzzy Coeffcent As ndcated, the salent features of the TFN are ts mode, ts left and rght spread, and ts support. When the two spreads are equal, the TFN s known as a symmetrcal TFN (STFN). The basc dea of the Tanaka approach, often referred to as possblstc regresson, was to mnmze the fuzzness of the model by mnmzng the total spread of the fuzzy coeffcents, subect to ncludng all the gven data. Key components of the Sánchez and Gómez methodology ncluded constructng a dscount functon from a lnear combnaton of quadratc or cubc splnes, the coeffcents of whch were assumed to be TFNs or STFNs, and usng the mnmum and maxmum negotated prce of fxed ncome assets to obtan the spreads of the dependent varable observatons. Gven the fuzzy dscount functons, the authors provded TFN approxmatons for the correspondng spot rates and forward rates. AFIR4_Shapro_5.doc

3 The purpose of ths artcle s to revst the fuzzy regresson portons of the foregong studes, and to dscuss ssues related to the Tanaka (possblstc) regresson model. Ths dscusson s not meant to be exhaustve but, rather, s ntended to pont out some of the maor consderatons. The outlne of the paper s as follows. We frst defne and conceptualze the general components of fuzzy regresson. Next, the essence of the Tanaka model s explored, ncludng a commentary on some of ts potental lmtatons. Then, fuzzy least-squares regresson models are dscussed as an alternatve to the Tanaka model. Throughout the paper, the same smple data set s used to show how the deas are mplemented. The paper ends wth a summary of the conclusons of the study and a suggeston for further study. Fuzzy Lnear Regresson Bascs Ths secton provdes an ntroducton to fuzzy lnear regresson. The topcs addressed nclude the motvaton for FR, the components of FR, fuzzy coeffcents, the h-certan factor, and fuzzy output.. Motvaton Standard (classcal) statstcal lnear regressons takes the form y = β + βx + L + β k xk + ε, =,,..., m () where the dependent (response) varable, y, the ndependent (explanatory) varables, x, and the coeffcents (parameters), β, are crsp values, and ε s a crsp random error term wth E(ε )=, varance σ (ε )=σ, and covarance σ(ε, ε ) =,,,. Although statstcal regresson has many applcatons, problems can occur n the followng stuatons: Number of observatons s nadequate (Small data set) Dffcultes verfyng dstrbuton assumptons Vagueness n the relatonshp between nput and output varables Ambguty of events or degree to whch they occur Inaccuracy and dstorton ntroduced by lnearalzaton Thus, statstcal regresson s problematc f the data set s too small, or there s dffculty verfyng that the error s normally dstrbuted, or f there s vagueness n the relatonshp between the ndependent and dependent varables, or f there s ambguty assocated wth the event or f the lnearty assumpton s napproprate. These are the very stuatons fuzzy regresson was meant to address. AFIR4_Shapro_5.doc 3

4 . The Components of Fuzzy Regresson There are two general ways (not mutually exclusve) to develop a fuzzy regresson model: () models where the relatonshp of the varables s fuzzy; and () models where the varables themselves are fuzzy. Both of these models are explored n the rest of ths artcle, but, for ths conceptualzaton, we focus on models where the data s crsp and the relatonshp of the varables s fuzzy. It s a smple matter to conceptualze fuzzy regresson. Consder for ths, and subsequent, examples the followng smple Ishbuch (99) data: Table : Data Pars x y Startng wth ths data, we ft a straght lne through two or more data ponts n such a way that t bounds the data ponts from above. Here, these ponts are determned heurstcally and OLS s used to compute the parameters of the lne labeled Y H, whch takes the values yˆ = x, as shown n Fgure (a). Fgure : Conceptualzng the upper and lower bound Smlarly, we ft a second straght lne through two or more data ponts n such a way that t bounds the data ponts from below. As shown n Fgure (b), the ftted lne n ths case s labeled Y L and takes the values yˆ = +. 5x. Assumng, for the purpose of ths example, that STFN are used for the MFs, the modes of the MFs fall mdway between the boundary lnes, as ndcated n Fgure 3. AFIR4_Shapro_5.doc 4

5 Fgure 3: Conceptualzng the mode For any gven data par, (x, y ), the foregong conceptualzatons can be summarzed by L U the fuzzy regresson nterval [Y,Y ] shown n Fgure 4. 3 Fgure 4: Fuzzy Regresson Interval h Y = h= U L s the mode of the MF and f a SFTN s assumed, Y = Y = (Y + Y )/. Gven the parameters, (Y U,Y L, Y h= ), whch characterze the fuzzy regresson model, the -th U L h= data par (x,y ), s assocated wth the model parameters (Y,Y,Y ). From a U L regresson perspectve, t s relevant to note that Y - y and y - Y are components of the h SST, y - Y = U h s a component of SSE, and Y - Y = h and Y = L - Y are components of the SSR, as dscussed by Wang and Tsaur (). AFIR4_Shapro_5.doc 5

6 In possblstc regresson based on STFN, only the data ponts nvolved n determnng the upper and lower bounds determne the structure of the model, as depcted n Fgure. The rest of the data ponts have no mpact on the structure. Ths problem s resolved by usng asymmetrc TFNs..3 The Fuzzy Coeffcents Combnng Equaton () and Fgure, and, for the present, restrctng the dscusson to STFNs, the MF of the -th coeffcent, may be defned as: a a µ A ( a) = max, (3) c where a s the mode and c s the spread, and represented as shown n Fgure 5. Defnng A Fgure 5: Symmetrcal fuzzy parameters { a, c } = { A : a c A a + c }, =,, L n = L, L and restrctng consderaton to the case where only the coeffcents are fuzzy, we can wrte (4) Y = A + x = = ( a n A n, c) L + ( a, c ) L = x (5) AFIR4_Shapro_5.doc 6

7 Ths s a useful formulaton because t explctly portrays the mode and spreads of the fuzzy parameters. In a subsequent secton, we explore fuzzy ndependent varables..4 The "h-certan" Factor If, as n Fgure 4, the supports 4 are ust suffcent to nclude all the data ponts of the sample, there would be only lmted confdence n out-of-sample proecton usng the estmated FRM. Ths s resolved for FR, ust as t s wth statstcal regresson, by extendng the supports. Consder the MF assocated wth the -th fuzzy coeffcent, a representaton of whch s shown n Fgure 6. Fgure 6: Estmatng A usng an "h-certan" factor For llustratve purposes, a non-symmetrc TFN s shown, where c L and c R represent the left and rght spread respectvely. Beyond that, what makes ths MF materally dfferent from the one shown n Fgure 5, s that t contans a pont "h" on the y-axs, called an "hcertan factor," whch, by controllng the sze of the feasble data nterval (the base of the shaded area), extends the support of the MF. 5 In partcular, as the h-factor ncreases, so L ncreases the spreads, c and c R..5 Observed Fuzzy Output An h-certan factor also can be appled to the observed output. Thus, the -th output data mght be represented by the STFN, Y (y,e ), where y s the mode and e s the spread, = AFIR4_Shapro_5.doc 7

8 as shown n Fgure 7. Here, the actual data ponts fall wthn the nterval y ± (-h) e, the base of the shaded porton of the graph. Fgure 7: Observed Fuzzy Output.6 Fttng the Fuzzy Regresson Model Gven the foregong, two general approaches are used to ft the fuzzy regresson model: The possblstc model. Mnmze the fuzzness of the model by mnmzng the total spreads of ts fuzzy coeffcents (see Fgure ), subect to ncludng the data ponts of each sample wthn a specfed feasble data nterval. The least-squares model. Mnmze the dstance between the output of the model and the observed output, based on ther modes and spreads. The detals of these approaches are addressed n the next two sectons of ths paper. 3 The Possblstc Regresson Model The possblstc regresson model s optmzed by mnmzng the spread, subect to adequate contanment of the data. The spread s mnmzed n mn c + c x, c = (6) AFIR4_Shapro_5.doc 8

9 Fgure 8 shows portrays the frst step n the contanment requrement, by showng how Fgure 6 can be easly extended to portray the fuzzy output of the model. Fgure 8: Fuzzy output of the model Puttng ths together wth the observed fuzzy output, Fgure 7, results n Fgure 9, whch shows a representaton of how the estmated fuzzy output may be ftted to the observed fuzzy data. Fgure 9: Fttng the estmated output to the observed output The key s that the observed fuzzy data, adusted for the h-certan factor, s contaned wthn the estmated fuzzy output, adusted for the h-certan factor. Formally, AFIR4_Shapro_5.doc 9

10 n n a + a x + ( h) c + c x > y + ( h) e = = (7) n n a + a x ( h) c + c x < y ( h) e = = c, =,,..., m, =,,..., n Fgure 6 shows the mpact of the h-factor on the sample data, gven h= and h=.7. Fgure : FLR and h-certan model The result s what one would expect. Increasng the h-factor expands the confdence nterval and, thus, ncreases the probablty that out-of-sample values wll fall wthn the model. Ths s comparable to ncreasng the confdence n statstcal regresson by ncreasng n the confdence nterval. The possblstc lnear regresson model, as depcted by equatons (6) and (7), s essentally the fuzzy regresson model used by Sánchez and Gómez (3a, 3b, 4) to nvestgate the TSIR. AFIR4_Shapro_5.doc

11 3. Crtcsms of the Possblstc Regresson Model There are a number of crtcsms of the possblstc regresson model. Some of the maor ones are the followng: Tanaka et al "used lnear programmng technques to develop a model superfcally resemblng lnear regresson, but t s unclear what the relaton s to a least-squares concept, or that any measure of best ft by resduals s present." [Damond (988: 4-)] The orgnal Tanaka model was extremely senstve to the outlers. [Peters (994)]. There s no proper nterpretaton about the fuzzy regresson nterval [Wang and Tsaur ()] Issue of forecastng have to be addressed [Savc and Pedrycz (99)] The fuzzy lnear regresson may tend to become multcollnear as more ndependent varables are collected [Km et al (996)]. The soluton s x pont-of-reference dependent, n the sense that the predcted functon wll be very dfferent f we frst subtract the mean of the ndependent varables, usng (x - x ) nstead of x. [Hoat (4), Bardossy (99) and Bardossy et al (99)] 4 The Fuzzy Least-Squares Regresson (FLSR) Model An obvous way to brng the FR more n lne wth statstcal regresson s to model the fuzzy regresson along the same lnes. In the case of a sngle explanatory varable, we start wth the standard lnear regresson model: [Kao and Chyu (3)] y = β + β x + ε, =,,..., m (8) whch n a comparable fuzzy model takes the form: Y X = β + β + ε, =,,...,m (9) Conceptually, the relatonshp between the fuzzy -th response and explanatory varables n (9) can be represented as shown n Fgure. AFIR4_Shapro_5.doc

12 Rearrangng the terms n (9), Fgure : fuzzy -th response and explanatory varables ε Y β β X =, =,,...,m () From a least squares perspectve, the problem then becomes n mn S = ( Y = There are a number of ways to mplement FLSR, but the two basc approaches are FLSR usng dstance measures and FLSR usng compatblty measures. A descrpton of these methods follows. () 4. FLSR usng Dstance Measures (Damond's Approach) Damond (988) was the frst to mplement the FLSR usng dstance measures and hs methodology s the most commonly used. Essentally, he defned an L - metrc d(.,.) between two TFNs by [Damond (988: 43) equaton ()] b b X ) d ( m, l, r, m, l, r ) = ( m m ) + (( m l ) ( m l )) + (( m + r ) ( m + r )) () Gven TFNs, t measures the dstance between two fuzzy numbers based on ther modes, left spread and rght spread. 7 The case most smlar to the Sánchez and Gómez model takes the form Y = β x + β + ε, =,,..., m (3) AFIR4_Shapro_5.doc

13 and requres the optmzaton of mn d( A + Bx, Y ) A, B The soluton follows from (), and f B s postve, t takes the form: (4) L L d ( A + xb, Y ) = ( a + bx y ) + ( a + bx ca cb x y + R R + ( a + bx + c + c x y + c c L Y ) (5) A smlar expresson holds when B s negatve. If the solutons exst, the parameters of A and B satsfy a system of sx equatons n the same number of unknowns, these equatons arsng from the dervatves assocated wth (5) beng set equal to zero. Of course, ths ftted model has the same general characterstcs as prevously shown, but now we can use the resdual sum of d-squares to gauge the effectveness of model. In the case most remnscent of statstcal regresson, the coeffcents are crsp and the task becomes the least-squares optmzaton problem A B R Y ) mn d( a + bx, Y ) a, b (6) Once agan, the soluton s gven by (), adusted to take nto account the sgn of b. Fnally, an nterestng problem when mplementng the Damond approach s assocated wth models of the form Y = β X + β + ε, =,,..., m (7) for whch there s no general soluton, snce the LHS, Y, s a TFN whle the RHS nvolves the fuzzy product β X, whose sdes are drumlke. One approach to ths ths problem (Hong et al ()) s to replace the t-norm mn(a,b) wth t-norm T w (a,b) = a, f b=; b, f a=;, otherwse. Snce T w (a,b) s a shape preservng operaton under multplcaton, t resolves the problem. Another approach s to use approxmate TFNs. 4. FLSR usng compatblty measures An alternate least-squares approach s based on the Celmņš (987) compatblty measure γ ( A, B) = max mn{ µ ( X ), µ ( X )} (8) x A B AFIR4_Shapro_5.doc 3

14 representatve examples of whch are shown n Fgure. 8 Fgure : Celmņš Compatblty Measure As ndcated, γ ranges from, when the MFs are mutually exclusve, to, when the modes of the MFs concde. Celmņš compatblty model, whch nvolved maxmzng the compatblty between the data and the ftted model, follows from ths measure. The obectve functon s m ( = Thus, for example, when there s a sngle crsp explanatory varable, [Chang and Ayyub (: 9)] γ ) (9) ˆ Y = A + A x = m + m x ± c + c x + c x () where m and m are determned usng weghted LS regresson, and c, c, and c are determned usng teraton and the desred compatblty measure. An example of the use of the Celmņš compatblty model appled to our sample data s shown n Fgure 3. 9 AFIR4_Shapro_5.doc 4

15 Fgure 3: FLS usng maxmum compatblty crteron The essental characterstcs of the model n ths case are the parabolc curves for the upper and lower bounds and that the hgher the compatblty level, the broader the wdth of the bounds. 5 Comment The studes of Sánchez and Gómez provde some nterestng nsghts nto the use of fuzzy regresson for the analyss of the TSIRs. However, ther methodology reles on possbltc regresson, whch has the potental lmtatons mentoned n secton 3.. Snce some of these lmtatons can be crcumvented by usng FLSR technques, t s mportant that researchers are famlar wth these technques as well. If ths artcle helps n ths regard, t wll have served ts purpose. References Bardossy, A. (99) "Note on fuzzy regresson," Fuzzy Sets and Systems 37, Bardossy, A., I. Bogard and L. Ducksten. (99) "Fuzzy regresson n hydrology," Water Resources Research 6, Celmņš, A. (987) "Least squares model fttng to fuzzy vector data," Fuzzy Sets and Systems, (3), Chang, Y.-H. O. and B. M. Ayyub. () "Fuzzy regresson methods a comparatve assessment," Fuzzy Sets and Systems, 9(), 87-3 AFIR4_Shapro_5.doc 5

16 Damond, P. (988) "Fuzzy least squares," Informaton Scences 46(3), 4-57 Hoat, M., C. R. Bector and K. Smmou. (4) "A smple method for computaton of fuzzy lnear regresson," European Journal of Operatonal Research (forthcomng) Hong, D. H., J-K. Song and H.Y. Do. () "Fuzzy least-squares lnear regresson analyss usng shape preservng operatons," Informaton Scences Ishbuch, H. (99) "Fuzzy regresson analyss," Fuzzy Theory and Systems, 4, Kao, C. and C-L Chyu. (3) "Least-squares estmates n fuzzy regresson analyss," European Journal of Operatonal Research 48, Km, K. J., H. Moskowtz and M. Koksalan. (996) "Fuzzy versus statstcal lnear regresson," European Journal of Operatonal Research, 9() McCauley-Bell, P. and H. Wang. (997) "Fuzzy lnear regresson models for assessng rsks of cumulatve trauma dsorders," Fuzzy Sets and Systems, 9(3), Peters, G. (994) "Fuzzy lnear regresson wth fuzzy ntervals," Fuzzy Sets and Systems, 63(), Sánchez, J. de A. and A. T. Gómez. (3a) "Applcatons Of Fuzzy Regresson In Actuaral Analyss," JRI 3, 7(4), Sánchez, J. de A. and A. T. Gómez. (3b) "Estmatng a term structure of nterest rates for fuzzy fnancal prcng by usng fuzzy regresson methods," Fuzzy Sets and Systems, 39(), Sánchez, J. de A. and A. T. Gómez. (4) "Estmatng a fuzzy term structure of nterest rates usng fuzzy regresson technques," European Journal of Operatonal Research 54, Savc, D. A. and W. Pedrycz. (99) "Evaluaton of fuzzy lnear regresson models," Fuzzy Sets and Systems, 39(), 5-63 Tanaka, H., Uema, S. and Asa, K. (98) "Lnear regresson analyss wth fuzzy model," IEEE Transactons on Systems, Man and Cybernetcs, (6), Wang, H.-F. and R.-C. Tsaur. () "Insght of a fuzzy regresson model," Fuzzy Sets and Systems, (3), Wünsche, A. and W. Näther. () "Least-squares fuzzy regresson wth fuzzy random varables," Fuzzy Sets and Systems, 3(), 43-5 AFIR4_Shapro_5.doc 6

17 Endnotes Snce the spot rates and forward rates are nonlnear functons of the dscount functon, they are not TFNs even though the dscount functon s a TFN. Ths approach to choosng the mode was dscussed by Wang and Tsaur () p Adapted from Wang and Tsaur (), Fgure. 4 Support functons are dscussed n Damond (988: 43) and Wünsche and Näther (: 47). 5 Note that the h-factor has the opposte purpose of an α-cut, n that the former s used to extend the support, whle the latter s used to reduce the support. 6 Adapted from Chang and Ayyub (), Fgure 4. 7 The methods of Damond's paper are rgorously ustfed by a proecton-type theorem for cones on a Banach space contanng the cone of trangular fuzzy numbers, where a Banach space s a normed vector space that s complete as a metrc space under the metrc d(x, y) = x-y nduced by the norm. 8 Adapted from Chang and Ayyub (), Fgure. 9 Adapted from Chang and Ayyub (), Fgure 5. AFIR4_Shapro_5.doc 7