Modelling Abrupt Shift in Time Series Using Indicator Variable: Evidence of Nigerian Insurance Stock

Transcription

1 Inernaonal Journal of Fnance and Accounng 5, 4(): 9-3 DOI:.593/.fa.54. Modellng Abru Shf n Tme Seres Usng Indcaor Varable: Evdence of Ngeran Insurance Sock H. G. Dkko, O. E. Asrbo, A. Samson,* Dearmen of Mahemacs, Ahmadu Bello Unversy, Zara, Ngera Dearmen of Sascs, Federal Unversy of Agrculure, Abeokua, Ngera Absrac Ths sudy models abru shf n me seres usng ndcaor varable. Seven symmerc and fve asymmerc models were consdered by ncororang an ndcaor varable n he varance euaon o monor he changes of some seleced Ngeran nsurance socks. The resuls showed ha he daly reurns were saonary bu no normally dsrbued and egh ou of en socks consdered for he sudy showed evdence of ARCH effec. The erformance of he dfferen models was evaluaed usng he RMSE, MAE and MAPE. The model ARCH () roved o be he mos suable among he welve comeng volaly models consdered. When he regme changes are ncororaed no he model, s found ha he hghly erssen volaly of he nsurance sock reurn rae s reduced for mos of he socks. Keywords Volaly, Heeroscedascy, Roo Mean Suare Error, Indcaor varable. Inroducon Dummy varables are varables ha can ake any values. They may be exlanaory or oucome varables; however, he focus of hs sudy s exlanaory dummy varable consrucon and usage. Tycally, dummy varables are used n he followng alcaons: me seres analyss wh seasonaly or regme swchng; analyss of ualave daa, such as survey resonses. Some scholars have argued based on sascal analyss of me seres ha ceran henomena do no corresond o regme shfs, [9]. Oulers, level shfs, and varance changes are common n aled me seres analyss. However, her exsence s ofen gnored and her mac s overlooked, for he lack of smle and useful mehods o deec and handle hose exraordnary evens. The roblem of deecng level shfs, and varance changes n a unvarae me seres s consdered. Three dfferen yes of regme shf (smooh, abru and dsconnuous) are denfed on he bass of dfferen aerns n he relaonsh beween he resonses. The smooh regme shfs s reresened by a uas-lnear relaonsh beween he resonse and conrol varables. The abru regme shf exhbs a nonlnear relaonsh beween he resonse and conrol varables, and he dsconnuous regme shf s characerzed by he raecory of he resonse varable dfferng when he forcng varable ncreases comared o when decreases see [5]. * Corresondng auhor: abuagboola@gmal.com (A. Samson) Publshed onlne a h://ournal.saub.org/fa Coyrgh 5 Scenfc & Academc Publshng. All Rghs Reserved In order o aly he conce o a arcular roblem, one has o conceually lm s range of dynamcs by fxng analycal caegores n me by consderng he even and caegorcally alyng n achevng he sgnfcan of he sudy. In hs sudy we model he abru shf n a me seres where he varable under sudy exhbs a nonlnear relaonsh beween he resonse and conrol varables usng some of he nsurance comany as a case sudy, ha s, from sable and unsable economc. Therefore our ndcaor varable wll ake n he value of for sable and for unsable economy n order o sudy he abru shf n me seres snce we are consderng a me seres daa o observe hese nonlnear relaonsh n each of he sock wh seven symmerc and fve asymmerc models ncororang an ndcaor varable n he varance euaon.. Leraure Revew ARCH and GARCH models, whch sand for auoregressve condonal heeroscedascy and generalzed auoregressve condonal heeroscedascy, have become wdesread ools for dealng wh heeroscedasc me seres. The goal of such models s o rovde a volaly measure lke a sandard devaon -- ha can be used n fnancal decsons concernng rsk analyss, orfolo selecon and dervave rcng. Alcaons of he ARCH/GARCH aroach are wdesread n suaons where volaly of reurns s a cenral ssue. Many banks and oher fnancal nsuons use he dea of value a rsk as a way o measure he rsks faced by her orfolos. [7] frs roosed he auoregressve condonal

2 H. G. Dkko e al.: Modellng Abru Shf n Tme Seres Usng Indcaor Varable: Evdence of Ngeran Insurance Sock heeroscedascy (ARCH) model for modelng he changng varance of a me seres; Engle used an ARCH model o sudy nflaon n he Uned Kngdom. [3] showed ha a GARCH model wh a small number of erms may be more effcen han an ARCH model wh many erms. Emrcal sudes n recen years have focused on volaly nvesgaon on he aern of fnancal asses such as ARCH effec, volaly cluserng, and erssence and leverage effec. For examle, [3], [8], [4], []. The use of dummy varables reures he moson of addonal consrans on he arameers of regresson euaons o oban esmaes for he model. Among he ossble consrans he mos useful are (a) o se he consan erm of he euaon o zero, or (b) o om one of he dummy varables from he euaon. In economercs me seres analyss, dummy varables may be used o ndcae he occurrence of wars or maor srkes. Dummy varables are used freuenly n me seres analyss wh regme swchng, seasonal analyss and ualave daa alcaons see [5]. Dummy varables are nvolved n sudes for economc forecasng, bo-medcal sudes, cred scorng, resonse modellng, ec. [] used dummy varable o comare he year nernally generaed revenue (IGR) and wage blls of he sx geoolcal zones n Ngera by caegorzng he geoolcal zones as dummy varables n a regresson model o fnd ou f he average nernally generaed revenue and wage blls of he geoolcal zone are sascally dfferen from each oher. From hs analyss, he concluded ha he norheas and norhwes zones are sascally dfferen. [9] used GARCH models wh dummes o sudy he mac of U.S moneary olcy on nflaon. From he analyss, he concluded ha he mac of U.S moneary olcy on nflaon s negave bu no sgnfcan on he arameer of he dummy varable he arameer. Sock reurn volaly reresens he varably of sock rce changes durng a erod of me. Ths henomenon has araced growng aenon of academa, olcy makers and oher layers n hs secor. Ths s because reurn s a maor measure of rsk assocaed wh asse nsead of rce because f you wan an nvesmen ha gves % of your reurn you nves on han n rce.e. s much beer o deal wh reurn han rce. Also, hgh volaly n socks, bonds and foregn exchange markes usually rase from moran ublc olcy ssues abou sably of fnancal marke and mac of sock volaly on he economy canno be sub esmaed. [7] used volaly o model four Ngeran frms lsed on he Ngeran Sock Exchange. [6] also conduced anoher sudy whch focused on he mac of he 5 recaalzaon of he bankng and nsurance ndusry on he sock marke. [6] carred ou a research on modellng and forecasng daly reurns of Ngeran nsurance sock usng [7] roosed model and [4] o esmae suable models n, from he sudy he researcher concluded ha he exonenal generalzed auoregressve condonal heeroscedasc (EGARCH) models s more suable n modellng sock rce reurns as ouerforms he oher models n goodness of f and ou-of-samle volaly forecasng. 3. Mehodology Daa for hs sudy were obaned from daly closng rces of nsurance socks raded on he floor of he Ngeran Sock Exchange (NSE) from nd January o 6 h May, 4. The en nsurance comany used for hs sudy are AIICO, GUINEAINS, GUINNESS, LASACO, LAWUNION, NEM, NIGERINS, PRESTIGE, UNIC AND WAPIC. Ls of Tess and Models Models secfcaon where Le denoe he reurns by P and P R = P ln, () P are he resen and revous closng rces and R been he connuously comounded reurn seres because s smly he sum of connuously comounded one-erod reurns nvolved Jarue-Bera Tes for normaly Jarue-Bera s a es sasc for esng wheher he seres s normally dsrbued. The es sasc measures he dfference of he skewness and kuross of he seres wh hose from he normal dsrbuon. The sasc s comued usng he exresson: N k JB = S 6 ( K 3) 4 where S s he skewness, K s he kuross, and k reresens he number of esmaed coeffcens Under he null hyohess of a normal dsrbuon, he Jarue-Bera sasc s dsrbued as a χ wh degrees of freedom. Saonary Tes (Augmened Dckey-Fuller es) Saonary of he reurn seres s one of he maor assumons n fnancal me seres modellng. Ths assumon can be checked usng a un roo es. The Augmened Dckey Fuller es (ADF) s a es for un roo n a me seres. Null hyohess s H : φ = and alernave hyohess s: The Tes Sasc (-rao): H : φ < n φ = = sd( φ ) T, P e = T σ P = The null hyohess s reeced f he calculaed value of s greaer han crcal value from nonsandard dsrbuons ()

3 Inernaonal Journal of Fnance and Accounng 5, 4(): 9-3 able. Tes for ARCH effec (Tes for heeroscedascy) One of he mos moran ssue before consder Heeroskedasc models s examne he resduals for evdence of heeroscedascy. To es for he resence of heeroscedascy n resduals of Ngeran nsurance sock reurn seres, he Lagrange Muller (LM) es for ARCH effecs roosed by Engle (98) s aled. In summary, he es rocedure s erformed by frs obanng he resduals e from he ordnary leas suares regresson of he condonal mean euaon whch mgh be an auoregressve (AR) rocess, movng average (MA) rocess or a combnaon of AR and MA rocesses; (ARMA) rocess usng EVews 7 sascal sofware. For examle, n ARMA (,) rocess he condonal mean euaon wll be as: r φ ε θ ε (3) = r Afer obanng he resduals e, he nex se s regress he suared resdual on a consan and s lags as n he followng euaon: e e... e v (4) = The null hyohess ha here s no ARCH effec u o order can be formulaed as: H... = (5) agans he alernave : = = H : for some {,.., m} a (6) The es sasc for he on sgnfcance of he -lagged suared resduals s he number of observaons mes he R-suared ( TR ) from he regresson. TR s esed agans χ () dsrbuon. Ths s asymocally locally mos owerful es. Volaly models These models nclude he symmerc and asymmerc volaly models. The models are, ARCH(),,, GARCH(, ), GARCH(, ), GARCH(, ) E GARCH (, ), EGARCH (, ), EGARCH (, ), EGARCH (, ) and TARCH(, ). In each model we ncororaed dummes varables. ARCH Models (Auoregressve Condonal Heeroskedasc Model) The ARCH() as roosed by Engle s gven by σ ε... ε r (7) = where >, for =,, are he arameers of he model. ARCH model wh dummy varable σ D r (8) = ε... ε δ Shf where δ DShf s he dummy varable added o he condonal varance model. GARCH Model (Generalze Auoregressve Condonal Heeroskedasc Model) The GARCH(,) as roosed by Nelson s gven by σ where > = ε β ε... ε... β ε and > r β for all and GARCH model wh dummy varable σ = ε... β ε... ε Shf δ D r β ε (9) () Where δ D Shf s he dummy varable added o he condonal varance model. TARCH (,) Threshold GARCH Model or TARCH (,), (Glosen e al.993) s = = ) γε d ( ε ( β σ σ () = d = f ε < and = ( ε < and bad news ( ε > ) where d oherwse. In hs model, good news ), have dfferen effecs on he condonal varance. TARCH (, ) wh dummy varable σ = = = ( β σ ( ε ) γε ) δ D Shf d ) () where δ D Shf s he dummy varable added o he condonal varance model. The E-GARCH (, ) s gven by as roosed n Nelson (99): ln( σ ) =,, γ, β = β ln( σ λε ) γ ε are he arameers of he model. π (3) If = and =, he model above reduces o EGARCH (, ) gven as

4 H. G. Dkko e al.: Modellng Abru Shf n Tme Seres Usng Indcaor Varable: Evdence of Ngeran Insurance Sock ε = z σ ε ε ln( σ = ) β lnσ γ (4) π σ σ, β, γ, where are he arameers of he model. EGARCH (, ) Model wh dummy varable s gven as where ln( σ ) = D Shf = ε = σ ε γ σ ( β ln( σ ) δ D Shf (5) δ s he added dummy varable o he condonal varance model. Goodness of fs cerera Akake Informaon Crera (AIC) and Schwarz Crera (SIC) are he mos commonly used model selecon crera where AIC = RSS K ln( LL) = K ln n (6) RSS = e s he resdual sum of suares. Forecas error sascs The forecas error sascs used n hs sudy are he roo mean suare error (RMSE), mean absolue error (MAE) and he mean absolue ercenage error (MAPE). These forecas error sascs are defned by: m RMSE = ( y m = y) (7) MAE = m T MAPE = m = m = y y ( ˆ σ σ ) σ (8) (9) where, =,..., m wh m, y, and denong he number of forecass, volaly value and he forecas, resecvely. The RMSE and MAE deend on he scale of he deenden varable and he dfferences beween volaly value and he forecased values. The smaller he error sasc s, he beer he forecasng ably of ha model n consderaon of ha measure. The MAPE s scale nvaran. The sasfacory forecasng model s execed o have MAPE close o % whch ndcae he bes forecasng erformance o he daa. 4. Analyss Resul 4.. Prelmnary Resul An nal descrve sascs analyss of he en Ngeran Insurance socks were carred ou and he resul shown n Table. The obaned resul as shown n Table, showed ha he Mean reurn seres for some of he nsurance were negave ndcang ha hese nsurances ncurred loss durng he erod under sudy. Dese hs loss, wo of he Insurances sll reored osve reurn. Also, he resul of Jarue-Bera sasc revealed ha he reurn seres for all he nsurance were no normally dsrbued as he -values were less han % and 5%. y Table. Descrve Sascs of he reurn of Ngeran Insurance socks Insurance Mean Maxmum Mnmum Sd. Dev. Skewness Kuross Jarue-Bera P-Value AIICO GUINEAINS GUINNESS LASACO LAWUNION NEM NIGERINS PRESTIGE UNIC WAPIC

5 Inernaonal Journal of Fnance and Accounng 5, 4(): Analyss of he Man Resul In Table below, he reurn seres were all saonary. Hence, here s no un roo. Therefore, here s no need for dfferencng. In he Tes for ARCH effec, he Lagrange Muller (LM) es roosed by Engle (98) was aled. The F Sasc and he obaned -values are summarzed n Table 3. The resuls of F Sasc were sgnfcan a % for egh nsurance sock reurns whle wo of he nsurance does no exhb heeroscedascy. Therefore we canno run he heeroscedascy model on hem because hey do no fulfll he condon of ARCH effec. Table. Augmened Dckey-Fuller Tes of saonary es (ADF) of he reurn seres of Ngeran Insurance socks Insurances ADF Tes Sasc Commen AIICO Saonary a level whou dfferencng GUINEAINS Saonary a level whou dfferencng GUINNESS Saonary a level whou dfferencng LASACO Saonary a level whou dfferencng LAWUNION Saonary a level whou dfferencng NEM Saonary a level whou dfferencng NIGERINS Saonary a level whou dfferencng PRESTIGE Saonary a level whou dfferencng UNIC Saonary a level whou dfferencng WAPIC Saonary a level whou dfferencng % crcal = -3.9 Table 3. Lagrange Muller es of he resence of ARCH effec Insurance F Sasc P-values AIICO GUINEAINS.3.97 GUINNESS LASACO LAWUNION NEM NIGERINS PRESTIGE UNIC.79. WAPIC Twelve dfferen heeroscedasc models were fed by addng a dummy varable o he condonal varance model o es he sgnfcance of he hyohess of he model on each of he model. For AIICO Insurance, all heeroscedasc models fed had all her arameers sgnfcan (<.5) exce ha some of he model n he abru shf showed a osve values wh sgnfcan level of.. Moreover, NEM, PRESTIGE and UNIC he arameers esmaed were sgnfcan exce he leverage effec of he TARCH (, ) model (>.5). For PRESTIGE Insurance, he ndcaor varable s osve hroughou he models ndcang ha he shf was osve. I.e. he global mel down dd no affec. Resuls are resened n Table 6 and Table 7 wh ohers nsurance socks. Ou of he welve comeng models, he selecon of he model ha could gve bes redcon was carred ou usng he Log lkelhood (LL), Akake Info Crera (AIC) and Schwarz Informaon Creron (SIC). Schwarz Informaon has been consdered o be he bes of hese crera o as SIC gve he heaves enales for loss of degrees of freedom (Afees and Ismal, ). Hence, was EGARCH (, ) for AIICO, NEM, WAPIC and EGARCH (, ) for GUINNESS, LAWUNION, UNIC and TARCH (, ) for NIGERINS and PRESTIGE. Resuls are resened n Table 6 and Table 7. Forecasng erformance of hese esmaed models were nvesgaed usng our samle daa and sascs lke Roo Mean Suare Error, Mean Absolue Error as well as he Mean Absolue Percenage error were comued. Model wh he smalles Mean Suare Error was consdered o he mos suable for forecasng. Hence, from he resuls obaned showed ha some nsurance socks are havng model han one model suable. Therefore we are gong o ado he Prncle of Parsmony ha he bes model s he smles model ha can caures he moran feaures of he daa. Hence EGARCH (, ) roved o mos suable for AIICO and NEM, LAWUNION s EGARCH (, ), GUINNESS s GARCH (, ) whle ARCH () for NIGERINS, UNIC and WAPIC whle ARCH () suable for PRESTIGE. The resuls are shown n Table 5 and Conclusons Ths sudy had examned he daly reurn volaly of Ngeran Insurance secor socks. The bes model was comued usng he AIC and SIC, he bolded models are consdered he bes fs model o be used n each of he socks. The forecasng erformance of several varans of condonal heeroscedascy volaly models were evaluaed usng model evaluaon erformance measures lke he Roo Mean Suare Error. The os esmaon evaluaon carred ou revealed varous condonal heeroscedascy models o be mos suable for modellng he reurn aern of he each nsurance. The EGARCH (, ) was suable for AIICO and NEM, LAWUNION s EGARCH (, ), GUINNESS s GARCH (, ) whle ARCH () for NIGERINS, UNIC and WAPIC whle ARCH () suable for PRESTIGE. Bu lookng a he nsurance and by evaluaon one can say ARCH () was mos suable followed by EARCH (, ). Ths fndng s very crucal and nformave o nvesors and nendng nvesors who mgh wan o nves n nsurance socks

6 4 H. G. Dkko e al.: Modellng Abru Shf n Tme Seres Usng Indcaor Varable: Evdence of Ngeran Insurance Sock Table 4. Parameer Esmaes of he heeroscedasc models of AIICO, GUINNESS, LAWUNION and NEM Parameers Esmaes Insurances Model 3 β β γ δ D Shf AIICO ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E-GARCH (, ) E-GARCH (, ) E-GARCH (, ) E-GARCH (, ) TARCH(, ).86**.7**.7** 4.8 x -5 **.843*.73**.74** -.49** -.355** ** **.868**.794**.436**.4557**.5864**.33**.4493**.444**.6546**.796**.45694**.38**.5785**.5657**.56548**.5755**.58978** ** **.4449** **.367**.849**.6985** ** -.76** **.35766**.97833** **.79694** -.35**.47** -.3 x -5 * -.36 x -5 ** -4.6 x -5 ** -9.5 x x -5 * -.9 x **.3** ** **.7** GUINNESS ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E-GARCH (, ) E-GARCH (, ) E-GARCH (, ) E-GARCH (, ) TARCH(, ).44**.38**.433**.363**.349** ** -4.85** ** **.363**.76437** **.4449** **.8545**.38673**.96576**.36534**.59584**.3339**.347** **.885** 6.57 x **.3377** -.746** ** ** **.53658**.577** **.6764** ** -.985**.7873** ** -.399** -7.9 x -5 ** -.368** -.** -.35** * ** ** -.983** -.** LAWUNION ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E-GARCH (, ) E-GARCH (, ) E-GARCH (, ) E-GARCH (, ) TARCH (, ).453**.37**.3**.3 x -5 **.49 x -5 ** 8.98 x -6 **.7 x -5 ** ** **.33975** **. x -5 **.5388**.4556**.47693**.683**.769**.8534**.8645**.6855**.379**.44794**.4677**.6677**.5574**.9474** **.9876** ** **.58386**.5593**.67966**.85563** **.7** -.795** **.83598**.49333** **.4563**.73875** **.4786**.96778**.6679**.5568** -.9** -.** -.94** x -6 ** -.9 x -5 ** x -6 ** -.48 x -5 ** ** ** ** -.479** x -6 ** NEM ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ).9**.**.9**.4**.49**.455**.33** ** ** ** **.437**.349**.849**.65545**.984**.97**.649**.876**.34893** ** **.5663**.54763**.5**.845** **.854** **.8974**.6336**.48**.59447** **.666** -.366** **.799**.9858** ** ** ** -.847** -.945** -.38** -.458** -.4** -.77** ** ** ** ** **sgnfcance a %, * sgnfcance a 5%

7 Inernaonal Journal of Fnance and Accounng 5, 4(): Table 5. Parameer Esmaes of he heeroscedasc models of NIGERINS, PRESTIGE, UNIC and WAPIC Parameers Esmaes Insurance s Model 3 β β γ δ D Shf NIGERINS ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ).569**.396**.8**.55 x -5 **.774** 3.7 x -5 ** 6.7 x -5 ** -.4** -.99** ** **.5 x -5 **.4643** **.4595**.7434**.956**.36994**.46357**.988**.375**.3956**.48555**.695**.79738**.934**.48764**.6567**.677**.56474**.778** **.3959**.69948**.47** **.399**.9969**.49586**.74445**.996**.49776**.4458**.754** **.986**.7369**.4773** * * -97** -.** -4.3 x -5 ** -.8 x -5 ** -.766** -.59 x -5 ** -5.3 x -5 ** ** -.738** -.664** ** -.6 x -5 ** PRESTIGE ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ).6**.399**.776*.4**.89**.43** ** ** ** -.43**.7**.388**.35**.5595**.3435**.36789**.8574**.7797**.458**.835**.5856**.5789** **.4446**.4493** -.4** ** **.76**.5** ** -.59** ** * * ** **.44435**.8779** **.536**.895**.86**.39**.57** **.686** **.635**.58** UNIC ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ).89**.737**.659**.68**.75**.77**.86** ** -.439** ** **.87**.9466** **.58534** **.973**.6564**.685**.9889**.855**.699**.784**.6833**.499**.4969**.78663**.8394* *.83**.8578**.5555**.5936**.545**.465** -.53** ** -.45** ** ** * **.9653**.4557**.9868** ** -.7** -.65** -.66** -.74** -.76** -.84** -.38**.7753**.48** ** -.86** WAPIC ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ).75**.66**.375**.79**.595**.799**.7 x -5 ** ** ** ** **.65** **.79699** **.47446**.9559**.88744**.68353**.5589**.6683**.6577**.355** 3.876**.5649**.676**.4375** **.65484**.7539** -.87**.558**.8577** -.78** ** **.5959*.633**.59773**.5865** * *.38837** ** * ** -.894* * -.55** -.9** -.35** -.68** -.7** -.488** -.4 x -5 ** ** -.97** -.4** **sgnfcance a %, * sgnfcance a 5%

8 6 H. G. Dkko e al.: Modellng Abru Shf n Tme Seres Usng Indcaor Varable: Evdence of Ngeran Insurance Sock Table 6. Model Selecon crera (Goodness of f crera and dagnosc checkng) of AIICO GUINNESS, LAWUNION and NEM Model selecon Crera Dagnosc check for ARCH Effec Insurances Model Log-Lkelhood AIC SIC F sasc P value AIICO ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) x GUINNESS ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) LAWUNION ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) NEM ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) AIC s he Akake Info Crera, SIC s he Schwarz nfo creron, Log s he log lkelhood Bolded AIC and SIC are he bes model selecon (Goodness of fs)

9 Inernaonal Journal of Fnance and Accounng 5, 4(): Table 7. Model Selecon crera (Goodness of f crera and dagnosc checkng) of NIGERINS, PRESTIGE, UNIC and WAPIC Model selecon Crera Insurances Model Log-Lkelhood AIC SIC F sasc P value NIGERINS ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) x x PRESTIGE ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) , UNIC ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) WAPIC ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) x AIC s he Akake Info Crera, SIC s he Schwarz nfo creron, Log s he log lkelhood Bolded AIC and SIC are he bes model selecon (Goodness of fs)

10 8 H. G. Dkko e al.: Modellng Abru Shf n Tme Seres Usng Indcaor Varable: Evdence of Ngeran Insurance Sock Table 8. Forecas Performance of esmaed model of AIICO, GUINNESS, LAWUNION and NEM Insurance AIICO GUINNESS LAWUNION NEM Heeroscedasc models ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) Sasc RMSE MAE MAPE Bolded values are he leas values of RMSE. RMSE s he Roo Mean Suare Error.

11 Inernaonal Journal of Fnance and Accounng 5, 4(): Table 9. Forecas Performance of esmaed model of NIGERINS, PRESTIGE, UNIC and WAPIC Insurance NIGERINS PRESTIGE UNIC WAPIC Heeroscedasc models ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) ARCH() GARCH(, ) GARCH(, ) GARCH(, ) E- E-GARCH(, ) E-GARCH(, ) E-GARCH(, ) TARCH(, ) Sasc RMSE MAE MAPE Bolded values are he leas values of RMSE. RMSE s he Roo Mean Suare Error.

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