Research Article Modeling Markov Switching ARMA-GARCH...

Research ArticleModeling Markov Switching ARMA-GARCH Neural NetworksModels and an Application to Forecasting Stock Returns

Melike Bildirici1 and Oumlzguumlr Ersin2

1 Yıldız Technical University Department of Economics Barbaros Bulvari Besiktas 34349 Istanbul Turkey2 Beykent University Department of Economics Ayazaga Sisli 34396 Istanbul Turkey

Correspondence should be addressed to Melike Bildirici melikebildiricigmailcom

Received 20 August 2013 Accepted 4 November 2013 Published 6 April 2014

Academic Editors T Chen Q Cheng and J Yang

Copyright copy 2014 M Bildirici and O Ersin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The study has two aims The first aim is to propose a family of nonlinear GARCH models that incorporate fractional integrationand asymmetric power properties to MS-GARCH processes The second purpose of the study is to augment the MS-GARCH typemodels with artificial neural networks to benefit from the universal approximation properties to achieve improved forecastingaccuracyTherefore the proposed Markov-switching MS-ARMA-FIGARCH APGARCH and FIAPGARCH processes are furtheraugmented with MLP Recurrent NN and Hybrid NN type neural networks The MS-ARMA-GARCH family and MS-ARMA-GARCH-NN family are utilized for modeling the daily stock returns in an emerging market the Istanbul Stock Index (ISE100)Forecast accuracy is evaluated in terms of MAE MSE and RMSE error criteria and Diebold-Mariano equal forecast accuracy testsThe results suggest that the fractionally integrated and asymmetric power counterparts of Grayrsquos MS-GARCH model providedpromising results while the best results are obtained for their neural network based counterparts Further among the modelsanalyzed the models based on the Hybrid-MLP and Recurrent-NN the MS-ARMA-FIAPGARCH-HybridMLP and MS-ARMA-FIAPGARCH-RNN provided the best forecast performances over the baseline single regime GARCHmodels and further over theGrayrsquos MS-GARCHmodel Therefore the models are promising for various economic applications

1 Introduction

In the light of the significant improvements in the economet-ric techniques and in the computer technologies modelingthe financial time series have been subject to acceleratedempirical investigation in the literature Accordingly follow-ing the developments in the nonlinear techniques analysesfocusing on the volatility in financial returns and economicvariables are observed to provide significant contributionsIt could be stated that important steps have been takenin terms of nonlinear measurement techniques focusing onthe instability or stability occurring vis-a-vis encounteredvolatility Further the determination of stability or insta-bility in terms of volatility in the financial markets gainsimportance especially for analyzing the risk encountered Inaddition to impact of the magnitude and the size of shocks

on volatility the financial returns are under the influenceof sudden or abrupt changes in the economy Hence thevolatility of economic data has been explored in econometricliterature as a result of the need of modelling uncertaintyand risk in the financial returns The relationship betweenthe financial returns and various important factors such asthe trade volume market price of financial assets and therelationship between volatility trade volume and financialreturns have been vigorously investigated [1ndash4]

TheARCHmodel introduced by Engle [5] and theGener-alized ARCH (GARCH) model introduced by Bollerslev [6]are generally accepted for measuring volatility in financialmodels GARCH models have been used intensively inacademic studies A tremendous amount of GARCHmodelsexist and various studies provide extended evaluation of thedevelopment

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 497941 21 pageshttpdxdoiorg1011552014497941

2 The Scientific World Journal

Among many Engle and Bollerslev [7] developed theIntegrated-GARCH (I-GARCH) process to incorporate inte-gration properties AGARCH model introduced by Engle[8] allows modeling asymmetric effects of negative andpositive innovations In terms of modeling asymmetriesGARCH models have been further developed by includingasymmetric impacts of the positive and negative shocksto capture the asymmetric effects of shocks on volatilityand return series which depend on the type of shocks ieeither negative or positive Following the generalization ofEGARCH model of Nelson [9] that allows modelling theasymmetries in the relationship between return and volatilitythe Glosten et al [10] noted the importance of asymmetrycaused by good and bad news in volatile series and proposed amodel that incorporates the past negative and positive inno-vations with an identity function that leads the conditionalvariance to follow different processes due to asymmetry Thefinding is a result of the empirical analyses which pointedat the fact that the negative shocks had a larger impact onvolatility Consequently the bad news have a larger impactcompared to the conditional volatility dynamics followedafter the good news Due to this effect asymmetric GARCHmodels have rapidly expanded The GJR-GARCHmodel wasdeveloped independently by Zakoian [11 12] and Glosten etal [10] It should be noted that in terms of asymmetry theThreshold GARCH (T-GARCH) of Zakoian [12] VGARCHand nonlinear asymmetric GARCH models (NAGARCH)of Engle and Ng [13] are closely related versions to modelasymmetry in financial asset returns The SQR-GARCHmodel of Heston andNandi [14] and the Aug-GARCHmodeldeveloped by Duan [15] nest several versions of the modelstaking asymmetry discussed above Further models suchas the Generalized Quadratic GARCH (GQARCH) modelof Sentana [16] utilize multiplicative error terms to capturevolatility more effectively The FIGARCH model of Baillie etal [17] benefits from an ARFIMA type fractional integrationrepresentation to better capture the long-run dynamics in theconditional variance (See for detailed information Bollerslev[18]) The APARCHAPGARCH model of Ding et al [19] isan asymmetric model that incorporates asymmetric powerterms which are allowed to be estimated directly from thedata The APGARCH model also nests several models suchas the TGARCH TSGARCH GJR and logGARCH TheFIAPGARCHmodel of Tse [20] combines the FIGARCHandthe APGARCH Hyperbolic GARCH (HYGARCH) modelof Davidson [21] nests the ARCH GARCH IGARCH andFIGARCHmodels (for an extended review GARCHmodelssee Bollerslev [18])

Even though the ARCHGARCH models can be appliedquickly for many time series the shortcomings in thesemodels were discussed by certain studies Perez-Quirosand Timmermann [22] focused on the conditional distri-butions of financial returns and showed that recessionaryand expansionary periods possess different characteristicswhile the parameters of a GARCH model are assumed tobe stable for the whole period Certain studies discussed thehigh volatility persistence inherited in the baseline GARCHand proposed early signs of regime switches Diebold [23]and Lamoureux and Lastrapes [24] are two of the highly

cited studies discussing high persistence in volatility due tostructural changes Lamoureux and Lastrapes [24] showedthat the encountered high persistence in volatility processesresulted from the volume effects that had not been taken intoaccount Qiao and Wong [25] followed a bivariate approachand confirmed that the Lamoureux and Lastrapes [24] effectexists due to the volume and turnover effects on conditionalvolatility and after the introduction of volumeturnover asexogenous variables it is possible to obtain a significantdecline in the the persistence Mikosch and Starica [26]showed that structural changes had an important impact thatleads to accepting an integrated GARCH process Bauwenset al [27 28] discussed that the persistence in the estimatedsingle regime GARCH processes could be considered asresulting from the misspecification which could be con-trolled by introducing an MS-GARCH specification wherethe regime switches are governed by a hidden Markov chain

Kramer [29] evaluated the autocorrelation in the squarederror terms and provided an important contribution Accord-ingly the observed empirical autocorrelations of the 120576

2

119905are

much larger than the theoretical autocorrelations implied bythe estimated parameters through evaluating anMS-GARCHmodel where the autocorrelation problem could be shown toaccelerate as the transition probabilities approached 1 (Fora proof see eg Francq and Zakoıan [30] Kramer [29])(In particular the empirical autocorrelations of the 120576

2

119905often

seem to indicate long memory which is not possible in theGARCH-model in fact in all standard GARCH-modelstheoretical autocorrelations must eventually decrease expo-nentially so longmemory is ruled out) Alexander and Lazaar[31] showed that leverage effects are due to asymmetry in thevolatility responses to the price shocks and the leverage effectaccelerates once the markets are in the more volatile regimeKramer and Tameze [32] showed that a single state GARCHmodel had only onemean reversion while by allowing regimeswitching in the GARCH processes mean reverting effectdiminishes In a perspective of volatility if these shifts arepersistent then there are two sources of volatility persistencedue to shocks and due to regime-switching in the parametersof the variance process By utilizing aMarkov transformationmodel it could be shown that the relationships among theregimes between the periods of 119905minus1 and 119905 could be explainedand the most important advantage of the MSGARCH modelexposes itself as there is no need for the researchers to observethe regime changes The model allows different regimes toreveal by itself [33]

The regime switching in light of the Markov switchingmodel has interesting properties to be examined such asthe stationarity by allowing the switching course of volatilityinherent in the asset prices The hidden Markov model(HMM) developed by Taylor [34] is a switching model thatbenefits from including an unobserved variable to capturevolatility to be modeled with transitions between the hiddenstates that possess different probability distributions attachedto each state Hidden Markov model has been appliedsuccessfully by Alexander and Dimitriu [35] Cheung andErlandsson [36] Francis and Owyang [37] and by Claridaet al [38] to capture the switching type of predictions in

The Scientific World Journal 3

stock returns interest rates and exchange rates Regimeswitching model has been used extensively for predictionof returns belonging to different stock market returns indifferent economies and by following the fact that the stockmarket indices are very sensitive to stock volatility whichaccelerates especially during periodswithmarket turbulences(see for detailed information Alexander and Kaeck [39])

The conventional statistical techniques for forecastingreached their limit in applications with nonlinearities fur-thermore recent results suggest that nonlinear models tendto perform better in models for stock returns forecasting[40] For this reason many researchers have used artificialneural networks methodologies for financial analysis on thestockmarket Lai andWong [41] contributed to the nonlineartime series modeling methodology by making use of single-layer neural network Further modeling of NN models forestimation and prediction for time series have importantcontributions Weigend et al [42] Weigend and Gershen-feld [43] White [44] Hutchinson et al [45] and Refeneset al [46] contributed to financial analyses stock marketreturns estimation pattern recognition and optimizationNN modeling methodology is applied successfully by Wanget al [47] and Wang [48] to forecast the value of stockindices Similarly Abhyankar et al [49] Castiglione [50]Freisleben [51] Kim and Chun [52] Liu and Yao [53] Phuaet al [54] Refenes et al [55] Resta [56] R Sitte and J Sitte[57] Tino et al [58] Yao and Poh [59] and Yao and Tan [60]are important investigations focusing on the relationshipsbetween stock prices and market volumes and volatilityFor similar applications see [1ndash4] Bildirici and Ersin [61]modeled NN-GARCH family models to forecast daily stockreturns for short and long run horizons and they showed thatGARCH models augmented with artificial neural networks(ANN) architectures and algorithms provided significantforecasting performances Ou and Wang [62] extended theNN-GARCH models to Support Vector Machines Azadehet al [63] evaluated NN-GARCH models and proposed theintegrated ANN models Bahrammirzaee [64] provided ananalysis based on financial markets to evaluate the artificialneural networks expert systems and hybrid intelligence sys-tems Further Kanas and Yannopoulos [65] and Kanas [40]used Markov switching and Neural Networks techniques forforecasting stock returns however their approaches departfrom the approach followed within this study

In this study the neural networks and Markov switchingstructures are aimed to be integrated to augment the ARMA-GARCH models by incorporating regime switching anddifferent neural networks structures The approach aims atformulations and estimations of MS-ARMA-GARCH-MLPMS-ARMA-APGARCH-MLP MS-ARMA-FIGARCH-MLPMS-ARMA-FIAPGARCH-MLP MS-ARMA-GARCH-RBFMS-ARMA-APGARCH-RBF MS-ARMA-FIGARCH-RBFand MS-ARMA-FIAPGARCH-RBF the recurrent neuralnetwork augmentations of the models are namely the MS-ARMA-GARCH-RNN MS-ARMA-APGARCH-RNN MS-ARMA-FIGARCH-RNN and MS-ARMA-FIAPGARCH-RNN And lastly the paper aims at providing Hybrid NN

versions the MS-ARMA-GARCH-HybridNN MS-ARMA-APGARCH-HybridNN MS-ARMA-FIGARCH-HybridNNand MS-ARMA-FIAPGARCH-HybridNN

2 The MS-GARCH Models

Over long periods there are many reasons why financialseries exhibit important breaks in behavior examples includedepression recession bankruptcies natural disasters andmarket panics as well as changes in government policiesinvestor expectations or the political instability resultingfrom regime change

Diebold [23] provided a throughout analysis on volatilitymodels One of the important findings is the fact that volatil-ity models that fail to adequately incorporate nonlinearity aresubject to an upward bias in the parameter estimates whichresults in strong forms of persistence that occurs especiallyin high volatility periods in financial time series As a resultof the bias in the parameter estimates one important resultof this fact is on the out-of-sample forecasts of single regimetype GARCH models Accordingly Schwert [66] proposed amodel that incorporates regime switching that is governedby a two state Markov process hence the model retainsdifferent characteristics in the regimes that are defined as highvolatility and low volatility regimes

Hamilton [67] proposed the early applications of HMCmodels within a Markov switching framework AccordinglyMS models were estimated by maximum likelihood (ML)where the regime probabilities are obtained by the proposedHamilton-filter [68ndash71] ML estimation of the model is basedon a version of the Expectation Maximization (EM) algo-rithm as discussed in Hamilton [72] Krolzig [73ndash76] In theMSmodels regime changes are unobserved and are a discretestate of a Markov chain which governs the endogenousswitches between different AR processes throughout timeBy inferring the probabilities of the unobserved regimeswhich are conditional on an information set it is possible toreconstruct the regime switches [77]

Furthermore certain studies aimed at the developmentof modeling techniques which incorporate both the proba-bilistic properties and the estimation of a Markov switchingARCH and GARCHmodels A condition for the stationarityof a natural path-dependent Markov switching GARCHmodel as in Francq et al [78] and a throughout analysisof the probabilistic structure of that model with conditionsfor the existence of moments of any order are developedand investigated in Francq and Zakoıan [30] Wong andLi [79] Alexander and Lazaar [80] and Haas et al [81ndash83] derived stationarity analysis for some mixing modelsof conditional heteroskedasticity [27 28] For the Markovswitching GARCH models that avoid the dependency of theconditional variance on the chainrsquos history the stationarityconditions are known for some special cases in the literature[84] Klaassen [85] developed the conditions for stationarityof the model as the special cases of the two regimesA necessary and sufficient stationarity condition has beendeveloped by Haas et al [81ndash83] for their Markov switchingGARCHmodel Furthermore Cai [86] showed the properties


of Bayesian estimation of a Markov switching ARCH modelwhere only the constant in the ARCH equation is allowedto have regime switches The approach has been investigatedby Kaufman and Fruhwirth-Schnatter [87] and Kaufmannand Scheicher [88] Das and Yoo [89] proposed an MCMCalgorithm for the same model (switches being allowed in theconstant term) with a single state GARCH term to show thatgains could be achieved to overcome path-dependence MS-GARCH models are studied by Francq and Zakoıan [30] toachieve their non-Bayesian estimation properties in light ofthe generalized method of moments Bauwens et al [27 28]proposed a Bayesian Markov chain Monte Carlo (MCMC)algorithm that is differentiated by including the state variablesin the parameter space to control the path-dependence byobtaining the parameter space with Gibbs sampling [90]

The high and low volatility probabilities of MS-GARCHmodels allow differentiating high and low volatility periodsBy observing the periods in which volatility is high it ispossible to investigate the economic and political reasons thatcaused increased volatility If a brief overview is to be pre-sented there are several models based on the idea of regimechanges which should be mentioned Schwert [66] exploresa model in which switches between these states that returnscan have a high or low variance are determined by a two-stateMarkov process Lamoureux and Lastrapes [24] suggest theuse of Markov switching models for a way of identifying thetiming of the shifts in the unconditional variance Hamiltonand Susmel [91] and Cai [86] proposed Markov switchingARCH model to capture the effects of sudden shifts in theconditional variance Further Hamilton and Susmel [91]extended the analysis to a model that allows three regimeswhich were differentiated between low moderate and highvolatility regimes where the high-volatility regime capturedthe economic recessions It is accepted that the proposals ofCai [86] andHamilton and Susmel [91] helped the researchersto control for the problem of path dependence which makesthe computation of the likelihood function impossible (Theconditional variance at time t depends on the entire sequenceof regimes up to time t due to the recursive nature of theGARCHprocess InMarkov switchingmodel the regimes areunobservable one needs to integrate over all possible regimepathsThenumber of possible paths grows exponentially witht which renders ML estimation intractable) (see for detailBauwens et al [27 28])

Gray [92] study is one of the important studies where aMarkov switching GARCH model is proposed to overcomethe path dependence problem According to Grayrsquos modelonce the conditional volatility processes are differentiatedbetween regimes an aggregation of the conditional variancesfor the regimes could be used to construct a single variancecoefficient to evaluate the path dependence A modificationis also conducted by Klaassen [85] Yang [93] Yao andAttali [94] Yao [95] and Francq and Zakoıan [96] derivedconditions for the asymptotic stationarity of some AR andARMA models with Markov switching regimes Haas et al[81ndash83] investigated a MS-GARCH model by which a finitestate-space Markov chain is assumed to govern the ARCHparameters whereas the autoregressive process followed bythe conditional variance is subject to the assumption that past

conditional variances are in the same regime (for details thereaders are referred to Bauwens et al [27 28] Klaassen [85]Haas et al [81ndash83] Francq and Zakoıan [30] Kramer [29]and Alexander and Kaeck [39])

Another area of analysis pioneered by Haas [97] andChang et al [98] allow different distributions in order togain forecast accuracy An important finding of these studiesshowed that by allowing the regime densities to follow skew-normal distributionwithGaussian tail characteristics severalreturn series could be modeled more efficiently in termsof forecast accuracy Liu [99] developed and discussed theconditions for stationarity in Markov switching GARCHstructure in Haas et al [81ndash83] and proved the existenceof the moments In addition Abramson and Cohen [100]discussed and further evaluated the stationarity conditionsin a Markov switching GARCH process and extended theanalysis to a general case with m-state Markov chains andGARCH(119901 119902) processes An evaluation and extension ofthe stationarity conditions for a class of nonlinear GARCHmodels are investigated in Abramson and Cohen [100]Francq and Zakoıan [30] derived the conditions for weakstationarity and existence of moments of any order MS-GARCH model Bauwens et al [27 28] showed that byenlarging the parameter space to include space variablesthough maximum likelihood estimation is not feasible theBayesian estimation of the extended process is feasible for amodel where the regime changes are governed with a hiddenMarkov chain Further Bauwens et al [27 28] acceptedmild regularity conditions under which the Markov chain isgeometrically ergodic and has finite moments and is strictlystationary

21 MS-ARMA-GARCH Models To avoid path-dependenceproblem Gray [92] suggests integrating out the unobservedregime path in the GARCH term by using the conditionalexpectation of the past variance Grayrsquos MS-GARCH modelis represented as follows

1205902

119905(119904119905)= 119908(119904119905)+

119902

sum

119894=1

120572119894(119904119905)1205762

119905minus119894+

119901

sum

119895=1

120573119895(119904119905)119864(

1205762

119905minus119895

119868119905minus119895minus1

)

= 119908(119904119905)+

119902

sum

119894=1

120572119894(119904119905)1205762

119905minus119894

+

119901

sum

119895=1

120573119895(119904119905)

119898

sum

119904119905minus119895=1

119875(119904119905minus119895

=

119904119905minus119895

119868119905minus119895minus1

)1205902

119905minus119895 119904119905minus119895

(1)

where 119908119904119905

gt 0 120572119894119904119905

ge 0 120573119895119904119905

ge 0 and 119894 = 1 119902119895 = 1 119901 119904

119905= 1 119898 The probabilistic structure of

the switching regime indicator 119904119905is defined as a first-order

Markov process with constant transition probabilities 1205871and

1205872 respectively (Pr119904

119905= 1 | 119904

119905minus1= 1 = 120587

1 Pr119904119905= 2 |

119904119905minus1

= 1 = 1 minus 1205871 Pr119904

119905= 2 | 119904

119905minus1= 2 = 120587

2 and

Pr119904119905= 1 | 119904

119905minus1= 2 = 1 minus 120587

2)

Although Dueker [101] accepts a collapsing procedure ofKimrsquos [102] algorithm to overcome path-dependence prob-lem Dueker [101] adopts the same framework of Gray [92]Accordingly the modified GARCH version of Dueker [101] is


accepted which governs the dispersion instead of traditionalGARCH(11) specification

Yang [103] Yao and Attali [94] Yao [95] and Francqand Zakoıan [96] derived conditions for the asymptoticstationarity of models with Markov switching regimes (seefor detailed information Bauwens and Rombouts [104]The major differences between Markov switching GARCHmodels are the specification of the variance process thatis the conditional variance 120590

2

119905= Var(120576

119905119878119905) To consider

the conditional variance as in the Bollerslevrsquos [105] GARCHmodel and to consider the regime dependent equation for theconditional variance in Frommel [106] are accepted that Thecoefficients 119908

119904119905 120572119904119905 120573119904119905correspond to respective coefficients

in the one-regime GARCHmodel but may differ dependingon the present state

Klaassen [85] (Klassen [85] model is defined as 1205902

119905(119904119905)=

119908(119904119905)+sum119902

119894=1120572119894(119904119905)1205762

119905minus119894+sum119901

119895=1120573119895(119904119905)sum119898

119904=1119875(119878119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905=

119904119905)1205902

119905minus119895 119904119905minus119895

) suggested to use the conditional expectation ofthe lagged conditional variance with a broader informationset than the model derived in Gray [92] AccordinglyKlaassen [85] suggested modifying Grayrsquos [92] model byreplacing 119901(119904

119905minus119895= 119904119905minus119895

| 119868119905minus119895minus1

) by 119901(119904119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905= 119904119905)

while evaluating 1205902

119905 119904119905

Another version of MS-GARCH model is developed byHaas et al [81ndash83] According to this model Markov chaincontrols the ARCH parameters at each regime (119908

119904 120572119894119904) and

the autoregressive behavior in each regime is subject to theassumption that the past conditional variances are in the sameregime as that of the current conditional variance [100]

In this study models will be derived following the MS-ARMA-GARCH specification in the spirit of Blazsek andDownarowicz [107] where the properties of MS-ARMA-GARCH processes were derived following Gray [92] andKlaassen [85] framework Henneke et al [108] developedan approach to investigate the model derived in Francqet al [78] for which the Bayesian framework was derivedThe stationarity of the model was evaluated by Francq andZakoıan [96] and an algorithm to compute the Bayesianestimator of the regimes and parameters was developed Itshould be noted that the MS-ARMA-GARCHmodels in thispaper were developed by following the models developedin the spirit of Gray [92] and Klaassen [85] similar to theframework of Blazsek and Downarowicz [107]

TheMS-ARMA-GARCHmodel with regime switching inthe conditional mean and variance are defined as a regimeswitching model where the regime switches are governed byan unobserved Markov chain in the conditional mean and inthe conditional variance processes as

119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119898

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119894=1

120572119894(119904119905)1205762

119905minus119894(119904119905)+

119902

sum

119895=1

120573(119904119905)120590119905minus119895(119904

119905)

(2)

where

120576119905minus119894minus1(119904

119905minus119894)= 119864 [120576

119905minus119894minus1(119904119905minus119894minus1)| 119904119905minus119894

119884119905minus119894minus1

]

120590119905minus119894minus1(119904

119905minus119894)= 119864 [120576


119884119905minus119894minus1

]

(3)

Thus the parameters have nonnegativity constraints120601 120579 120593 119908 120572 120573 gt 0 and the regimes are determined by 119904

119905

119871 =

119879

prod

119905=1

119891 (119910119905| 119904119905= 119894 119884119905minus1

)Pr [119904119905= 119894 | 119884

119905minus1] (4)

and the probability Pr[119904119905= 119894 | 119884

119905minus1] is calculated through

iteration

120587119895119905

= Pr [119904119905= 119895 | 119884

119905minus1]

=

1

sum

119894=0

Pr [119904119905= 119895 | 119904

119905minus1= 119894]Pr [119904

119905= 119895 | 119884

119905minus1]

1

sum

119894=0

120578119895119894120587lowast

119894119905minus1

(5)

Accordingly the twomodels the Henneke et al [108] and theFrancq et al [78] approaches could be easily differentiatedthrough the definitions of 1205762

119905minus1and 120590

119905minus1 Further asymmetric

power terms and fractional integration will be introduced tothe derived model in the following sections

22 MS-ARMA-APGARCH Model Liu [99] provided a gen-eralization of the Markov switching GARCH model of Haaset al [81ndash83] and derived the conditions for stationarityand for the existence of moments Liu [99] proposes amodel which allowed for a nonlinear relation between pastshocks and future volatility as well as for the leverage effectsThe leverage effect is an outcome of the observation thatthe reaction of stock market volatility differed significantlyto the positive and the negative innovations Haas et al[109 110] complements Liursquos [99] work in two ways Firstlythe representation of the model developed by Haas [109]allows computational ease for obtaining the unconditionalmoments Secondly the dynamic autocorrelation structure ofthe power-transformed absolute returns (residuals)was takenas a measure of volatility

Haas [109] model assumes that time series 120576119905 119905 isin Z

follows a k regime MS-APGARCH process

120576119905= 120578119905120590Δ119905119905

119905 isin Z (6)

with 120578119905 119905 isin Z being iid sequence and Δ

119905 119905 isin Z is a

Markov chain with finite state space 119878 = 1 119896 and 119875 isthe irreducible and aperiodic transition matrix with typicalelement 119901

119894119895= 119901(Δ

119905= 119895 | Δ

119905minus1= 119894) so that

119875 = [119901119894119895] = [119901 (Δ

119905= 119895 | Δ

119905minus1= 119894)] 119894 119895 = 1 119896 (7)

The stationary distribution of Markov-chain is shown as120587infin

= (1205871infin

120587119896infin

)1015840

According to the Liu [99] notation of MS-APGARCHmodel the conditional variance 120590

2

119895119905of jth regime follows a

univariate APGARCH process as follows

120590120575

119895119905= 119908119895+ 1205721119895

1003816100381610038161003816120576+

119905minus1

1003816100381610038161003816

120575

+ 1205722119895

1003816100381610038161003816120576minus

119905minus1

1003816100381610038161003816

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (8)


where 119908119895gt 0 120572

1119895 1205722119895 120573119895ge 0 119895 = 1 119896 For the power

term 120575 = 2 and for 1205721119895

= 1205722119895 the model in (8) reduces toMS-

GARCHmodel Similar to theDing et al [19] the asymmetrywhich is called ldquoleverage effectrdquo is captured by 120572

1119895= 1205722119895[109]

If the past negative shocks have deeper impact parameters areexpected to be 120572

1119895lt 1205722119895so that the leverage effect becomes

strongerAnother approach that is similar to Liu [99] model is

the Haas [109] model where the asymmetry terms have adifferentiated form as

120590120575

119895119905= 119908119895+ 120572119895(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (9)

with the restrictions 0 lt 119908119895 120572119895 120573119895

ge 0 120574119895

isin [minus1 1] withregimes 119895 = 1 119896 The MS-APGARCH model of Haas[109] reduces to Ding et al [19] single regime APGARCHmodel if 119895 = 1 Equation (9) reduces to Liu [99] MS-APGARCH specification if 120572

1119895= 120572119895(1 minus 120574

119895)120575 and 120573

119895=

120572119904

119895(1 + 120574

119895)120575

The MS-ARMA-GARCH type model specification inthis study assumes that the conditional mean follows MS-ARMA process whereas the conditional variance followsregime switching in the GARCH architecture AccordinglyMS-ARMA-APGARCH architecture nests several models byapplying certain restrictions The MS-ARMA-APGARCHmodel is derived by moving from MS-ARMA process inthe conditional mean and MS-APGARCH(119897 119898) conditionalvariance process as follows

120590

120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119903

sum

119897=1

120572119897(119904119905)(1003816100381610038161003816120576119905minus119897

1003816100381610038161003816minus 120574119897(119904119905)120576119905minus119897

)

120575(119904119905)

+

119902

sum

119898=1

120573119898(119904119905)120590

120575(119904119905)

119905minus119898(119904119905) 120575(119904119905)gt 0

(10)

where the regime switches are governed by (119904119905) and the

parameters are restricted as 119908(119904119905)

gt 0 120572119897(119904119905) 120573119898(119904119905)

ge

0 with 120574119897(119904119905)

isin (minus1 1) 119897 = 1 119903 One importantdifference is thatMS-ARMA-APGARCHmodel in (10) allowsthe power parameters to vary across regimes Further if thefollowing restrictions are applied 119897 = 1 119895 = 1 120575

(119904119905)= 120575 the

model reduces to the model of Haas [109] given in (9)In applied economics literature it is shown that many

financial time series possess longmemory which can be frac-tionally integrated Fractional integration will be introducedto the MS-ARMA-APGARCHmodel given above

23 MS-ARMA-FIAPGARCH Model Andersen and Boller-slev [111] Baillie et al [17] Tse [112] and Ding et al [19]provided interesting applications in which the attention hadbeen directed on long memory Long memory could beincorporated to the model above by introducing fractionalintegration in the conditional mean and the conditionalvariance processes

MS-ARMA-FIAPGARCH derived is a fractional integra-tion augmented model as follows

(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 120596 + ((1 minus 120573(119904119905)119871) minus (1 minus 120572

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(119904119905)120576119905minus1

)

120575(119904119905)

(11)

where the lag operator is denoted by 119871 autoregressive param-eters are 120573

(119904119905) and 120572

(119904119905)shows themoving average parameters

120575(1)

gt 0 denotes the optimal power transformation thefractional differentiation parameter varies between 0 le

119889(119904119905)

le 1 and allows long memory to be integrated to themodel Regime states (119904

119905) are defined with 119898 regimes as

119894 = 1 119898 The asymmetry term |120574(119904119905)| lt 1 ensures that

positive and negative innovations of the same size may haveasymmetric effects on the conditional variance in differentregimes

Similar to the MS-ARMA-APGARCH model the MS-ARMA-FIAPGARCH model nests several models By apply-ing 120575

(119904119905)

= 2 the model reduces to Markov switchingfractionally integrated asymmetric GARCH (MS-ARMA-FIAGARCH) if 120575

(119904119905)

= 2 restriction is applied with 120574(119904119905)

=

0 the model reduces to Markov switching FIGARCH (MS-ARMA-FIGARCH) For 119889

(119904119905)= 0 model reduces to the short

memory version the MS-ARMA-APGARCH model if theadditional constraint 120575

(119904119905)= 2 is applied the model reduces

to MS-Asymmetric GARCH (MS-AGARCH) Lastly for allthe models mentioned above if 119894 = 1 all models reduceto single regime versions of the relevant models namelythe FIAPGARCH FIAGARCH FIGARCH and AGARCHmodels their relevant single regime variants For a typicalwith the constraints 119894 = 1 and 120575

(119904119905)= 2 120574

(119904119905)= 0 the model

reduces to single regime FIGARCHmodel of Baillie et al [17]To differentiate between the GARCH specifications forecastperformance criteria comparisons are assumed

3 Neural Network andMS-ARMA-GARCH Models

In this section of the study the MultiLayer Perceptron Rad-ical Basis Function and Recurrent Neural Network modelsthat belong to the ANN family will be combinedwithMarkovswitching and GARCH models In this respect Spezia andParoli [113] is another study that merged the Neural Networkand MS-ARCHmodels

31 Multilayer Perceptron (MLP) Models

311 MS-ARMA-GARCH-MLP Model Artificial NeuralNetwork models have many applications in modeling offunctional forms in various fields In economics literaturethe early studies such as Dutta and Shektar [114] Tom andKiang [115] Do and Grudinsky [116] Freisleben [51] andRefenes et al [55] utilize ANN models to option pricingreal estates bond ratings and prediction of banking failuresamong many whereas studies such as Kanas [40] Kanas and


Yannopoulos [65] and Shively [117] applied ANN models tostock return forecasting and Donaldson and Kamstra [118]proposed hybrid modeling to combine GARCH GJR andEGARCHmodels with ANN architecture

The MLP an important class of neural networks consistsof a set of sensory units that constitute the input layer oneor more hidden layers and an output layer The additionallinear input which is connected to the MLP network is calledthe Hybrid MLP Hamilton model can also be consideredas a nonlinear mixture of autoregressive functions such asthe multilayer perceptron and thus the Hamilton model iscalled Hybrid MLP-HMC models [119] Accordingly in theHMCmodel the regime changes are dominated by aMarkovchain without making a priori assumptions in light of thenumber of regimes [119] In fact Hybrid MLP accepts thenetwork inputs to be connected to the output nodes withweighted connections to form a linear model that is parallelwith nonlinear Multilayer Perceptron

In the study the MS-ARMA-GARCH-MLP model tobe proposed allows Markov switching type regime changesboth in the conditional mean and conditional varianceprocesses augmented with MLP type neural networks toachieve improvement in terms of in-sample and out-of-sample forecast accuracy

The MS-ARMA-GARCH-MLP model is defined of theform

119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119899

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905) (12)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(13)

where the regimes are governed by unobservable Markovprocess

119898

sum

119894=1

1205902

119905(119894)119875 (119878119905= 119894 | 119911

119905minus1) 119894 = 1 119898 (14)

In the MLP type neural network the logistic type sigmoidfunction is defined as

120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

= [1+exp(minus120591ℎ(119904119905)(

119897

sum

119897=1

[

ℎ

sum

ℎ=1

120582ℎ119897(119904119905)119911ℎ

119905minus119897(119904119905)+ 120579ℎ(119904119905)]))]

minus1

(15)

(

1

2

) 120582ℎ119889

sim uniform [minus1 +1] (16)

and119875(119878119905= 119894 | 119911

119905minus1) the filtered probability with the following

representation

(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (17)

if 119899119895119894transition probability 119875(119904

119905= 119894 | 119904

119905minus1= 119895) is accepted

119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(18)

119904 rarr max119901 119902 recursive procedure is started by con-structing 119875(119911

119904= 119894 | 119911

119904minus1) where 120595(119911

119905120582ℎ) is of the form

1(1 + exp(minus119909)) a twice-differentiable continuous functionbounded between [0 1] The weight vector 120585 = 119908 120595 = 119892

logistic activation function and input variables are defined as119911119905120582ℎ= 119909119894 where 120582

ℎis defined as in (16)

If 119899119895119894

transition probability 119875(119911119905

= 119894 | 119911119905minus1

= 119895) isaccepted

119891 (119910119905| 119909119905 119911119905= 119894)

=

1

radic2120587ℎ119905(119894)

exp

minus(119910119905minus 1199091015840

119905120593 minus sum

119867

119895=1120573119895119901 (1199091015840

119905120574119895))

2

2ℎ119905(119895)

(19)

119904 rarr max119901 119902 recursive procedure is started by construct-ing 119875(119911

119904= 119894 | 119911

119904minus1)

312 MS-ARMA-APGARCH-MLP Model Asymmetricpower GARCH (APGARCH) model has interesting featuresIn the construction of the model the APGARCH structureof Ding et al [19] is followed The model given in (13)is modified to obtain the Markov switching APGARCHMultilayer Perceptron (MS-ARMA-APGARCH-MLP)model of the form

120590120575(119904119905)

119905(119904119905)

= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119901

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(20)

where regimes are governed by unobservable Markovprocess The model is closed as defining the conditionalmean as in (12) and conditional variance of the formequationrsquos (14)ndash(19) and (20) to augment the MS-ARMA-GARCH-MLP model with asymmetric power terms toobtain MS-ARMA-APGARCH-MLP Note that modelnest several specifications Equation (20) reduces to theMS-ARMA-GARCH-MLP model in (13) if the powerterm 120575 = 2 and 120574

119901(119904119905)

= 0 Similarly the model nestsMS-GJR-MLP if 120575 = 2 and 0 le 120574

119901(119904119905)

le 1 are imposedThe model may be shown as MSTGARCH-MLP modelif 120575 = 1 and 0 le 120574

119901(119904119905)

le 1 Similarly by applying asingle regime restriction 119904

119905= 119904 = 1 the quoted models

reduce to their respective single regime variants namelythe ARMA-APGARCH-MLP ARMA-GARCH-MLPARMA-NGARCH-MLP ARMA-GJRGARCH-MLP andARMA-GARCH-MLP models (for further discussion inNN-GARCH family models see Bildirici and Ersin [61])


313 MS-ARMA-FIAPGARCH-MLP Model Following themethodology discussed in the previous section MS-ARMA-APGARCH-MLP model is augmented with neural networkmodeling architecture and that accounts for fractional inte-gration to achieve long memory characteristics to obtainMS-ARMA-FIAPGARCH-MLP Following the MS-ARMA-FIAPGARCH represented in (11) the MLP type neuralnetwork augmented MS-ARMA-FIAPGARCH-MLP modelrepresentation is achieved

(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(21)

where ℎ are neurons defined with sigmoid type logisticfunctions 119894 = 1 119898 regime states governed byunobservable variable following Markov process Equation(21) defines the MS-ARMA-FIAPGARCH-MLP modelthe fractionally integration variant of the MSAGARCH-MLP model modified with the ANN and the logisticactivation function 120595(120591

ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905)) defined as

in (15) Bildirici and Ersin [61] proposes a class of NN-GARCH models including the NN-APGARCH Similarlythe MS-ARMA-FIAPGARCH-MLP model reduces tothe MS-FIGARCH-MLP model for restrictions on thepower term 120575

(119904119905)

= 2 and 120574(119904119905)

= 0 Further the modelreduces to MS-FINGARCH-MLP model for 120574

(119904119905)

= 0 andto the MS-FI-GJRGARCH-MLP model if 120575

(119904119905)

= 2 and120574(119904119905)is restricted to be in the range of 0 le 120574

(119904119905)

le 1 Themodel reduces to MS-TGARCH-MLP model if 120575

(119904119905)

= 1

in addition to the 0 le 120574(119904119905)

le 1 restriction On thecontrary if single regime restriction is imposed modelsdiscussed above namely MS-ARMA-FIAPGARCH-MLPMSFIGARCH-NN MSFIGARCH-NN MSFINGARCH-MLP MSFIGJRGARCH-MLP and MSFITGARCH-MLPmodels reduce to NN-FIAPGARCH NN-FIGARCH NN-FIGARCH NN-FINGARCH NN-FIGJRGARCH andNN-FITGARCH models which are single regime neuralnetwork augmented GARCH family models of the formBildirici and Ersin [61] that do not possess Markov switchingtype asymmetry (Bildirici and Ersin [61]) The model alsonests model variants that do not possess long memorycharacteristics By imposing 119889

(119904119905)

= 0 to the fractionalintegration parameter which may take different values under119894 = 1 2 119898 different regimes the model in (21) reducesto MS-ARMA-APGARCH-MLP model the short memorymodel variant In addition to the restrictions appliedabove application of 119889

(119904119905)

= 0 results in models withoutlong memory characteristics MS-ARMA-FIAPGARCH-MLP MS-ARMA-GARCH-MLP MS-ARMA-GARCH-MLP MSNGARCH-MLP MS-GJR-GARCH-MLP andMSTGARCH-MLP

For a typical example consider a MS-ARMA-FIAPGARCH-MLP model representation with two regimes

(1 minus 120573(1)

119871) 120590

120575(1)

119905(1)

= 119908(1)

+ ((1 minus 120573(1)

119871) minus (1 minus 120601(1)

119871) (1 minus 119871)119889(1))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(1)

120576119905minus1

)120575(1)

+

ℎ

sum

ℎ=1

120585ℎ(1)

120595 (120591ℎ(1)

119885119905(1)

120582ℎ(1)

120579ℎ(1)

)

(1 minus 120573(2)

119871) 120590

120575(2)

119905(2)

= 119908(2)

+ ((1 minus 120573(2)

119871) minus (1 minus 120601(2)

119871) (1 minus 119871)119889(2))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(2)

120576119905minus1

)120575(2)

+

ℎ

sum

ℎ=1

120585ℎ(2)

120595 (120591ℎ(2)

119885119905(2)

120582ℎ(2)

120579ℎ(2)

)

(22)

Following the division of regression space into two sub-spaces withMarkov switching themodel allows two differentasymmetric power terms 120575

(1)and 120575

(2) and two different

fractional differentiation parameters 119889(1)

and 119889(2) as a result

different long memory and asymmetric power structures areallowed in two distinguished regimes

It is possible to show the model as a single regime NN-FIAPGARCHmodel if 119894 = 1

(1 minus 120573119871) 120590120575

119905

= 120596 + ((1 minus 120573119871) minus (1 minus 120601119871) (1 minus 119871)119889)

times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(23)

Further the model reduces to NN-FIGARCH if 119894 = 1 and120575(1199041)= 120575 = 2 in the fashion of Bildirici and Ersin [61]

(1 minus 120573119871) 1205902

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

2

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(24)

32 Radial Basis Function Model Radial Basis Functions areone of the most commonly applied neural network modelsthat aim at solving the interpolation problem encounteredin nonlinear curve fittingLiu and Zhang [120] utilized theRadial Basis Function Neural Networks (RBF) and Markovregime-switching regressionsto divide the regression spaceinto two sub-spaces to overcome the difficulty in estimatingthe conditional volatility inherent in stock returns FurtherSantos et al [121] developed a RBF-NN-GARCH model thatbenefit from the RBF type neural networks Liu and Zhang[120] combined RBF neural networkmodels with theMarkovSwitchingmodel tomergeMarkov switchingNeural Networkmodel based on RBF models RBF neural networks in their


models are trained to generate both time series forecastsand certainty factors Accordingly RBF neural network isrepresented as a composition of three layers of nodes firstthe input layer that feeds the input data to each of the nodesin the second or hidden layer the second layer that differsfrom other neural networks in that each node represents adata cluster which is centered at a particular point and has agiven radius and in the third layer consisting of one node

321 MS-ARMA-GARCH-RBF Model MS-GARCH-RBFmodel is defined as

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)

where 119894 = 1 119898 regimes are governed by unobservableMarkov process

119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)

A Gaussian basis function for the hidden units given as 120601(119909)for 119909 = 1 2 119883where the activation function is defined as

120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)

With 119901 defining the width of each function 119885119905is a vector

of lagged explanatory variables 120572 + 120573 lt 1 is essential toensure stationarity Networks of this type can generate anyreal-valued output but in their applications where they havea priori knowledge of the range of the desired outputs itis computationally more efficient to apply some nonlineartransfer function to the outputs to reflect that knowledge

119875(119878119905

= 119894 | 119911119905minus1

) is the filtered probability with thefollowing representation

(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)

If 119899119895119894transition probability 119875(119904

119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)

322 MS-ARMA-APGARCH-RBFModel Radial basis func-tions are three-layer neural network models with linearoutput functions and nonlinear activation functions definedas Gaussian functions in hidden layer utilized to the inputs

in light of modeling a radial function of the distance betweenthe inputs and calculated value in the hidden unitThe outputunit produces a linear combination of the basis functions toprovide a mapping between the input and output vectors

120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)

where 119894 = 1 119898 regime model and regimes are governedby unobservable Markov process Equations (26)ndash(29) with(30) define the MS-ARMA-APGARCH-RBF model Similarto the MS-ARMA-APGARCH-MLP model the MS-ARMA-APGARCH-RBF model nests several models Equation (30)reduces to the MS-ARMA-GARCH-RBF model if the powerterm 120575 = 2 and 120574

119901(119904119905)= 0 to the MSGARCH-RBF model

for 120574119901(119904119905)

= 0 and to the MSGJRGARCH-RBF model if120575 = 2 and 0 le 120574

119901(119904119905)

le 1 restrictions are allowed Themodel may be shown as MSTGARCH-RBF model if 120575 =

1 and 0 le 120574119901(119904119905)

le 1 Further single regime modelsnamely NN-APGARCH NN-GARCH NN-GARCH NN-NGARCH NN-GJRGARCH and NN-TGARCH modelsmay be obtained if 119905 = 1 (for further discussion in NN-GARCH family models see Bildirici and Ersin [61])

323 MS-ARMA-FIAPGARCH-RBF Model MS-FIAPGARCH-RBF model is defined as

(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)

where ℎ are neurons defined with Gaussian functionsThe MS-ARMA-FIAPGARCH-RBF model is a variant ofthe MSAGARCH-RBF model with fractional integrationaugmented with ANN architecture Similarly the MS-ARMA-FIAPGARCH-RBFmodel reduces to theMS-ARMA-FIGARCH-RBF model with restrictions on the power term120575(119904119905)

= 2 and 120574(119904119905)

= 0 The model nests MSFINGARCH-RBF model for 120574

(119904119905)= 0 and MSFIGJRGARCH-RBF model

if 120575(119904119905)= 2 and 120574

(119904119905)varies between 0 le 120574

(119904119905)le 1 Further the

model may be shown as MSTGARCH-RBF model if 120575(119904119905)= 1

and 0 le 120574(119904119905)le 1 With single regime restriction 119894 = 1 dis-

cussed models reduce to NN-FIAPGARCH NN-FIGARCHNN-FIGARCH NN-FINGARCH NN-FIGJRGARCH andNN-FITGARCH models which do not possess Markovswitching type asymmetry To obtain the model with short


memory characteristics 119889(sdot)

= 0 restriction on fractionalintegration parameters should be imposed and the modelreduces to MSAPGARCH-RBF model the short memorymodel variant Additionally by applying 119889

(sdot)= 0 with

the restrictions discussed above models without long mem-ory characteristics MSFIAPGARCH-RBF MSGARCH-RBFMSGARCH-RBF MSNGARCH-RBF MSGJRGARCH-RBFand MSTGARCH-RBF models could be obtained

33 Recurrent Neural Network MS-GARCH Models TheRNN model includes the feed-forward system howeverit distinguishes itself from standard feed-forward networkmodels in the activation characteristics within the layersThe activations are allowed to provide a feedback to unitswithin the same or preceding layer(s) This forms an internalmemory system that enables a RNN to construct sensitiveinternal representations in response to temporal featuresfound within a data set

The Jordan [122] and Elmanrsquos [123] networks are simplerecurrent networks to obtain forecasts Jordan and Elmannetworks extend the multilayer perceptron with contextunits which are processing elements (PEs) that rememberpast activity Context units provide the network with the abil-ity to extract temporal information from the data The RNNmodel employs back propagation-through-time an efficientgradient-descent learning algorithm for recurrent networksIt was used as a standard variant of cross-validation referredto as the leave-one-out method and as a stopping criterionsuitable for estimation problems with sparse data and so it isidentified the onset of overfitting during training The RNNwas functionally equivalent to a nonlinear regression modelused for time-series forecasting (Zhang et al [124] Binneret al [125]) Tino et al [126] merged the RNN and GARCHmodels

331 MS-ARMA-GARCH-RNN Models The model isdefined as

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)

Similar to the models above (32) is shown for 119894 = 1 119898

regimes which are governed by unobservable Markov pro-cess Activation function is taken as the logistic function

332 MS-ARMA-APGARCH-RNN Markov switchingAPGARCH Recurrent Neural Network Model is representedas

120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)

119894 = 1 119898 regimes are governed by unobservable Markovprocess 120579

119896ℎ(119904119905)is the weights of connection from pre to

postsynaptic nodesΠ(119909) is a logistic sigmoid function of theform given in (15) 120594

119905minus119896ℎ(119904119905)is a variable vector corresponding

to the activations of postsynaptic nodes the output vector ofthe hidden units and 120579

119896ℎ(119904119905)are the bias parameters of the

presynaptic nodes and 120585119894(119904119905)are the weights of each hidden

unit for ℎ hidden neurons 119894 = 1 ℎ The parameters areestimated by minimizing the sum of the squared-error lossmin 120582 = sum

119879

119905minus1[120590119905

minus 119905]

2

Themodel is estimated by recurrentback-propagation algorithm and by the recurrent Newtonalgorithm By imposing several restrictions similar to theMS-ARMA-APGARCH-RBF model several representations areshown under certain restrictions Equation (33) reduces toMS-ARMA-GARCH-RNN model with 120575 = 2 and 120574

119901(119904119905)

=

0 to the MSGARCH-RNN model for 120574119901(119904119905)

= 0 andto the MSGJRGARCH-RNN model if 120575 = 2 and 0 le

120574119901(119904119905)

le 1 restrictions are imposed MSTGARCH-RNNmodel is obtained if 120575 = 1 and 0 le 120574

119901(119904119905)le 1 In addition

to the restrictions above if the single regime restriction119894 = 1 is implied the model given in Equation (33) reducesto their single regime variants namely the APGARCH-RNN GARCH-RNN GJRGARCH-RNN and TGARCH-RNN models respectively

333 MS-ARMA-FIAPGARCH-RNN Markov SwitchingFractionally Integrated APGARCH Recurrent NeuralNetwork Model is defined as

(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)

where ℎ are neurons defined as sigmoid type logisticfunctions and 119894 = 1 119898 regime states the followingMarkov process The MS-ARMA-FIAPGARCH-RNNmodelis the fractionally integrated variant of the MS-ARMA-APGARCH-RNN model The MS-ARMA-FIAPGARCH-RNN model reduces to the MS-ARMA-FIGARCH-RNNmodel with restrictions on the power term 120575

(119904119905)= 2 and 120574

(119904119905)=

0 Further themodel reduces toMSFINGARCH-RNNmodelfor 120574(119904119905)= 0 to the MSFIGJRGARCH-RNNmodel if 120575

(119904119905)= 2


and 0 le 120574(119904119905)

le 1 The model reduces to MSFITGARCH-RNN model for 0 le 120574

(119904119905)le 1 and 120575

(119904119905)= 1 Single regime

restriction 119894 = 1 leads to the NN-FIAPGARCH-RNN NN-FIGARCH-RNN NN-FIGARCH-RNN NN-FINGARCH-RNN NN-FIGJRGARCH-RNN andNN-FITGARCH-RNNmodels without Markov switching

4 Data and Econometric Results

41 The Data In order to test forecasting performance of theabovementioned models stock return in Turkey is calculatedby using the daily closing prices of Istanbul Stock Index ISE100 covering the 07121986ndash13122010 period correspondingto 5852 observations To obtain return series the data iscalculated as follows 119910

119905= ln(119875

119905119875119905minus1

) where ln(sdot) is thenatural logarithms 119875

119905is the ISE 100 index and 119910

119905is taken

as a measure of stock returns In the process of training themodels the sample is divided between training test and out-of-sample subsamples with the percentages of 80 10 and10 Further we took the sample size for the training sampleas the first 4680 observations whereas the sample sizes for thetest and out-of-sample samples are 585 and 587 The statisticsof daily returns calculated from ISE 100 Index are given inTable 1

In order to provide out-of-sample forecasts of the ISE100daily returns two competing nonlinear model structures areused the univariate Markov switching model and NeuralNetwork Models MLP RBF and RNN In order to assess thepredictability of models models are compared for their out-of-sample forecasting performance Firstly by calculatingRMSE and MSE error criteria the forecast comparisons areobtained

42 Econometric Results At the first stage selected GARCHfamily models taken as baseline models are estimated forevaluation purposes Results are given in Table 2 Includedmodels have different characteristics to be evaluated namelyfractional integration asymmetric power and fractionallyintegrated asymmetric power models namely GARCHAPGARCH FIGARCH and FIAPGARCHmodels Randomwalk (RW) model is estimated for comparison purposeFurthermore the models given in Table 2 will provide basisfor nonlinear models to be estimated

It is observed that though all volatility models performbetter than the RWmodel in light of Log Likelihood criteriaas we move from the GARCH model to asymmetric powerGARCH (APGARCH) model the fit of the models improveaccordingly The sum of ARCH and GARCH parametersis calculated as 0987 and less than 1 The results for theAPGARCH model show that the calculated power termis 135 and the asymmetry is present Further similar tothe findings of McKenzie and Mitchell [127] it is observedthat the addition of the leverage and power terms improvesgeneralization power and thus show that squared power termmay not necessarily be optimal as Ding et al [19] studysuggested

In Table 3 transition matrix and the MS model wereestimatedThe standard deviation takes the values of 005287

and 0014572 for regime 1 and regime 2 It lasts approximately7587 months in regime 1 and 10761 months in regime 2 Byusing maximum likelihood approach MS-GARCH modelsare tested by assuming that the error terms follow student-119905 distribution with the help of BFGS algorithm Number ofregimes are taken as 2 and 3 GARCH effect in the residualsis tested and at 1 significance level the hypothesis thatthere are no GARCH effects is rejected Additionally thenormality in the residuals is tested with Jacque-Berra testat 1 significance level it is detected that the residuals arenot normally distributed As a result MS-GARCH modelis estimated under the 119905 distribution assumption In theMS-GARCH model the transition probability results arecalculated as Prob(119904

119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1

= 2) = 051 and show that the persistence is low in theMS-GARCHmodel

Statistical inference regarding the empirical validity oftwo-regime switching process was carried out by usingnonstandard LR tests [128] The nonstandard LR test isstatistically significant and this suggests that linearity isstrongly rejected

The volatility values tend to be calculated at higher valuesthan they actually are in GARCH models As the persistencecoefficients obtained for the MS-GARCH MS-PGARCHand MS-APGARCH models are compared to those obtainedfor the GARCH models the persistence in the GARCHmodels are comparatively higher and for theGARCHmodelsa shock in volatility is persistent and shows continuing effectThis situation occurs as a result of omitting the importanceof structural change in the ARCH process In the MS-GARCH MS-PGARCH and MS-APGARCH models whichtake this situation into consideration the value of persistenceparameter decreases

Power terms are reported comparatively lower for devel-oped countries than the less developed countries Haas et al[109 110] calculated three state RS-GARCH RS-PGARCHand RS-APGARCH models for the daily returns in NYSEand estimated the power terms for the RS-APGARCHmodelas 125 109 and 108 For Turkey Ural [129] estimated RS-APGARCH models for returns in ISE100 index in Turkey inaddition to United Kingdom FTSE100 CAC40 in France andNIKKEI 225 indices in Japan and reported highest powerestimates (184) compared to the power terms calculated as126 131 and 124 for FTSE100 NIKKEI 225 and CAC40Further power terms obtained for returns calculated forstock indices in many developing economies are calculatedcomparatively higher than those obtained for the variousindices in developed countries Ane and Ureche-Rangau[130] estimated single regimeGARCH andAPGARCHmod-els in addition to RS-GARCH and RS-APGARCH modelsfollowing Gray [92] model Power terms in single regimeAPGARCH models were calculated for daily returns as 157in Nikkei 225 Index as 181 in Hang Seng Index as 169 inKuala Lumpur Composite Index and as 241 in SingaporeSES-ALL Index whereas for regime switching APGARCHmodels power terms are calculated as 120 in regime 1 and183 in regime 2 for Nikkei 225 Index 216 in regime 1 and 231in regime 2 for Heng Seng Index 195 and 217 for regimes1 and 2 in Singapore SES-ALL Index and 171 and 225 in


Table 1 Daily returns in ISE 100 index basic statistics

Mean Median Mode Minimum Maximum Standard deviation Skewness Kurtosis0001725 0001421 00000001 minus019979 0217108 0028795 0071831 4207437

Table 2 Baseline volatility models

Baseline GARCHmodels single regime(1) RW C log 119871

0001873lowastlowastlowast(0000391) 1224156

(2) GARCH ARCH GARCH C log 11987101591259lowastlowastlowast(00068625)

08280324lowastlowastlowast(00056163)

00000206lowastlowastlowast(180119890 minus 06) 1317466

(3) APGARCH APARCH APARCH E PGARCH POWER C log 11987102153934lowastlowastlowast(00076925)

minus00319876lowastlowastlowast(00160775)




(4) FIAPGARCH ARCH(Phi1)

GARCH(Beta1) D-FIGARCH APARCH

(Gamma1)APARCH(Delta)

C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630

regimes 1 and 2 for the model calculated for returns in KualaLumpur Composite Index Teletar and Binay [131] estimatedAPARCH models for Turkey and 10 national stock indicesandnoted that power terms reported for developing countriestend to be high and varying though those reported for thedeveloped countries are estimated with low and close valuesThey estimated power terms for ISE100 index as 1960 for1987ndash2001 period and 148 for 1989ndash1996 period whereaspower terms for SampP FTSE NIKKEI Hong Kong NZSE40DAX index of Germany CAC40 of France Singaporersquos SESTSE ALORAI and MSCI indices as 121 143 122 135 137091 117 248 145 101 and 137 respectively

On the other hand though the improvement by shiftingto modeling the conditional volatility with regime switchingis noteworthy the desired results are still not obtainedtherefore MS-GARCH models are extended with MLP RBFand RNN models and their modeling performances aretested

421 MS-ARMA-GARCH-Neural Networks Results In thestudy model estimation is gathered through utilizing back-propagation algorithm and the parameters are updatedwith respect to a quadratic loss function whereas theweights are iteratively calculated with weight decay methodto achieve the lowest error Alternative methods includeGenetic Algorithms [132ndash134] and 2nd order derivativebased optimization algorithms such as Conjugate Gradi-ent Descent Quasi-Newton Quick Propagation Delta-Bar-Delta and Levenberg-Marquardt which are fast and effectivealgorithms but may be subject to overfitting (see [135ndash137])In the study we followed a two-step methodology Firstlyall models were trained over a given training sample vis-a-vis checking for generalization accuracy in light of RMSEcriteria in test sampleThe approach is repeated for estimatingeach model for 100 times with different number of sigmoid

activation functions in the hidden layer Hence to obtainparsimony in models best model is further selected withrespect to the AIC information criterion For estimatingNN-GARCH models with early stopping combined withalgorithm corporation readers are referred to Bildirici andErsin [61]

The estimated models are reported in Table 4 For com-parative purposesMSE and RMSE values for training sampleare given Among the MS-ARMA-GARCH-NN models andfor the training sample the lowest RMSE value is achievedas 0114 by the MS-ARMA-GARCH-RBF model followed byMS-ARMA-GARCH-MLPmodel with a RMSE value of 0186as the second best model among the models with GARCHspecifications noted at the first part of Table 4 Furthermorethe above mentioned models are followed by MS-ARMA-GARCH-HYBRID MLP model that deserves the 3rd placewith RMSE = 0187 though the value is almost equal tothe value obtained for the MS-ARMA-GARCH-MLP modelhaving the 2nd place Further MS-ARMA-GARCH-RNNandMS-ARMA-GARCH-ELMAN RNNmodels take the 4thand 5th place among the GARCH type competing models interms of in-sample accuracy only

The Markov-switching models with asymmetric powerterms in the conditional volatility namely the MS-ARMA-APGARCH-NN models are reported in thesecond part of Table 4 The model group consists ofMS-ARMA-APGARCH-RNN MS-ARMA-APGARCH-RBF MS-ARMA-APGARCH-ELMAN RNN and MS-ARMA-APGARCH-MLP models It is observed amongthe group that the lowest RMSE value is attained for theMS-ARMA-APGARCH-MLP model with a value of 016307followed by MS-ARMA-APGARCH-RNN model FurtherMS-ARMA-APGARCH-ELMAN RNN and MS-ARMA-APGARCH-RNN models possess the 2nd and 3rd placeswith RMSE values equal to 01663 and 01776 respectively


Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664

83(001307)lowastlowastlowast

000

0333727

(1916119890minus006)lowastlowastlowast

003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124

(001307)lowastlowastlowast

000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


Table 4 Markov switching GARCH neural network models training results

MSE RMSEModel Group 1 MS-ARMA-GARCH-neural network models

MS-ARMA-GARCH-RNN 0035103533 0187359368 (4th)MS-ARMA-GARCH-RBF 0013009577 0114059534 (1st)MS-ARMA-GARCH-ELMAN RNN 0051024351 0225885704 (5th)MS-ARMA-GARCH-HYBRID MLP 0034996508 0187073536 (3rd)MS-ARMA-GARCH-MLP 0034665716 0186187315 (2nd)

Model Group 2 MS-ARMA-APGARCH-neural network modelsMS-ARMA-APGARCH-RNN 0031530707 0177568881 (4th)MS-ARMA-APGARCH-RBF 0056680761 0238077218 (5th)MS-ARMA-APGARCH-ELMAN RNN 0027644691 0166266929 (3rd)MS-ARMA-APGARCH-HYBRID MLP 0026504969 0162803470 (1st)MS-ARMA-APGARCH-MLP 0026591934 0163070336 (2nd)

Model Group 3 MS-ARMA-FIAPGARCH-neural network modelsMS-ARMA-FIAPGARCH-RNN 0029951509 0173065042 (1st)MS-ARMA-ARMA-FIAPGARCH-RBF 0045969206 0214404305 (5th)MS-ARMA-FIAPGARCH-ELMAN RNN 0033859273 0184008896 (4th)MS-ARMA-FIAPGARCH-HYBRID MLP 0031056323 0176228044 (2nd)MS-ARMA-FIAPGARCH-MLP 0031221803 0176696926 (3rd)

Lastly the lowest RMSE value (=02381) is attained for theMS-APGARCH-RBF model among the group

Models in the third model group namely MS-ARMA-FIAPGARCH-RNN MS-ARMA-FIAPGARCH-RBFMS-ARMA-FIAPGARCH-RNN MS-ARMA-FIAPGARCH-HYBRID MLP and MS-ARMA-FIAPGARCH-MLP modelsare modified versions of MS-ARMA-APGARCH-NNmodels with fractional integration According to the resultsobtained in Table 4 the lowest RMSE value is obtainedfor the MS-ARMA-FIAPGARCH-RNN model (RMSE =01731) followed by the MS-ARMA-APGARCH-HYBRIDMLP model (RMSE = 01762) The RMSE is calculated as01767 for the MS-ARMA-FIAPGARCH-MLP model whichobtained the 3rd place though it is observed that the modelsthat deserved the first three places have very close RMSEvalues Further MS-ARMA-FIAPGARCH-ELMAN RNNand MS-ARMA-FIAPGARCH-RBF models possess the 4thand 5th places with RMSE values of 01840 and 02144 withrespect to the results obtained for the training sample

The results obtained for the test sample and the hold-out sample are given in Tables 5 and 6 If an overlookis to be provided it is observed that holding the trainingsample results on one side the real improvement occurs forforecasting When we move from MS-ARMA-GARCH-NNmodels (denoted as Model Group 1) towards MS-ARMA-APGARCH (denoted as Model Group 2) and fractionallyintegrated models of MS-ARMA-FIAPGARCH-NN (ModelGroup 3) models though certain amount of in-sampleforecast gain is achieved the MSE and RMSE values are thelowest for test sample and most importantly for the out-of-sample forecasts

To evaluate the test sample performance of the estimatedmodelsMSE and RMSE error criteria are reported in Table 5Accordingly among the MS-ARMA-GARCH-NN models

the lowest RMSE value for the test sample is 01238 andis calculated for the MS-ARMA-GARCH-MLP model thatis followed by the MS-ARMA-GARCH-RNN model with aRMSE of 01242 The MS-ARMA-GARCH-HYBRID MLPmodel takes the 3rd place in the test sample similar to theresults obtained for the training sample with RMSE = 01243though the value is almost equal to the models taking thefirst and the second places Compared with the first threemodels we mentioned with RMSE values that are achievedas low as around 0124 MS-ARMA-GARCH-ELMAN RNNand MS-ARMA-GARCH-RBF models take the 4th and 5thplaces with RMSE values equal to 01538 and 01546 Theresults show that for the test sample MS-ARMA-GARCH-MLP MS-ARMA-GARCH-RNN and MS-ARMA-GARCH-HYBRID MLP models have very similar results and wellgeneralization capabilities though the last 2 competingmod-els also have promising results Furthermore as we evaluatethe APGARCH type models the generalization capabilitiesincrease significantly Among the MS-ARMA-APGARCH-NN models given in the table MS-ARMA-APGARCH-HYBRID MLP model has the lowest RMSE (=000011539)followed by the MS-ARMA-APGARCH-MLP model (RMSE= 000011548) The third best model is the MS-ARMA-APGARCH-ELMAN RNN model with RMSE = 000011789followed by the 4th and 5th modelsMS-ARMA-APGARCH-RNN (RMSE = 00001276) and MS-ARMA-APGARCH-RBF(RMSE = 00001631) respectively As discussed models inthe third group are MS-ARMA-FIAPGARCH-NN modelsAccording to the results obtained in Table 5 the lowestRMSE values are obtained for the first three models MS-ARMA-FIAPGARCH-RNN (RMSE= 0001164)MS-ARMA-APGARCH-HYBRID MLP (RMSE = 00001177) and MS-ARMA-FIAPGARCH-MLP (RMSE = 00001178) modelsSimilar to the results in the test sample the models that


Table 5 Markov switching ARMA-GARCH neural network models test sample results


MS-ARMA-GARCH-RNN 0015436917 0124245392 (2nd)MS-ARMA-GARCH-RBF 0023914916 0154644483 (5th)MS-ARMA-GARCH-ELMAN RNN 0023667136 0153841268 (4th)MS-ARMA-GARCH-HYBRID MLP 0015461623 0124344774 (3rd)MS-ARMA-GARCH-MLP 0015333378 0123828017 (1st)


Model Group 3 MS-ARMA-FIAPGARCH-neural network modelsMS-ARMA-FIAPGARCH-RNN 000000001354729 000011639284333 (1st)MS-ARMA-FIAPGARCH-RBF 000000002576468 000016051379977 (5th)MS-ARMA-FIAPGARCH-ELMAN RNN 000000001521219 000012333770222 (4th)MS-ARMA-FIAPGARCH-HYBRID MLP 000000001385396 000011770286713 (2nd)MS-ARMA-FIAPGARCH-MLP 000000001389543 000011787886146 (3rd)

Table 6 Markov switching ARMA-GARCH neural network models out of sample results


MS-ARMA-GARCH 0210599000 0458911000 (6th)MS-ARMA-GARCH-RNN 0015436917 0124245392 (2nd)MS-ARMA-GARCH-RBF 0023914916 0154644483 (5th)MS-ARMA-GARCH-ELMAN RNN 0023667136 0153841268 (4th)MS-ARMA-GARCH-HYBRID MLP 0015461623 0124344774 (3rd)MS-ARMA-GARCH-MLP 0015333378 0123828017 (1st)

Model Group 2 MS-ARMA-APGARCH-neural network modelsMS-ARMA-APGARCH 017740000000000 042111000000000 (6th)MS-ARMA-APGARCH-RNN 000000001628449 000012761067957 (4th)MS-ARMA-APGARCH-RBF 000000002662825 000016318165387 (5th)MS-ARMA-APGARCH-ELMAN RNN 000000001389858 000011789225071 (3rd)MS-ARMA-APGARCH-HYBRID MLP 000000001331575 000011539391162 (1st)MS-ARMA-APGARCH-MLP 000000001333791 000011548986313 (2nd)

Model Group 3 MS-ARMA-FIAPGARCH-neural network modelsMS-ARMA-FIAPGARCH 017814000000000 042220660000000 (6th)MS-ARMA-FIAPGARCH-RNN 000000001354729 000011639284333 (1st)MS-ARMA-FIAPGARCH-RBF 000000002576468 000016051379977 (5th)MS-ARMA-FIAPGARCH-ELMAN RNN 000000001521219 000012333770222 (4th)MS-ARMA-FIAPGARCH-HYBRID MLP 000000001385396 000011770286713 (2nd)MS-ARMA-FIAPGARCH-MLP 000000001389543 000011787886146 (3rd)

deserved the first three places have very close RMSE valuesSimilarly MS-ARMA-FIAPGARCH-ELMAN RNN (RMSE= 0000123) and MS-ARMA-FIAPGARCH-RBF (RMSE =000016) models took the 4th and 5th places

In Table 6 the models are compared for their respectiveout-of-sample forecast performances At the first part ofTable 6 MS-ARMA-GARCH-MLP has the lowest RMSE(=01238) in the MS-GARCH-NN group followed by

MS-ARMA-GARCH-RNN (RMSE = 01242) and MS-ARMA-GARCH-HYBRID MLP (RMSE = 01243) modelsSimilar to the results obtained for the test sample theMS-ARMA-APGARCH-NN and MS-ARMA-FIAPGARCH-NN specifications show significant improvement in lightof MSE and RMSE criteria Among the MS-ARMA-APGARCH-NN models the lowest RMSE is achievedby MS-ARMA-APGARCH-HYBRID MLP (RMSE =


000011539) followed by the MS-ARMA-APGARCH-MLP(RMSE = 000011548) model MS-ARMA-APGARCH-ELMAN RNN and MS-ARMA-APGARCH-RNN modelstook the 4th and 5th places in out-of-sample forecastinghowever though significant improvement is reportedMS-ARMA-APGARCH-RBF model took the 5th placeamong other MS-ARMA-APGARCH-NN models inthe group If MS-ARMA-FIAPGARCH-NN modelsare evaluated the MS-ARMA-FIAPGARCH-RNN andMS-ARMA-FIAPGARCH-HYBRID MLP models areobserved to take the 1st and 2nd places with respectiveRMSE values of 00001163 and 00001177 Further theMS-ARMA-FIAPGARCH-MLP model is the 3rd withRMSE = 00001178 and the 4th and the 5th are MS-ARMA-FIAPGARCH-ELMAN RNN (RMSE = 00001233) andMS-FIAPGARCH-RBF (RMSE = 00001605) models It isnoteworthy that though all RMSE values are significantlylow RBF based model has become the last model amongthe MS-ARMA-FIAPGARCH-NN group members FurtherDiebold-Mariano equal forecast accuracy tests will beapplied to evaluate the models

To evaluate the forecast accuracy of the estimatedmodelsDiebold-Mariano test results are reported in Table 7 Theforecasting sample corresponds to the last 587 observationsof ISE100 daily return series For interpretation purposes [r]and [c] given in brackets denote that the selectedmodel is therow or the columnmodel with respect to the 119875 value given inparentheses for the calculated Diebold-Mariano test statistic

Firstly among the models in Group 1 the MS-ARMA-GARCH-RNN model is compared with the MS-ARMA-GARCH-RBF model in the first column The calculated DMtest statistic is minus4020 and significant at the 1 significancelevel thus the null hypothesis of equal forecast accuracy isrejected in favor of MS-ARMA-GARCH-RNN model TheMS-ARMA-GARCH-RBF model is compared with the MS-ARMA-GARCH-RNNmodel The DM statistic is minus3176 andshows that at 5 significance level MS-ARMA-GARCH-RNN model is selected over MS-ARMA-GARCH-ELMANRNNmodel IfMS-ARMA-GARCH-HYBRIDMLP andMS-ARMA-GARCH-RNNmodels are compared we fail to rejectthe null hypothesis of equal forecast accuracy If MS-ARMA-GARCH-MLP and MS-ARMA-GARCH-RNN models arecompared DM test statistic is calculated as 0574 with 119875

value = 056 therefore the null hypothesis of equal forecastaccuracy cannot be rejected The results show that MS-ARMA-GARCH-RNN model is preferred over MS-ARMA-GARCH-RBF andMS-ARMA-GARCH-ELMANRNNmod-els If the MS-ARMA-GARCH-RBF model at the sec-ond row is analyzed the null of equal forecast accuracycannot be rejected between MS-ARMA-GARCH-RBF andMS-ARMA-GARCH-ELMAN RNN models whereas MS-ARMA-GARCH-HYBRID MLP (DM = 4011) and MS-ARMA-GARCH-MLP (DM = 4142) models are selectedover MS-ARMA-GARCH-RBF model The equal forecastaccuracy of MS-ARMA-GARCH-ELMAN RNN and MS-ARMA-GARCH-HYBRID MLP models is rejected and thealternative hypothesis that MS-ARMA-GARCH-HYBRIDMLP model has better forecast accuracy is accepted at 1significance level (DM = 3309) The DM statistic calculated

for MS-ARMA-GARCH-ELMAN RNN and MS-ARMA-GARCH-MLP models is 3396 suggesting that MS-GARCH-MLP model is selected

The asymmetric power GARCH architecture based mod-els in Group 2 are evaluated at the second part of Table 7MS-ARMA-APGARCH-RNN is compared with the othermodels at the first rowDM test statistic is calculated asminus2932suggesting that the null of equal forecast accuracy betweenMS-ARMA-APGARCH-RNN and MS-ARMA-APGARCH-RBF is rejected in favor of MS-ARMA-APGARCH-RNNmodel Further the null hypotheses for the test of equalforecast accuracy betweenMS-ARMA-APGARCH-RNNandthe other models are rejected and better forecast accu-racy of MS-ARMA-APGARCH-ELMAN RNN MS-ARMA-APGARCH-HYBRID MLP and MS-ARMA-APGARCH-MLP are accepted at 1 significance level Similar to the anal-ysis conducted before RBF based MS-ARMA-APGARCH-RBFmodel fail to have better forecast accuracy than the othermodels at the second row MS-ARMA-APGARCH-HYBRIDMLP model proved to have better forecast accuracy than allother models whereas MS-ARMA-APGARCH-MLP modelshow to provide better forecast accuracy than all othermodelsexcept for the MS-ARMA-APGARCH-HYBRIDMLPmodelat 1 significance level

MS-ARMA-FIAPGARCH-NN results are given in thethird section of Table 7TheMS-ARMA-FIAPGARCH-RNNmodel is compared with the MS-ARMA-FIAPGARCH-RBFmodel in the first column The calculated DM test statistic isminus2519 and suggests that the null hypothesis of equal forecastaccuracy is rejected in favor of MS-ARMA-FIAPGARCH-RNN model The DM statistic is minus1717 and shows thatMS-ARMA-FIAPGARCH-RNN model is selected over MS-ARMA-FIAPGARCH-ELMAN RNN model only at 10 sig-nificance level If we compare the MS-ARMA-FIAPGARCH-HYBRID MLP and MS-ARMA-GARCH-RNN models wefail to reject the null hypothesis On the other hand MS-FIAPGARCH-HYBRID MLP is preferred over MS-ARMA-FIAPGARCH-ELMAN RNN at 5 significance level (DM= 2154) and over MS-ARMA-FIAPGARCH-RBF model at1 significance level (DM= 2608) Accordingly MS-ARMA-FIAPGARCH-RBF model fail to beat other models thoughRMSE values show significant improvement theMS-ARMA-FIAPGARCH-RNNmodel is selected over the other modelswhereas theMS-ARMA-FIAPGARCH-HYBRIDMLPmodelis the second best model for the analyzed time series andanalyzed sample

5 Conclusions

In the study a family of regime switching neural networkaugmented volatility models is analyzed to achieve anevaluation of forecast accuracy and an application to dailyreturns in an emerging market stock index presentedIn this respect GARCH neural network models whichallow the generalization of MS type RS-GARCH modelsto MS-ARMA-GARCH-NN models are incorporated withneural networks models based on Multilayer Perceptron(MLP) Radial Basis Function (RBF) Elman Recurrent NN


Table 7 Diebold Mariano equal forecast accuracy test results out of sample

Model Group 1 MS-ARMA-GARCH-neural network modelsMS-ARMA-GARCH-RNN

MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP

MS-ARMA-GARCH-RNN mdash minus4020lowastlowastlowast

(0000) [r]minus3176lowastlowast(0002) [r]

minus0645(0518) [r]

0574(0566) [c]

MS-ARMA-GARCH-RBF mdash 0585

(0558) [c]4011lowastlowastlowast(0000) [c]

4142lowastlowastlowast(0000) [c]

MS-ARMA-GARCH-ELMAN RNN mdash 3309lowastlowastlowast(0001) [c]


MS-ARMA-GARCH-HYBRIDMLP mdash 3355lowastlowastlowast

(001) [c]MS-ARMA-GARCH-MLP mdash

Model Group 2 MS-ARMA-APGARCH-neural network modelsMS-ARMA-APGARCH-

RNN

MS-ARMA-APGARCH-

RBF

MS-ARMA-APGARCH-ELMAN RNN

MS-ARMA-APGARCH-HYBRID MLP

MS-ARMA-APGARCH-

MLP

MS-ARMA-APGARCH-RNN mdash minus2932lowastlowastlowast(0003) [r]




MS-ARMA-APGARCH-RBF mdash 3188lowastlowastlowast



MS-ARMA-APGARCH-ELMANRNN mdash 2797lowastlowastlowast


MS-ARMA-APGARCH-HYBRIDMLP mdash minus1835lowastlowast

(0066) [r]MS-ARMA-APGARCH-MLP mdash

Model Group 3 MS-ARMA-FIAPGARCH-neural network modelsMS-ARMA-

FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF

MS-ARMA-FIAPGARCH-ELMAN RNN

MS-ARMA-FIAPGARCH-HYBRID MLP

MS-ARMA-FIAPGARCH-

MLP

MS-ARMA-FIAPGARCH-RNN mdash minus2519lowastlowast(0011) [r]

minus1717lowast(0086) [r]

minus0588(0556) [r]

minus0836(0403) [r]

MS-ARMA-FIAPGARCH-RBF mdash 2214lowastlowast(0027) [c]


2526lowastlowast(0012) [c]

MS-ARMA-FIAPGARCH-ELMANRNN mdash 2154lowastlowast

(0031) [c]minus2214lowastlowast(0026) [r]

MS-ARMA-FIAPGARCH-HYBRIDMLP

mdash minus0716(0473) [r]

MS-ARMA-FIAPGARCH-MLP mdashNotes Statistical significances of the relevant tests are denoted with lowastlowastlowastshow statistical significance at 1 significance level while lowastlowast119886119899119889lowastshow significance at5 and 10 respectively D-M test allows for reporting the selected model Accordingly [r] shows that the model reported in the ldquorowrdquo is selected over themodel in the column Similarly [c] stands for the column model being accepted over the row model

(Elman RNN) Time Lag (delay) Recurrent NN (RNN)and Hybrid MLP models Gray [92] RS-GARCH modelis taken as the baseline modeling structure in the studyGray [92] utilizes the Hamilton [67] Markov-switchingapproach however the regime-switching is allowed in theconditional volatility processes of a time series On theother hand the study aimed at Hamilton [67] type regime-switching between GARCH-Neural Network approaches

developed by Donaldson and Kamstra [118] and furtherextanded to a class of GARCH-NN models by Bildiriciand Ersin [61] The model within the family analyzed areMS-ARMA-GARCH-MLP MS-ARMA-GARCH-RBF MS-ARMA-GARCH-ElmanRNN MS-ARMA-GARCH-RNNand MS-ARMA-GARCH-Hybrid MLP which are furtherextended to account for asymmetric power terms based onthe APGARCH architecture MS-ARMA-APGARCH-MLP


MS-ARMA-APGARCH-RBF MS-ARMA-APGARCH-ElmanRNNMS-ARMA-APGARCH-RNN andMS-ARMA-APGARCH-Hybrid MLP

Further by accounting for fractional integration (FI)in GARCH specification the models are generalized asMS-ARMA-FIAPGARCH-MLP MS-ARMA-FIAPGARCH-RBF MS-ARMA-FIAPGARCH-ElmanRNN MS-ARMA-FIAPGARCH-RNN and MS-ARMA-FIAPGARCH-HybridMLP models that account for fractionally integrated modelswith asymmetric power transformations and generalizedto neural network models with possibly improved forecastcapabilities

Models are evaluated with MSE and RMSE error crite-ria To evaluate equal forecast accuracy modified Diebold-Mariano tests are applied It is observed that holding thetraining sample results on one side the real improve-ment occurs for the test sample and most importantly forthe out-of-sample By evaluating MS-GARCH-NN modelsthough in-sample performance is noticeablemoving towardsMS-ARMA-APGARCH-NN models and fractionally inte-grated models of MS-ARMA-FIAPGARCH-NN significantimprovement is noticed in light of theMSE andRMSE criteriaand in terms of Diebold-Mariano equal forecast accuracytests

Among the models analyzed asymmetric power mod-eling with fractional integration generalized to Time LagRecurrent Neural Network architecture and Hybrid Multi-layer Perceptron are shown to provide significant forecastand modeling capabilities Thus the models with regimeswitching further augmented with neural network modelingtechniques promise significant achievements in return andvolatility modeling and forecasting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[3] C Hiemstra and J D Jones ldquoTesting for linear and nonlinearGranger causality in the stock price-volume relationrdquo TheJournal of Finance vol 49 pp 1639ndash1664 1994

[4] X Wang P H Phua and W Lin ldquoStock market predictionusing neural networks does trading volume help in short-term predictionrdquo in Proceedings of IEEE International JointConference on Neural Networks vol 4 pp 2438ndash2442 2003

[5] R F Engle ldquoAutoregressive conditional heteroskedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 pp 987ndash1007 1982

[6] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

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[10] L R Glosten R Jagannathan and D Runkle ldquoOn the relationbetween the expected value and the volatility of the nominalexcess return on stocksrdquo Journal of Finance vol 48 pp 1779ndash1801 1993

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[12] J M Zakoian ldquoThreshold heteroscedastic modelsrdquo DiscussionPaper INSE 1991

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[97] M Haas ldquoSkew-normal mixture and Markov-switchingGARCH processesrdquo Studies in Nonlinear Dynamics andEconometrics vol 14 no 4 pp 1ndash56 2010

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[102] C-J Kim ldquoDynamic linear models with Markov-switchingrdquoJournal of Econometrics vol 60 no 1-2 pp 1ndash22 1994


[104] L Bauwens and J V K Rombouts ldquoBayesian inference forthe mixed conditional heteroskedasticity modelrdquo EconometricsJournal vol 10 no 2 pp 408ndash425 2007



[106] M Frommel ldquoVolatility regimes in central and Eastern Euro-pean countriesrsquo exchange ratesrdquo Faculteit Economie en Bedrijf-skunde Working Paper 2007487

[107] S I Blazsek and A Downarowicz ldquoRegime switching modelsof hedge fund returnsrdquo Working Paper 2008

[108] J S Henneke S T Rachev F J Fabozzi and M NikoloveldquoMCMC-based estimation of Markov Switching ARMA-GARCH modelsrdquo Applied Economics vol 43 no 3 pp259ndash271 2011

[109] M Haas ldquoThe autocorrelation structure of the Markov-switching asymmetric power GARCH processrdquo Statistics andProbability Letters vol 78 no 12 pp 1480ndash1489 2008

[110] M Haas S Mittnik andM S Paolella ldquoAsymmetric multivari-ate normal mixture GARCH Forthcomingrdquo in ComputationalStatistics and Data Analysis 2008

[111] T G Andersen and T Bollerslev ldquoHeterogeneous informationarrivals and return volatility dynamics uncovering the long-runin high frequency returnsrdquo Journal of Finance vol 52 no 3 pp975ndash1005 1997


[113] L Spezia and R Paroli ldquoBayesian inference and forecasting indynamic neural networks with fully Markov switching ARCHnoisesrdquo Communications in Statistics vol 37 no 13 pp 2079ndash2094 2008

[114] S Dutta and S Shektar ldquoBond ratings a non-conservativeapplication of neural networkrdquo in Proceedings of the IEEEInternational Conference on Neural Networks vol 2 pp 43ndash4501998

[115] K Y Tom and M Y Kiang ldquoManagerial applications of neuralnetworks the case of bank failure predictionsrdquo ManagementScience vol 38 no 7 pp 926ndash947 1992

[116] A Do and G Grudinsky ldquoA neural network approach toresidential property appraisalrdquo in The Real Estate Appraiser(December 1992) pp 38ndash45 1992

[117] P A Shively ldquoThe nonlinear dynamics of stock pricesrdquo Quar-terly Review of Economics and Finance vol 43 no 3 pp 505ndash517 2003

[118] R G Donaldson and M Kamstra ldquoForecast combining withneural networksrdquo Journal of Forecasting vol 15 no 1 pp 49ndash611996

[119] MOlteanu J Rynkiewicz andBMaillet ldquoNonlinear analysis ofshockswhenfinancialmarkets are subject to changes in regimerdquoin ESANN pp 28ndash30 April 2004

[120] D Liu and L Zhang ldquoChina stock market regimes predictionwith artificial neural network andmarkov regime switchingrdquo inWorld Congress on Engineering (WCE rsquo10) vol 1 pp 378ndash383London UK July 2010

[121] A A P Santos L D S Coelho and C E Klein ldquoForecastingelectricity prices using a RBF neural network with GARCHerrorsrdquo in Proceedings of the International Joint Conference onNeural Networks (IJCNN rsquo10) pp 1ndash8 July 2010

[122] M Jordan ldquoAttractor dynamics and parallelism in a connec-tionist sequential machinerdquo in Proceedings of the 8th AnnualConference of the Cognitive Science Society pp 531ndash545 1986

[123] J L Elman ldquoFinding structure in timerdquo Cognitive Science vol14 no 2 pp 179ndash211 1990

[124] G Zhang B E Patuwo and M Y Hu ldquoForecasting withartificial neural networks the state of the artrdquo InternationalJournal of Forecasting vol 14 no 1 pp 35ndash62 1998

[125] J M Binner C T Elger B Nilsson and J A Tepper ldquoPre-dictable non-linearities inUS Inflationrdquo Economics Letters vol93 no 3 pp 323ndash328 2006


[127] M McKenzie and H Mitchell ldquoGeneralized asymmetric powerARCH modelling of exchange rate volatilityrdquo Applied FinancialEconomics vol 12 no 8 pp 555ndash564 2002

[128] R B Davies ldquoHypothesis testing when a nuisance parameter ispresent only under the alternativerdquo Biometrika vol 74 no 1 pp33ndash43 1987

[129] M Ural ldquoGeneralized Asymmetric Power ARCH Modellingof National Stock Market Returns SU IIBF Sosyal veEkonomik Arastırmalar Dergisirdquo pp 575ndash590 2009httpwwwiibfselcukedutriibf dergidosyalar851348078522pdf

[130] T Ane and L Ureche-Rangau ldquoStock market dynamics in aregime-switching asymmetric power GARCHmodelrdquo Interna-tional Review of Financial Analysis vol 15 no 2 pp 109ndash1292006

[131] E Teletar and S Binay ldquoIMKB Endeksinin Uslu OtoregresifKosullu Degisken Varyans (PARCH) ile Modellenmesirdquo 2001httpidaricuedutrsempozyumbil6htm

[132] D E Goldberg ldquoGenetic algorithms and Walsh functions partII deception and its analysisrdquo Complex System vol 3 pp 153ndash171 1989

[133] D E Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[134] C D E Goldberg ldquoSizing populations for serial and parallelgenetic algorithmsrdquo in Proceedings of the 3rd InternationalConference on Genetic Algorithms pp 70ndash79 1989

[135] D Patterson Artifical Neural Networks Prentice Hall Singop-ure 1996

[136] S Haykin Neural Networks A Comprehensive FoundationMacmillan New York NY USA 1994

[137] L Fausett Fundamentals of Neural Networks Prentice HallEnglewood Cliffs NJ USA 1994

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Among many Engle and Bollerslev [7] developed theIntegrated-GARCH (I-GARCH) process to incorporate inte-gration properties AGARCH model introduced by Engle[8] allows modeling asymmetric effects of negative andpositive innovations In terms of modeling asymmetriesGARCH models have been further developed by includingasymmetric impacts of the positive and negative shocksto capture the asymmetric effects of shocks on volatilityand return series which depend on the type of shocks ieeither negative or positive Following the generalization ofEGARCH model of Nelson [9] that allows modelling theasymmetries in the relationship between return and volatilitythe Glosten et al [10] noted the importance of asymmetrycaused by good and bad news in volatile series and proposed amodel that incorporates the past negative and positive inno-vations with an identity function that leads the conditionalvariance to follow different processes due to asymmetry Thefinding is a result of the empirical analyses which pointedat the fact that the negative shocks had a larger impact onvolatility Consequently the bad news have a larger impactcompared to the conditional volatility dynamics followedafter the good news Due to this effect asymmetric GARCHmodels have rapidly expanded The GJR-GARCHmodel wasdeveloped independently by Zakoian [11 12] and Glosten etal [10] It should be noted that in terms of asymmetry theThreshold GARCH (T-GARCH) of Zakoian [12] VGARCHand nonlinear asymmetric GARCH models (NAGARCH)of Engle and Ng [13] are closely related versions to modelasymmetry in financial asset returns The SQR-GARCHmodel of Heston andNandi [14] and the Aug-GARCHmodeldeveloped by Duan [15] nest several versions of the modelstaking asymmetry discussed above Further models suchas the Generalized Quadratic GARCH (GQARCH) modelof Sentana [16] utilize multiplicative error terms to capturevolatility more effectively The FIGARCH model of Baillie etal [17] benefits from an ARFIMA type fractional integrationrepresentation to better capture the long-run dynamics in theconditional variance (See for detailed information Bollerslev[18]) The APARCHAPGARCH model of Ding et al [19] isan asymmetric model that incorporates asymmetric powerterms which are allowed to be estimated directly from thedata The APGARCH model also nests several models suchas the TGARCH TSGARCH GJR and logGARCH TheFIAPGARCHmodel of Tse [20] combines the FIGARCHandthe APGARCH Hyperbolic GARCH (HYGARCH) modelof Davidson [21] nests the ARCH GARCH IGARCH andFIGARCHmodels (for an extended review GARCHmodelssee Bollerslev [18])

Even though the ARCHGARCH models can be appliedquickly for many time series the shortcomings in thesemodels were discussed by certain studies Perez-Quirosand Timmermann [22] focused on the conditional distri-butions of financial returns and showed that recessionaryand expansionary periods possess different characteristicswhile the parameters of a GARCH model are assumed tobe stable for the whole period Certain studies discussed thehigh volatility persistence inherited in the baseline GARCHand proposed early signs of regime switches Diebold [23]and Lamoureux and Lastrapes [24] are two of the highly

cited studies discussing high persistence in volatility due tostructural changes Lamoureux and Lastrapes [24] showedthat the encountered high persistence in volatility processesresulted from the volume effects that had not been taken intoaccount Qiao and Wong [25] followed a bivariate approachand confirmed that the Lamoureux and Lastrapes [24] effectexists due to the volume and turnover effects on conditionalvolatility and after the introduction of volumeturnover asexogenous variables it is possible to obtain a significantdecline in the the persistence Mikosch and Starica [26]showed that structural changes had an important impact thatleads to accepting an integrated GARCH process Bauwenset al [27 28] discussed that the persistence in the estimatedsingle regime GARCH processes could be considered asresulting from the misspecification which could be con-trolled by introducing an MS-GARCH specification wherethe regime switches are governed by a hidden Markov chain

Kramer [29] evaluated the autocorrelation in the squarederror terms and provided an important contribution Accord-ingly the observed empirical autocorrelations of the 120576

2

119905are

much larger than the theoretical autocorrelations implied bythe estimated parameters through evaluating anMS-GARCHmodel where the autocorrelation problem could be shown toaccelerate as the transition probabilities approached 1 (Fora proof see eg Francq and Zakoıan [30] Kramer [29])(In particular the empirical autocorrelations of the 120576

2

119905often

seem to indicate long memory which is not possible in theGARCH-model in fact in all standard GARCH-modelstheoretical autocorrelations must eventually decrease expo-nentially so longmemory is ruled out) Alexander and Lazaar[31] showed that leverage effects are due to asymmetry in thevolatility responses to the price shocks and the leverage effectaccelerates once the markets are in the more volatile regimeKramer and Tameze [32] showed that a single state GARCHmodel had only onemean reversion while by allowing regimeswitching in the GARCH processes mean reverting effectdiminishes In a perspective of volatility if these shifts arepersistent then there are two sources of volatility persistencedue to shocks and due to regime-switching in the parametersof the variance process By utilizing aMarkov transformationmodel it could be shown that the relationships among theregimes between the periods of 119905minus1 and 119905 could be explainedand the most important advantage of the MSGARCH modelexposes itself as there is no need for the researchers to observethe regime changes The model allows different regimes toreveal by itself [33]

The regime switching in light of the Markov switchingmodel has interesting properties to be examined such asthe stationarity by allowing the switching course of volatilityinherent in the asset prices The hidden Markov model(HMM) developed by Taylor [34] is a switching model thatbenefits from including an unobserved variable to capturevolatility to be modeled with transitions between the hiddenstates that possess different probability distributions attachedto each state Hidden Markov model has been appliedsuccessfully by Alexander and Dimitriu [35] Cheung andErlandsson [36] Francis and Owyang [37] and by Claridaet al [38] to capture the switching type of predictions in


















1205902

119905(119904119905)= 119908(119904119905)+

119902

sum

119894=1

120572119894(119904119905)1205762

119905minus119894+

119901

sum

119895=1

120573119895(119904119905)119864(

1205762

119905minus119895

119868119905minus119895minus1

)

= 119908(119904119905)+

119902

sum

119894=1

120572119894(119904119905)1205762

119905minus119894

+

119901

sum

119895=1

120573119895(119904119905)

119898

sum

119904119905minus119895=1

119875(119904119905minus119895

=

119904119905minus119895

119868119905minus119895minus1

)1205902

119905minus119895 119904119905minus119895

(1)

where 119908119904119905

gt 0 120572119894119904119905

ge 0 120573119895119904119905

ge 0 and 119894 = 1 119902119895 = 1 119901 119904





119905= 1 | 119904

119905minus1= 1 = 120587

1 Pr119904119905= 2 |

119904119905minus1

= 1 = 1 minus 1205871 Pr119904

119905= 2 | 119904

119905minus1= 2 = 120587

2 and

Pr119904119905= 1 | 119904

119905minus1= 2 = 1 minus 120587

2)





2

119905= Var(120576

119905119878119905) To consider





119905(119904119905)=

119908(119904119905)+sum119902

119894=1120572119894(119904119905)1205762

119905minus119894+sum119901

119895=1120573119895(119904119905)sum119898

119904=1119875(119878119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905=

119904119905)1205902

119905minus119895 119904119905minus119895


119905minus119895= 119904119905minus119895

| 119868119905minus119895minus1

) by 119901(119904119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905= 119904119905)


119905 119904119905


119904 120572119894119904) and




119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119898

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119894=1

120572119894(119904119905)1205762

119905minus119894(119904119905)+

119902

sum

119895=1

120573(119904119905)120590119905minus119895(119904

119905)

(2)

where

120576119905minus119894minus1(119904

119905minus119894)= 119864 [120576


119884119905minus119894minus1

]

120590119905minus119894minus1(119904

119905minus119894)= 119864 [120576


119884119905minus119894minus1

]

(3)


119905

119871 =

119879

prod

119905=1

119891 (119910119905| 119904119905= 119894 119884119905minus1

)Pr [119904119905= 119894 | 119884

119905minus1] (4)



iteration

120587119895119905

= Pr [119904119905= 119895 | 119884

119905minus1]

=

1

sum

119894=0

Pr [119904119905= 119895 | 119904

119905minus1= 119894]Pr [119904

119905= 119895 | 119884

119905minus1]

1

sum

119894=0

120578119895119894120587lowast

119894119905minus1

(5)


119905minus1and 120590






120576119905= 120578119905120590Δ119905119905

119905 isin Z (6)


119905 119905 isin Z is a


119894119895= 119901(Δ

119905= 119895 | Δ

119905minus1= 119894) so that

119875 = [119901119894119895] = [119901 (Δ

119905= 119895 | Δ

119905minus1= 119894)] 119894 119895 = 1 119896 (7)


= (1205871infin

120587119896infin

)1015840


2



120590120575

119895119905= 119908119895+ 1205721119895

1003816100381610038161003816120576+

119905minus1

1003816100381610038161003816

120575

+ 1205722119895

1003816100381610038161003816120576minus

119905minus1

1003816100381610038161003816

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (8)


where 119908119895gt 0 120572

1119895 1205722119895 120573119895ge 0 119895 = 1 119896 For the power

term 120575 = 2 and for 1205721119895



1119895= 1205722119895[109]





120590120575

119895119905= 119908119895+ 120572119895(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (9)


ge 0 120574119895


1119895= 120572119895(1 minus 120574

119895)120575 and 120573

119895=

120572119904

119895(1 + 120574

119895)120575


120590

120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119903

sum

119897=1

120572119897(119904119905)(1003816100381610038161003816120576119905minus119897

1003816100381610038161003816minus 120574119897(119904119905)120576119905minus119897

)

120575(119904119905)

+

119902

sum

119898=1

120573119898(119904119905)120590

120575(119904119905)

119905minus119898(119904119905) 120575(119904119905)gt 0

(10)



gt 0 120572119897(119904119905) 120573119898(119904119905)

ge

0 with 120574119897(119904119905)


(119904119905)= 120575 the





(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)


(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(119904119905)120576119905minus1

)

120575(119904119905)

(11)


(119904119905) and 120572


120575(1)


119889(119904119905)






(119904119905)


(119904119905)


=






(119904119905)= 2 120574

(119904119905)= 0 the model











119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119899

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905) (12)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(13)


119898

sum

119894=1

1205902

119905(119894)119875 (119878119905= 119894 | 119911

119905minus1) 119894 = 1 119898 (14)


120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

= [1+exp(minus120591ℎ(119904119905)(

119897

sum

119897=1

[

ℎ

sum

ℎ=1

120582ℎ119897(119904119905)119911ℎ

119905minus119897(119904119905)+ 120579ℎ(119904119905)]))]

minus1

(15)

(

1

2

) 120582ℎ119889


and119875(119878119905= 119894 | 119911


representation

(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (17)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(18)


119904= 119894 | 119911

119904minus1) where 120595(119911





If 119899119895119894


= 119894 | 119911119905minus1


119891 (119910119905| 119909119905 119911119905= 119894)

=

1

radic2120587ℎ119905(119894)

exp

minus(119910119905minus 1199091015840

119905120593 minus sum

119867

119895=1120573119895119901 (1199091015840

119905120574119895))

2

2ℎ119905(119895)

(19)


119904= 119894 | 119911

119904minus1)


120590120575(119904119905)

119905(119904119905)

= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119901

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(20)


119901(119904119905)


119901(119904119905)


119901(119904119905)






(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(21)


ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905)) defined as


(119904119905)

= 2 and 120574(119904119905)


(119904119905)


(119904119905)


(119904119905)


(119904119905)

= 1



(119904119905)


(119904119905)



(1 minus 120573(1)

119871) 120590

120575(1)

119905(1)

= 119908(1)

+ ((1 minus 120573(1)

119871) minus (1 minus 120601(1)

119871) (1 minus 119871)119889(1))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(1)

120576119905minus1

)120575(1)

+

ℎ

sum

ℎ=1

120585ℎ(1)

120595 (120591ℎ(1)

119885119905(1)

120582ℎ(1)

120579ℎ(1)

)

(1 minus 120573(2)

119871) 120590

120575(2)

119905(2)

= 119908(2)

+ ((1 minus 120573(2)

119871) minus (1 minus 120601(2)

119871) (1 minus 119871)119889(2))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(2)

120576119905minus1

)120575(2)

+

ℎ

sum

ℎ=1

120585ℎ(2)

120595 (120591ℎ(2)

119885119905(2)

120582ℎ(2)

120579ℎ(2)

)

(22)


(1)and 120575






(1 minus 120573119871) 120590120575

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(23)


(1 minus 120573119871) 1205902

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

2

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(24)





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)


119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)


120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)



119875(119878119905

= 119894 | 119911119905minus1


(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)



120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)



for 120574119901(119904119905)


119901(119904119905)


1 and 0 le 120574119901(119904119905)



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)


= 2 and 120574(119904119905)



if 120575(119904119905)= 2 and 120574









(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




















1205902

119905(119904119905)= 119908(119904119905)+

119902

sum

119894=1

120572119894(119904119905)1205762

119905minus119894+

119901

sum

119895=1

120573119895(119904119905)119864(

1205762

119905minus119895

119868119905minus119895minus1

)

= 119908(119904119905)+

119902

sum

119894=1

120572119894(119904119905)1205762

119905minus119894

+

119901

sum

119895=1

120573119895(119904119905)

119898

sum

119904119905minus119895=1

119875(119904119905minus119895

=

119904119905minus119895

119868119905minus119895minus1

)1205902

119905minus119895 119904119905minus119895

(1)

where 119908119904119905

gt 0 120572119894119904119905

ge 0 120573119895119904119905

ge 0 and 119894 = 1 119902119895 = 1 119901 119904





119905= 1 | 119904

119905minus1= 1 = 120587

1 Pr119904119905= 2 |

119904119905minus1

= 1 = 1 minus 1205871 Pr119904

119905= 2 | 119904

119905minus1= 2 = 120587

2 and

Pr119904119905= 1 | 119904

119905minus1= 2 = 1 minus 120587

2)





2

119905= Var(120576

119905119878119905) To consider





119905(119904119905)=

119908(119904119905)+sum119902

119894=1120572119894(119904119905)1205762

119905minus119894+sum119901

119895=1120573119895(119904119905)sum119898

119904=1119875(119878119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905=

119904119905)1205902

119905minus119895 119904119905minus119895


119905minus119895= 119904119905minus119895

| 119868119905minus119895minus1

) by 119901(119904119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905= 119904119905)


119905 119904119905


119904 120572119894119904) and




119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119898

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119894=1

120572119894(119904119905)1205762

119905minus119894(119904119905)+

119902

sum

119895=1

120573(119904119905)120590119905minus119895(119904

119905)

(2)

where

120576119905minus119894minus1(119904

119905minus119894)= 119864 [120576


119884119905minus119894minus1

]

120590119905minus119894minus1(119904

119905minus119894)= 119864 [120576


119884119905minus119894minus1

]

(3)


119905

119871 =

119879

prod

119905=1

119891 (119910119905| 119904119905= 119894 119884119905minus1

)Pr [119904119905= 119894 | 119884

119905minus1] (4)



iteration

120587119895119905

= Pr [119904119905= 119895 | 119884

119905minus1]

=

1

sum

119894=0

Pr [119904119905= 119895 | 119904

119905minus1= 119894]Pr [119904

119905= 119895 | 119884

119905minus1]

1

sum

119894=0

120578119895119894120587lowast

119894119905minus1

(5)


119905minus1and 120590






120576119905= 120578119905120590Δ119905119905

119905 isin Z (6)


119905 119905 isin Z is a


119894119895= 119901(Δ

119905= 119895 | Δ

119905minus1= 119894) so that

119875 = [119901119894119895] = [119901 (Δ

119905= 119895 | Δ

119905minus1= 119894)] 119894 119895 = 1 119896 (7)


= (1205871infin

120587119896infin

)1015840


2



120590120575

119895119905= 119908119895+ 1205721119895

1003816100381610038161003816120576+

119905minus1

1003816100381610038161003816

120575

+ 1205722119895

1003816100381610038161003816120576minus

119905minus1

1003816100381610038161003816

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (8)


where 119908119895gt 0 120572

1119895 1205722119895 120573119895ge 0 119895 = 1 119896 For the power

term 120575 = 2 and for 1205721119895



1119895= 1205722119895[109]





120590120575

119895119905= 119908119895+ 120572119895(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (9)


ge 0 120574119895


1119895= 120572119895(1 minus 120574

119895)120575 and 120573

119895=

120572119904

119895(1 + 120574

119895)120575


120590

120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119903

sum

119897=1

120572119897(119904119905)(1003816100381610038161003816120576119905minus119897

1003816100381610038161003816minus 120574119897(119904119905)120576119905minus119897

)

120575(119904119905)

+

119902

sum

119898=1

120573119898(119904119905)120590

120575(119904119905)

119905minus119898(119904119905) 120575(119904119905)gt 0

(10)



gt 0 120572119897(119904119905) 120573119898(119904119905)

ge

0 with 120574119897(119904119905)


(119904119905)= 120575 the





(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)


(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(119904119905)120576119905minus1

)

120575(119904119905)

(11)


(119904119905) and 120572


120575(1)


119889(119904119905)






(119904119905)


(119904119905)


=






(119904119905)= 2 120574

(119904119905)= 0 the model











119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119899

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905) (12)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(13)


119898

sum

119894=1

1205902

119905(119894)119875 (119878119905= 119894 | 119911

119905minus1) 119894 = 1 119898 (14)


120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

= [1+exp(minus120591ℎ(119904119905)(

119897

sum

119897=1

[

ℎ

sum

ℎ=1

120582ℎ119897(119904119905)119911ℎ

119905minus119897(119904119905)+ 120579ℎ(119904119905)]))]

minus1

(15)

(

1

2

) 120582ℎ119889


and119875(119878119905= 119894 | 119911


representation

(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (17)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(18)


119904= 119894 | 119911

119904minus1) where 120595(119911





If 119899119895119894


= 119894 | 119911119905minus1


119891 (119910119905| 119909119905 119911119905= 119894)

=

1

radic2120587ℎ119905(119894)

exp

minus(119910119905minus 1199091015840

119905120593 minus sum

119867

119895=1120573119895119901 (1199091015840

119905120574119895))

2

2ℎ119905(119895)

(19)


119904= 119894 | 119911

119904minus1)


120590120575(119904119905)

119905(119904119905)

= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119901

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(20)


119901(119904119905)


119901(119904119905)


119901(119904119905)






(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(21)


ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905)) defined as


(119904119905)

= 2 and 120574(119904119905)


(119904119905)


(119904119905)


(119904119905)


(119904119905)

= 1



(119904119905)


(119904119905)



(1 minus 120573(1)

119871) 120590

120575(1)

119905(1)

= 119908(1)

+ ((1 minus 120573(1)

119871) minus (1 minus 120601(1)

119871) (1 minus 119871)119889(1))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(1)

120576119905minus1

)120575(1)

+

ℎ

sum

ℎ=1

120585ℎ(1)

120595 (120591ℎ(1)

119885119905(1)

120582ℎ(1)

120579ℎ(1)

)

(1 minus 120573(2)

119871) 120590

120575(2)

119905(2)

= 119908(2)

+ ((1 minus 120573(2)

119871) minus (1 minus 120601(2)

119871) (1 minus 119871)119889(2))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(2)

120576119905minus1

)120575(2)

+

ℎ

sum

ℎ=1

120585ℎ(2)

120595 (120591ℎ(2)

119885119905(2)

120582ℎ(2)

120579ℎ(2)

)

(22)


(1)and 120575






(1 minus 120573119871) 120590120575

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(23)


(1 minus 120573119871) 1205902

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

2

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(24)





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)


119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)


120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)



119875(119878119905

= 119894 | 119911119905minus1


(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)



120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)



for 120574119901(119904119905)


119901(119904119905)


1 and 0 le 120574119901(119904119905)



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)


= 2 and 120574(119904119905)



if 120575(119904119905)= 2 and 120574









(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






2

119905= Var(120576

119905119878119905) To consider





119905(119904119905)=

119908(119904119905)+sum119902

119894=1120572119894(119904119905)1205762

119905minus119894+sum119901

119895=1120573119895(119904119905)sum119898

119904=1119875(119878119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905=

119904119905)1205902

119905minus119895 119904119905minus119895


119905minus119895= 119904119905minus119895

| 119868119905minus119895minus1

) by 119901(119904119905minus119895

= 119904119905minus119895

| 119868119905minus1

119878119905= 119904119905)


119905 119904119905


119904 120572119894119904) and




119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119898

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119894=1

120572119894(119904119905)1205762

119905minus119894(119904119905)+

119902

sum

119895=1

120573(119904119905)120590119905minus119895(119904

119905)

(2)

where

120576119905minus119894minus1(119904

119905minus119894)= 119864 [120576


119884119905minus119894minus1

]

120590119905minus119894minus1(119904

119905minus119894)= 119864 [120576


119884119905minus119894minus1

]

(3)


119905

119871 =

119879

prod

119905=1

119891 (119910119905| 119904119905= 119894 119884119905minus1

)Pr [119904119905= 119894 | 119884

119905minus1] (4)



iteration

120587119895119905

= Pr [119904119905= 119895 | 119884

119905minus1]

=

1

sum

119894=0

Pr [119904119905= 119895 | 119904

119905minus1= 119894]Pr [119904

119905= 119895 | 119884

119905minus1]

1

sum

119894=0

120578119895119894120587lowast

119894119905minus1

(5)


119905minus1and 120590






120576119905= 120578119905120590Δ119905119905

119905 isin Z (6)


119905 119905 isin Z is a


119894119895= 119901(Δ

119905= 119895 | Δ

119905minus1= 119894) so that

119875 = [119901119894119895] = [119901 (Δ

119905= 119895 | Δ

119905minus1= 119894)] 119894 119895 = 1 119896 (7)


= (1205871infin

120587119896infin

)1015840


2



120590120575

119895119905= 119908119895+ 1205721119895

1003816100381610038161003816120576+

119905minus1

1003816100381610038161003816

120575

+ 1205722119895

1003816100381610038161003816120576minus

119905minus1

1003816100381610038161003816

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (8)


where 119908119895gt 0 120572

1119895 1205722119895 120573119895ge 0 119895 = 1 119896 For the power

term 120575 = 2 and for 1205721119895



1119895= 1205722119895[109]





120590120575

119895119905= 119908119895+ 120572119895(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (9)


ge 0 120574119895


1119895= 120572119895(1 minus 120574

119895)120575 and 120573

119895=

120572119904

119895(1 + 120574

119895)120575


120590

120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119903

sum

119897=1

120572119897(119904119905)(1003816100381610038161003816120576119905minus119897

1003816100381610038161003816minus 120574119897(119904119905)120576119905minus119897

)

120575(119904119905)

+

119902

sum

119898=1

120573119898(119904119905)120590

120575(119904119905)

119905minus119898(119904119905) 120575(119904119905)gt 0

(10)



gt 0 120572119897(119904119905) 120573119898(119904119905)

ge

0 with 120574119897(119904119905)


(119904119905)= 120575 the





(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)


(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(119904119905)120576119905minus1

)

120575(119904119905)

(11)


(119904119905) and 120572


120575(1)


119889(119904119905)






(119904119905)


(119904119905)


=






(119904119905)= 2 120574

(119904119905)= 0 the model











119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119899

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905) (12)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(13)


119898

sum

119894=1

1205902

119905(119894)119875 (119878119905= 119894 | 119911

119905minus1) 119894 = 1 119898 (14)


120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

= [1+exp(minus120591ℎ(119904119905)(

119897

sum

119897=1

[

ℎ

sum

ℎ=1

120582ℎ119897(119904119905)119911ℎ

119905minus119897(119904119905)+ 120579ℎ(119904119905)]))]

minus1

(15)

(

1

2

) 120582ℎ119889


and119875(119878119905= 119894 | 119911


representation

(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (17)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(18)


119904= 119894 | 119911

119904minus1) where 120595(119911





If 119899119895119894


= 119894 | 119911119905minus1


119891 (119910119905| 119909119905 119911119905= 119894)

=

1

radic2120587ℎ119905(119894)

exp

minus(119910119905minus 1199091015840

119905120593 minus sum

119867

119895=1120573119895119901 (1199091015840

119905120574119895))

2

2ℎ119905(119895)

(19)


119904= 119894 | 119911

119904minus1)


120590120575(119904119905)

119905(119904119905)

= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119901

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(20)


119901(119904119905)


119901(119904119905)


119901(119904119905)






(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(21)


ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905)) defined as


(119904119905)

= 2 and 120574(119904119905)


(119904119905)


(119904119905)


(119904119905)


(119904119905)

= 1



(119904119905)


(119904119905)



(1 minus 120573(1)

119871) 120590

120575(1)

119905(1)

= 119908(1)

+ ((1 minus 120573(1)

119871) minus (1 minus 120601(1)

119871) (1 minus 119871)119889(1))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(1)

120576119905minus1

)120575(1)

+

ℎ

sum

ℎ=1

120585ℎ(1)

120595 (120591ℎ(1)

119885119905(1)

120582ℎ(1)

120579ℎ(1)

)

(1 minus 120573(2)

119871) 120590

120575(2)

119905(2)

= 119908(2)

+ ((1 minus 120573(2)

119871) minus (1 minus 120601(2)

119871) (1 minus 119871)119889(2))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(2)

120576119905minus1

)120575(2)

+

ℎ

sum

ℎ=1

120585ℎ(2)

120595 (120591ℎ(2)

119885119905(2)

120582ℎ(2)

120579ℎ(2)

)

(22)


(1)and 120575






(1 minus 120573119871) 120590120575

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(23)


(1 minus 120573119871) 1205902

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

2

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(24)





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)


119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)


120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)



119875(119878119905

= 119894 | 119911119905minus1


(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)



120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)



for 120574119901(119904119905)


119901(119904119905)


1 and 0 le 120574119901(119904119905)



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)


= 2 and 120574(119904119905)



if 120575(119904119905)= 2 and 120574









(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




where 119908119895gt 0 120572

1119895 1205722119895 120573119895ge 0 119895 = 1 119896 For the power

term 120575 = 2 and for 1205721119895



1119895= 1205722119895[109]





120590120575

119895119905= 119908119895+ 120572119895(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+ 120573119895120590120575

119895119905minus1 120575 gt 0 (9)


ge 0 120574119895


1119895= 120572119895(1 minus 120574

119895)120575 and 120573

119895=

120572119904

119895(1 + 120574

119895)120575


120590

120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119903

sum

119897=1

120572119897(119904119905)(1003816100381610038161003816120576119905minus119897

1003816100381610038161003816minus 120574119897(119904119905)120576119905minus119897

)

120575(119904119905)

+

119902

sum

119898=1

120573119898(119904119905)120590

120575(119904119905)

119905minus119898(119904119905) 120575(119904119905)gt 0

(10)



gt 0 120572119897(119904119905) 120573119898(119904119905)

ge

0 with 120574119897(119904119905)


(119904119905)= 120575 the





(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)


(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(119904119905)120576119905minus1

)

120575(119904119905)

(11)


(119904119905) and 120572


120575(1)


119889(119904119905)






(119904119905)


(119904119905)


=






(119904119905)= 2 120574

(119904119905)= 0 the model











119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119899

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905) (12)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(13)


119898

sum

119894=1

1205902

119905(119894)119875 (119878119905= 119894 | 119911

119905minus1) 119894 = 1 119898 (14)


120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

= [1+exp(minus120591ℎ(119904119905)(

119897

sum

119897=1

[

ℎ

sum

ℎ=1

120582ℎ119897(119904119905)119911ℎ

119905minus119897(119904119905)+ 120579ℎ(119904119905)]))]

minus1

(15)

(

1

2

) 120582ℎ119889


and119875(119878119905= 119894 | 119911


representation

(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (17)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(18)


119904= 119894 | 119911

119904minus1) where 120595(119911





If 119899119895119894


= 119894 | 119911119905minus1


119891 (119910119905| 119909119905 119911119905= 119894)

=

1

radic2120587ℎ119905(119894)

exp

minus(119910119905minus 1199091015840

119905120593 minus sum

119867

119895=1120573119895119901 (1199091015840

119905120574119895))

2

2ℎ119905(119895)

(19)


119904= 119894 | 119911

119904minus1)


120590120575(119904119905)

119905(119904119905)

= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119901

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(20)


119901(119904119905)


119901(119904119905)


119901(119904119905)






(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(21)


ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905)) defined as


(119904119905)

= 2 and 120574(119904119905)


(119904119905)


(119904119905)


(119904119905)


(119904119905)

= 1



(119904119905)


(119904119905)



(1 minus 120573(1)

119871) 120590

120575(1)

119905(1)

= 119908(1)

+ ((1 minus 120573(1)

119871) minus (1 minus 120601(1)

119871) (1 minus 119871)119889(1))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(1)

120576119905minus1

)120575(1)

+

ℎ

sum

ℎ=1

120585ℎ(1)

120595 (120591ℎ(1)

119885119905(1)

120582ℎ(1)

120579ℎ(1)

)

(1 minus 120573(2)

119871) 120590

120575(2)

119905(2)

= 119908(2)

+ ((1 minus 120573(2)

119871) minus (1 minus 120601(2)

119871) (1 minus 119871)119889(2))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(2)

120576119905minus1

)120575(2)

+

ℎ

sum

ℎ=1

120585ℎ(2)

120595 (120591ℎ(2)

119885119905(2)

120582ℎ(2)

120579ℎ(2)

)

(22)


(1)and 120575






(1 minus 120573119871) 120590120575

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(23)


(1 minus 120573119871) 1205902

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

2

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(24)





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)


119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)


120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)



119875(119878119905

= 119894 | 119911119905minus1


(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)



120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)



for 120574119901(119904119905)


119901(119904119905)


1 and 0 le 120574119901(119904119905)



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)


= 2 and 120574(119904119905)



if 120575(119904119905)= 2 and 120574









(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








119910119905= 119888(119904119905)+

119903

sum

119894=1

120579119894(119904119905)119910119905minus119894

+ 120576119905(119904119905)+

119899

sum

119895=1

120593119895(119904119905)120576119905minus119895(119904

119905) (12)

1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(13)


119898

sum

119894=1

1205902

119905(119894)119875 (119878119905= 119894 | 119911

119905minus1) 119894 = 1 119898 (14)


120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

= [1+exp(minus120591ℎ(119904119905)(

119897

sum

119897=1

[

ℎ

sum

ℎ=1

120582ℎ119897(119904119905)119911ℎ

119905minus119897(119904119905)+ 120579ℎ(119904119905)]))]

minus1

(15)

(

1

2

) 120582ℎ119889


and119875(119878119905= 119894 | 119911


representation

(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (17)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(18)


119904= 119894 | 119911

119904minus1) where 120595(119911





If 119899119895119894


= 119894 | 119911119905minus1


119891 (119910119905| 119909119905 119911119905= 119894)

=

1

radic2120587ℎ119905(119894)

exp

minus(119910119905minus 1199091015840

119905120593 minus sum

119867

119895=1120573119895119901 (1199091015840

119905120574119895))

2

2ℎ119905(119895)

(19)


119904= 119894 | 119911

119904minus1)


120590120575(119904119905)

119905(119904119905)

= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119901

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(20)


119901(119904119905)


119901(119904119905)


119901(119904119905)






(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(21)


ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905)) defined as


(119904119905)

= 2 and 120574(119904119905)


(119904119905)


(119904119905)


(119904119905)


(119904119905)

= 1



(119904119905)


(119904119905)



(1 minus 120573(1)

119871) 120590

120575(1)

119905(1)

= 119908(1)

+ ((1 minus 120573(1)

119871) minus (1 minus 120601(1)

119871) (1 minus 119871)119889(1))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(1)

120576119905minus1

)120575(1)

+

ℎ

sum

ℎ=1

120585ℎ(1)

120595 (120591ℎ(1)

119885119905(1)

120582ℎ(1)

120579ℎ(1)

)

(1 minus 120573(2)

119871) 120590

120575(2)

119905(2)

= 119908(2)

+ ((1 minus 120573(2)

119871) minus (1 minus 120601(2)

119871) (1 minus 119871)119889(2))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(2)

120576119905minus1

)120575(2)

+

ℎ

sum

ℎ=1

120585ℎ(2)

120595 (120591ℎ(2)

119885119905(2)

120582ℎ(2)

120579ℎ(2)

)

(22)


(1)and 120575






(1 minus 120573119871) 120590120575

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(23)


(1 minus 120573119871) 1205902

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

2

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(24)





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)


119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)


120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)



119875(119878119905

= 119894 | 119911119905minus1


(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)



120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)



for 120574119901(119904119905)


119901(119904119905)


1 and 0 le 120574119901(119904119905)



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)


= 2 and 120574(119904119905)



if 120575(119904119905)= 2 and 120574









(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120595 (120591ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905))

(21)


ℎ(119904119905) 119885119905(119904119905)120582ℎ(119904119905) 120579ℎ(119904119905)) defined as


(119904119905)

= 2 and 120574(119904119905)


(119904119905)


(119904119905)


(119904119905)


(119904119905)

= 1



(119904119905)


(119904119905)



(1 minus 120573(1)

119871) 120590

120575(1)

119905(1)

= 119908(1)

+ ((1 minus 120573(1)

119871) minus (1 minus 120601(1)

119871) (1 minus 119871)119889(1))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(1)

120576119905minus1

)120575(1)

+

ℎ

sum

ℎ=1

120585ℎ(1)

120595 (120591ℎ(1)

119885119905(1)

120582ℎ(1)

120579ℎ(1)

)

(1 minus 120573(2)

119871) 120590

120575(2)

119905(2)

= 119908(2)

+ ((1 minus 120573(2)

119871) minus (1 minus 120601(2)

119871) (1 minus 119871)119889(2))

times(1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574(2)

120576119905minus1

)120575(2)

+

ℎ

sum

ℎ=1

120585ℎ(2)

120595 (120591ℎ(2)

119885119905(2)

120582ℎ(2)

120579ℎ(2)

)

(22)


(1)and 120575






(1 minus 120573119871) 120590120575

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

120575

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(23)


(1 minus 120573119871) 1205902

119905


times (1003816100381610038161003816120576119905minus1

1003816100381610038161003816minus 120574119895120576119905minus1

)

2

+

ℎ

sum

ℎ=1

120585ℎ120595 (120591ℎ 119885119905120582ℎ 120579ℎ)

(24)





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)


119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)


120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)



119875(119878119905

= 119894 | 119911119905minus1


(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)



120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)



for 120574119901(119904119905)


119901(119904119905)


1 and 0 le 120574119901(119904119905)



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)


= 2 and 120574(119904119905)



if 120575(119904119905)= 2 and 120574









(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(25)


119898

sum

119894=1

1205902

119905(119904119905)119875 (119878119905= 119894 | 119911

119905minus1) (26)


120601 (ℎ (119904119905) 119885119905) = exp(

minus

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817

2

21205882

) (27)



119875(119878119905

= 119894 | 119911119905minus1


(119875 (119878119905= 119894 | 119911

119905minus1) 120572119891 (119875 (120590

119905minus1| 119911119905minus1

119904119905minus1

= 1))) (28)


119905= 119894 | 119904


119911119905minus119889

=

[120576119905minus119889

minus 119864 (120576)]

radic119864 (1205762)

(29)


119904= 119894 | 119911

119904minus1)



120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(30)



for 120574119901(119904119905)


119901(119904119905)


1 and 0 le 120574119901(119904119905)



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120601ℎ(119904119905)(

10038171003817100381710038171003817119885119905(119904119905)minus 120583ℎ(119904119905)

10038171003817100381710038171003817)

(31)


= 2 and 120574(119904119905)



if 120575(119904119905)= 2 and 120574









(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






(sdot)= 0 with





1205902

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)1205762

119905minus119901(119904119905)+

119902

sum

119902=1

120573119902(119904119905)120590119905minus119902(119904

119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)120587ℎ(119904119905)(119908119896ℎ(119904119905)120579119905minus119896

+ 120579119896ℎ(119904119905))

(32)




120590120575(119904119905)

119905(119904119905)= 119908(119904119905)+

119901

sum

119901=1

120572119901(119904119905)(

10038161003816100381610038161003816120576119905minus119895

10038161003816100381610038161003816minus 120574119901(119904119905)120576119905minus119901(119904

119905))

120575(119904119905)

+

119902

sum

119902=1

120573119902(119904119905)120590120575(119904119905)

119905minus119902(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(33)









119879

119905minus1[120590119905

minus 119905]

2


119901(119904119905)

=



120574119901(119904119905)


119901(119904119905)le 1 In addition



(1 minus 120573(119904119905)119871) 120590

120575(119904119905)

119905(119904119905)

= 119908(119904119905)+ ((1 minus 120573

(119904119905)119871) minus (1 minus 120601

(119904119905)119871) (1 minus 119871)

119889(119904119905))

times (

10038161003816100381610038161003816120576119905minus1(119904

119905)

10038161003816100381610038161003816minus 120574(119904119905)120576119905minus1(119904

119905))

120575(119904119905)

+

ℎ

sum

ℎ=1

120585ℎ(119904119905)Π(120579119896ℎ(119904119905)120594119905minus119896ℎ(119904

119905)+ 120579119896ℎ(119904119905))

(34)


(119904119905)= 2 and 120574

(119904119905)=


(119904119905)= 2


and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




and 0 le 120574(119904119905)


(119904119905)le 1 and 120575

(119904119905)= 1 Single regime




119905= ln(119875

119905119875119905minus1



119905is taken







119905= 1 | 119904

119905minus1= 1) = 050 and Prob(119904

119905= 2 |

119904119905minus1






















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




















C(MEAN)

C(VAR) log 119871




0027382(0033565)



8054283(66685) 1319630








Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table3MS-ARM

A-GARC

Hmod

els

MS-ARM

A-GARC

Hmod

els

(1)M

S-ARM

A-GARC

HGarch

Sigm

aArch

Con

stant

1199010|0

1198751|1

log119871

RMSE

Regime1

09664


000

0333727


003351

(0005)

623008119890minus005


0500244

05102

38509

0458911

Regime2

0436124


000

04356


056387

(00098)

629344119890minus005


(2)M

S-ARM

A-APG

ARC

HGarch

sigma

Arch

Con

stant

Mean

1198750|0

1199011|1

Power

log119871

RMSE

Regime1

0616759


000

0679791


0383241

(00102)

813394119890minus005


000

021042

0500227

050300

080456(

000546)

17565

042111

Regime2

0791950


00012381


020805

(00201)

820782119890minus005


283119864

minus03

060567(00234)

(3)M

S-ARM

A-FIAPG

ARC

HARC

H(Phi1)

GARC

H(Beta1)

d-Figarch

APA

RCH

(Gam

ma1)

APA

RCH

(Delta)

Cst

1199010|0

1199010|1

log119871

RMSE

Regime1

0277721

(000)

067848

(000)

02761233

(00266)

0220157

(00106)

0123656

(0001)

000135

(0001)

050212

0510021

18779

0422206

6

Regime2

0309385

(0002)

0680615

(00001)

0181542

(0000

05)

021083

(00299)

01448

(00234)

000112

(000984)


































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




































5 Conclusions





MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






MS-ARMA-GARCH-

RBF

MS-ARMA-GARCH-

ELMAN RNN

MS-ARMA-GARCH-

HYBRID MLP

MS-ARMA-GARCH-MLP



minus0645(0518) [r]

0574(0566) [c]









RNN

MS-ARMA-APGARCH-

RBF



MS-ARMA-APGARCH-

MLP













FIAPGARCH-RNN

MS-ARMA-FIAPGARCH-

RBF



MS-ARMA-FIAPGARCH-

MLP



minus0588(0556) [r]

minus0836(0403) [r]


















References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










References




















































































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



























































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article Modeling Markov Switching ARMA-GARCH...

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