Mathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol 3, No.3, 2013

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1 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 The ffec of Inverse Transformaion on he Uni Mean and Consan Variance Assumpions of a Muliplicaive rror Model Whose rror Componen has a Gamma Disribuion. By Deparmen of Ohawe J. Mahemaicsa and Saisics/ Compuer Science/Physics, Faculy of Science, Federal Universiy Ouoe, P.M.B. 6, Yenagoa, Bayelsa Sae, Nigeria. mail: ohawe.johnson@yahoo.com ; Phone No.: Absrac In his paper, he effec of inverse ransformaion on he uni mean and consan variance assumpions of a muliplicaive error model whose error componen is Gamma disribued was sudied. From he resuls of he sudy, i was discovered ha he uni mean assumpion is violaed afer inverse ransformaion. The mean and variance of he inverse-ransformed gamma error componen were found o be smaller han hose of he unransformed error. Furhermore his change in mean, µ was modeled and was found o increase per uni increase in, he shape parameer while ha of he variance was found o decrease per uni increase in he shape parameer and heir relaionships (predicive equaions) were deermined. Finally, i was discovered ha in order o achieve he uni mean condiion afer inverse ransformaion, he condiion = is unavoidable where and are respecively he shape and locaion parameers of he Gamma densiy funcion. Key words: Muliplicaive rror Model; Gamma disribuion; Inverse Transformaion; Mean; Variance. Inroducion Suppose he model of ineres is a muliplicaive error model given as ( ) X = Ψ X () where,, N X N is a discree ime series process defined on [0, ), Ψ ( X ), he informaion available for forecasing X, N and, a random variable defined over a [0, + ) suppor wih uni mean and unnown consan variance, + ~ V, ( ). Tha is () By he definiion given in () i is very clear ha in () can be specified by means of any probabiliy densiy funcion (pdf) having he characerisics in (). xamples are Gamma, Log-Normal, Weibull, and mixures of hem (Brownlees e al., (0)). ngle and Gallo (006) favor a Gamma ( φ, φ ) (which implies = φ ); Bauwens and Gio (000), in Auoregressive Condiional Duraion (ACD) model

2 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 framewor considered a Weilbull Γ ( + φ ), φ (in his case, ( φ ) ( φ ) = Γ + Γ + ). In saisical modeling, he familiar applicaion of he normal linear model involves a response variable ha is assumed normally disribued wih consan variance. In oher applicaions a response variable may occur in a form ha suppresses an underlying normal linear srucure (Fraser (967)). Someimes in hese applicaions he conex may sugges a logarihm or inverse or square roo ransformaion and so on, which reveals he normal linear form. Transformaion may also be necessary o eiher sabilize he variance componen of a model or o normalize i. Deails on he reasons for ransformaions are found in; Box and Jenings (96); Iwueze e al., (0). Recenly here are various sudies on he effecs of ransformaion on he error componen of he muliplicaive error model whose error componen is classified under he characerisics given in () of which he muliplicaive ime series model is a subclass. The aim of such sudies is o esablish he condiions for successful ransformaion. A successful ransformaion is achieved when he desirable properies of a daa se remains unchanged afer ransformaion. These basic properies or assumpions of ineres for his sudy are; (i) Uni mean and (ii) consan variance. In his area of research, Iwueze (007) invesigaed he effec of logarihmic ransformaion on he error componen (e ) of a muliplicaive ime series model where ( ~ (, ) e N ) and discovered ha he logarihm ransform; Y = Log e can be assumed o be normally disribued wih mean, zero and he same variance, for < 0.. Similarly Nwosu e al., (00) and Ouonye e al., (0) had sudied he effecs of inverse and square roo ransformaion on he error componen of he same model. Nwosu e al., (00) discovered ha he inverse ransform Y = can be assumed o be normally disribued wih mean, one and he same variance provided e Similarly Ouonye e al., (0) discovered ha he square roo ransform; Y = e can be assumed o be normally disribued wih uni mean and variance, for 0.59, where variance of he original error componen before ransformaion. where The applicaion of inverse ransformaion o model () gives ( ), N is he X = Ψ X () X =, ( X ), N X, N Ψ = and Ψ ( X ) =. Model () is sill a muliplicaive error model and herefore mus also be characerized wih uni mean and some consan variance, which may or may no be equal o. In his paper we wan o sudy he effec of inverse ransformaion on a non-normal disribued error componen of a muliplicaive error model whose disribuional characerisics belong o he Gamma disribuion. The purpose is o deermine if he assumed 5

3 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 fundamenal srucure (uni mean and consan variance) is mainained afer inverse ransformaion and also o invesigae wha happens o and in erms of equaliy or non-equaliy. The overall reason for concenraing on he error componen of model () is as plane as he nose on he face: he reason is ha he assumpions for model analysis are always placed on he error componen, ~ V, ( ) +, hus The paper is organized ino six Secions. The inroducion is conained in secion one while some of he basic disribuional characerisics of his sudy are given in Secion wo. The relaionship beween he means and variances of he ransformed and unransformed Gamma disribued error componen would be deermined in Secion while he Summary of he resuls and conclusion are conained in secion four. Finally he references and Appendix are conained in Secions five and six respecively. ().0 Some Basic Disribuional Characerisics of he Sudy Given ha in () has a gamma disribuion, is probabiliy densiy funcion (Freund (000)) is given by f wih e =, > 0, Γ > 0 Γ ( ) = µ = where ( ) rue ( ) ( ) Γ + Γ is he h momen of he disribuion ( =,,,...). From (6), he following resuls are ( ) = µ = (7) ( ) = + (8) = (9) µ = (0) ( ) µ = + () where ( µ ) and ( µ ) are measures of sewness and urosis. Suppose inverse ransformaion is deemed appropriae in a daa se whose model can be suiably represened by model () o eiher sabilize he variance or remedy he presence of oulier(s) in a daa se, 6 (5) (6)

4 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 we herefore obain model () whose error componen, has he probabiliy densiy funcion given by f e ( ) =, > 0, Γ > 0 Γ as obained by Coo (008). The disribuional characerisics of () is given in (). Given (), he following resuls are rue = µ =, =,,,... ( ) = µ = ( ) ( )( )...( ) ( ) = Var = ( ) ( ) () () () (5) ( µ ) = ( µ ) ( ) ( )( ) = ( + 5) ( ) ( )( )( ) In pracice, he required general assumpions for modeling () are uni mean and consan variance and hese would be he major focus of his sudy. Considering ha () (Model () afer inverse ransformaion) is also a muliplicaive error model, he uni mean and consan variance assumpions sill remain valid even hough he variances of he wo models may or may no be equal. Given (), he condiion for uni mean from (7) is eiher = (8) or = (9) However considering ha is he shape parameer, we shall be ineresed in (8). On applying he uni mean condiion of (8) in he resuls of (7) hrough () we obain he resuls given in Table. Also included in Table is he raio of he momens of he unransformed Gamma o hose of he inverse-ransformed disribuion subjec o he applicaion of he uni mean condiion. Having obained he momens of he inverse-ransformed Gamma disribued error componen subjec o he uni-mean condiion, he nex as would be o model he relaionship beween he unransformed and ransformed Gamma disribued error componen in erms of he mean and variance and hese would be 7 (6) (7)

5 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 invesigaed in Secion..0 Relaionship beween he Mean and Variance of he Gamma rror Componen before and afer Inverse Transformaion In his Secion, he relaionship beween he means and variances of he ransformed and unransformed Gamma disribued error componen would be deermined wih a view o ascerain he uni increase or decrease in he mean and variance of he inverse ransformed error componen per uni increase in he value of, he shape parameer. For his purpose, he differences ( ) ( ) Var ( ) Var ( ) = µ and = are compued and he relaionships beween he compued differences and he shape parameer, are obained. The resuls of he compuaions of ( ), ( ) ( ) Var and Var using he expressions in Table are given in Table. and Furhermore µ and would be regressed on o obain a predicive funcion given by µ = f (0) = () g ( ) where f ( ) and g ( ) are finie funcions of. The reasons for deermining he predicive funcions are o enable an analys deermine he increase/decrease in he mean/variance of a gamma disribued error componen afer an inverse ransformaion. For he purpose of he regression analysis, considering ha in Table, Var ( ) 0 = = for all values of >, he regression analysis would be consrained o he values of =,,,...,. Goodness of he regression fi would be assessed using he coefficien of deerminaion, R (Draper and Smih (98)). The resuls of he regression analysis are given in Figures a, b and. From Figures a and b, he cubic predicive equaion given by µ = () whose R = 9.5% is a beer predicive equaion of he increase in mean of a Gamma disribuion afer inverse ransformaion han ha of he quadraic, whose R = 79.8%. However from Figure, he decrease in variance is given by = () whose R = 99.7%..0: Summary and Conclusion In his paper, he effec of inverse ransformaion on he uni mean and consan variance assumpions of a muliplicaive error model whose error componen is Gamma disribued was sudied. From he resuls of he sudy, i was discovered ha he uni mean assumpion is violaed afer inverse ransformaion. The 8

6 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 mean and variance of he inverse-ransformed gamma error componen were found o be smaller han hose of he unransformed error. Furhermore he decrease in mean, µ was found o increase per uni increase in, he shape parameer while ha of he variance 9 was found o decrease per uni increase in he shape parameer and he relaionships (predicive equaions) are respecively given by (i) (ii) µ = , =,,..., and = = ,,,..., Furhermore, he uni mean condiion of he unransformed gamma error componen is achieved when =, however, for he inverse ransformed gamma error erm, =. In conclusion, inasmuch as here is a decreased error variance afer inverse ransformaion, he uni mean violaion where he mean of he ransformed error componen is µ 0.5 ( µ is he mean afer he inverse ransformaion) is of a major concern. In order o achieve he uni mean condiion afer inverse ransformaion, he condiion = is unavoidable. Finally based of he resuls of his sudy I recommend ha inverse ransformaion is no appropriae for a muliplicaive error model wih a Gamma disribued error componen. 5.0 References Bauwens, L. and Gio, P. (000). The logarihmic acd model: An applicaion o he bid-as quoe Process of hree nyse socs. Annales d conomie e de Saisique, 60, 7 9. Box, G.. P. and Cox, D. R. (96). An analysis of ransformaions. J. Roy. Sais. Soc., B-6, -, discussion -5. Brownlees C. T, Cipollini F., and Gallo G. M. (0). Muliplicaive rror Models. Woring paper 0/0,Universia degli sudi di Firenze. Coo J. D. (008). Inverse Gamma Disribuion. De Luca, G. and Gallo, G. M. (00). A ime-varying mixing muliplicaive error model for Realized volailiy. conomerics Woring PapersArchive wp00 0, Universia degli Sudi di Firenze, Diparimeno di Saisica G. Pareni. Draper N. R. and H. Smih(98). Applied Regression Analysis, nd d., Johns Wiley and Sons Inc., New Yor. ngle, R. F. and Gallo, G. M. (006). A muliple indicaors model for volailiy using inra-daily daa. Journal of conomerics,, 7. Fraser D. A. S. (967). Daa ransformaions and he linear model. Ann. Mah. Sais. Vol. 8, N0. 5, 967. Freund J.. (000). Mahemaical Saisics, 5 h ed. Prenice Hall of India privae limied, New Delhi 000, 000. Iwueze Iheanyi S. (007). Some Implicaions of Truncaing he N (, ) Disribuion o he lef a Zero. Journal of Applied Sciences. 7() (007) pp Iwueze I.S, Nwogu.C., Ohawe J and Ajaraogu J.C, (0).New Uses of Buys-Ballo Table. Applied Mahemaics, 0, DOI: 0.6/am Nwosu C. R, Iwueze I.S. and Ohawe J. (00). Disribuion of he rror Term of he Muliplicaive Time Series Model Under Inverse Transformaion. Advances and Applicaions in Mahemaical Sciences. Volume 7, Issue, 00, pp. 9 9 Ouony. L, Iwueze I.S. and Ohawe J. (0). The ffec of Square Roo Transformaion on he rror Componen of he Muliplicaive Time Series Model. Inernaional Journal of Saisics and Sysems.. Vol. 6, No., 0, pp ISSN:

7 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., Appendix µ.. 0 F i e d L i n e P l o S R - S q % R - S q ( a d j ) 7. 8 % µ = Figure a: Quadraic Predicive quaion of he Change in Mean of he Gamma Disribued rror Componen afer Inverse ransformaion 0 µ. 0 F i e d L i n e P l o S R - S q 9. 5 % R - S q ( a d j ) % µ = Figure b: Cubic Predicive quaion of he Change in Mean of he Gamma Disribued rror Componen afer Inverse ransformaion F i e d L i n e P l o S R - S q % R - S q ( a d j ) % 0. 0 = Figure : Predicive quaion of he Change in Variance of he Gamma Disribued rror Componen afer Inverse ransformaion 50

8 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 Momen Table : The implicaion of he Uni Mean Condiion on he Gamma Disribuion and is counerpar Under Inverse Transformaion Mean.0 Variance Sewness ( ) ( ) ( ), >, > ( )( ) Kurosis ( + ), > ( ) ( )( ) ( ) ( )( )( ) >, / ( ) ( )( ) ( ) ( )( ) ( + )( ) ( )( )( ), > 5

9 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 Table : Compuaions of he Means and Variances of he Transformed and he Unransformed Gamma Disribued rror Componen ( ) ( ) ( ) Var ( ) Var ( ) Var ( ) Var ( )