Vol. 5 (1), S/No11, February, 2016: ISSN: (Print) ISSN (Online) DOI:

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1 STECH VOL 5 () FEBRUARY, 06 7 Vol. 5 (), S/No, Februar, 06: 7-39 ISSN: (Prin) ISSN (Online) DOI: p://dx.doi.org/0.434/sec.v5i.3 Inerval Forecas for Smoo Transiion Auoregressive Model Ekosuei, Nosa Deparmen of Maemaics, Universi of Benin, Benin Ci, Edo Sae, Nigeria of corresponding auor: ekosu@aoo.com Omosigo, S. E. Deparmen of Maemaics, Universi of Benin, Benin Ci, Edo Sae, Nigeria of corresponding auor: ekosu@aoo.com & Absrac In is paper, we propose a simple meod for consrucing inerval forecas for smoo ransiion auoregressive (STAR) model. Tis inerval forecas is based on boosrapping e residual error of e esimaed STAR model for eac forecas orizon and compuing various Akaike informaion crierion (AIC) funcion. Tis new inerval forecas sugges definie and beer coverage o e fuure sample pa an e convenional meod of using a muliple of sandard error of e forecas disribuion using boosrap meod. Simulaion sudies are used o illusrae e proposed meod. Ke words: AIC; boosrap; Inerval forecas; STAR Coprig IAARR, 0-06: Indexed African Journals Online:

2 STECH VOL 5 () FEBRUARY, Inroducion One of e major reasons for ime series modelling is o forecas fuure values. Forecasing fuure values of ime series daa ma ake e form of poin forecas or an inerval forecas. For non-linear ime series model, e consrucion of poin and inerval forecas is problemaic. I is known a in linear auoregressive moving average (ARMA(p, q)) model, forecas inerval can be consruced eoreicall using a weig funcion derived from e esimaed model, see Box and Jenkins (976). In non-linear ime series models, a eoreical forecas inerval is no eas o consruc. Cafield (993) presened a road map for consrucing inerval forecas for saionar and non-saionar models and Crisofferson (998) gave a general meod for evaluaing inerval forecas wile oer work on inerval forecas was done using Sieve boosrap meod, see Bülmann (997), Alonso e.al (00, 003) for examples. Tese auors onl acieve good predicion inerval resuls for linear ARMA models wereas eir predicion inervals failed for non-linear models. Te simple reason for is is a linear ARMA model can be represened as an infinie linear process wile non-linear model canno be represened so. Giordano e.al (007) used a neural nework sieve boosrap o consruc inerval forecas for STAR model. Tis was also made possible b using a class of neural nework o approximae e original non-linear model. Teir approac is raer complicaed since e neural nework requires raining. A e momen, ere is no sandard meod of obaining inerval forecas for non-linear ime series model. In is sud we propose a simple forecas inerval for non-linear ime series model wi paricular aenion given o logisic smoo ransiion auoregressive (LSTAR) model. Our meod makes use of boosrapping e residuals generaed from e esimaed STAR model and compuing various AIC funcion, due o Akaike (969). Te res of is work is organized as follows. In Secion, we describe briefl e basic represenaion of STAR model and poin forecas. In Secion 3, we presen e proposed meodolog for consrucing inerval forecas for STAR model. Secion 4 is on simulaion experimens and resuls o illusrae e proposed meod. In Secion 5, e resuls of e simulaion sudies are presened. Secion 6 concludes e paper.. Represenaion of e basic STAR model and poin forecas We firs consider e auoregressive (AR(p)) model of order p before considering e basic STAR model. An AR(p) model of order p, for some posiive ineger p, can be wrien as: ' (.) w Coprig IAARR, 0-06: Indexed African Journals Online:

3 STECH VOL 5 () FEBRUARY, 06 9 were w a e zeroes of ' and,,, are real parameers suc ' (,,,, p ) ( 0 p ) p ( z ) z z p lie ouside e uni disk and normall disribued wi mean zero and variance. auoregressive (STAR) model for a univariae ime series,,,n is given b: ) is Te smoo ransiion wic is observed a ' ' w wg( s ;, c (.) ' were (,, ), i,, 0, i i, 0 i, p w and are as defined in (.). G( s ;, c) is called e ransiion funcion and lies beween 0 and.te ransiion funcion could eier be of e logisic pe or e exponenial pe. Te variable s is called e ransiion variable wic could be assumed o be a lagged endogenous variable a is s d for cerain ineger d 0 or linear rend ( s ), wic gives rise o smool canging parameers. If e ransiion funcion is of e Logisic pe, we ave G(, c; s ) (.4) ( exp{ ( s c)}) Afer esimaion of e parameers of an of e model given b (.) and (.) and e model found o be adequae, en i can be used for forecasing. We illusrae ow o generae poin forecas using a general model of order, F( ; ), (.5) for some linear or non-linear funcion of F( ; ). Te opimal poin forecas of fuure values of e ime series are given b eir condiional expecaions. See Box and Jenkins (976), Franses and Van Dijk (000) for exposiion. Te -sep-aead forecas of e fuure values a ime is given b: ˆ E[ / ] (.6) were denoes e pas isor of e ime series up o and including e observaion a ime. An opimal -sep-aead forecas, using e fac a E[ / ] 0 is obained as: ˆ E[ / ] F( ; ) (.7) Coprig IAARR, 0-06: Indexed African Journals Online:

4 STECH VOL 5 () FEBRUARY, Tis opimal -sep-aead forecas for F( ; ) is e same for linear and non-linear model. For -sep-aead greaer an or equal forecass can be generaed recursivel wiou difficul for linear model, wile is is no longer eas o andle for non-linear model. In fac, an opimal -sep-aead forecas using (.6) and (.7) for non-linear model is given b: ˆ E[ / ] E[ F( ; ) / ] (.8) Te linear condiional expecaion operaor E canno be inercanged wi e nonlinear operaor F; ence (.8) can be expressed using e relaionsip a exiss beween - and- -seps aead forecass wic is now wrien as: ˆ E[ F( F( ; ) ; ) / ] E[ F( ˆ ; ) / ] (.9) Te exac meod of forecas from (.9), is ˆ F( ; ) d( ) d (.0) (.0) requires numerical inegraion and e dimension of inegraion increases as e forecas orizon increases. As a resul of is, is exac meod is usuall replaced b simulaion meod, given b: ˆ M (/ M ) F( i ; ) i (.) (.) is called e Mone-Carlo meod if an assumed error disribuion is used or e boosrap meod if e error generaed from e fied meod suc as e one in (.) is used. 3 Meodolog for Consrucing Inerval Forecas for STAR Model Given a ime series{ },,,..., n ; Franses and Van Dijk (000) lised ree meods of consrucing inerval forecas for STAR models. Tese are: 3. An inerval smmeric around e mean, i is defined b: S ( ˆ w, ˆ w), P were w is cosen suc a ( S / ), ˆ is forecas made a ime origin for a specific orizon, and is e expeced fuure value. Coprig IAARR, 0-06: Indexed African Journals Online:

5 STECH VOL 5 () FEBRUARY, 06 3 For linear models, w is given b w z ( ( )) / / FMSE, wile for non-linear models, w M(sde) were sde is e sandard error of e forecas disribuion a eac forecas orizon, M 0 is an ineger. Tus, we mus generae e forecas before we can consruc e forecas inerval as suggesed b Hndman (995). 3. An inerval beween / and ( / ) quaniles of e forecas disribuion denoed q and q / respecivel. Tis inerval is given b: b / Q ( q /, q / ) 3.3 Te iges densi region (HDR) a is, were HDR g is suc a { / g( / ) g} P HDR / ) ( / Hndman (996), assered a e ree meods of consrucing forecas region using (3.)-(3.3) as been found o be e same wen e forecas disribuion is normal. Ta for non-normal forecas disribuions; e regions are all differen and recommend a iges-densi forecas regions be used. For linear models, w is given b w z ( FMSE ( )) /, were FMSE () can be obained eoreicall. For nonlinear models, w M(sde) were sde is e sandard error given b / sde ( FMSE ( )) wic canno be obained eoreicall and M (an ineger) is used o consruc inerval forecas. For non-linear model w is se equal o a muliple of sandard error of e forecas disribuion. Tis meod of inerval forecas in nonlinear model does no sugges an appropriae or opimal inerval forecas. given b: Now, for an forecas made ere exiss a forecas error or predicion error e ˆ (3.4) e were is e fuure values and ˆ is e forecas made a ime. Franses and Van Dijk (000) saed a i is desirable o coose e forecas ˆ a minimizes e forecas mean squared error (FMSE) given b: FMSE ( ) E( e ) E[( ˆ ) ] (3.5) Coprig IAARR, 0-06: Indexed African Journals Online:

6 STECH VOL 5 () FEBRUARY, 06 3 I as been sowed in Box-Jenkins (976) a e forecas a minimizes (3.5) is e condiional expecaion of a ime, a is, ˆ E[ /. ] Assuming normali, a 00( )% forecasing inerval for ˆ in ARMA(p, q) is given b e following inerval: ˆ / Z /.( FMSE ( )) and / ˆ Z /.( FMSE ( )) (3.6) were FMSE () can be obained eoreicall and i is given b: j0 FMSE ( ) (3.7) j were 0, j for j,,..., are e forecas weigs wic can be compued recursivel from infinie represenaion of ARMA(p, q) model and is e residual error variance esimaed from a fied ARMA(p, q) model. Te AIC funcion ma be given b an of e following: AIC ( k) nlog( ˆ ) k (3.8) AIC ( k) nlog( ˆ ) k (3.9) Te expressions for AIC in (3.8) and (3.9) can eier be posiive or negaive. Tese values are used indiscriminael in e lieraure. In order o obain posiive values for AIC, Ekosuei (00) adoped e following expression: n log( ˆ ) k, log( ˆ ) 0 AIC ( k) (3.0) n log( ˆ ) k, log( ˆ ) 0 were k denoe e number of parameers in e model, n is e sample size and n ˆ n ˆ, were ˆ is e residuals generaed from e fied model. Ekosuei and Omosigo (0), sow a e sampling disribuion of e AIC funcion represened b (3.8) and (3.9) as e normal disribuion, wile e one given b (3.0) as a Ci-square disribuion. Te specificaion given b (3.0) is a i permis posiive values of AIC a all imes. In a simulaion experimen, Ekosuei and Coprig IAARR, 0-06: Indexed African Journals Online:

7 STECH VOL 5 () FEBRUARY, Omosigo (00) esablised a close relaionsip beween FMSE () given in (3.7) and e AIC given in (3.0) namel: ( FMSE ( )) / AIC( k) (3.) n Our meod for obaining w is based on e boosrap meod o esimae e FMSE for LSTAR model. Tis meod is acieved b consrucing B number of AIC funcion from e re-sampled error sequence, and en compues e sandard error. Te idea is similar o e Sieve boosrap for imes series proposed b Bülmann (997) and Sieve boosrap inerval b Alonso e.al (00, 003). Te AIC funcion a is used in is regard is e one defined b (3.8) or (3.9) since e ield e same variance wen eir boosrap is conduced. Te procedure for consrucing e AIC predicion inerval is given b e following seps: Sep. Given a ime series observaions { }, use JMULTI (a ime series sofware package) o model an appropriae LSTAR model. Sep. Generae e residuals ˆ from e fied LSTAR model for p,, n Sep4. Compue e empirical disribuion funcion of e sandardized residuals, n ~ ˆ * F *( x) ( n p) { * x}, were, ~ n ˆ ( n p) ˆ p ˆ p and n ( n p) ˆ p * Sep5. Draw a resample of i.i.d observaions from e sandardized residuals o obain AIC forecas values of leng using e boosrap meod and also compue e funcion for eac boosrap replicaion. Sep6. Compue e sandard error of e boosrap AIC. Were e sandard error is given as: Coprig IAARR, 0-06: Indexed African Journals Online:

8 STECH VOL 5 () FEBRUARY, were / B * b s eˆ B( AIC ) ( AIC AIC *) (3.) ( B ) b AIC* B B b AIC * b Sep7. Consruc e inerval forecas using S ( ˆ, ˆ w w) were w ( ˆ ) 3 se B( AIC ) 4 Simulaion Experimen We consider e following daa generaing process (DGP) using e logisic smoo ransiion auoregressive (LSTAR) model. Model (I) is an LSTAR() model wile Model (II) is an LSTAR() model. Model (I) ( exp( 0 )) Model (II).8.06 ( ) G( ) were G ( ) ( exp{ 0( 0.0)}). Te sequence of error,, wic is normall disribued wi mean 0 and variance, was generaed using e random number generaor in MATLAB We generaed ( 300 n) sample sizes using Models (I) and (II). Onl e las n observaions are kep, wile e firs 300 are discarded o minimize iniializaion effec. Among e n generaed arificial ime series observaions, e firs ( n 0) observaions were used for modeling wile e remaining 0 observaions are kep for ou-of-sample performance. Te modeling ccle wic involves esing for nonlineari, model specificaion and evaluaion suc as in Luukonnen e al. (988a), Luukonnen e al. (988b), Teräsvira (994), Eireim and Teräsvira (996), Van Dijk e al. (00) were execued using a Time series sofware package called JMULTI. Afer esimaion of e model, i is en used for forecasing and for consrucing forecas inervals. Te forecasing meod emploed is e boosrap meod using e idea of Efron and Tibsirani (998), Marinez and Marinez (00). Tis sage of compuing poin forecas and consrucing forecas inerval in non-linear LSTAR model was wrien and execued in MATLAB7.5.0, since JMULTI does no provide forecas opion in eir package. Te forecasing orizon ( ) is aken from o 0 0, so as o agree wi e number of observaions kep for e ou-of-sample performance. Coprig IAARR, 0-06: Indexed African Journals Online:

9 STECH VOL 5 () FEBRUARY, Tis process was replicaed man imes. We owever, presen few resuls in e nex secion o illusrae is new inerval forecas. Figures and sows e upper and lower forecas limis using e AIC boosrap wi a muliple of e sandard error compued from e forecas mean square error using Model (I) for sample size n = 00 and 00 respecivel, wile Figures 3 and 4 sows e upper and lower forecas limis using e AIC boosrap wi a muliple of e sandard error compued from e forecas mean square error using Model (II) for sample size n = 00 and 00 respecivel. All e graps also conain e acual observaions kep for ou-of-sample performance and e poin forecas made. Te muliple of e sandard error of e forecas disribuion used for consrucing e inerval forecas for e boosrap meod ranges from o 5. Figure : Represen forecas inerval using plus or minus a muliple of sandard deviaion as forecas inerval and e proposed AIC forecas inerval using rice boosrap sandard error for model (I) for sample size n- 0 = 00, 0 = 0. Coprig IAARR, 0-06: Indexed African Journals Online:

10 STECH VOL 5 () FEBRUARY, Figure : Represen forecas inerval using plus or minus a muliple of sandard deviaion as forecas inerval and e proposed AIC forecas inerval using rice boosrap sandard error for model (I) for sample size n- 0 = 00, 0 = 0. Figure 3: Represen forecas inerval using plus or minus a muliple of sandard deviaion as forecas inerval and e proposed AIC forecas inerval using rice boosrap sandard error for model (II) for sample size n- 0 = 00, 0 = 0. Coprig IAARR, 0-06: Indexed African Journals Online:

11 STECH VOL 5 () FEBRUARY, Figure 4: Represen forecas inerval using plus or minus a muliple of sandard deviaion as forecas inerval and e proposed AIC forecas inerval using rice boosrap sandard error for Model (II) for sample size n- 0 = 00, 0 = 0. 5 Discussion of Resuls Te inerval forecas in Figures and are consruced using plus or minus a muliple of sandard deviaion for e forecas disribuion wen boosrap meod is applied and e AIC boosrap forecas inerval for sample sizes n = 00 and 00 respecivel using Model (I). Similarl, Figures 3 and 4 are consruced using plus or minus a muliple of sandard deviaion for e forecas disribuion wen boosrap meod is applied and e AIC boosrap forecas inerval for sample sizes n = 00 and 00 respecivel using Model (II). In Figures and, i is observed a e larges inerval of e muliple of sandard deviaion, wic is 5 imes sandard deviaion gives 70% and 80% coverage o e expeced fuure values respecivel wile e boosrap AIC forecas inerval gives a 90% and 00% coverage o e expeced fuure values respecivel. In Figure 3 and 4, e smalles inerval of e muliple of sandard deviaion, wic imes sandard deviaion gives 00% coverage o e expeced fuure values respecivel wile e boosrap AIC forecas inerval gives 70% and 00% coverage o e expeced fuure values. Coprig IAARR, 0-06: Indexed African Journals Online:

12 STECH VOL 5 () FEBRUARY, Conclusion In is paper, a new meod of consrucing inerval forecas for non-linear LSTAR model using e boosrap AIC funcion is presened. Tis new inerval forecas is consruced using e boosrap sandard error of AIC obained b re-sampling from e generaed residuals of e esimaed LSTAR model. Simulaion resuls sow a e inerval prediced b is meod gives a beer coverage compared o e convenional meod of using a muliple of e sandard deviaion compued from e forecas disribuion wic does no indicae or sugges an appropriae forecas inerval. References Akaike, H. (969) Fiing auoregressive models for predicion. Annals of e insiue of saisical Maemaics,, Alonso, A. M, Peña, D., and Romo, J. (00). Forecasing ime series wi sieve boosrap. J. Sais. Planning inference 00, - Alonso, A. M., Peña, D., & Romo, J. (003). On sieve boosrap predicion inervals. Saisics & Probabili Leers 65, 3-0. Box, G. E. P & Jenkins, G. M. (976). Time series analsis, forecasing and conrol. San Francisco:Holden-Da. Bülmann, P. (997). Sieve boosrap for ime series. Bernoulli, 3() pp Cafield, C. (993). Calculaing inerval forecass. Journal of Business Saisics, Vol., No. pp Crisofferson, P. F. (998). Evaluaing inerval forecass. Inernaional Economic Review, Vol. 39, No. 4 pp Efron, B. & Tibsirani, R. J. (998). An inroducion o e Boosrap. London: Capman and Hall. Ekosuei, N. (00). Modelling and forecasing wi linear and nonlinear Time series daa. P.D Tesis submied o scool of posgraduae sudies, Universi of Benin, Benin Ci, Nigeria. Ekosuei, N. & Omosigo, S. E. (00). New inerval forecas for saionar Auoregressive models. Inernaional Journal of Naural and Applied Sciences, 6(4): pp Ekosuei, N. & Omosigo, S. E. (0). On e sampling disribuion of Akaike informaion crierion. Nigerian Journal of Applied Science. Vol. 30 pp Eireim, O. & Teräsvira, T. (996). Tesing e adequac of smoo ransiion auoregressive models. Journal of Economerics 74, Coprig IAARR, 0-06: Indexed African Journals Online:

13 STECH VOL 5 () FEBRUARY, Giordano, F., M. La Rocca & Perna, C. (007). Forecasing nonlinear ime series wi neural nework sieve boosrap. Compuaional saisics and Daa Analsis, 5, pp Hndman, R. J. (995). Higes densi forecas regions for non-linear and non-normal ime series models. Journal of Forecasing 4, pp Hndman, R. J. (996). Compuing and graping iges densi regions. Te American Saisician, 50, 0-6. Lundberg, S. & T. Teräsvira (00). Forecasing wi Smoo Transiion Auoregressive models, in Clemens, M. P. & Hendr, D. F. (eds.), A Companion o Economic Forecasing. Oxford: Blackwell, pp Luukonnen, R., Saikkonen, P. & Teräsvira, T., (988a). Tesing lineari in univariae ime series models. Scandinavian Journal of Saisics, 5, Luukonnen, R., Saikkonen, P. and Teräsvira, T., (988b). Tesing lineari agains smoo ransiion auoregressive models. Biomerika, 75, Marinez, W. L. & Marinez, A. R. (00). Compuaional Saisics Handbook wi MATLAB. London: Capman and Hall. Teräsvira, T., (994). Specificaion, esimaion, and evaluaion of smoo ransiion auoregressive models, Journal of e American Saisical Associaion, 89, Van Dijk, D., Teräsvira, T., & Frances, P. H. (00). Smoo ransiion auorergressive models-a surve of recen developmens. Economeric Reviews,, Coprig IAARR, 0-06: Indexed African Journals Online: