Smooth Transition Autoregressive-GARCH Model in Forecasting Non-linear Economic Time Series Data

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Journal of Saisical and conomeric Mehods, vol., no., 03, -9 ISSN: 05-5057 (prin version), 05-5065(online) Scienpress d, 03 Smooh Transiion Auoregressive-GARCH Model in Forecasing Non-linear conomic Time Series Daa Akinunde Muairu Oyewale, Shangodoyin Dahud Kehinde and Kgosi Phazamile 3 Absrac The regime swiching models are paricularly popular in he comiy of non-linear models; i is of inereso invesigae regime swiching models wih GARCH specificaion. GARCH model was augmened wih STAR model vis-a vis xponenial auoregressive GARCH (AR-GARCH), xponenial smooh ransiion auoregressive GARCH (STAR- GARCH) model and ogisic smooh ransiion auoregressive GARCH (STAR-GARCH) model. The properies of he new models were derived and compared wih convenional GARCH model which shows hahe variance obained for STAR- GARCH model was minimum compared o classical GARCH model, he new mehodology proposed is illusraed wih foreign exchange rae daa from Grea Briain (Pound) and Boswana (Pula) agains Unied Saes of America (Dollar). I is evidenha all STAR-GARCH ouperformed he classical GARCH model, however, STAR- GARCH performed bes and closely followed by STAR-GARCH, his is followed by AR-GARCH. The implicaion is hahe use of STAR GARCH produces he bes resul; however STAR may be uilized in some occasions. Keywords: GARCH models, STAR-GARCH models, AR-GARCH. STAR-GARCH STAR-GARCH, foreign exchange daa. Deparmen of Saisics, faculy of social sciences, Universiy of Boswana, Gaborone Universiy of Boswana, Gaborone. Deparmen of Saisics, faculy of social sciences, Universiy of Boswana, Gaborone Universiy of Boswana, Gaborone. 3 Deparmen of Saisics, faculy of social sciences, Universiy of Boswana, Gaborone Universiy of Boswana, Gaborone. Aricle Info: Received : January 9, 03. Revised : March, 03. Published online : June, 03

Akinunde Muairu Oyewale, Shangodoyin Dahud Kehinde and Kgosi Phazamile Inroducion Non-linear ime series models are increasingly becoming very popular his is because several financial asses canno be modeled by pure linear processes. I seems o be generally acceped ha many economic variables follow non-linear processes. The regime swiching models are paricularly popular in he commiee of non-linear models, i is of inereso sudy regime swiching models wih GARCH specificaion, in his paper GARCH model will be augmened wih STAR model vis-a vis xponenial auoregressive GARCH (AR-GARCH), ogisic smooh ransiion auoregressive GARCH (STAR-GARCH) model and xponenial smooh ransiion auoregressive (STAR- GARCH) model. Disinc feaures of GARCH model and is exension lies on he fachahey have abiliy o capure volailiy clusering, for insance, if he shock from he previous period is high or low, large or small he values of will cerainly have an effec on is variance. Smooh ransiion auoregressive (STAR) models are applied o ime series daa as an exension of auoregressive models, in order o allow for higher degree of flexibiliy in model parameers hrough a smooh ransiion. So also STAR models are inroduced according o Terasvira and Anderson (99), Granger and Terasvira (994) and Terasvira (994) because of he exisence of wo disinc regimes, wih poenially differen dynamic properies and hahe ransiion beween he regimes is smooh. STAR models allow economic variables o follow a given number of regimes wih swiches beween regimes achieved in a smooh and coninuous fashion and governed by he value of a paricular variable or group of variables. The ransiion parameer denoed by s,, c is a slope of parameer ha deermines he speed of ransiion beween he wo exreme regimes wih low absolue values resuling in slower ransiion. I should be noed ha s,, c are generaed by daa series. Two commonly used ransiion funcions are he logisic auoregressive (STAR) model and exponenial auoregressive (STAR) model. However, exponenial auoregressive (AR) model from where he STAR was generalized will be sudied along wih hese wo radiionally sudied models. In he non-linear GARCH model, he condiional variance is expressed as a non linear of lagged residuals. In he STAR models, he non-lineariy is inroduced via eiher logisic or an exponenial ransiion funcion. The non-lineariy in his paper is linked o exisence of bid-ask spread in he currencies being exchanged. General Represenaion We define he general represenaion of STAR model as: y x G y,, c x G y,, c (),,,...,,,,...,,,, is he error erm Where x y y y p i0 i ip i disribued as independenly and idenically wih mean zero and variance. Gs,, c is he ransiion funcion bounded beween zero and uniy and hus allowing for a smooh ransiion beween regimes.

Smooh Transiion Auoregressive-GARCH Model in Forecasing 3 Now, using he lagged endogenous variable, he various forms of STAR models are as follows: The logisic STAR model is expressed as ( exp + y exp y c y y y c j d j j j d = y G y G () The exponenial STAR model is of he form + y y c y y j exp y d c j j j exp d = y G y G (3) The exponenial auoregressive STAR (AR-STAR) model is of he form y y exp y y y exp y + j R R j j j = y G y G (4) For large values of he parameer, he logisic funcion G converges o one when y d c 0 when 0 he STAR converges o an auoregressive model of order p. The STAR shows slighly differen paerns wih respeco. For large values of, he exponenial funcion G converges o one for values of y d below or above hreshold parameer c. The AR-STAR is a modified form of STAR wih d 0. Base on he aforemenioned condiions, he STAR model offers he possibiliy o invesigae he presence of non lineariy in ime series daa which may accoun for he weakness of GARCH model menioned in chaper four, and wihou loss of generaliy we can srenghen he GARCH model wih STAR models by adjusing he error erms. The STAR GARCH model is proposed by combining equaion y wih equaion (), (3) and (4) o ge y = y G y G (5) j j y = y G y G (6) j j y = y G y G (7) R R j j Which are AR-GARCH (5), STAR-GARCH (6) and STAR-GARCH (7) In general, he STAR-GARCH model proposed for he sudy is of he form y = y G y G (8) Where j j G is he varying smooh ransiion funcions defined in equaions ( hrough 4)

4 Akinunde Muairu Oyewale, Shangodoyin Dahud Kehinde and Kgosi Phazamile 3 Properies of STAR-GARCH Model Suppose hahe general STAR-GARCH is of he form y y G y G y y G y G (9) S G j j j j j Assume ha and have he same number of parameer such ha and V y jg j,,..., p and Z, hen equaion (9) reduces o y y S G j V Z (0) e us assume ha y j, V and Z are independen wih zero co-variances and he esimaes of model is:- and are respecively ˆ and ˆ. The mean of general STAR-GARCH y y V Z j Z and y j Since 0 y j y y V, which reduces o, hen he las expression is of he form S G () y y V ˆ To derive he variance of S G y, consider he expression for j, his reduces o j y y V Z y y V Z ˆ From equaions () and () we have 0 Var y S G V ˆ i j y as ˆ V In order o relae STAR-GARCH model wih he GARCH model; if in equaion (5.) V 0 hen S G y reduce o j 0 V ˆ i y and he variance of S G () (3) Var y in equaion (3) will

Smooh Transiion Auoregressive-GARCH Model in Forecasing 5 4 mpirical Resuls/Daa Analysis wih xchange Rae Daa This secion examines he empirical resuls obained for Smooh Transiion Auoregressive GARCH models (STAR-GARCH) for four ses of exchange raes daa namely Briish (Pounds), Japanese (Yen), Nigerian (Naira) and Baswana (Pula) agains American (Dollar). Here he Parameers of xponenial auoregressive GARCH models (AR-GARCH), ogisic smooh ransiion auoregressive GARCH models (STAR- GARCH) and ha of xponenial smooh ransiion auoregressive GARCH (STAR- GARCH) models were obained using he derived equaions for all he series. The following values of variances were obained for classical GARCH models: Table : The GARCH model fied for all series COFFICINT (S.) SRIS 0 NAIRA 3.8580.679-0.99980 (0.345) (0.598) (0.0006) POUND 0.0007 0.979-0.0004 (0.00005) (0.3370) (0.0657) PUA 0.04763.9036-0.906 (0.0084) (0.30959) (0.0367) YN 0.67948.0088-0.03 (0.640) (0.647) (0.408) MOD VARIANC 4949.04 0.6586.444 546.605 In esimaing and c as required in equaions (5) hrough (7) simaion of γ and C: Saring values needed for he nonlinear opimizaion algorihm can be obained using wo dimensional grid search over and c, and selechose ha give smalles esimaor for he residual variance. The wo dimensional grid give hree possible values are ables and 3. Table : Values of grid of C SRIS I II III NAIRA 0.35 55.76 30 POUND 0.48.4 30 PUA 0.74 7.97 30 YN 0.74 7.97 30 Table 3: Values of grid of SRIS I II III NAIRA 0.50 0.00 30 POUND 0.50 0.00 30 PUA 0.50 0.00 30 YN 0.50 0.00 30

6 Akinunde Muairu Oyewale, Shangodoyin Dahud Kehinde and Kgosi Phazamile In he ables ( and 3) all he aserisk values are seleced because hey have minimum values and are subsequenly used in equaions (5)-(7). We fied models described in equaions (5)-(7) as follows: We can now illusrae he empirical implicaion of hese heories here under: 4. ARSTAR-GARCH Model for all Series exp exp y y y y y ( SG) (i) ynaira ( SG) 5.38446 y G 0.003043* y G.97588 0.005048 wih he variance 64.8983 (ii) ypound SG 0.097709 y G 0.4858 y G 0.00740 0.003444 wih he variance 0.0037 (iii) ypula ( SG) 9.844 y G 5.36047* y G 0.96958 0.084636 wih he variance 45.838 (iv) yyen ( SG) 3.3970 y G.6550* Q y G 0.0338.45775 wih he variance 6.495 ( ) 4. STAR-GARCH for all Series exp exp 0.003033 4.738773* y y y c y y c ( SG) (i) ynaira ( SG) y G y G 0.005034.070787 wih he variance 64.5584 (ii) ypound ( SG) 0.5449 y G 0.899484* y G 0.00507 0.04678 wih he variance 0.0030 (iii) ypula ( SG) 5.70756 y G.76754* y G 0.08334 0.63386 wih he variance 43.476 (iv) yyen ( SG).877449 y G.0408* y G 0.00004 0.004776 wih he variance 5.6686 4.3 STAR-GARCH for all Series exp exp 6.86795 0.00040* y y y c y y c ( SG) (i) ynaira ( SG) y G y G.37383 0.00083 wih he variance 9.7358 (ii) ypound ( SG) 0.585333 y G 0.9089* y G 0.0606 0.04403

Smooh Transiion Auoregressive-GARCH Model in Forecasing 7 wih he variance 0.000 (iii) ypula ( SG) 4.35766 y G 0.3639* y G 0.579097 0.00680 wih he variance 9.886 (iv) yyen ( SG).64506 y G.04076* y G 0.00578 0.007704 wih he variance.3990. 4.4 AR-GARCH Model Tables for all Series Table 4: Fied model for AR-GARCH series SRIS COFFICINT (S) MOD VARIANC C() C() NAIRA 0.00304.9759 0.00505 64.8983 POUND 0.4853 0.00740 0.00344 0.0037 PUA 5.3605 0.9696 0.08464 45.838 YN.65500 6.495 3.397 0.030 0.0037 4.5 STAR-GARCH Model Tables for all Series SRIS NAIRA POUND PUA YN Table 5: Fied model for STAR-GARCH series COFFICINT (S) MOD VARIANC C() C() 4.73877 0.00503.07078 64.5584 0.89948 0.0030 0.544 0.005 0.0468 5.7076.7675 43.476 0.0833 0.6339.87745.0408 5.6686 0.0000 0.00478 4.6 ogisic-garch Model Tables for all Series SRIS NAIRA POUND PUA YN Table 6: Fied model for STAR-GARCH series COFFICINT (S) MOD VARIANC C() C() 4.73877 9.7358 0.00303 0.00503.07078 0.544 0.89948 0.000 0.005 0.0468 5.7076.7675 9.886 0.0833 0.6339.87745.0408 0.0000 0.00478.3990 4.7 Table Comparing he Variances of all Series wih GARCH Model Table (7) shows he variances of all STAR-GARCH models wih GARCH, i is quie evidenha all STAR-GARCH acually ouperformed he classical GARCH model,

8 Akinunde Muairu Oyewale, Shangodoyin Dahud Kehinde and Kgosi Phazamile however, he STAR-GARCH performed bes and closely followed by STAR-GARCH, his is followed by AR-GARCH, he implicaion of his is ha for would be forecaser, he use of STAR-GARCH produced he bes resul. However, researcher can equally make do wih STAR as is performance could be considered as well. STAR-GARCH is srongly recommended for opimum resul. Table 7: Variances of all series wih GARCH model SRIS GARCH MOD AR-GARCH STAR-GARCH STAR-GARCH NAIRA 4949.04 64.8983 64.5584 9.7358 POUND 0.658 0.0037 0.0030 0.000 PUA.444 45.838 43.476 9.886 YN 546.603 6.495 5.6686.3990 5 Conclusion In able 7, he variances of all STAR-GARCH models wih GARCH are displayed, i is quie evidenha all STAR-GARCH ouperformed he classical GARCH model, however, he STAR-GARCH performed bes and closely followed by STAR-GARCH, his is followed by AR-GARCH. The implicaion is hahe use of STAR GARCH produces he bes resul; however STAR may be uilized in some occasions. Bu STAR would produce opimal resul. References [] Ahdi, N.A., anouar, C (0). The Tunisia sock marke: A regime swiching approach. Asian journal of Bossiness and Managemen sciences (3), 43-55. [] Andersen, T.G., and T. Bollerslev (998). Answering he skepics: Yes, sandard volailiy models do provide accurae forecass. Inernaional conomic Review 39(4), 885-905. [3] Baillie, R. and T. Bollerslev (989). The message in daily exchange raes: A condiional variance ale. Journal of Business and conomic Saisics 7(3), 97-305. [4] Bollerslev, T. (986). Generalized auoregressive condiional heeroskedasiciy. Journal of conomerics 3, 307-37 [5] Bonilla, C., Romero-Meza, R. and Hinich, M. J. (006) pisodic nonlineariies in he ain American sock marke indices, Applied conomics eers, 3, 95 9. [6] Brooks, C. and M. Hinich, 998, pisodic nonsaionariy in exchange raes, Applied conomics eers 5, 79-7. [7] Bonilla, C.; R. Romero-Meza; and M. Hinich. 005. pisodic nonlineariies in he ain American Sock Marke indices, Applied conomics eers, 3, 95-99 [8] ngle, R. F. (98). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaions. conomerical 50, 987-007.

Smooh Transiion Auoregressive-GARCH Model in Forecasing 9 [9] Granger and Teräsvira T. (994) Specificaion, esimaion, and evaluaion of smooh ransiion auoregressive models. Journal of he American Saisical Associaion, 89: 08 8. [0] iew, e.al 003. The inadequacy of linear auoregressive models for real exchange raes: empirical evidence from Asian economies, Applied conomics 35, 387 39. [] im, K. P., Hinich, M. J. and iew, V. (004) Adequacy of GARCH models for ASAN exchange raes reurn series, Inernaional Journal of Business and Sociey, 5, 7 3. [] im, K. P., Hinich, M. J. and iew, V. (005) Saisical inadequacy of GARCH models for Asian sock markes: evidence and implicaions, Inernaional Journal of merging Marke Finance, 4, 63 79. [3] Teräsvira T. (998) Modelling economic relaionships wih smooh ransiion regressions. In A. Ullah and D. Giles, ediors, Handbook of Applied conomic Saisics, pages 507 55. Marcel Dekker, New York. [4] Teräsvira T. and Anderson H. (99) Characerizing nonlineariies in business cycles using smooh ransiion auoregressive models. Journal of Applied conomerics 7: S9 S36, December 99.

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